Notation by work
Stratified Universality
All notation · Glossary · Source coverage
2009 symbol spellings in 4032 mathematical passages across 66 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 2009 entries are available below.
*Definition passage · 38 passages
Search aliases: *
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 352
d ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a monoidal structure. A tag is not itself an existence theorem. item Cert is a list of named residual and certificate types. Each certificate type specif…
Exact mathematical passage
C^*
+Definition passage · 332 passages
Search aliases: +
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
\mathbf{1}Definition passage · 4 passages
Search aliases: 1 mathbf
Definition passage: Divisor--ghost transform · Stratified_Universality_Traditional.tex, line 5179
[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).
Exact mathematical passage
\mathfrak G(b) = \mathbf{1}\mathbin{\ast_D}(\operatorname{id}\cdot b),
=Definition passage · 1096 passages
Search aliases: =
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
ADefinition passage · 253 passages
Search aliases: A
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 862
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A=(A_1,\dots,A_m)
\mathcal ADefinition passage · 36 passages
Search aliases: A mathcal
Definition passage: Linear grammar category · Stratified_Universality_Traditional.tex, line 853
[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.
Exact mathematical passage
\mathcal A_{\mathbb K}:=\kk Q/I.
\mathfrak ADefinition passage · 17 passages
Search aliases: A mathfrak
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 44
[Certified Global synthesis atlas atlas] Fix a prime p, a characteristic-zero field K, and a finite Galois extension L/K for the Galois component. The certified atlas A_13 is the finite coproduct of the following five connected profiled atlases.
Exact mathematical passage
\mathfrak A_{13}
\mathfrak aDefinition passage · 4 passages
Search aliases: a mathfrak
Definition passage: Root-filter alias and address-cancellation defects · block09_section.tex, line 325
[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.
Exact mathematical passage
\mathfrak a_N(F)=Nr-r_N^{\rm orb}, \qquad \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
\mathsf ADefinition passage · 1 passage
Search aliases: A mathsf
Definition passage: Multiplicative Euler-overlap category · Stratified_Universality_Traditional.tex, line 5270
[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.
Exact mathematical passage
\mathsf B(b)=b, \qquad \mathsf A(b)=A_b(z), \qquad \mathsf C(b)=\mathfrak G(b), \qquad \mathsf P(b)=\bigl(b(p^e),c(p^e)\bigr)_{p,e}.
aDefinition passage · 500 passages
Search aliases: a
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 362
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
M_a:M_{s(a)}\longrightarrow M_{t(a)}
A^r_{U,i}Definition passage · 3 passages
Search aliases: A^r_U,i
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 55
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i}, \iota^r_{A,U,i},\pi^r_{A,U,i}\right)_{i\in I_{A,U,r}}
A^r_{V,\lambda^A_{\gamma,r}(i)}Definition passage · 1 passage
Search aliases: A^r_V,λ^A_γ,r(i) lambda gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 65
ject A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π…
Exact mathematical passage
\beta^A_{\gamma,r,i}: F_\gamma(A^r_{U,i})\xrightarrow{\sim} A^r_{V,\lambda^A_{\gamma,r}(i)}.
\mathcal A_{\mathbb K}Definition passage · 6 passages
Search aliases: A_ K mathcal mathbb
Definition passage: Linear grammar category · Stratified_Universality_Traditional.tex, line 853
[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.
Exact mathematical passage
\mathcal A_{\mathbb K}:=\kk Q/I.
A_0Definition passage · 5 passages
Search aliases: A_0
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3549
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
a_0Definition passage · 15 passages
Search aliases: a_0
Definition passage: Branch-separation index · block10_section.tex, line 43
[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\boxed{ J_{\rm alg}(F) = (a_0,a_1,\ldots,a_{\sigma-1}), \qquad F(z)=\sum_{n\geq0}a_nz^n.}
A_1Definition passage · 6 passages
Search aliases: A_1
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 862
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A=(A_1,\dots,A_m)
a_1Definition passage · 2 passages
Search aliases: a_1
Definition passage: Branch-separation index · block10_section.tex, line 43
[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\boxed{ J_{\rm alg}(F) = (a_0,a_1,\ldots,a_{\sigma-1}), \qquad F(z)=\sum_{n\geq0}a_nz^n.}
\mathfrak A_{13}Definition passage · 5 passages
Search aliases: A_13 mathfrak
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 44
[Certified Global synthesis atlas atlas] Fix a prime p, a characteristic-zero field K, and a finite Galois extension L/K for the Galois component. The certified atlas A_13 is the finite coproduct of the following five connected profiled atlases.
Exact mathematical passage
\mathfrak A_{13}
a_{[u]_k+k^{|u|}[v]_k}Definition passage · 1 passage
Search aliases: a_[u]_k+k^|u|[v]_k
Definition passage: Word-Hankel matrix · Stratified_Universality_Traditional.tex, line 3649
[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.
Exact mathematical passage
H_a(u,v) = a_{[u]_k+k^{|u|}[v]_k} = (\Lambda_u a)([v]_k), \qquad u,v\in\Sigma_k^*. \label{eq:block05-word-hankel}
a_{[w]_k}Definition passage · 4 passages
Search aliases: a_[w]_k
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3553
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
a_{[w]_k} = \lambda A_wv, \qquad A_w=A_{d_{\ell-1}}\cdots A_{d_0}. \label{eq:block05-digit-linear-semantics}
A_bDefinition passage · 15 passages
Search aliases: A_b
Definition passage: Multiplicative Euler-overlap category · Stratified_Universality_Traditional.tex, line 5270
[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.
Exact mathematical passage
\mathsf B(b)=b, \qquad \mathsf A(b)=A_b(z), \qquad \mathsf C(b)=\mathfrak G(b), \qquad \mathsf P(b)=\bigl(b(p^e),c(p^e)\bigr)_{p,e}.
A_dDefinition passage · 5 passages
Search aliases: A_d
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
A_d\in\End_K(W)
A_{d_0}Definition passage · 1 passage
Search aliases: A_d_0
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3553
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
a_{[w]_k} = \lambda A_wv, \qquad A_w=A_{d_{\ell-1}}\cdots A_{d_0}. \label{eq:block05-digit-linear-semantics}
A_{d_{\ell-1}}Definition passage · 1 passage
Search aliases: A_d_ℓ-1 ell
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3553
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
a_{[w]_k} = \lambda A_wv, \qquad A_w=A_{d_{\ell-1}}\cdots A_{d_0}. \label{eq:block05-digit-linear-semantics}
\mathfrak A_{\mathrm{DRLM},p}Definition passage · 2 passages
Search aliases: A_DRLM,p mathfrak mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 47
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{DRLM},p}
\mathfrak A_{\mathrm{EM}}Definition passage · 1 passage
Search aliases: A_EM mathfrak mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 50
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{EM}}
\mathfrak A_{\mathrm{Fuss}}Definition passage · 1 passage
Search aliases: A_Fuss mathfrak mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 49
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{Fuss}}
\mathfrak A_{\mathrm{Gal},L/K}Definition passage · 1 passage
Search aliases: A_Gal,L/K mathfrak mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 51
component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{Gal},L/K}
A_jDefinition passage · 1 passage
Search aliases: A_j
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 865
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
\mathcal A_{\mathbb K}(A_j,B_i)
A_{k-1}Definition passage · 1 passage
Search aliases: A_k-1
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3549
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
\mathfrak A_{\mathrm{Loc},p}Definition passage · 2 passages
Search aliases: A_Loc,p mathfrak mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 48
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{Loc},p}
A_mDefinition passage · 10 passages
Search aliases: A_m
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 862
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A=(A_1,\dots,A_m)
\mathfrak a_NDefinition passage · 2 passages
Search aliases: a_N mathfrak
Definition passage: Root-filter alias and address-cancellation defects · block09_section.tex, line 325
[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.
Exact mathematical passage
\mathfrak a_N(F)=Nr-r_N^{\rm orb}, \qquad \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
a_nDefinition passage · 29 passages
Search aliases: a_n
Definition passage: Minimal tail recurrence · block09_section.tex, line 44
[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.
Exact mathematical passage
a=(a_n)_{n\geq0}
A_UDefinition passage · 10 passages
Search aliases: A_U
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 53
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
A_U
A_wDefinition passage · 2 passages
Search aliases: A_w
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3553
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
a_{[w]_k} = \lambda A_wv, \qquad A_w=A_{d_{\ell-1}}\cdots A_{d_0}. \label{eq:block05-digit-linear-semantics}
a_{\sigma-1}Definition passage · 2 passages
Search aliases: a_σ-1 sigma
Definition passage: Branch-separation index · block10_section.tex, line 43
[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\boxed{ J_{\rm alg}(F) = (a_0,a_1,\ldots,a_{\sigma-1}), \qquad F(z)=\sum_{n\geq0}a_nz^n.}
\AddDefinition passage · 4 passages
Search aliases: Add operatorname
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 861
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
\Add(\mathcal A_{\mathbb K})
\AnalysisDefinition passage · 6 passages
Search aliases: Analysis Analysis A mathcal
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 435
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Analysis_{\mathcal P} := (\pi_i)_{i\in I}: X\longrightarrow\bigoplus_{i\in I}X_i,
\Analysis_{\mathcal P}Definition passage · 3 passages
Search aliases: Analysis_ P A_ P Analysis mathcal mathcal mathcal
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 435
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Analysis_{\mathcal P} := (\pi_i)_{i\in I}: X\longrightarrow\bigoplus_{i\in I}X_i,
\operatorname{Ann}Definition passage · 2 passages
Search aliases: Ann operatorname
Definition passage: Minimal differential annihilator · block09_section.tex, line 27
[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).
Exact mathematical passage
\operatorname{Ann}_{\mathcal D}(F) = \{L\in\mathcal D_K:LF=0\}.
\operatorname{Ann}_{\mathcal D}Definition passage · 1 passage
Search aliases: Ann_ D operatorname mathcal
Definition passage: Minimal differential annihilator · block09_section.tex, line 27
[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).
Exact mathematical passage
\operatorname{Ann}_{\mathcal D}(F) = \{L\in\mathcal D_K:LF=0\}.
\operatorname{Ann}_{\rm tail}Definition passage · 1 passage
Search aliases: Ann_rm tail operatorname rm
Definition passage: Minimal tail recurrence · block09_section.tex, line 45
[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.
Exact mathematical passage
\operatorname{Ann}_{\rm tail}(a) = \{R\in\mathcal S_K:(Ra)_n=0\text{ for all sufficiently large }n\}.
BDefinition passage · 152 passages
Search aliases: B
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 865
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A\to B=(B_1,\dots,B_n)
\mathfrak bDefinition passage · 2 passages
Search aliases: b mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\mathsf BDefinition passage · 4 passages
Search aliases: B mathsf
Definition passage: Multiplicative Euler-overlap category · Stratified_Universality_Traditional.tex, line 5270
[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.
Exact mathematical passage
\mathsf B(b)=b, \qquad \mathsf A(b)=A_b(z), \qquad \mathsf C(b)=\mathfrak G(b), \qquad \mathsf P(b)=\bigl(b(p^e),c(p^e)\bigr)_{p,e}.
bDefinition passage · 175 passages
Search aliases: b
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
B_\mathfrakDefinition passage · 2 passages
Search aliases: B_ mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
B_1Definition passage · 1 passage
Search aliases: B_1
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 865
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A\to B=(B_1,\dots,B_n)
b_dDefinition passage · 2 passages
Search aliases: b_d
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4857
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
c_n(b)=\sum_{d\mid n}d\,b_d, \qquad n\geq1.
B_iDefinition passage · 1 passage
Search aliases: B_i
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 865
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
\mathcal A_{\mathbb K}(A_j,B_i)
b_iDefinition passage · 2 passages
Search aliases: b_i
Definition passage: Greedy shortlex kernel basis · Stratified_Universality_Traditional.tex, line 3731
[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.
Exact mathematical passage
b_i=\Lambda_{u_i}a.
b_mDefinition passage · 11 passages
Search aliases: b_m
Definition passage: Euler-coordinate map · Stratified_Universality_Traditional.tex, line 4798
[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.
Exact mathematical passage
b=(b_m)_{m\geq1}\in\mathcal E_{\mathbb Z}
B_nDefinition passage · 1 passage
Search aliases: B_n
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 865
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A\to B=(B_1,\dots,B_n)
\bbKDefinition passage · 6 passages
Search aliases: bbK K mathbb
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1142
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
\Rep_{\mathbf d}(\bbK)
\BehDefinition passage · 2 passages
Search aliases: Beh
Definition passage: Resolved observational system · block17_semantic_quotients_final.tex, line 40
lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.
Exact mathematical passage
\Beh_{\mathsf S}:X\longrightarrow \prod_{(m,j)\in M\times J}Y_j, \qquad x\longmapsto\bigl(o_j(\rho(m)x)\bigr)_{m,j}.
\Beh_{\mathsf S}Definition passage · 2 passages
Search aliases: Beh_ S Beh mathsf
Definition passage: Resolved observational system · block17_semantic_quotients_final.tex, line 40
lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.
Exact mathematical passage
\Beh_{\mathsf S}:X\longrightarrow \prod_{(m,j)\in M\times J}Y_j, \qquad x\longmapsto\bigl(o_j(\rho(m)x)\bigr)_{m,j}.
\bmodDefinition passage · 8 passages
Search aliases: bmod
Definition passage: Prime-power section automaton · Stratified_Universality_Traditional.tex, line 4382
[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).
Exact mathematical passage
\mathcal K_{k,p^r}^{\mathrm{sem}}(a) = \left\{ \Lambda_w a\bmod p^rL_k(a): w\in\Sigma_k^* \right\} \subseteq \overline M_{k,p^r}(a).
\operatorname{Br}Definition passage · 4 passages
Search aliases: Br operatorname
Definition passage: Branch-separation index · block10_section.tex, line 32
[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},
Exact mathematical passage
F\in\operatorname{Br}_0(P_F)
\operatorname{Br}_0Definition passage · 3 passages
Search aliases: Br_0 operatorname
Definition passage: Branch-separation index · block10_section.tex, line 32
[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},
Exact mathematical passage
F\in\operatorname{Br}_0(P_F)
\bulletDefinition passage · 1 passage
Search aliases: bullet
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1652
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
D:C^\bullet\to E^\bullet
CDefinition passage · 179 passages
Search aliases: C
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 352
d ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a monoidal structure. A tag is not itself an existence theorem. item Cert is a list of named residual and certificate types. Each certificate type specif…
Exact mathematical passage
C^*
\mathcal CDefinition passage · 54 passages
Search aliases: C mathcal
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 361
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
\mathcal C
\mathfrak CDefinition passage · 3 passages
Search aliases: C mathfrak
Definition passage: Constructor universe · Stratified_Universality_Traditional.tex, line 741
[Constructor universe] Fix a declared universe C of constructors and universal properties. Typical entries are 0, 1, finite sums, composition, scalar action, biproducts,
Exact mathematical passage
\mathfrak C
\mathfrak cDefinition passage · 6 passages
Search aliases: c mathfrak
Definition passage: Root-filter alias and address-cancellation defects · block09_section.tex, line 325
[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.
Exact mathematical passage
\mathfrak a_N(F)=Nr-r_N^{\rm orb}, \qquad \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
\mathsf CDefinition passage · 10 passages
Search aliases: C mathsf
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
cDefinition passage · 160 passages
Search aliases: c
Definition passage: Relational support product · Stratified_Universality_Traditional.tex, line 666
[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.
Exact mathematical passage
F\circ E := \{(c,i):\text{there exists }a\in I_Y \text{ with }(c,a)\in F\text{ and }(a,i)\in E\}.
C^*Definition passage · 4 passages
Search aliases: C^*
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 352
d ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a monoidal structure. A tag is not itself an existence theorem. item Cert is a list of named residual and certificate types. Each certificate type specif…
Exact mathematical passage
C^*
C^\bulletDefinition passage · 1 passage
Search aliases: C^bullet bullet
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1652
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
D:C^\bullet\to E^\bullet
C^{n+1}Definition passage · 1 passage
Search aliases: C^n+1
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
\mathfrak c^{\rm tuple}_{N,j}Definition passage · 2 passages
Search aliases: c^rm tuple_N,j mathfrak rm
Definition passage: Algebraic packet collision defects · block10_section.tex, line 295
[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
Exact mathematical passage
\mathfrak c^{\rm tuple}_{N,j} = d^N-|\mathcal C_{N,j}|,
c_0Definition passage · 15 passages
Search aliases: c_0
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5454
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\mathcal M_w = \{c\in c_0(\mathbb Z_p):v_p(c_r)\geq w(r)\}, \qquad \mathcal C_w=\mathcal M_p(\mathcal M_w).
c_dDefinition passage · 4 passages
Search aliases: c_d
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4862
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
\frac1n\sum_{d\mid n}\mu\!\left(\frac nd\right)c_d\in\mathbb Z \qquad(n\geq1).
C_EDefinition passage · 13 passages
Search aliases: C_E
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 633
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
C_E(f):=\sum_{(a,i)\in E}q_afp_i, \qquad R_E(f):=f-C_E(f).
C_kDefinition passage · 20 passages
Search aliases: C_k
Definition passage: Integral saturation defect · Stratified_Universality_Traditional.tex, line 4095
[Integral saturation defect] The finite abelian group C_k(a) = L_k(a)/M_k(a) is the integral saturation defect. Its order δ_k(a) = [L_k(a):M_k(a)] is the saturation index.
Exact mathematical passage
C_k(a) = L_k(a)/M_k(a)
\mathsf C_kDefinition passage · 5 passages
Search aliases: C_k mathsf
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3802
[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathcal J_{k,K}^{\mathrm{reg}}(a) = \bigl( r_k(a), \mathsf C_k(a), \mathsf{SL}_{k,K}(a), [H_a[U,V]] \bigr), \label{eq:block05-regular-invariant}
c_nDefinition passage · 11 passages
Search aliases: c_n
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4857
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
c_n(b)=\sum_{d\mid n}d\,b_d, \qquad n\geq1.
\mathcal C_{N,j}Definition passage · 5 passages
Search aliases: C_N,j mathcal
Definition passage: Algebraic packet collision defects · block10_section.tex, line 295
[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
Exact mathematical passage
\mathfrak c^{\rm tuple}_{N,j} = d^N-|\mathcal C_{N,j}|,
\mathfrak c_{N,j}Definition passage · 2 passages
Search aliases: c_N,j mathfrak
Definition passage: Root-filter alias and address-cancellation defects · block09_section.tex, line 325
[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.
Exact mathematical passage
\mathfrak a_N(F)=Nr-r_N^{\rm orb}, \qquad \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
\mathsf C_pDefinition passage · 4 passages
Search aliases: C_p mathsf
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
c_rDefinition passage · 17 passages
Search aliases: c_r
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5450
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\nu_f(r)=v_p(c_r)\in\mathbb Z_{\geq0}\cup\{\infty\}.
\mathcal C_UDefinition passage · 3 passages
Search aliases: C_U mathcal
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
\mathcal C_wDefinition passage · 2 passages
Search aliases: C_w mathcal
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5454
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\mathcal M_w = \{c\in c_0(\mathbb Z_p):v_p(c_r)\geq w(r)\}, \qquad \mathcal C_w=\mathcal M_p(\mathcal M_w).
\mathcal C_\alphaDefinition passage · 3 passages
Search aliases: C_α mathcal alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\CatDefinition passage · 2 passages
Search aliases: Cat mathbf
Definition passage: Finite profiled resolved atlas · Stratified_Universality_Traditional.tex, line 1876
[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Se…
Exact mathematical passage
\mathfrak R:\check{\mathcal A}^{\mathrm{op}}\to\Cat;
\operatorname{cdim}Definition passage · 3 passages
Search aliases: cdim operatorname
Definition passage: Address-preserving cyclic dimension · block11_section.tex, line 283
[Address-preserving cyclic dimension] The cyclic packet dimension of the marked system is cdim_ P(Φ) = ∑_C cyclic SCC|C|. tag11.7 It measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.
Exact mathematical passage
\boxed{ \operatorname{cdim}_{\mathcal P}(\Phi) = \sum_{C\text{ cyclic SCC}}|C|.} \tag{11.7}
\operatorname{cdim}_{\mathcal P}Definition passage · 3 passages
Search aliases: cdim_ P operatorname mathcal
Definition passage: Address-preserving cyclic dimension · block11_section.tex, line 283
[Address-preserving cyclic dimension] The cyclic packet dimension of the marked system is cdim_ P(Φ) = ∑_C cyclic SCC|C|. tag11.7 It measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.
Exact mathematical passage
\boxed{ \operatorname{cdim}_{\mathcal P}(\Phi) = \sum_{C\text{ cyclic SCC}}|C|.} \tag{11.7}
\mathsf{Cert}Definition passage · 2 passages
Search aliases: Cert mathsf
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
\checkDefinition passage · 3 passages
Search aliases: check
Definition passage: Finite profiled resolved atlas · Stratified_Universality_Traditional.tex, line 1874
[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Se…
Exact mathematical passage
\check{\mathcal A}
\ClDefinition passage · 2 passages
Search aliases: Cl
Definition passage: Extensional task · block17_semantic_quotients_final.tex, line 235
[Extensional task] An extensional task is a closed query theory, equivalently a query family modulo equality of the closure Cl. When the admitted query universe separates the desired quotients, an extensional task can equivalently be represented by its observationally definable congruence.
Exact mathematical passage
\Cl
\operatorname{Coim}Definition passage · 1 passage
Search aliases: Coim operatorname
Definition passage: Resolved observational system · block17_semantic_quotients_final.tex, line 47
with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.
Exact mathematical passage
Q_{\mathsf S}:=\operatorname{Coim}(\Beh_{\mathsf S})
\cokerDefinition passage · 15 passages
Search aliases: coker coker operatorname
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1542
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D=V/\im D.
\operatorname{coker}Definition passage · 1 passage
Search aliases: coker operatorname
Definition passage: Ghost and missing reduction modules · Stratified_Universality_Traditional.tex, line 4188
[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.
Exact mathematical passage
Q_{k,m}(a) = \operatorname{coker} \left( M_k(a)/mM_k(a) \longrightarrow L_k(a)/mL_k(a) \right).
\operatorname{Coll}Definition passage · 2 passages
Search aliases: Coll operatorname
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
\operatorname{Coll}_BDefinition passage · 2 passages
Search aliases: Coll_B operatorname
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
\mathsf{Comp}Definition passage · 2 passages
Search aliases: Comp mathsf
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
\ConeDefinition passage · 3 passages
Search aliases: Cone operatorname
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
\CoreDefinition passage · 3 passages
Search aliases: Core operatorname
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1037
realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the native semantics functor. item I_α:Core( N_α)⟶ D_α is a declared invariant functor, when one has been constructed. Its codomain D_α and classification target are part of the record. The original four-field stratum ( C_α,Π_α, M_α,Sem_α) remains the…
Exact mathematical passage
\mathcal I_\alpha:\Core(\mathcal N_\alpha)\longrightarrow\mathcal D_\alpha
\operatorname{csupp}Definition passage · 2 passages
Search aliases: csupp operatorname
Definition passage: Proof-support certificate · Stratified_Universality_Traditional.tex, line 761
[Proof-support certificate] A proof-support certificate for a finite proof P is a finite subset csupp( P)⊆ C containing every constructor and universal property used in the proof. The certificate need not be minimal. It must be sufficient and auditable.
Exact mathematical passage
\operatorname{csupp}(\mathcal P)\subseteq\mathfrak C
DDefinition passage · 137 passages
Search aliases: D
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1538
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
D:U\longrightarrow V
\mathbf dDefinition passage · 8 passages
Search aliases: d mathbf
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1138
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
\mathbf d=(d_X)_{X\in\mathsf S}
\mathcal DDefinition passage · 15 passages
Search aliases: D mathcal
Definition passage: Verified preservation profile · Stratified_Universality_Traditional.tex, line 755
[Verified preservation profile] A verified preservation profile Π_F for a functor F: C→ D consists of a subset of C together with a proof or declared comparison isomorphism showing that F preserves each listed constructor. We write c∈Π_F only after the corresponding preservation claim has been verified.
Exact mathematical passage
F:\mathcal C\to\mathcal D
dDefinition passage · 225 passages
Search aliases: d
Definition passage: Karoubi envelope · Stratified_Universality_Traditional.tex, line 871
[Karoubi envelope] The Karoubi envelope Kar(Add( A_ K)) has pairs (A,e), where e:A→ A is idempotent, as objects. A morphism f:(A,e)⟶(B,d) is a morphism f:A→ B satisfying f=dfe. The identity on (A,e) is e.
Exact mathematical passage
f:(A,e)\longrightarrow(B,d)
D'Definition passage · 5 passages
Search aliases: D'
Definition passage: Doctrine-indexed task fiber · block19_admissibility_lifting_final.tex, line 327
[Doctrine-indexed task fiber] For a doctrine D and represented object X, let Task_D(X) denote the ordered family of continuation-stable semantic quotients admitted by the declared query universe. Inside one fiber, kernel or congruence calculus governs task change. A change of doctrine D⟶ D' requires a bridge passport specifying carrier transport, preserved constructions, comparison maps, and any lifting obstruction.
Exact mathematical passage
D\longrightarrow D'
D^jDefinition passage · 6 passages
Search aliases: D^j
Definition passage: Ordinary-point locus · block09_section.tex, line 181
[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.
Exact mathematical passage
L_F = D^r+\sum_{j=0}^{r-1}\ell_j(z)D^j
d^NDefinition passage · 23 passages
Search aliases: d^N
Definition passage: Algebraic packet collision defects · block10_section.tex, line 295
[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
Exact mathematical passage
\mathfrak c^{\rm tuple}_{N,j} = d^N-|\mathcal C_{N,j}|,
D^rDefinition passage · 5 passages
Search aliases: D^r
Definition passage: Ordinary-point locus · block09_section.tex, line 181
[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.
Exact mathematical passage
L_F = D^r+\sum_{j=0}^{r-1}\ell_j(z)D^j
d_0Definition passage · 15 passages
Search aliases: d_0
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
w=d_0\cdots d_{\ell-1}
d_1Definition passage · 12 passages
Search aliases: d_1
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4699
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathsf{SF}_k(a) = \left( d_1,\ldots,d_\rho; \operatorname{Fitt}_0(C_k(a)),\ldots, \operatorname{Fitt}_{\rho-1}(C_k(a)) \right).
d_CDefinition passage · 2 passages
Search aliases: d_C
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
d_{\Cone}Definition passage · 1 passage
Search aliases: d_Cone Cone operatorname
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
d_EDefinition passage · 2 passages
Search aliases: d_E
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
\mathcal D_KDefinition passage · 2 passages
Search aliases: D_K mathcal
Definition passage: Minimal differential annihilator · block09_section.tex, line 27
[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).
Exact mathematical passage
\operatorname{Ann}_{\mathcal D}(F) = \{L\in\mathcal D_K:LF=0\}.
d_XDefinition passage · 2 passages
Search aliases: d_X
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1138
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
\mathbf d=(d_X)_{X\in\mathsf S}
\mathcal D_\alphaDefinition passage · 2 passages
Search aliases: D_α mathcal alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1037
realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the native semantics functor. item I_α:Core( N_α)⟶ D_α is a declared invariant functor, when one has been constructed. Its codomain D_α and classification target are part of the record. The original four-field stratum ( C_α,Π_α, M_α,Sem_α) remains the…
Exact mathematical passage
\mathcal I_\alpha:\Core(\mathcal N_\alpha)\longrightarrow\mathcal D_\alpha
d_\rhoDefinition passage · 6 passages
Search aliases: d_ρ rho
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4699
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathsf{SF}_k(a) = \left( d_1,\ldots,d_\rho; \operatorname{Fitt}_0(C_k(a)),\ldots, \operatorname{Fitt}_{\rho-1}(C_k(a)) \right).
d_{\ell-1}Definition passage · 12 passages
Search aliases: d_ℓ-1 ell
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
w=d_0\cdots d_{\ell-1}
\DescDefinition passage · 22 passages
Search aliases: Desc Desc operatorname
Definition passage: Descent category · Stratified_Universality_Traditional.tex, line 1891
[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.
Exact mathematical passage
\Desc(\mathfrak R)
\dimDefinition passage · 13 passages
Search aliases: dim
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3094
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-rank} r_p(F)=\dim_{\mathbb F_p}\mathcal V_p(F)
\dim_{\mathbb F_p}Definition passage · 1 passage
Search aliases: dim_ F_p dim mathbb
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3094
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-rank} r_p(F)=\dim_{\mathbb F_p}\mathcal V_p(F)
\dim_KDefinition passage · 3 passages
Search aliases: dim_K dim
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3536
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
r_k(a)=\dim_K\mathcal V_k(a).
\mathfrak{Dir}Definition passage · 1 passage
Search aliases: Dir mathfrak
Definition passage: Typed directed transformation category · Stratified_Universality_Traditional.tex, line 1178
[Typed directed transformation category] The directed transformation category Dir( K) has pairs (α,M), with M∈ M_α, as objects. A morphism (α,M)⟶(β,N) is an admitted bridge instance equipped with a bridge passport as in [source reference]. Composition is defined only when the codomain profile of the first bridge satisfies the domain requirements of the second. The union of the representation groupoids embeds as the wide invertible subcategory.
Exact mathematical passage
\mathfrak{Dir}(\bbK)
\mathrm{DRLM}Definition passage · 10 passages
Search aliases: DRLM mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 47
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{DRLM},p}
EDefinition passage · 138 passages
Search aliases: E
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 632
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
E\subseteq I_{\mathcal Q}\times I_{\mathcal P}
\mathcal EDefinition passage · 9 passages
Search aliases: E mathcal
Definition passage: Euler-coordinate map · Stratified_Universality_Traditional.tex, line 4798
[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.
Exact mathematical passage
b=(b_m)_{m\geq1}\in\mathcal E_{\mathbb Z}
eDefinition passage · 159 passages
Search aliases: e
Definition passage: Karoubi envelope · Stratified_Universality_Traditional.tex, line 870
[Karoubi envelope] The Karoubi envelope Kar(Add( A_ K)) has pairs (A,e), where e:A→ A is idempotent, as objects. A morphism f:(A,e)⟶(B,d) is a morphism f:A→ B satisfying f=dfe. The identity on (A,e) is e.
Exact mathematical passage
(A,e)
E^\bulletDefinition passage · 1 passage
Search aliases: E^bullet bullet
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1652
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
D:C^\bullet\to E^\bullet
E^nDefinition passage · 2 passages
Search aliases: E^n
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
\mathcal E_{\mathbb Z}Definition passage · 5 passages
Search aliases: E_ Z mathcal mathbb
Definition passage: Euler-coordinate map · Stratified_Universality_Traditional.tex, line 4798
[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.
Exact mathematical passage
b=(b_m)_{m\geq1}\in\mathcal E_{\mathbb Z}
\mathrm{EM}Definition passage · 5 passages
Search aliases: EM mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 50
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{EM}}
\EndDefinition passage · 7 passages
Search aliases: End End operatorname
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
A_d\in\End_K(W)
\operatorname{End}Definition passage · 1 passage
Search aliases: End operatorname
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 405
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
\mathcal P=(p_i)_{i\in I}\subseteq\operatorname{End}(X)
\End_{\mathcal C}Definition passage · 1 passage
Search aliases: End_ C End mathcal
Definition passage: Finite profiled lane system · block15_section.tex, line 36
[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system …
Exact mathematical passage
\mathcal P_X^r=(p_{X,i}^r)_{i\in I_{X,r}} \subseteq\End_{\mathcal C}(X)
\End_KDefinition passage · 1 passage
Search aliases: End_K End operatorname
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
A_d\in\End_K(W)
\operatorname{EssIm}Definition passage · 2 passages
Search aliases: EssIm operatorname
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
\mathbf{EulMult}Definition passage · 4 passages
Search aliases: EulMult mathbf
Definition passage: Multiplicative Euler-overlap category · Stratified_Universality_Traditional.tex, line 5264
[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.
Exact mathematical passage
\mathbf{EulMult}_{\mathbb Z}
\mathbf{EulMult}_{\mathbb Z}Definition passage · 4 passages
Search aliases: EulMult_ Z mathbf mathbb
Definition passage: Multiplicative Euler-overlap category · Stratified_Universality_Traditional.tex, line 5264
[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.
Exact mathematical passage
\mathbf{EulMult}_{\mathbb Z}
\operatorname{ev}Definition passage · 2 passages
Search aliases: ev operatorname
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4685
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathcal I_k(a) = \left( V_{\mathbb Q}(a), M_k(a), L_k(a), C_k(a), (\Lambda_d)_{0\leq d<k}, a, \operatorname{ev}_0 \right).
\operatorname{ev}_0Definition passage · 1 passage
Search aliases: ev_0 operatorname
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4685
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathcal I_k(a) = \left( V_{\mathbb Q}(a), M_k(a), L_k(a), C_k(a), (\Lambda_d)_{0\leq d<k}, a, \operatorname{ev}_0 \right).
FDefinition passage · 277 passages
Search aliases: F
Definition passage: Relational support product · Stratified_Universality_Traditional.tex, line 665
[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.
Exact mathematical passage
F\subseteq I_Z\times I_Y
\mathbb FDefinition passage · 61 passages
Search aliases: F mathbb
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3094
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-rank} r_p(F)=\dim_{\mathbb F_p}\mathcal V_p(F)
fDefinition passage · 186 passages
Search aliases: f
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
f^{-1}Definition passage · 4 passages
Search aliases: f^-1
Definition passage: Task-compatible bridge · block18_carrier_observer_change_final.tex, line 93
[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.
Exact mathematical passage
N_X\subseteq f^{-1}(N_Y).
f_{ai}Definition passage · 5 passages
Search aliases: f_ai
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 517
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
f_{ai}:=\varpi_a f\iota_i:X_i\longrightarrow Y_a.
F_{f,p}Definition passage · 7 passages
Search aliases: F_f,p
Definition passage: Prime-power seed series · Stratified_Universality_Traditional.tex, line 5017
[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].
Exact mathematical passage
F_{f,p}(T) = \sum_{e\geq0}f(p^e)T^e \in1+T\mathbb Z[[T]].
F_mDefinition passage · 2 passages
Search aliases: F_m
Definition passage: Coefficientwise filtered product · Stratified_Universality_Traditional.tex, line 4763
[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.
Exact mathematical passage
F_m(z)\in 1+z^{w(m)}R[[z]], \qquad w(m)\longrightarrow\infty.
\mathbb F_pDefinition passage · 31 passages
Search aliases: F_p mathbb
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3094
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-rank} r_p(F)=\dim_{\mathbb F_p}\mathcal V_p(F)
F_\gammaDefinition passage · 10 passages
Search aliases: F_γ gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 65
ject A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π…
Exact mathematical passage
\beta^A_{\gamma,r,i}: F_\gamma(A^r_{U,i})\xrightarrow{\sim} A^r_{V,\lambda^A_{\gamma,r}(i)}.
\operatorname{Fitt}Definition passage · 3 passages
Search aliases: Fitt operatorname
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4699
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathsf{SF}_k(a) = \left( d_1,\ldots,d_\rho; \operatorname{Fitt}_0(C_k(a)),\ldots, \operatorname{Fitt}_{\rho-1}(C_k(a)) \right).
\operatorname{Fitt}_0Definition passage · 1 passage
Search aliases: Fitt_0 operatorname
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4699
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathsf{SF}_k(a) = \left( d_1,\ldots,d_\rho; \operatorname{Fitt}_0(C_k(a)),\ldots, \operatorname{Fitt}_{\rho-1}(C_k(a)) \right).
\operatorname{Fitt}_{\rho-1}Definition passage · 1 passage
Search aliases: Fitt_ρ-1 operatorname rho
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4699
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathsf{SF}_k(a) = \left( d_1,\ldots,d_\rho; \operatorname{Fitt}_0(C_k(a)),\ldots, \operatorname{Fitt}_{\rho-1}(C_k(a)) \right).
\FunDefinition passage · 2 passages
Search aliases: Fun operatorname
Definition passage: Profile-indexed resolved envelope · Stratified_Universality_Traditional.tex, line 964
[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functors.
Exact mathematical passage
\Mod_\Pi(\mathbb K,\mathcal C) \simeq \Fun_\Pi(\Syn_\Pi(\mathbb K),\mathcal C)
\Fun_\PiDefinition passage · 1 passage
Search aliases: Fun_Π Fun Pi operatorname Pi
Definition passage: Profile-indexed resolved envelope · Stratified_Universality_Traditional.tex, line 964
[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functors.
Exact mathematical passage
\Mod_\Pi(\mathbb K,\mathcal C) \simeq \Fun_\Pi(\Syn_\Pi(\mathbb K),\mathcal C)
\mathrm{Fuss}Definition passage · 8 passages
Search aliases: Fuss mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 49
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{Fuss}}
GDefinition passage · 144 passages
Search aliases: G
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1139
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
G_{\mathbf d}=\prod_{X\in\mathsf S}GL_{d_X}(\kk)
\mathcal GDefinition passage · 11 passages
Search aliases: G mathcal
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4861
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
\mathcal G_{\mathbb Z}
\mathfrak GDefinition passage · 5 passages
Search aliases: G mathfrak
Definition passage: Divisor--ghost transform · Stratified_Universality_Traditional.tex, line 5179
[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).
Exact mathematical passage
\mathfrak G(b) = \mathbf{1}\mathbin{\ast_D}(\operatorname{id}\cdot b),
gDefinition passage · 106 passages
Search aliases: g
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
G_{\mathbf d}Definition passage · 5 passages
Search aliases: G_ d mathbf
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1139
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
G_{\mathbf d}=\prod_{X\in\mathsf S}GL_{d_X}(\kk)
\mathcal G_{\mathbb Z}Definition passage · 5 passages
Search aliases: G_ Z mathcal mathbb
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4861
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
\mathcal G_{\mathbb Z}
G_{k,m}Definition passage · 3 passages
Search aliases: G_k,m
Definition passage: Ghost and missing reduction modules · Stratified_Universality_Traditional.tex, line 4177
[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.
Exact mathematical passage
G_{k,m}(a) = \ker \left( M_k(a)/mM_k(a) \longrightarrow L_k(a)/mL_k(a) \right)
G_{N,j}^{\lambda,\alpha}Definition passage · 4 passages
Search aliases: G_N,j^λ,α lambda alpha
Definition passage: Marked binomial--Fuchsian packet · block09_section.tex, line 747
[Marked binomial--Fuchsian packet] A marked packet is a tuple b=(α,λ,N,j;G), where α≠0, N≥1, 0≤ j<N, and G(x)=G_N,j^λ,α(x). The parameters (α,λ,N,j) are part of the semantic object rather than metadata to be forgotten.
Exact mathematical passage
G(x)=G_{N,j}^{\lambda,\alpha}(x).
\mathrm{Gal}Definition passage · 6 passages
Search aliases: Gal mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 51
component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{Gal},L/K}
HDefinition passage · 152 passages
Search aliases: H
Definition passage: Word-Hankel matrix · Stratified_Universality_Traditional.tex, line 3649
[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.
Exact mathematical passage
H_a(u,v) = a_{[u]_k+k^{|u|}[v]_k} = (\Lambda_u a)([v]_k), \qquad u,v\in\Sigma_k^*. \label{eq:block05-word-hankel}
hDefinition passage · 53 passages
Search aliases: h
Definition passage: Root-filter alias and address-cancellation defects · block09_section.tex, line 325
[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.
Exact mathematical passage
\mathfrak a_N(F)=Nr-r_N^{\rm orb}, \qquad \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
H_aDefinition passage · 13 passages
Search aliases: H_a
Definition passage: Word-Hankel matrix · Stratified_Universality_Traditional.tex, line 3649
[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.
Exact mathematical passage
H_a(u,v) = a_{[u]_k+k^{|u|}[v]_k} = (\Lambda_u a)([v]_k), \qquad u,v\in\Sigma_k^*. \label{eq:block05-word-hankel}
h_{a i}Definition passage · 2 passages
Search aliases: h_a i
Definition passage: Lane matrix category at a fixed resolution · block15_section.tex, line 510
y Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, as morphisms. Composition is matrix contraction and identities are diagonal identity matrices.
Exact mathematical passage
(h_{a i}), \qquad h_{a i}:X_i^r\to Y_a^r,
h_{ai}Definition passage · 2 passages
Search aliases: h_ai
Definition passage: Matrix local system · block16_section_final.tex, line 104
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
h=(h_{ai})\in\mathscr M^{u,r}_{Y,X}(U)
h_UDefinition passage · 7 passages
Search aliases: h_U
Definition passage: Matrix local system · block16_section_final.tex, line 115
⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right) := \left\{(h_U)_U:\gamma_*h_U=h_V \text{ for every }\gamma:U\to V\right\}.
h_VDefinition passage · 3 passages
Search aliases: h_V
Definition passage: Matrix local system · block16_section_final.tex, line 115
⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right) := \left\{(h_U)_U:\gamma_*h_U=h_V \text{ for every }\gamma:U\to V\right\}.
\mathbf{Hol}Definition passage · 1 passage
Search aliases: Hol mathbf
Definition passage: Ordinary-point locus · block09_section.tex, line 177
[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.
Exact mathematical passage
\mathbf{Hol}^{\rm ord}_K
\mathbf{Hol}^{\rm ord}_KDefinition passage · 1 passage
Search aliases: Hol^rm ord_K mathbf rm
Definition passage: Ordinary-point locus · block09_section.tex, line 177
[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.
Exact mathematical passage
\mathbf{Hol}^{\rm ord}_K
\HomDefinition passage · 11 passages
Search aliases: Hom Hom operatorname
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
\Hom_{\mathcal C_U}Definition passage · 1 passage
Search aliases: Hom_ C_U Hom mathcal
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
\Hom_{\mathcal U}Definition passage · 1 passage
Search aliases: Hom_ U Hom mathcal operatorname mathcal
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
IDefinition passage · 99 passages
Search aliases: I
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
\mathcal IDefinition passage · 9 passages
Search aliases: I mathcal
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
iDefinition passage · 177 passages
Search aliases: i
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
I_{\mathcal P}Definition passage · 1 passage
Search aliases: I_ P mathcal
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 632
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
E\subseteq I_{\mathcal Q}\times I_{\mathcal P}
I_{\mathcal Q}Definition passage · 1 passage
Search aliases: I_ Q mathcal
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 632
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
E\subseteq I_{\mathcal Q}\times I_{\mathcal P}
I_{A,U,r}Definition passage · 3 passages
Search aliases: I_A,U,r
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 55
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i}, \iota^r_{A,U,i},\pi^r_{A,U,i}\right)_{i\in I_{A,U,r}}
I_{A,V,r}Definition passage · 1 passage
Search aliases: I_A,V,r
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 61
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
\lambda^A_{\gamma,r}:I_{A,U,r}\xrightarrow{\sim}I_{A,V,r}
\mathcal I_kDefinition passage · 4 passages
Search aliases: I_k mathcal
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4685
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathcal I_k(a) = \left( V_{\mathbb Q}(a), M_k(a), L_k(a), C_k(a), (\Lambda_d)_{0\leq d<k}, a, \operatorname{ev}_0 \right).
I_rDefinition passage · 2 passages
Search aliases: I_r
Definition passage: Finite refinement · Stratified_Universality_Traditional.tex, line 587
[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.
Exact mathematical passage
\mathcal P^r=(p^r_i)_{i\in I_r}
I_sDefinition passage · 2 passages
Search aliases: I_s
Definition passage: Finite refinement · Stratified_Universality_Traditional.tex, line 587
[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.
Exact mathematical passage
\mathcal P^s=(p^s_j)_{j\in I_s}
I_XDefinition passage · 2 passages
Search aliases: I_X
Definition passage: Relational support product · Stratified_Universality_Traditional.tex, line 665
[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.
Exact mathematical passage
E\subseteq I_Y\times I_X
I_{X,r}Definition passage · 10 passages
Search aliases: I_X,r
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 343
ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such a…
Exact mathematical passage
\rho_{sr}:I_{X,s}\twoheadrightarrow I_{X,r}
I_{X,s}Definition passage · 3 passages
Search aliases: I_X,s
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 343
ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such a…
Exact mathematical passage
\rho_{sr}:I_{X,s}\twoheadrightarrow I_{X,r}
I_{X,U,r}Definition passage · 2 passages
Search aliases: I_X,U,r
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
I_YDefinition passage · 3 passages
Search aliases: I_Y
Definition passage: Relational support product · Stratified_Universality_Traditional.tex, line 665
[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.
Exact mathematical passage
E\subseteq I_Y\times I_X
I_{Y,U,u}Definition passage · 3 passages
Search aliases: I_Y,U,u
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
I_ZDefinition passage · 1 passage
Search aliases: I_Z
Definition passage: Relational support product · Stratified_Universality_Traditional.tex, line 665
[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.
Exact mathematical passage
F\subseteq I_Z\times I_Y
\mathcal I_\alphaDefinition passage · 4 passages
Search aliases: I_α mathcal alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
I_{\lambda,a}Definition passage · 3 passages
Search aliases: I_λ,a lambda
Definition passage: Coherent invariant family · block13_section.tex, line 140
[Coherent invariant family] The family (I_λ,a,κ_ab) is coherent when the two induced comparison maps from a to c agree on every triple overlap and the corresponding quadruple-overlap equation holds.
Exact mathematical passage
(I_{\lambda,a},\kappa_{ab})
\idDefinition passage · 4 passages
Search aliases: id mathrm
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\operatorname{id}Definition passage · 5 passages
Search aliases: id operatorname
Definition passage: Divisor--ghost transform · Stratified_Universality_Traditional.tex, line 5179
[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).
Exact mathematical passage
\mathfrak G(b) = \mathbf{1}\mathbin{\ast_D}(\operatorname{id}\cdot b),
\imDefinition passage · 30 passages
Search aliases: im im operatorname
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1542
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D=V/\im D.
\infDefinition passage · 1 passage
Search aliases: inf
Definition passage: Class-relative defect · block19_admissibility_lifting_final.tex, line 16
[Class-relative defect] Let X be a represented system, let Γ be a declared class of admissible auxiliary structures, and let d(X,g)∈[0,∞] be a defect functional. Define δ_Γ(X):=inf_g∈Γd(X,g). The pair consisting of the infimum and its attainment status is the class-relative defect profile.
Exact mathematical passage
\delta_\Gamma(X):=\inf_{g\in\Gamma}d(X,g).
\inf_{g\in\Gamma}Definition passage · 1 passage
Search aliases: inf_g∈Γ inf in Gamma
Definition passage: Class-relative defect · block19_admissibility_lifting_final.tex, line 16
[Class-relative defect] Let X be a represented system, let Γ be a declared class of admissible auxiliary structures, and let d(X,g)∈[0,∞] be a defect functional. Define δ_Γ(X):=inf_g∈Γd(X,g). The pair consisting of the infimum and its attainment status is the class-relative defect profile.
Exact mathematical passage
\delta_\Gamma(X):=\inf_{g\in\Gamma}d(X,g).
\mathrm{int}Definition passage · 1 passage
Search aliases: int mathrm
Definition passage: Three effectivity obstructions · Stratified_Universality_Traditional.tex, line 4560
[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even after integral effectivity holds, the lift need not be an actual radix section.
Exact mathematical passage
\epsilon_{\mathrm{int}}(x) = [x]\in M_{k,p}(a)/M_k(a).
JDefinition passage · 61 passages
Search aliases: J
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1341
abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\sigma_{\alpha\beta}: J_{\alpha\beta}\Sem_\alpha \Longrightarrow \Sem_\beta\Phi_{\alpha\beta}
\mathcal JDefinition passage · 27 passages
Search aliases: J mathcal
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3802
[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathcal J_{k,K}^{\mathrm{reg}}(a) = \bigl( r_k(a), \mathsf C_k(a), \mathsf{SL}_{k,K}(a), [H_a[U,V]] \bigr), \label{eq:block05-regular-invariant}
jDefinition passage · 257 passages
Search aliases: j
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 337
nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ …
Exact mathematical passage
p_{X,r,i}p_{X,r,j}=\delta_{ij}p_{X,r,i}, \qquad \sum_i p_{X,r,i}=1_X.
\mathcal J_{k,K}^{\mathrm{reg}}Definition passage · 1 passage
Search aliases: J_k,K^reg mathcal mathrm
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3802
[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathcal J_{k,K}^{\mathrm{reg}}(a) = \bigl( r_k(a), \mathsf C_k(a), \mathsf{SL}_{k,K}(a), [H_a[U,V]] \bigr), \label{eq:block05-regular-invariant}
J_{\rm alg}Definition passage · 4 passages
Search aliases: J_rm alg rm
Definition passage: Branch-separation index · block10_section.tex, line 43
[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\boxed{ J_{\rm alg}(F) = (a_0,a_1,\ldots,a_{\sigma-1}), \qquad F(z)=\sum_{n\geq0}a_nz^n.}
J_{s,t}Definition passage · 6 passages
Search aliases: J_s,t
Definition passage: Joint semantic completion · block18_carrier_observer_change_final.tex, line 47
[Joint semantic completion] Define j=(s,t): R⟶ S× T, J_s,t:=im(j). The projection p_s:J_s,t⟶im(s) is the relative completion map from the joint semantics to the source semantics.
Exact mathematical passage
j=(s,t):\mathcal R\longrightarrow S\times T, \qquad J_{s,t}:=\im(j).
J_{\alpha\beta}Definition passage · 2 passages
Search aliases: J_αβ alpha beta
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1341
abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\sigma_{\alpha\beta}: J_{\alpha\beta}\Sem_\alpha \Longrightarrow \Sem_\beta\Phi_{\alpha\beta}
\jmathDefinition passage · 3 passages
Search aliases: jmath
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
(\jmath_a,\varpi_a)
\jmath_aDefinition passage · 3 passages
Search aliases: jmath_a jmath
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
(\jmath_a,\varpi_a)
KDefinition passage · 208 passages
Search aliases: K
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\mathbb KDefinition passage · 20 passages
Search aliases: K mathbb
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
\mathcal KDefinition passage · 23 passages
Search aliases: K mathcal
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
kDefinition passage · 284 passages
Search aliases: k
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3525
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
k
k^{|u|}Definition passage · 1 passage
Search aliases: k^|u|
Definition passage: Word-Hankel matrix · Stratified_Universality_Traditional.tex, line 3649
[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.
Exact mathematical passage
H_a(u,v) = a_{[u]_k+k^{|u|}[v]_k} = (\Lambda_u a)([v]_k), \qquad u,v\in\Sigma_k^*. \label{eq:block05-word-hankel}
\mathcal K_kDefinition passage · 8 passages
Search aliases: K_k mathcal
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\mathcal K_{k,p^r}^{\mathrm{sem}}Definition passage · 3 passages
Search aliases: K_k,p^r^sem mathcal mathrm
Definition passage: Prime-power section automaton · Stratified_Universality_Traditional.tex, line 4382
[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).
Exact mathematical passage
\mathcal K_{k,p^r}^{\mathrm{sem}}(a) = \left\{ \Lambda_w a\bmod p^rL_k(a): w\in\Sigma_k^* \right\} \subseteq \overline M_{k,p^r}(a).
\mathbb K_{\rm fin}Definition passage · 2 passages
Search aliases: K_rm fin mathbb rm
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
\mathbb K_{\rm lin}Definition passage · 6 passages
Search aliases: K_rm lin mathbb rm
Definition passage: Linear grammar category · Stratified_Universality_Traditional.tex, line 852
[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.
Exact mathematical passage
\mathbb K_{\rm lin}=(Q,I,\mathsf P,\mathsf{Ref})
\KarDefinition passage · 3 passages
Search aliases: Kar operatorname
Definition passage: Karoubi envelope · Stratified_Universality_Traditional.tex, line 870
[Karoubi envelope] The Karoubi envelope Kar(Add( A_ K)) has pairs (A,e), where e:A→ A is idempotent, as objects. A morphism f:(A,e)⟶(B,d) is a morphism f:A→ B satisfying f=dfe. The identity on (A,e) is e.
Exact mathematical passage
\Kar(\Add(\mathcal A_{\mathbb K}))
\kerDefinition passage · 33 passages
Search aliases: ker
Definition passage: Ghost and missing reduction modules · Stratified_Universality_Traditional.tex, line 4177
[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.
Exact mathematical passage
G_{k,m}(a) = \ker \left( M_k(a)/mM_k(a) \longrightarrow L_k(a)/mL_k(a) \right)
\kkDefinition passage · 36 passages
Search aliases: kk kk 𝕜 Bbbk
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 326
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\kk
LDefinition passage · 191 passages
Search aliases: L
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1139
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
G_{\mathbf d}=\prod_{X\in\mathsf S}GL_{d_X}(\kk)
\mathcal LDefinition passage · 6 passages
Search aliases: L mathcal
Definition passage: Finite profiled lane system · block15_section.tex, line 34
[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system …
Exact mathematical passage
\mathcal L_X
\mathsf LDefinition passage · 1 passage
Search aliases: L mathsf
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3549
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
lDefinition passage · 22 passages
Search aliases: l
Definition passage: Linear grammar category · Stratified_Universality_Traditional.tex, line 852
[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.
Exact mathematical passage
\mathbb K_{\rm lin}=(Q,I,\mathsf P,\mathsf{Ref})
\mathcal L_{\mathsf R}Definition passage · 5 passages
Search aliases: L_ R mathcal mathsf
Definition passage: Lane matrix category at a fixed resolution · block15_section.tex, line 503
[Lane matrix category at a fixed resolution] Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, as morphisms. Composition is matrix contraction and identities are diagonal identity matrices.
Exact mathematical passage
\mathcal L_{\mathsf R}
L_{d_X}Definition passage · 1 passage
Search aliases: L_d_X
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1139
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
G_{\mathbf d}=\prod_{X\in\mathsf S}GL_{d_X}(\kk)
L_FDefinition passage · 8 passages
Search aliases: L_F
Definition passage: Minimal differential annihilator · block09_section.tex, line 33
[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).
Exact mathematical passage
\boxed{L_F.}
L_kDefinition passage · 31 passages
Search aliases: L_k
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
\mathcal L_XDefinition passage · 1 passage
Search aliases: L_X mathcal
Definition passage: Finite profiled lane system · block15_section.tex, line 34
[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system …
Exact mathematical passage
\mathcal L_X
\mathrm{Loc}Definition passage · 8 passages
Search aliases: Loc mathrm
Definition passage: Certified Global synthesis atlas atlas · block13_section.tex, line 48
[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple Galois component for the fixed extension L/K.
Exact mathematical passage
\mathfrak A_{\mathrm{Loc},p}
\LongleftrightarrowDefinition passage · 10 passages
Search aliases: Longleftrightarrow
Definition passage: Complete invariant on a declared target · Stratified_Universality_Traditional.tex, line 1065
[Complete invariant on a declared target] An invariant I_α is complete on N_α when I_α(X)≅ I_α(Y) Longleftrightarrow X≅ Y for all X,Y∈ N_α. This statement classifies the objects of N_α up to the isomorphisms allowed there. It does not classify arbitrary presentations outside that subcategory.
Exact mathematical passage
\mathcal I_\alpha(X)\cong\mathcal I_\alpha(Y) \quad\Longleftrightarrow\quad X\cong Y
\LongrightarrowDefinition passage · 8 passages
Search aliases: Longrightarrow
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1054
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
\nu_\alpha:\Sem_\alpha N_\alpha \Longrightarrow \Sem_\alpha|_{\mathcal U_\alpha}.
MDefinition passage · 189 passages
Search aliases: M
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 361
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
M_X
\mathcal MDefinition passage · 30 passages
Search aliases: M mathcal
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
mDefinition passage · 164 passages
Search aliases: m
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 862
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A=(A_1,\dots,A_m)
M^{u,r}_{Y,X}Definition passage · 8 passages
Search aliases: M^u,r_Y,X
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
M_aDefinition passage · 3 passages
Search aliases: M_a
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 362
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
M_a:M_{s(a)}\longrightarrow M_{t(a)}
M_kDefinition passage · 37 passages
Search aliases: M_k
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
M_{k,p}Definition passage · 14 passages
Search aliases: M_k,p
Definition passage: Three effectivity obstructions · Stratified_Universality_Traditional.tex, line 4556
[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even after integral effectivity holds, the lift need not be an actual radix section.
Exact mathematical passage
x\in M_{k,p}(a)
\mathcal M_pDefinition passage · 14 passages
Search aliases: M_p mathcal
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5449
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
f=\mathcal M_p(c)
M_{s(a)}Definition passage · 1 passage
Search aliases: M_s(a)
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 362
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
M_a:M_{s(a)}\longrightarrow M_{t(a)}
M_{t(a)}Definition passage · 1 passage
Search aliases: M_t(a)
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 362
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
M_a:M_{s(a)}\longrightarrow M_{t(a)}
\mathcal M_wDefinition passage · 1 passage
Search aliases: M_w mathcal
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5454
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\mathcal M_w = \{c\in c_0(\mathbb Z_p):v_p(c_r)\geq w(r)\}, \qquad \mathcal C_w=\mathcal M_p(\mathcal M_w).
M_XDefinition passage · 3 passages
Search aliases: M_X
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 361
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
M_X
M_\alphaDefinition passage · 1 passage
Search aliases: M_α alpha
Definition passage: Descent category · Stratified_Universality_Traditional.tex, line 1892
[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.
Exact mathematical passage
(M_\alpha,\theta_{\alpha\beta})
\mathcal M_\alphaDefinition passage · 11 passages
Search aliases: M_α mathcal alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\mathcal M_{\alpha\beta}Definition passage · 2 passages
Search aliases: M_αβ mathcal alpha beta
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1329
[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\mathcal M_{\alpha\beta}\subseteq\mathcal M_\alpha, \qquad \mathcal M_{\beta\alpha}\subseteq\mathcal M_\beta,
\mathcal M_\betaDefinition passage · 2 passages
Search aliases: M_β mathcal beta
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1208
from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the information-loss profile; item ν_ b is the sema…
Exact mathematical passage
\mathcal V_\mathfrak b\subseteq\mathcal M_\beta
\mathcal M_{\beta\alpha}Definition passage · 2 passages
Search aliases: M_βα mathcal beta alpha
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1329
[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\mathcal M_{\alpha\beta}\subseteq\mathcal M_\alpha, \qquad \mathcal M_{\beta\alpha}\subseteq\mathcal M_\beta,
\operatorname{Mat}Definition passage · 4 passages
Search aliases: Mat operatorname
Definition passage: Lane matrix category at a fixed resolution · block15_section.tex, line 507
[Lane matrix category at a fixed resolution] Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, as morphisms. Composition is matrix contraction and identities are diagonal identity matrices.
Exact mathematical passage
\operatorname{Mat}_r(\mathcal L_{\mathsf R})
\operatorname{Mat}_rDefinition passage · 3 passages
Search aliases: Mat_r operatorname
Definition passage: Lane matrix category at a fixed resolution · block15_section.tex, line 507
[Lane matrix category at a fixed resolution] Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, as morphisms. Composition is matrix contraction and identities are diagonal identity matrices.
Exact mathematical passage
\operatorname{Mat}_r(\mathcal L_{\mathsf R})
\operatorname{Mat}_sDefinition passage · 1 passage
Search aliases: Mat_s operatorname
Definition passage: Lane matrix category at a fixed resolution · block15_section.tex, line 518
For ssucceq r, define Γ_sr:Mat_s( L_ R) ⟶Mat_r( L_ R) by replacing each object tag with its r-tag and sending a fine matrix to the certified source-and-target coordinate change of [source reference].
Exact mathematical passage
\Gamma_{sr}:\operatorname{Mat}_s(\mathcal L_{\mathsf R}) \longrightarrow\operatorname{Mat}_r(\mathcal L_{\mathsf R})
\mathbinDefinition passage · 14 passages
Search aliases: mathbin
Definition passage: Divisor--ghost transform · Stratified_Universality_Traditional.tex, line 5179
[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).
Exact mathematical passage
\mathfrak G(b) = \mathbf{1}\mathbin{\ast_D}(\operatorname{id}\cdot b),
\mathscrDefinition passage · 21 passages
Search aliases: mathscr
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
\maxDefinition passage · 15 passages
Search aliases: max
Definition passage: Branch-separation index · block10_section.tex, line 33
[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F}, [3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\sigma(P_F,F) = \begin{cases} 0,&\operatorname{Br}_0(P_F)=\{F\},\\[3pt] 1+\displaystyle\max_{G\in\operatorname{Br}_0(P_F),\,G\neq F} \operatorname{ord}_z(F-G),&\text{otherwise}. \end{cases}
\max_{G\in\operatorname{Br}_0(P_F),\,G\neq F}Definition passage · 1 passage
Search aliases: max_G∈Br_0(P_F), G≠ F max in operatorname neq
Definition passage: Branch-separation index · block10_section.tex, line 33
[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F}, [3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\sigma(P_F,F) = \begin{cases} 0,&\operatorname{Br}_0(P_F)=\{F\},\\[3pt] 1+\displaystyle\max_{G\in\operatorname{Br}_0(P_F),\,G\neq F} \operatorname{ord}_z(F-G),&\text{otherwise}. \end{cases}
\midDefinition passage · 41 passages
Search aliases: mid
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4857
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
c_n(b)=\sum_{d\mid n}d\,b_d, \qquad n\geq1.
\ModDefinition passage · 2 passages
Search aliases: Mod operatorname
Definition passage: Profile-indexed resolved envelope · Stratified_Universality_Traditional.tex, line 964
[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functors.
Exact mathematical passage
\Mod_\Pi(\mathbb K,\mathcal C) \simeq \Fun_\Pi(\Syn_\Pi(\mathbb K),\mathcal C)
\mathbf{Mod}Definition passage · 3 passages
Search aliases: Mod mathbf
Definition passage: Semantic slice · block17_semantic_quotients_final.tex, line 19
athcal V, O) consisting of: [label=textup(roman*)] item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family O=(o_j)_j∈ J of observations in V. The slice belongs to the stratum passport. It does not replace the preservation profile Π. The profile specifies whic…
Exact mathematical passage
R\text{-}\mathbf{Mod}
\Mod_\PiDefinition passage · 1 passage
Search aliases: Mod_Π Mod Pi operatorname Pi
Definition passage: Profile-indexed resolved envelope · Stratified_Universality_Traditional.tex, line 964
[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functors.
Exact mathematical passage
\Mod_\Pi(\mathbb K,\mathcal C) \simeq \Fun_\Pi(\Syn_\Pi(\mathbb K),\mathcal C)
\operatorname{MorKer}Definition passage · 3 passages
Search aliases: MorKer operatorname
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
\operatorname{MorKer}_BDefinition passage · 3 passages
Search aliases: MorKer_B operatorname
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
\mathbf{Mult}Definition passage · 2 passages
Search aliases: Mult mathbf
Definition passage: Integral multiplicative category · Stratified_Universality_Traditional.tex, line 4991
[Integral multiplicative category] Let Mult_ Z be the discrete category whose objects are multiplicative arithmetic functions f: N_≥1→ Z satisfying f(1)=1, f(mn)=f(m)f(n) when (m,n)=1. The monoidal operation is Dirichlet convolution.
Exact mathematical passage
\mathbf{Mult}_{\mathbb Z}
\mathbf{Mult}_{\mathbb Z}Definition passage · 2 passages
Search aliases: Mult_ Z mathbf mathbb
Definition passage: Integral multiplicative category · Stratified_Universality_Traditional.tex, line 4991
[Integral multiplicative category] Let Mult_ Z be the discrete category whose objects are multiplicative arithmetic functions f: N_≥1→ Z satisfying f(1)=1, f(mn)=f(m)f(n) when (m,n)=1. The monoidal operation is Dirichlet convolution.
Exact mathematical passage
\mathbf{Mult}_{\mathbb Z}
NDefinition passage · 413 passages
Search aliases: N
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1050
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
N_\alpha:\mathcal U_\alpha\longrightarrow\mathcal N_\alpha
\mathbb NDefinition passage · 18 passages
Search aliases: N mathbb
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
\mathcal NDefinition passage · 11 passages
Search aliases: N mathcal
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
nDefinition passage · 258 passages
Search aliases: n
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
N_{\mathsf S}Definition passage · 8 passages
Search aliases: N_ S mathsf
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 65
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
N_{\mathsf S} := \bigcap_{m\in M}\bigcap_{j\in J} \ker(o_j\rho(m)).
N_XDefinition passage · 8 passages
Search aliases: N_X
Definition passage: Task-compatible bridge · block18_carrier_observer_change_final.tex, line 89
[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.
Exact mathematical passage
f(N_X)\subseteq N_Y.
N_YDefinition passage · 12 passages
Search aliases: N_Y
Definition passage: Task-compatible bridge · block18_carrier_observer_change_final.tex, line 89
[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.
Exact mathematical passage
f(N_X)\subseteq N_Y.
N_\alphaDefinition passage · 3 passages
Search aliases: N_α alpha
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1050
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
N_\alpha:\mathcal U_\alpha\longrightarrow\mathcal N_\alpha
\mathcal N_\alphaDefinition passage · 11 passages
Search aliases: N_α mathcal alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\mathbb N_{\geq1}Definition passage · 5 passages
Search aliases: N_≥1 mathbb geq
Definition passage: Integral multiplicative category · Stratified_Universality_Traditional.tex, line 4991
[Integral multiplicative category] Let Mult_ Z be the discrete category whose objects are multiplicative arithmetic functions f: N_≥1→ Z satisfying f(1)=1, f(mn)=f(m)f(n) when (m,n)=1. The monoidal operation is Dirichlet convolution.
Exact mathematical passage
f:\mathbb N_{\geq1}\to\mathbb Z
\mathcal ODefinition passage · 38 passages
Search aliases: O mathcal
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1032
tegory in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the native semantics functor. item I_α:Core( N_α)⟶ D_α is a declared invariant functor, when one has been constructed. Its codomain D_α and classification …
Exact mathematical passage
\Sem_\alpha:\mathcal M_\alpha\longrightarrow\mathcal O_\alpha
\mathfrak ODefinition passage · 1 passage
Search aliases: O mathfrak
Definition passage: Finite profiled resolved atlas · Stratified_Universality_Traditional.tex, line 1882
hcal A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Sem: R⇒ O into a common observable doctrine; item optional pseudonatural invariant families with explicitly typed comparison cells.
Exact mathematical passage
\Sem:\mathfrak R\Rightarrow\mathfrak O
oDefinition passage · 49 passages
Search aliases: o
Definition passage: Ordinary-point locus · block09_section.tex, line 177
[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.
Exact mathematical passage
\mathbf{Hol}^{\rm ord}_K
o_jDefinition passage · 12 passages
Search aliases: o_j
Definition passage: Semantic slice · block17_semantic_quotients_final.tex, line 20
item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family O=(o_j)_j∈ J of observations in V. The slice belongs to the stratum passport. It does not replace the preservation profile Π. The profile specifies which constructions a bridge preserves, while the slice…
Exact mathematical passage
\mathcal O=(o_j)_{j\in J}
\mathcal O_kDefinition passage · 3 passages
Search aliases: O_k mathcal
Definition passage: Three effectivity obstructions · Stratified_Universality_Traditional.tex, line 4567
effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even after integral effectivity holds, the lift need not be an actual radix section.
Exact mathematical passage
M_k(a)\setminus\mathcal O_k(a).
\mathcal O_\alphaDefinition passage · 1 passage
Search aliases: O_α mathcal alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1032
tegory in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the native semantics functor. item I_α:Core( N_α)⟶ D_α is a declared invariant functor, when one has been constructed. Its codomain D_α and classification …
Exact mathematical passage
\Sem_\alpha:\mathcal M_\alpha\longrightarrow\mathcal O_\alpha
\ObDefinition passage · 4 passages
Search aliases: Ob operatorname
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1542
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D=V/\im D.
\operatorname{Ob}Definition passage · 1 passage
Search aliases: Ob operatorname
Definition passage: Information-loss profile · Stratified_Universality_Traditional.tex, line 1246
[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) &= \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\ \operatorname{Coll}_B &= \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\ \operatorname{EssIm}(B) &\subseteq\mathcal V,
\Ob_DDefinition passage · 4 passages
Search aliases: Ob_D Ob operatorname
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1542
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D=V/\im D.
\mathrm{op}Definition passage · 2 passages
Search aliases: op mathrm
Definition passage: Finite profiled resolved atlas · Stratified_Universality_Traditional.tex, line 1876
[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Se…
Exact mathematical passage
\mathfrak R:\check{\mathcal A}^{\mathrm{op}}\to\Cat;
\operatorname{ord}Definition passage · 10 passages
Search aliases: ord operatorname
Definition passage: Minimal differential annihilator · block09_section.tex, line 37
[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).
Exact mathematical passage
r_D(F)=\operatorname{ord}_D L_F
\operatorname{ord}_DDefinition passage · 1 passage
Search aliases: ord_D operatorname
Definition passage: Minimal differential annihilator · block09_section.tex, line 37
[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).
Exact mathematical passage
r_D(F)=\operatorname{ord}_D L_F
\operatorname{ord}_zDefinition passage · 1 passage
Search aliases: ord_z operatorname
Definition passage: Branch-separation index · block10_section.tex, line 33
[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F}, [3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\sigma(P_F,F) = \begin{cases} 0,&\operatorname{Br}_0(P_F)=\{F\},\\[3pt] 1+\displaystyle\max_{G\in\operatorname{Br}_0(P_F),\,G\neq F} \operatorname{ord}_z(F-G),&\text{otherwise}. \end{cases}
\overline fDefinition passage · 13 passages
Search aliases: overline f overline
Definition passage: Task-compatible bridge · block18_carrier_observer_change_final.tex, line 97
[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.
Exact mathematical passage
\overline f:X/N_X\longrightarrow Y/N_Y.
\overline MDefinition passage · 9 passages
Search aliases: overline M overline
Definition passage: Prime-power section automaton · Stratified_Universality_Traditional.tex, line 4382
[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).
Exact mathematical passage
\mathcal K_{k,p^r}^{\mathrm{sem}}(a) = \left\{ \Lambda_w a\bmod p^rL_k(a): w\in\Sigma_k^* \right\} \subseteq \overline M_{k,p^r}(a).
\overline M_{k,p^r}Definition passage · 7 passages
Search aliases: overline M_k,p^r overline
Definition passage: Prime-power section automaton · Stratified_Universality_Traditional.tex, line 4382
[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).
Exact mathematical passage
\mathcal K_{k,p^r}^{\mathrm{sem}}(a) = \left\{ \Lambda_w a\bmod p^rL_k(a): w\in\Sigma_k^* \right\} \subseteq \overline M_{k,p^r}(a).
\oversetDefinition passage · 5 passages
Search aliases: overset
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1335
[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\Phi_{\alpha\beta}:\mathcal M_{\alpha\beta} \overset{\sim}{\longrightarrow} \mathcal M_{\beta\alpha},
PDefinition passage · 99 passages
Search aliases: P
Definition passage: Branch-separation index · block10_section.tex, line 32
[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},
Exact mathematical passage
F\in\operatorname{Br}_0(P_F)
\mathcal PDefinition passage · 39 passages
Search aliases: P mathcal
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 405
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
\mathcal P=(p_i)_{i\in I}\subseteq\operatorname{End}(X)
\mathsf PDefinition passage · 15 passages
Search aliases: P mathsf
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
pDefinition passage · 547 passages
Search aliases: p
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 336
item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r,…
Exact mathematical passage
p_{X,r,i}
p'_{B(X),a}Definition passage · 1 passage
Search aliases: p'_B(X),a
Definition passage: Bridge-law certificate · Stratified_Universality_Traditional.tex, line 1362
cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under which B(p_Y,bfp_X,a) ≅ p'_B(Y),b B(f) p'_B(X),a, and refinement naturality requires the image of each coarse packet to equal the declared sum of its re…
Exact mathematical passage
B(p_{Y,b}fp_{X,a}) \cong p'_{B(Y),b}\,B(f)\,p'_{B(X),a},
p'_{B(Y),b}Definition passage · 1 passage
Search aliases: p'_B(Y),b
Definition passage: Bridge-law certificate · Stratified_Universality_Traditional.tex, line 1362
cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under which B(p_Y,bfp_X,a) ≅ p'_B(Y),b B(f) p'_B(X),a, and refinement naturality requires the image of each coarse packet to equal the declared sum of its re…
Exact mathematical passage
B(p_{Y,b}fp_{X,a}) \cong p'_{B(Y),b}\,B(f)\,p'_{B(X),a},
p^eDefinition passage · 17 passages
Search aliases: p^e
Definition passage: Prime-power seed series · Stratified_Universality_Traditional.tex, line 5017
[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].
Exact mathematical passage
F_{f,p}(T) = \sum_{e\geq0}f(p^e)T^e \in1+T\mathbb Z[[T]].
\mathcal P^rDefinition passage · 1 passage
Search aliases: P^r mathcal
Definition passage: Finite refinement · Stratified_Universality_Traditional.tex, line 587
[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.
Exact mathematical passage
\mathcal P^r=(p^r_i)_{i\in I_r}
p^rDefinition passage · 30 passages
Search aliases: p^r
Definition passage: Prime-power section automaton · Stratified_Universality_Traditional.tex, line 4381
[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).
Exact mathematical passage
p^r
p^{r}_{A,U,i}Definition passage · 1 passage
Search aliases: p^r_A,U,i
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 55
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i}, \iota^r_{A,U,i},\pi^r_{A,U,i}\right)_{i\in I_{A,U,r}}
p^r_iDefinition passage · 4 passages
Search aliases: p^r_i
Definition passage: Finite refinement · Stratified_Universality_Traditional.tex, line 587
[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.
Exact mathematical passage
\mathcal P^r=(p^r_i)_{i\in I_r}
\mathcal P^sDefinition passage · 1 passage
Search aliases: P^s mathcal
Definition passage: Finite refinement · Stratified_Universality_Traditional.tex, line 587
[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.
Exact mathematical passage
\mathcal P^s=(p^s_j)_{j\in I_s}
p^s_jDefinition passage · 4 passages
Search aliases: p^s_j
Definition passage: Finite refinement · Stratified_Universality_Traditional.tex, line 587
[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.
Exact mathematical passage
\mathcal P^s=(p^s_j)_{j\in I_s}
P_FDefinition passage · 27 passages
Search aliases: P_F
Definition passage: Branch-separation index · block10_section.tex, line 32
[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},
Exact mathematical passage
F\in\operatorname{Br}_0(P_F)
p_iDefinition passage · 13 passages
Search aliases: p_i
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 405
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
\mathcal P=(p_i)_{i\in I}\subseteq\operatorname{End}(X)
p_i^2Definition passage · 1 passage
Search aliases: p_i^2
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 409
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
p_i^2=p_i, \qquad p_ip_j=0\quad(i\ne j), \qquad \sum_{i\in I}p_i=1_X.
p_jDefinition passage · 2 passages
Search aliases: p_j
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 409
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
p_i^2=p_i, \qquad p_ip_j=0\quad(i\ne j), \qquad \sum_{i\in I}p_i=1_X.
p_sDefinition passage · 11 passages
Search aliases: p_s
Definition passage: Joint semantic completion · block18_carrier_observer_change_final.tex, line 53
[Joint semantic completion] Define j=(s,t): R⟶ S× T, J_s,t:=im(j). The projection p_s:J_s,t⟶im(s) is the relative completion map from the joint semantics to the source semantics.
Exact mathematical passage
p_s:J_{s,t}\longrightarrow\im(s)
p_{X,a}Definition passage · 1 passage
Search aliases: p_X,a
Definition passage: Bridge-law certificate · Stratified_Universality_Traditional.tex, line 1362
cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under which B(p_Y,bfp_X,a) ≅ p'_B(Y),b B(f) p'_B(X),a, and refinement naturality requires the image of each coarse packet to equal the declared sum of its re…
Exact mathematical passage
B(p_{Y,b}fp_{X,a}) \cong p'_{B(Y),b}\,B(f)\,p'_{B(X),a},
p_{X,i}^rDefinition passage · 9 passages
Search aliases: p_X,i^r
Definition passage: Finite profiled lane system · block15_section.tex, line 36
[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system …
Exact mathematical passage
\mathcal P_X^r=(p_{X,i}^r)_{i\in I_{X,r}} \subseteq\End_{\mathcal C}(X)
p_{X,j}^rDefinition passage · 1 passage
Search aliases: p_X,j^r
Definition passage: Finite profiled lane system · block15_section.tex, line 41
block15-finite-profiled-lane-system Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the s…
Exact mathematical passage
(p_{X,i}^r)^2=p_{X,i}^r, \qquad p_{X,i}^rp_{X,j}^r=0\quad(i\ne j), \qquad \sum_{i\in I_{X,r}}p_{X,i}^r=1_X,
p_{X,j}^sDefinition passage · 2 passages
Search aliases: p_X,j^s
Definition passage: Finite profiled lane system · block15_section.tex, line 62
p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address sets and refinement maps are declared data; they are not inferred from their cardinalities. Its verified profile contains finite biproducts, addition of …
Exact mathematical passage
p_{X,i}^r = \sum_{\rho^X_{sr}(j)=i}p_{X,j}^s, \qquad \rho^X_{tr}=\rho^X_{sr}\rho^X_{ts}.
p_{X,r,i}Definition passage · 3 passages
Search aliases: p_X,r,i
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 336
item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r,…
Exact mathematical passage
p_{X,r,i}
p_{X,r,j}Definition passage · 1 passage
Search aliases: p_X,r,j
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 337
nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ …
Exact mathematical passage
p_{X,r,i}p_{X,r,j}=\delta_{ij}p_{X,r,i}, \qquad \sum_i p_{X,r,i}=1_X.
p_{X,s,j}Definition passage · 1 passage
Search aliases: p_X,s,j
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 347
is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a m…
Exact mathematical passage
p_{X,r,i} = \sum_{\rho_{sr}(j)=i}p_{X,s,j}.
\mathcal P_X^rDefinition passage · 1 passage
Search aliases: P_X^r mathcal
Definition passage: Finite profiled lane system · block15_section.tex, line 36
[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system …
Exact mathematical passage
\mathcal P_X^r=(p_{X,i}^r)_{i\in I_{X,r}} \subseteq\End_{\mathcal C}(X)
p_{Y,b}Definition passage · 1 passage
Search aliases: p_Y,b
Definition passage: Bridge-law certificate · Stratified_Universality_Traditional.tex, line 1362
cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under which B(p_Y,bfp_X,a) ≅ p'_B(Y),b B(f) p'_B(X),a, and refinement naturality requires the image of each coarse packet to equal the declared sum of its re…
Exact mathematical passage
B(p_{Y,b}fp_{X,a}) \cong p'_{B(Y),b}\,B(f)\,p'_{B(X),a},
\mathrm{pres}Definition passage · 2 passages
Search aliases: pres mathrm
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
QDefinition passage · 110 passages
Search aliases: Q
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
\mathbb QDefinition passage · 32 passages
Search aliases: Q mathbb
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
\mathcal QDefinition passage · 14 passages
Search aliases: Q mathcal
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
\mathcal Q=(q_a)
qDefinition passage · 167 passages
Search aliases: q
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
\mathcal Q=(q_a)
\mathbb Q^{\mathbb N}Definition passage · 4 passages
Search aliases: Q^ N mathbb mathbb
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4052
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
\mathbb Q^{\mathbb N}
Q_{\mathsf S}Definition passage · 6 passages
Search aliases: Q_ S mathsf
Definition passage: Resolved observational system · block17_semantic_quotients_final.tex, line 47
with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.
Exact mathematical passage
Q_{\mathsf S}:=\operatorname{Coim}(\Beh_{\mathsf S})
Q_0Definition passage · 3 passages
Search aliases: Q_0
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 334
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
Q=(Q_0,Q_1,s,t)
Q_1Definition passage · 2 passages
Search aliases: Q_1
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 334
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
Q=(Q_0,Q_1,s,t)
q_aDefinition passage · 7 passages
Search aliases: q_a
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
\mathcal Q=(q_a)
Q_{k,m}Definition passage · 3 passages
Search aliases: Q_k,m
Definition passage: Ghost and missing reduction modules · Stratified_Universality_Traditional.tex, line 4188
[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.
Exact mathematical passage
Q_{k,m}(a) = \operatorname{coker} \left( M_k(a)/mM_k(a) \longrightarrow L_k(a)/mL_k(a) \right).
RDefinition passage · 103 passages
Search aliases: R
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 633
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
C_E(f):=\sum_{(a,i)\in E}q_afp_i, \qquad R_E(f):=f-C_E(f).
\mathcal RDefinition passage · 18 passages
Search aliases: R mathcal
Definition passage: Joint semantic completion · block18_carrier_observer_change_final.tex, line 47
[Joint semantic completion] Define j=(s,t): R⟶ S× T, J_s,t:=im(j). The projection p_s:J_s,t⟶im(s) is the relative completion map from the joint semantics to the source semantics.
Exact mathematical passage
j=(s,t):\mathcal R\longrightarrow S\times T, \qquad J_{s,t}:=\im(j).
\mathfrak RDefinition passage · 30 passages
Search aliases: R mathfrak
Definition passage: Finite profiled resolved atlas · Stratified_Universality_Traditional.tex, line 1876
[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Se…
Exact mathematical passage
\mathfrak R:\check{\mathcal A}^{\mathrm{op}}\to\Cat;
\mathsf RDefinition passage · 15 passages
Search aliases: R mathsf
Definition passage: Finite resolution shape · block15_section.tex, line 25
[Finite resolution shape] A finite resolution shape is a finite partially ordered set R. We write ssucceq r when s is finer than r. The shape contains no address sets: object-specific addresses and their refinement maps are declared in the lane system below.
Exact mathematical passage
\mathsf R
rDefinition passage · 397 passages
Search aliases: r
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 336
item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r,…
Exact mathematical passage
p_{X,r,i}
R_aDefinition passage · 9 passages
Search aliases: R_a
Definition passage: Minimal tail recurrence · block09_section.tex, line 51
[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.
Exact mathematical passage
\boxed{R_a.}
r_DDefinition passage · 11 passages
Search aliases: r_D
Definition passage: Minimal differential annihilator · block09_section.tex, line 37
[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).
Exact mathematical passage
r_D(F)=\operatorname{ord}_D L_F
R_EDefinition passage · 8 passages
Search aliases: R_E
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 633
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
C_E(f):=\sum_{(a,i)\in E}q_afp_i, \qquad R_E(f):=f-C_E(f).
r_kDefinition passage · 24 passages
Search aliases: r_k
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3536
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
r_k(a)=\dim_K\mathcal V_k(a).
r_{N,j}^{\rm char}Definition passage · 6 passages
Search aliases: r_N,j^rm char rm
Definition passage: Root-filter alias and address-cancellation defects · block09_section.tex, line 325
[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.
Exact mathematical passage
\mathfrak a_N(F)=Nr-r_N^{\rm orb}, \qquad \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
r_N^{\rm orb}Definition passage · 5 passages
Search aliases: r_N^rm orb rm
Definition passage: Root-filter alias and address-cancellation defects · block09_section.tex, line 325
[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.
Exact mathematical passage
\mathfrak a_N(F)=Nr-r_N^{\rm orb}, \qquad \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
r_pDefinition passage · 6 passages
Search aliases: r_p
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3094
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-rank} r_p(F)=\dim_{\mathbb F_p}\mathcal V_p(F)
R_{T\mid S}Definition passage · 4 passages
Search aliases: R_Tmid S mid
Definition passage: Relative semantic residual · block17_semantic_quotients_final.tex, line 362
[Relative semantic residual] The information needed to augment S so that it also retains T is R_Tmid S := ker(J→ S) ≅ N/(N∩ M) ≅ (N+M)/M.
Exact mathematical passage
R_{T\mid S} := \ker(J\to S) \cong N/(N\cap M) \cong (N+M)/M.
\mathsf{Ref}Definition passage · 3 passages
Search aliases: Ref mathsf
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
\mathrm{refl}Definition passage · 2 passages
Search aliases: refl mathrm
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\mathbf{Reg}Definition passage · 5 passages
Search aliases: Reg mathbf
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3813
ular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathbf{Reg}_k(K)
\mathrm{reg}Definition passage · 1 passage
Search aliases: reg mathrm
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3802
[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathcal J_{k,K}^{\mathrm{reg}}(a) = \bigl( r_k(a), \mathsf C_k(a), \mathsf{SL}_{k,K}(a), [H_a[U,V]] \bigr), \label{eq:block05-regular-invariant}
\mathbf{Reg}_kDefinition passage · 3 passages
Search aliases: Reg_k mathbf
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3813
ular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathbf{Reg}_k(K)
\RepDefinition passage · 3 passages
Search aliases: Rep operatorname
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1142
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
\Rep_{\mathbf d}(\bbK)
\Rep_{\mathbf d}Definition passage · 3 passages
Search aliases: Rep_ d Rep mathbf operatorname mathbf
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1142
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
\Rep_{\mathbf d}(\bbK)
\rmDefinition passage · 56 passages
Search aliases: rm
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 327
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
\mathbb K_{\rm fin} = (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
SDefinition passage · 67 passages
Search aliases: S
Definition passage: Relative semantic residual · block17_semantic_quotients_final.tex, line 361
[Relative semantic residual] The information needed to augment S so that it also retains T is R_Tmid S := ker(J→ S) ≅ N/(N∩ M) ≅ (N+M)/M.
Exact mathematical passage
S
\mathbf SDefinition passage · 1 passage
Search aliases: S mathbf
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\mathcal SDefinition passage · 3 passages
Search aliases: S mathcal
Definition passage: Minimal tail recurrence · block09_section.tex, line 45
[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.
Exact mathematical passage
\operatorname{Ann}_{\rm tail}(a) = \{R\in\mathcal S_K:(Ra)_n=0\text{ for all sufficiently large }n\}.
\mathfrak sDefinition passage · 4 passages
Search aliases: s mathfrak
Definition passage: Algebraic packet collision defects · block10_section.tex, line 301
[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
Exact mathematical passage
\mathfrak s^{\rm Gal}_{N,j} = |\mathcal C_{N,j}|-|\Gamma\cdot U_j|.
\mathsf SDefinition passage · 25 passages
Search aliases: S mathsf
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1138
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
\mathbf d=(d_X)_{X\in\mathsf S}
sDefinition passage · 211 passages
Search aliases: s
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 334
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
Q=(Q_0,Q_1,s,t)
\mathfrak s^{\rm Gal}_{N,j}Definition passage · 2 passages
Search aliases: s^rm Gal_N,j mathfrak rm
Definition passage: Algebraic packet collision defects · block10_section.tex, line 301
[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
Exact mathematical passage
\mathfrak s^{\rm Gal}_{N,j} = |\mathcal C_{N,j}|-|\Gamma\cdot U_j|.
\mathcal S_KDefinition passage · 2 passages
Search aliases: S_K mathcal
Definition passage: Minimal tail recurrence · block09_section.tex, line 45
[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.
Exact mathematical passage
\operatorname{Ann}_{\rm tail}(a) = \{R\in\mathcal S_K:(Ra)_n=0\text{ for all sufficiently large }n\}.
\mathbf S_\alphaDefinition passage · 1 passage
Search aliases: S_α mathbf alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\SemDefinition passage · 8 passages
Search aliases: Sem operatorname
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\mathrm{sem}Definition passage · 5 passages
Search aliases: sem mathrm
Definition passage: Prime-power section automaton · Stratified_Universality_Traditional.tex, line 4382
[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).
Exact mathematical passage
\mathcal K_{k,p^r}^{\mathrm{sem}}(a) = \left\{ \Lambda_w a\bmod p^rL_k(a): w\in\Sigma_k^* \right\} \subseteq \overline M_{k,p^r}(a).
\Sem_\alphaDefinition passage · 6 passages
Search aliases: Sem_α Sem alpha operatorname alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\Sem_\betaDefinition passage · 1 passage
Search aliases: Sem_β Sem beta operatorname beta
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1341
abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\sigma_{\alpha\beta}: J_{\alpha\beta}\Sem_\alpha \Longrightarrow \Sem_\beta\Phi_{\alpha\beta}
\mathbf{Set}Definition passage · 4 passages
Search aliases: Set mathbf
Definition passage: Semantic slice · block17_semantic_quotients_final.tex, line 19
al A, G, V, O) consisting of: [label=textup(roman*)] item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family O=(o_j)_j∈ J of observations in V. The slice belongs to the stratum passport. It does not replace the preservation profile Π. The profile specifies wh…
Exact mathematical passage
\mathbf{Set}
\mathsf{SF}Definition passage · 1 passage
Search aliases: SF mathsf
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4699
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathsf{SF}_k(a) = \left( d_1,\ldots,d_\rho; \operatorname{Fitt}_0(C_k(a)),\ldots, \operatorname{Fitt}_{\rho-1}(C_k(a)) \right).
\mathsf{SF}_kDefinition passage · 1 passage
Search aliases: SF_k mathsf
Definition passage: Canonical integral regular record · Stratified_Universality_Traditional.tex, line 4699
[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).
Exact mathematical passage
\mathsf{SF}_k(a) = \left( d_1,\ldots,d_\rho; \operatorname{Fitt}_0(C_k(a)),\ldots, \operatorname{Fitt}_{\rho-1}(C_k(a)) \right).
\mathsf{SL}Definition passage · 5 passages
Search aliases: SL mathsf
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3802
[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathcal J_{k,K}^{\mathrm{reg}}(a) = \bigl( r_k(a), \mathsf C_k(a), \mathsf{SL}_{k,K}(a), [H_a[U,V]] \bigr), \label{eq:block05-regular-invariant}
\mathsf{SL}_{k,K}Definition passage · 3 passages
Search aliases: SL_k,K mathsf
Definition passage: Fixed-radix regular invariant · Stratified_Universality_Traditional.tex, line 3802
[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).
Exact mathematical passage
\mathcal J_{k,K}^{\mathrm{reg}}(a) = \bigl( r_k(a), \mathsf C_k(a), \mathsf{SL}_{k,K}(a), [H_a[U,V]] \bigr), \label{eq:block05-regular-invariant}
\SpanDefinition passage · 6 passages
Search aliases: Span Span span operatorname
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\operatorname{span}Definition passage · 6 passages
Search aliases: span operatorname
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
\operatorname{span}_{\mathbb Z}Definition passage · 1 passage
Search aliases: span_ Z operatorname mathbb
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
\Span_KDefinition passage · 4 passages
Search aliases: Span_K Span Span_K span_K Span operatorname
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\succeqDefinition passage · 15 passages
Search aliases: succeq
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 342
ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine …
Exact mathematical passage
s\succeq r
\SynDefinition passage · 7 passages
Search aliases: Syn operatorname
Definition passage: Profile-indexed resolved envelope · Stratified_Universality_Traditional.tex, line 963
[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functors.
Exact mathematical passage
\Syn_\Pi(\mathbb K)
\Syn_\PiDefinition passage · 2 passages
Search aliases: Syn_Π Syn Pi operatorname Pi
Definition passage: Profile-indexed resolved envelope · Stratified_Universality_Traditional.tex, line 963
[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functors.
Exact mathematical passage
\Syn_\Pi(\mathbb K)
\SynthesisDefinition passage · 5 passages
Search aliases: Synthesis Synthesis S mathcal
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 441
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Synthesis_{\mathcal P} := (\iota_i)_{i\in I}: \bigoplus_{i\in I}X_i\longrightarrow X.
\Synthesis_{\mathcal P}Definition passage · 3 passages
Search aliases: Synthesis_ P S_ P Synthesis mathcal mathcal mathcal
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 441
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Synthesis_{\mathcal P} := (\iota_i)_{i\in I}: \bigoplus_{i\in I}X_i\longrightarrow X.
TDefinition passage · 127 passages
Search aliases: T
Definition passage: Prime-power seed series · Stratified_Universality_Traditional.tex, line 5017
[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].
Exact mathematical passage
F_{f,p}(T) = \sum_{e\geq0}f(p^e)T^e \in1+T\mathbb Z[[T]].
tDefinition passage · 97 passages
Search aliases: t
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 334
[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declara…
Exact mathematical passage
Q=(Q_0,Q_1,s,t)
T^eDefinition passage · 3 passages
Search aliases: T^e
Definition passage: Prime-power seed series · Stratified_Universality_Traditional.tex, line 5017
[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].
Exact mathematical passage
F_{f,p}(T) = \sum_{e\geq0}f(p^e)T^e \in1+T\mathbb Z[[T]].
\mathsf{Task}Definition passage · 1 passage
Search aliases: Task mathsf
Definition passage: Doctrine-indexed task fiber · block19_admissibility_lifting_final.tex, line 323
[Doctrine-indexed task fiber] For a doctrine D and represented object X, let Task_D(X) denote the ordered family of continuation-stable semantic quotients admitted by the declared query universe. Inside one fiber, kernel or congruence calculus governs task change. A change of doctrine D⟶ D' requires a bridge passport specifying carrier transport, preserved constructions, comparison maps, and any lifting obstruction.
Exact mathematical passage
\mathsf{Task}_D(X)
\mathsf{Task}_DDefinition passage · 1 passage
Search aliases: Task_D mathsf
Definition passage: Doctrine-indexed task fiber · block19_admissibility_lifting_final.tex, line 323
[Doctrine-indexed task fiber] For a doctrine D and represented object X, let Task_D(X) denote the ordered family of continuation-stable semantic quotients admitted by the declared query universe. Inside one fiber, kernel or congruence calculus governs task change. A change of doctrine D⟶ D' requires a bridge passport specifying carrier transport, preserved constructions, comparison maps, and any lifting obstruction.
Exact mathematical passage
\mathsf{Task}_D(X)
\triangleleftDefinition passage · 3 passages
Search aliases: triangleleft
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 335
labeldef:finite-resolved-signature Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A …
Exact mathematical passage
I\triangleleft \kk Q
UDefinition passage · 167 passages
Search aliases: U
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1538
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
D:U\longrightarrow V
\mathcal UDefinition passage · 14 passages
Search aliases: U mathcal
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1049
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
\mathcal U_\alpha\subseteq\mathcal M_\alpha
uDefinition passage · 119 passages
Search aliases: u
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1541
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
u\in U
\mathcal U_\mathfrakDefinition passage · 3 passages
Search aliases: U_ mathcal mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
u_1Definition passage · 1 passage
Search aliases: u_1
Definition passage: Greedy shortlex kernel basis · Stratified_Universality_Traditional.tex, line 3727
[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.
Exact mathematical passage
U_a=(u_1,\ldots,u_{r_k(a)})
U_aDefinition passage · 2 passages
Search aliases: U_a
Definition passage: Greedy shortlex kernel basis · Stratified_Universality_Traditional.tex, line 3727
[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.
Exact mathematical passage
U_a=(u_1,\ldots,u_{r_k(a)})
u_iDefinition passage · 4 passages
Search aliases: u_i
Definition passage: Greedy shortlex kernel basis · Stratified_Universality_Traditional.tex, line 3731
[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.
Exact mathematical passage
b_i=\Lambda_{u_i}a.
U_jDefinition passage · 19 passages
Search aliases: U_j
Definition passage: Algebraic packet collision defects · block10_section.tex, line 301
[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
Exact mathematical passage
\mathfrak s^{\rm Gal}_{N,j} = |\mathcal C_{N,j}|-|\Gamma\cdot U_j|.
u_{r_k(a)}Definition passage · 1 passage
Search aliases: u_r_k(a)
Definition passage: Greedy shortlex kernel basis · Stratified_Universality_Traditional.tex, line 3727
[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.
Exact mathematical passage
U_a=(u_1,\ldots,u_{r_k(a)})
\mathcal U_\alphaDefinition passage · 4 passages
Search aliases: U_α mathcal alpha
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1049
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
\mathcal U_\alpha\subseteq\mathcal M_\alpha
VDefinition passage · 104 passages
Search aliases: V
Definition passage: Linear correction datum · Stratified_Universality_Traditional.tex, line 1538
[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
D:U\longrightarrow V
\mathcal VDefinition passage · 39 passages
Search aliases: V mathcal
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
vDefinition passage · 92 passages
Search aliases: v
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3549
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
\mathcal V_\mathfrakDefinition passage · 3 passages
Search aliases: V_ mathcal mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
V_{\mathbb Q}Definition passage · 9 passages
Search aliases: V_ Q mathbb
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
\mathcal V_kDefinition passage · 15 passages
Search aliases: V_k mathcal
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\mathcal V_pDefinition passage · 12 passages
Search aliases: V_p mathcal
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
v_pDefinition passage · 16 passages
Search aliases: v_p
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5450
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\nu_f(r)=v_p(c_r)\in\mathbb Z_{\geq0}\cup\{\infty\}.
\mathbf{Vect}Definition passage · 2 passages
Search aliases: Vect mathbf
Definition passage: Semantic slice · block17_semantic_quotients_final.tex, line 19
ting of: [label=textup(roman*)] item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family O=(o_j)_j∈ J of observations in V. The slice belongs to the stratum passport. It does not replace the preservation profile Π. The profile specifies which constructions a b…
Exact mathematical passage
K\text{-}\mathbf{Vect}
WDefinition passage · 31 passages
Search aliases: W
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3549
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
wDefinition passage · 74 passages
Search aliases: w
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
W^*Definition passage · 3 passages
Search aliases: W^*
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\lambda\in W^*
XDefinition passage · 216 passages
Search aliases: X
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 336
numerate[label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refineme…
Exact mathematical passage
X
xDefinition passage · 180 passages
Search aliases: x
Definition passage: Three effectivity obstructions · Stratified_Universality_Traditional.tex, line 4556
[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even after integral effectivity holds, the lift need not be an actual radix section.
Exact mathematical passage
x\in M_{k,p}(a)
X^r_{U,i}Definition passage · 1 passage
Search aliases: X^r_U,i
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
X_iDefinition passage · 8 passages
Search aliases: X_i
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 435
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Analysis_{\mathcal P} := (\pi_i)_{i\in I}: X\longrightarrow\bigoplus_{i\in I}X_i,
X_i^rDefinition passage · 10 passages
Search aliases: X_i^r
Definition passage: Finite profiled lane system · block15_section.tex, line 49
finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_…
Exact mathematical passage
X_i^r\xrightarrow{\iota_{X,i}^r}X \xrightarrow{\pi_{X,i}^r}X_i^r, \qquad \iota_{X,i}^r\pi_{X,i}^r=p_{X,i}^r, \qquad \pi_{X,i}^r\iota_{X,i}^r=1_{X_i^r}.
\xrightarrowDefinition passage · 25 passages
Search aliases: xrightarrow
Definition passage: Finite profiled lane system · block15_section.tex, line 49
finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_…
Exact mathematical passage
X_i^r\xrightarrow{\iota_{X,i}^r}X \xrightarrow{\pi_{X,i}^r}X_i^r, \qquad \iota_{X,i}^r\pi_{X,i}^r=p_{X,i}^r, \qquad \pi_{X,i}^r\iota_{X,i}^r=1_{X_i^r}.
YDefinition passage · 145 passages
Search aliases: Y
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
Y
yDefinition passage · 24 passages
Search aliases: y
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 58
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
x\equiv_{\mathsf S}y \quad\Longleftrightarrow\quad o_j(\rho(m)x)=o_j(\rho(m)y) \quad\text{for every }m\in M,\ j\in J.
Y^u_{U,a}Definition passage · 1 passage
Search aliases: Y^u_U,a
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
Y_aDefinition passage · 1 passage
Search aliases: Y_a
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 517
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
f_{ai}:=\varpi_a f\iota_i:X_i\longrightarrow Y_a.
Y_a^rDefinition passage · 1 passage
Search aliases: Y_a^r
Definition passage: Lane matrix category at a fixed resolution · block15_section.tex, line 510
y Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, as morphisms. Composition is matrix contraction and identities are diagonal identity matrices.
Exact mathematical passage
(h_{a i}), \qquad h_{a i}:X_i^r\to Y_a^r,
Y_jDefinition passage · 2 passages
Search aliases: Y_j
Definition passage: Resolved observational system · block17_semantic_quotients_final.tex, line 32
[Resolved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.
Exact mathematical passage
o_j:X\longrightarrow Y_j\qquad(j\in J).
ZDefinition passage · 18 passages
Search aliases: Z
Definition passage: Relational support product · Stratified_Universality_Traditional.tex, line 665
[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.
Exact mathematical passage
F\subseteq I_Z\times I_Y
\mathbb ZDefinition passage · 136 passages
Search aliases: Z mathbb
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
zDefinition passage · 265 passages
Search aliases: z
Definition passage: Coefficientwise filtered product · Stratified_Universality_Traditional.tex, line 4763
[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.
Exact mathematical passage
F_m(z)\in 1+z^{w(m)}R[[z]], \qquad w(m)\longrightarrow\infty.
\mathbb Z^{\mathbb N}Definition passage · 2 passages
Search aliases: Z^ N mathbb mathbb
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
z^mDefinition passage · 13 passages
Search aliases: z^m
Definition passage: Euler-coordinate map · Stratified_Universality_Traditional.tex, line 4799
[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.
Exact mathematical passage
\mathcal E(b)(z) = \prod_{m\geq1}(1-z^m)^{-b_m}.
z^NDefinition passage · 18 passages
Search aliases: z^N
Definition passage: Coefficientwise filtered product · Stratified_Universality_Traditional.tex, line 4772
[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.
Exact mathematical passage
z^N
z^nDefinition passage · 24 passages
Search aliases: z^n
Definition passage: Branch-separation index · block10_section.tex, line 43
[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\boxed{ J_{\rm alg}(F) = (a_0,a_1,\ldots,a_{\sigma-1}), \qquad F(z)=\sum_{n\geq0}a_nz^n.}
z^{w(m)}Definition passage · 1 passage
Search aliases: z^w(m)
Definition passage: Coefficientwise filtered product · Stratified_Universality_Traditional.tex, line 4763
[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.
Exact mathematical passage
F_m(z)\in 1+z^{w(m)}R[[z]], \qquad w(m)\longrightarrow\infty.
\mathbb Z_pDefinition passage · 44 passages
Search aliases: Z_p mathbb
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5454
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\mathcal M_w = \{c\in c_0(\mathbb Z_p):v_p(c_r)\geq w(r)\}, \qquad \mathcal C_w=\mathcal M_p(\mathcal M_w).
\mathbb Z_{\geq0}Definition passage · 4 passages
Search aliases: Z_≥0 mathbb geq
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5450
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\nu_f(r)=v_p(c_r)\in\mathbb Z_{\geq0}\cup\{\infty\}.
|Definition passage · 45 passages
Search aliases: |
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1054
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
\nu_\alpha:\Sem_\alpha N_\alpha \Longrightarrow \Sem_\alpha|_{\mathcal U_\alpha}.
|_{\mathcal U_\alpha}Definition passage · 1 passage
Search aliases: |_ U_α mathcal alpha
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1054
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
\nu_\alpha:\Sem_\alpha N_\alpha \Longrightarrow \Sem_\alpha|_{\mathcal U_\alpha}.
|_{\mathcal V_p(F)}Definition passage · 1 passage
Search aliases: |_ V_p(F) mathcal
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
\cdotDefinition passage · 13 passages
Search aliases: · cdot
Definition passage: Divisor--ghost transform · Stratified_Universality_Traditional.tex, line 5179
[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).
Exact mathematical passage
\mathfrak G(b) = \mathbf{1}\mathbin{\ast_D}(\operatorname{id}\cdot b),
\timesDefinition passage · 68 passages
Search aliases: × times
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 632
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
E\subseteq I_{\mathcal Q}\times I_{\mathcal P}
\alphaDefinition passage · 125 passages
Search aliases: α alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\betaDefinition passage · 51 passages
Search aliases: β beta
Definition passage: Typed directed transformation category · Stratified_Universality_Traditional.tex, line 1179
[Typed directed transformation category] The directed transformation category Dir( K) has pairs (α,M), with M∈ M_α, as objects. A morphism (α,M)⟶(β,N) is an admitted bridge instance equipped with a bridge passport as in [source reference]. Composition is defined only when the codomain profile of the first bridge satisfies the domain requirements of the second. The union of the representation groupoids embeds as the wide invertible subcategory.
Exact mathematical passage
(\alpha,M)\longrightarrow(\beta,N)
\beta^A_{\gamma,r,i}Definition passage · 3 passages
Search aliases: β^A_γ,r,i beta gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 65
ject A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π…
Exact mathematical passage
\beta^A_{\gamma,r,i}: F_\gamma(A^r_{U,i})\xrightarrow{\sim} A^r_{V,\lambda^A_{\gamma,r}(i)}.
\beta^X_{\gamma,r,i}Definition passage · 2 passages
Search aliases: β^X_γ,r,i beta gamma
Definition passage: Matrix local system · block16_section_final.tex, line 106
atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\left(\gamma_*h\right)_{ \lambda^Y_{\gamma,u}(a),\lambda^X_{\gamma,r}(i)} := \beta^Y_{\gamma,u,a} F_\gamma(h_{ai}) \left(\beta^X_{\gamma,r,i}\right)^{-1}.
\beta^Y_{\gamma,u,a}Definition passage · 2 passages
Search aliases: β^Y_γ,u,a beta gamma
Definition passage: Matrix local system · block16_section_final.tex, line 106
atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\left(\gamma_*h\right)_{ \lambda^Y_{\gamma,u}(a),\lambda^X_{\gamma,r}(i)} := \beta^Y_{\gamma,u,a} F_\gamma(h_{ai}) \left(\beta^X_{\gamma,r,i}\right)^{-1}.
\GammaDefinition passage · 44 passages
Search aliases: Γ Gamma
Definition passage: Algebraic packet collision defects · block10_section.tex, line 301
[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
Exact mathematical passage
\mathfrak s^{\rm Gal}_{N,j} = |\mathcal C_{N,j}|-|\Gamma\cdot U_j|.
\gammaDefinition passage · 58 passages
Search aliases: γ gamma
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\gamma_*Definition passage · 9 passages
Search aliases: γ_* gamma
Definition passage: Matrix local system · block16_section_final.tex, line 106
atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\left(\gamma_*h\right)_{ \lambda^Y_{\gamma,u}(a),\lambda^X_{\gamma,r}(i)} := \beta^Y_{\gamma,u,a} F_\gamma(h_{ai}) \left(\beta^X_{\gamma,r,i}\right)^{-1}.
\Gamma_{sr}Definition passage · 4 passages
Search aliases: Γ_sr Gamma
Definition passage: Lane matrix category at a fixed resolution · block15_section.tex, line 518
For ssucceq r, define Γ_sr:Mat_s( L_ R) ⟶Mat_r( L_ R) by replacing each object tag with its r-tag and sending a fine matrix to the certified source-and-target coordinate change of [source reference].
Exact mathematical passage
\Gamma_{sr}:\operatorname{Mat}_s(\mathcal L_{\mathsf R}) \longrightarrow\operatorname{Mat}_r(\mathcal L_{\mathsf R})
\deltaDefinition passage · 38 passages
Search aliases: δ delta
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 337
nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ …
Exact mathematical passage
p_{X,r,i}p_{X,r,j}=\delta_{ij}p_{X,r,i}, \qquad \sum_i p_{X,r,i}=1_X.
\delta_{ij}Definition passage · 2 passages
Search aliases: δ_ij delta
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 337
nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ …
Exact mathematical passage
p_{X,r,i}p_{X,r,j}=\delta_{ij}p_{X,r,i}, \qquad \sum_i p_{X,r,i}=1_X.
\delta_kDefinition passage · 2 passages
Search aliases: δ_k delta
Definition passage: Integral saturation defect · Stratified_Universality_Traditional.tex, line 4101
[Integral saturation defect] The finite abelian group C_k(a) = L_k(a)/M_k(a) is the integral saturation defect. Its order δ_k(a) = [L_k(a):M_k(a)] is the saturation index.
Exact mathematical passage
\delta_k(a) = [L_k(a):M_k(a)]
\delta_\GammaDefinition passage · 7 passages
Search aliases: δ_Γ delta Gamma
Definition passage: Class-relative defect · block19_admissibility_lifting_final.tex, line 16
[Class-relative defect] Let X be a represented system, let Γ be a declared class of admissible auxiliary structures, and let d(X,g)∈[0,∞] be a defect functional. Define δ_Γ(X):=inf_g∈Γd(X,g). The pair consisting of the infimum and its attainment status is the class-relative defect profile.
Exact mathematical passage
\delta_\Gamma(X):=\inf_{g\in\Gamma}d(X,g).
\epsilonDefinition passage · 10 passages
Search aliases: ϵ epsilon
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
\epsilon_0Definition passage · 9 passages
Search aliases: ϵ_0 epsilon
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
\epsilon_{\mathrm{int}}Definition passage · 1 passage
Search aliases: ϵ_int epsilon mathrm
Definition passage: Three effectivity obstructions · Stratified_Universality_Traditional.tex, line 4560
[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even after integral effectivity holds, the lift need not be an actual radix section.
Exact mathematical passage
\epsilon_{\mathrm{int}}(x) = [x]\in M_{k,p}(a)/M_k(a).
\etaDefinition passage · 5 passages
Search aliases: η eta
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\eta_{\alpha\beta\gamma}Definition passage · 2 passages
Search aliases: η_αβγ eta alpha beta gamma
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\eta_{\alpha\beta\delta}Definition passage · 1 passage
Search aliases: η_αβδ eta alpha beta delta
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\eta_{\alpha\gamma\delta}Definition passage · 1 passage
Search aliases: η_αγδ eta alpha gamma delta
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\eta_{\beta\gamma\delta}Definition passage · 1 passage
Search aliases: η_βγδ eta beta gamma delta
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\ThetaDefinition passage · 33 passages
Search aliases: Θ Theta
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 64
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
\Theta_{\mathsf S}
\thetaDefinition passage · 15 passages
Search aliases: θ theta
Definition passage: Descent category · Stratified_Universality_Traditional.tex, line 1892
[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.
Exact mathematical passage
(M_\alpha,\theta_{\alpha\beta})
\theta^A_\gammaDefinition passage · 5 passages
Search aliases: θ^A_γ theta gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 72
profiled-lane-system. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No glo…
Exact mathematical passage
\theta^A_\gamma F_\gamma(\iota^r_{A,U,i}) = \iota^r_{A,V,\lambda^A_{\gamma,r}(i)} \beta^A_{\gamma,r,i},
\Theta_{\mathsf S}Definition passage · 7 passages
Search aliases: Θ_ S Theta mathsf
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 64
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
\Theta_{\mathsf S}
\theta_{\alpha\beta}Definition passage · 1 passage
Search aliases: θ_αβ theta alpha beta
Definition passage: Descent category · Stratified_Universality_Traditional.tex, line 1892
[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.
Exact mathematical passage
(M_\alpha,\theta_{\alpha\beta})
\iotaDefinition passage · 28 passages
Search aliases: ι iota
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 441
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Synthesis_{\mathcal P} := (\iota_i)_{i\in I}: \bigoplus_{i\in I}X_i\longrightarrow X.
\iota^r_{A,U,i}Definition passage · 2 passages
Search aliases: ι^r_A,U,i iota
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 55
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i}, \iota^r_{A,U,i},\pi^r_{A,U,i}\right)_{i\in I_{A,U,r}}
\iota^r_{A,V,\lambda^A_{\gamma,r}(i)}Definition passage · 1 passage
Search aliases: ι^r_A,V,λ^A_γ,r(i) iota lambda gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 72
profiled-lane-system. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No glo…
Exact mathematical passage
\theta^A_\gamma F_\gamma(\iota^r_{A,U,i}) = \iota^r_{A,V,\lambda^A_{\gamma,r}(i)} \beta^A_{\gamma,r,i},
\iota_iDefinition passage · 6 passages
Search aliases: ι_i iota
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 441
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Synthesis_{\mathcal P} := (\iota_i)_{i\in I}: \bigoplus_{i\in I}X_i\longrightarrow X.
\iota_{X,i}^rDefinition passage · 6 passages
Search aliases: ι_X,i^r iota
Definition passage: Finite profiled lane system · block15_section.tex, line 49
finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_…
Exact mathematical passage
X_i^r\xrightarrow{\iota_{X,i}^r}X \xrightarrow{\pi_{X,i}^r}X_i^r, \qquad \iota_{X,i}^r\pi_{X,i}^r=p_{X,i}^r, \qquad \pi_{X,i}^r\iota_{X,i}^r=1_{X_i^r}.
\kappaDefinition passage · 37 passages
Search aliases: κ kappa
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3540
nd its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\kappa_k(a)=|\mathcal K_k(a)|.
\kappa_{ab}Definition passage · 2 passages
Search aliases: κ_ab kappa
Definition passage: Coherent invariant family · block13_section.tex, line 140
[Coherent invariant family] The family (I_λ,a,κ_ab) is coherent when the two induced comparison maps from a to c agree on every triple overlap and the corresponding quadruple-overlap equation holds.
Exact mathematical passage
(I_{\lambda,a},\kappa_{ab})
\kappa_kDefinition passage · 2 passages
Search aliases: κ_k kappa
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3540
nd its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\kappa_k(a)=|\mathcal K_k(a)|.
\kappa_{k,p^r}^{\mathrm{sem}}Definition passage · 2 passages
Search aliases: κ_k,p^r^sem kappa mathrm
Definition passage: Prime-power section automaton · Stratified_Universality_Traditional.tex, line 4393
[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).
Exact mathematical passage
\kappa_{k,p^r}^{\mathrm{sem}}(a).
\LambdaDefinition passage · 59 passages
Search aliases: Λ Lambda
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\lambdaDefinition passage · 94 passages
Search aliases: λ lambda
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3549
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
\lambda^A_{\gamma,r}Definition passage · 7 passages
Search aliases: λ^A_γ,r lambda gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 61
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
\lambda^A_{\gamma,r}:I_{A,U,r}\xrightarrow{\sim}I_{A,V,r}
\lambda^A_{\gamma,s}Definition passage · 1 passage
Search aliases: λ^A_γ,s lambda gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 84
V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No global address set and no global trivialization of the address bijections is assumed.
Exact mathematical passage
\rho^{A,V}_{sr}\lambda^A_{\gamma,s} = \lambda^A_{\gamma,r}\rho^{A,U}_{sr}.
\lambda^X_{\gamma,r}Definition passage · 3 passages
Search aliases: λ^X_γ,r lambda gamma
Definition passage: Matrix local system · block16_section_final.tex, line 106
atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\left(\gamma_*h\right)_{ \lambda^Y_{\gamma,u}(a),\lambda^X_{\gamma,r}(i)} := \beta^Y_{\gamma,u,a} F_\gamma(h_{ai}) \left(\beta^X_{\gamma,r,i}\right)^{-1}.
\lambda^Y_{\gamma,u}Definition passage · 4 passages
Search aliases: λ^Y_γ,u lambda gamma
Definition passage: Matrix local system · block16_section_final.tex, line 106
atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\left(\gamma_*h\right)_{ \lambda^Y_{\gamma,u}(a),\lambda^X_{\gamma,r}(i)} := \beta^Y_{\gamma,u,a} F_\gamma(h_{ai}) \left(\beta^X_{\gamma,r,i}\right)^{-1}.
\Lambda_\mathfrakDefinition passage · 2 passages
Search aliases: Λ_ Lambda mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\Lambda_dDefinition passage · 23 passages
Search aliases: Λ_d Lambda
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
\Lambda_uDefinition passage · 2 passages
Search aliases: Λ_u Lambda
Definition passage: Word-Hankel matrix · Stratified_Universality_Traditional.tex, line 3649
[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.
Exact mathematical passage
H_a(u,v) = a_{[u]_k+k^{|u|}[v]_k} = (\Lambda_u a)([v]_k), \qquad u,v\in\Sigma_k^*. \label{eq:block05-word-hankel}
\Lambda_{u_i}Definition passage · 1 passage
Search aliases: Λ_u_i Lambda
Definition passage: Greedy shortlex kernel basis · Stratified_Universality_Traditional.tex, line 3731
[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.
Exact mathematical passage
b_i=\Lambda_{u_i}a.
\Lambda_wDefinition passage · 21 passages
Search aliases: Λ_w Lambda
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\muDefinition passage · 38 passages
Search aliases: μ mu
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4862
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
\frac1n\sum_{d\mid n}\mu\!\left(\frac nd\right)c_d\in\mathbb Z \qquad(n\geq1).
\nuDefinition passage · 30 passages
Search aliases: ν nu
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1054
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
\nu_\alpha:\Sem_\alpha N_\alpha \Longrightarrow \Sem_\alpha|_{\mathcal U_\alpha}.
\nu_\mathfrakDefinition passage · 2 passages
Search aliases: ν_ nu mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\nu_fDefinition passage · 1 passage
Search aliases: ν_f nu
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5450
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\nu_f(r)=v_p(c_r)\in\mathbb Z_{\geq0}\cup\{\infty\}.
\nu_\alphaDefinition passage · 1 passage
Search aliases: ν_α nu alpha
Definition passage: Normalization witness · Stratified_Universality_Traditional.tex, line 1054
[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in N_α. A normalization may be partial, domain-restricted, or defined only after auxiliary choices. Those restrictions belong to the passport.
Exact mathematical passage
\nu_\alpha:\Sem_\alpha N_\alpha \Longrightarrow \Sem_\alpha|_{\mathcal U_\alpha}.
\PiDefinition passage · 19 passages
Search aliases: Π Pi
Definition passage: Verified preservation profile · Stratified_Universality_Traditional.tex, line 755
[Verified preservation profile] A verified preservation profile Π_F for a functor F: C→ D consists of a subset of C together with a proof or declared comparison isomorphism showing that F preserves each listed constructor. We write c∈Π_F only after the corresponding preservation claim has been verified.
Exact mathematical passage
\Pi_F
\piDefinition passage · 43 passages
Search aliases: π pi
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 435
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Analysis_{\mathcal P} := (\pi_i)_{i\in I}: X\longrightarrow\bigoplus_{i\in I}X_i,
\varpiDefinition passage · 2 passages
Search aliases: ϖ varpi
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
(\jmath_a,\varpi_a)
\Pi^{\mathrm{pres}}_\mathfrakDefinition passage · 2 passages
Search aliases: Π^pres_ Pi mathrm mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\pi^r_{A,U,i}Definition passage · 2 passages
Search aliases: π^r_A,U,i pi
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 55
For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_…
Exact mathematical passage
\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i}, \iota^r_{A,U,i},\pi^r_{A,U,i}\right)_{i\in I_{A,U,r}}
\pi^r_{A,V,\lambda^A_{\gamma,r}(i)}Definition passage · 1 passage
Search aliases: π^r_A,V,λ^A_γ,r(i) pi lambda gamma
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 78
I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No global address set and no global trivialization of the address bijections is assumed.
Exact mathematical passage
\beta^A_{\gamma,r,i}F_\gamma(\pi^r_{A,U,i}) = \pi^r_{A,V,\lambda^A_{\gamma,r}(i)}\theta^A_\gamma.
\Pi^{\mathrm{refl}}_\mathfrakDefinition passage · 2 passages
Search aliases: Π^refl_ Pi mathrm mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\varpi_aDefinition passage · 2 passages
Search aliases: ϖ_a varpi
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
(\jmath_a,\varpi_a)
\Pi_FDefinition passage · 5 passages
Search aliases: Π_F Pi
Definition passage: Verified preservation profile · Stratified_Universality_Traditional.tex, line 755
[Verified preservation profile] A verified preservation profile Π_F for a functor F: C→ D consists of a subset of C together with a proof or declared comparison isomorphism showing that F preserves each listed constructor. We write c∈Π_F only after the corresponding preservation claim has been verified.
Exact mathematical passage
\Pi_F
\pi_iDefinition passage · 9 passages
Search aliases: π_i pi
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 435
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Analysis_{\mathcal P} := (\pi_i)_{i\in I}: X\longrightarrow\bigoplus_{i\in I}X_i,
\Pi_UDefinition passage · 1 passage
Search aliases: Π_U Pi
Definition passage: Finite profiled resolved atlas · Stratified_Universality_Traditional.tex, line 1879
e profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Sem: R⇒ …
Exact mathematical passage
\Pi_U
\pi_{X,i}^rDefinition passage · 7 passages
Search aliases: π_X,i^r pi
Definition passage: Finite profiled lane system · block15_section.tex, line 49
finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_…
Exact mathematical passage
X_i^r\xrightarrow{\iota_{X,i}^r}X \xrightarrow{\pi_{X,i}^r}X_i^r, \qquad \iota_{X,i}^r\pi_{X,i}^r=p_{X,i}^r, \qquad \pi_{X,i}^r\iota_{X,i}^r=1_{X_i^r}.
\Pi_\alphaDefinition passage · 3 passages
Search aliases: Π_α Pi alpha
Definition passage: Profiled stratum passport · Stratified_Universality_Traditional.tex, line 1019
[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theore…
Exact mathematical passage
\mathbf S_\alpha = (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha, \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
\rhoDefinition passage · 74 passages
Search aliases: ρ rho
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 343
ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such a…
Exact mathematical passage
\rho_{sr}:I_{X,s}\twoheadrightarrow I_{X,r}
\rho^{A,U}_{sr}Definition passage · 1 passage
Search aliases: ρ^A,U_sr rho
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 84
V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No global address set and no global trivialization of the address bijections is assumed.
Exact mathematical passage
\rho^{A,V}_{sr}\lambda^A_{\gamma,s} = \lambda^A_{\gamma,r}\rho^{A,U}_{sr}.
\rho^{A,V}_{sr}Definition passage · 1 passage
Search aliases: ρ^A,V_sr rho
Definition passage: Holonomy-profiled finite lane datum · block16_section_final.tex, line 84
V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No global address set and no global trivialization of the address bijections is assumed.
Exact mathematical passage
\rho^{A,V}_{sr}\lambda^A_{\gamma,s} = \lambda^A_{\gamma,r}\rho^{A,U}_{sr}.
\rho^X_{rr}Definition passage · 1 passage
Search aliases: ρ^X_rr rho
Definition passage: Finite profiled lane system · block15_section.tex, line 69
ittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address sets and refinement maps are declared data; they are not inferred from their cardinalities. Its verified profile contains finite biproducts, addition of morphisms, composition, the displayed splittings, and these projector-su…
Exact mathematical passage
\rho^X_{rr}=1_{I_{X,r}}
\rho^X_{sr}Definition passage · 3 passages
Search aliases: ρ^X_sr rho
Definition passage: Finite profiled lane system · block15_section.tex, line 58
_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address sets and refinement maps are declared data; they are not inferred from their cardinalities. Its verified profile contain…
Exact mathematical passage
\rho^X_{sr}:I_{X,s}\twoheadrightarrow I_{X,r}
\rho^X_{tr}Definition passage · 1 passage
Search aliases: ρ^X_tr rho
Definition passage: Finite profiled lane system · block15_section.tex, line 62
p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address sets and refinement maps are declared data; they are not inferred from their cardinalities. Its verified profile contains finite biproducts, addition of …
Exact mathematical passage
p_{X,i}^r = \sum_{\rho^X_{sr}(j)=i}p_{X,j}^s, \qquad \rho^X_{tr}=\rho^X_{sr}\rho^X_{ts}.
\rho^X_{ts}Definition passage · 1 passage
Search aliases: ρ^X_ts rho
Definition passage: Finite profiled lane system · block15_section.tex, line 62
p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address sets and refinement maps are declared data; they are not inferred from their cardinalities. Its verified profile contains finite biproducts, addition of …
Exact mathematical passage
p_{X,i}^r = \sum_{\rho^X_{sr}(j)=i}p_{X,j}^s, \qquad \rho^X_{tr}=\rho^X_{sr}\rho^X_{ts}.
\rho_{sr}Definition passage · 8 passages
Search aliases: ρ_sr rho
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 343
ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such a…
Exact mathematical passage
\rho_{sr}:I_{X,s}\twoheadrightarrow I_{X,r}
\SigmaDefinition passage · 19 passages
Search aliases: Σ Sigma
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\sigmaDefinition passage · 34 passages
Search aliases: σ sigma
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1341
abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\sigma_{\alpha\beta}: J_{\alpha\beta}\Sem_\alpha \Longrightarrow \Sem_\beta\Phi_{\alpha\beta}
\Sigma_k^*Definition passage · 10 passages
Search aliases: Σ_k^* Sigma
Definition passage: Fixed-radix kernel and section module · Stratified_Universality_Traditional.tex, line 3526
[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.
Exact mathematical passage
\mathcal K_k(a) = \{\Lambda_w a:w\in\Sigma_k^*\}, \qquad \mathcal V_k(a) = \Span_K\mathcal K_k(a).
\sigma_{\alpha\beta}Definition passage · 1 passage
Search aliases: σ_αβ sigma alpha beta
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1341
abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\sigma_{\alpha\beta}: J_{\alpha\beta}\Sem_\alpha \Longrightarrow \Sem_\beta\Phi_{\alpha\beta}
\tauDefinition passage · 42 passages
Search aliases: τ tau
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\tau_\mathfrakDefinition passage · 2 passages
Search aliases: τ_ tau mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\PhiDefinition passage · 50 passages
Search aliases: Φ Phi
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1335
[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\Phi_{\alpha\beta}:\mathcal M_{\alpha\beta} \overset{\sim}{\longrightarrow} \mathcal M_{\beta\alpha},
\Phi_{\alpha\beta}Definition passage · 4 passages
Search aliases: Φ_αβ Phi alpha beta
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1335
[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\Phi_{\alpha\beta}:\mathcal M_{\alpha\beta} \overset{\sim}{\longrightarrow} \mathcal M_{\beta\alpha},
\Phi_{\alpha\delta}Definition passage · 2 passages
Search aliases: Φ_αδ Phi alpha delta
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1410
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Phi_{\alpha\delta}
\chiDefinition passage · 10 passages
Search aliases: χ chi
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\chi_\mathfrakDefinition passage · 2 passages
Search aliases: χ_ chi mathfrak
Definition passage: Bridge passport · Stratified_Universality_Traditional.tex, line 1196
[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal properties; item Π^refl_ b records properties or isomorphisms reflected by the bridge; item Λ_ b is the informatio…
Exact mathematical passage
\mathfrak b = (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b, B_\mathfrak b,\tau_\mathfrak b, \Pi^{\mathrm{pres}}_\mathfrak b, \Pi^{\mathrm{refl}}_\mathfrak b, \Lambda_\mathfrak b, \nu_\mathfrak b,\chi_\mathfrak b),
\OmegaDefinition passage · 10 passages
Search aliases: Ω Omega
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\Omega_{\alpha\beta\gamma\delta}Definition passage · 3 passages
Search aliases: Ω_αβγδ Omega alpha beta gamma delta
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1395
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\Omega_{\alpha\beta\gamma\delta} = \Bigl[ \eta_{\alpha\gamma\delta} \circ (\id\ast\eta_{\alpha\beta\gamma}) \Bigr] \circ \Bigl[ \eta_{\alpha\beta\delta} \circ (\eta_{\beta\gamma\delta}\ast\id) \Bigr]^{-1}.
\dotsDefinition passage · 4 passages
Search aliases: … dots
Definition passage: Finite additive hull · Stratified_Universality_Traditional.tex, line 862
[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.
Exact mathematical passage
A=(A_1,\dots,A_m)
\ldotsDefinition passage · 78 passages
Search aliases: … ldots
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3549
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
\mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
\ellDefinition passage · 50 passages
Search aliases: ℓ ell
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
w=d_0\cdots d_{\ell-1}
\ell_jDefinition passage · 4 passages
Search aliases: ℓ_j ell
Definition passage: Ordinary-point locus · block09_section.tex, line 181
[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.
Exact mathematical passage
L_F = D^r+\sum_{j=0}^{r-1}\ell_j(z)D^j
\toDefinition passage · 106 passages
Search aliases: → to
Definition passage: Mixed transfer blocks · Stratified_Universality_Traditional.tex, line 516
[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.
Exact mathematical passage
f:X\to Y
\twoheadrightarrowDefinition passage · 11 passages
Search aliases: ↠ twoheadrightarrow
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 343
ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such a…
Exact mathematical passage
\rho_{sr}:I_{X,s}\twoheadrightarrow I_{X,r}
\RightarrowDefinition passage · 6 passages
Search aliases: ⇒ Rightarrow
Definition passage: Finite profiled resolved atlas · Stratified_Universality_Traditional.tex, line 1882
hcal A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Sem: R⇒ O into a common observable doctrine; item optional pseudonatural invariant families with explicitly typed comparison cells.
Exact mathematical passage
\Sem:\mathfrak R\Rightarrow\mathfrak O
\inDefinition passage · 280 passages
Search aliases: ∈ in
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 361
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
X\in Q_0
\prodDefinition passage · 37 passages
Search aliases: ∏ prod
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1139
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
G_{\mathbf d}=\prod_{X\in\mathsf S}GL_{d_X}(\kk)
\prod_{(m,j)\in M\times J}Definition passage · 1 passage
Search aliases: ∏_(m,j)∈ M× J prod in times
Definition passage: Resolved observational system · block17_semantic_quotients_final.tex, line 40
lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.
Exact mathematical passage
\Beh_{\mathsf S}:X\longrightarrow \prod_{(m,j)\in M\times J}Y_j, \qquad x\longmapsto\bigl(o_j(\rho(m)x)\bigr)_{m,j}.
\prod_{m\geq1}Definition passage · 6 passages
Search aliases: ∏_m≥1 prod geq
Definition passage: Coefficientwise filtered product · Stratified_Universality_Traditional.tex, line 4769
[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.
Exact mathematical passage
\prod_{m\geq1}F_m(z),
\prod_{X\in\mathsf S}Definition passage · 1 passage
Search aliases: ∏_X∈ S prod in mathsf
Definition passage: Finite coordinate action groupoid · Stratified_Universality_Traditional.tex, line 1139
[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.
Exact mathematical passage
G_{\mathbf d}=\prod_{X\in\mathsf S}GL_{d_X}(\kk)
\sumDefinition passage · 164 passages
Search aliases: ∑ sum
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 337
nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ …
Exact mathematical passage
p_{X,r,i}p_{X,r,j}=\delta_{ij}p_{X,r,i}, \qquad \sum_i p_{X,r,i}=1_X.
\sum_{(a,i)\in E}Definition passage · 2 passages
Search aliases: ∑_(a,i)∈ E sum in
Definition passage: Support compression and residual · Stratified_Universality_Traditional.tex, line 633
[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.
Exact mathematical passage
C_E(f):=\sum_{(a,i)\in E}q_afp_i, \qquad R_E(f):=f-C_E(f).
\sum_{C\text{ cyclic SCC}}Definition passage · 1 passage
Search aliases: ∑_C cyclic SCC sum text
Definition passage: Address-preserving cyclic dimension · block11_section.tex, line 283
[Address-preserving cyclic dimension] The cyclic packet dimension of the marked system is cdim_ P(Φ) = ∑_C cyclic SCC|C|. tag11.7 It measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.
Exact mathematical passage
\boxed{ \operatorname{cdim}_{\mathcal P}(\Phi) = \sum_{C\text{ cyclic SCC}}|C|.} \tag{11.7}
\sum_{d\mid n}Definition passage · 8 passages
Search aliases: ∑_dmid n sum mid
Definition passage: Euler ghost coordinates · Stratified_Universality_Traditional.tex, line 4857
[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).
Exact mathematical passage
c_n(b)=\sum_{d\mid n}d\,b_d, \qquad n\geq1.
\sum_{e\geq0}Definition passage · 1 passage
Search aliases: ∑_e≥0 sum geq
Definition passage: Prime-power seed series · Stratified_Universality_Traditional.tex, line 5017
[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].
Exact mathematical passage
F_{f,p}(T) = \sum_{e\geq0}f(p^e)T^e \in1+T\mathbb Z[[T]].
\sum_iDefinition passage · 6 passages
Search aliases: ∑_i sum
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 337
nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ …
Exact mathematical passage
p_{X,r,i}p_{X,r,j}=\delta_{ij}p_{X,r,i}, \qquad \sum_i p_{X,r,i}=1_X.
\sum_{i\in I}Definition passage · 2 passages
Search aliases: ∑_i∈ I sum in
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 409
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
p_i^2=p_i, \qquad p_ip_j=0\quad(i\ne j), \qquad \sum_{i\in I}p_i=1_X.
\sum_{i\in I_{X,r}}Definition passage · 1 passage
Search aliases: ∑_i∈ I_X,r sum in
Definition passage: Finite profiled lane system · block15_section.tex, line 41
block15-finite-profiled-lane-system Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the s…
Exact mathematical passage
(p_{X,i}^r)^2=p_{X,i}^r, \qquad p_{X,i}^rp_{X,j}^r=0\quad(i\ne j), \qquad \sum_{i\in I_{X,r}}p_{X,i}^r=1_X,
\sum_{j=0}^{r-1}Definition passage · 2 passages
Search aliases: ∑_j=0^r-1 sum
Definition passage: Ordinary-point locus · block09_section.tex, line 181
[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.
Exact mathematical passage
L_F = D^r+\sum_{j=0}^{r-1}\ell_j(z)D^j
\sum_{n\geq0}Definition passage · 4 passages
Search aliases: ∑_n≥0 sum geq
Definition passage: Branch-separation index · block10_section.tex, line 43
[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.
Exact mathematical passage
\boxed{ J_{\rm alg}(F) = (a_0,a_1,\ldots,a_{\sigma-1}), \qquad F(z)=\sum_{n\geq0}a_nz^n.}
\sum_{\rho^X_{sr}(j)=i}Definition passage · 2 passages
Search aliases: ∑_ρ^X_sr(j)=i sum rho
Definition passage: Finite profiled lane system · block15_section.tex, line 62
p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address sets and refinement maps are declared data; they are not inferred from their cardinalities. Its verified profile contains finite biproducts, addition of …
Exact mathematical passage
p_{X,i}^r = \sum_{\rho^X_{sr}(j)=i}p_{X,j}^s, \qquad \rho^X_{tr}=\rho^X_{sr}\rho^X_{ts}.
\sum_{\rho_{sr}(j)=i}Definition passage · 4 passages
Search aliases: ∑_ρ_sr(j)=i sum rho
Definition passage: Finite resolved signature · Stratified_Universality_Traditional.tex, line 347
is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a m…
Exact mathematical passage
p_{X,r,i} = \sum_{\rho_{sr}(j)=i}p_{X,s,j}.
\setminusDefinition passage · 3 passages
Search aliases: ∖ setminus
Definition passage: Three effectivity obstructions · Stratified_Universality_Traditional.tex, line 4567
effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even after integral effectivity holds, the lift need not be an actual radix section.
Exact mathematical passage
M_k(a)\setminus\mathcal O_k(a).
\astDefinition passage · 17 passages
Search aliases: ∗ ast
Definition passage: Quadruple-overlap coherence defect · Stratified_Universality_Traditional.tex, line 1394
[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.
Exact mathematical passage
\ast
\ast_DDefinition passage · 15 passages
Search aliases: ∗_D ast
Definition passage: Divisor--ghost transform · Stratified_Universality_Traditional.tex, line 5179
[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).
Exact mathematical passage
\mathfrak G(b) = \mathbf{1}\mathbin{\ast_D}(\operatorname{id}\cdot b),
\circDefinition passage · 7 passages
Search aliases: ∘ circ
Definition passage: Relational support product · Stratified_Universality_Traditional.tex, line 666
[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.
Exact mathematical passage
F\circ E := \{(c,i):\text{there exists }a\in I_Y \text{ with }(c,a)\in F\text{ and }(a,i)\in E\}.
\inftyDefinition passage · 18 passages
Search aliases: ∞ infty
Definition passage: Coefficientwise filtered product · Stratified_Universality_Traditional.tex, line 4763
[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.
Exact mathematical passage
F_m(z)\in 1+z^{w(m)}R[[z]], \qquad w(m)\longrightarrow\infty.
\capDefinition passage · 21 passages
Search aliases: ∩ cap
Definition passage: Raw and saturated section lattices · Stratified_Universality_Traditional.tex, line 4043
[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.
Exact mathematical passage
M_k(a) = \operatorname{span}_{\mathbb Z}\mathcal K_k(a), \qquad L_k(a) = V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
\cupDefinition passage · 6 passages
Search aliases: ∪ cup
Definition passage: Exact valuation profile and certified class · Stratified_Universality_Traditional.tex, line 5450
[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local analyticity.
Exact mathematical passage
\nu_f(r)=v_p(c_r)\in\mathbb Z_{\geq0}\cup\{\infty\}.
\simDefinition passage · 16 passages
Search aliases: ∼ sim
Definition passage: Typed overlap · Stratified_Universality_Traditional.tex, line 1335
[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized when the equivalence restricts to the selected normalized subcategories.
Exact mathematical passage
\Phi_{\alpha\beta}:\mathcal M_{\alpha\beta} \overset{\sim}{\longrightarrow} \mathcal M_{\beta\alpha},
\simeqDefinition passage · 16 passages
Search aliases: ≃ simeq
Definition passage: Profile-indexed resolved envelope · Stratified_Universality_Traditional.tex, line 964
[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functors.
Exact mathematical passage
\Mod_\Pi(\mathbb K,\mathcal C) \simeq \Fun_\Pi(\Syn_\Pi(\mathbb K),\mathcal C)
\congDefinition passage · 45 passages
Search aliases: ≅ cong
Definition passage: Complete invariant on a declared target · Stratified_Universality_Traditional.tex, line 1065
[Complete invariant on a declared target] An invariant I_α is complete on N_α when I_α(X)≅ I_α(Y) Longleftrightarrow X≅ Y for all X,Y∈ N_α. This statement classifies the objects of N_α up to the isomorphisms allowed there. It does not classify arbitrary presentations outside that subcategory.
Exact mathematical passage
\mathcal I_\alpha(X)\cong\mathcal I_\alpha(Y) \quad\Longleftrightarrow\quad X\cong Y
\neDefinition passage · 12 passages
Search aliases: ≠ ne
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 409
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
p_i^2=p_i, \qquad p_ip_j=0\quad(i\ne j), \qquad \sum_{i\in I}p_i=1_X.
\neqDefinition passage · 22 passages
Search aliases: ≠ neq
Definition passage: Marked binomial--Fuchsian packet · block09_section.tex, line 746
[Marked binomial--Fuchsian packet] A marked packet is a tuple b=(α,λ,N,j;G), where α≠0, N≥1, 0≤ j<N, and G(x)=G_N,j^λ,α(x). The parameters (α,λ,N,j) are part of the semantic object rather than metadata to be forgotten.
Exact mathematical passage
\alpha\neq0
\equivDefinition passage · 19 passages
Search aliases: ≡ equiv
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 58
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
x\equiv_{\mathsf S}y \quad\Longleftrightarrow\quad o_j(\rho(m)x)=o_j(\rho(m)y) \quad\text{for every }m\in M,\ j\in J.
\equiv_{\mathsf S}Definition passage · 2 passages
Search aliases: ≡_ S equiv mathsf
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 58
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
x\equiv_{\mathsf S}y \quad\Longleftrightarrow\quad o_j(\rho(m)x)=o_j(\rho(m)y) \quad\text{for every }m\in M,\ j\in J.
\leqDefinition passage · 86 passages
Search aliases: ≤ leq
Definition passage: Canonical Cartier module and Cartier rank · Stratified_Universality_Traditional.tex, line 3082
[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.
Exact mathematical passage
\label{eq:block04-cartier-module} \mathsf C_p(F) = \left( \mathcal V_p(F), (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p}, F, \epsilon_0 \right).
\geqDefinition passage · 127 passages
Search aliases: ≥ geq
Definition passage: Coefficientwise filtered product · Stratified_Universality_Traditional.tex, line 4769
[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.
Exact mathematical passage
\prod_{m\geq1}F_m(z),
\subseteqDefinition passage · 61 passages
Search aliases: ⊆ subseteq
Definition passage: Finite packet system · Stratified_Universality_Traditional.tex, line 405
[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.
Exact mathematical passage
\mathcal P=(p_i)_{i\in I}\subseteq\operatorname{End}(X)
\bigoplusDefinition passage · 34 passages
Search aliases: ⊕ bigoplus
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 435
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Analysis_{\mathcal P} := (\pi_i)_{i\in I}: X\longrightarrow\bigoplus_{i\in I}X_i,
\oplusDefinition passage · 18 passages
Search aliases: ⊕ oplus
Definition passage: Mapping cone · Stratified_Universality_Traditional.tex, line 1653
[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}, \qquad d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
\bigoplus_{a\in I_{Y,U,u}}Definition passage · 1 passage
Search aliases: ⊕_a∈ I_Y,U,u bigoplus in
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
\bigoplus_{i\in I}Definition passage · 3 passages
Search aliases: ⊕_i∈ I bigoplus in
Definition passage: Packet analysis and synthesis · Stratified_Universality_Traditional.tex, line 435
[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.
Exact mathematical passage
\Analysis_{\mathcal P} := (\pi_i)_{i\in I}: X\longrightarrow\bigoplus_{i\in I}X_i,
\bigoplus_{i\in I_{X,U,r}}Definition passage · 1 passage
Search aliases: ⊕_i∈ I_X,U,r bigoplus in
Definition passage: Matrix local system · block16_section_final.tex, line 97
[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.
Exact mathematical passage
\mathscr M^{u,r}_{Y,X}(U) := \bigoplus_{a\in I_{Y,U,u}} \bigoplus_{i\in I_{X,U,r}} \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
\bigcapDefinition passage · 12 passages
Search aliases: ⋂ bigcap
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 65
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
N_{\mathsf S} := \bigcap_{m\in M}\bigcap_{j\in J} \ker(o_j\rho(m)).
\bigcap_{j\in J}Definition passage · 1 passage
Search aliases: ⋂_j∈ J bigcap in
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 65
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
N_{\mathsf S} := \bigcap_{m\in M}\bigcap_{j\in J} \ker(o_j\rho(m)).
\bigcap_{m\in M}Definition passage · 1 passage
Search aliases: ⋂_m∈ M bigcap in
Definition passage: Behavioral congruence and linear behavioral kernel · block17_semantic_quotients_final.tex, line 65
[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.
Exact mathematical passage
N_{\mathsf S} := \bigcap_{m\in M}\bigcap_{j\in J} \ker(o_j\rho(m)).
\cdotsDefinition passage · 38 passages
Search aliases: ⋯ cdots
Definition passage: Digit-linear realization · Stratified_Universality_Traditional.tex, line 3552
[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equation [source reference] includes the required leading-zero consistency.
Exact mathematical passage
w=d_0\cdots d_{\ell-1}
\longrightarrowDefinition passage · 103 passages
Search aliases: ⟶ longrightarrow
Definition passage: Realization of a finite resolved signature · Stratified_Universality_Traditional.tex, line 362
[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
Exact mathematical passage
M_a:M_{s(a)}\longrightarrow M_{t(a)}
\longmapstoDefinition passage · 20 passages
Search aliases: ⟼ longmapsto
Definition passage: Resolved observational system · block17_semantic_quotients_final.tex, line 40
lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.
Exact mathematical passage
\Beh_{\mathsf S}:X\longrightarrow \prod_{(m,j)\in M\times J}Y_j, \qquad x\longmapsto\bigl(o_j(\rho(m)x)\bigr)_{m,j}.
a_cIntroduction passage · 3 passages
Search aliases: a_c
Introduction passage · block19_admissibility_lifting_final.tex, line 61
Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not attained anywhere in Γ, a_c>0. Every near-minimizing sequence eventually has defect below min{1,a_c}, and therefore cannot belong to Γ_c. Since c was…
Exact mathematical passage
a_c=\min_{g\in K_c}d(X,g).
\mathfrak A_{\mathrm{cert}}Introduction passage · 4 passages
Search aliases: A_cert mathfrak mathrm
Introduction passage · block14_section.tex, line 5
Let A_cert:= A_13 denote the finite certified atlas constructed in [source reference]. Its reversible core is the disjoint union of five connected semantic components, while its directional satellites retain explicit loss or obstruction passports.
Exact mathematical passage
\mathfrak A_{\mathrm{cert}}:=\mathfrak A_{13}
A_FIntroduction passage · 2 passages
Search aliases: A_F
Introduction passage · block10_section.tex, line 90
Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.
Exact mathematical passage
A_F=K(z)[Y]/(P_F),
a_{n+T}Introduction passage · 1 passage
Search aliases: a_n+T
Introduction passage · Stratified_Universality_Traditional.tex, line 2214
Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For exampl…
Exact mathematical passage
a_0=1 \qquad\text{and}\qquad a_{n+T}=a_n \quad(n\geq 0)
a_rIntroduction passage · 1 passage
Search aliases: a_r
Introduction passage · Stratified_Universality_Traditional.tex, line 2225
p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3,…
Exact mathematical passage
W_a(z)=\sum_{r=0}^{T(a)-1}a_rz^r.
a_{T(a)-1}Introduction passage · 1 passage
Search aliases: a_T(a)-1
Introduction passage · Stratified_Universality_Traditional.tex, line 2221
Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For exampl…
Exact mathematical passage
w(a)=(a_0,\ldots,a_{T(a)-1})
\bar YIntroduction passage · 4 passages
Search aliases: bar Y bar
Introduction passage · block10_section.tex, line 93
Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.
Exact mathematical passage
\bar Y
\bar Y^{d-1}Introduction passage · 1 passage
Search aliases: bar Y^d-1 bar
Introduction passage · block10_section.tex, line 94
Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.
Exact mathematical passage
1,\bar Y,\ldots,\bar Y^{d-1},
\mathrm{cert}Introduction passage · 5 passages
Search aliases: cert mathrm
Introduction passage · block14_section.tex, line 5
Let A_cert:= A_13 denote the finite certified atlas constructed in [source reference]. Its reversible core is the disjoint union of five connected semantic components, while its directional satellites retain explicit loss or obstruction passports.
Exact mathematical passage
\mathfrak A_{\mathrm{cert}}:=\mathfrak A_{13}
\mathsf DIntroduction passage · 7 passages
Search aliases: D mathsf
Introduction passage · block13_bridge_matrix.tex, line 24
parbox0.96linewidthscriptsize Legend. E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form E_Fuss/ D means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahler coefficient equations an…
Exact mathematical passage
\mathsf D
\mathsf EIntroduction passage · 23 passages
Search aliases: E mathsf
Introduction passage · block13_bridge_matrix.tex, line 24
parbox0.96linewidthscriptsize Legend. E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form E_Fuss/ D means equivalence on the marked Fuss slice and directional loss on the ambi
Exact mathematical passage
\mathsf E
\mathsf E_{\mathrm{Fuss}}Introduction passage · 1 passage
Search aliases: E_Fuss mathsf mathrm
Introduction passage · block13_bridge_matrix.tex, line 24
E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form E_Fuss/ D means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahler coefficient equations and Newton--Mahler interpolation remain …
Exact mathematical passage
\mathsf E_{\mathrm{Fuss}}/\mathsf D
\mathbb F_p^{\mathbb N}Introduction passage · 1 passage
Search aliases: F_p^ N mathbb mathbb
Introduction passage · Stratified_Universality_Traditional.tex, line 2210
Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For exampl…
Exact mathematical passage
a=(a_n)_{n\geq 0}\in \mathbb F_p^{\mathbb N}
\mathsf{FF}Introduction passage · 3 passages
Search aliases: FF mathsf
Introduction passage · block13_bridge_matrix.tex, line 24
parbox0.96linewidthscriptsize Legend. E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form E_Fuss/ D means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahle
Exact mathematical passage
\mathsf{FF}
K_cIntroduction passage · 4 passages
Search aliases: K_c
Introduction passage · block19_admissibility_lifting_final.tex, line 55
Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not attained anywhere in Γ, a_c>0. Every near-minimizing sequence eventually has defect below min{1,a_c}, and therefore cannot belong to Γ_c. Since c was…
Exact mathematical passage
K_c=\{g\in\Gamma_c:d(X,g)\leq 1\}.
M_FIntroduction passage · 2 passages
Search aliases: M_F
Introduction passage · block10_section.tex, line 97
Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.
Exact mathematical passage
M_F
\minIntroduction passage · 25 passages
Search aliases: min
Introduction passage · block19_admissibility_lifting_final.tex, line 61
Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not attained anywhere in Γ, a_c>0. Every near-minimizing sequence eventually has defect below min{1,a_c}, and therefore cannot belong to Γ_c. Since c was…
Exact mathematical passage
a_c=\min_{g\in K_c}d(X,g).
\min_{g\in K_c}Introduction passage · 1 passage
Search aliases: min_g∈ K_c min in
Introduction passage · block19_admissibility_lifting_final.tex, line 61
Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not attained anywhere in Γ, a_c>0. Every near-minimizing sequence eventually has defect below min{1,a_c}, and therefore cannot belong to Γ_c. Since c was…
Exact mathematical passage
a_c=\min_{g\in K_c}d(X,g).
\PerIntroduction passage · 8 passages
Search aliases: Per mathbf
Introduction passage · Stratified_Universality_Traditional.tex, line 2209
Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For exampl…
Exact mathematical passage
\Per_p^{\times,\mathrm{per}}
\mathrm{per}Introduction passage · 24 passages
Search aliases: per mathrm
Introduction passage · Stratified_Universality_Traditional.tex, line 2209
Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For exampl…
Exact mathematical passage
\Per_p^{\times,\mathrm{per}}
\Per_p^{\times,\mathrm{per}}Introduction passage · 8 passages
Search aliases: Per_p^×,per Per times mathrm mathbf times mathrm
Introduction passage · Stratified_Universality_Traditional.tex, line 2209
Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For exampl…
Exact mathematical passage
\Per_p^{\times,\mathrm{per}}
\mathfrak R_p^{\mathrm{per}}Introduction passage · 5 passages
Search aliases: R_p^per mathfrak mathrm
Introduction passage · Stratified_Universality_Traditional.tex, line 2648
Let R_p^per denote the four-chart pseudofunctor defined by the preceding passports and comparison cells.
Exact mathematical passage
\mathfrak R_p^{\mathrm{per}}
\mathfrak R_p^{\mathrm{rm}}Introduction passage · 2 passages
Search aliases: R_p^rm mathfrak mathrm
Introduction passage · Stratified_Universality_Traditional.tex, line 3277
Let R_p^rm denote the three-chart pseudofunctor defined by the passports above.
Exact mathematical passage
\mathfrak R_p^{\mathrm{rm}}
\mathrm{rm}Introduction passage · 8 passages
Search aliases: rm mathrm
Introduction passage · Stratified_Universality_Traditional.tex, line 3277
Let R_p^rm denote the three-chart pseudofunctor defined by the passports above.
Exact mathematical passage
\mathfrak R_p^{\mathrm{rm}}
\mathcal TIntroduction passage · 8 passages
Search aliases: T mathcal
Introduction passage · Stratified_Universality_Traditional.tex, line 5582
Let T_M,s denote this coefficient operator. Conjugating the clopen packet theorem by M_p gives the following normalized coefficient atlas.
Exact mathematical passage
\mathcal T_{M,s}
\mathcal T_{M,s}Introduction passage · 2 passages
Search aliases: T_M,s mathcal
Introduction passage · Stratified_Universality_Traditional.tex, line 5582
Let T_M,s denote this coefficient operator. Conjugating the clopen packet theorem by M_p gives the following normalized coefficient atlas.
Exact mathematical passage
\mathcal T_{M,s}
W_aIntroduction passage · 2 passages
Search aliases: W_a
Introduction passage · Stratified_Universality_Traditional.tex, line 2225
p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3,…
Exact mathematical passage
W_a(z)=\sum_{r=0}^{T(a)-1}a_rz^r.
z^rIntroduction passage · 5 passages
Search aliases: z^r
Introduction passage · Stratified_Universality_Traditional.tex, line 2225
p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3,…
Exact mathematical passage
W_a(z)=\sum_{r=0}^{T(a)-1}a_rz^r.
\Gamma_cIntroduction passage · 4 passages
Search aliases: Γ_c Gamma
Introduction passage · block19_admissibility_lifting_final.tex, line 53
Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not attained anywhere in Γ, a_c>0. Every near-minimizing sequence eventually has defect below min{1,a_c}, and therefore cannot belong to Γ_c. Since c was…
Exact mathematical passage
\Gamma_c
\sum_{r=0}^{T(a)-1}Introduction passage · 1 passage
Search aliases: ∑_r=0^T(a)-1 sum
Introduction passage · Stratified_Universality_Traditional.tex, line 2225
p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3,…
Exact mathematical passage
W_a(z)=\sum_{r=0}^{T(a)-1}a_rz^r.
\mathbf1Use in source · 1 passage
Search aliases: 1 mathbf
Use in source · block07_named_local_table.tex, line 10
Constant-one function 1 · β_1=1 · 1,1,1,1,1,1,… · 1,1,1,1,1,1,…
Exact mathematical passage
\mathbf 1
\mathbf aUse in source · 2 passages
Search aliases: a mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2040
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
(\chi,\mathbf a_0),
A'Use in source · 3 passages
Search aliases: A'
Use in source · Stratified_Universality_Traditional.tex, line 4889
[Ghost-coordinate equivalence] The divisor ghost map W: E_ Z ⟶ G_ Z, b⟼ c(b), is an additive group isomorphism. Its inverse is b_n = frac1n∑_dmid n μ (frac nd)c_d. If A(z)= E(b)(z), then fraczA'(z)A(z) = ∑_n≥1c_nz^n.
Exact mathematical passage
\boxed{ \frac{zA'(z)}{A(z)} = \sum_{n\geq1}c_nz^n. }
a'_tUse in source · 1 passage
Search aliases: a'_t
Use in source · block18_carrier_observer_change_final.tex, line 187
ed-observer Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(s,b)=tL_b(a_s). If a finite family of raw histories has already been aggregated by their state using ⊕, then this update gives exactly the same result as advancing every raw histor…
Exact mathematical passage
a'_t := \bigoplus_{\delta(s,b)=t}L_b(a_s).
A^{\mathsf T}Use in source · 1 passage
Search aliases: A^ T mathsf
Use in source · block18_carrier_observer_change_final.tex, line 362
For a linear map A, h_A^ TC(x)=h_C(Ax). If the incoming fiber is empty, both sides of the formula are -∞. Otherwise, the support function of the convex hull of a finite union is the maximum of the component support functions, with the same conventions for empty components. These identities give the formula. Finite unions of compact sets have compact convex hulls in finite dimension, so every updated packet stays in the declared carrier. Iteration follows from composition of the linear maps and o…
Exact mathematical passage
h_{A^{\mathsf T}C}(x)=h_C(Ax).
\mathfrak a^{u,r}_{X,Y}Use in source · 1 passage
Search aliases: a^u,r_X,Y mathfrak
Use in source · block15_section.tex, line 396
[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contribute. The inverse is the corresponding fine analysis after coarse synthesis. Consequently, changing a complete lane resolution loses no information. Lo…
Exact mathematical passage
\Gamma^{(u,r)}_{(v,s)} := \mathfrak a^{u,r}_{X,Y} \bigl(\mathfrak a^{v,s}_{X,Y}\bigr)^{-1}.
\mathfrak A^{u,r}_{Y,X}Use in source · 2 passages
Search aliases: A^u,r_Y,X mathfrak
Use in source · block16_section_final.tex, line 141
item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix record. It changes the global type of that record from a constantly labeled matrix to a compatible section of a matrix local system.
Exact mathematical passage
\mathfrak A^{u,r}_{Y,X}: \Hom_{\Desc(\mathscr G)}(X,Y) \xrightarrow{\sim} \Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right), \qquad f\longmapsto \left(\bigl([f_U]^{u,r}_{ai}\bigr)_{a,i}\right)_U.
\mathfrak A^{v,r}_{Z,X}Use in source · 1 passage
Search aliases: A^v,r_Z,X mathfrak
Use in source · block16_section_final.tex, line 177
item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, maths…
Exact mathematical passage
\mathfrak A^{v,r}_{Z,X}(gf) = \mathfrak A^{v,u}_{Z,Y}(g) \star \mathfrak A^{u,r}_{Y,X}(f).
\mathfrak a^{v,s}_{X,Y}Use in source · 2 passages
Search aliases: a^v,s_X,Y mathfrak
Use in source · block15_section.tex, line 385
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathfrak a^{v,s}_{X,Y}: \Hom_{\mathcal C}(X,Y)\xrightarrow{\sim} \mathsf B^{v,s}(X,Y),
\mathfrak A^{v,u}_{Z,Y}Use in source · 1 passage
Search aliases: A^v,u_Z,Y mathfrak
Use in source · block16_section_final.tex, line 177
item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, maths…
Exact mathematical passage
\mathfrak A^{v,r}_{Z,X}(gf) = \mathfrak A^{v,u}_{Z,Y}(g) \star \mathfrak A^{u,r}_{Y,X}(f).
\mathbf a_0Use in source · 2 passages
Search aliases: a_0 mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2040
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
(\chi,\mathbf a_0),
A_2Use in source · 7 passages
Search aliases: A_2
Use in source · block11_section.tex, line 104
[The selected object is a marked system] Two different polynomial systems can have the same fixed point. Conversely, a single polynomial system can have several fixed points outside the contraction ball. The normalized object in this section is therefore (s,A_2,…,A_D; P;certificate), not an unmarked semantic vector u.
Exact mathematical passage
(s,A_2,\ldots,A_D;\mathcal P;\text{certificate}),
A_3Use in source · 2 passages
Search aliases: A_3
Use in source · block12_section.tex, line 166
[The S_3 nonnormal atlas] Let G=S_3 and let H be generated by one transposition. There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts. Their subgroup core is trivial, while their normal closure is all of S_3. By contrast, the order-three subgroup A_3 is normal and its fixed chart carries the quotient symmetry S_3/A_3≅ C_2.
Exact mathematical passage
A_3
A_4Use in source · 1 passage
Search aliases: A_4
Use in source · block12_section.tex, line 589
The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra. It audits cyclic groups, V_4, S_3, D_4, and A_4; every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commuting and noncommuting subgroup pairs; double-coset partitions and the Mackey dimension law; cyclic ramification-style filtrations; explicit Hilbert 90 co…
Exact mathematical passage
A_4
A_b'Use in source · 1 passage
Search aliases: A_b'
Use in source · Stratified_Universality_Traditional.tex, line 5213
heorem[Exact divisor--ghost bridge] The map b↦ c= G(b) is injective, with inverse b(n) = frac1n (μmathbin∗_Dc)(n). Its integral image consists exactly of the arithmetic functions c satisfying nmid(μmathbin∗_Dc)(n) (n≥1). For the Euler product A_b(z)=∏_n≥1(1-z^n)^-b(n), one has fraczA_b'(z)A_b(z) = ∑_n≥1c(n)z^n.
Exact mathematical passage
\frac{zA_b'(z)}{A_b(z)} = \sum_{n\geq1}c(n)z^n.
A_b^{\mathsf T}Use in source · 2 passages
Search aliases: A_b^ T mathsf
Use in source · block18_carrier_observer_change_final.tex, line 207
Two important realizations have the same state skeleton but different carriers. item For convex value semantics, A is the family of compact convex sets, including the empty set, ⊕ is convex hull of union with identity ∅, and L_b(C)=A_b^ TC. item For linear polyphase routing, A is an additive convolution algebra, ⊕ is coefficient addition, and L_b multiplies route kernels by the declared filter or transition weight. The theorem transports the observer skeleton only after the carrier law and the h…
Exact mathematical passage
L_b(C)=A_b^{\mathsf T}C
A_DUse in source · 2 passages
Search aliases: A_D
Use in source · block11_section.tex, line 104
[The selected object is a marked system] Two different polynomial systems can have the same fixed point. Conversely, a single polynomial system can have several fixed points outside the contraction ball. The normalized object in this section is therefore (s,A_2,…,A_D; P;certificate), not an unmarked semantic vector u.
Exact mathematical passage
(s,A_2,\ldots,A_D;\mathcal P;\text{certificate}),
a_{d-1}Use in source · 2 passages
Search aliases: a_d-1
Use in source · Stratified_Universality_Traditional.tex, line 2048
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\mathbf a_0=(a_0,\dots,a_{d-1})
A_{\operatorname{dig}_k(n)}Use in source · 3 passages
Search aliases: A_dig_k(n) operatorname
Use in source · Stratified_Universality_Traditional.tex, line 3598
Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs to B(W), so dim_K V_k(a) ≤ dim_K W. Thus a is regular. The realization in [source reference] has dimension exactly r_k(a).
Exact mathematical passage
\mathcal B(x)(n) = \lambda A_{\operatorname{dig}_k(n)}x,
a_eUse in source · 1 passage
Search aliases: a_e
Use in source · block18_carrier_observer_change_final.tex, line 266
[Observer base-change exact sequence] The evaluation map π:K^(E)⟶Span_K(E), ∑_e∈ Ea_e[e] ⟼ ∑_e∈ Ea_e e is surjective and fits into the exact sequence 0⟶Rel_K(E) ⟶ K^(E) xrightarrowπSpan_K(E) ⟶0, where Rel_K(E):=kerπ is the module of linear relations among distinct discrete residual states. Hence Set minimization commutes with free linearization exactly when E is linearly independent.
Exact mathematical passage
\pi:K^{(E)}\longrightarrow\Span_K(E), \qquad \sum_{e\in E}a_e[e] \longmapsto \sum_{e\in E}a_e e
A_{h j,h i}Use in source · 1 passage
Search aliases: A_h j,h i
Use in source · block16_section_final.tex, line 520
Write an arbitrary kk-linear map as a matrix A=(A_ji). The intertwining equations for the permutation actions are equivalent to A_h j,h i=A_j i (h∈ H). Hence the entries are constant on diagonal orbits. The orbit-indicator matrices have disjoint supports, so they are linearly independent over every commutative ring and span all invariant matrices.
Exact mathematical passage
A_{h j,h i}=A_{j i} \qquad(h\in H).
a_{H_ax+j}Use in source · 1 passage
Search aliases: a_H_ax+j
Use in source · Stratified_Universality_Traditional.tex, line 5845
[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.
Exact mathematical passage
x\mapsto a_{H_ax+j}
a_{Hx+j}Use in source · 1 passage
Search aliases: a_Hx+j
Use in source · Stratified_Universality_Traditional.tex, line 5813
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r\geq0} p^{Lr}\binom xr w^{\mathsf T}C^jB^rv_0,}
A_{j i}Use in source · 1 passage
Search aliases: A_j i
Use in source · block16_section_final.tex, line 520
Write an arbitrary kk-linear map as a matrix A=(A_ji). The intertwining equations for the permutation actions are equivalent to A_h j,h i=A_j i (h∈ H). Hence the entries are constant on diagonal orbits. The orbit-indicator matrices have disjoint supports, so they are linearly independent over every commutative ring and span all invariant matrices.
Exact mathematical passage
A_{h j,h i}=A_{j i} \qquad(h\in H).
A_{ji}Use in source · 1 passage
Search aliases: A_ji
Use in source · block16_section_final.tex, line 518
Write an arbitrary kk-linear map as a matrix A=(A_ji). The intertwining equations for the permutation actions are equivalent to A_h j,h i=A_j i (h∈ H). Hence the entries are constant on diagonal orbits. The orbit-indicator matrices have disjoint supports, so they are linearly independent over every commutative ring and span all invariant matrices.
Exact mathematical passage
A=(A_{ji})
a_{kn+d}Use in source · 1 passage
Search aliases: a_kn+d
Use in source · block15_section.tex, line 821
[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
\Lambda_d\left(\sum_{n\ge0}a_nz^n\right) :=\sum_{n\ge0}a_{kn+d}z^n, \qquad S_d(G):=z^dG(z^k), \quad 0\le d<k.
a_mUse in source · 2 passages
Search aliases: a_m
Use in source · block15_section.tex, line 664
For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0
Exact mathematical passage
F(z)=\sum_{m\ge0}a_mz^m\in A[[z]]
A_{M,s}^{(d)}Use in source · 3 passages
Search aliases: A_M,s^(d)
Use in source · Stratified_Universality_Traditional.tex, line 5645
The same integer kernel matrix has different categorical status after changing the coefficient doctrine. Fix d≥0 and write M_d(R)=R^d+1 for the formal truncated Mahler coefficient module. Let A_M,s^(d) =[Θ_M,s(r,k)]_0≤ k,r≤ d, where the target degree k is the row index.
Exact mathematical passage
A_{M,s}^{(d)} =\bigl[\Theta_{M,s}(r,k)\bigr]_{0\leq k,r\leq d},
A_{m,s}^{(d)}Use in source · 2 passages
Search aliases: A_m,s^(d)
Use in source · Stratified_Universality_Traditional.tex, line 1966
Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with trivial holonomy on those diamonds.
Exact mathematical passage
A_{m,s}^{(d)}
A_{m;i\mid j_1,\ldots,j_m}Use in source · 5 passages
Search aliases: A_m;imid j_1,…,j_m mid ldots
Use in source · block11_section.tex, line 114
For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
A_{m;i\mid j_1,\ldots,j_m} = P_iA_m(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
A_{mn,mt+s}^{(d)}Use in source · 1 passage
Search aliases: A_mn,mt+s^(d)
Use in source · Stratified_Universality_Traditional.tex, line 1967
Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with trivial holonomy on those diamonds.
Exact mathematical passage
A_{n,t}^{(d)}A_{m,s}^{(d)}=A_{mn,mt+s}^{(d)}.
A_NUse in source · 7 passages
Search aliases: A_N
Use in source · block09_section.tex, line 605
Define the alias and cancellation deficits by A_N(λ)=N-r_rm orb(N;λ), C_N,j(λ) = r_rm orb(N;λ) -r_D(N,j;λ). Then N-r_D(N,j;λ) = A_N(λ)+C_N,j(λ).
Exact mathematical passage
\boxed{ A_N(\lambda)=N-r_{\rm orb}(N;\lambda),}
a_NUse in source · 1 passage
Search aliases: a_N
Use in source · Stratified_Universality_Traditional.tex, line 4845
[Finite-prefix triangularity] For each N, the coefficient prefix (a_1,…,a_N) of A(z)=1+∑_n≥1a_nz^n and the exponent prefix (b_1,…,b_N) determine one another by integral triangular recursions. No exponent b_m with m>N affects the coefficient prefix through degree N.
Exact mathematical passage
(a_1,\ldots,a_N) \quad\text{of}\quad A(z)=1+\sum_{n\geq1}a_nz^n
a_{n+1}Use in source · 1 passage
Search aliases: a_n+1
Use in source · Stratified_Universality_Traditional.tex, line 2049
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
a_{n+d}=c_{d-1}a_{n+d-1}+\cdots+c_1a_{n+1}+c_0a_n.
a_{n+d}Use in source · 1 passage
Search aliases: a_n+d
Use in source · Stratified_Universality_Traditional.tex, line 2049
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
a_{n+d}=c_{d-1}a_{n+d-1}+\cdots+c_1a_{n+1}+c_0a_n.
a_{n+d-1}Use in source · 1 passage
Search aliases: a_n+d-1
Use in source · Stratified_Universality_Traditional.tex, line 2049
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
a_{n+d}=c_{d-1}a_{n+d-1}+\cdots+c_1a_{n+1}+c_0a_n.
a_{n+s}Use in source · 3 passages
Search aliases: a_n+s
Use in source · block09_section.tex, line 134
Conversely, suppose ∑_s=0^qp_s(n)a_n+s=0 (n≥ N), where p_s∈ K[n], and put θ=zD.
Exact mathematical passage
\sum_{s=0}^{q}p_s(n)a_{n+s}=0 \qquad(n\geq N),
A_{N,j}Use in source · 8 passages
Search aliases: A_N,j
Use in source · block09_section.tex, line 364
Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j measures cancellation relative to that construction. It is not intrinsic until the construction has been fixed.
Exact mathematical passage
Q^{\rm cl}_{N,j}=A_{N,j}L_{N,j}
A_{n,t}^{(d)}Use in source · 1 passage
Search aliases: A_n,t^(d)
Use in source · Stratified_Universality_Traditional.tex, line 1967
Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with trivial holonomy on those diamonds.
Exact mathematical passage
A_{n,t}^{(d)}A_{m,s}^{(d)}=A_{mn,mt+s}^{(d)}.
a_{n-j}Use in source · 1 passage
Search aliases: a_n-j
Use in source · Stratified_Universality_Traditional.tex, line 4917
[Exact coefficient recurrence] Let A(z)=1+∑_n≥1a_nz^n and let c_n be its Euler ghost coordinates. Then na_n = ∑_j=1^nc_j a_n-j (n≥1), with a_0=1. Conversely, any c∈ G_ Z and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.
Exact mathematical passage
\boxed{ na_n = \sum_{j=1}^{n}c_j a_{n-j} \qquad(n\geq1), }
a_{n-m+j}Use in source · 2 passages
Search aliases: a_n-m+j
Use in source · block09_section.tex, line 108
[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.
Exact mathematical passage
\boxed{ [z^n]LF = \sum_{j=0}^{r} \sum_{m=0}^{\min(d_j,n)} q_{j,m} (n-m+j)^{\underline{j}} a_{n-m+j},}
a_{N_a+s-1}Use in source · 2 passages
Search aliases: a_N_a+s-1
Use in source · block09_section.tex, line 64
Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
Exact mathematical passage
\mathfrak R(a) = \bigl(R_a,N_a,a_0,\ldots,a_{N_a+s-1}\bigr)
a_{Nm+j}Use in source · 4 passages
Search aliases: a_Nm+j
Use in source · Stratified_Universality_Traditional.tex, line 2164
Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [source reference].
Exact mathematical passage
a_{Nm+j}
a_{Nn+j}Use in source · 1 passage
Search aliases: a_Nn+j
Use in source · block15_section.tex, line 791
[Typed polyphase realization] Let K be a field, let N∈ Z with N≥1 be invertible in K, and suppose K contains a primitive N-th root of unity. For F(z)=∑_n≥0a_nz^n∈ K[[z]], set F_j(w):=∑_n≥0a_Nn+jw^n. Then F(z)=∑_j=0^N-1z^jF_j(z^N) is exact analysis and synthesis. The Fourier residue projectors and the polyphase components satisfy the explicit coordinate relation P_N,jF(z)=z^jF_j(z^N). For every integer M≥1, refinement from N to NM is the projector-sum law of [source reference].
Exact mathematical passage
F_j(w):=\sum_{n\ge0}a_{Nn+j}w^n.
\mathcal A_pUse in source · 4 passages
Search aliases: A_p mathcal
Use in source · Stratified_Universality_Traditional.tex, line 2683
For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a),
Exact mathematical passage
\label{eq:periodic-invariant-package} \mathcal J_p(a) = \bigl( T(a), w(a), m_a(x), P_a(z)/Q_a(z), \mathcal K_p(a), \mathcal A_p(a), R_a(z) \bigr),
a_{p^\ell n+r}Use in source · 2 passages
Search aliases: a_p^ℓ n+r ell
Use in source · Stratified_Universality_Traditional.tex, line 2346
The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.
Exact mathematical passage
\Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a = \bigl(n\longmapsto a_{p^\ell n+r}\bigr), \qquad r=\sum_{j=0}^{\ell-1}d_jp^j.
a_{pn+d}Use in source · 1 passage
Search aliases: a_pn+d
Use in source · Stratified_Universality_Traditional.tex, line 2322
For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.
Exact mathematical passage
(\Lambda_da)_n=a_{pn+d}.
A_qUse in source · 3 passages
Search aliases: A_q
Use in source · block13_section.tex, line 296
Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value c_0=1.
Exact mathematical passage
A_q(n) =(n+1)\prod_{r=2}^{q}\bigl((q-1)n+r\bigr),
a_sUse in source · 2 passages
Search aliases: a_s
Use in source · block18_carrier_observer_change_final.tex, line 186
belthm:block18-carrier-indexed-observer Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(s,b)=tL_b(a_s). If a finite family of raw histories has already been aggregated by their state using ⊕, then this update gives exactly the same result a…
Exact mathematical passage
a=(a_s)_{s\in S}
A_uUse in source · 1 passage
Search aliases: A_u
Use in source · Stratified_Universality_Traditional.tex, line 3689
For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.
Exact mathematical passage
H_a(u,v) = \lambda A_vA_uv_{\mathrm{in}}.
A_VUse in source · 4 passages
Search aliases: A_V
Use in source · block15_section.tex, line 1106
Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global re…
Exact mathematical passage
\lambda_{UV}^{A,r}: I_{A_U,r}\xrightarrow{\sim}I_{A_V,r}
A_vUse in source · 1 passage
Search aliases: A_v
Use in source · Stratified_Universality_Traditional.tex, line 3689
For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.
Exact mathematical passage
H_a(u,v) = \lambda A_vA_uv_{\mathrm{in}}.
A_\sigmaUse in source · 5 passages
Search aliases: A_σ sigma
Use in source · block12_section.tex, line 412
A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.
Exact mathematical passage
A_\sigma\in\operatorname{GL}_n(L).
a_\sigmaUse in source · 5 passages
Search aliases: a_σ sigma
Use in source · Stratified_Universality_Traditional.tex, line 1691
[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
a_{\sigma\tau}=a_\sigma\,\sigma(a_\tau), \qquad a_\sigma\in GL_n(L),
a_{\sigma\tau}Use in source · 2 passages
Search aliases: a_στ sigma tau
Use in source · Stratified_Universality_Traditional.tex, line 1691
[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
a_{\sigma\tau}=a_\sigma\,\sigma(a_\tau), \qquad a_\sigma\in GL_n(L),
A_{\sigma\tau}^{-1}Use in source · 1 passage
Search aliases: A_στ^-1 sigma tau
Use in source · block12_section.tex, line 416
A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.
Exact mathematical passage
\boxed{ c(\sigma,\tau) =A_\sigma\sigma(A_\tau)A_{\sigma\tau}^{-1} \in L^\times.} \tag{12.18}
A_\tauUse in source · 1 passage
Search aliases: A_τ tau
Use in source · block12_section.tex, line 416
A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.
Exact mathematical passage
\boxed{ c(\sigma,\tau) =A_\sigma\sigma(A_\tau)A_{\sigma\tau}^{-1} \in L^\times.} \tag{12.18}
a_\tauUse in source · 3 passages
Search aliases: a_τ tau
Use in source · Stratified_Universality_Traditional.tex, line 1691
[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
a_{\sigma\tau}=a_\sigma\,\sigma(a_\tau), \qquad a_\sigma\in GL_n(L),
\acuteUse in source · 5 passages
Search aliases: acute
Use in source · block12_section.tex, line 4
cal descent, profinite refinement, and the tensor-fiber boundary at which a symmetry group can be reconstructed. The strongest Galois source already contains Reynolds projectors, the subgroup--fixed-field correspondence, normality and quotient compression, conjugate projector atlases, Mackey double-coset packets, ramification details, Haar projectors, Hilbert 90, the Brauer obstruction, finite acuteetale permutation resolutions, and neutral Tannakian reconstruction [source reference]. The purpos…
Exact mathematical passage
\acute{e}
\mathrm{ak}Use in source · 5 passages
Search aliases: ak mathrm
Use in source · Stratified_Universality_Traditional.tex, line 884
[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right side is the category of typed realizations satisfying all declared relations and carrying every declared packet idempotent to a split idempotent.
Exact mathematical passage
\boxed{ \Syn_{\mathrm{ak}}(\mathbb K_{\rm lin}) := \Kar\!\bigl(\Add(\kk Q/I)\bigr)}
\Analysis_{p^a}Use in source · 1 passage
Search aliases: Analysis_p^a Analysis
Use in source · block15_section.tex, line 892
[Typed local clopen realization] Fix a prime p and an integer a≥0. For f∈ C( Z_p, Z_p), with q∈ Z_p the branch coordinate, define Analysis_p^a(f) := (q↦ f(p^aq+s))_0≤ s<p^a. Piecewise synthesis on the disjoint clopen cosets s+p^a Z_p is its inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix composition law. Let M:= M_p^-1:C( Z_p, Z_p) xrightarrow∼c_0( Z_p) be Mahler analysis from [source reference], and write R_p^a,sf(q):=f(p^aq+s). Conjugat…
Exact mathematical passage
\Analysis_{p^a}(f) := \bigl(q\mapsto f(p^aq+s)\bigr)_{0\le s<p^a}.
\Analysis_X^rUse in source · 2 passages
Search aliases: Analysis_X^r Analysis
Use in source · block15_section.tex, line 78
At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.
Exact mathematical passage
\Analysis_X^r :=(\pi_{X,i}^r)_i: X\longrightarrow\bigoplus_iX_i^r, \qquad \Synthesis_X^r :=(\iota_{X,i}^r)_i: \bigoplus_iX_i^r\longrightarrow X.
\AutUse in source · 10 passages
Search aliases: Aut Aut operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1170
Intrinsic symmetry · Aut_ M_α(M) · automorphisms of one resolved object inside the stratum · none beyond the native category
Exact mathematical passage
\Aut_{\mathcal M_\alpha}(M)
\mathbf{Aut}Use in source · 3 passages
Search aliases: Aut mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3219
Minimal digital chart. An object of Aut_p^min,rat is a reachable minimal least-significant-digit-first p-DFAO whose coefficient series belongs to RMah_p^×. Its canonical representative has state set K_p(F), transition G↦Λ_dG, start state F, and output G↦ G(0).
Exact mathematical passage
\mathbf{Aut}_p^{\mathrm{min,rat}}
\operatorname{Aut}Use in source · 2 passages
Search aliases: Aut operatorname
Use in source · block12_section.tex, line 494
[Neutral Tannakian reconstruction] Let C be a neutral Tannakian category over k, and let ω: C⟶Vect_k be a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) tag12.20 is an affine group scheme, and C≃Rep_k( G) tag12.21 as tensor categories. Declared packet idempotents and refinement maps preserved by ω become tensor-compatible concrete projector resolutions in every fiber.
Exact mathematical passage
\boxed{ \mathcal G=\operatorname{Aut}^{\otimes}(\omega)} \tag{12.20}
\Aut^{\otimes}Use in source · 2 passages
Search aliases: Aut^⊗ Aut otimes operatorname otimes
Use in source · Stratified_Universality_Traditional.tex, line 1171
Reconstructed symmetry · Aut^⊗(ω) · tensor automorphisms of a fiber functor and the corresponding affine group scheme · faithful exact tensor fiber functor
Exact mathematical passage
\Aut^{\otimes}(\omega)
\operatorname{Aut}^{\otimes}Use in source · 2 passages
Search aliases: Aut^⊗ operatorname otimes
Use in source · block12_section.tex, line 494
[Neutral Tannakian reconstruction] Let C be a neutral Tannakian category over k, and let ω: C⟶Vect_k be a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) tag12.20 is an affine group scheme, and C≃Rep_k( G) tag12.21 as tensor categories. Declared packet idempotents and refinement maps preserved by ω become tensor-compatible concrete projector resolutions in every fiber.
Exact mathematical passage
\boxed{ \mathcal G=\operatorname{Aut}^{\otimes}(\omega)} \tag{12.20}
\Aut_{\mathcal M_\alpha}Use in source · 1 passage
Search aliases: Aut_ M_α Aut mathcal alpha operatorname mathcal alpha
Use in source · Stratified_Universality_Traditional.tex, line 1170
Intrinsic symmetry · Aut_ M_α(M) · automorphisms of one resolved object inside the stratum · none beyond the native category
Exact mathematical passage
\Aut_{\mathcal M_\alpha}(M)
\mathbf{Aut}_kUse in source · 1 passage
Search aliases: Aut_k mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3856
k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
\mathbf{Aut}_k(K) \simeq \mathbf{FinOrbSec}_k(K) \simeq \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}} \simeq \mathbf{DFAO}_{k,K}^{\min}. \label{eq:block05-automatic-finite-orbit-atlas}
\Aut_{\mathscr G}Use in source · 4 passages
Search aliases: Aut_mathscr G Aut mathscr
Use in source · block16_section_final.tex, line 304
[Base-fiber invariant normal form] Fix a base chart U_0. Evaluation at U_0 induces an isomorphism Γ (mathscr G,mathscr M^u,r_Y,X) xrightarrow∼ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0), where the right-hand side is the subgroup fixed by every loop-transport map. Consequently, Hom_Desc(mathscr G)(X,Y) ≅ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0). After choosing one arrow τ_U:U_0→ U for each chart, the inverse sends a fixed base matrix h_0 to the section h_U=(τ_U)_*h_0. The result is independent of…
Exact mathematical passage
\Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right) \xrightarrow{\sim} \mathscr M^{u,r}_{Y,X}(U_0)^{\Aut_{\mathscr G}(U_0)},
\mathbf{Aut}_p^{\mathrm{min,rat}}Use in source · 2 passages
Search aliases: Aut_p^min,rat mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3219
Minimal digital chart. An object of Aut_p^min,rat is a reachable minimal least-significant-digit-first p-DFAO whose coefficient series belongs to RMah_p^×. Its canonical representative has state set K_p(F), transition G↦Λ_dG, start state F, and output G↦ G(0).
Exact mathematical passage
\mathbf{Aut}_p^{\mathrm{min,rat}}
\Aut_p^{\times,\mathrm{per}}Use in source · 2 passages
Search aliases: Aut_p^×,per Aut times mathrm operatorname times mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2415
Let Aut_p^×,per be the groupoid of reachable minimal least-significant-digit-first p-DFAOs whose output sequence lies in Per_p^×,per. Morphisms are automaton isomorphisms preserving the start state, transitions, and outputs. Reachability forces every automorphism to be the identity.
Exact mathematical passage
\Aut_p^{\times,\mathrm{per}}
\mathbf bUse in source · 3 passages
Search aliases: b mathbf
Use in source · block15_section.tex, line 717
K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
\mathbf b_{N,j} := \left\{ \frac{j+r}{N}:1\le r\le N, \ r\ne N-j \right\},
\mathcal BUse in source · 12 passages
Search aliases: B mathcal
Use in source · Stratified_Universality_Traditional.tex, line 3598
Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs to B(W), so dim_K V_k(a) ≤ dim_K W. Thus a is regular. The realization in [source reference] has dimension exactly r_k(a).
Exact mathematical passage
\mathcal B(x)(n) = \lambda A_{\operatorname{dig}_k(n)}x,
b'Use in source · 1 passage
Search aliases: b'
Use in source · Stratified_Universality_Traditional.tex, line 4827
Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, the…
Exact mathematical passage
\mathcal E(b+b')=\mathcal E(b)\mathcal E(b').
B^{-1}Use in source · 3 passages
Search aliases: B^-1
Use in source · Stratified_Universality_Traditional.tex, line 1697
[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
a_\sigma=B^{-1}\sigma(B).
B^{p-2}Use in source · 1 passage
Search aliases: B^p-2
Use in source · Stratified_Universality_Traditional.tex, line 2575
Let R(z)=fracA(z)B(z), A(0)=B(0)=1, with a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source reference]. It computes the coefficient sequence exactly.
Exact mathematical passage
P\longmapsto\Lambda_d(PAB^{p-2}), \qquad 0\leq d<p.
B^rUse in source · 1 passage
Search aliases: B^r
Use in source · Stratified_Universality_Traditional.tex, line 5813
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r\geq0} p^{Lr}\binom xr w^{\mathsf T}C^jB^rv_0,}
\mathsf B^{u,r}Use in source · 1 passage
Search aliases: B^u,r mathsf
Use in source · block15_section.tex, line 383
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathsf B^{u,r}(X,Y)
\mathsf B^{v,s}Use in source · 2 passages
Search aliases: B^v,s mathsf
Use in source · block15_section.tex, line 376
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathsf B^{v,s}(X,Y) := \bigoplus_{b\in I_{Y,v}} \bigoplus_{j\in I_{X,s}} \Hom_{\mathcal C}(X_j^s,Y_b^v),
B_{\mathcal O}Use in source · 2 passages
Search aliases: B_ O mathcal
Use in source · block16_section_final.tex, line 478
[Permutation-module orbital lane calculus] Let kk be a commutative ring, let a finite group H act on finite sets I,J,K, and let kk[I],kk[J],kk[K] be the corresponding permutation modules. For an orbit O⊆ J× I under the diagonal action, let B_ O:kk[I]→kk[J] be the matrix whose (j,i)-entry is one when (j,i)∈ O and zero otherwise. Then Hom_kk H(kk[I],kk[J]) = ⊕_ O∈ Hbackslash(J× I) kk B_ O. In particular, an equivariant matrix is exactly a matrix constant on every pair orbit.
Exact mathematical passage
B_{\mathcal O}:\kk[I]\to\kk[J]
B_{\mathcal P}Use in source · 2 passages
Search aliases: B_ P mathcal
Use in source · block16_section_final.tex, line 507
Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.
Exact mathematical passage
B_{\mathcal Q}B_{\mathcal P} = \sum_{\mathcal R\in H\backslash(K\times I)} c^{\mathcal R}_{\mathcal Q,\mathcal P}B_{\mathcal R}.
B_{\mathcal Q}Use in source · 2 passages
Search aliases: B_ Q mathcal
Use in source · block16_section_final.tex, line 507
Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.
Exact mathematical passage
B_{\mathcal Q}B_{\mathcal P} = \sum_{\mathcal R\in H\backslash(K\times I)} c^{\mathcal R}_{\mathcal Q,\mathcal P}B_{\mathcal R}.
B_{\mathcal R}Use in source · 1 passage
Search aliases: B_ R mathcal
Use in source · block16_section_final.tex, line 507
Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.
Exact mathematical passage
B_{\mathcal Q}B_{\mathcal P} = \sum_{\mathcal R\in H\backslash(K\times I)} c^{\mathcal R}_{\mathcal Q,\mathcal P}B_{\mathcal R}.
b_0Use in source · 3 passages
Search aliases: b_0
Use in source · Stratified_Universality_Traditional.tex, line 2335
e p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.
Exact mathematical passage
b_0
b_1Use in source · 2 passages
Search aliases: b_1
Use in source · Stratified_Universality_Traditional.tex, line 4830
Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, the…
Exact mathematical passage
b_1,\ldots,b_{n-1}
\mathcal B_2Use in source · 1 passage
Search aliases: B_2 mathcal
Use in source · block10_section.tex, line 478
[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.
Exact mathematical passage
|\mathcal B_2|=4, \qquad |\mathcal C_{2,0}|=3, \qquad |\Gamma\cdot E|=2.
B_aUse in source · 3 passages
Search aliases: B_a
Use in source · Stratified_Universality_Traditional.tex, line 5841
[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.
Exact mathematical passage
B_a\in M_d(\mathbb Z_p)
\mathcal B_aUse in source · 2 passages
Search aliases: B_a mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5480
For a≥1 and 0≤ s<p^a, define the branch B_a,sf(q)=f(p^a q+s), q∈ Z_p, and the complete packet B_a f=( B_a,sf)_0≤ s<p^a.
Exact mathematical passage
\mathcal B_a f=\bigl(\mathcal B_{a,s}f\bigr)_{0\leq s<p^a}.
\mathcal B_{a,s}Use in source · 3 passages
Search aliases: B_a,s mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5475
For a≥1 and 0≤ s<p^a, define the branch B_a,sf(q)=f(p^a q+s), q∈ Z_p, and the complete packet B_a f=( B_a,sf)_0≤ s<p^a.
Exact mathematical passage
\mathcal B_{a,s}f(q)=f(p^a q+s), \qquad q\in\mathbb Z_p,
B_dUse in source · 2 passages
Search aliases: B_d
Use in source · Stratified_Universality_Traditional.tex, line 4318
Choose Smith bases for M_k(a) and L_k(a), and let D=diag(d_1,…,d_ρ) be the inclusion matrix. Let A_d and B_d be the integer matrices of Λ_d on the raw and saturated lattices.
Exact mathematical passage
B_d
B_eUse in source · 12 passages
Search aliases: B_e
Use in source · block15_section.tex, line 211
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
B_e:=\Hom_{\mathcal C}(X_i^r,Y_a^u) \quad\text{for }e=(a,i)\in E.
b_eUse in source · 9 passages
Search aliases: b_e
Use in source · block15_section.tex, line 219
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
\Phi_{\kappa,\tau}: \bigoplus_{e\in E}B_e\longrightarrow C, \qquad (b_e)_e\longmapsto \sum_{e\in E}j_{\kappa(e)}\tau_e(b_e)
B_{e'}Use in source · 3 passages
Search aliases: B_e'
Use in source · block15_section.tex, line 232
Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of Φ_κ,τ, so the fold is split lossless.
Exact mathematical passage
b_{e'}\in B_{e'}
b_{e'}Use in source · 3 passages
Search aliases: b_e'
Use in source · block15_section.tex, line 232
Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of Φ_κ,τ, so the fold is split lossless.
Exact mathematical passage
b_{e'}\in B_{e'}
b_jUse in source · 1 passage
Search aliases: b_j
Use in source · Stratified_Universality_Traditional.tex, line 3742
[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).
Exact mathematical passage
\Lambda_db_j = \sum_{i=1}^{r_k(a)}(M_d(a))_{ij}b_i, \qquad a=\sum_i v_i(a)b_i, \qquad \ell_i(a)=b_i(0).
B_NUse in source · 2 passages
Search aliases: B_N
Use in source · Stratified_Universality_Traditional.tex, line 5695
For a fixed modulus N≥2, set B_N(M) = max({1}∪ {p^v_p(M):pmidgcd(M,N)}). The least packet size for a left inverse over Z/N Z is L_N(M,d)=min(d+1,B_N(M)). The exponents v_p(N) do not change the minimum; the first residue layer already detects the obstruction.
Exact mathematical passage
B_N(M) = \max\left(\{1\}\cup \{p^{v_p(M)}:p\mid\gcd(M,N)\}\right).
\mathcal B_NUse in source · 3 passages
Search aliases: B_N mathcal
Use in source · block10_section.tex, line 275
Let Ω be a normal splitting field of the generic packet scheme over K(x), and let Γ=Gal(Ω/K(x)). After splitting E_N, the geometric packet branches form a set B_N of cardinality d^N. Let π_j: B_N→Ω be the jth packet coordinate, and set C_N,j=π_j( B_N). For the selected semantic packet U_j, separability gives [K(x)(U_j):K(x)] = |Γ· U_j|.
Exact mathematical passage
\mathcal B_N
b_NUse in source · 1 passage
Search aliases: b_N
Use in source · Stratified_Universality_Traditional.tex, line 4850
[Finite-prefix triangularity] For each N, the coefficient prefix (a_1,…,a_N) of A(z)=1+∑_n≥1a_nz^n and the exponent prefix (b_1,…,b_N) determine one another by integral triangular recursions. No exponent b_m with m>N affects the coefficient prefix through degree N.
Exact mathematical passage
(b_1,\ldots,b_N)
b_nUse in source · 6 passages
Search aliases: b_n
Use in source · Stratified_Universality_Traditional.tex, line 4839
Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, the…
Exact mathematical passage
b_n=r_n
b_n'Use in source · 1 passage
Search aliases: b_n'
Use in source · Stratified_Universality_Traditional.tex, line 4839
iv 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, their quotient would have nonzero coefficient (b_n-b_n')z^n modulo z^n+1, contradicting equality. Hence the representation is unique.
Exact mathematical passage
(b_n-b_n')z^n
B_{N,F}Use in source · 2 passages
Search aliases: B_N,F
Use in source · block10_section.tex, line 206
Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.
Exact mathematical passage
B_{N,F}=E_N[Y]/(P_F).
B_{N,j}Use in source · 6 passages
Search aliases: B_N,j
Use in source · block09_section.tex, line 466
Its coefficients are a_n = frac(λ)_nn!α^n. For 0≤ j<N, put G_N,j^λ,α(x) = ∑_m≥0 frac(λ)_Nm+j(Nm+j)! α^Nm+jx^m. Define the upper and lower parameter packets A_N,j(λ) = { fracλ+j+rN:0≤ r<N }, B_N,j = { fracj+1+rN:0≤ r<N }∖{1}.
Exact mathematical passage
B_{N,j} = \left\{ \frac{j+1+r}{N}:0\leq r<N \right\}\setminus\{1\}.
\mathbf b_{N,j}Use in source · 3 passages
Search aliases: b_N,j mathbf
Use in source · block15_section.tex, line 717
K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
\mathbf b_{N,j} := \left\{ \frac{j+r}{N}:1\le r\le N, \ r\ne N-j \right\},
b_{n-1}Use in source · 1 passage
Search aliases: b_n-1
Use in source · Stratified_Universality_Traditional.tex, line 4830
Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, the…
Exact mathematical passage
b_1,\ldots,b_{n-1}
B_QUse in source · 3 passages
Search aliases: B_Q
Use in source · block19_admissibility_lifting_final.tex, line 75
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
B_Q(T):=R_tTR_s^{-1}.
B_qUse in source · 3 passages
Search aliases: B_q
Use in source · block13_section.tex, line 300
x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value c_0=1.
Exact mathematical passage
B_q(n) =\prod_{r=1}^{q}(qn+r).
b_{u,r}Use in source · 2 passages
Search aliases: b_u,r
Use in source · Stratified_Universality_Traditional.tex, line 2357
[Finite periodic kernel and exact state model] Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives by equality of the sequences in [source reference] gives the canonical minimal least-significant-digit-first p-DFAO for a.
Exact mathematical passage
\label{eq:periodic-kernel-state} b_{u,r}(n)=w_{un+r\bmod T}, \qquad u\in U_p(T),\quad r\in\mathbb Z/T\mathbb Z,
\mathcal B_WUse in source · 1 passage
Search aliases: B_W mathcal
Use in source · Stratified_Universality_Traditional.tex, line 3697
An infinite matrix of finite rank r_k(a) has a nonsingular finite r_k(a)× r_k(a) minor, which gives the first assertion. For the remaining assertions, define the behavior map of any realization by B_W(x)(n)=λ A_dig_k(n)x. Observability makes this map injective. Reachability and equality of the represented sequence make its image equal to V_k(a). In minimal dimension it is therefore an isomorphism intertwining the digit operators, initial vector, and output. Composing the two behavior isomorphism…
Exact mathematical passage
\mathcal B_W(x)(n)=\lambda A_{\operatorname{dig}_k(n)}x.
B_\DeltaUse in source · 1 passage
Search aliases: B_Δ Delta
Use in source · block16_section_final.tex, line 567
0 · 1. The individual projectors do not descend because Sp_1S^-1=p_2, while their orbit sum p_1+p_2=1 does descend. The pair-address set has two orbits, Δ={(1,1),(2,2)}, Ω={(1,2),(2,1)}. Accordingly, End_kk C_2(kk^2) = { a · b b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivariant identity matrix to p_1, which is not equivariant. The orbit selectors C_Δ and C_Ω are compatible and split every equivariant endomorphism into its d…
Exact mathematical passage
\End_{\kk C_2}(\kk^2) = \left\{ \begin{pmatrix}a&b\\b&a\end{pmatrix}:a,b\in\kk \right\} = \kk B_\Delta\oplus\kk B_\Omega.
B_\OmegaUse in source · 1 passage
Search aliases: B_Ω Omega
Use in source · block16_section_final.tex, line 567
0 · 1. The individual projectors do not descend because Sp_1S^-1=p_2, while their orbit sum p_1+p_2=1 does descend. The pair-address set has two orbits, Δ={(1,1),(2,2)}, Ω={(1,2),(2,1)}. Accordingly, End_kk C_2(kk^2) = { a · b b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivariant identity matrix to p_1, which is not equivariant. The orbit selectors C_Δ and C_Ω are compatible and split every equivariant endomorphism into its d…
Exact mathematical passage
\End_{\kk C_2}(\kk^2) = \left\{ \begin{pmatrix}a&b\\b&a\end{pmatrix}:a,b\in\kk \right\} = \kk B_\Delta\oplus\kk B_\Omega.
\backslashUse in source · 11 passages
Search aliases: backslash
Use in source · block12_section.tex, line 173
Restriction and induction connect subgroup charts that need not commute. Their exact address set is the double-coset space Kbackslash G/H.
Exact mathematical passage
K\backslash G/H.
\bar aUse in source · 1 passage
Search aliases: bar a bar
Use in source · Stratified_Universality_Traditional.tex, line 5892
A residue sequence (bar a_n) admits many integral lifts. If (a_n) is one lift, then (a_n+pn) is another with the same reduction modulo p. Their Newton--Mahler coefficients differ by the nonzero coefficient sequence of pn. Thus the residue-series or functional-mask presentation cannot select one local interpolation without a declared lifting rule.
Exact mathematical passage
(\bar a_n)
\bar a_nUse in source · 1 passage
Search aliases: bar a_n bar
Use in source · Stratified_Universality_Traditional.tex, line 5892
A residue sequence (bar a_n) admits many integral lifts. If (a_n) is one lift, then (a_n+pn) is another with the same reduction modulo p. Their Newton--Mahler coefficients differ by the nonzero coefficient sequence of pn. Thus the residue-series or functional-mask presentation cannot select one local interpolation without a declared lifting rule.
Exact mathematical passage
(\bar a_n)
\bar eUse in source · 1 passage
Search aliases: bar e bar
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
T_{\bar e}=T_e^{-1}
\bar xUse in source · 2 passages
Search aliases: bar x bar
Use in source · block12_section.tex, line 483
For a connected scheme X with geometric point bar x, the fiber functor on finite acuteetale covers identifies that category with finite continuous π_1^acuteet(X,bar x)-sets. Every finite fiber S produces a permutation resolution on ℓ^2(S): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms give equivariant packet transfers. The profinite fundamental group is the automorphism group of this fiber functor.
Exact mathematical passage
\bar x
\BlockFifteenValidationAssertionsUse in source · 1 passage
Search aliases: BlockFifteenValidationAssertions
Use in source · Stratified_Universality_Traditional.tex, line 6012
ted replay assertions already counted in the preceding records and contribute BlockThirteenNewChecks new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is BlockThirteenCumulative exact assertions with no failures; the raw executed total, including replay, is BlockThirteenCumulativeExecuted. The Block 15 verifier separately executes BlockFifteenValidationAssertions exact finite regression assertions for selected lane-analysis, block-composition, re…
Exact mathematical passage
\BlockFifteenValidationAssertions
\BlockSixteenValidationAssertionsUse in source · 1 passage
Search aliases: BlockSixteenValidationAssertions
Use in source · Stratified_Universality_Traditional.tex, line 6012
kThirteenCumulative exact assertions with no failures; the raw executed total, including replay, is BlockThirteenCumulativeExecuted. The Block 15 verifier separately executes BlockFifteenValidationAssertions exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes BlockSixteenValidationAssertions exact assertions for finite group actions, object and pair orbits, orbital struct…
Exact mathematical passage
\BlockSixteenValidationAssertions
\BlockSixteenValidationFailuresUse in source · 1 passage
Search aliases: BlockSixteenValidationFailures
Use in source · Stratified_Universality_Traditional.tex, line 6012
ne-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes BlockSixteenValidationAssertions exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit projectors, and theorem-source contracts, with BlockSixteenValidationFailures recorded failures. Detailed tables and machine-readable JSON records are inc…
Exact mathematical passage
\BlockSixteenValidationFailures
\BlockThirteenBridgeChecksUse in source · 1 passage
Search aliases: BlockThirteenBridgeChecks
Use in source · Stratified_Universality_Traditional.tex, line 6012
or fixed-radix regular systems, 697,958 for integral saturation and prime-power towers, 595,964 for Euler-product and multiplicative coordinates, 229,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagged-…
Exact mathematical passage
\BlockThirteenBridgeChecks
\BlockThirteenChecksUse in source · 1 passage
Search aliases: BlockThirteenChecks
Use in source · block13_section.tex, line 540
The two suites execute BlockThirteenChecks exact assertions with no failures. Of these, BlockThirteenRegressionChecks authenticate or replay assertions already represented in the preceding cumulative record. The global synthesis therefore contributes BlockThirteenNewChecks nonduplicative comparison, coherence, packet, and loss-control assertions. The blockwise cumulative record is BlockThirteenCumulative exact assertions with no recorded failures. Counting every regression assertion executed dur…
Exact mathematical passage
\BlockThirteenChecks
\BlockThirteenCumulativeUse in source · 2 passages
Search aliases: BlockThirteenCumulative
Use in source · Stratified_Universality_Traditional.tex, line 6012
BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagged-invariant assertions. They include BlockThirteenRegressionChecks authenticated replay assertions already counted in the preceding records and contribute BlockThirteenNewChecks new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is BlockThirteenCumulative exact assertions with no failures; the raw executed total, including replay, is BlockThirteenCu…
Exact mathematical passage
\BlockThirteenCumulative
\BlockThirteenCumulativeExecutedUse in source · 2 passages
Search aliases: BlockThirteenCumulativeExecuted
Use in source · Stratified_Universality_Traditional.tex, line 6012
invariant assertions. They include BlockThirteenRegressionChecks authenticated replay assertions already counted in the preceding records and contribute BlockThirteenNewChecks new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is BlockThirteenCumulative exact assertions with no failures; the raw executed total, including replay, is BlockThirteenCumulativeExecuted. The Block 15 verifier separately executes BlockFifteenValidationAssertions exact fin…
Exact mathematical passage
\BlockThirteenCumulativeExecuted
\BlockThirteenInvariantChecksUse in source · 1 passage
Search aliases: BlockThirteenInvariantChecks
Use in source · Stratified_Universality_Traditional.tex, line 6012
saturation and prime-power towers, 595,964 for Euler-product and multiplicative coordinates, 229,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagged-invariant assertions. They include BlockThirteenRegre…
Exact mathematical passage
\BlockThirteenInvariantChecks
\BlockThirteenNewChecksUse in source · 2 passages
Search aliases: BlockThirteenNewChecks
Use in source · Stratified_Universality_Traditional.tex, line 6012
ts, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagged-invariant assertions. They include BlockThirteenRegressionChecks authenticated replay assertions already counted in the preceding records and contribute BlockThirteenNewChecks new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative r…
Exact mathematical passage
\BlockThirteenNewChecks
\BlockThirteenRegressionChecksUse in source · 2 passages
Search aliases: BlockThirteenRegressionChecks
Use in source · Stratified_Universality_Traditional.tex, line 6012
29,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagged-invariant assertions. They include BlockThirteenRegressionChecks authenticated replay assertions already counted in the preceding records and contr…
Exact mathematical passage
\BlockThirteenRegressionChecks
\BrUse in source · 2 passages
Search aliases: Br operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1725
[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is naturally a module over a central simple K-algebra split by L, rather than an ordinary K-vector space [source reference].
Exact mathematical passage
\delta(\alpha) \in H^2(G,L^\times) = \Br(L/K).
\mathbf{Branch}Use in source · 1 passage
Search aliases: Branch mathbf
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\mathbf{Branch}^{\mathrm{cts}}_{p,a}Use in source · 1 passage
Search aliases: Branch^cts_p,a mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\mathbf{BranchMah}Use in source · 1 passage
Search aliases: BranchMah mathbf
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\mathbf{BranchMah}^{c_0}_{p,a}Use in source · 1 passage
Search aliases: BranchMah^c_0_p,a mathbf
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\mathbb CUse in source · 8 passages
Search aliases: C mathbb
Use in source · block09_section.tex, line 6
Throughout this section, K is a characteristic-zero field. When local monodromy is discussed, K is embedded in C and all local exponents are taken over an algebraic closure.
Exact mathematical passage
\mathbb C
C'Use in source · 2 passages
Search aliases: C'
Use in source · Stratified_Universality_Traditional.tex, line 2092
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
S:(C,v,\lambda)\longrightarrow(C',v',\lambda')
C'_tUse in source · 2 passages
Search aliases: C'_t
Use in source · block18_carrier_observer_change_final.tex, line 342
ite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.
Exact mathematical passage
C'_t = \operatorname{conv} \bigcup_{\delta(s,b)=t}A_b^{\mathsf T}C_s.
c^{\mathcal R}_{\mathcal Q,\mathcal P}Use in source · 2 passages
Search aliases: c^ R_ Q, P mathcal mathcal mathcal
Use in source · block16_section_final.tex, line 498
Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.
Exact mathematical passage
c^{\mathcal R}_{\mathcal Q,\mathcal P} := \#\left\{ j\in J:(k,j)\in\mathcal Q, (j,i)\in\mathcal P \right\}.
C^{12}Use in source · 1 passage
Search aliases: C^12
Use in source · Stratified_Universality_Traditional.tex, line 5862
1 · 0, one has C^12 =I+829 · 18 18 · 11. Hence Δ_12·2^a^ rF_n ≡0pmod2^(a+3)r.
Exact mathematical passage
C^{12} =I+8\begin{pmatrix}29&18\\18&11\end{pmatrix}.
C^2Use in source · 2 passages
Search aliases: C^2
Use in source · block10_section.tex, line 380
Substitute C=E+zO into C=1+zC^2 and separate even and odd powers. The two resultants give the displayed quartics.
Exact mathematical passage
C=1+zC^2
c^2Use in source · 2 passages
Search aliases: c^2
Use in source · block19_admissibility_lifting_final.tex, line 191
Write Q=a · c c · b, d=ab-c^2>0. The squared induced norms are u:= A_1 _Q^2 =fracb(a+b+2c)d, v:= A_2 _Q^2 =fraca(a+b-2c)d. A direct calculation gives ab((a+b)^2-4c^2)-4(ab-c^2)^2 =ab(a-b)^2+4c^2(ab-c^2)≥0. Thus uv≥4, and therefore max{sqrt u,sqrt v}≥(uv)^1/4≥sqrt2. Equality forces a=b and c=0, and then u=v=2.
Exact mathematical passage
Q=\begin{pmatrix}a&c\\c&b\end{pmatrix}, \qquad d=ab-c^2>0.
C^HUse in source · 1 passage
Search aliases: C^H
Use in source · Stratified_Universality_Traditional.tex, line 5808
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
C^H=I+p^LB, \qquad L\geq1.
C^{Hx+j}Use in source · 1 passage
Search aliases: C^Hx+j
Use in source · Stratified_Universality_Traditional.tex, line 5833
Use C^Hx+j=C^j(I+p^LB)^x and expand the principal-unit matrix power by the convergent binomial series. One forward difference in x contributes the factor p^LB.
Exact mathematical passage
C^{Hx+j}=C^j(I+p^LB)^x
C^jUse in source · 2 passages
Search aliases: C^j
Use in source · Stratified_Universality_Traditional.tex, line 5813
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r\geq0} p^{Lr}\binom xr w^{\mathsf T}C^jB^rv_0,}
C^nUse in source · 5 passages
Search aliases: C^n
Use in source · Stratified_Universality_Traditional.tex, line 1669
The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.
Exact mathematical passage
c\in C^n
C^{p-1}Use in source · 1 passage
Search aliases: C^p-1
Use in source · Stratified_Universality_Traditional.tex, line 2961
Conversely, suppose all valuation residues vanish. Since A and B are coprime, every irreducible exponent in each polynomial is divisible by p-1. Factoring into monic irreducibles therefore gives A=C^p-1, B=D^p-1 for polynomials C,D after a constant normalization. The unit conditions A(0)=B(0)=1 remove the remaining scalar obstruction. Thus B/A=(D/C)^p-1. Multiplying D/C by its inverse constant term does not change its (p-1)-st power, and gives the unique root with constant term one. The formal s…
Exact mathematical passage
A=C^{p-1}, \qquad B=D^{p-1}
C^qUse in source · 1 passage
Search aliases: C^q
Use in source · block13_section.tex, line 281
Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value c_0=1.
Exact mathematical passage
C=1+\alpha xC^q. \label{eq:block13-fuss-compressed}
\mathfrak c^{\rm tuple}_{2,0}Use in source · 1 passage
Search aliases: c^rm tuple_2,0 mathfrak rm
Use in source · block10_section.tex, line 486
[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.
Exact mathematical passage
\mathfrak c^{\rm tuple}_{2,0}=1, \qquad \mathfrak s^{\rm Gal}_{2,0}=1.
\mathfrak c^{\rm tuple}_{2,1}Use in source · 1 passage
Search aliases: c^rm tuple_2,1 mathfrak rm
Use in source · block10_section.tex, line 498
r:block10-antisymmetric-collision If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.
Exact mathematical passage
\mathfrak c^{\rm tuple}_{2,1}=1, \qquad \mathfrak s^{\rm Gal}_{2,1}=2.
\mathbb C^\timesUse in source · 1 passage
Search aliases: C^× mathbb times
Use in source · block12_section.tex, line 463
-1 · 0 . Then A_σσ(A_σ)=-I. The projective cocycle condition holds because -I is scalar. Its boundary is represented by -1 in R^×/N_ C/ R( C^×). Every complex norm is positive, so -1 is nontrivial. The descended central simple algebra is the Hamilton quaternion algebra. Replacing A_σ by 0 · 1
Exact mathematical passage
\mathbb R^\times/N_{\mathbb C/\mathbb R}(\mathbb C^\times).
C_{(p)}Use in source · 1 passage
Search aliases: C_(p)
Use in source · Stratified_Universality_Traditional.tex, line 4472
The Z-module Z_p is torsion free and hence flat. Tensor the integral short exact sequence with Z_p. For a finite abelian group C, C⊗_ Z Z_p≅ C_(p). The Smith-coordinate formula follows by tensoring the diagonal inclusion.
Exact mathematical passage
C\otimes_{\mathbb Z}\mathbb Z_p\cong C_{(p)}.
c_1Use in source · 4 passages
Search aliases: c_1
Use in source · Stratified_Universality_Traditional.tex, line 2044
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\chi(x)=x^d-c_{d-1}x^{d-1}-\cdots-c_1x-c_0, \qquad c_0\ne0,
C_2Use in source · 3 passages
Search aliases: C_2
Use in source · block12_section.tex, line 166
[The S_3 nonnormal atlas] Let G=S_3 and let H be generated by one transposition. There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts. Their subgroup core is trivial, while their normal closure is all of S_3. By contrast, the order-three subgroup A_3 is normal and its fixed chart carries the quotient symmetry S_3/A_3≅ C_2.
Exact mathematical passage
S_3/A_3\cong C_2
c_2Use in source · 1 passage
Search aliases: c_2
Use in source · Stratified_Universality_Traditional.tex, line 5913
For f(x)=x^2, the source Mahler vector is (c_0,c_1,c_2)=(0,1,2). The two dyadic branches are f(2q)=4binom q1+8binom q2, and f(2q+1)=1+8binom q1+8binom q2. This example exhibits exact finite-window coefficient transport.
Exact mathematical passage
(c_0,c_1,c_2)=(0,1,2).
\mathcal C_{2,0}Use in source · 1 passage
Search aliases: C_2,0 mathcal
Use in source · block10_section.tex, line 478
[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.
Exact mathematical passage
|\mathcal B_2|=4, \qquad |\mathcal C_{2,0}|=3, \qquad |\Gamma\cdot E|=2.
\mathcal C_{2,1}Use in source · 1 passage
Search aliases: C_2,1 mathcal
Use in source · block10_section.tex, line 492
[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.
Exact mathematical passage
|\mathcal C_{2,1}|=3, \qquad |\Gamma\cdot O|=1,
c_{a,j,r}Use in source · 1 passage
Search aliases: c_a,j,r
Use in source · Stratified_Universality_Traditional.tex, line 5846
[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.
Exact mathematical passage
\boxed{v_p(c_{a,j,r})\geq(L+a)r.}
c_{d-1}Use in source · 4 passages
Search aliases: c_d-1
Use in source · Stratified_Universality_Traditional.tex, line 2044
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\chi(x)=x^d-c_{d-1}x^{d-1}-\cdots-c_1x-c_0, \qquad c_0\ne0,
C_E^2Use in source · 2 passages
Search aliases: C_E^2
Use in source · Stratified_Universality_Traditional.tex, line 644
[Boolean algebra of block selectors] As endomorphisms of the 𝕜-module Hom(X,Y), the selectors satisfy C_E^2=C_E, C_EC_F=C_E∩ F, R_E=I-C_E. In particular, C_ER_E=R_EC_E=0, f=C_E(f)+R_E(f).
Exact mathematical passage
C_E^2=C_E, \qquad C_EC_F=C_{E\cap F}, \qquad R_E=I-C_E.
C_E^{u,r}Use in source · 1 passage
Search aliases: C_E^u,r
Use in source · block15_section.tex, line 164
item Selectors and named residuals. For E⊆ I_Y,u× I_X,r, put C_E^u,r(f) :=∑_(a,i)∈ Ep_Y,a^ufp_X,i^r, R_E^u,r(f):=f-C_E^u,r(f). As additive endomorphisms of Hom_ C(X,Y), C_E^2=C_E, C_EC_F=C_E∩ F, R_E=1-C_E, f=C_E(f)+R_E(f). Thus deletion of a lane block is never silent. Its complement is an exact, named residual.
Exact mathematical passage
C_E^{u,r}(f) :=\sum_{(a,i)\in E}p_{Y,a}^ufp_{X,i}^r, \qquad R_E^{u,r}(f):=f-C_E^{u,r}(f).
C_{E_U}Use in source · 2 passages
Search aliases: C_E_U
Use in source · block16_section_final.tex, line 213
item Holonomy-compatible selectors and residuals. Let E_U⊆ I_Y,U,u× I_X,U,r be a family satisfying E_V= (λ^Y_γ,u×λ^X_γ,r)(E_U) for every arrow γ:U→ V. Then the local selectors C_E_U(h_U)_ai = h_U,ai, · (a,i)∈ E_U, 0, · (a,i)∉ E_U assemble to an idempotent endomorphism C_E of the section group. For two compatible support families E,F, C_EC_F=C_E∩ F, R_E:=1-C_E, h=C_Eh+R_Eh. Hence named residuals remain exact under arbitrary finite address holonomy, provided the retained support itself is holonomy…
Exact mathematical passage
C_{E_U}(h_U)_{ai} = \begin{cases} h_{U,ai},&(a,i)\in E_U,\\ 0,&(a,i)\notin E_U \end{cases}
C_{E_V}Use in source · 1 passage
Search aliases: C_E_V
Use in source · block16_section_final.tex, line 296
For (H5), support compatibility implies γ_*C_E_U=C_E_Vγ_*, so the selectors preserve compatible sections. Every Boolean and residual identity is then checked chartwise.
Exact mathematical passage
\gamma_*C_{E_U}=C_{E_V}\gamma_*
C_{E\cap F}Use in source · 3 passages
Search aliases: C_E∩ F cap
Use in source · Stratified_Universality_Traditional.tex, line 644
[Boolean algebra of block selectors] As endomorphisms of the 𝕜-module Hom(X,Y), the selectors satisfy C_E^2=C_E, C_EC_F=C_E∩ F, R_E=I-C_E. In particular, C_ER_E=R_EC_E=0, f=C_E(f)+R_E(f).
Exact mathematical passage
C_E^2=C_E, \qquad C_EC_F=C_{E\cap F}, \qquad R_E=I-C_E.
C_FUse in source · 7 passages
Search aliases: C_F
Use in source · Stratified_Universality_Traditional.tex, line 644
[Boolean algebra of block selectors] As endomorphisms of the 𝕜-module Hom(X,Y), the selectors satisfy C_E^2=C_E, C_EC_F=C_E∩ F, R_E=I-C_E. In particular, C_ER_E=R_EC_E=0, f=C_E(f)+R_E(f).
Exact mathematical passage
C_E^2=C_E, \qquad C_EC_F=C_{E\cap F}, \qquad R_E=I-C_E.
C_{F\circ E}Use in source · 2 passages
Search aliases: C_F∘ E circ
Use in source · Stratified_Universality_Traditional.tex, line 677
[Support propagation under composition] If f=C_E(f) and g=C_F(g), then gf=C_F∘ E(gf). Thus exact zero patterns propagate by relational composition.
Exact mathematical passage
gf=C_{F\circ E}(gf).
c_jUse in source · 2 passages
Search aliases: c_j
Use in source · Stratified_Universality_Traditional.tex, line 4917
[Exact coefficient recurrence] Let A(z)=1+∑_n≥1a_nz^n and let c_n be its Euler ghost coordinates. Then na_n = ∑_j=1^nc_j a_n-j (n≥1), with a_0=1. Conversely, any c∈ G_ Z and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.
Exact mathematical passage
\boxed{ na_n = \sum_{j=1}^{n}c_j a_{n-j} \qquad(n\geq1), }
c_{j,r}Use in source · 1 passage
Search aliases: c_j,r
Use in source · Stratified_Universality_Traditional.tex, line 5822
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
\boxed{v_p(c_{j,r})\geq Lr,}
C_NUse in source · 8 passages
Search aliases: C_N
Use in source · block16_section_final.tex, line 590
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(
Exact mathematical passage
C_N=\langle g\rangle
c_NUse in source · 2 passages
Search aliases: c_N
Use in source · Stratified_Universality_Traditional.tex, line 4598
[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
Exact mathematical passage
c_N=\sum_{j=0}^{N}p^{2j}
c_{n+1}Use in source · 1 passage
Search aliases: c_n+1
Use in source · block13_section.tex, line 305
ssed Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value c_0=1.
Exact mathematical passage
\boxed{ A_q(n)c_{n+1} = \alpha B_q(n)c_n.} \label{eq:block13-fuss-recurrence}
c_{n+r}Use in source · 2 passages
Search aliases: c_n+r
Use in source · block09_section.tex, line 216
Write F(z)=∑ c_nz^n. At coefficient degree n, the monic term D^rF contributes frac(n+r)!n!c_n+r, while every lower derivative involves only c_0,…,c_n+r-1. Characteristic zero makes the displayed coefficient invertible. The equation therefore determines c_n+r recursively from the jet.
Exact mathematical passage
\frac{(n+r)!}{n!}c_{n+r},
c_{n+r-1}Use in source · 1 passage
Search aliases: c_n+r-1
Use in source · block09_section.tex, line 219
Write F(z)=∑ c_nz^n. At coefficient degree n, the monic term D^rF contributes frac(n+r)!n!c_n+r, while every lower derivative involves only c_0,…,c_n+r-1. Characteristic zero makes the displayed coefficient invertible. The equation therefore determines c_n+r recursively from the jet.
Exact mathematical passage
c_0,\ldots,c_{n+r-1}
C_{N,j}Use in source · 7 passages
Search aliases: C_N,j
Use in source · block09_section.tex, line 609
Define the alias and cancellation deficits by A_N(λ)=N-r_rm orb(N;λ), C_N,j(λ) = r_rm orb(N;λ) -r_D(N,j;λ). Then N-r_D(N,j;λ) = A_N(λ)+C_N,j(λ).
Exact mathematical passage
\boxed{ C_{N,j}(\lambda) = r_{\rm orb}(N;\lambda) -r_D(N,j;\lambda).}
C_{n,M}Use in source · 1 passage
Search aliases: C_n,M
Use in source · block11_section.tex, line 235
[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.
Exact mathematical passage
\boxed{ C_{n,M} =[x^n]\prod_{r\ge1}(1-x^r)^{-M}.} \tag{11.6}
C_oUse in source · 7 passages
Search aliases: C_o
Use in source · Stratified_Universality_Traditional.tex, line 1479
a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
C_o = \bigcap_{e\notin S}\ker(H_e-I) = \Fix\langle H_e:e\notin S\rangle, \qquad C_v=G_vC_o.
\mathcal C_pUse in source · 4 passages
Search aliases: C_p mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5416
This section treats the second channel. Fix a prime p. Let C_p=C( Z_p, Z_p) with the uniform norm, and let c_0( Z_p) = {c=(c_r)_r≥0:c_r∈ Z_p, c_r⟶0} with the coefficient sup norm.
Exact mathematical passage
\mathcal C_p=C(\mathbb Z_p,\mathbb Z_p)
c_{p^ak}Use in source · 1 passage
Search aliases: c_p^ak
Use in source · Stratified_Universality_Traditional.tex, line 5781
[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{d_{0,k}\equiv c_{p^ak}\pmod p.}
c_{p^ak+u}Use in source · 1 passage
Search aliases: c_p^ak+u
Use in source · Stratified_Universality_Traditional.tex, line 5773
[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{ d_{s,k} \equiv \sum_{u=0}^{s}\binom su c_{p^ak+u} \pmod p.}
c_qUse in source · 3 passages
Search aliases: c_q
Use in source · block16_section_final.tex, line 623
, where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
\boxed{ E_q = \frac1N\sum_{\ell=0}^{N-1}c_q(\ell)g^\ell,}
C_{q,\alpha}Use in source · 2 passages
Search aliases: C_q,α alpha
Use in source · block13_section.tex, line 275
Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value
Exact mathematical passage
x=z^{q-1}, \qquad U_{q,\alpha}(z)=zC_{q,\alpha}(x).
C_sUse in source · 3 passages
Search aliases: C_s
Use in source · block18_carrier_observer_change_final.tex, line 333
Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.
Exact mathematical passage
C_s\subseteq\mathbb R^d
C_tUse in source · 11 passages
Search aliases: C_t
Use in source · block15_section.tex, line 215
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
C_t
C_vUse in source · 3 passages
Search aliases: C_v
Use in source · Stratified_Universality_Traditional.tex, line 1479
a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
C_o = \bigcap_{e\notin S}\ker(H_e-I) = \Fix\langle H_e:e\notin S\rangle, \qquad C_v=G_vC_o.
\mathcal C_VUse in source · 1 passage
Search aliases: C_V mathcal
Use in source · block16_section_final.tex, line 25
Let mathscr G be a finite connected groupoid of charts. For an arrow γ:U→ V, let F_γ: C_U⟶ C_V be an additive equivalence between additive idempotent-complete categories. The transition functors carry specified identity and composition comparisons. To keep the formulas readable, those comparison isomorphisms are suppressed; every displayed cocycle equation is understood after inserting the declared comparisons.
Exact mathematical passage
F_\gamma:\mathcal C_U\longrightarrow\mathcal C_V
C_\DeltaUse in source · 1 passage
Search aliases: C_Δ Delta
Use in source · block16_section_final.tex, line 578
b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivariant identity matrix to p_1, which is not equivariant. The orbit selectors C_Δ and C_Ω are compatible and split every equivariant endomorphism into its diagonal-orbit and off-diagonal-orbit parts.
Exact mathematical passage
C_\Delta
C_{\kappa(e)}Use in source · 2 passages
Search aliases: C_κ(e) kappa
Use in source · block15_section.tex, line 216
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
\tau_e:B_e\to C_{\kappa(e)}
\mathcal C_{\lambda,a}Use in source · 5 passages
Search aliases: C_λ,a mathcal lambda
Use in source · block13_section.tex, line 85
Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.
Exact mathematical passage
\mathcal C_{\lambda,a}
C_\OmegaUse in source · 1 passage
Search aliases: C_Ω Omega
Use in source · block16_section_final.tex, line 578
b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivariant identity matrix to p_1, which is not equivariant. The orbit selectors C_Δ and C_Ω are compatible and split every equivariant endomorphism into its diagonal-orbit and off-diagonal-orbit parts.
Exact mathematical passage
C_\Omega
c_\inftyUse in source · 2 passages
Search aliases: c_∞ infty
Use in source · Stratified_Universality_Traditional.tex, line 4602
For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
Exact mathematical passage
c_\infty = \frac{1}{1-p^2},
\mathrm{can}Use in source · 2 passages
Search aliases: can mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3782
[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence. Then Reg_k(K) ≃ Sec_k^can(K) ≃ Lin_k,K^sl. Every normalized fiber over a fixed sequence is contractible. Consequently the associated finite presenta…
Exact mathematical passage
\mathbf{Sec}_k^{\mathrm{can}}(K)
\mathbf{Cart}Use in source · 2 passages
Search aliases: Cart mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3212
Minimal Cartier-module chart. An object of Cart_p^min,rat is a finite-dimensional tuple (V,(L_d)_d=0^p-1,v,λ) that is reachable and observable, whose digit coefficient function belongs to RMah_p^×. Morphisms are similarities preserving every L_d, the initial vector, and the output functional. Its normalized representative is C_p(F).
Exact mathematical passage
\mathbf{Cart}_p^{\mathrm{min,rat}}
\mathbf{Cart}_p^{\mathrm{min,rat}}Use in source · 2 passages
Search aliases: Cart_p^min,rat mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3212
Minimal Cartier-module chart. An object of Cart_p^min,rat is a finite-dimensional tuple (V,(L_d)_d=0^p-1,v,λ) that is reachable and observable, whose digit coefficient function belongs to RMah_p^×. Morphisms are similarities preserving every L_d, the initial vector, and the output functional. Its normalized representative is C_p(F).
Exact mathematical passage
\mathbf{Cart}_p^{\mathrm{min,rat}}
\operatorname{codim}Use in source · 2 passages
Search aliases: codim operatorname
Use in source · block12_section.tex, line 274
The classical conductor formula uses codimV^G_i. Since P_i projects onto V^G_i, this codimension is Tr(I-P_i). Also codimV^G_i=∑_j≥ id_j. Reversing the finite sums gives the detail formula.
Exact mathematical passage
\operatorname{codim}V^{G_i}
\mathrm{col}Use in source · 2 passages
Search aliases: col mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3328
The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside RMah_p,col^×.
Exact mathematical passage
\RMah_p^{\times} = \RMah_{p,\mathrm{col}}^{\times} \sqcup \RMah_{p,\mathrm{ncol}}^{\times}.
\operatorname{Coll}_{CB}Use in source · 1 passage
Search aliases: Coll_CB operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1288
position law for bridge loss] Let UxrightarrowB VxrightarrowC W be certified bridges. [label=textup(roman*)] item Every constructor or universal property preserved by both B and C is preserved by CB. item MorKer_B(X,Y) ⊆ MorKer_CB(X,Y). item Coll_B ⊆ Coll_CB. item EssIm(CB) ⊆ EssIm(C). item If both bridges are conservative, then their composite is conservative. If both are classification-faithful, then their composite is classification-faithful. If both are equivalences, then their composite is …
Exact mathematical passage
\operatorname{Coll}_B \subseteq \operatorname{Coll}_{CB}.
\operatorname{conv}Use in source · 2 passages
Search aliases: conv operatorname
Use in source · block18_carrier_observer_change_final.tex, line 335
Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.
Exact mathematical passage
h_{\varnothing}(x)=-\infty,\qquad \operatorname{conv}(\varnothing)=\varnothing,\qquad \max\varnothing=-\infty.
\operatorname{Core}Use in source · 4 passages
Search aliases: Core operatorname
Use in source · block12_section.tex, line 136
If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.
Exact mathematical passage
\operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1}, \qquad \operatorname{Ncl}_G(H)=\left\langle gHg^{-1}:g\in G\right\rangle.
\operatorname{Core}_GUse in source · 4 passages
Search aliases: Core_G operatorname
Use in source · block12_section.tex, line 136
If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.
Exact mathematical passage
\operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1}, \qquad \operatorname{Ncl}_G(H)=\left\langle gHg^{-1}:g\in G\right\rangle.
\coshUse in source · 1 passage
Search aliases: cosh
Use in source · block09_section.tex, line 811
For 2cosh z, the even packet is nonzero and the odd packet is zero. There is no finite singularity, so the cancellation occurs at the level of the character-projected solution rather than by finite singularity collision.
Exact mathematical passage
2\cosh z,
\crUse in source · 1 passage
Search aliases: cr
Use in source · Stratified_Universality_Traditional.tex, line 257
The proposed architecture has six layers: resolved syntax⟶normalized strata ⟶typed overlaps⟶coherencecr ⟶effective descent ⟶compatible complete invariants. Each arrow carries a proof obligation. A semantic analogy is not a bridge. A bridge is not reversible merely because formulas exist in both directions. Local completeness is not global completeness until the local isomorphisms themselves satisfy descent.
Exact mathematical passage
\boxed{ \begin{gathered} \text{resolved syntax}\longrightarrow\text{normalized strata} \longrightarrow\text{typed overlaps}\longrightarrow\text{coherence}\cr \longrightarrow\text{effective descent} \longrightarrow\text{compatible complete invariants}. \end{gathered}}
\crefUse in source · 2 passages
Search aliases: cref
Use in source · Stratified_Universality_Traditional.tex, line 2538
[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart [4pt] Reduced OGF ↔ rational mask · equivalence on the unit-periodic essential image · F(0)=1, periodic rational F, reduced regular mask R · exact se…
Exact mathematical passage
4pt] Reduced OGF \(\leftrightarrow\) minimal DFAO & equivalence on the periodic essential image & unit purely periodic series and reachable minimal LSD-first \(p\)-DFAOs & coefficient semantics, minimal period, canonical \(p\)-kernel, output behavior & equality of represented sequences & a nonminimal residue-pair automaton is quotiented before entering the normalized chart\\[4pt] Reduced OGF \(\leftrightarrow\) rational mask & equivalence on the unit-periodic essential image & \(F(0)=1\), periodic rational \(F\), reduced regular mask \(R\) & exact series, functional equation, reduced rational mask, minimal period & equality of series and masks & undefined for nonunit series; unrestricted rational masks may have nonperiodic products\\[4pt] Mask \(\to\) Cartier numerator representation & constructor & reduced \(A/B\) over \(\mathbb F_p\) & exact coefficient semantics, digit matrices, theorem degree bound & not generally reflective & introduces basis and denominator coordinates and may contain unreachable ambient states\\[4pt] Ambient Cartier compiler \(\to\) minimal DFAO & reachable restriction & full finite numerator-state compiler & every reachable coefficient section and every future digit behavior & equality of the represented \(p\)-kernel states & removes only ambient states outside the orbit of the initial numerator\\[4pt] Unrestricted automatic \(\dashrightarrow\) rational mask & partial bridge only & automatic sequences whose unit inverse mask is rational and whose product lies in the declared periodic class & coefficient semantics on the domain of definition & no unrestricted reflection theorem & no claim that every automatic sequence has a rational periodic mask\\ \end{longtable} \end{landscape} \subsubsection{The rational-to-digital construction} For a periodic word \(w\) of length \(T\), construct the finite pair automaton from \cref{eq:pair-transition}, then quotient by equality of the induced subsequences. This gives the canonical minimal DFAO directly. Conversely, a minimal DFAO in the selected chart is promised to produce a purely periodic sequence. Its minimal period can be recovered exactly by enumerating \(N=1,2,\ldots\), constructing the finite-state transducer for addition by \(N\), and testing equivalence between the original output and its \(N\)-shift. Equivalence of finite deterministic output automata is a finite product-graph calculation. The promise of pure periodicity guarantees termination. Once \(T\) is known, \cref{eq:periodic-ogf-unreduced} gives the rational series, which is then reduced. This inverse is effective but not claimed to be complexity optimal. The point is effectivity, not a sharp automata algorithm. \subsubsection{The rational-mask compiler lane} Let \[ R(z)=\frac{A(z)}{B(z)}, \qquad A(0)=B(0)=1,
\mathrm{cts}Use in source · 2 passages
Search aliases: cts mathrm
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
D^2Use in source · 1 passage
Search aliases: D^2
Use in source · block09_section.tex, line 74
[Derivative-module rank] For a nonzero D-finite series F, r_D(F) = dim_K(z)span_K(z) {F,DF,D^2F,…}.
Exact mathematical passage
\boxed{ r_D(F) = \dim_{K(z)}\operatorname{span}_{K(z)} \{F,DF,D^2F,\ldots\}.}
D^{k(\mu)}Use in source · 1 passage
Search aliases: D^k(μ) mu
Use in source · block11_section.tex, line 206
[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.
Exact mathematical passage
\boxed{ \begin{aligned} &P_\delta D^n(F\circ G)(x)[h,\ldots,h]\\ &=\sum_{\mu} \frac{n!} {\displaystyle\prod_{r,\gamma} (r!)^{\mu_{r,\gamma}}\mu_{r,\gamma}!} \,P_\delta D^{k(\mu)}F(Gx) \bigl[ (g_{r,\gamma})^{\mu_{r,\gamma}} \bigr]_{r,\gamma}. \end{aligned}} \tag{11.5}
D^nUse in source · 1 passage
Search aliases: D^n
Use in source · block11_section.tex, line 206
[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.
Exact mathematical passage
\boxed{ \begin{aligned} &P_\delta D^n(F\circ G)(x)[h,\ldots,h]\\ &=\sum_{\mu} \frac{n!} {\displaystyle\prod_{r,\gamma} (r!)^{\mu_{r,\gamma}}\mu_{r,\gamma}!} \,P_\delta D^{k(\mu)}F(Gx) \bigl[ (g_{r,\gamma})^{\mu_{r,\gamma}} \bigr]_{r,\gamma}. \end{aligned}} \tag{11.5}
D^{N+q}Use in source · 1 passage
Search aliases: D^N+q
Use in source · block09_section.tex, line 161
[Recurrence residual as an Euler-operator polynomial] Define Q = ∑_s=0^q z^q-sp_s(θ-s). Then [z^n+q]QF = ∑_s=0^qp_s(n)a_n+s. Hence QF is a polynomial of degree less than N+q, and D^N+qQF=0.
Exact mathematical passage
\boxed{D^{N+q}QF=0.}
d^{N/g}Use in source · 1 passage
Search aliases: d^N/g
Use in source · block15_section.tex, line 1014
If, in addition, their distinct Kummer classes generate a subgroup isomorphic to ( Z/d Z)^N/g in the Kummer quotient K(z)^×/K(z)^× d, then the compositum of the corresponding radical branch fields has degree d^N/g over K(z). Without that independence hypothesis, the lane orbit count remains exact but the degree formula is only an upper bound.
Exact mathematical passage
d^{N/g}
D^{p-1}Use in source · 1 passage
Search aliases: D^p-1
Use in source · Stratified_Universality_Traditional.tex, line 2961
Conversely, suppose all valuation residues vanish. Since A and B are coprime, every irreducible exponent in each polynomial is divisible by p-1. Factoring into monic irreducibles therefore gives A=C^p-1, B=D^p-1 for polynomials C,D after a constant normalization. The unit conditions A(0)=B(0)=1 remove the remaining scalar obstruction. Thus B/A=(D/C)^p-1. Multiplying D/C by its inverse constant term does not change its (p-1)-st power, and gives the unique root with constant term one. The formal s…
Exact mathematical passage
A=C^{p-1}, \qquad B=D^{p-1}
d^{r+1}Use in source · 1 passage
Search aliases: d^r+1
Use in source · Stratified_Universality_Traditional.tex, line 5329
[Power bridge] For b(n)=n^r, which is completely multiplicative, the divisor ghost is c(n)=∑_dmid nd^r+1=σ_r+1(n). At a prime power, σ_r+1(p^e) = 1+p^r+1+⋯+p^e(r+1). Thus one simple multiplicative exponent rule simultaneously produces a filtered ordinary product, a multiplicative divisor-sum ghost, and a one-dimensional geometric local rule at every prime.
Exact mathematical passage
c(n)=\sum_{d\mid n}d^{r+1}=\sigma_{r+1}(n).
d_{\mathcal A}Use in source · 11 passages
Search aliases: d_ A mathcal
Use in source · block11_section.tex, line 332
Define the arity period d_ A = gcd{m-1:m∈ A}. tag11.9 Every tree with internal vertices v has leaf count ℓ(T)=1+∑_v(deg(v)-1).
Exact mathematical passage
\boxed{ d_{\mathcal A} = \gcd\{m-1:m\in\mathcal A\}.} \tag{11.9}
D_0Use in source · 4 passages
Search aliases: D_0
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
d_{0,k}Use in source · 1 passage
Search aliases: d_0,k
Use in source · Stratified_Universality_Traditional.tex, line 5781
[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{d_{0,k}\equiv c_{p^ak}\pmod p.}
d_2Use in source · 2 passages
Search aliases: d_2
Use in source · Stratified_Universality_Traditional.tex, line 4110
By the Smith normal form, there are uniquely determined integers 1≤ d_1(a)mid d_2(a)mid⋯mid d_ρ(a) such that C_k(a) ≅ ⊕_i=1^ρ Z/d_i(a) Z. Factors with d_i=1 are harmless but are retained so that the list has length ρ.
Exact mathematical passage
1\leq d_1(a)\mid d_2(a)\mid\cdots\mid d_{\rho}(a)
D_4Use in source · 1 passage
Search aliases: D_4
Use in source · block12_section.tex, line 589
The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra. It audits cyclic groups, V_4, S_3, D_4, and A_4; every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commuting and noncommuting subgroup pairs; double-coset partitions and the Mackey dimension law; cyclic ramification-style filtrations; explicit Hilbert 90 co…
Exact mathematical passage
D_4
D_aUse in source · 1 passage
Search aliases: D_a
Use in source · Stratified_Universality_Traditional.tex, line 1592
e finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summands and D is block diagonal, this becomes coker D≅⊕_acoker D_a. Off-diagonal correction blocks can couple packet defects, so the block-diagonal hypothesis cannot be omitted.
Exact mathematical passage
\coker D\cong\bigoplus_a\coker D_a.
d_{a-1}Use in source · 3 passages
Search aliases: d_a-1
Use in source · block15_section.tex, line 834
elprop:block15-cartier-specialization Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
w=d_0d_1\cdots d_{a-1}
D_cUse in source · 4 passages
Search aliases: D_c
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
D_fUse in source · 4 passages
Search aliases: D_f
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
D_iUse in source · 3 passages
Search aliases: D_i
Use in source · block12_section.tex, line 232
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
D_i=P_{i+1}-P_i.
d_iUse in source · 11 passages
Search aliases: d_i
Use in source · Stratified_Universality_Traditional.tex, line 2326
For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.
Exact mathematical passage
\label{eq:p-kernel} \mathcal K_p(a) = \left\{ \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a: \ell\geq 0,\ 0\leq d_i<p \right\}.
D_jUse in source · 1 passage
Search aliases: D_j
Use in source · block12_section.tex, line 264
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
d_j=\operatorname{rank}D_j
d_jUse in source · 7 passages
Search aliases: d_j
Use in source · Stratified_Universality_Traditional.tex, line 2346
The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.
Exact mathematical passage
\Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a = \bigl(n\longmapsto a_{p^\ell n+r}\bigr), \qquad r=\sum_{j=0}^{\ell-1}d_jp^j.
d_kUse in source · 3 passages
Search aliases: d_k
Use in source · block08_examples_table.tex, line 12
(1+2)^x, branch 2q · c_r=2^r, 0≤ r≤4 · d_k=8^k, 0≤ k≤4
Exact mathematical passage
d_k=8^k,\ 0\leq k\leq4
D_QUse in source · 4 passages
Search aliases: D_Q
Use in source · block19_admissibility_lifting_final.tex, line 79
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
D_Q(T) := \frac{\sigma_{\max}(B_Q(T))}{\sigma_{\min}(B_Q(T))} \geq1, \qquad \Delta_Q(T):=\log D_Q(T).
D_{Q_s}Use in source · 1 passage
Search aliases: D_Q_s
Use in source · block19_admissibility_lifting_final.tex, line 232
0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.
Exact mathematical passage
D_{Q_s}(H)\longrightarrow1 \qquad(s\downarrow0),
d_{s,k}Use in source · 5 passages
Search aliases: d_s,k
Use in source · Stratified_Universality_Traditional.tex, line 5560
[Affine Newton--Mahler transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion and put g_s(q)=f(Mq+s). Then g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k =∑_r=k^Mk+sc_rΘ_M,s(r,k). Every output coefficient depends on a finite source window.
Exact mathematical passage
g_s(q)=\sum_{k\geq0}d_{s,k}\binom qk,
D_tUse in source · 1 passage
Search aliases: D_t
Use in source · block09_section.tex, line 314
The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r…
Exact mathematical passage
L_q\in E(t)\langle D_t\rangle
D_xUse in source · 1 passage
Search aliases: D_x
Use in source · block09_section.tex, line 507
_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\theta=xD_x
d_{\rho}Use in source · 1 passage
Search aliases: d_ρ rho
Use in source · Stratified_Universality_Traditional.tex, line 4110
By the Smith normal form, there are uniquely determined integers 1≤ d_1(a)mid d_2(a)mid⋯mid d_ρ(a) such that C_k(a) ≅ ⊕_i=1^ρ Z/d_i(a) Z. Factors with d_i=1 are harmless but are retained so that the list has length ρ.
Exact mathematical passage
1\leq d_1(a)\mid d_2(a)\mid\cdots\mid d_{\rho}(a)
d_{\rho-j}Use in source · 1 passage
Search aliases: d_ρ-j rho
Use in source · Stratified_Universality_Traditional.tex, line 4287
[Fitting-ideal certificate] For 0≤ j<ρ, Fitt_j(C_k(a)) = ( d_1(a)d_2(a)⋯ d_ρ-j(a) ), and Fitt_j(C_k(a))= Z (j≥ρ). Thus the Smith profile and the Fitting filtration contain the same reduction-defect data at different resolutions.
Exact mathematical passage
\operatorname{Fitt}_j(C_k(a)) = \left( d_1(a)d_2(a)\cdots d_{\rho-j}(a) \right),
\dashrightarrowUse in source · 4 passages
Search aliases: dashrightarrow
Use in source · Stratified_Universality_Traditional.tex, line 2538
[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart [4pt] Reduced OGF ↔ rational mask · equivalence on the unit-periodic essential image · F(0)=1, periodic rational F, reduced regular mask R · exact se…
Exact mathematical passage
4pt] Reduced OGF \(\leftrightarrow\) minimal DFAO & equivalence on the periodic essential image & unit purely periodic series and reachable minimal LSD-first \(p\)-DFAOs & coefficient semantics, minimal period, canonical \(p\)-kernel, output behavior & equality of represented sequences & a nonminimal residue-pair automaton is quotiented before entering the normalized chart\\[4pt] Reduced OGF \(\leftrightarrow\) rational mask & equivalence on the unit-periodic essential image & \(F(0)=1\), periodic rational \(F\), reduced regular mask \(R\) & exact series, functional equation, reduced rational mask, minimal period & equality of series and masks & undefined for nonunit series; unrestricted rational masks may have nonperiodic products\\[4pt] Mask \(\to\) Cartier numerator representation & constructor & reduced \(A/B\) over \(\mathbb F_p\) & exact coefficient semantics, digit matrices, theorem degree bound & not generally reflective & introduces basis and denominator coordinates and may contain unreachable ambient states\\[4pt] Ambient Cartier compiler \(\to\) minimal DFAO & reachable restriction & full finite numerator-state compiler & every reachable coefficient section and every future digit behavior & equality of the represented \(p\)-kernel states & removes only ambient states outside the orbit of the initial numerator\\[4pt] Unrestricted automatic \(\dashrightarrow\) rational mask & partial bridge only & automatic sequences whose unit inverse mask is rational and whose product lies in the declared periodic class & coefficient semantics on the domain of definition & no unrestricted reflection theorem & no claim that every automatic sequence has a rational periodic mask\\ \end{longtable} \end{landscape} \subsubsection{The rational-to-digital construction} For a periodic word \(w\) of length \(T\), construct the finite pair automaton from \cref{eq:pair-transition}, then quotient by equality of the induced subsequences. This gives the canonical minimal DFAO directly. Conversely, a minimal DFAO in the selected chart is promised to produce a purely periodic sequence. Its minimal period can be recovered exactly by enumerating \(N=1,2,\ldots\), constructing the finite-state transducer for addition by \(N\), and testing equivalence between the original output and its \(N\)-shift. Equivalence of finite deterministic output automata is a finite product-graph calculation. The promise of pure periodicity guarantees termination. Once \(T\) is known, \cref{eq:periodic-ogf-unreduced} gives the rational series, which is then reduced. This inverse is effective but not claimed to be complexity optimal. The point is effectivity, not a sharp automata algorithm. \subsubsection{The rational-mask compiler lane} Let \[ R(z)=\frac{A(z)}{B(z)}, \qquad A(0)=B(0)=1,
\DefUse in source · 4 passages
Search aliases: Def
Use in source · block17_semantic_quotients_final.tex, line 204
Fix a universe mathscr Q(X) of admissible queries on X. For Q⊆mathscr Q(X), define Ind( Q) := ⋂_q∈ Qkerpair(q). For an equivalence relation or congruence R, define Def(R) := {q∈mathscr Q(X):R⊆kerpair(q)}.
Exact mathematical passage
\Def(R) := \{q\in\mathscr Q(X):R\subseteq\kerpair(q)\}.
\degUse in source · 17 passages
Search aliases: deg
Use in source · Stratified_Universality_Traditional.tex, line 2057
An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.
Exact mathematical passage
A(z)=\frac{P(z)}{Q(z)}, \qquad Q(0)=1, \qquad \gcd(P,Q)=1, \qquad \deg P<\deg Q,
\deg_TUse in source · 1 passage
Search aliases: deg_T deg
Use in source · block10_section.tex, line 252
[Constructive packet eliminant] For each j, one can compute an annihilating polynomial Q_N,j(x,T)∈ K(x)[T] with deg_TQ_N,j≤ d^N by multiplication matrices, resultants, or a Gr obner elimination of the exact packet system.
Exact mathematical passage
\deg_TQ_{N,j}\leq d^N
\deg_YUse in source · 2 passages
Search aliases: deg_Y deg
Use in source · block10_section.tex, line 18
Let F∈ K[[z]] be algebraic over K(z). Its monic irreducible polynomial is P_F(z,Y)∈ K(z)[Y], P_F(z,F(z))=0. Write d(F)=deg_Y P_F. After multiplying by a common denominator and dividing by the greatest common divisor of the polynomial coefficients, one obtains a primitive representative in K[z,Y], unique up to a nonzero scalar. We normalize that scalar by fixing the first nonzero coefficient in graded lexicographic order to be one.
Exact mathematical passage
d(F)=\deg_Y P_F.
\detUse in source · 7 passages
Search aliases: det
Use in source · Stratified_Universality_Traditional.tex, line 5660
[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac…
Exact mathematical passage
\boxed{\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}
\mathbf{DFAO}Use in source · 1 passage
Search aliases: DFAO mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3856
k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
\mathbf{Aut}_k(K) \simeq \mathbf{FinOrbSec}_k(K) \simeq \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}} \simeq \mathbf{DFAO}_{k,K}^{\min}. \label{eq:block05-automatic-finite-orbit-atlas}
\mathbf{DFAO}_{k,K}^{\min}Use in source · 1 passage
Search aliases: DFAO_k,K^min mathbf min
Use in source · Stratified_Universality_Traditional.tex, line 3856
k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
\mathbf{Aut}_k(K) \simeq \mathbf{FinOrbSec}_k(K) \simeq \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}} \simeq \mathbf{DFAO}_{k,K}^{\min}. \label{eq:block05-automatic-finite-orbit-atlas}
\mathrm{DG}Use in source · 3 passages
Search aliases: DG mathrm
Use in source · Stratified_Universality_Traditional.tex, line 5284
[Effective descent on the divisor--ghost overlap] The exponent, fixed Euler-product, multiplicative ghost, and prime-power seed charts form a finite normalized atlas R_DG. Every bridge is reversible on the declared overlap, every multiple-overlap comparison is the unique identity over the represented exponent function, and Desc( R_DG) ≃ EulMult_ Z. Descent is effective for objects and morphisms.
Exact mathematical passage
\mathfrak R_{\mathrm{DG}}.
\operatorname{diag}Use in source · 3 passages
Search aliases: diag operatorname
Use in source · Stratified_Universality_Traditional.tex, line 4278
Choose Smith bases in which the inclusion M_k(a)↪ L_k(a) is the diagonal matrix D=diag(d_1,…,d_ρ). Modulo m, the kernel and cokernel of multiplication by d_i are both cyclic of order gcd(m,d_i), and the image has order m/gcd(m,d_i). The map is invertible in every coordinate exactly when every d_i is a unit modulo m, which is equivalent to gcd(m,d_ρ)=1.
Exact mathematical passage
D=\operatorname{diag}(d_1,\ldots,d_\rho).
\operatorname{dig}Use in source · 3 passages
Search aliases: dig operatorname
Use in source · Stratified_Universality_Traditional.tex, line 3598
Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs to B(W), so dim_K V_k(a) ≤ dim_K W. Thus a is regular. The realization in [source reference] has dimension exactly r_k(a).
Exact mathematical passage
\mathcal B(x)(n) = \lambda A_{\operatorname{dig}_k(n)}x,
\operatorname{dig}_kUse in source · 3 passages
Search aliases: dig_k operatorname
Use in source · Stratified_Universality_Traditional.tex, line 3598
Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs to B(W), so dim_K V_k(a) ≤ dim_K W. Thus a is regular. The realization in [source reference] has dimension exactly r_k(a).
Exact mathematical passage
\mathcal B(x)(n) = \lambda A_{\operatorname{dig}_k(n)}x,
\dim_{\mathbb Q}Use in source · 1 passage
Search aliases: dim_ Q dim mathbb
Use in source · Stratified_Universality_Traditional.tex, line 4033
Throughout this section, k≥2 is fixed and a: N⟶ Z is an integer-valued sequence whose rational section space V_ Q(a) = span_ Q K_k(a) ⊆ Q^ N has finite dimension ρ_k(a)=dim_ QV_ Q(a). Thus a is k-regular after extension of scalars to Q.
Exact mathematical passage
\rho_k(a)=\dim_{\mathbb Q}V_{\mathbb Q}(a).
\dim_{K(z)}Use in source · 1 passage
Search aliases: dim_K(z) dim
Use in source · block09_section.tex, line 74
[Derivative-module rank] For a nonzero D-finite series F, r_D(F) = dim_K(z)span_K(z) {F,DF,D^2F,…}.
Exact mathematical passage
\boxed{ r_D(F) = \dim_{K(z)}\operatorname{span}_{K(z)} \{F,DF,D^2F,\ldots\}.}
\downarrowUse in source · 3 passages
Search aliases: downarrow
Use in source · Stratified_Universality_Traditional.tex, line 1554
rem[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V downarrow a · · downarrow b U' · xrightarrowD' · V', bD=D'a, there is a unique induced map overline b:coker D⟶coker D' with overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \downarrow a & & \downarrow b\\ U' & \xrightarrow{D'} & V', \end{array} \qquad bD=D'a,
e'Use in source · 8 passages
Search aliases: e'
Use in source · Stratified_Universality_Traditional.tex, line 1669
The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.
Exact mathematical passage
e'\in E^{n-1}
e^{-2\pi i\lambda}Use in source · 1 passage
Search aliases: e^-2π iλ pi lambda
Use in source · block09_section.tex, line 719
item the monodromy eigenvalues at zero form the complete group of Nth roots of unity; item the ratios of monodromy eigenvalues at infinity are Nth roots of unity; item the distinguished nontrivial eigenvalue at α^-N is e^-2π iλ.
Exact mathematical passage
e^{-2\pi i\lambda}.
E^0Use in source · 1 passage
Search aliases: E^0
Use in source · Stratified_Universality_Traditional.tex, line 1675
The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.
Exact mathematical passage
E^0/\im D
E^2Use in source · 9 passages
Search aliases: E^2
Use in source · Stratified_Universality_Traditional.tex, line 1447
[Split bridge rectification] Let A be an additive idempotent-complete category and suppose XxrightarrowAYxrightarrowLX satisfies LA=id_X. Put E=AL. Then: [label=textup(roman*)] item E^2=E; item Y≅im E⊕im(1-E); item A:X→im E is an isomorphism with inverse induced by L; item L(1-E)=0, so im(1-E) is a named residual invisible to the decoder; item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core im E. In a module or vector-space realizatio…
Exact mathematical passage
E^2=E
e^{2\pi i a\ell/q}Use in source · 1 passage
Search aliases: e^2π i aℓ/q pi ell
Use in source · block16_section_final.tex, line 630
Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projectors (E_q)_qmid N form the finest Q-rational coarsening obtainable by summing the split Fourier idempotents, and Q[C_N]E_q≅ Q(ζ_q).
Exact mathematical passage
c_q(\ell) := \sum_{\substack{1\le a\le q\\\gcd(a,q)=1}} e^{2\pi i a\ell/q} = \sum_{d\mid\gcd(q,\ell)}d\,\mu(q/d) \in\mathbb Z
e^{2\pi i\rho}Use in source · 1 passage
Search aliases: e^2π iρ pi rho
Use in source · block09_section.tex, line 409
finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
e^{2\pi i\rho}.
E^4Use in source · 2 passages
Search aliases: E^4
Use in source · block10_section.tex, line 365
[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.
Exact mathematical passage
\boxed{4xE^4-E^2+1=0}
E^NUse in source · 1 passage
Search aliases: E^N
Use in source · Stratified_Universality_Traditional.tex, line 2270
i). The orbit of v under an invertible map of the finite set F_p^d is a cycle containing its initial state, so (ii) ⇒ (iii). A sequence of period T is annihilated by E^T-I, where E is the forward shift, giving (iii) ⇒ (i). The minimal annihilator divides every annihilator, hence m_amid x^T-1. Conversely, if m_amid x^N-1, then E^N-I annihilates the sequence. Minimality of T proves [source reference].
Exact mathematical passage
E^N-I
E^{n-1}Use in source · 1 passage
Search aliases: E^n-1
Use in source · Stratified_Universality_Traditional.tex, line 1669
The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.
Exact mathematical passage
e'\in E^{n-1}
E^TUse in source · 1 passage
Search aliases: E^T
Use in source · Stratified_Universality_Traditional.tex, line 2270
A recurrence with nonzero constant term has an invertible companion matrix, giving (i) ⇒ (ii). The orbit of v under an invertible map of the finite set F_p^d is a cycle containing its initial state, so (ii) ⇒ (iii). A sequence of period T is annihilated by E^T-I, where E is the forward shift, giving (iii) ⇒ (i). The minimal annihilator divides every annihilator, hence m_amid x^T-1. Conversely, if m_amid x^N-1, then E^N-I annihilates the sequence. Minimality of T proves [source reference].
Exact mathematical passage
E^T-I
e^zUse in source · 1 passage
Search aliases: e^z
Use in source · block15_section.tex, line 711
uppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
E_{N,j}(z) :=\mathcal P_{N,j}e^z =\sum_{n\ge0}\frac{z^{Nn+j}}{(Nn+j)!}.
E_{\mathbf n}Use in source · 1 passage
Search aliases: E_ n mathbf
Use in source · Stratified_Universality_Traditional.tex, line 4064
[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.
Exact mathematical passage
E_{\mathbf n}:V\longrightarrow\mathbb Q^\rho, \qquad f\longmapsto \bigl(f(n_1),\ldots,f(n_\rho)\bigr)
E_0Use in source · 3 passages
Search aliases: E_0
Use in source · block09_section.tex, line 686
For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.
Exact mathematical passage
\boxed{ E_0 = \{0\}\cup\{1-b:b\in B_{N,j}\},}
e_0Use in source · 5 passages
Search aliases: e_0
Use in source · block18_carrier_observer_change_final.tex, line 294
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
S_z e_j=e_{j+1}\quad(j<N-1), \qquad S_z e_{N-1}=z e_0,
e_0^{\mathsf T}Use in source · 4 passages
Search aliases: e_0^ T mathsf
Use in source · block18_carrier_observer_change_final.tex, line 300
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
M_N=I+S_z, \qquad f_N(n)=e_0^{\mathsf T}M_N^n e_0.
e_{aj}Use in source · 2 passages
Search aliases: e_aj
Use in source · block16_section_final.tex, line 602
rier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
\sigma_a(e_j)=e_{aj}.
E_fUse in source · 2 passages
Search aliases: E_f
Use in source · block18_carrier_observer_change_final.tex, line 121
t sequence] For every task-compatible linear bridge, there is an exact sequence 0⟶ fracf^-1(N_Y)N_X ⟶ fracXN_X xrightarrowoverline f fracYN_Y ⟶ fracYN_Y+f(X) ⟶0. Thus L_f:=fracf^-1(N_Y)N_X measures source distinctions collapsed by the bridge, while E_f:=fracYN_Y+f(X) measures target semantic states outside the essential image.
Exact mathematical passage
E_f:=\frac{Y}{N_Y+f(X)}
e_{gHg^{-1}}Use in source · 1 passage
Search aliases: e_gHg^-1
Use in source · block12_section.tex, line 128
Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.
Exact mathematical passage
e_{gHg^{-1}}
e_HUse in source · 9 passages
Search aliases: e_H
Use in source · block12_section.tex, line 41
In the faithful regular realization, or in the field realization supplied by a normal basis, the projector determines the subgroup. Indeed, the support of the group-algebra idempotent e_H=frac1|H|∑_h∈ Hh is exactly H.
Exact mathematical passage
e_H=\frac1{|H|}\sum_{h\in H}h
e_iUse in source · 1 passage
Search aliases: e_i
Use in source · Stratified_Universality_Traditional.tex, line 4459
[p-adic saturation sequence] There is a digit-equivariant short exact sequence 0 ⟶ M_k,p(a) ⟶ L_k,p(a) ⟶ C_k(a)_(p) ⟶ 0, where C_k(a)_(p) is the p-primary subgroup of C_k(a). In Smith coordinates, M_k,p(a) = ⊕_i=1^ρp^ν_i Z_p e_i ⊆ ⊕_i=1^ρ Z_p e_i = L_k,p(a).
Exact mathematical passage
M_{k,p}(a) = \bigoplus_{i=1}^{\rho}p^{\nu_i}\mathbb Z_p e_i \subseteq \bigoplus_{i=1}^{\rho}\mathbb Z_p e_i = L_{k,p}(a).
e_JUse in source · 2 passages
Search aliases: e_J
Use in source · block12_section.tex, line 107
If HJ=JH, every element of HJ has |H∩ J| representations as hj. Multiplying the two group averages therefore gives the average over HJ. Conversely, the support of e_He_J is HJ, while the support of e_Je_H is JH. Equality in the regular group algebra forces equality of the product sets.
Exact mathematical passage
e_He_J
e_jUse in source · 7 passages
Search aliases: e_j
Use in source · block16_section_final.tex, line 592
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, whe…
Exact mathematical passage
e_j := \frac1N\sum_{\ell=0}^{N-1} \zeta_N^{-j\ell}g^\ell, \qquad j\in\mathbb Z/N\mathbb Z.
e_{j+1}Use in source · 1 passage
Search aliases: e_j+1
Use in source · block18_carrier_observer_change_final.tex, line 294
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
S_z e_j=e_{j+1}\quad(j<N-1), \qquad S_z e_{N-1}=z e_0,
E_{\mathrm{mark}}Use in source · 1 passage
Search aliases: E_mark mathrm
Use in source · block13_section.tex, line 35
The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate from neutral Tannakian reconstruction.
Exact mathematical passage
E_{\mathrm{mark}}
E_NUse in source · 6 passages
Search aliases: E_N
Use in source · block10_section.tex, line 200
Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.
Exact mathematical passage
R=K(x), \qquad E_N=R[z]/(z^N-x).
E_{N,j}Use in source · 2 passages
Search aliases: E_N,j
Use in source · block15_section.tex, line 711
uppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
E_{N,j}(z) :=\mathcal P_{N,j}e^z =\sum_{n\ge0}\frac{z^{Nn+j}}{(Nn+j)!}.
e_{N-1}Use in source · 2 passages
Search aliases: e_N-1
Use in source · block18_carrier_observer_change_final.tex, line 294
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
S_z e_j=e_{j+1}\quad(j<N-1), \qquad S_z e_{N-1}=z e_0,
e_{N-k}^{\mathsf T}Use in source · 1 passage
Search aliases: e_N-k^ T mathsf
Use in source · block18_carrier_observer_change_final.tex, line 318
Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving the determinant formula.
Exact mathematical passage
z e_{N-k}^{\mathsf T}
E_{\mathrm{per}}Use in source · 1 passage
Search aliases: E_per mathrm
Use in source · block13_section.tex, line 35
The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate from neutral Tannakian reconstruction.
Exact mathematical passage
E_{\mathrm{per}}
E_qUse in source · 8 passages
Search aliases: E_q
Use in source · block16_section_final.tex, line 613
ll=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
E_q:=\sum_{j\in\mathcal O_q}e_j
E_{q'}Use in source · 1 passage
Search aliases: E_q'
Use in source · block16_section_final.tex, line 617
are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
E_qE_{q'}=\delta_{q,q'}E_q, \qquad \sum_{q\mid N}E_q=1.
E_{\mathrm{rm}}Use in source · 1 passage
Search aliases: E_rm mathrm
Use in source · block13_section.tex, line 35
The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate from neutral Tannakian reconstruction.
Exact mathematical passage
E_{\mathrm{rm}}
\mathsf E_{\rm Fuss}Use in source · 6 passages
Search aliases: E_rm Fuss mathsf rm
Use in source · block13_bridge_matrix.tex, line 13
HOL · P · E · -- · E_rm Fuss/ P · E_rm Fuss/ P · -- · -- · --
Exact mathematical passage
\mathsf E_{\rm Fuss}/\mathsf P
\mathsf E_{\rm per}Use in source · 4 passages
Search aliases: E_rm per mathsf rm
Use in source · block13_bridge_matrix.tex, line 12
LIN · E · FF · E_rm per · -- · -- · -- · E_rm per · --
Exact mathematical passage
\mathsf E_{\rm per}
\mathsf E_{\rm rm}Use in source · 2 passages
Search aliases: E_rm rm mathsf rm
Use in source · block13_bridge_matrix.tex, line 14
DIG · E_rm per · P · E · P · -- · -- · E_rm rm · --
Exact mathematical passage
\mathsf E_{\rm rm}
E_UUse in source · 4 passages
Search aliases: E_U
Use in source · block16_section_final.tex, line 204
item Holonomy-compatible selectors and residuals. Let E_U⊆ I_Y,U,u× I_X,U,r be a family satisfying E_V= (λ^Y_γ,u×λ^X_γ,r)(E_U) for every arrow γ:U→ V. Then the local selectors C_E_U(h_U)_ai = h_U,ai, · (a,i)∈ E_U,
Exact mathematical passage
E_U\subseteq I_{Y,U,u}\times I_{X,U,r}
E_VUse in source · 2 passages
Search aliases: E_V
Use in source · block16_section_final.tex, line 208
item Holonomy-compatible selectors and residuals. Let E_U⊆ I_Y,U,u× I_X,U,r be a family satisfying E_V= (λ^Y_γ,u×λ^X_γ,r)(E_U) for every arrow γ:U→ V. Then the local selectors C_E_U(h_U)_ai = h_U,ai, · (a,i)∈ E_U,
Exact mathematical passage
E_V= (\lambda^Y_{\gamma,u}\times\lambda^X_{\gamma,r})(E_U)
e_XUse in source · 3 passages
Search aliases: e_X
Use in source · Stratified_Universality_Traditional.tex, line 374
[Typed realization and module equivalence, imported] Let Λ_ K:=𝕜 Q/I. In 𝕜-modules, the category of typed realizations of the linear grammar (Q,I) is equivalent to the category of unital left Λ_ K-modules. Under the equivalence, the vertex idempotent e_X acts as the projector onto the component of sort X, and each arrow acts in its declared source-target block.
Exact mathematical passage
e_X
e_YUse in source · 1 passage
Search aliases: e_Y
Use in source · Stratified_Universality_Traditional.tex, line 382
A typed realization defines the action of a path by composition and then extends 𝕜-linearly. Every element of I acts by zero, so the action factors through Λ_ K. Conversely, for a unital module M, the pairwise orthogonal vertex idempotents give M=⊕_X∈ Q_0e_XM, and an arrow a:X→ Y maps e_XM into e_YM. These constructions are inverse up to their canonical natural isomorphisms. This is the typed-module theorem of Paper I [source reference].
Exact mathematical passage
e_YM
E_{\alpha^{-N}}Use in source · 2 passages
Search aliases: E_α^-N alpha
Use in source · block09_section.tex, line 692
For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.
Exact mathematical passage
\boxed{ E_{\alpha^{-N}} = \{0,1,\ldots,N-2,-\lambda\},}
e_\chiUse in source · 3 passages
Search aliases: e_χ chi
Use in source · block15_section.tex, line 1035
[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map is block diagonal in these isotypic lanes, while an arbitrary K-linear map has the full lane matrix of [source reference].
Exact mathematical passage
e_\chi = \frac{\chi(1)}{|G|} \sum_{g\in G}\chi(g^{-1})g \in K[G]
E_\inftyUse in source · 3 passages
Search aliases: E_∞ infty
Use in source · block09_section.tex, line 699
For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.
Exact mathematical passage
\boxed{ E_\infty = A_{N,j}(\lambda).}
\End_{\kk C_2}Use in source · 1 passage
Search aliases: End_kk C_2 End kk
Use in source · block16_section_final.tex, line 567
0 · 1. The individual projectors do not descend because Sp_1S^-1=p_2, while their orbit sum p_1+p_2=1 does descend. The pair-address set has two orbits, Δ={(1,1),(2,2)}, Ω={(1,2),(2,1)}. Accordingly, End_kk C_2(kk^2) = { a · b b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivariant identity matrix to p_1, which is not equivariant. The orbit selectors C_Δ and C_Ω are compatible and split every equivariant endomorphism into its d…
Exact mathematical passage
\End_{\kk C_2}(\kk^2) = \left\{ \begin{pmatrix}a&b\\b&a\end{pmatrix}:a,b\in\kk \right\} = \kk B_\Delta\oplus\kk B_\Omega.
\End_{\kk H}Use in source · 1 passage
Search aliases: End_kk H End kk
Use in source · block16_section_final.tex, line 538
[Orbital endomorphism algebra] For one finite H-set I, the pair-orbit matrices form a basis of End_kk H(kk[I]). Matrix multiplication in this basis is exactly the finite compatible-path contraction of [source reference], with the preceding intersection numbers as structure constants.
Exact mathematical passage
\End_{\kk H}(\kk[I])
\mathbf FUse in source · 1 passage
Search aliases: F mathbf
Use in source · Stratified_Universality_Traditional.tex, line 1319
item A selected algebraic branch can produce a D-finite annihilator, but the operator may forget the selected branch and its minimal polynomial. The bridge is directional unless that data are retained. item A rational functional-Mahler mask over F_p produces a Cartier representation and finite automaton under the stated hypotheses, but this does not identify it with a Newton--Mahler interpolation chart [source reference]. item A faithful Hilbert-space corner representation realizes packet projec…
Exact mathematical passage
\mathbf F_p
F'Use in source · 3 passages
Search aliases: F'
Use in source · block09_section.tex, line 190
The normalized jet is J_0(F) = (F(0),F'(0),…,F^(r-1)(0)).
Exact mathematical passage
J_0(F) = \bigl(F(0),F'(0),\ldots,F^{(r-1)}(0)\bigr).
F^{(d)}Use in source · 1 passage
Search aliases: F^(d)
Use in source · block10_section.tex, line 545
Differentiate P_F(z,F(z))=0. Separability makes ∂_YP_F(z,F) nonzero in the function field. The field is closed under the induced derivation. The d+1 elements F,F',…,F^(d) are linearly dependent over K(z).
Exact mathematical passage
F,F',\ldots,F^{(d)}
F^{(j)}Use in source · 1 passage
Search aliases: F^(j)
Use in source · block09_section.tex, line 207
[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.
Exact mathematical passage
LF=0, \qquad F^{(j)}(0)=y_j.
F^{(r-1)}Use in source · 1 passage
Search aliases: F^(r-1)
Use in source · block09_section.tex, line 190
The normalized jet is J_0(F) = (F(0),F'(0),…,F^(r-1)(0)).
Exact mathematical passage
J_0(F) = \bigl(F(0),F'(0),\ldots,F^{(r-1)}(0)\bigr).
f^{-1_D}Use in source · 2 passages
Search aliases: f^-1_D
Use in source · Stratified_Universality_Traditional.tex, line 5002
[Dirichlet inverse and multiplicative closure] If f(1)=1, then f has a unique integral Dirichlet inverse. If f and g are multiplicative, then fmathbin∗_Dg and f^-1_D are multiplicative.
Exact mathematical passage
f^{-1_D}
F^{p-1}Use in source · 2 passages
Search aliases: F^p-1
Use in source · Stratified_Universality_Traditional.tex, line 2866
[Frobenius-radical characterization] Let F∈1+z F_p[[z]]. Then R_F(z)=F(z)^1-p. Consequently, the following are equivalent. [label=textup(roman*)] item The canonical mask R_F is rational. item The power F^p-1 belongs to F_p(z). item There are coprime polynomials A,B∈ F_p[z] with A(0)=B(0)=1 and B(z)F(z)=A(z)F(z^p). For the reduced mask R_F=A/B, one has F(z)^p-1=fracB(z)A(z). The ratio and product constructions are strict inverses: F⟼ R_F, R⟼ M_p[R]. In particular, every object of RMah_p^× is alge…
Exact mathematical passage
F^{p-1}
f_1Use in source · 2 passages
Search aliases: f_1
Use in source · Stratified_Universality_Traditional.tex, line 559
[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.
Exact mathematical passage
X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m
\mathbb F_2Use in source · 12 passages
Search aliases: F_2 mathbb
Use in source · Stratified_Universality_Traditional.tex, line 2740
Even indicator over F_2. For the word 10, T=2, m(x)=x^2+1=(x+1)^2, F(z)=frac11+z^2, R(z)=1+z^2. The repeated recurrence root is compatible with pure periodicity because the field has characteristic two. The canonical 2-kernel has three states: the original sequence, the constant-one sequence, and the constant-zero sequence. The rational-mask compiler gives the safe ambient bound 2^3=8 before reachability restriction.
Exact mathematical passage
\mathbb F_2
f_2Use in source · 1 passage
Search aliases: f_2
Use in source · Stratified_Universality_Traditional.tex, line 559
[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.
Exact mathematical passage
X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m
\mathbb F_3Use in source · 12 passages
Search aliases: F_3 mathbb
Use in source · Stratified_Universality_Traditional.tex, line 2235
a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3, the period word 12 has frac1+2z1-z^2=frac11+z.
Exact mathematical passage
\mathbb F_3
F_aUse in source · 5 passages
Search aliases: F_a
Use in source · Stratified_Universality_Traditional.tex, line 2229
screte category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3, the period word 12 has frac1+2z…
Exact mathematical passage
\label{eq:periodic-ogf-unreduced} F_a(z)=\sum_{n\geq 0}a_nz^n = \frac{W_a(z)}{1-z^{T(a)}}.
f_{a+b,s+p^a t}Use in source · 1 passage
Search aliases: f_a+b,s+p^a t
Use in source · Stratified_Universality_Traditional.tex, line 5509
[Strict nested-branch law] For a,b≥1, 0≤ s<p^a, and 0≤ t<p^b, (f_a,s)_b,t=f_a+b,s+p^a t. All multiple-overlap comparison cells for the nested p-power branch atlas are identities. Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.
Exact mathematical passage
\boxed{(f_{a,s})_{b,t}=f_{a+b,s+p^a t}.}
f_{a,s}Use in source · 2 passages
Search aliases: f_a,s
Use in source · Stratified_Universality_Traditional.tex, line 5509
[Strict nested-branch law] For a,b≥1, 0≤ s<p^a, and 0≤ t<p^b, (f_a,s)_b,t=f_a+b,s+p^a t. All multiple-overlap comparison cells for the nested p-power branch atlas are identities. Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.
Exact mathematical passage
\boxed{(f_{a,s})_{b,t}=f_{a+b,s+p^a t}.}
f_bUse in source · 2 passages
Search aliases: f_b
Use in source · block13_section.tex, line 391
Let A_b(z)=∏_m≥1(1-z^m)^-b_m be an integral unit series in the fixed Euler basis, and let c_n=∑_dmid nd b_d be its divisor-ghost sequence. Define the prime-uniform multiplicative function f_b(n)=∏_p^eparallel n[z^e]A_b(z). Every prime has the same local factor A_b(T). By [source reference], the coefficient, exponent, ghost, and prime-uniform multiplicative charts are reversible on this declared locus.
Exact mathematical passage
f_b(n)=\prod_{p^e\parallel n}[z^e]A_b(z).
f_EUse in source · 7 passages
Search aliases: f_E
Use in source · Stratified_Universality_Traditional.tex, line 690
[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .
Exact mathematical passage
f_E=C_E(f),\quad r_f=R_E(f), \qquad g_F=C_F(g),\quad r_g=R_F(g).
F_{f,p}'Use in source · 1 passage
Search aliases: F_f,p'
Use in source · Stratified_Universality_Traditional.tex, line 5130
[Valuation logarithmic derivative] Fix a prime p. Let Γ_f,p be the arithmetic function supported on powers of p with Γ_f,p(1)=0, Γ_f,p(p^e)=γ_p,e(f) (e≥1). Then (v_p· f)(n) = (Γ_f,pmathbin∗_Df)(n) (n≥1). Equivalently, fracT F_f,p'(T)F_f,p(T) = ∑_e≥1γ_p,e(f)T^e.
Exact mathematical passage
\frac{T F_{f,p}'(T)}{F_{f,p}(T)} = \sum_{e\geq1}\gamma_{p,e}(f)T^e.
F_{f\ast_D g,p}Use in source · 1 passage
Search aliases: F_f∗_D g,p ast
Use in source · Stratified_Universality_Traditional.tex, line 5046
[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).
Exact mathematical passage
F_{f\ast_D g,p}(T) = F_{f,p}(T)F_{g,p}(T).
F_{g,p}Use in source · 1 passage
Search aliases: F_g,p
Use in source · Stratified_Universality_Traditional.tex, line 5046
[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).
Exact mathematical passage
F_{f\ast_D g,p}(T) = F_{f,p}(T)F_{g,p}(T).
F_jUse in source · 3 passages
Search aliases: F_j
Use in source · block15_section.tex, line 791
[Typed polyphase realization] Let K be a field, let N∈ Z with N≥1 be invertible in K, and suppose K contains a primitive N-th root of unity. For F(z)=∑_n≥0a_nz^n∈ K[[z]], set F_j(w):=∑_n≥0a_Nn+jw^n. Then F(z)=∑_j=0^N-1z^jF_j(z^N) is exact analysis and synthesis. The Fourier residue projectors and the polyphase components satisfy the explicit coordinate relation P_N,jF(z)=z^jF_j(z^N). For every integer M≥1, refinement from N to NM is the projector-sum law of [source reference].
Exact mathematical passage
F_j(w):=\sum_{n\ge0}a_{Nn+j}w^n.
f_mUse in source · 2 passages
Search aliases: f_m
Use in source · Stratified_Universality_Traditional.tex, line 559
[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.
Exact mathematical passage
X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m
F_nUse in source · 2 passages
Search aliases: F_n
Use in source · Stratified_Universality_Traditional.tex, line 2187
Fibonacci F_n · x^2-x-1 · z/(1-z-z^2) · 2 · The root ratio is not torsion. Every arithmetic subsequence retains rank 2.
Exact mathematical passage
F_n
f_NUse in source · 4 passages
Search aliases: f_N
Use in source · block18_carrier_observer_change_final.tex, line 300
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
M_N=I+S_z, \qquad f_N(n)=e_0^{\mathsf T}M_N^n e_0.
F_{N-1}Use in source · 3 passages
Search aliases: F_N-1
Use in source · block09_section.tex, line 477
[Exact hypergeometric packet form] If (λ)_j≠0, then G_N,j^λ,α(x) = frac(λ)_jα^jj! _NF_N-1 ( A_N,j(λ) B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ G_{N,j}^{\lambda,\alpha}(x) = \frac{(\lambda)_j\alpha^j}{j!} {}_NF_{N-1} \!\left( \begin{matrix}A_{N,j}(\lambda)\\ B_{N,j}\end{matrix} ;\alpha^Nx \right).}
F_{n-1}Use in source · 1 passage
Search aliases: F_n-1
Use in source · Stratified_Universality_Traditional.tex, line 6060
bibitemBeukersHeckman1989 F. Beukers and G. Heckman, ``Monodromy for the hypergeometric function _nF_n-1,'' Inventiones Mathematicae 95 (1989), 325--354.
Exact mathematical passage
{}_nF_{n-1}
F_oUse in source · 5 passages
Search aliases: F_o
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
G_v:F_o\to F_v
\mathbf F_pUse in source · 1 passage
Search aliases: F_p mathbf
Use in source · Stratified_Universality_Traditional.tex, line 1319
item A selected algebraic branch can produce a D-finite annihilator, but the operator may forget the selected branch and its minimal polynomial. The bridge is directional unless that data are retained. item A rational functional-Mahler mask over F_p produces a Cartier representation and finite automaton under the stated hypotheses, but this does not identify it with a Newton--Mahler interpolation chart [source reference]. item A faithful Hilbert-space corner representation realizes packet projec…
Exact mathematical passage
\mathbf F_p
\mathbb F_p^dUse in source · 1 passage
Search aliases: F_p^d mathbb
Use in source · Stratified_Universality_Traditional.tex, line 2270
A recurrence with nonzero constant term has an invertible companion matrix, giving (i) ⇒ (ii). The orbit of v under an invertible map of the finite set F_p^d is a cycle containing its initial state, so (ii) ⇒ (iii). A sequence of period T is annihilated by E^T-I, where E is the forward shift, giving (iii) ⇒ (i). The minimal annihilator divides every annihilator, hence m_amid x^T-1. Conversely, if m_amid x^N-1, then E^N-I annihilates the sequence. Minimality of T proves [source reference].
Exact mathematical passage
\mathbb F_p^d
\mathbb F_p^{D+1}Use in source · 1 passage
Search aliases: F_p^D+1 mathbb
Use in source · Stratified_Universality_Traditional.tex, line 2579
a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source reference]. It computes the coefficient sequence exactly.
Exact mathematical passage
\mathbb F_p^{D+1}
F_qUse in source · 2 passages
Search aliases: F_q
Use in source · block09_section.tex, line 309
The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r…
Exact mathematical passage
F_q(t)=F(\omega_N^qt), \qquad G_{N,j}(t)=\frac1N\sum_{q=0}^{N-1}\omega_N^{-jq}F_q(t).
\mathbb F_qUse in source · 3 passages
Search aliases: F_q mathbb
Use in source · Stratified_Universality_Traditional.tex, line 3866
a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
K=\mathbb F_q
F_uUse in source · 1 passage
Search aliases: F_u
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
T_e:F_u\to F_v
f_UUse in source · 10 passages
Search aliases: f_U
Use in source · block15_section.tex, line 1095
s whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
f=(f_U):X\longrightarrow Y, \qquad g=(g_U):Y\longrightarrow Z
F_{UV}Use in source · 3 passages
Search aliases: F_UV
Use in source · block15_section.tex, line 1084
[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
F_{UV}:\mathfrak R(U)\to\mathfrak R(V)
F_vUse in source · 3 passages
Search aliases: F_v
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
F_v
f_VUse in source · 4 passages
Search aliases: f_V
Use in source · block16_section_final.tex, line 44
A descent object A=(A_U,θ^A_γ) consists of objects A_U∈ C_U and isomorphisms θ^A_γ:F_γ(A_U)xrightarrow∼A_V satisfying the identity and cocycle equations. A descended morphism f:X→ Y is a compatible family (f_U)_U satisfying θ^Y_γ F_γ(f_U)=f_Vθ^X_γ.
Exact mathematical passage
\theta^Y_\gamma F_\gamma(f_U)=f_V\theta^X_\gamma.
F_{\alpha,\alpha}Use in source · 1 passage
Search aliases: F_α,α alpha alpha
Use in source · block10_section.tex, line 514
If α=β, the quadratic equation Y^2-(1-α z)^2=0 is not the normalized source polynomial. The selected branch is rational: F_α,α(z)=1-α z, so the correct source degree is one. This is a negative control against using a nonminimal defining equation in the degree ledger.
Exact mathematical passage
F_{\alpha,\alpha}(z)=1-\alpha z,
F_{\alpha,\beta}Use in source · 2 passages
Search aliases: F_α,β alpha beta
Use in source · block10_section.tex, line 403
Let F_α,β(z) = sqrt(1-α z)(1-β z), F_α,β(0)=1, and assume first that α≠β, so the normalized source degree is two. Write F_α,β(z)=E(x)+zO(x), x=z^2, and put s=α+β, p=αβ.
Exact mathematical passage
F_{\alpha,\beta}(z) = \sqrt{(1-\alpha z)(1-\beta z)}, \qquad F_{\alpha,\beta}(0)=1,
F_{\lambda,\alpha}Use in source · 2 passages
Search aliases: F_λ,α lambda alpha
Use in source · block09_section.tex, line 436
The general minimal multisection problem remains open. The family F_λ,α(z) = (1-α z)^-λ provides a nontrivial exact model that contains irreducible Fuchsian packets, rational collapse, finite-support aliasing, and zero-address cancellation in one parameter space.
Exact mathematical passage
\boxed{ F_{\lambda,\alpha}(z) = (1-\alpha z)^{-\lambda}}
\mathbf{FinOrbLin}Use in source · 1 passage
Search aliases: FinOrbLin mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3856
k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
\mathbf{Aut}_k(K) \simeq \mathbf{FinOrbSec}_k(K) \simeq \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}} \simeq \mathbf{DFAO}_{k,K}^{\min}. \label{eq:block05-automatic-finite-orbit-atlas}
\mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}}Use in source · 1 passage
Search aliases: FinOrbLin_k,K^sl mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3856
k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
\mathbf{Aut}_k(K) \simeq \mathbf{FinOrbSec}_k(K) \simeq \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}} \simeq \mathbf{DFAO}_{k,K}^{\min}. \label{eq:block05-automatic-finite-orbit-atlas}
\mathbf{FinOrbSec}Use in source · 1 passage
Search aliases: FinOrbSec mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3856
k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
\mathbf{Aut}_k(K) \simeq \mathbf{FinOrbSec}_k(K) \simeq \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}} \simeq \mathbf{DFAO}_{k,K}^{\min}. \label{eq:block05-automatic-finite-orbit-atlas}
\mathbf{FinOrbSec}_kUse in source · 1 passage
Search aliases: FinOrbSec_k mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3856
k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.
Exact mathematical passage
\mathbf{Aut}_k(K) \simeq \mathbf{FinOrbSec}_k(K) \simeq \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}} \simeq \mathbf{DFAO}_{k,K}^{\min}. \label{eq:block05-automatic-finite-orbit-atlas}
\operatorname{Fitt}_jUse in source · 2 passages
Search aliases: Fitt_j operatorname
Use in source · Stratified_Universality_Traditional.tex, line 4287
[Fitting-ideal certificate] For 0≤ j<ρ, Fitt_j(C_k(a)) = ( d_1(a)d_2(a)⋯ d_ρ-j(a) ), and Fitt_j(C_k(a))= Z (j≥ρ). Thus the Smith profile and the Fitting filtration contain the same reduction-defect data at different resolutions.
Exact mathematical passage
\operatorname{Fitt}_j(C_k(a)) = \left( d_1(a)d_2(a)\cdots d_{\rho-j}(a) \right),
\FixUse in source · 1 passage
Search aliases: Fix operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1479
a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
C_o = \bigcap_{e\notin S}\ker(H_e-I) = \Fix\langle H_e:e\notin S\rangle, \qquad C_v=G_vC_o.
\Fun_{\kk}^{\oplus}Use in source · 1 passage
Search aliases: Fun_kk^⊕ Fun_𝕜^⊕ Fun kk oplus operatorname Bbbk oplus
Use in source · Stratified_Universality_Traditional.tex, line 891
gintheorem[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right side is the category of typed realizations satisfying all declared relations and carrying every declared packet idempotent to a split idempotent.
Exact mathematical passage
\Fun_{\kk}^{\oplus} \bigl(\Syn_{\mathrm{ak}}(\mathbb K_{\rm lin}),\mathcal C\bigr) \simeq \Mod_{\mathrm{ak}}(\mathbb K_{\rm lin},\mathcal C),
g^{-1}Use in source · 4 passages
Search aliases: g^-1
Use in source · block12_section.tex, line 128
Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.
Exact mathematical passage
e_{gHg^{-1}}
G^{p-1}Use in source · 2 passages
Search aliases: G^p-1
Use in source · Stratified_Universality_Traditional.tex, line 2949
iterion] Let F= M_p[A/B]. The following are equivalent. [label=textup(roman*)] item F∈ F_p(z). item B/A is a (p-1)-st power in F_p(z). item ω_p-1(A/B)=0. When these conditions hold, there is a unique rational function G∈ F_p(z) with G(0)=1 and G^p-1=B/A, and F=G.
Exact mathematical passage
G^{p-1}=B/A,
g^\ellUse in source · 3 passages
Search aliases: g^ℓ ell
Use in source · block16_section_final.tex, line 592
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, whe…
Exact mathematical passage
e_j := \frac1N\sum_{\ell=0}^{N-1} \zeta_N^{-j\ell}g^\ell, \qquad j\in\mathbb Z/N\mathbb Z.
G_0Use in source · 5 passages
Search aliases: G_0
Use in source · block12_section.tex, line 220
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
G_0\supseteq G_1\supseteq G_2\supseteq\cdots
G_1Use in source · 1 passage
Search aliases: G_1
Use in source · block12_section.tex, line 220
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
G_0\supseteq G_1\supseteq G_2\supseteq\cdots
G_2Use in source · 1 passage
Search aliases: G_2
Use in source · block12_section.tex, line 220
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
G_0\supseteq G_1\supseteq G_2\supseteq\cdots
g_{ca}Use in source · 2 passages
Search aliases: g_ca
Use in source · Stratified_Universality_Traditional.tex, line 538
[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.
Exact mathematical passage
\boxed{ (gf)_{ci} = \sum_a g_{ca}f_{ai}.}
g_FUse in source · 7 passages
Search aliases: g_F
Use in source · Stratified_Universality_Traditional.tex, line 690
[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .
Exact mathematical passage
f_E=C_E(f),\quad r_f=R_E(f), \qquad g_F=C_F(g),\quad r_g=R_F(g).
G_iUse in source · 7 passages
Search aliases: G_i
Use in source · block12_section.tex, line 224
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
P_i=\frac1{|G_i|}\sum_{g\in G_i}U_g.
g_iUse in source · 3 passages
Search aliases: g_i
Use in source · Stratified_Universality_Traditional.tex, line 4253
[Exact Smith formulas] Let g_i(m)=gcd(m,d_i(a)). Then G_k,m(a) ≅ Q_k,m(a) ≅ ⊕_i=1^ρ Z/g_i(m) Z, and |overline M_k,m(a)| = ∏_i=1^ρ fracmg_i(m). The reduction map is an isomorphism if and only if gcd(m,d_ρ(a))=1. Equivalently, every prime dividing a reduction defect divides the saturation index.
Exact mathematical passage
g_i(m)=\gcd(m,d_i(a))
G_{k,p^r}Use in source · 2 passages
Search aliases: G_k,p^r
Use in source · Stratified_Universality_Traditional.tex, line 4341
[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.
Exact mathematical passage
G_{k,p^r}(a) \cong Q_{k,p^r}(a) \cong \bigoplus_{i=1}^{\rho} \mathbb Z/p^{\min(r,\nu_i)}\mathbb Z,
g_NUse in source · 10 passages
Search aliases: g_N
Use in source · block11_section.tex, line 347
[Finite tree-address signature] Fix a packet width N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10
Exact mathematical passage
g_N=\gcd(N,d_{\mathcal A}).
G_{N,j}Use in source · 4 passages
Search aliases: G_N,j
Use in source · block09_section.tex, line 309
The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r…
Exact mathematical passage
F_q(t)=F(\omega_N^qt), \qquad G_{N,j}(t)=\frac1N\sum_{q=0}^{N-1}\omega_N^{-jq}F_q(t).
G_{N,j}^{-M,\alpha}Use in source · 1 passage
Search aliases: G_N,j^-M,α alpha
Use in source · block09_section.tex, line 538
item If λ∉ Z, then r_D(N,j;λ)=N. item If λ=L∈ Z_>0, then every packet is a nonzero rational function with exact denominator (1-α^Nx)^L, and r_D(N,j;L)=1. item If λ=-M∈ Z_≤0, then G_N,j^-M,α is zero when j>M. Otherwise it is a nonzero polynomial of degree lfloorfracM-jNrfloor and r_D(N,j;-M) = 1, · j≤ M,
Exact mathematical passage
G_{N,j}^{-M,\alpha}
g_{r,\gamma}Use in source · 3 passages
Search aliases: g_r,γ gamma
Use in source · block11_section.tex, line 185
Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.
Exact mathematical passage
g_{r,\gamma} = Q_\gamma D^rG(x)[h,\ldots,h].
g_sUse in source · 3 passages
Search aliases: g_s
Use in source · Stratified_Universality_Traditional.tex, line 5493
[Complete clopen packet equivalence] The analysis map B_a: C_p ⟶ ∏_s=0^p^a-1 C_p is an isometric isomorphism. Its inverse is the piecewise synthesis ( S_a g)(x) =g_s (fracx-sp^a) for x∈ s+p^a Z_p. Thus the clopen residue cover of Z_p has effective descent for both objects and equality morphisms.
Exact mathematical passage
\boxed{ (\mathcal S_a g)(x) =g_s\!\left(\frac{x-s}{p^a}\right) \quad\text{for }x\in s+p^a\mathbb Z_p.}
G_uUse in source · 2 passages
Search aliases: G_u
Use in source · Stratified_Universality_Traditional.tex, line 1475
el in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
H_e=G_v^{-1}T_eG_u\in GL(F_o).
g_UUse in source · 3 passages
Search aliases: g_U
Use in source · block15_section.tex, line 1095
s whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
f=(f_U):X\longrightarrow Y, \qquad g=(g_U):Y\longrightarrow Z
G_vUse in source · 4 passages
Search aliases: G_v
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
G_v:F_o\to F_v
G_v^{-1}Use in source · 3 passages
Search aliases: G_v^-1
Use in source · Stratified_Universality_Traditional.tex, line 1475
el in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
H_e=G_v^{-1}T_eG_u\in GL(F_o).
G_\PhiUse in source · 2 passages
Search aliases: G_Φ Phi
Use in source · block11_section.tex, line 271
For the packet system textup(11.4), define the directed dependency graph G_Φ on I by placing an edge j⟶ i when some active block contributing to the equation for u_i contains u_j as an input. Let SCC(G_Φ) be its strongly connected components. A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.
Exact mathematical passage
G_\Phi
\GalUse in source · 1 passage
Search aliases: Gal
Use in source · block16_section_final.tex, line 600
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, whe…
Exact mathematical passage
\sigma_a\in\Gal(K/\mathbb Q)
\operatorname{Gal}Use in source · 5 passages
Search aliases: Gal operatorname
Use in source · block10_section.tex, line 271
Let Ω be a normal splitting field of the generic packet scheme over K(x), and let Γ=Gal(Ω/K(x)). After splitting E_N, the geometric packet branches form a set B_N of cardinality d^N. Let π_j: B_N→Ω be the jth packet coordinate, and set C_N,j=π_j( B_N). For the selected semantic packet U_j, separability gives [K(x)(U_j):K(x)] = |Γ· U_j|.
Exact mathematical passage
\Gamma=\operatorname{Gal}(\Omega/K(x)).
\gcdUse in source · 23 passages
Search aliases: gcd
Use in source · Stratified_Universality_Traditional.tex, line 2057
An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.
Exact mathematical passage
A(z)=\frac{P(z)}{Q(z)}, \qquad Q(0)=1, \qquad \gcd(P,Q)=1, \qquad \deg P<\deg Q,
\operatorname{GL}Use in source · 8 passages
Search aliases: GL operatorname
Use in source · block12_section.tex, line 327
The image is a compact totally disconnected subgroup of GL_n( C). After conjugation it lies in a compact Lie group. A compact totally disconnected Lie group is finite.
Exact mathematical passage
\operatorname{GL}_n(\mathbb C)
\operatorname{GL}_nUse in source · 8 passages
Search aliases: GL_n operatorname
Use in source · block12_section.tex, line 327
The image is a compact totally disconnected subgroup of GL_n( C). After conjugation it lies in a compact Lie group. A compact totally disconnected Lie group is finite.
Exact mathematical passage
\operatorname{GL}_n(\mathbb C)
\mathcal HUse in source · 5 passages
Search aliases: H mathcal
Use in source · block09_section.tex, line 497
B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ \mathcal H_{N,j}^{\lambda,\alpha} = \theta \prod_{b\in B_{N,j}}(\theta+b-1) - \alpha^Nx \prod_{a\in A_{N,j}(\lambda)}(\theta+a),}
H^*Use in source · 2 passages
Search aliases: H^*
Use in source · block19_admissibility_lifting_final.tex, line 227
0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.
Exact mathematical passage
H^*QH=Q
H^0Use in source · 1 passage
Search aliases: H^0
Use in source · Stratified_Universality_Traditional.tex, line 1663
[Cohomological correction criterion] For a cocycle e∈ E^n, the class of (e,0) in H^n(Cone(D)) vanishes exactly when [e] lies in the image of H^n(D). When the complexes are concentrated in degree zero, H^0(Cone(D))≅coker D.
Exact mathematical passage
H^0(\Cone(D))\cong\coker D.
H^1Use in source · 1 passage
Search aliases: H^1
Use in source · Stratified_Universality_Traditional.tex, line 1714
[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is naturally a module over a central simple K-algebra split by L, rather than an ordinary K-vector space [source reference].
Exact mathematical passage
\alpha\in H^1(G,PGL_n(L)).
H^2Use in source · 3 passages
Search aliases: H^2
Use in source · Stratified_Universality_Traditional.tex, line 1725
[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is naturally a module over a central simple K-algebra split by L, rather than an ordinary K-vector space [source reference].
Exact mathematical passage
\delta(\alpha) \in H^2(G,L^\times) = \Br(L/K).
\mathcal H^HUse in source · 1 passage
Search aliases: H^H mathcal
Use in source · block12_section.tex, line 297
Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.
Exact mathematical passage
\mathcal H^H
H^nUse in source · 3 passages
Search aliases: H^n
Use in source · Stratified_Universality_Traditional.tex, line 1662
[Cohomological correction criterion] For a cocycle e∈ E^n, the class of (e,0) in H^n(Cone(D)) vanishes exactly when [e] lies in the image of H^n(D). When the complexes are concentrated in degree zero, H^0(Cone(D))≅coker D.
Exact mathematical passage
H^n(\Cone(D))
h_0Use in source · 2 passages
Search aliases: h_0
Use in source · block16_section_final.tex, line 317
,mathscr M^u,r_Y,X) xrightarrow∼ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0), where the right-hand side is the subgroup fixed by every loop-transport map. Consequently, Hom_Desc(mathscr G)(X,Y) ≅ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0). After choosing one arrow τ_U:U_0→ U for each chart, the inverse sends a fixed base matrix h_0 to the section h_U=(τ_U)_*h_0. The result is independent of the chosen arrows because two choices differ by loop transport at U_0.
Exact mathematical passage
h_0
h_{A^{\mathsf T}C}Use in source · 1 passage
Search aliases: h_A^ TC mathsf
Use in source · block18_carrier_observer_change_final.tex, line 362
For a linear map A, h_A^ TC(x)=h_C(Ax). If the incoming fiber is empty, both sides of the formula are -∞. Otherwise, the support function of the convex hull of a finite union is the maximum of the component support functions, with the same conventions for empty components. These identities give the formula. Finite unions of compact sets have compact convex hulls in finite dimension, so every updated packet stays in the declared carrier. Iteration follows from composition of the linear maps and o…
Exact mathematical passage
h_{A^{\mathsf T}C}(x)=h_C(Ax).
H_A^rUse in source · 6 passages
Search aliases: H_A^r
Use in source · block16_section_final.tex, line 344
Fix a base chart U_0. Every loop γ∈Aut_mathscr G(U_0) acts on an address set by i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the direct product of the two separate monodromy groups.
Exact mathematical passage
H_A^r := \operatorname{im}\!\left( \Aut_{\mathscr G}(U_0) \longrightarrow \operatorname{Sym}(I_{A,U_0,r}) \right)
h_{bj}Use in source · 2 passages
Search aliases: h_bj
Use in source · block15_section.tex, line 402
[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contribute. The inverse is the corresponding fine analysis after coarse synthesis. Consequently, changing a complete lane resolution loses no information. Lo…
Exact mathematical passage
h=(h_{bj})
h_CUse in source · 1 passage
Search aliases: h_C
Use in source · block18_carrier_observer_change_final.tex, line 362
For a linear map A, h_A^ TC(x)=h_C(Ax). If the incoming fiber is empty, both sides of the formula are -∞. Otherwise, the support function of the convex hull of a finite union is the maximum of the component support functions, with the same conventions for empty components. These identities give the formula. Finite unions of compact sets have compact convex hulls in finite dimension, so every updated packet stays in the declared carrier. Iteration follows from composition of the linear maps and o…
Exact mathematical passage
h_{A^{\mathsf T}C}(x)=h_C(Ax).
h_{C'_t}Use in source · 1 passage
Search aliases: h_C'_t
Use in source · block18_carrier_observer_change_final.tex, line 352
[Observer-conditioned convex packet transport] For every state t∈ S and direction x, h_C'_t(x) = max_δ(s,b)=th_C_s(A_bx). Repeated updates compose exactly with branch words. Within each fixed observer state, replacing a finite atom family by any other finite generating family of the same convex hull preserves all future value queries generated by linear branch actions.
Exact mathematical passage
h_{C'_t}(x) = \max_{\delta(s,b)=t}h_{C_s}(A_bx).
h_{C_s}Use in source · 1 passage
Search aliases: h_C_s
Use in source · block18_carrier_observer_change_final.tex, line 352
[Observer-conditioned convex packet transport] For every state t∈ S and direction x, h_C'_t(x) = max_δ(s,b)=th_C_s(A_bx). Repeated updates compose exactly with branch words. Within each fixed observer state, replacing a finite atom family by any other finite generating family of the same convex hull preserves all future value queries generated by linear branch actions.
Exact mathematical passage
h_{C'_t}(x) = \max_{\delta(s,b)=t}h_{C_s}(A_bx).
H_eUse in source · 8 passages
Search aliases: H_e
Use in source · Stratified_Universality_Traditional.tex, line 1475
el in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
H_e=G_v^{-1}T_eG_u\in GL(F_o).
H_e^*Use in source · 1 passage
Search aliases: H_e^*
Use in source · block19_admissibility_lifting_final.tex, line 143
Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible family directly. A group preserving a positive-definite metric is conjugate into a compact orthogonal or unitary group. Conversely, averaging an inner p…
Exact mathematical passage
H_e^*Q_oH_e=|\det H_e|^{2/n}Q_o.
H_{ij}Use in source · 1 passage
Search aliases: H_ij
Use in source · block18_carrier_observer_change_final.tex, line 306
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
H_{ij}=f_N(i+j), \qquad 0\leq i,j<N,
h_NUse in source · 1 passage
Search aliases: h_N
Use in source · block11_section.tex, line 424
[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arity signature determines both the number of active packet addresses and the size of the complete dominant-singularity orbit.
Exact mathematical passage
\boxed{ \rho^N\,\mu_{h_N}, \qquad h_N=\frac{d_{\mathcal A}}{\gcd(N,d_{\mathcal A})}.} \tag{11.13}
\mathcal H_{N,j}^{\lambda,\alpha}Use in source · 2 passages
Search aliases: H_N,j^λ,α mathcal lambda alpha
Use in source · block09_section.tex, line 497
B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ \mathcal H_{N,j}^{\lambda,\alpha} = \theta \prod_{b\in B_{N,j}}(\theta+b-1) - \alpha^Nx \prod_{a\in A_{N,j}(\lambda)}(\theta+a),}
h_{U,ai}Use in source · 4 passages
Search aliases: h_U,ai
Use in source · block16_section_final.tex, line 151
item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix record. It changes the global type of that record from a constantly labeled matrix to a compatible section of a matrix local system.
Exact mathematical passage
(h_U)_U\longmapsto \left( \sum_{a,i} \iota^u_{Y,U,a}h_{U,ai}\pi^r_{X,U,i} \right)_U.
H_{Y,X}^{u,r}Use in source · 3 passages
Search aliases: H_Y,X^u,r
Use in source · block16_section_final.tex, line 362
[ H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the direct product of the two separate monodromy groups.
Exact mathematical passage
H_{Y,X}^{u,r}
h_{\varnothing}Use in source · 1 passage
Search aliases: h_∅ varnothing
Use in source · block18_carrier_observer_change_final.tex, line 335
Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.
Exact mathematical passage
h_{\varnothing}(x)=-\infty,\qquad \operatorname{conv}(\varnothing)=\varnothing,\qquad \max\varnothing=-\infty.
\hlineUse in source · 1 passage
Search aliases: hline
Use in source · Stratified_Universality_Traditional.tex, line 4945
[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plane-partition series. More generally, the power family converts the exponent rule n^r into the divisor-sum ghost σ_r+1.
Exact mathematical passage
\begin{array}{c|c|c} b_n & A_b(z) & c_n\\ \hline 1 & \displaystyle\prod_{n\geq1}(1-z^n)^{-1} & \sigma_1(n)\\[4pt] n & \displaystyle\prod_{n\geq1}(1-z^n)^{-n} & \sigma_2(n)\\[4pt] n^r & \displaystyle\prod_{n\geq1}(1-z^n)^{-n^r} & \sigma_{r+1}(n). \end{array}
\operatorname{Hom}Use in source · 2 passages
Search aliases: Hom operatorname
Use in source · Stratified_Universality_Traditional.tex, line 502
[Universal mapping property of packet coordinates] For every object Z, restriction to the packet objects gives natural bijections Hom(X,Z) ≅ ∏_iHom(X_i,Z), Hom(Z,X) ≅ ∏_iHom(Z,X_i), where the products are finite. Packetwise maps therefore assemble uniquely into a global map.
Exact mathematical passage
\operatorname{Hom}(X,Z) \cong \prod_i\operatorname{Hom}(X_i,Z), \qquad \operatorname{Hom}(Z,X) \cong \prod_i\operatorname{Hom}(Z,X_i),
\Hom_{\mathcal C}Use in source · 6 passages
Search aliases: Hom_ C Hom mathcal
Use in source · block15_section.tex, line 136
item Unique lane matrix. Every morphism f:X→ Y is recovered uniquely from its lane blocks: f = ∑_a,ip_Y,a^u f p_X,i^r = ∑_a,iι_Y,a^u [f]^u,r_a iπ_X,i^r. Equivalently, the assignment f⟼ ([f]^u,r_a i)_a,i is a bijection from Hom_ C(X,Y) to the group of rectangular matrices with entries in Hom_ C(X_i^r,Y_a^u).
Exact mathematical passage
\Hom_{\mathcal C}(X,Y)
\Hom_{\Desc(\mathscr G)}Use in source · 2 passages
Search aliases: Hom_Desc(mathscr G) Hom Desc mathscr
Use in source · block16_section_final.tex, line 141
item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix record. It changes the global type of that record from a constantly labeled matrix to a compatible section of a matrix local system.
Exact mathematical passage
\mathfrak A^{u,r}_{Y,X}: \Hom_{\Desc(\mathscr G)}(X,Y) \xrightarrow{\sim} \Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right), \qquad f\longmapsto \left(\bigl([f_U]^{u,r}_{ai}\bigr)_{a,i}\right)_U.
\Hom_{\kk H}Use in source · 1 passage
Search aliases: Hom_kk H Hom kk
Use in source · block16_section_final.tex, line 480
ion-orbital-calculus Let kk be a commutative ring, let a finite group H act on finite sets I,J,K, and let kk[I],kk[J],kk[K] be the corresponding permutation modules. For an orbit O⊆ J× I under the diagonal action, let B_ O:kk[I]→kk[J] be the matrix whose (j,i)-entry is one when (j,i)∈ O and zero otherwise. Then Hom_kk H(kk[I],kk[J]) = ⊕_ O∈ Hbackslash(J× I) kk B_ O. In particular, an equivariant matrix is exactly a matrix constant on every pair orbit.
Exact mathematical passage
\Hom_{\kk H}(\kk[I],\kk[J]) = \bigoplus_{\mathcal O\in H\backslash(J\times I)} \kk B_{\mathcal O}.
\mathrm{HS}Use in source · 1 passage
Search aliases: HS mathrm
Use in source · Stratified_Universality_Traditional.tex, line 721
[Hilbert--Schmidt optimality, imported] Assume X,Y are Hilbert spaces, the packet idempotents are orthogonal projectors, and f:X→ Y is Hilbert--Schmidt. Then C_E is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and R_E(f) _HS^2 = ∑_(a,i)∉ E q_afp_i _HS^2. Consequently, C_E(f) is the unique best Hilbert--Schmidt approximation to f among operators supported on E.
Exact mathematical passage
\|R_E(f)\|_{\mathrm{HS}}^2 = \sum_{(a,i)\notin E}\|q_afp_i\|_{\mathrm{HS}}^2.
i'Use in source · 1 passage
Search aliases: i'
Use in source · block16_section_final.tex, line 530
The (k,i)-entry of B_ QB_ P is the displayed cardinality. If (k',i')=(hk,hi), then j↦ hj is a bijection between the two sets being counted. The entry is therefore constant on R, which gives the orbit-basis expansion.
Exact mathematical passage
(k',i')=(hk,hi)
i_1Use in source · 2 passages
Search aliases: i_1
Use in source · block15_section.tex, line 594
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T_{a;i_1,\ldots,i_d} := \pi_{Y,a}T (\iota_{1,i_1}-,\ldots,\iota_{d,i_d}-).
\mathcal I_{13}Use in source · 1 passage
Search aliases: I_13 mathcal
Use in source · block13_section.tex, line 152
nvariant-descent Let A_13 be the atlas of [source reference]. Suppose that, in every connected component: item the local invariant is complete on every normalized chart; item the bridge comparisons are coherent; item invariant-compatible local isomorphisms satisfy effective descent. Then the local invariants induce a functor I_13: Desc( A_13) ⟶ Desc( V_13) that reflects isomorphisms. If each semantic component is discrete, equality of descended invariants is equivalent to equality of the represe…
Exact mathematical passage
\mathcal I_{13}: \Desc(\mathfrak A_{13}) \longrightarrow \Desc(\mathfrak V_{13})
I_{A,r}Use in source · 5 passages
Search aliases: I_A,r
Use in source · block15_section.tex, line 1124
V^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local pro…
Exact mathematical passage
I_{A,r}
I_{A,s}Use in source · 1 passage
Search aliases: I_A,s
Use in source · block15_section.tex, line 1125
I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U…
Exact mathematical passage
\rho_{sr}^{A}:I_{A,s}\twoheadrightarrow I_{A,r}
I_{A,U_0,r}Use in source · 2 passages
Search aliases: I_A,U_0,r
Use in source · block16_section_final.tex, line 344
Fix a base chart U_0. Every loop γ∈Aut_mathscr G(U_0) acts on an address set by i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the direct product of the two separate monodromy groups.
Exact mathematical passage
H_A^r := \operatorname{im}\!\left( \Aut_{\mathscr G}(U_0) \longrightarrow \operatorname{Sym}(I_{A,U_0,r}) \right)
I_{A_U,r}Use in source · 2 passages
Search aliases: I_A_U,r
Use in source · block15_section.tex, line 1106
Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global re…
Exact mathematical passage
\lambda_{UV}^{A,r}: I_{A_U,r}\xrightarrow{\sim}I_{A_V,r}
I_{A_V,r}Use in source · 1 passage
Search aliases: I_A_V,r
Use in source · block15_section.tex, line 1106
Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global re…
Exact mathematical passage
\lambda_{UV}^{A,r}: I_{A_U,r}\xrightarrow{\sim}I_{A_V,r}
i_dUse in source · 2 passages
Search aliases: i_d
Use in source · block15_section.tex, line 594
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T_{a;i_1,\ldots,i_d} := \pi_{Y,a}T (\iota_{1,i_1}-,\ldots,\iota_{d,i_d}-).
I_{X,U_0,r}Use in source · 1 passage
Search aliases: I_X,U_0,r
Use in source · block16_section_final.tex, line 361
i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the direct product of the two separate monodromy groups.
Exact mathematical passage
I_{Y,U_0,u}\times I_{X,U_0,r}
I_{Y,u}Use in source · 3 passages
Search aliases: I_Y,u
Use in source · block15_section.tex, line 142
item Composition and path law. If f:X→ Y and g:Y→ Z, with the same resolution u used on the intermediate object, then [gf]^v,r_c i = ∑_a∈ I_Y,u[g]^v,u_c a[f]^u,r_a i. More generally, a block of a composable string is the finite sum of the products along all compatible intermediate address paths.
Exact mathematical passage
[gf]^{v,r}_{c i} = \sum_{a\in I_{Y,u}}[g]^{v,u}_{c a}[f]^{u,r}_{a i}.
I_{Y,U_0,u}Use in source · 1 passage
Search aliases: I_Y,U_0,u
Use in source · block16_section_final.tex, line 361
i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the direct product of the two separate monodromy groups.
Exact mathematical passage
I_{Y,U_0,u}\times I_{X,U_0,r}
I_{Y,v}Use in source · 1 passage
Search aliases: I_Y,v
Use in source · block15_section.tex, line 376
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathsf B^{v,s}(X,Y) := \bigoplus_{b\in I_{Y,v}} \bigoplus_{j\in I_{X,s}} \Hom_{\mathcal C}(X_j^s,Y_b^v),
I_{\alpha,L}Use in source · 1 passage
Search aliases: I_α,L alpha
Use in source · block09_section.tex, line 401
Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
\boxed{ I_{\alpha,L}(\rho) = \sum_{v_j-j=\mu} q_{j,v_j}\rho^{\underline j}.}
I_{\lambda,b}Use in source · 1 passage
Search aliases: I_λ,b lambda
Use in source · block13_section.tex, line 131
For every chart C_λ,a, let I_λ,a: C_λ,a⟶ V_λ,a be its native normalized invariant. A bridge comparison is a natural isomorphism κ_ab: J_abI_λ,a Longrightarrow I_λ,bΦ_ab, where J_ab transports invariant coordinates between their chart-specific target categories.
Exact mathematical passage
\kappa_{ab}: J_{ab}I_{\lambda,a} \Longrightarrow I_{\lambda,b}\Phi_{ab},
I_{\xi,L_G}Use in source · 1 passage
Search aliases: I_ξ,L_G xi
Use in source · block09_section.tex, line 760
ther with its finite initial record; item the monic minimal differential operator from [source reference], together with the distinguished power-series solution selected by the marked coefficient record; item the hypergeometric parameter packet (A_N,j(λ),B_N,j;α^N) and leading coefficient; item the primitive minimal-operator singular and indicial record mathscr E(G) = ( Σ_rm op(G), (I_ξ,L_G)_ξ∈Σ_rm op(G) ), together with the orbit-rank and defect data (r_rm orb,r_D,A_N,C_N,j). In the nonintegral…
Exact mathematical passage
\mathscr E(G) = \left( \Sigma_{\rm op}(G), \bigl(I_{\xi,L_G}\bigr)_{\xi\in\Sigma_{\rm op}(G)} \right),
i_\ellUse in source · 1 passage
Search aliases: i_ℓ ell
Use in source · block15_section.tex, line 632
Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement is then a finite sum of the specified homogeneous pieces.
Exact mathematical passage
x_\ell=\sum_{i_\ell}p_{\ell,i_\ell}x_\ell
\id_XUse in source · 2 passages
Search aliases: id_X id mathrm
Use in source · Stratified_Universality_Traditional.tex, line 1445
[Split bridge rectification] Let A be an additive idempotent-complete category and suppose XxrightarrowAYxrightarrowLX satisfies LA=id_X. Put E=AL. Then: [label=textup(roman*)] item E^2=E; item Y≅im E⊕im(1-E); item A:X→im E is an isomorphism with inverse induced by L; item L(1-E)=0, so im(1-E) is a named residual invisible to the decoder; item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core im E. In a module or vector-space realizatio…
Exact mathematical passage
LA=\id_X
\operatorname{im}Use in source · 2 passages
Search aliases: im operatorname
Use in source · Stratified_Universality_Traditional.tex, line 4162
For an integer m≥2, the raw coefficient reduction is M_k(a)/mM_k(a), while the semantic reduction inside the saturated lattice is overline M_k,m(a) = im ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). These need not agree.
Exact mathematical passage
\overline M_{k,m}(a) = \operatorname{im} \left( M_k(a)/mM_k(a) \longrightarrow L_k(a)/mL_k(a) \right).
\mathrm{in}Use in source · 2 passages
Search aliases: in mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3688
For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.
Exact mathematical passage
(W,A_d,v_{\mathrm{in}},\lambda)
\IndUse in source · 4 passages
Search aliases: Ind
Use in source · block17_semantic_quotients_final.tex, line 198
Fix a universe mathscr Q(X) of admissible queries on X. For Q⊆mathscr Q(X), define Ind( Q) := ⋂_q∈ Qkerpair(q). For an equivalence relation or congruence R, define Def(R) := {q∈mathscr Q(X):R⊆kerpair(q)}.
Exact mathematical passage
\Ind(\mathcal Q) := \bigcap_{q\in\mathcal Q}\kerpair(q).
\operatorname{Ind}Use in source · 1 passage
Search aliases: Ind operatorname
Use in source · block12_section.tex, line 180
[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.
Exact mathematical passage
\boxed{ \operatorname{Res}_K^G\operatorname{Ind}_H^GW \cong \bigoplus_{x\in K\backslash G/H} \operatorname{Ind}_{K\cap xHx^{-1}}^K \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).} \tag{12.8}
\operatorname{Ind}_H^GUse in source · 1 passage
Search aliases: Ind_H^G operatorname
Use in source · block12_section.tex, line 180
[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.
Exact mathematical passage
\boxed{ \operatorname{Res}_K^G\operatorname{Ind}_H^GW \cong \bigoplus_{x\in K\backslash G/H} \operatorname{Ind}_{K\cap xHx^{-1}}^K \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).} \tag{12.8}
\operatorname{Ind}_{K\cap xHx^{-1}}^KUse in source · 1 passage
Search aliases: Ind_K∩ xHx^-1^K operatorname cap
Use in source · block12_section.tex, line 180
[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.
Exact mathematical passage
\boxed{ \operatorname{Res}_K^G\operatorname{Ind}_H^GW \cong \bigoplus_{x\in K\backslash G/H} \operatorname{Ind}_{K\cap xHx^{-1}}^K \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).} \tag{12.8}
\mathbf{Inv}Use in source · 1 passage
Search aliases: Inv mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2721
[Invariant descent and global completeness on the periodic atlas] The native invariants in the recurrence, rational, digital, and mask charts admit coherent comparison maps to J_p. These maps descend to J_p: Desc( R_p^per) ⟶ Inv_p^per. The descended invariant is complete on the declared semantic category.
Exact mathematical passage
\mathcal J_p: \Desc(\mathfrak R_p^{\mathrm{per}}) \longrightarrow \mathbf{Inv}_p^{\mathrm{per}}.
\mathbf{Inv}_p^{\mathrm{per}}Use in source · 1 passage
Search aliases: Inv_p^per mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2721
[Invariant descent and global completeness on the periodic atlas] The native invariants in the recurrence, rational, digital, and mask charts admit coherent comparison maps to J_p. These maps descend to J_p: Desc( R_p^per) ⟶ Inv_p^per. The descended invariant is complete on the declared semantic category.
Exact mathematical passage
\mathcal J_p: \Desc(\mathfrak R_p^{\mathrm{per}}) \longrightarrow \mathbf{Inv}_p^{\mathrm{per}}.
\operatorname{Irr}Use in source · 2 passages
Search aliases: Irr operatorname
Use in source · block15_section.tex, line 1042
[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map is block diagonal in these isotypic lanes, while an arbitrary K-linear map has the full lane matrix of [source reference].
Exact mathematical passage
\chi\in\operatorname{Irr}_K(G)
\operatorname{Irr}_KUse in source · 2 passages
Search aliases: Irr_K operatorname
Use in source · block15_section.tex, line 1042
[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map is block diagonal in these isotypic lanes, while an arbitrary K-linear map has the full lane matrix of [source reference].
Exact mathematical passage
\chi\in\operatorname{Irr}_K(G)
J_0Use in source · 2 passages
Search aliases: J_0
Use in source · block09_section.tex, line 190
The normalized jet is J_0(F) = (F(0),F'(0),…,F^(r-1)(0)).
Exact mathematical passage
J_0(F) = \bigl(F(0),F'(0),\ldots,F^{(r-1)}(0)\bigr).
j_1Use in source · 7 passages
Search aliases: j_1
Use in source · block11_section.tex, line 113
For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
i,j_1,\ldots,j_m\in I
\mathcal J_{13}Use in source · 4 passages
Search aliases: J_13 mathcal
Use in source · block13_section.tex, line 456
Let Λ_13 = {DRLM,Loc,Fuss,EM,Gal} be the component-tag set. Define J_13(X) = (λ(X), J_λ(X)(X)).
Exact mathematical passage
\mathcal J_{13}(X) = \bigl(\lambda(X),\mathcal J_{\lambda(X)}(X)\bigr).
J_{ab}Use in source · 2 passages
Search aliases: J_ab
Use in source · block13_section.tex, line 131
For every chart C_λ,a, let I_λ,a: C_λ,a⟶ V_λ,a be its native normalized invariant. A bridge comparison is a natural isomorphism κ_ab: J_abI_λ,a Longrightarrow I_λ,bΦ_ab, where J_ab transports invariant coordinates between their chart-specific target categories.
Exact mathematical passage
\kappa_{ab}: J_{ab}I_{\lambda,a} \Longrightarrow I_{\lambda,b}\Phi_{ab},
\mathcal J_{\mathrm{cert}}Use in source · 2 passages
Search aliases: J_cert mathcal mathrm
Use in source · block14_section.tex, line 19
whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object. item The semantic normalizers induce the comparison 2-cells on all nonempty triple overlaps and force the quadruple-overlap coherence equation. item The compatible native invariant family descends to a tagged global invariant J_cert: Desc( A_cert) ⟶ Desc( V_cert). item The descended invariant is complete on the declared normaliz…
Exact mathematical passage
\mathcal J_{\mathrm{cert}}: \Desc(\mathfrak A_{\mathrm{cert}}) \longrightarrow \Desc(\mathcal V_{\mathrm{cert}}).
\mathcal J_{\mathrm{DG}}Use in source · 1 passage
Search aliases: J_DG mathcal mathrm
Use in source · Stratified_Universality_Traditional.tex, line 5305
[Complete descended prime-power invariant] The prime-power package J_DG(b) = ( b(p^e), c(p^e) )_p prime, e≥0 subject to p^e b(p^e)=c(p^e)-c(p^e-1) (e≥1) is complete on EulMult_ Z. It reconstructs b, c, the Euler product A_b, and the coefficient recurrence of A_b.
Exact mathematical passage
\boxed{ \mathcal J_{\mathrm{DG}}(b) = \left( b(p^e), c(p^e) \right)_{p\text{ prime},\ e\geq0} }
\mathcal J_{\mathrm{DRLM},p}Use in source · 3 passages
Search aliases: J_DRLM,p mathcal mathrm
Use in source · block13_section.tex, line 189
For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise contains the primitive period word, exact period, minimal recurrence, reduced OGF, and periodic packet profile.
Exact mathematical passage
\mathcal J_{\mathrm{DRLM},p}(a) = \left( \mathsf{SL}_{p,\mathbb F_p}(a), \mathcal P_{\mathrm{rm}}(a), \mathcal P_{\mathrm{per}}(a) \right),
\mathcal J_{\mathrm{EM}}Use in source · 1 passage
Search aliases: J_EM mathcal mathrm
Use in source · block13_section.tex, line 399
[Euler--multiplicative invariant comparison] The exponent sequence J_EM(A_b)=b is a complete invariant of the certified Euler--multiplicative component. In every chart it is recovered by the same M obius formula b_n = frac1n ∑_dmid nμ(n/d)c_d, and Dirichlet convolution corresponds to exponent addition.
Exact mathematical passage
\mathcal J_{\mathrm{EM}}(A_b)=b
\mathcal J_{\mathrm{Fuss}}Use in source · 2 passages
Search aliases: J_Fuss mathcal mathrm
Use in source · block13_section.tex, line 351
Define J_Fuss(q,α) = ( q, α, P_q,α, {c_n(q,α)}_n≥0, {g_N}_N≥1, {fracq-1g_N}_N≥1 ).
Exact mathematical passage
\mathcal J_{\mathrm{Fuss}}(q,\alpha) = \left( q, \alpha, P_{q,\alpha}, \{c_n(q,\alpha)\}_{n\geq0}, \{g_N\}_{N\geq1}, \left\{\frac{q-1}{g_N}\right\}_{N\geq1} \right).
\mathcal J_{\mathrm{Gal}}Use in source · 2 passages
Search aliases: J_Gal mathcal mathrm
Use in source · block13_section.tex, line 422
Fix a finite Galois extension L/K in the semisimple setting. For each subgroup H≤ G, define J_Gal(H) = ( H, L^H, P_H, Core_G(H), Ncl_G(H), {Jbackslash G/H}_J≤ G ).
Exact mathematical passage
\mathcal J_{\mathrm{Gal}}(H) = \left( H, L^H, P_H, \operatorname{Core}_G(H), \operatorname{Ncl}_G(H), \{J\backslash G/H\}_{J\leq G} \right).
\mathcal J_{L/K}Use in source · 1 passage
Search aliases: J_L/K mathcal
Use in source · block12_section.tex, line 572
A complete chart record on the finite subgroup locus is J_L/K(H) = ( H, L^H, P_H, Core_G(H), Ncl_G(H), {Kbackslash G/H}_K≤ G ). The first three entries already classify the normalized object, while the remaining entries record its atlas position and transition structure. For a framed semilinear vector object, the descended K-space with its frame is complete. For a projective object, the Brauer class is an obstruction coordinate, not a complete invariant of all central simple algebras without the…
Exact mathematical passage
\mathcal J_{L/K}(H) = \left( H, L^H, P_H, \operatorname{Core}_G(H), \operatorname{Ncl}_G(H), \{K\backslash G/H\}_{K\leq G} \right).
\mathcal J_{\mathrm{Loc},p}Use in source · 2 passages
Search aliases: J_Loc,p mathcal mathrm
Use in source · block13_section.tex, line 242
[Continuous local invariant comparison] The Mahler coefficient sequence J_Loc,p(f) = (c_r(f))_r≥0 is a complete invariant of A_Loc,p. Complete branch packets, branchwise Mahler coordinates, affine coefficient kernels, and nested refinements admit coherent comparison maps to J_Loc,p.
Exact mathematical passage
\mathcal J_{\mathrm{Loc},p}(f) = (c_r(f))_{r\ge0}
j_mUse in source · 7 passages
Search aliases: j_m
Use in source · block11_section.tex, line 113
For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
i,j_1,\ldots,j_m\in I
J_{N,j}Use in source · 2 passages
Search aliases: J_N,j
Use in source · block09_section.tex, line 290
[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of order at most s.
Exact mathematical passage
\boxed{ w_{m+1}=J_{N,j}(m)w_m,}
\mathcal J_pUse in source · 5 passages
Search aliases: J_p mathcal
Use in source · Stratified_Universality_Traditional.tex, line 2683
For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a),
Exact mathematical passage
\label{eq:periodic-invariant-package} \mathcal J_p(a) = \bigl( T(a), w(a), m_a(x), P_a(z)/Q_a(z), \mathcal K_p(a), \mathcal A_p(a), R_a(z) \bigr),
\mathcal J_p^{\mathrm{rm}}Use in source · 2 passages
Search aliases: J_p^rm mathcal mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3302
For F∈RMah_p^×, define J_p^rm(F) = ( R_F, ω_p-1(R_F), D(R_F), V_p(F), r_p(F), K_p(F), κ_p(F), A_p(F) ).
Exact mathematical passage
\label{eq:block04-invariant-package} \mathcal J_p^{\mathrm{rm}}(F) = \left( R_F, \omega_{p-1}(R_F), D(R_F), \mathcal V_p(F), r_p(F), \mathcal K_p(F), \kappa_p(F), \mathcal A_p(F) \right).
\mathcal J_{\rm BF}Use in source · 1 passage
Search aliases: J_rm BF mathcal rm
Use in source · block09_section.tex, line 782
[Effective marked binomial--Fuchsian atlas] The five normalized presentations above determine one another on the declared marked family. With equality morphisms they form equivalent discrete groupoids, every normalized fiber is contractible, and objects and morphisms descend effectively. The package J_rm BF = ( α,λ,N,j, R_N,j,L_N,j, A_N,j(λ),B_N,j, mathscr E(G), r_rm orb,r_D,A_N,C_N,j ) is complete on this marked category.
Exact mathematical passage
\boxed{ \mathcal J_{\rm BF} = \left( \alpha,\lambda,N,j, R_{N,j},L_{N,j}, A_{N,j}(\lambda),B_{N,j}, \mathscr E(G), r_{\rm orb},r_D,A_N,C_{N,j} \right)}
j_tUse in source · 3 passages
Search aliases: j_t
Use in source · block15_section.tex, line 218
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
j_t:C_t\to C
j_{\kappa(e')}Use in source · 2 passages
Search aliases: j_κ(e') kappa
Use in source · block15_section.tex, line 233
Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of Φ_κ,τ, so the fold is split lossless.
Exact mathematical passage
Q_ej_{\kappa(e')}\tau_{e'}(b_{e'}) = \delta_{e,e'}j_{\kappa(e')}\tau_{e'}(b_{e'}).
j_{\kappa(e)}Use in source · 7 passages
Search aliases: j_κ(e) kappa
Use in source · block15_section.tex, line 219
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
\Phi_{\kappa,\tau}: \bigoplus_{e\in E}B_e\longrightarrow C, \qquad (b_e)_e\longmapsto \sum_{e\in E}j_{\kappa(e)}\tau_e(b_e)
\mathcal J_{\lambda(X)}Use in source · 1 passage
Search aliases: J_λ(X) mathcal lambda
Use in source · block13_section.tex, line 456
Let Λ_13 = {DRLM,Loc,Fuss,EM,Gal} be the component-tag set. Define J_13(X) = (λ(X), J_λ(X)(X)).
Exact mathematical passage
\mathcal J_{13}(X) = \bigl(\lambda(X),\mathcal J_{\lambda(X)}(X)\bigr).
k'Use in source · 1 passage
Search aliases: k'
Use in source · block16_section_final.tex, line 530
The (k,i)-entry of B_ QB_ P is the displayed cardinality. If (k',i')=(hk,hi), then j↦ hj is a bijection between the two sets being counted. The entry is therefore constant on R, which gives the orbit-basis expansion.
Exact mathematical passage
(k',i')=(hk,hi)
K^{\mathbb N}Use in source · 2 passages
Search aliases: K^ N mathbb
Use in source · Stratified_Universality_Traditional.tex, line 3505
For a sequence a=(a_n)_n≥0∈ K^ N, define the digit-section operators (Λ_d a)(n)=a(kn+d), 0≤ d<k. For a word w=d_0⋯ d_ℓ-1, put Λ_w = Λ_d_ℓ-1⋯Λ_d_1Λ_d_0. Then (Λ_w a)(n) = a(k^|w|n+[w]_k).
Exact mathematical passage
a=(a_n)_{n\geq0}\in K^{\mathbb N}
K^{(E)}Use in source · 3 passages
Search aliases: K^(E)
Use in source · block18_carrier_observer_change_final.tex, line 261
Let E⊆ K^M be the set of distinct residual future functions of a scalar series or observational system. Set minimization retains the elements of E as distinct states. Free linearization produces the vector space K^(E), while linear minimization retains only their span.
Exact mathematical passage
K^{(E)}
K^2Use in source · 1 passage
Search aliases: K^2
Use in source · block15_section.tex, line 1247
[Endpoint selection does not implement refinement] Let K be a field and let X=K^2. At the coarse resolution use the single projector p=1_X. At the fine resolution use p_1= 1 · 0
Exact mathematical passage
X=K^2
k^aUse in source · 1 passage
Search aliases: k^a
Use in source · block15_section.tex, line 839
it-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
k^an+[w]_k, \qquad [w]_k=d_0+kd_1+\cdots+k^{a-1}d_{a-1}.
k^{a-1}Use in source · 1 passage
Search aliases: k^a-1
Use in source · block15_section.tex, line 839
it-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
k^an+[w]_k, \qquad [w]_k=d_0+kd_1+\cdots+k^{a-1}d_{a-1}.
K^dUse in source · 1 passage
Search aliases: K^d
Use in source · Stratified_Universality_Traditional.tex, line 2082
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
(C,v,\lambda), \qquad C\in M_d(K),\quad v\in K^d,\quad \lambda\in(K^d)^*,
k^jUse in source · 1 passage
Search aliases: k^j
Use in source · Stratified_Universality_Traditional.tex, line 3498
Fix an integer radix k≥2 and a field K. Let Σ_k={0,1,…,k-1}. Words are read from least significant digit to most significant digit. For w=d_0d_1⋯ d_ℓ-1∈Σ_k^*, define |w|=ℓ, [w]_k=∑_j=0^ℓ-1d_jk^j. The empty word ε has length zero and value zero.
Exact mathematical passage
|w|=\ell, \qquad [w]_k=\sum_{j=0}^{\ell-1}d_jk^j.
K^MUse in source · 1 passage
Search aliases: K^M
Use in source · block18_carrier_observer_change_final.tex, line 261
Let E⊆ K^M be the set of distinct residual future functions of a scalar series or observational system. Set minimization retains the elements of E as distinct states. Free linearization produces the vector space K^(E), while linear minimization retains only their span.
Exact mathematical passage
E\subseteq K^M
K^NUse in source · 1 passage
Search aliases: K^N
Use in source · block18_carrier_observer_change_final.tex, line 293
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
K^N
K^rUse in source · 1 passage
Search aliases: K^r
Use in source · block09_section.tex, line 203
[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.
Exact mathematical passage
(y_0,\ldots,y_{r-1})\in K^r
K^sUse in source · 1 passage
Search aliases: K^s
Use in source · block09_section.tex, line 289
[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of order at most s.
Exact mathematical passage
v_n\in K^s
k^{|w|}Use in source · 2 passages
Search aliases: k^|w|
Use in source · Stratified_Universality_Traditional.tex, line 3517
For a sequence a=(a_n)_n≥0∈ K^ N, define the digit-section operators (Λ_d a)(n)=a(kn+d), 0≤ d<k. For a word w=d_0⋯ d_ℓ-1, put Λ_w = Λ_d_ℓ-1⋯Λ_d_1Λ_d_0. Then (Λ_w a)(n) = a(k^|w|n+[w]_k).
Exact mathematical passage
(\Lambda_w a)(n) = a\bigl(k^{|w|}n+[w]_k\bigr). \label{eq:block05-word-section}
K^\timesUse in source · 3 passages
Search aliases: K^× times
Use in source · block13_section.tex, line 259
Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.
Exact mathematical passage
\alpha\in K^\times
k^\ellUse in source · 5 passages
Search aliases: k^ℓ ell
Use in source · Stratified_Universality_Traditional.tex, line 3924
Choose ℓ with k^ℓ>m and consider the m+1 sections P(k^ℓ n+r), 0≤ r≤ m. When expanded in the basis 1,n,…,n^m, their coefficient functions as functions of r have successive degrees m,m-1,…,0, with nonzero leading coefficients. Evaluation at the distinct points 0,1,…,m therefore gives an invertible generalized Vandermonde matrix. These sections are linearly independent, proving rank m+1.
Exact mathematical passage
k^\ell>m
K_0Use in source · 14 passages
Search aliases: K_0
Use in source · block15_section.tex, line 328
kappa(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈Γ:overline b_t=0}. Then |S_0|-|supp(overline b)| = |A_0| +(|S_1|-|Γ|) +|K_0|. The three terms record, respectively, annihilation by the local maps τ_e, collision of surviving lane addresses under κ, and cancellation among folded values at a common address. If every τ_e is injective, then A_0=∅. If globally typed selectors satisfy the strict separation law of (L8), then K_0=∅.
Exact mathematical passage
\operatorname{supp}(\overline b) :=\{t\in T:\overline b_t\ne0\}, \qquad K_0:=\{t\in\Gamma:\overline b_t=0\}.
\mathcal K_{k,p^{r+1}}^{\mathrm{sem}}Use in source · 1 passage
Search aliases: K_k,p^r+1^sem mathcal mathrm
Use in source · Stratified_Universality_Traditional.tex, line 4409
ower-automata For every r≥1, the semantic p^r-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence abmod p^r. Moreover, κ_k,p^r^sem(a) ≤ |overline M_k,p^r(a)| = p^∑_i(r-min(r,ν_i)). Reduction induces surjective digit-equivariant maps K_k,p^r+1^sem(a) ↠ K_k,p^r^sem(a).
Exact mathematical passage
\mathcal K_{k,p^{r+1}}^{\mathrm{sem}}(a) \twoheadrightarrow \mathcal K_{k,p^r}^{\mathrm{sem}}(a).
\mathcal K_pUse in source · 12 passages
Search aliases: K_p mathcal
Use in source · Stratified_Universality_Traditional.tex, line 2326
For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.
Exact mathematical passage
\label{eq:p-kernel} \mathcal K_p(a) = \left\{ \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a: \ell\geq 0,\ 0\leq d_i<p \right\}.
K_UUse in source · 3 passages
Search aliases: K_U
Use in source · block15_section.tex, line 1167
quare remains a projector-sum refinement square in the descent category; item every fold whose source objects, target objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K…
Exact mathematical passage
K_U=\ker(\Phi_U)
k_UUse in source · 1 passage
Search aliases: k_U
Use in source · block16_section_final.tex, line 279
For (H3), the local path law gives the displayed contraction formula in every chart. Applying γ_* to a contracted block distributes over the finite sum and carries each product to the product of the transported blocks. Since λ^Y_γ,u is a bijection, reindexing the intermediate address shows γ_*(k_U⋆ h_U) = (γ_*k_U)⋆(γ_*h_U) = k_V⋆ h_V. The identity section is compatible by the splitting cocycle. The equivalence then follows from (H2).
Exact mathematical passage
\gamma_*(k_U\star h_U) = (\gamma_*k_U)\star(\gamma_*h_U) = k_V\star h_V.
k_{U,ca}Use in source · 1 passage
Search aliases: k_U,ca
Use in source · block16_section_final.tex, line 170
item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, maths…
Exact mathematical passage
(k\star h)_{U,ci} := \sum_{a\in I_{Y,U,u}}k_{U,ca}h_{U,ai}.
K_VUse in source · 1 passage
Search aliases: K_V
Use in source · block15_section.tex, line 1169
et objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K_U)xrightarrow∼K_V compatible with the kernel inclusions and satisfying the triple-overlap cocycle. Then the global…
Exact mathematical passage
\eta_{UV}:F_{UV}(K_U)\xrightarrow{\sim}K_V
k_VUse in source · 1 passage
Search aliases: k_V
Use in source · block16_section_final.tex, line 279
For (H3), the local path law gives the displayed contraction formula in every chart. Applying γ_* to a contracted block distributes over the finite sum and carries each product to the product of the transported blocks. Since λ^Y_γ,u is a bijection, reindexing the intermediate address shows γ_*(k_U⋆ h_U) = (γ_*k_U)⋆(γ_*h_U) = k_V⋆ h_V. The identity section is compatible by the splitting cocycle. The equivalence then follows from (H2).
Exact mathematical passage
\gamma_*(k_U\star h_U) = (\gamma_*k_U)\star(\gamma_*h_U) = k_V\star h_V.
\ker_{\mathfrak R(U)}Use in source · 1 passage
Search aliases: ker_ R(U) ker mathfrak
Use in source · block15_section.tex, line 1174
cends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K_U)xrightarrow∼K_V compatible with the kernel inclusions and satisfying the triple-overlap cocycle. Then the global kernel exists and is computed chartwise: (ker_Desc( R)Φ)_U ≅ ker_ R(U)(Φ_U).
Exact mathematical passage
\bigl(\ker_{\Desc(\mathfrak R)}\Phi\bigr)_U \cong \ker_{\mathfrak R(U)}(\Phi_U).
\ker_{\Desc(\mathfrak R)}Use in source · 1 passage
Search aliases: ker_Desc( R) ker Desc mathfrak
Use in source · block15_section.tex, line 1174
cends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K_U)xrightarrow∼K_V compatible with the kernel inclusions and satisfying the triple-overlap cocycle. Then the global kernel exists and is computed chartwise: (ker_Desc( R)Φ)_U ≅ ker_ R(U)(Φ_U).
Exact mathematical passage
\bigl(\ker_{\Desc(\mathfrak R)}\Phi\bigr)_U \cong \ker_{\mathfrak R(U)}(\Phi_U).
\kerpairUse in source · 4 passages
Search aliases: kerpair
Use in source · block17_semantic_quotients_final.tex, line 108
Let K_0:=⋂_jkerpair(o_j) in the relational setting and N_0:=⋂_jker(o_j) in the linear setting.
Exact mathematical passage
K_0:=\bigcap_j\kerpair(o_j)
\kk^2Use in source · 4 passages
Search aliases: kk^2 kk kk^2 𝕜^2 kk Bbbk
Use in source · Stratified_Universality_Traditional.tex, line 579
0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.
Exact mathematical passage
\kk^2
L^{\operatorname{Core}_G(H)}Use in source · 1 passage
Search aliases: L^Core_G(H) operatorname
Use in source · block12_section.tex, line 145
[Canonical envelopes of a nonnormal Galois atlas] Let M=L^H. Then L^Core_G(H) =∏_g∈ Gg(M) tag12.6 is the smallest Galois extension inside L containing every conjugate chart, while L^Ncl_G(H) =⋂_g∈ Gg(M) tag12.7 is the largest Galois subfield contained in every chart. The core and normal-closure projectors are the two canonical global envelopes of the conjugate atlas.
Exact mathematical passage
\boxed{ L^{\operatorname{Core}_G(H)} =\prod_{g\in G}g(M)} \tag{12.6}
L^{\operatorname{Gal}(L/E)}Use in source · 1 passage
Search aliases: L^Gal(L/E) operatorname
Use in source · block12_section.tex, line 71
The finite Galois correspondence gives H=Gal(L/L^H) and E=L^Gal(L/E). The range of P_H is L^H after faithful scalar extension, and the support of e_H recovers H. All three assignments are therefore mutually inverse on the declared charts.
Exact mathematical passage
E=L^{\operatorname{Gal}(L/E)}
L^HUse in source · 10 passages
Search aliases: L^H
Use in source · block12_section.tex, line 49
[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P_H≥ P_J. Thus the subgroup poset, the intermediate-field poset, and the faithful fixed-projector poset form three normalized charts of one finite semisimple Ga…
Exact mathematical passage
\boxed{ H \quad\longleftrightarrow\quad L^H \quad\longleftrightarrow\quad P_H } \tag{12.3}
L^JUse in source · 1 passage
Search aliases: L^J
Use in source · block12_section.tex, line 60
intheorem[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P_H≥ P_J. Thus the subgroup poset, the intermediate-field poset, and the faithful fixed-projector poset form three normalized charts of one finite semi…
Exact mathematical passage
H\leq J \quad\Longleftrightarrow\quad L^H\supseteq L^J \quad\Longleftrightarrow\quad P_H\geq P_J.
L^nUse in source · 2 passages
Search aliases: L^n
Use in source · Stratified_Universality_Traditional.tex, line 1701
[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
W=(L^n)^{G,\star}
L^{\operatorname{Ncl}_G(H)}Use in source · 1 passage
Search aliases: L^Ncl_G(H) operatorname
Use in source · block12_section.tex, line 152
[Canonical envelopes of a nonnormal Galois atlas] Let M=L^H. Then L^Core_G(H) =∏_g∈ Gg(M) tag12.6 is the smallest Galois extension inside L containing every conjugate chart, while L^Ncl_G(H) =⋂_g∈ Gg(M) tag12.7 is the largest Galois subfield contained in every chart. The core and normal-closure projectors are the two canonical global envelopes of the conjugate atlas.
Exact mathematical passage
\boxed{ L^{\operatorname{Ncl}_G(H)} =\bigcap_{g\in G}g(M)} \tag{12.7}
L^\timesUse in source · 6 passages
Search aliases: L^× times
Use in source · Stratified_Universality_Traditional.tex, line 1718
[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is naturally a module over a central simple K-algebra split by L, rather than an ordinary K-vector space [source reference].
Exact mathematical passage
1\longrightarrow L^\times \longrightarrow GL_n(L) \longrightarrow PGL_n(L) \longrightarrow1
L_{\mathbb Z}Use in source · 4 passages
Search aliases: L_ Z mathbb
Use in source · Stratified_Universality_Traditional.tex, line 1997
Addendum 04 proves that the prefix S_*={0,1,…,L_ Z(m,d)-1} with L_ Z(m,d)=min(d+1,B(m)), B(m)=max({1}∪{p^v_p(m):pmid m}), always admits an integer decoder and is cardinality-minimal [source reference].
Exact mathematical passage
S_*=\{0,1,\dots,L_{\mathbb Z}(m,d)-1\}
L_0Use in source · 1 passage
Search aliases: L_0
Use in source · block09_section.tex, line 315
der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r_N,j^rm char. For a zero character packet put r_N,j^rm c…
Exact mathematical passage
L_N^{\rm orb}=\operatorname{LCLM}(L_0,\ldots,L_{N-1}), \qquad r_N^{\rm orb}=\operatorname{ord}L_N^{\rm orb},
L_bUse in source · 5 passages
Search aliases: L_b
Use in source · block18_carrier_observer_change_final.tex, line 183
[Carrier-indexed observer packet theorem] Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(s,b)=tL_b(a_s). If a finite family of raw histories has already been aggregated by their state using ⊕, then this update gives exactly the same result…
Exact mathematical passage
L_b:A\longrightarrow A.
L_dUse in source · 4 passages
Search aliases: L_d
Use in source · Stratified_Universality_Traditional.tex, line 2257
[Finite-field persistence equals pure periodicity] For a sequence over F_p, the following are equivalent. [label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0. If m_a(x) is the monic minimal recurrence polynomial and T=T(a), then m_a(x)mid x^T-1, T=min{N≥ 1:m_a(x)mid x^N-1}.
Exact mathematical passage
C\in GL_d(\mathbb F_p)
L_fUse in source · 3 passages
Search aliases: L_f
Use in source · block18_carrier_observer_change_final.tex, line 117
[Bridge-loss exact sequence] For every task-compatible linear bridge, there is an exact sequence 0⟶ fracf^-1(N_Y)N_X ⟶ fracXN_X xrightarrowoverline f fracYN_Y ⟶ fracYN_Y+f(X) ⟶0. Thus L_f:=fracf^-1(N_Y)N_X measures source distinctions collapsed by the bridge, while E_f:=fracYN_Y+f(X) measures target semantic states outside the essential image.
Exact mathematical passage
L_f:=\frac{f^{-1}(N_Y)}{N_X}
L_GUse in source · 1 passage
Search aliases: L_G
Use in source · block09_section.tex, line 760
ther with its finite initial record; item the monic minimal differential operator from [source reference], together with the distinguished power-series solution selected by the marked coefficient record; item the hypergeometric parameter packet (A_N,j(λ),B_N,j;α^N) and leading coefficient; item the primitive minimal-operator singular and indicial record mathscr E(G) = ( Σ_rm op(G), (I_ξ,L_G)_ξ∈Σ_rm op(G) ), together with the orbit-rank and defect data (r_rm orb,r_D,A_N,C_N,j). In the nonintegral…
Exact mathematical passage
\mathscr E(G) = \left( \Sigma_{\rm op}(G), \bigl(I_{\xi,L_G}\bigr)_{\xi\in\Sigma_{\rm op}(G)} \right),
L_gUse in source · 1 passage
Search aliases: L_g
Use in source · block18_carrier_observer_change_final.tex, line 156
Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.
Exact mathematical passage
0\longrightarrow L_f \longrightarrow L_{gf} \longrightarrow L_g \longrightarrow0.
L_{gf}Use in source · 1 passage
Search aliases: L_gf
Use in source · block18_carrier_observer_change_final.tex, line 156
Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.
Exact mathematical passage
0\longrightarrow L_f \longrightarrow L_{gf} \longrightarrow L_g \longrightarrow0.
L_{k,p}Use in source · 6 passages
Search aliases: L_k,p
Use in source · Stratified_Universality_Traditional.tex, line 4438
Define M_k,p(a)=M_k(a)⊗_ Z Z_p, L_k,p(a)=L_k(a)⊗_ Z Z_p.
Exact mathematical passage
M_{k,p}(a)=M_k(a)\otimes_{\mathbb Z}\mathbb Z_p, \qquad L_{k,p}(a)=L_k(a)\otimes_{\mathbb Z}\mathbb Z_p.
L_NUse in source · 1 passage
Search aliases: L_N
Use in source · Stratified_Universality_Traditional.tex, line 5702
For a fixed modulus N≥2, set B_N(M) = max({1}∪ {p^v_p(M):pmidgcd(M,N)}). The least packet size for a left inverse over Z/N Z is L_N(M,d)=min(d+1,B_N(M)). The exponents v_p(N) do not change the minimum; the first residue layer already detects the obstruction.
Exact mathematical passage
\boxed{L_N(M,d)=\min(d+1,B_N(M)).}
L_nUse in source · 6 passages
Search aliases: L_n
Use in source · Stratified_Universality_Traditional.tex, line 1691
[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
a_{\sigma\tau}=a_\sigma\,\sigma(a_\tau), \qquad a_\sigma\in GL_n(L),
L_{N,j}Use in source · 5 passages
Search aliases: L_N,j
Use in source · block09_section.tex, line 363
Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j measures cancellation relative to that construction. It is not intrinsic until the construction has been fixed.
Exact mathematical passage
L_{N,j}
L_{N,j}^{\rm char}Use in source · 2 passages
Search aliases: L_N,j^rm char rm
Use in source · block09_section.tex, line 320
qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r_N,j^rm char. For a zero character packet put r_N,j^rm char=0.
Exact mathematical passage
L_{N,j}^{\rm char}
L_{N-1}Use in source · 1 passage
Search aliases: L_N-1
Use in source · block09_section.tex, line 315
der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r_N,j^rm char. For a zero character packet put r_N,j^rm c…
Exact mathematical passage
L_N^{\rm orb}=\operatorname{LCLM}(L_0,\ldots,L_{N-1}), \qquad r_N^{\rm orb}=\operatorname{ord}L_N^{\rm orb},
L_N^{\rm orb}Use in source · 2 passages
Search aliases: L_N^rm orb rm
Use in source · block09_section.tex, line 315
der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r_N,j^rm char. For a zero character packet put r_N,j^rm c…
Exact mathematical passage
L_N^{\rm orb}=\operatorname{LCLM}(L_0,\ldots,L_{N-1}), \qquad r_N^{\rm orb}=\operatorname{ord}L_N^{\rm orb},
L_qUse in source · 2 passages
Search aliases: L_q
Use in source · block09_section.tex, line 314
The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r…
Exact mathematical passage
L_q\in E(t)\langle D_t\rangle
L_wUse in source · 1 passage
Search aliases: L_w
Use in source · Stratified_Universality_Traditional.tex, line 3119
For uniqueness, let (V,(L_d),v,λ) be another reachable-observable realization of the same digit function. Send the reachable vector L_wv to Λ_wF. If a linear combination of reachable vectors vanishes, applying every continuation and then λ shows that the corresponding combination of Cartier sections has every coefficient zero. Observability makes the assignment well defined. Reachability makes it surjective, and the symmetric construction gives an inverse. The intertwining, initial-vector, and o…
Exact mathematical passage
L_wv
\lceilUse in source · 2 passages
Search aliases: lceil
Use in source · Stratified_Universality_Traditional.tex, line 2570
Let R(z)=fracA(z)B(z), A(0)=B(0)=1, with a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source reference]. It computes the coefficient sequence exactly.
Exact mathematical passage
\label{eq:cartier-degree-bound-block03} D=\max\left\{b,\left\lceil\frac{c}{p-1}\right\rceil\right\}.
\operatorname{LCLM}Use in source · 1 passage
Search aliases: LCLM operatorname
Use in source · block09_section.tex, line 315
der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r_N,j^rm char. For a zero character packet put r_N,j^rm c…
Exact mathematical passage
L_N^{\rm orb}=\operatorname{LCLM}(L_0,\ldots,L_{N-1}), \qquad r_N^{\rm orb}=\operatorname{ord}L_N^{\rm orb},
\lfloorUse in source · 4 passages
Search aliases: lfloor
Use in source · Stratified_Universality_Traditional.tex, line 3051
If deg P≤ D, then deg(PH) ≤ D+a+(p-2)b. Cartier selection divides the retained degrees by p. The definition of D gives deg T_d(P) ≤ lfloorfracD+a+(p-2)bprfloor ≤ D. This proves (i).
Exact mathematical passage
\deg T_d(P) \leq \left\lfloor\frac{D+a+(p-2)b}{p}\right\rfloor \leq D.
\mathbf{Lin}Use in source · 2 passages
Search aliases: Lin mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3782
[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence. Then Reg_k(K) ≃ Sec_k^can(K) ≃ Lin_k,K^sl. Every normalized fiber over a fixed sequence is contractible. Consequently the associated finite presenta…
Exact mathematical passage
\mathbf{Lin}_{k,K}^{\mathrm{sl}}
\mathbf{Lin}_{k,K}^{\mathrm{sl}}Use in source · 2 passages
Search aliases: Lin_k,K^sl mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3782
[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence. Then Reg_k(K) ≃ Sec_k^can(K) ≃ Lin_k,K^sl. Every normalized fiber over a fixed sequence is contractible. Consequently the associated finite presenta…
Exact mathematical passage
\mathbf{Lin}_{k,K}^{\mathrm{sl}}
\linrankUse in source · 1 passage
Search aliases: linrank operatorname
Use in source · Stratified_Universality_Traditional.tex, line 2155
For a persistent C-finite sequence, the minimal dimension ρ(a)=linrank(a)=rank H(a) is simultaneously the minimal recurrence order, the reduced denominator degree, and the dimension of a minimal reachable-observable state.
Exact mathematical passage
\rho(a)=\linrank(a)=\rank H(a)
\mathbf{Loc}Use in source · 2 passages
Search aliases: Loc mathbf
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\mathbf{Loc}^{\mathrm{cts}}_pUse in source · 2 passages
Search aliases: Loc^cts_p mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\logUse in source · 1 passage
Search aliases: log
Use in source · block19_admissibility_lifting_final.tex, line 79
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
D_Q(T) := \frac{\sigma_{\max}(B_Q(T))}{\sigma_{\min}(B_Q(T))} \geq1, \qquad \Delta_Q(T):=\log D_Q(T).
\longleftrightarrowUse in source · 1 passage
Search aliases: longleftrightarrow
Use in source · block12_section.tex, line 49
[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P_H≥ P_J. Thus the subgroup poset, the intermediate-field poset, and the faithful fixed-projector poset form three normalized charts of one finite semisimple Ga…
Exact mathematical passage
\boxed{ H \quad\longleftrightarrow\quad L^H \quad\longleftrightarrow\quad P_H } \tag{12.3}
\lVertUse in source · 3 passages
Search aliases: lVert
Use in source · block11_section.tex, line 35
Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.
Exact mathematical passage
a=\lVert s\rVert, \qquad b_m=\lVert A_m\rVert, \qquad \varphi(t)=\sum_{m=2}^{D}b_mt^m.
\mathfrak MUse in source · 9 passages
Search aliases: M mathfrak
Use in source · block09_section.tex, line 254
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\mathfrak M_{N,j}F(x) = \sum_{m\geq0}a_{Nm+j}x^m.
M'Use in source · 2 passages
Search aliases: M'
Use in source · block17_semantic_quotients_final.tex, line 241
[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.
Exact mathematical passage
M\subseteq M',\qquad \mathcal O\subseteq\mathcal O' \quad\Longrightarrow\quad \Theta_{M',\mathcal O'}\subseteq\Theta_{M,\mathcal O},
\mathcal M^{-1}Use in source · 1 passage
Search aliases: M^-1 mathcal
Use in source · block15_section.tex, line 909
s inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix composition law. Let M:= M_p^-1:C( Z_p, Z_p) xrightarrow∼c_0( Z_p) be Mahler analysis from [source reference], and write R_p^a,sf(q):=f(p^aq+s). Conjugation gives the exact coefficient operator T_p^a,s := M R_p^a,s M^-1: c_0( Z_p)⟶ c_0( Z_p), whose affine kernel is computed in [source reference]. Its mixed-radix composition is [source reference].
Exact mathematical passage
\mathcal T_{p^a,s} :=\mathcal M R_{p^a,s}\mathcal M^{-1}: c_0(\mathbb Z_p)\longrightarrow c_0(\mathbb Z_p),
M^2Use in source · 1 passage
Search aliases: M^2
Use in source · Stratified_Universality_Traditional.tex, line 5656
[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac…
Exact mathematical passage
1,M,M^2,\ldots,M^d.
M^dUse in source · 1 passage
Search aliases: M^d
Use in source · Stratified_Universality_Traditional.tex, line 5656
[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac…
Exact mathematical passage
1,M,M^2,\ldots,M^d.
M^{d(d+1)/2}Use in source · 2 passages
Search aliases: M^d(d+1)/2
Use in source · Stratified_Universality_Traditional.tex, line 5660
[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac…
Exact mathematical passage
\boxed{\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}
M^kUse in source · 1 passage
Search aliases: M^k
Use in source · block11_section.tex, line 242
[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.
Exact mathematical passage
N_{n,M}=\sum_{k=1}^{n}S(n,k)M^k.
M^{r_B,r_A}_{B,A}Use in source · 1 passage
Search aliases: M^r_B,r_A_B,A
Use in source · block16_section_final.tex, line 187
_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, mathscr M^r_B,r_A_B,A), and composition ⋆ is isomorphic, by analysis and synthesis, to the corresponding category of lane-decorated descent objects and their underlying descend…
Exact mathematical passage
\Gamma\!\left(\mathscr G, \mathscr M^{r_B,r_A}_{B,A}\right),
M^{v,r}_{Z,X}Use in source · 1 passage
Search aliases: M^v,r_Z,X
Use in source · block16_section_final.tex, line 176
item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, maths…
Exact mathematical passage
\mathscr M^{v,r}_{Z,X}
M^{v,u}_{Z,Y}Use in source · 1 passage
Search aliases: M^v,u_Z,Y
Use in source · block16_section_final.tex, line 164
item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, maths…
Exact mathematical passage
h\in\Gamma(\mathscr G,\mathscr M^{u,r}_{Y,X}), \qquad k\in\Gamma(\mathscr G,\mathscr M^{v,u}_{Z,Y}),
M_0Use in source · 1 passage
Search aliases: M_0
Use in source · Stratified_Universality_Traditional.tex, line 3752
[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).
Exact mathematical passage
\mathsf{SL}_{k,K}(a) = \bigl( U_a, M_0(a),\ldots,M_{k-1}(a), v(a), \ell(a) \bigr). \label{eq:block05-shortlex-record}
m_aUse in source · 8 passages
Search aliases: m_a
Use in source · Stratified_Universality_Traditional.tex, line 2260
nt-periodic For a sequence over F_p, the following are equivalent. [label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0. If m_a(x) is the monic minimal recurrence polynomial and T=T(a), then m_a(x)mid x^T-1, T=min{N≥ 1:m_a(x)mid x^N-1}.
Exact mathematical passage
m_a(x)
M_dUse in source · 4 passages
Search aliases: M_d
Use in source · Stratified_Universality_Traditional.tex, line 2082
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
(C,v,\lambda), \qquad C\in M_d(K),\quad v\in K^d,\quad \lambda\in(K^d)^*,
\mathcal M_dUse in source · 1 passage
Search aliases: M_d mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5641
The same integer kernel matrix has different categorical status after changing the coefficient doctrine. Fix d≥0 and write M_d(R)=R^d+1 for the formal truncated Mahler coefficient module. Let A_M,s^(d) =[Θ_M,s(r,k)]_0≤ k,r≤ d, where the target degree k is the row index.
Exact mathematical passage
\mathcal M_d(R)=R^{d+1}
M_{k-1}Use in source · 1 passage
Search aliases: M_k-1
Use in source · Stratified_Universality_Traditional.tex, line 3752
[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).
Exact mathematical passage
\mathsf{SL}_{k,K}(a) = \bigl( U_a, M_0(a),\ldots,M_{k-1}(a), v(a), \ell(a) \bigr). \label{eq:block05-shortlex-record}
M_NUse in source · 1 passage
Search aliases: M_N
Use in source · block18_carrier_observer_change_final.tex, line 300
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
M_N=I+S_z, \qquad f_N(n)=e_0^{\mathsf T}M_N^n e_0.
M_nUse in source · 1 passage
Search aliases: M_n
Use in source · block12_section.tex, line 381
A useful constructive form is obtained by choosing any matrix C∈ M_n(L) and setting T(C)=∑_σ∈ Ga_σσ(C). Then a_ττ(T(C))=T(C). Whenever T(C) is invertible, B=T(C)^-1 is a Hilbert 90 trivialization. Standard descent guarantees that such a C exists.
Exact mathematical passage
C\in M_n(L)
\mathfrak M_{N,j}Use in source · 6 passages
Search aliases: M_N,j mathfrak
Use in source · block09_section.tex, line 254
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\mathfrak M_{N,j}F(x) = \sum_{m\geq0}a_{Nm+j}x^m.
M_N^iUse in source · 1 passage
Search aliases: M_N^i
Use in source · block18_carrier_observer_change_final.tex, line 318
Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving the determinant formula.
Exact mathematical passage
e_0^{\mathsf T}M_N^i
M_N^jUse in source · 1 passage
Search aliases: M_N^j
Use in source · block18_carrier_observer_change_final.tex, line 318
Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving the determinant formula.
Exact mathematical passage
M_N^j e_0
M_N^nUse in source · 1 passage
Search aliases: M_N^n
Use in source · block18_carrier_observer_change_final.tex, line 300
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
M_N=I+S_z, \qquad f_N(n)=e_0^{\mathsf T}M_N^n e_0.
\mathcal M_p^{-1}Use in source · 1 passage
Search aliases: M_p^-1 mathcal
Use in source · block15_section.tex, line 902
). For f∈ C( Z_p, Z_p), with q∈ Z_p the branch coordinate, define Analysis_p^a(f) := (q↦ f(p^aq+s))_0≤ s<p^a. Piecewise synthesis on the disjoint clopen cosets s+p^a Z_p is its inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix composition law. Let M:= M_p^-1:C( Z_p, Z_p) xrightarrow∼c_0( Z_p) be Mahler analysis from [source reference], and write R_p^a,sf(q):=f(p^aq+s). Conjugation gives the exact coefficient operator T_p^a,s := M R_p^a,s M^…
Exact mathematical passage
\mathcal M:=\mathcal M_p^{-1}:C(\mathbb Z_p,\mathbb Z_p) \xrightarrow{\sim}c_0(\mathbb Z_p)
\mathfrak M_rUse in source · 3 passages
Search aliases: M_r mathfrak
Use in source · block15_section.tex, line 530
[Fixed-resolution matrix-category equivalence] The functor M_r: L_ R⟶ Mat_r( L_ R), X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences are compatible with every certified change of resolution.
Exact mathematical passage
\mathfrak M_r:\mathcal L_{\mathsf R}\longrightarrow \operatorname{Mat}_r(\mathcal L_{\mathsf R}), \qquad X\longmapsto[X;(X_i^r)_i], \qquad f\longmapsto([f]_{a i}^{r,r}),
\mathfrak M_sUse in source · 1 passage
Search aliases: M_s mathfrak
Use in source · block15_section.tex, line 544
L_ R⟶ Mat_r( L_ R), X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences are compatible with every certified change of resolution.
Exact mathematical passage
\Gamma_{sr}\mathfrak M_s=\mathfrak M_r.
M_wUse in source · 1 passage
Search aliases: M_w
Use in source · Stratified_Universality_Traditional.tex, line 3768
[Strict fixed-radix normal form] The record SL_k,K(a) is uniquely determined by a and reconstructs a exactly through a_[w]_k = ℓ(a)M_w(a)v(a). It has dimension r_k(a). Two regular sequences have the same strict shortlex record if and only if they are equal.
Exact mathematical passage
a_{[w]_k} = \ell(a)M_w(a)v(a).
M_YUse in source · 2 passages
Search aliases: M_Y
Use in source · Stratified_Universality_Traditional.tex, line 393
[Two-sort transfer grammar] Take two sorts X,Y, one arrow a:X→ Y, and no nontrivial relation. A realization is a single typed map M_XxrightarrowM_aM_Y. After collapsing to M_X⊕ M_Y, the vertex idempotents are coordinate packets and M_a is one off-diagonal transfer block. The smallest typed grammar therefore already contains the distinction between object packets and transfer channels.
Exact mathematical passage
M_X\xrightarrow{M_a}M_Y.
\mathbf{Mah}Use in source · 1 passage
Search aliases: Mah mathbf
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\mathbf{Mah}^{c_0}_pUse in source · 1 passage
Search aliases: Mah^c_0_p mathbf
Use in source · Stratified_Universality_Traditional.tex, line 5624
Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet. Every normalized fiber is contractible. The descent category is therefore equivalent to the discrete semantic category Loc^…
Exact mathematical passage
\boxed{ \mathbf{Loc}^{\mathrm{cts}}_p \simeq \mathbf{Mah}^{c_0}_p \simeq \mathbf{Branch}^{\mathrm{cts}}_{p,a} \simeq \mathbf{BranchMah}^{c_0}_{p,a}.}
\mathrm{mark}Use in source · 1 passage
Search aliases: mark mathrm
Use in source · block13_section.tex, line 35
The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate from neutral Tannakian reconstruction.
Exact mathematical passage
E_{\mathrm{mark}}
\MaskUse in source · 8 passages
Search aliases: Mask mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2456
Define Mask_p^×,per to be the groupoid of such reduced rational masks whose normalized product solution [source reference] has purely periodic coefficients. Morphisms preserve the reduced mask.
Exact mathematical passage
\Mask_p^{\times,\mathrm{per}}
\Mask_2^{\times,\mathrm{per}}Use in source · 1 passage
Search aliases: Mask_2^×,per Mask times mathrm mathbf times mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2797
Negative control: a nonunit periodic sequence. Over F_2, the word 01 gives F(z)=fracz1+z^2. Then fracF(z)F(z^2) = frac1+z^2z, which is Laurent rather than regular at the origin. The sequence remains purely periodic, C-finite, and automatic, but it is not an object of Mask_2^×,per. This verifies that the unit overlap condition is structural rather than cosmetic.
Exact mathematical passage
\Mask_2^{\times,\mathrm{per}}
\Mask_p^{\mathrm{rat}}Use in source · 2 passages
Search aliases: Mask_p^rat Mask mathrm mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3203
Reduced mask chart. An object of Mask_p^rat is a reduced rational function R(z)=fracA(z)B(z), A(0)=B(0)=1. Morphisms preserve the rational function. The semantics is R↦ M_p[R]. The inverse chart map is the canonical ratio F↦ F/F(z^p).
Exact mathematical passage
\Mask_p^{\mathrm{rat}}
\Mask_p^{\times,\mathrm{per}}Use in source · 4 passages
Search aliases: Mask_p^×,per Mask times mathrm mathbf times mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2456
Define Mask_p^×,per to be the groupoid of such reduced rational masks whose normalized product solution [source reference] has purely periodic coefficients. Morphisms preserve the reduced mask.
Exact mathematical passage
\Mask_p^{\times,\mathrm{per}}
\operatorname{Mat}_dUse in source · 1 passage
Search aliases: Mat_d operatorname
Use in source · block15_section.tex, line 862
[Typed holonomic realization] Let K be a characteristic-zero field and let the K-valued sequence (u_n) satisfy a linear recurrence with polynomial coefficients. Beyond a finite singular prefix, write the recurrence as a finite-dimensional rational nonautonomous state model v_n+1=A(n)v_n, A(n)∈Mat_d(K(n)). For a fixed integer N≥1 and residue 0≤ j<N, the lane state w_m=v_Nm+j evolves by the exact N-step rational transfer w_m+1 = A(Nm+j+N-1)⋯ A(Nm+j)w_m whenever the displayed factors are defined. E…
Exact mathematical passage
v_{n+1}=A(n)v_n, \qquad A(n)\in\operatorname{Mat}_d(K(n)).
\max_{\delta(s,b)=t}Use in source · 1 passage
Search aliases: max_δ(s,b)=t max delta
Use in source · block18_carrier_observer_change_final.tex, line 352
[Observer-conditioned convex packet transport] For every state t∈ S and direction x, h_C'_t(x) = max_δ(s,b)=th_C_s(A_bx). Repeated updates compose exactly with branch words. Within each fixed observer state, replacing a finite atom family by any other finite generating family of the same convex hull preserves all future value queries generated by linear branch actions.
Exact mathematical passage
h_{C'_t}(x) = \max_{\delta(s,b)=t}h_{C_s}(A_bx).
\max_{\lambda\in\operatorname{spec}(H)}Use in source · 1 passage
Search aliases: max_λ∈spec(H) max lambda in operatorname
Use in source · block19_admissibility_lifting_final.tex, line 103
[Coordinate covariance and composition] Simultaneous coordinate transport of T,Q_s,Q_t leaves D_Q(T) invariant. For composable invertible maps, D_Q(T_2T_1) ≤ D_Q(T_2)D_Q(T_1), so Δ_Q(T_2T_1) ≤ Δ_Q(T_2)+Δ_Q(T_1). For a closed word H, D_Q(H) ≥ fracmax_λ∈spec(H)|λ| min_λ∈spec(H)|λ|.
Exact mathematical passage
D_Q(H) \geq \frac{\max_{\lambda\in\operatorname{spec}(H)}|\lambda|} {\min_{\lambda\in\operatorname{spec}(H)}|\lambda|}.
\mathrm{min}Use in source · 4 passages
Search aliases: min mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2039
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\Rec_K^{\mathrm{min}}
\mathrm{min,rat}Use in source · 3 passages
Search aliases: min,rat mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3212
Minimal Cartier-module chart. An object of Cart_p^min,rat is a finite-dimensional tuple (V,(L_d)_d=0^p-1,v,λ) that is reachable and observable, whose digit coefficient function belongs to RMah_p^×. Morphisms are similarities preserving every L_d, the initial vector, and the output functional. Its normalized representative is C_p(F).
Exact mathematical passage
\mathbf{Cart}_p^{\mathrm{min,rat}}
\min_jUse in source · 1 passage
Search aliases: min_j min
Use in source · block09_section.tex, line 395
Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
v_j=v_\alpha(q_j), \qquad \mu=\min_j(v_j-j).
\min_{k\leq r\leq Mk+s}Use in source · 2 passages
Search aliases: min_k≤ r≤ Mk+s min leq leq
Use in source · Stratified_Universality_Traditional.tex, line 5734
[Exact branchwise valuation ledger] For every refined coefficient, v_p(d_s,k) ≥ min_k≤ r≤ Mk+s (v_p(c_r)+v_p(Θ_M,s(r,k))). If v_p(c_r)≥ w(r) for a nondecreasing weight w, then v_p(d_s,k)≥ w(k). Thus every declared monotone Mahler precision ladder survives affine polyphase refinement.
Exact mathematical passage
\boxed{ v_p(d_{s,k}) \geq \min_{k\leq r\leq Mk+s} \left(v_p(c_r)+v_p(\Theta_{M,s}(r,k))\right).}
\min_{\lambda\in\operatorname{spec}(H)}Use in source · 1 passage
Search aliases: min_λ∈spec(H) min lambda in operatorname
Use in source · block19_admissibility_lifting_final.tex, line 103
[Coordinate covariance and composition] Simultaneous coordinate transport of T,Q_s,Q_t leaves D_Q(T) invariant. For composable invertible maps, D_Q(T_2T_1) ≤ D_Q(T_2)D_Q(T_1), so Δ_Q(T_2T_1) ≤ Δ_Q(T_2)+Δ_Q(T_1). For a closed word H, D_Q(H) ≥ fracmax_λ∈spec(H)|λ| min_λ∈spec(H)|λ|.
Exact mathematical passage
D_Q(H) \geq \frac{\max_{\lambda\in\operatorname{spec}(H)}|\lambda|} {\min_{\lambda\in\operatorname{spec}(H)}|\lambda|}.
\mathrm{mod}Use in source · 2 passages
Search aliases: mod mathrm
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
\Mod_{\mathrm{ak}}Use in source · 1 passage
Search aliases: Mod_ak Mod mathrm operatorname mathrm
Use in source · Stratified_Universality_Traditional.tex, line 891
gintheorem[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right side is the category of typed realizations satisfying all declared relations and carrying every declared packet idempotent to a split idempotent.
Exact mathematical passage
\Fun_{\kk}^{\oplus} \bigl(\Syn_{\mathrm{ak}}(\mathbb K_{\rm lin}),\mathcal C\bigr) \simeq \Mod_{\mathrm{ak}}(\mathbb K_{\rm lin},\mathcal C),
\operatorname{MorKer}_{CB}Use in source · 1 passage
Search aliases: MorKer_CB operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1282
[Composition law for bridge loss] Let UxrightarrowB VxrightarrowC W be certified bridges. [label=textup(roman*)] item Every constructor or universal property preserved by both B and C is preserved by CB. item MorKer_B(X,Y) ⊆ MorKer_CB(X,Y). item Coll_B ⊆ Coll_CB. item EssIm(CB) ⊆ EssIm(C). item If both bridges are conservative, then their composite is conservative. If both are classification-faithful, then their composite is classification-faithful. If both are equivalences, then their composite…
Exact mathematical passage
\operatorname{MorKer}_B(X,Y) \subseteq \operatorname{MorKer}_{CB}(X,Y).
\mathbf nUse in source · 2 passages
Search aliases: n mathbf
Use in source · Stratified_Universality_Traditional.tex, line 4064
[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.
Exact mathematical passage
E_{\mathbf n}:V\longrightarrow\mathbb Q^\rho, \qquad f\longmapsto \bigl(f(n_1),\ldots,f(n_\rho)\bigr)
N^{(0)}Use in source · 2 passages
Search aliases: N^(0)
Use in source · block17_semantic_quotients_final.tex, line 178
[Finite certification] For finite X, descending partition refinement stabilizes after finitely many strict refinements. For finite-dimensional X over a field, the chain N^(0)=N_0, N^(r+1)= N_0∩⋂_a∈ Aρ(a)^-1(N^(r)) stabilizes after at most dim X strict dimension drops. At stabilization it equals N_ S.
Exact mathematical passage
N^{(0)}=N_0, \qquad N^{(r+1)}= N_0\cap\bigcap_{a\in A}\rho(a)^{-1}(N^{(r)})
N^{(r)}Use in source · 1 passage
Search aliases: N^(r)
Use in source · block17_semantic_quotients_final.tex, line 178
[Finite certification] For finite X, descending partition refinement stabilizes after finitely many strict refinements. For finite-dimensional X over a field, the chain N^(0)=N_0, N^(r+1)= N_0∩⋂_a∈ Aρ(a)^-1(N^(r)) stabilizes after at most dim X strict dimension drops. At stabilization it equals N_ S.
Exact mathematical passage
N^{(0)}=N_0, \qquad N^{(r+1)}= N_0\cap\bigcap_{a\in A}\rho(a)^{-1}(N^{(r)})
N^{(r+1)}Use in source · 1 passage
Search aliases: N^(r+1)
Use in source · block17_semantic_quotients_final.tex, line 178
[Finite certification] For finite X, descending partition refinement stabilizes after finitely many strict refinements. For finite-dimensional X over a field, the chain N^(0)=N_0, N^(r+1)= N_0∩⋂_a∈ Aρ(a)^-1(N^(r)) stabilizes after at most dim X strict dimension drops. At stabilization it equals N_ S.
Exact mathematical passage
N^{(0)}=N_0, \qquad N^{(r+1)}= N_0\cap\bigcap_{a\in A}\rho(a)^{-1}(N^{(r)})
n^2Use in source · 3 passages
Search aliases: n^2
Use in source · Stratified_Universality_Traditional.tex, line 3935
For a_n=n, one has r_k(a)=2 and exactly k^ℓ distinct sections at depth ℓ. For a_n=n^2, one has r_k(a)=3 and again exactly k^ℓ distinct sections at depth ℓ.
Exact mathematical passage
a_n=n^2
n^mUse in source · 3 passages
Search aliases: n^m
Use in source · Stratified_Universality_Traditional.tex, line 3929
Choose ℓ with k^ℓ>m and consider the m+1 sections P(k^ℓ n+r), 0≤ r≤ m. When expanded in the basis 1,n,…,n^m, their coefficient functions as functions of r have successive degrees m,m-1,…,0, with nonzero leading coefficients. Evaluation at the distinct points 0,1,…,m therefore gives an invertible generalized Vandermonde matrix. These sections are linearly independent, proving rank m+1.
Exact mathematical passage
1,n,\ldots,n^m
N^NUse in source · 1 passage
Search aliases: N^N
Use in source · block15_section.tex, line 726
P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( - b_N,j; fracz^NN^N ). Thus coefficient-residue, root-of-unity, and hypergeometric descriptions are exact additive coordinate realizations of one residue-address decomposition. The step-factorial identity is an exa…
Exact mathematical passage
E_{N,j}(z) = \frac{z^j}{j!} {}_0F_{N-1} \left( \begin{matrix}-\\ \mathbf b_{N,j}\end{matrix}; \frac{z^N}{N^N} \right).
N^{Nn}Use in source · 1 passage
Search aliases: N^Nn
Use in source · block15_section.tex, line 752
The projector laws follow coefficient by coefficient. The residue classes modulo NM lying over jpmod N are precisely j,j+N,…,j+(M-1)N, which proves refinement. Character orthogonality gives the Fourier formula. Finally, (Nn+j)! = j! N^Nn ∏_r=1^N(fracj+rN)_n. Exactly one factor in the product is (1)_n=n!. Removing it leaves the parameter set b_N,j and gives the hypergeometric series.
Exact mathematical passage
(Nn+j)! = j!\,N^{Nn} \prod_{r=1}^{N}\left(\frac{j+r}{N}\right)_n.
n^rUse in source · 4 passages
Search aliases: n^r
Use in source · Stratified_Universality_Traditional.tex, line 4945
[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plane-partition series. More generally, the power family converts the exponent rule n^r into the divisor-sum ghost σ_r+1.
Exact mathematical passage
\begin{array}{c|c|c} b_n & A_b(z) & c_n\\ \hline 1 & \displaystyle\prod_{n\geq1}(1-z^n)^{-1} & \sigma_1(n)\\[4pt] n & \displaystyle\prod_{n\geq1}(1-z^n)^{-n} & \sigma_2(n)\\[4pt] n^r & \displaystyle\prod_{n\geq1}(1-z^n)^{-n^r} & \sigma_{r+1}(n). \end{array}
N_{\mathbb C/\mathbb R}Use in source · 1 passage
Search aliases: N_ C/ R mathbb mathbb
Use in source · block12_section.tex, line 463
-1 · 0 . Then A_σσ(A_σ)=-I. The projective cocycle condition holds because -I is scalar. Its boundary is represented by -1 in R^×/N_ C/ R( C^×). Every complex norm is positive, so -1 is nontrivial. The descended central simple algebra is the Hamilton quaternion algebra. Replacing A_σ by 0 · 1
Exact mathematical passage
\mathbb R^\times/N_{\mathbb C/\mathbb R}(\mathbb C^\times).
N_0Use in source · 4 passages
Search aliases: N_0
Use in source · block17_semantic_quotients_final.tex, line 112
Let K_0:=⋂_jkerpair(o_j) in the relational setting and N_0:=⋂_jker(o_j) in the linear setting.
Exact mathematical passage
N_0:=\bigcap_j\ker(o_j)
N_1Use in source · 2 passages
Search aliases: N_1
Use in source · block17_semantic_quotients_final.tex, line 426
[Composition of task loss] For N_1⊆ N_2⊆ N_3, there is an exact sequence 0⟶ N_2/N_1 ⟶ N_3/N_1 ⟶ N_3/N_2 ⟶0. Hence task loss composes as extension data rather than as a canonically split sum.
Exact mathematical passage
N_1\subseteq N_2\subseteq N_3,
n_1Use in source · 4 passages
Search aliases: n_1
Use in source · Stratified_Universality_Traditional.tex, line 4060
[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.
Exact mathematical passage
0\leq n_1<\cdots<n_\rho
N_2Use in source · 2 passages
Search aliases: N_2
Use in source · block17_semantic_quotients_final.tex, line 426
[Composition of task loss] For N_1⊆ N_2⊆ N_3, there is an exact sequence 0⟶ N_2/N_1 ⟶ N_3/N_1 ⟶ N_3/N_2 ⟶0. Hence task loss composes as extension data rather than as a canonically split sum.
Exact mathematical passage
N_1\subseteq N_2\subseteq N_3,
N_3Use in source · 2 passages
Search aliases: N_3
Use in source · block17_semantic_quotients_final.tex, line 426
[Composition of task loss] For N_1⊆ N_2⊆ N_3, there is an exact sequence 0⟶ N_2/N_1 ⟶ N_3/N_1 ⟶ N_3/N_2 ⟶0. Hence task loss composes as extension data rather than as a canonically split sum.
Exact mathematical passage
N_1\subseteq N_2\subseteq N_3,
N_aUse in source · 3 passages
Search aliases: N_a
Use in source · block09_section.tex, line 63
Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
Exact mathematical passage
N_a
n_jUse in source · 1 passage
Search aliases: n_j
Use in source · Stratified_Universality_Traditional.tex, line 4074
The coordinate functionals on Q^ N separate points of V. Choose n_1 for which some element of V has nonzero value. If n_1,…,n_j have been chosen and the joint evaluation map has rank j<ρ, its kernel contains a nonzero f. Choose n_j+1 with f(n_j+1)≠0. The rank increases. After ρ steps, the evaluation map has full rank.
Exact mathematical passage
n_1,\ldots,n_j
n_{j+1}Use in source · 2 passages
Search aliases: n_j+1
Use in source · Stratified_Universality_Traditional.tex, line 4074
The coordinate functionals on Q^ N separate points of V. Choose n_1 for which some element of V has nonzero value. If n_1,…,n_j have been chosen and the joint evaluation map has rank j<ρ, its kernel contains a nonzero f. Choose n_j+1 with f(n_j+1)≠0. The rank increases. After ρ steps, the evaluation map has full rank.
Exact mathematical passage
n_{j+1}
N_{n,M}Use in source · 1 passage
Search aliases: N_n,M
Use in source · block11_section.tex, line 242
[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.
Exact mathematical passage
N_{n,M}=\sum_{k=1}^{n}S(n,k)M^k.
N_sUse in source · 5 passages
Search aliases: N_s
Use in source · block17_semantic_quotients_final.tex, line 405
[Task-change exact sequence] Let a stronger task have behavioral kernel N_s, and let a weaker task have behavioral kernel N_w, with N_s⊆ N_w. Then 0⟶ N_w/N_s ⟶ X/N_s ⟶ X/N_w ⟶0 is exact. Thus N_w/N_s is the exact semantic information discarded when the task is weakened.
Exact mathematical passage
N_s
N_{\mathrm{strong}}Use in source · 1 passage
Search aliases: N_strong mathrm
Use in source · block19_admissibility_lifting_final.tex, line 307
Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders and should not be merged into one scalar notion of a stronger contract.
Exact mathematical passage
N_{\mathrm{strong}}\subseteq N_{\mathrm{weak}}, \qquad Q_{\mathrm{strong}}\twoheadrightarrow Q_{\mathrm{weak}}.
N_wUse in source · 5 passages
Search aliases: N_w
Use in source · block17_semantic_quotients_final.tex, line 405
[Task-change exact sequence] Let a stronger task have behavioral kernel N_s, and let a weaker task have behavioral kernel N_w, with N_s⊆ N_w. Then 0⟶ N_w/N_s ⟶ X/N_s ⟶ X/N_w ⟶0 is exact. Thus N_w/N_s is the exact semantic information discarded when the task is weakened.
Exact mathematical passage
N_w
N_{\mathrm{weak}}Use in source · 1 passage
Search aliases: N_weak mathrm
Use in source · block19_admissibility_lifting_final.tex, line 307
Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders and should not be merged into one scalar notion of a stronger contract.
Exact mathematical passage
N_{\mathrm{strong}}\subseteq N_{\mathrm{weak}}, \qquad Q_{\mathrm{strong}}\twoheadrightarrow Q_{\mathrm{weak}}.
N_{\lambda,a}Use in source · 4 passages
Search aliases: N_λ,a lambda
Use in source · block13_section.tex, line 86
Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.
Exact mathematical passage
N_{\lambda,a}:\mathcal C_{\lambda,a}\longrightarrow\mathcal O_\lambda
n_\rhoUse in source · 2 passages
Search aliases: n_ρ rho
Use in source · Stratified_Universality_Traditional.tex, line 4060
[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.
Exact mathematical passage
0\leq n_1<\cdots<n_\rho
\operatorname{Ncl}Use in source · 4 passages
Search aliases: Ncl operatorname
Use in source · block12_section.tex, line 136
If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.
Exact mathematical passage
\operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1}, \qquad \operatorname{Ncl}_G(H)=\left\langle gHg^{-1}:g\in G\right\rangle.
\operatorname{Ncl}_GUse in source · 4 passages
Search aliases: Ncl_G operatorname
Use in source · block12_section.tex, line 136
If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.
Exact mathematical passage
\operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1}, \qquad \operatorname{Ncl}_G(H)=\left\langle gHg^{-1}:g\in G\right\rangle.
\mathrm{ncol}Use in source · 1 passage
Search aliases: ncol mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3328
The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside RMah_p,col^×.
Exact mathematical passage
\RMah_p^{\times} = \RMah_{p,\mathrm{col}}^{\times} \sqcup \RMah_{p,\mathrm{ncol}}^{\times}.
\nmidUse in source · 1 passage
Search aliases: nmid
Use in source · Stratified_Universality_Traditional.tex, line 4422
[Good-prime transfer] If pnmidδ_k(a), then for every r≥1, M_k(a)/p^rM_k(a) overset∼⟶ overline M_k,p^r(a) = L_k(a)/p^rL_k(a). At a good prime, the raw and saturated prime-power towers agree exactly.
Exact mathematical passage
p\nmid\delta_k(a)
\notUse in source · 1 passage
Search aliases: not
Use in source · block11_section.tex, line 470
Take u=z+u^q, q≥2. tag11.14 Lagrange inversion gives the exact coefficient formula [z^n]u = displaystyle frac1(q-1)m+1binomqmm, · n=1+(q-1)m, [9pt] 0, · nnot≡1pmodq-1. tag11.15 The nonzero coefficients are the Fuss--Catalan numbers of full plane q-ary trees.
Exact mathematical passage
\boxed{ [z^n]u = \begin{cases} \displaystyle \frac{1}{(q-1)m+1}\binom{qm}{m}, &n=1+(q-1)m,\\[9pt] 0,&n\not\equiv1\pmod{q-1}. \end{cases}} \tag{11.15}
OUse in source · 18 passages
Search aliases: O
Use in source · Stratified_Universality_Traditional.tex, line 2538
[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart [4pt] Reduced OGF ↔ rational mask · equivalence on the unit-periodic essential image · F(0)=1, periodic rational F, reduced regular mask R · exact se…
Exact mathematical passage
4pt] Reduced OGF \(\leftrightarrow\) minimal DFAO & equivalence on the periodic essential image & unit purely periodic series and reachable minimal LSD-first \(p\)-DFAOs & coefficient semantics, minimal period, canonical \(p\)-kernel, output behavior & equality of represented sequences & a nonminimal residue-pair automaton is quotiented before entering the normalized chart\\[4pt] Reduced OGF \(\leftrightarrow\) rational mask & equivalence on the unit-periodic essential image & \(F(0)=1\), periodic rational \(F\), reduced regular mask \(R\) & exact series, functional equation, reduced rational mask, minimal period & equality of series and masks & undefined for nonunit series; unrestricted rational masks may have nonperiodic products\\[4pt] Mask \(\to\) Cartier numerator representation & constructor & reduced \(A/B\) over \(\mathbb F_p\) & exact coefficient semantics, digit matrices, theorem degree bound & not generally reflective & introduces basis and denominator coordinates and may contain unreachable ambient states\\[4pt] Ambient Cartier compiler \(\to\) minimal DFAO & reachable restriction & full finite numerator-state compiler & every reachable coefficient section and every future digit behavior & equality of the represented \(p\)-kernel states & removes only ambient states outside the orbit of the initial numerator\\[4pt] Unrestricted automatic \(\dashrightarrow\) rational mask & partial bridge only & automatic sequences whose unit inverse mask is rational and whose product lies in the declared periodic class & coefficient semantics on the domain of definition & no unrestricted reflection theorem & no claim that every automatic sequence has a rational periodic mask\\ \end{longtable} \end{landscape} \subsubsection{The rational-to-digital construction} For a periodic word \(w\) of length \(T\), construct the finite pair automaton from \cref{eq:pair-transition}, then quotient by equality of the induced subsequences. This gives the canonical minimal DFAO directly. Conversely, a minimal DFAO in the selected chart is promised to produce a purely periodic sequence. Its minimal period can be recovered exactly by enumerating \(N=1,2,\ldots\), constructing the finite-state transducer for addition by \(N\), and testing equivalence between the original output and its \(N\)-shift. Equivalence of finite deterministic output automata is a finite product-graph calculation. The promise of pure periodicity guarantees termination. Once \(T\) is known, \cref{eq:periodic-ogf-unreduced} gives the rational series, which is then reduced. This inverse is effective but not claimed to be complexity optimal. The point is effectivity, not a sharp automata algorithm. \subsubsection{The rational-mask compiler lane} Let \[ R(z)=\frac{A(z)}{B(z)}, \qquad A(0)=B(0)=1,
\mathcal O'Use in source · 2 passages
Search aliases: O' mathcal
Use in source · block17_semantic_quotients_final.tex, line 241
[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.
Exact mathematical passage
M\subseteq M',\qquad \mathcal O\subseteq\mathcal O' \quad\Longrightarrow\quad \Theta_{M',\mathcal O'}\subseteq\Theta_{M,\mathcal O},
O^2Use in source · 4 passages
Search aliases: O^2
Use in source · block10_section.tex, line 358
[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.
Exact mathematical passage
\boxed{ E=1+2xEO, \qquad O=E^2+xO^2.}
O^3Use in source · 1 passage
Search aliases: O^3
Use in source · block10_section.tex, line 369
[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.
Exact mathematical passage
\boxed{4x^3O^4-8x^2O^3+5xO^2-O+1=0.}
O^4Use in source · 2 passages
Search aliases: O^4
Use in source · block10_section.tex, line 369
[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.
Exact mathematical passage
\boxed{4x^3O^4-8x^2O^3+5xO^2-O+1=0.}
\mathcal O^{\mathrm{raw}}_{k,p^r}Use in source · 3 passages
Search aliases: O^raw_k,p^r mathcal mathrm
Use in source · Stratified_Universality_Traditional.tex, line 4523
Let O_k(a)={Λ_w a:w∈Σ_k^*}⊆ M_k(a) be the actual section orbit, and let O^raw_k,p^r(a) be its image in M_k(a)/p^rM_k(a).
Exact mathematical passage
\mathcal O^{\mathrm{raw}}_{k,p^r}(a)
\mathcal O^{\mathrm{raw}}_{k,p^{r+1}}Use in source · 1 passage
Search aliases: O^raw_k,p^r+1 mathcal mathrm
Use in source · Stratified_Universality_Traditional.tex, line 4531
[Raw orbit-closure descent] The reduction maps O^raw_k,p^r+1(a) ↠ O^raw_k,p^r(a) are surjective, and overline O_k(a)^ p overset∼⟶ varprojlim_r O^raw_k,p^r(a), where the closure is taken in M_k,p(a).
Exact mathematical passage
\mathcal O^{\mathrm{raw}}_{k,p^{r+1}}(a) \twoheadrightarrow \mathcal O^{\mathrm{raw}}_{k,p^r}(a)
\mathcal O_{13}Use in source · 2 passages
Search aliases: O_13 mathcal
Use in source · block13_section.tex, line 67
The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category. The tag records which semantic doctrine is in force.
Exact mathematical passage
\mathcal O_{13} = \mathcal O_{\mathrm{DRLM}} \sqcup \mathcal O_{\mathrm{Loc}} \sqcup \mathcal O_{\mathrm{Fuss}} \sqcup \mathcal O_{\mathrm{EM}} \sqcup \mathcal O_{\mathrm{Gal}}.
\mathcal O_{\mathrm{DRLM}}Use in source · 4 passages
Search aliases: O_DRLM mathcal mathrm
Use in source · block13_section.tex, line 67
The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category. The tag records which semantic doctrine is in force.
Exact mathematical passage
\mathcal O_{13} = \mathcal O_{\mathrm{DRLM}} \sqcup \mathcal O_{\mathrm{Loc}} \sqcup \mathcal O_{\mathrm{Fuss}} \sqcup \mathcal O_{\mathrm{EM}} \sqcup \mathcal O_{\mathrm{Gal}}.
\mathcal O_{\mathrm{EM}}Use in source · 2 passages
Search aliases: O_EM mathcal mathrm
Use in source · block13_section.tex, line 67
The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category. The tag records which semantic doctrine is in force.
Exact mathematical passage
\mathcal O_{13} = \mathcal O_{\mathrm{DRLM}} \sqcup \mathcal O_{\mathrm{Loc}} \sqcup \mathcal O_{\mathrm{Fuss}} \sqcup \mathcal O_{\mathrm{EM}} \sqcup \mathcal O_{\mathrm{Gal}}.
\mathcal O_{\mathrm{Fuss}}Use in source · 2 passages
Search aliases: O_Fuss mathcal mathrm
Use in source · block13_section.tex, line 67
The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category. The tag records which semantic doctrine is in force.
Exact mathematical passage
\mathcal O_{13} = \mathcal O_{\mathrm{DRLM}} \sqcup \mathcal O_{\mathrm{Loc}} \sqcup \mathcal O_{\mathrm{Fuss}} \sqcup \mathcal O_{\mathrm{EM}} \sqcup \mathcal O_{\mathrm{Gal}}.
\mathcal O_{\mathrm{Gal}}Use in source · 2 passages
Search aliases: O_Gal mathcal mathrm
Use in source · block13_section.tex, line 67
The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category. The tag records which semantic doctrine is in force.
Exact mathematical passage
\mathcal O_{13} = \mathcal O_{\mathrm{DRLM}} \sqcup \mathcal O_{\mathrm{Loc}} \sqcup \mathcal O_{\mathrm{Fuss}} \sqcup \mathcal O_{\mathrm{EM}} \sqcup \mathcal O_{\mathrm{Gal}}.
\mathcal O_{\mathrm{Loc}}Use in source · 3 passages
Search aliases: O_Loc mathcal mathrm
Use in source · block13_section.tex, line 67
The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category. The tag records which semantic doctrine is in force.
Exact mathematical passage
\mathcal O_{13} = \mathcal O_{\mathrm{DRLM}} \sqcup \mathcal O_{\mathrm{Loc}} \sqcup \mathcal O_{\mathrm{Fuss}} \sqcup \mathcal O_{\mathrm{EM}} \sqcup \mathcal O_{\mathrm{Gal}}.
\mathcal O_NUse in source · 1 passage
Search aliases: O_N mathcal
Use in source · block09_section.tex, line 569
Let ω_N be primitive and consider the rotated orbit O_N(F_λ,α) = {(1-αω_N^rt)^-λ:0≤ r<N}. Let r_rm orb be its dimension over the constant field.
Exact mathematical passage
\mathcal O_N(F_{\lambda,\alpha}) = \left \{(1-\alpha\omega_N^rt)^{-\lambda}:0\leq r<N\right\}.
\mathcal O_qUse in source · 4 passages
Search aliases: O_q mathcal
Use in source · block16_section_final.tex, line 606
=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
\mathcal O_q = \left\{j\in\mathbb Z/N\mathbb Z: \frac{N}{\gcd(N,j)}=q\right\}.
\mathcal O_UUse in source · 4 passages
Search aliases: O_U mathcal
Use in source · block16_section_final.tex, line 374
[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.
Exact mathematical passage
\mathcal O_U\subseteq I_{A,U,r}
\mathcal O_\lambdaUse in source · 2 passages
Search aliases: O_λ mathcal lambda
Use in source · block13_section.tex, line 86
Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.
Exact mathematical passage
N_{\lambda,a}:\mathcal C_{\lambda,a}\longrightarrow\mathcal O_\lambda
\Ob_{D'}Use in source · 1 passage
Search aliases: Ob_D' Ob operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1567
U' · xrightarrowD' · V', bD=D'a, there is a unique induced map overline b:coker D⟶coker D' with overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\overline b\,\Ob_D(r)=\Ob_{D'}(br).
\operatorname{ord}_1Use in source · 1 passage
Search aliases: ord_1 operatorname
Use in source · Stratified_Universality_Traditional.tex, line 2391
The pair model gives a useful finite bound. Write T=p^ν M, gcd(M,p)=1, and set ord_1(p)=1.
Exact mathematical passage
\operatorname{ord}_1(p)=1
\operatorname{ord}_MUse in source · 3 passages
Search aliases: ord_M operatorname
Use in source · Stratified_Universality_Traditional.tex, line 2396
[Exact multiplier-orbit size] The set U_p(T) has cardinality |U_p(T)| = ν+ord_M(p). Consequently, | K_p(a)| ≤ T(ν+ord_M(p)).
Exact mathematical passage
\label{eq:u-size} |U_p(T)| = \nu+\operatorname{ord}_M(p).
\operatorname{ord}_{M}Use in source · 1 passage
Search aliases: ord_M operatorname
Use in source · Stratified_Universality_Traditional.tex, line 2698
For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)| ≤ T(a)(ν_p(T(a))+ord_M(p)), R_a(z) · =fracF_a(z)F_a(z^p), T(a)=p^ν_p(T(a))M, gcd(M,p)=1.
Exact mathematical passage
\label{eq:periodic-invariant-identities} \begin{aligned} \deg m_a &=\deg Q_a =\rank H(a) =\dim \State_{\min}(a),\\ T(a) &=\min\{N:m_a\mid x^N-1\} =\min\{N:Q_a\mid 1-z^N\},\\ |\mathcal A_p(a)| &=|\mathcal K_p(a)| \leq T(a)\bigl(\nu_p(T(a))+\operatorname{ord}_{M}(p)\bigr),\\ R_a(z) &=\frac{F_a(z)}{F_a(z^p)}, \qquad T(a)=p^{\nu_p(T(a))}M, \quad \gcd(M,p)=1. \end{aligned}
\overline AUse in source · 1 passage
Search aliases: overline A overline
Use in source · Stratified_Universality_Traditional.tex, line 5666
e det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac dβ_prfloor+1. A local channel loses residual rank exactly at primes dividing M. item A complete neighboring packet can recover the source…
Exact mathematical passage
\boxed{ \rank_{\mathbb F_p}\overline A_{M,s}^{(d)} =\left\lfloor\frac d{\beta_p}\right\rfloor+1.}
\overline A_{M,s}^{(d)}Use in source · 1 passage
Search aliases: overline A_M,s^(d) overline
Use in source · Stratified_Universality_Traditional.tex, line 5666
e det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac dβ_prfloor+1. A local channel loses residual rank exactly at primes dividing M. item A complete neighboring packet can recover the source…
Exact mathematical passage
\boxed{ \rank_{\mathbb F_p}\overline A_{M,s}^{(d)} =\left\lfloor\frac d{\beta_p}\right\rfloor+1.}
\overline BUse in source · 3 passages
Search aliases: overline B overline
Use in source · block11_section.tex, line 56
[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applied to the ordered values of its children.
Exact mathematical passage
\overline B_R
\overline bUse in source · 9 passages
Search aliases: overline b overline
Use in source · Stratified_Universality_Traditional.tex, line 1563
U' · xrightarrowD' · V', bD=D'a, there is a unique induced map overline b:coker D⟶coker D' with overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\overline b:\coker D\longrightarrow\coker D'
\overline B_RUse in source · 3 passages
Search aliases: overline B_R overline
Use in source · block11_section.tex, line 56
[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applied to the ordered values of its children.
Exact mathematical passage
\overline B_R
\overline b_tUse in source · 3 passages
Search aliases: overline b_t overline
Use in source · block15_section.tex, line 321
[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1 κ(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈Γ:overline b_t=0}. Then |S_0|-|supp(overline b)| = |A_0| +(|S_1|-|Γ|) +|K_0|. The three terms record, respectively, annihilation by the local maps τ_e…
Exact mathematical passage
\overline b_t := \sum_{\substack{e\in S_1\\\kappa(e)=t}}\tau_e(b_e) \in C_t,
\overline gUse in source · 5 passages
Search aliases: overline g overline
Use in source · block18_carrier_observer_change_final.tex, line 145
[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.
Exact mathematical passage
Q_X\xrightarrow{\overline f}Q_Y\xrightarrow{\overline g}Q_Z,
\overline KUse in source · 3 passages
Search aliases: overline K overline
Use in source · block10_section.tex, line 6
Throughout this section, K is a characteristic-zero field and overline K is an algebraic closure. Algebraic degree always means degree over the displayed rational-function field. All assertions about formal branches are made inside overline K[[z]] unless a different local field is named.
Exact mathematical passage
\overline K
\overline M_{k,m}Use in source · 2 passages
Search aliases: overline M_k,m overline
Use in source · Stratified_Universality_Traditional.tex, line 4162
For an integer m≥2, the raw coefficient reduction is M_k(a)/mM_k(a), while the semantic reduction inside the saturated lattice is overline M_k,m(a) = im ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). These need not agree.
Exact mathematical passage
\overline M_{k,m}(a) = \operatorname{im} \left( M_k(a)/mM_k(a) \longrightarrow L_k(a)/mL_k(a) \right).
\overline{\mathcal O_k(a)}Use in source · 1 passage
Search aliases: overline O_k(a) overline mathcal
Use in source · Stratified_Universality_Traditional.tex, line 4537
[Raw orbit-closure descent] The reduction maps O^raw_k,p^r+1(a) ↠ O^raw_k,p^r(a) are surjective, and overline O_k(a)^ p overset∼⟶ varprojlim_r O^raw_k,p^r(a), where the closure is taken in M_k,p(a).
Exact mathematical passage
\overline{\mathcal O_k(a)}^{\,p} \overset{\sim}{\longrightarrow} \varprojlim_r \mathcal O^{\mathrm{raw}}_{k,p^r}(a),
\overline{\mathcal O_k(a)}^{\,p}Use in source · 1 passage
Search aliases: overline O_k(a)^ p overline mathcal
Use in source · Stratified_Universality_Traditional.tex, line 4537
[Raw orbit-closure descent] The reduction maps O^raw_k,p^r+1(a) ↠ O^raw_k,p^r(a) are surjective, and overline O_k(a)^ p overset∼⟶ varprojlim_r O^raw_k,p^r(a), where the closure is taken in M_k,p(a).
Exact mathematical passage
\overline{\mathcal O_k(a)}^{\,p} \overset{\sim}{\longrightarrow} \varprojlim_r \mathcal O^{\mathrm{raw}}_{k,p^r}(a),
\overline{\mathcal O_k(s_k)}Use in source · 1 passage
Search aliases: overline O_k(s_k) overline mathcal
Use in source · Stratified_Universality_Traditional.tex, line 4592
[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
Exact mathematical passage
\overline{\mathcal O_k(s_k)}^{\,p} = \{s_k+c:c\in\mathbb Z_p\}.
\overline{\mathcal O_k(s_k)}^{\,p}Use in source · 1 passage
Search aliases: overline O_k(s_k)^ p overline mathcal
Use in source · Stratified_Universality_Traditional.tex, line 4592
[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
Exact mathematical passage
\overline{\mathcal O_k(s_k)}^{\,p} = \{s_k+c:c\in\mathbb Z_p\}.
\overline QUse in source · 2 passages
Search aliases: overline Q overline
Use in source · Stratified_Universality_Traditional.tex, line 1583
[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summands and D is block diagonal, this becomes coker D≅⊕_acoker D_a. Off-diagonal correction blocks can couple packet defects, so the block-diagonal hypoth…
Exact mathematical passage
\overline Q_a
\overline qUse in source · 2 passages
Search aliases: overline q overline
Use in source · block17_semantic_quotients_final.tex, line 84
[Universal behavioral coimage] Assume the relevant effective quotient or module quotient exists. Then Q_ S preserves every declared future observation. Moreover, if q:X↠ Z is any surjective representation through which every map o_jρ(m) factors, there is a unique map overline q:Z⟶ Q_ S with overline q q:X→ Q_ S equal to the canonical quotient map. Thus Q_ S is the coarsest exact representation for the declared semantic slice.
Exact mathematical passage
\overline q:Z\longrightarrow Q_{\mathsf S}
\overline Q_aUse in source · 2 passages
Search aliases: overline Q_a overline
Use in source · Stratified_Universality_Traditional.tex, line 1583
[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summands and D is block diagonal, this becomes coker D≅⊕_acoker D_a. Off-diagonal correction blocks can couple packet defects, so the block-diagonal hypoth…
Exact mathematical passage
\overline Q_a
p^2Use in source · 1 passage
Search aliases: p^2
Use in source · Stratified_Universality_Traditional.tex, line 4602
For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
Exact mathematical passage
c_\infty = \frac{1}{1-p^2},
p^{2j}Use in source · 1 passage
Search aliases: p^2j
Use in source · Stratified_Universality_Traditional.tex, line 4598
[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
Exact mathematical passage
c_N=\sum_{j=0}^{N}p^{2j}
p^aUse in source · 32 passages
Search aliases: p^a
Use in source · Stratified_Universality_Traditional.tex, line 5474
For a≥1 and 0≤ s<p^a, define the branch B_a,sf(q)=f(p^a q+s), q∈ Z_p, and the complete packet B_a f=( B_a,sf)_0≤ s<p^a.
Exact mathematical passage
0\leq s<p^a
p^{a+b}Use in source · 3 passages
Search aliases: p^a+b
Use in source · Stratified_Universality_Traditional.tex, line 5517
Direct substitution gives f_a,s(p^bq+t) =f(p^a(p^bq+t)+s) =f(p^a+bq+s+p^a t). The address is uniquely determined in the interval 0≤ s+p^a t<p^a+b.
Exact mathematical passage
f_{a,s}(p^bq+t) =f(p^a(p^bq+t)+s) =f(p^{a+b}q+s+p^a t).
p^bUse in source · 2 passages
Search aliases: p^b
Use in source · Stratified_Universality_Traditional.tex, line 5508
[Strict nested-branch law] For a,b≥1, 0≤ s<p^a, and 0≤ t<p^b, (f_a,s)_b,t=f_a+b,s+p^a t. All multiple-overlap comparison cells for the nested p-power branch atlas are identities. Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.
Exact mathematical passage
0\leq t<p^b
p^{D(A,B)+1}Use in source · 1 passage
Search aliases: p^D(A,B)+1
Use in source · Stratified_Universality_Traditional.tex, line 3169
[Exact state-complexity hierarchy] For a reduced rational mask A/B, r_p(F) ≤ κ_p(F) ≤ p^r_p(F) ≤ p^D(A,B)+1. Moreover, r_p(F)=dim W(A,B), κ_p(F)=| R(A,B)|.
Exact mathematical passage
\label{eq:block04-state-hierarchy} r_p(F) \leq \kappa_p(F) \leq p^{r_p(F)} \leq p^{D(A,B)+1}.
p^{D+1}Use in source · 1 passage
Search aliases: p^D+1
Use in source · Stratified_Universality_Traditional.tex, line 2580
n D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source reference]. It computes the coefficient sequence exactly.
Exact mathematical passage
\label{eq:compiler-state-bound-block03} p^{D+1}
p^{e(r+1)}Use in source · 1 passage
Search aliases: p^e(r+1)
Use in source · Stratified_Universality_Traditional.tex, line 5333
[Power bridge] For b(n)=n^r, which is completely multiplicative, the divisor ghost is c(n)=∑_dmid nd^r+1=σ_r+1(n). At a prime power, σ_r+1(p^e) = 1+p^r+1+⋯+p^e(r+1). Thus one simple multiplicative exponent rule simultaneously produces a filtered ordinary product, a multiplicative divisor-sum ghost, and a one-dimensional geometric local rule at every prime.
Exact mathematical passage
\sigma_{r+1}(p^e) = 1+p^{r+1}+\cdots+p^{e(r+1)}.
p^{e-1}Use in source · 2 passages
Search aliases: p^e-1
Use in source · Stratified_Universality_Traditional.tex, line 5244
theorem[Multiplicativity is preserved and reflected] Assume b(1)=1 and c= G(b). Then b is multiplicative Longleftrightarrow c is multiplicative. On this locus, both b and c are reconstructed from prime-power seeds. For every prime p and e≥1, c(p^e)=∑_j=0^ep^j b(p^j), b(p^e)=fracc(p^e)-c(p^e-1)p^e.
Exact mathematical passage
\boxed{ b(p^e)=\frac{c(p^e)-c(p^{e-1})}{p^e}.}
p^{e-j}Use in source · 1 passage
Search aliases: p^e-j
Use in source · Stratified_Universality_Traditional.tex, line 5139
The local logarithmic-derivative recurrence gives e f(p^e) = ∑_j=1^eγ_p,j(f)f(p^e-j). Write n=p^e u with (u,p)=1 and multiply by f(u). Since Γ_f,p is supported on p-powers, the resulting sum is exactly the Dirichlet convolution at n.
Exact mathematical passage
e f(p^e) = \sum_{j=1}^{e}\gamma_{p,j}(f)f(p^{e-j}).
p^{e-r}Use in source · 1 passage
Search aliases: p^e-r
Use in source · Stratified_Universality_Traditional.tex, line 5055
Unique factorization writes every positive integer as a finite product of pairwise coprime prime powers, so multiplicativity gives the reconstruction formula. Conversely, an arbitrary family of local unit series defines a multiplicative function by the same finite product. For n=p^e, Dirichlet convolution becomes (fmathbin∗_Dg)(p^e) = ∑_r=0^ef(p^r)g(p^e-r), which is exactly the coefficient of T^e in the local Cauchy product.
Exact mathematical passage
(f\mathbin{\ast_D}g)(p^e) = \sum_{r=0}^{e}f(p^r)g(p^{e-r}),
p^jUse in source · 6 passages
Search aliases: p^j
Use in source · Stratified_Universality_Traditional.tex, line 2346
The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.
Exact mathematical passage
\Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a = \bigl(n\longmapsto a_{p^\ell n+r}\bigr), \qquad r=\sum_{j=0}^{\ell-1}d_jp^j.
p^{j+1}Use in source · 1 passage
Search aliases: p^j+1
Use in source · Stratified_Universality_Traditional.tex, line 2477
Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.
Exact mathematical passage
\prod_{j=0}^{N}\frac{F(z^{p^j})}{F(z^{p^{j+1}})} = \frac{F(z)}{F(z^{p^{N+1}})}.
p^LUse in source · 4 passages
Search aliases: p^L
Use in source · Stratified_Universality_Traditional.tex, line 5808
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
C^H=I+p^LB, \qquad L\geq1.
p^{L+a}Use in source · 2 passages
Search aliases: p^L+a
Use in source · Stratified_Universality_Traditional.tex, line 5842
[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.
Exact mathematical passage
\boxed{X^{p^a}=I+p^{L+a}B_a.}
p^{L+a+1}Use in source · 1 passage
Search aliases: p^L+a+1
Use in source · Stratified_Universality_Traditional.tex, line 5853
Induct on a. If X_a=I+p^L+aB_a, the binomial expansion of X_a^p has linear term divisible by p^L+a+1, and every higher term has at least the same valuation. Apply [source reference] at the refined jump.
Exact mathematical passage
p^{L+a+1}
p^{Lr}Use in source · 2 passages
Search aliases: p^Lr
Use in source · Stratified_Universality_Traditional.tex, line 5813
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r\geq0} p^{Lr}\binom xr w^{\mathsf T}C^jB^rv_0,}
p^{\min(r,\nu_i)}Use in source · 3 passages
Search aliases: p^min(r,ν_i) min nu
Use in source · Stratified_Universality_Traditional.tex, line 4341
[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.
Exact mathematical passage
G_{k,p^r}(a) \cong Q_{k,p^r}(a) \cong \bigoplus_{i=1}^{\rho} \mathbb Z/p^{\min(r,\nu_i)}\mathbb Z,
p^{N+1}Use in source · 1 passage
Search aliases: p^N+1
Use in source · Stratified_Universality_Traditional.tex, line 2477
Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.
Exact mathematical passage
\prod_{j=0}^{N}\frac{F(z^{p^j})}{F(z^{p^{j+1}})} = \frac{F(z)}{F(z^{p^{N+1}})}.
p^{r+1}Use in source · 3 passages
Search aliases: p^r+1
Use in source · Stratified_Universality_Traditional.tex, line 4409
ower-automata For every r≥1, the semantic p^r-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence abmod p^r. Moreover, κ_k,p^r^sem(a) ≤ |overline M_k,p^r(a)| = p^∑_i(r-min(r,ν_i)). Reduction induces surjective digit-equivariant maps K_k,p^r+1^sem(a) ↠ K_k,p^r^sem(a).
Exact mathematical passage
\mathcal K_{k,p^{r+1}}^{\mathrm{sem}}(a) \twoheadrightarrow \mathcal K_{k,p^r}^{\mathrm{sem}}(a).
P^r_{A,\mathcal O}Use in source · 1 passage
Search aliases: P^r_A, O mathcal
Use in source · block16_section_final.tex, line 408
item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.
Exact mathematical passage
P^r_{A,\mathcal O} = \sum_{\substack{\mathcal Q\text{ a fine orbit}\\ \mathcal Q\subseteq(\rho^A_{sr})^{-1}(\mathcal O)}} P^s_{A,\mathcal Q}.
P^r_{A,U,\mathcal O}Use in source · 2 passages
Search aliases: P^r_A,U, O mathcal
Use in source · block16_section_final.tex, line 375
[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.
Exact mathematical passage
P^r_{A,U,\mathcal O} := \sum_{i\in\mathcal O_U}p^r_{A,U,i}
p^r_{A,U,i}Use in source · 2 passages
Search aliases: p^r_A,U,i
Use in source · block16_section_final.tex, line 375
[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.
Exact mathematical passage
P^r_{A,U,\mathcal O} := \sum_{i\in\mathcal O_U}p^r_{A,U,i}
P^r_{A,V,\mathcal O}Use in source · 1 passage
Search aliases: P^r_A,V, O mathcal
Use in source · block16_section_final.tex, line 424
Choose a path from U_0 to U and transport O along its address bijection. Two path choices differ by a loop at U_0, which preserves O setwise. Hence O_U is path independent. Projector compatibility gives θ^A_γ F_γ(P^r_A,U, O) = P^r_A,V, Oθ^A_γ, so the orbit sums descend. Their local splittings are the finite direct sums ⊕_i∈ O_UA^r_U,i; the direct sums of the β-maps, with the induced permutation of summands, provide their descent data. This proves (O1).
Exact mathematical passage
\theta^A_\gamma F_\gamma(P^r_{A,U,\mathcal O}) = P^r_{A,V,\mathcal O}\theta^A_\gamma,
p^{r_p(F)}Use in source · 2 passages
Search aliases: p^r_p(F)
Use in source · Stratified_Universality_Traditional.tex, line 3169
[Exact state-complexity hierarchy] For a reduced rational mask A/B, r_p(F) ≤ κ_p(F) ≤ p^r_p(F) ≤ p^D(A,B)+1. Moreover, r_p(F)=dim W(A,B), κ_p(F)=| R(A,B)|.
Exact mathematical passage
\label{eq:block04-state-hierarchy} r_p(F) \leq \kappa_p(F) \leq p^{r_p(F)} \leq p^{D(A,B)+1}.
P^s_{A,\mathcal Q}Use in source · 1 passage
Search aliases: P^s_A, Q mathcal
Use in source · block16_section_final.tex, line 408
item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.
Exact mathematical passage
P^r_{A,\mathcal O} = \sum_{\substack{\mathcal Q\text{ a fine orbit}\\ \mathcal Q\subseteq(\rho^A_{sr})^{-1}(\mathcal O)}} P^s_{A,\mathcal Q}.
p^{v_p(M)}Use in source · 4 passages
Search aliases: p^v_p(M)
Use in source · Stratified_Universality_Traditional.tex, line 5665
upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac dβ_prfloor+1. A local channel loses residual rank exactly at primes dividing M. item A …
Exact mathematical passage
\beta_p=p^{v_p(M)}
p^{v_p(m)}Use in source · 1 passage
Search aliases: p^v_p(m)
Use in source · Stratified_Universality_Traditional.tex, line 2001
Addendum 04 proves that the prefix S_*={0,1,…,L_ Z(m,d)-1} with L_ Z(m,d)=min(d+1,B(m)), B(m)=max({1}∪{p^v_p(m):pmid m}), always admits an integer decoder and is cardinality-minimal [source reference].
Exact mathematical passage
L_{\mathbb Z}(m,d)=\min(d+1,B(m)), \qquad B(m)=\max\bigl(\{1\}\cup\{p^{v_p(m)}:p\mid m\}\bigr),
p^\nuUse in source · 3 passages
Search aliases: p^ν nu
Use in source · Stratified_Universality_Traditional.tex, line 2386
The pair model gives a useful finite bound. Write T=p^ν M, gcd(M,p)=1, and set ord_1(p)=1.
Exact mathematical passage
T=p^\nu M, \qquad \gcd(M,p)=1,
p^{\nu_i}Use in source · 4 passages
Search aliases: p^ν_i nu
Use in source · Stratified_Universality_Traditional.tex, line 4459
[p-adic saturation sequence] There is a digit-equivariant short exact sequence 0 ⟶ M_k,p(a) ⟶ L_k,p(a) ⟶ C_k(a)_(p) ⟶ 0, where C_k(a)_(p) is the p-primary subgroup of C_k(a). In Smith coordinates, M_k,p(a) = ⊕_i=1^ρp^ν_i Z_p e_i ⊆ ⊕_i=1^ρ Z_p e_i = L_k,p(a).
Exact mathematical passage
M_{k,p}(a) = \bigoplus_{i=1}^{\rho}p^{\nu_i}\mathbb Z_p e_i \subseteq \bigoplus_{i=1}^{\rho}\mathbb Z_p e_i = L_{k,p}(a).
p^{\nu_p(T(a))}Use in source · 1 passage
Search aliases: p^ν_p(T(a)) nu
Use in source · Stratified_Universality_Traditional.tex, line 2698
For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)| ≤ T(a)(ν_p(T(a))+ord_M(p)), R_a(z) · =fracF_a(z)F_a(z^p), T(a)=p^ν_p(T(a))M, gcd(M,p)=1.
Exact mathematical passage
\label{eq:periodic-invariant-identities} \begin{aligned} \deg m_a &=\deg Q_a =\rank H(a) =\dim \State_{\min}(a),\\ T(a) &=\min\{N:m_a\mid x^N-1\} =\min\{N:Q_a\mid 1-z^N\},\\ |\mathcal A_p(a)| &=|\mathcal K_p(a)| \leq T(a)\bigl(\nu_p(T(a))+\operatorname{ord}_{M}(p)\bigr),\\ R_a(z) &=\frac{F_a(z)}{F_a(z^p)}, \qquad T(a)=p^{\nu_p(T(a))}M, \quad \gcd(M,p)=1. \end{aligned}
p^\ellUse in source · 3 passages
Search aliases: p^ℓ ell
Use in source · Stratified_Universality_Traditional.tex, line 2346
The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.
Exact mathematical passage
\Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a = \bigl(n\longmapsto a_{p^\ell n+r}\bigr), \qquad r=\sum_{j=0}^{\ell-1}d_jp^j.
p^{\ell-1}Use in source · 1 passage
Search aliases: p^ℓ-1 ell
Use in source · Stratified_Universality_Traditional.tex, line 3026
m the linear map Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.
Exact mathematical passage
n=d_0+d_1p+\cdots+d_{\ell-1}p^{\ell-1}
p^{\sum_i(r-\min(r,\nu_i))}Use in source · 1 passage
Search aliases: p^∑_i(r-min(r,ν_i)) sum min nu
Use in source · Stratified_Universality_Traditional.tex, line 4401
[Finite prime-power automata] For every r≥1, the semantic p^r-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence abmod p^r. Moreover, κ_k,p^r^sem(a) ≤ |overline M_k,p^r(a)| = p^∑_i(r-min(r,ν_i)). Reduction induces surjective digit-equivariant maps K_k,p^r+1^sem(a) ↠ K_k,p^r^sem(a).
Exact mathematical passage
\kappa_{k,p^r}^{\mathrm{sem}}(a) \leq \left|\overline M_{k,p^r}(a)\right| = p^{\sum_i(r-\min(r,\nu_i))}.
p^{\sum_{i=1}^{\rho}(r-\min(r,\nu_i))}Use in source · 1 passage
Search aliases: p^∑_i=1^ρ(r-min(r,ν_i)) sum rho min nu
Use in source · Stratified_Universality_Traditional.tex, line 4350
[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.
Exact mathematical passage
\left|\overline M_{k,p^r}(a)\right| = p^{\sum_{i=1}^{\rho}(r-\min(r,\nu_i))}.
P_0Use in source · 2 passages
Search aliases: P_0
Use in source · Stratified_Universality_Traditional.tex, line 3025
d preserves P_D; item the linear map Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.
Exact mathematical passage
P_0=B
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · block12_section.tex, line 228
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
P_0\leq P_1\leq P_2\leq\cdots\leq I.
p_1Use in source · 11 passages
Search aliases: p_1
Use in source · Stratified_Universality_Traditional.tex, line 568
[Exact oblique packets] Over any field, set p_1= 1 · 1 0 · 0 , p_2= 0 · -1 0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.
Exact mathematical passage
p_1= \begin{pmatrix} 1&1\\0&0 \end{pmatrix}, \qquad p_2= \begin{pmatrix} 0&-1\\0&1 \end{pmatrix}.
p_1^2Use in source · 1 passage
Search aliases: p_1^2
Use in source · Stratified_Universality_Traditional.tex, line 579
0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.
Exact mathematical passage
p_1^2=p_1
P_2Use in source · 1 passage
Search aliases: P_2
Use in source · block12_section.tex, line 228
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
P_0\leq P_1\leq P_2\leq\cdots\leq I.
p_2Use in source · 10 passages
Search aliases: p_2
Use in source · Stratified_Universality_Traditional.tex, line 568
[Exact oblique packets] Over any field, set p_1= 1 · 1 0 · 0 , p_2= 0 · -1 0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.
Exact mathematical passage
p_1= \begin{pmatrix} 1&1\\0&0 \end{pmatrix}, \qquad p_2= \begin{pmatrix} 0&-1\\0&1 \end{pmatrix}.
p_2^2Use in source · 1 passage
Search aliases: p_2^2
Use in source · Stratified_Universality_Traditional.tex, line 579
0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.
Exact mathematical passage
p_2^2=p_2
P_aUse in source · 3 passages
Search aliases: P_a
Use in source · Stratified_Universality_Traditional.tex, line 2274
Let F_a(z)=fracP_a(z)Q_a(z) be reduced with P_a(0)=Q_a(0)=1. If d=deg m_a, then Q_a(z)=z^dm_a(z^-1), with the evident coefficient normalization. Taking reciprocal polynomials in [source reference] gives Q_a(z)mid 1-z^T, T=min{N≥ 1:Q_a(z)mid 1-z^N}. Thus pure periodicity is visible both in the linear annihilator and in the reduced rational denominator.
Exact mathematical passage
F_a(z)=\frac{P_a(z)}{Q_a(z)}
p_aUse in source · 1 passage
Search aliases: p_a
Use in source · block15_section.tex, line 1270
The displayed matrix multiplication gives the two values immediately. The full refinement identity recovers f=∑_a,i∈{1,2}p_afp_i, whose only nonzero term here is p_2fp_2.
Exact mathematical passage
f=\sum_{a,i\in\{1,2\}}p_afp_i
p_{A_U,i}^rUse in source · 1 passage
Search aliases: p_A_U,i^r
Use in source · block15_section.tex, line 1137
there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the verified transition profiles preserve the chosen splittings, finite sums, composition, and refinement equations, with coherent splitting comparisons on …
Exact mathematical passage
\theta^A_{UV} F_{UV}(p_{A_U,i}^r) = p_{A_V,\lambda_{UV}^{A,r}(i)}^r \theta^A_{UV},
p_{A_V,\lambda_{UV}^{A,r}(i)}^rUse in source · 1 passage
Search aliases: p_A_V,λ_UV^A,r(i)^r lambda
Use in source · block15_section.tex, line 1137
there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the verified transition profiles preserve the chosen splittings, finite sums, composition, and refinement equations, with coherent splitting comparisons on …
Exact mathematical passage
\theta^A_{UV} F_{UV}(p_{A_U,i}^r) = p_{A_V,\lambda_{UV}^{A,r}(i)}^r \theta^A_{UV},
\mathcal P_DUse in source · 4 passages
Search aliases: P_D mathcal
Use in source · Stratified_Universality_Traditional.tex, line 2998
Write a=deg A, b=deg B, H(z)=A(z)B(z)^p-2, and define D(A,B) = max{ b, lceilfraca+(p-2)bp-1rceil }. Let P_D= F_p[z]_≤ D. For 0≤ d<p, define T_d(P)=Λ_d(PH), where Λ_d selects the coefficients in degrees congruent to d modulo p.
Exact mathematical passage
\mathcal P_D=\mathbb F_p[z]_{\leq D}
P_{gHg^{-1}}Use in source · 1 passage
Search aliases: P_gHg^-1
Use in source · block12_section.tex, line 129
Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.
Exact mathematical passage
U_gP_HU_g^{-1}=P_{gHg^{-1}}
P_HUse in source · 18 passages
Search aliases: P_H
Use in source · block12_section.tex, line 16
Let L/K be a finite Galois extension with group G=Gal(L/K). Let V be a finite-dimensional representation of G over a characteristic-zero field, or a unitary complex realization obtained after scalar extension. For every subgroup H≤ G, define P_H=frac1|H|∑_h∈ HU_h. tag12.1
Exact mathematical passage
\boxed{ P_H=\frac1{|H|}\sum_{h\in H}U_h.} \tag{12.1}
P_H^2Use in source · 1 passage
Search aliases: P_H^2
Use in source · block12_section.tex, line 37
The average is invariant under left and right multiplication by elements of H. Therefore P_H^2=P_H, and its range is exactly the fixed subspace. If H≤ J, then V^J⊆ V^H. The two projections are nested and multiply to the projector with smaller range.
Exact mathematical passage
P_H^2=P_H
P_{HJ}Use in source · 1 passage
Search aliases: P_HJ
Use in source · block12_section.tex, line 99
[Permutable subgroups and exact common refinement] In a faithful regular realization, the following are equivalent: item HJ=JH; item HJ is a subgroup of G; item P_HP_J=P_JP_H. When these conditions hold, P_HP_J=P_HJ. tag12.4 If they fail, P_HP_J and P_JP_H are mixed atlas corners rather than projectors.
Exact mathematical passage
\boxed{P_HP_J=P_{HJ}.} \tag{12.4}
P_iUse in source · 9 passages
Search aliases: P_i
Use in source · block11_section.tex, line 9
Throughout the analytic part, X is a Banach space over R or C. A finite packet system on X is a family of bounded projections P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. All sums over packet addresses are finite. The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.
Exact mathematical passage
P_iP_j=\delta_{ij}P_i, \qquad \sum_{i\in I}P_i=I_X.
P_{i+1}Use in source · 1 passage
Search aliases: P_i+1
Use in source · block12_section.tex, line 232
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
D_i=P_{i+1}-P_i.
P_JUse in source · 6 passages
Search aliases: P_J
Use in source · block12_section.tex, line 29
[Reynolds fixed packet and reverse order] The operator P_H is the projector onto the fixed packet V^H={v:U_hv=v for all h∈ H}. If H≤ J, then P_HP_J=P_JP_H=P_J. tag12.2 Consequently, subgroup inclusion reverses the fixed-projector order.
Exact mathematical passage
\boxed{P_HP_J=P_JP_H=P_J.} \tag{12.2}
P_jUse in source · 1 passage
Search aliases: P_j
Use in source · block11_section.tex, line 9
Throughout the analytic part, X is a Banach space over R or C. A finite packet system on X is a family of bounded projections P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. All sums over packet addresses are finite. The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.
Exact mathematical passage
P_iP_j=\delta_{ij}P_i, \qquad \sum_{i\in I}P_i=I_X.
P_{j_1}Use in source · 1 passage
Search aliases: P_j_1
Use in source · block11_section.tex, line 114
For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
A_{m;i\mid j_1,\ldots,j_m} = P_iA_m(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
P_{j_m}Use in source · 1 passage
Search aliases: P_j_m
Use in source · block11_section.tex, line 114
For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
A_{m;i\mid j_1,\ldots,j_m} = P_iA_m(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
P_{m,S}^{(d)}Use in source · 1 passage
Search aliases: P_m,S^(d)
Use in source · Stratified_Universality_Traditional.tex, line 1979
Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.
Exact mathematical passage
A=P_{m,S}^{(d)}:\mathbb Z^{d+1}\to\mathbb Z^{|S|(d+1)}
P_NUse in source · 1 passage
Search aliases: P_N
Use in source · block12_section.tex, line 310
12-shrinking-subgroups Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.
Exact mathematical passage
P_{U_n}\to P_N
\mathcal P_{N,j}Use in source · 8 passages
Search aliases: P_N,j mathcal
Use in source · block15_section.tex, line 665
For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0 m≡ j (mod N)a_mz^m, 0≤ j<N.
Exact mathematical passage
\mathcal P_{N,j}F(z) := \sum_{\substack{m\ge0\\m\equiv j\ (\mathrm{mod}\ N)}}a_mz^m, \qquad 0\le j<N.
\mathcal P_{N,j}^2Use in source · 1 passage
Search aliases: P_N,j^2 mathcal
Use in source · block15_section.tex, line 686
ion] Let N,c,n∈ Z with N≥1, c≥0, and n≥0. Then frac(Nn+c)!c! = ∏_r=c+1^c+N(r)_n;N. Each integer factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_N,j = ∑_ℓ=0^M-1 P_NM,j+ℓ N.
Exact mathematical passage
\mathcal P_{N,j}^2=\mathcal P_{N,j}, \qquad \mathcal P_{N,j}\mathcal P_{N,k}=0\ (j\ne k), \qquad \sum_{j=0}^{N-1}\mathcal P_{N,j}=1.
P_{N,k}Use in source · 1 passage
Search aliases: P_N,k
Use in source · Stratified_Universality_Traditional.tex, line 5149
[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.
Exact mathematical passage
\boxed{ P_{N,k}(f\mathbin{\ast_D}g) = \sum_{rs\equiv k\, (N)} (P_{N,r}f)\mathbin{\ast_D}(P_{N,s}g). }
\mathcal P_{N,k}Use in source · 1 passage
Search aliases: P_N,k mathcal
Use in source · block15_section.tex, line 686
ion] Let N,c,n∈ Z with N≥1, c≥0, and n≥0. Then frac(Nn+c)!c! = ∏_r=c+1^c+N(r)_n;N. Each integer factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_N,j = ∑_ℓ=0^M-1 P_NM,j+ℓ N.
Exact mathematical passage
\mathcal P_{N,j}^2=\mathcal P_{N,j}, \qquad \mathcal P_{N,j}\mathcal P_{N,k}=0\ (j\ne k), \qquad \sum_{j=0}^{N-1}\mathcal P_{N,j}=1.
P_{N,r}Use in source · 2 passages
Search aliases: P_N,r
Use in source · Stratified_Universality_Traditional.tex, line 5148
[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.
Exact mathematical passage
P_{N,r}f
P_{N,s}Use in source · 1 passage
Search aliases: P_N,s
Use in source · Stratified_Universality_Traditional.tex, line 5149
[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.
Exact mathematical passage
\boxed{ P_{N,k}(f\mathbin{\ast_D}g) = \sum_{rs\equiv k\, (N)} (P_{N,r}f)\mathbin{\ast_D}(P_{N,s}g). }
\mathcal P_{NM,j+\ell N}Use in source · 1 passage
Search aliases: P_NM,j+ℓ N mathcal ell
Use in source · block15_section.tex, line 694
factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_N,j = ∑_ℓ=0^M-1 P_NM,j+ℓ N.
Exact mathematical passage
\mathcal P_{N,j} = \sum_{\ell=0}^{M-1}\mathcal P_{NM,j+\ell N}.
\mathcal P_{\mathrm{per}}Use in source · 2 passages
Search aliases: P_per mathcal mathrm
Use in source · block13_section.tex, line 189
For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise contains the primitive period word, exact period, minimal recurrence, reduced OGF, and periodic packet profile.
Exact mathematical passage
\mathcal J_{\mathrm{DRLM},p}(a) = \left( \mathsf{SL}_{p,\mathbb F_p}(a), \mathcal P_{\mathrm{rm}}(a), \mathcal P_{\mathrm{per}}(a) \right),
P_{q,\alpha}Use in source · 3 passages
Search aliases: P_q,α alpha
Use in source · block13_section.tex, line 269
Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.
Exact mathematical passage
P_{q,\alpha}(z,Y)=Y-z-\alpha Y^q
\mathcal P_{\mathrm{rm}}Use in source · 2 passages
Search aliases: P_rm mathcal mathrm
Use in source · block13_section.tex, line 189
For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise contains the primitive period word, exact period, minimal recurrence, reduced OGF, and periodic packet profile.
Exact mathematical passage
\mathcal J_{\mathrm{DRLM},p}(a) = \left( \mathsf{SL}_{p,\mathbb F_p}(a), \mathcal P_{\mathrm{rm}}(a), \mathcal P_{\mathrm{per}}(a) \right),
p_s^{-1}Use in source · 1 passage
Search aliases: p_s^-1
Use in source · block18_carrier_observer_change_final.tex, line 78
The pair (a,b) factors j through c, and the image universal property gives the map to J_s,t. If t=φ s, then J_s,t is the graph of φ, so p_s is invertible. Conversely, if p_s is invertible, define φ=p_t p_s^-1.
Exact mathematical passage
\varphi=p_t p_s^{-1}
p_tUse in source · 1 passage
Search aliases: p_t
Use in source · block18_carrier_observer_change_final.tex, line 78
The pair (a,b) factors j through c, and the image universal property gives the map to J_s,t. If t=φ s, then J_s,t is the graph of φ, so p_s is invertible. Conversely, if p_s is invertible, define φ=p_t p_s^-1.
Exact mathematical passage
\varphi=p_t p_s^{-1}
P_{U_n}Use in source · 2 passages
Search aliases: P_U_n
Use in source · block12_section.tex, line 306
[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.
Exact mathematical passage
\boxed{P_{U_n}\longrightarrow I} \tag{12.15}
p_{Y,a}Use in source · 1 passage
Search aliases: p_Y,a
Use in source · block15_section.tex, line 634
Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement is then a finite sum of the specified homogeneous pieces.
Exact mathematical passage
1_Y=\sum_a p_{Y,a}
p_{Y,a}^uUse in source · 5 passages
Search aliases: p_Y,a^u
Use in source · block15_section.tex, line 96
ion r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.
Exact mathematical passage
p_{Y,a}^u f p_{X,i}^r = \iota_{Y,a}^u[f]^{u,r}_{a i}\pi_{X,i}^r.
p_{Y,b}^vUse in source · 2 passages
Search aliases: p_Y,b^v
Use in source · block15_section.tex, line 152
item Projector-sum refinement. If ssucceq r on X and vsucceq u on Y, then p_Y,a^u f p_X,i^r = ∑_ρ^Y_vu(b)=a ∑_ρ^X_sr(j)=i p_Y,b^v f p_X,j^s. Coarsening a fine composite gives the same ambient morphism as composing at the coarse level.
Exact mathematical passage
p_{Y,a}^u f p_{X,i}^r = \sum_{\rho^Y_{vu}(b)=a} \sum_{\rho^X_{sr}(j)=i} p_{Y,b}^v f p_{X,j}^s.
P_\betaUse in source · 1 passage
Search aliases: P_β beta
Use in source · block11_section.tex, line 184
Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.
Exact mathematical passage
h\in P_\beta X
P_\deltaUse in source · 1 passage
Search aliases: P_δ delta
Use in source · block11_section.tex, line 206
[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.
Exact mathematical passage
\boxed{ \begin{aligned} &P_\delta D^n(F\circ G)(x)[h,\ldots,h]\\ &=\sum_{\mu} \frac{n!} {\displaystyle\prod_{r,\gamma} (r!)^{\mu_{r,\gamma}}\mu_{r,\gamma}!} \,P_\delta D^{k(\mu)}F(Gx) \bigl[ (g_{r,\gamma})^{\mu_{r,\gamma}} \bigr]_{r,\gamma}. \end{aligned}} \tag{11.5}
P_\lambdaUse in source · 1 passage
Search aliases: P_λ lambda
Use in source · Stratified_Universality_Traditional.tex, line 2161
Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [source reference].
Exact mathematical passage
a_n=\sum_{\lambda\in\Lambda}P_\lambda(n)\lambda^n.
p_{\ell,i_\ell}Use in source · 1 passage
Search aliases: p_ℓ,i_ℓ ell ell
Use in source · block15_section.tex, line 632
Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement is then a finite sum of the specified homogeneous pieces.
Exact mathematical passage
x_\ell=\sum_{i_\ell}p_{\ell,i_\ell}x_\ell
\parallelUse in source · 2 passages
Search aliases: parallel
Use in source · Stratified_Universality_Traditional.tex, line 5041
[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).
Exact mathematical passage
\boxed{ f(n)=\prod_{p^e\parallel n}f(p^e).}
\mathrm{pers}Use in source · 3 passages
Search aliases: pers mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2035
Let Seq_K^pers be the discrete groupoid of persistent C-finite sequences over K.
Exact mathematical passage
\Seq_K^{\mathrm{pers}}
\operatorname{PGL}Use in source · 3 passages
Search aliases: PGL operatorname
Use in source · block12_section.tex, line 411
A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.
Exact mathematical passage
\operatorname{PGL}_n(L)
\operatorname{PGL}_nUse in source · 2 passages
Search aliases: PGL_n operatorname
Use in source · block12_section.tex, line 411
A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.
Exact mathematical passage
\operatorname{PGL}_n(L)
\pmodUse in source · 17 passages
Search aliases: pmod
Use in source · Stratified_Universality_Traditional.tex, line 2368
belprop:periodic-kernel Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives by equality of the sequences in [source reference] gives the canonical minimal least-significant-digit-first p-DFAO for a.
Exact mathematical passage
\label{eq:pair-transition} (u,r)\xrightarrow{d}(pu,r+ud)\pmod T,
q'Use in source · 2 passages
Search aliases: q'
Use in source · block09_section.tex, line 12
Let D_K=K(z)⟨ D⟩, Dq=qD+q', be the differential Ore algebra, and let S_K=K(n)⟨ S⟩, Sq(n)=q(n+1)S, be the forward-shift Ore algebra. Both are Euclidean skew-polynomial rings in their operator variable, so their nonzero left ideals have unique monic generators [source reference].
Exact mathematical passage
\mathcal D_K=K(z)\langle D\rangle, \qquad Dq=qD+q',
q^{-1/(q-1)}Use in source · 1 passage
Search aliases: q^-1/(q-1)
Use in source · block11_section.tex, line 518
r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,N/g,h). tag11.17
Exact mathematical passage
\tau=q^{-1/(q-1)}, \qquad \rho=(q-1)q^{-q/(q-1)};
q^{-q/(q-1)}Use in source · 1 passage
Search aliases: q^-q/(q-1)
Use in source · block11_section.tex, line 518
r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,N/g,h). tag11.17
Exact mathematical passage
\tau=q^{-1/(q-1)}, \qquad \rho=(q-1)q^{-q/(q-1)};
Q^2Use in source · 1 passage
Search aliases: Q^2
Use in source · block19_admissibility_lifting_final.tex, line 197
c · b, d=ab-c^2>0. The squared induced norms are u:= A_1 _Q^2 =fracb(a+b+2c)d, v:= A_2 _Q^2 =fraca(a+b-2c)d. A direct calculation gives ab((a+b)^2-4c^2)-4(ab-c^2)^2 =ab(a-b)^2+4c^2(ab-c^2)≥0. Thus uv≥4, and therefore max{sqrt u,sqrt v}≥(uv)^1/4≥sqrt2. Equality forces a=b and c=0, and then u=v=2.
Exact mathematical passage
u:=\|A_1\|_Q^2 =\frac{b(a+b+2c)}{d}, \qquad v:=\|A_2\|_Q^2 =\frac{a(a+b-2c)}{d}.
q^{r_k(a)}Use in source · 3 passages
Search aliases: q^r_k(a)
Use in source · Stratified_Universality_Traditional.tex, line 3870
statement written in semantic, intrinsic-module, and coordinate form. The kernel automaton in [source reference] recovers the deterministic chart, while its state-to-section map recovers the finite orbit inside the canonical module. These constructions are inverse on normalized objects. If K= F_q and r_k(a)<∞, the vector space V_k(a) has only q^r_k(a) elements, so the section orbit is automatically finite.
Exact mathematical passage
q^{r_k(a)}
Q^{\rm cl}_{N,j}Use in source · 3 passages
Search aliases: Q^rm cl_N,j rm
Use in source · block09_section.tex, line 363
Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j measures cancellation relative to that construction. It is not intrinsic until the construction has been fixed.
Exact mathematical passage
Q^{\rm cl}_{N,j}
q^u_aUse in source · 2 passages
Search aliases: q^u_a
Use in source · Stratified_Universality_Traditional.tex, line 614
[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.
Exact mathematical passage
q^u_a=\sum_{\sigma_{vu}(b)=a}q^v_b
q^v_bUse in source · 2 passages
Search aliases: q^v_b
Use in source · Stratified_Universality_Traditional.tex, line 614
[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.
Exact mathematical passage
q^u_a=\sum_{\sigma_{vu}(b)=a}q^v_b
\mathbb Q^\rhoUse in source · 1 passage
Search aliases: Q^ρ mathbb rho
Use in source · Stratified_Universality_Traditional.tex, line 4064
[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.
Exact mathematical passage
E_{\mathbf n}:V\longrightarrow\mathbb Q^\rho, \qquad f\longmapsto \bigl(f(n_1),\ldots,f(n_\rho)\bigr)
Q_aUse in source · 11 passages
Search aliases: Q_a
Use in source · Stratified_Universality_Traditional.tex, line 1579
[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summands and D is block diagonal, this becomes coker D≅⊕_acoker D_a. Off-diagonal correction blocks can couple packet defects, so the block-diagonal hypoth…
Exact mathematical passage
(Q_a)
q_bUse in source · 1 passage
Search aliases: q_b
Use in source · Stratified_Universality_Traditional.tex, line 660
Applying C_E to a block q_bfp_j retains it exactly when (b,j)∈ E. Pairwise annihilation removes every mismatched source or target index. The formulas follow block by block.
Exact mathematical passage
q_bfp_j
Q_eUse in source · 11 passages
Search aliases: Q_e
Use in source · block15_section.tex, line 231
Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of Φ_κ,τ, so the fold is split lossless.
Exact mathematical passage
Q_e\in\End(C)
q_eUse in source · 2 passages
Search aliases: q_e
Use in source · block15_section.tex, line 442
[Abelian fold-loss exact sequence] Let A be an abelian category. Suppose a finite family of lane objects (B_e)_e∈ E, target objects (C_t)_t∈ T, and morphisms τ_e:B_e→ C_κ(e) is given. Let q_e:⊕_e'∈ EB_e'→ B_e be the coordinate projection and define Φ:=∑_e∈ Ej_κ(e)τ_eq_e: ⊕_e∈ EB_e⟶ ⊕_t∈ TC_t. Then there are canonical short exact sequences 0⟶kerΦ ⟶⊕_eB_e ⟶imΦ ⟶0 and 0⟶imΦ ⟶⊕_tC_t ⟶cokerΦ ⟶0. The first sequence is the exact fold-loss certificate. Suppose every τ_e is monic and there are endomorphi…
Exact mathematical passage
q_e:\bigoplus_{e'\in E}B_{e'}\to B_e
q_jUse in source · 4 passages
Search aliases: q_j
Use in source · block09_section.tex, line 97
Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.
Exact mathematical passage
L = \sum_{j=0}^{r}q_j(z)D^j, \qquad q_j(z)=\sum_{m=0}^{d_j}q_{j,m}z^m.
q_{j,m}Use in source · 3 passages
Search aliases: q_j,m
Use in source · block09_section.tex, line 97
Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.
Exact mathematical passage
L = \sum_{j=0}^{r}q_j(z)D^j, \qquad q_j(z)=\sum_{m=0}^{d_j}q_{j,m}z^m.
q_{j,v_j}Use in source · 2 passages
Search aliases: q_j,v_j
Use in source · block09_section.tex, line 400
Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
q_{j,v_j}
Q_{k,p^r}Use in source · 2 passages
Search aliases: Q_k,p^r
Use in source · Stratified_Universality_Traditional.tex, line 4341
[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.
Exact mathematical passage
G_{k,p^r}(a) \cong Q_{k,p^r}(a) \cong \bigoplus_{i=1}^{\rho} \mathbb Z/p^{\min(r,\nu_i)}\mathbb Z,
Q_{M',\mathcal O'}Use in source · 1 passage
Search aliases: Q_M', O' mathcal
Use in source · block17_semantic_quotients_final.tex, line 247
[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.
Exact mathematical passage
Q_{M',\mathcal O'}\longrightarrow Q_{M,\mathcal O}.
Q_{M,\mathcal O}Use in source · 1 passage
Search aliases: Q_M, O mathcal
Use in source · block17_semantic_quotients_final.tex, line 247
[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.
Exact mathematical passage
Q_{M',\mathcal O'}\longrightarrow Q_{M,\mathcal O}.
Q_{N,j}Use in source · 2 passages
Search aliases: Q_N,j
Use in source · block10_section.tex, line 248
[Constructive packet eliminant] For each j, one can compute an annihilating polynomial Q_N,j(x,T)∈ K(x)[T] with deg_TQ_N,j≤ d^N by multiplication matrices, resultants, or a Gr obner elimination of the exact packet system.
Exact mathematical passage
Q_{N,j}(x,T)\in K(x)[T]
Q_oUse in source · 5 passages
Search aliases: Q_o
Use in source · block19_admissibility_lifting_final.tex, line 134
A compatible family makes every closed root holonomy H conformal for the root metric Q_o. Taking determinants fixes its positive scale as |det H|^1/n, so widehat H preserves Q_o.
Exact mathematical passage
Q_o
q_rUse in source · 1 passage
Search aliases: q_r
Use in source · block09_section.tex, line 387
Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of this operator coordinate until an explicit desingularization has been performed.
Exact mathematical passage
\Sigma_{\rm op}(F) = \{q_r(z)=0\}\cup\{\infty\}.
Q_sUse in source · 7 passages
Search aliases: Q_s
Use in source · block19_admissibility_lifting_final.tex, line 74
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
T:(V,Q_s)\to(W,Q_t)
Q_{s(e)}Use in source · 1 passage
Search aliases: Q_s(e)
Use in source · block19_admissibility_lifting_final.tex, line 126
raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge scales satisfying T_e^*Q_t(e)T_e=λ_e^2Q_s(e) exist if and only if the determinant-normalized holonomy representation preserves a positive-definite metr…
Exact mathematical passage
T_e^*Q_{t(e)}T_e=\lambda_e^2Q_{s(e)}
Q_{\mathrm{strong}}Use in source · 1 passage
Search aliases: Q_strong mathrm
Use in source · block19_admissibility_lifting_final.tex, line 307
Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders and should not be merged into one scalar notion of a stronger contract.
Exact mathematical passage
N_{\mathrm{strong}}\subseteq N_{\mathrm{weak}}, \qquad Q_{\mathrm{strong}}\twoheadrightarrow Q_{\mathrm{weak}}.
Q_tUse in source · 3 passages
Search aliases: Q_t
Use in source · block19_admissibility_lifting_final.tex, line 74
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
T:(V,Q_s)\to(W,Q_t)
Q_{t(e)}Use in source · 1 passage
Search aliases: Q_t(e)
Use in source · block19_admissibility_lifting_final.tex, line 126
raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge scales satisfying T_e^*Q_t(e)T_e=λ_e^2Q_s(e) exist if and only if the determinant-normalized holonomy representation preserves a positive-definite metr…
Exact mathematical passage
T_e^*Q_{t(e)}T_e=\lambda_e^2Q_{s(e)}
Q_UUse in source · 1 passage
Search aliases: Q_U
Use in source · block16_section_final.tex, line 387
item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.
Exact mathematical passage
Q_U=\sum_{i\in S_U}p^r_{A,U,i}
Q_uUse in source · 1 passage
Search aliases: Q_u
Use in source · block19_admissibility_lifting_final.tex, line 147
Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible family directly. A group preserving a positive-definite metric is conjugate into a compact orthogonal or unitary group. Conversely, averaging an inner p…
Exact mathematical passage
T_e^*Q_vT_e=|\det H_e|^{2/n}Q_u, \qquad \lambda_e=|\det H_e|^{1/n}>0.
Q_vUse in source · 2 passages
Search aliases: Q_v
Use in source · block19_admissibility_lifting_final.tex, line 139
Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible family directly. A group preserving a positive-definite metric is conjugate into a compact orthogonal or unitary group. Conversely, averaging an inner p…
Exact mathematical passage
Q_v=(G_v^{-1})^*Q_oG_v^{-1}.
Q_{\mathrm{weak}}Use in source · 1 passage
Search aliases: Q_weak mathrm
Use in source · block19_admissibility_lifting_final.tex, line 307
Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders and should not be merged into one scalar notion of a stronger contract.
Exact mathematical passage
N_{\mathrm{strong}}\subseteq N_{\mathrm{weak}}, \qquad Q_{\mathrm{strong}}\twoheadrightarrow Q_{\mathrm{weak}}.
Q_XUse in source · 1 passage
Search aliases: Q_X
Use in source · block18_carrier_observer_change_final.tex, line 145
[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.
Exact mathematical passage
Q_X\xrightarrow{\overline f}Q_Y\xrightarrow{\overline g}Q_Z,
Q_YUse in source · 1 passage
Search aliases: Q_Y
Use in source · block18_carrier_observer_change_final.tex, line 145
[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.
Exact mathematical passage
Q_X\xrightarrow{\overline f}Q_Y\xrightarrow{\overline g}Q_Z,
Q_ZUse in source · 1 passage
Search aliases: Q_Z
Use in source · block18_carrier_observer_change_final.tex, line 145
[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.
Exact mathematical passage
Q_X\xrightarrow{\overline f}Q_Y\xrightarrow{\overline g}Q_Z,
Q_\gammaUse in source · 1 passage
Search aliases: Q_γ gamma
Use in source · block11_section.tex, line 185
Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.
Exact mathematical passage
g_{r,\gamma} = Q_\gamma D^rG(x)[h,\ldots,h].
Q_{\lambda,a}Use in source · 1 passage
Search aliases: Q_λ,a lambda
Use in source · block13_section.tex, line 90
Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.
Exact mathematical passage
Q_{\lambda,a}:\mathcal O_\lambda\longrightarrow\mathcal C_{\lambda,a}.
Q_{\lambda,b}Use in source · 1 passage
Search aliases: Q_λ,b lambda
Use in source · block13_section.tex, line 94
has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.
Exact mathematical passage
\Phi_{ab}=Q_{\lambda,b}N_{\lambda,a}.
\mathbb RUse in source · 6 passages
Search aliases: R mathbb
Use in source · block11_section.tex, line 8
Throughout the analytic part, X is a Banach space over R or C. A finite packet system on X is a family of bounded projections P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. All sums over packet addresses are finite. The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.
Exact mathematical passage
\mathbb R
\mathbb R^2Use in source · 1 passage
Search aliases: R^2 mathbb
Use in source · block19_admissibility_lifting_final.tex, line 174
[Sharp geometry-class separation] The norm x _∞ satisfies max{ A_1x _∞, A_2x _∞} = x _∞ for every x∈ R^2. In contrast, for every positive-definite quadratic metric Q, max{ A_1 _Q, A_2 _Q}≥sqrt2. Equality holds for scalar multiples of the Euclidean metric. Hence the exact optimal convex/Finsler bound is 1, while the exact optimal quadratic bound is sqrt2.
Exact mathematical passage
x\in\mathbb R^2
\mathbb R^dUse in source · 1 passage
Search aliases: R^d mathbb
Use in source · block18_carrier_observer_change_final.tex, line 333
Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.
Exact mathematical passage
C_s\subseteq\mathbb R^d
R^{d+1}Use in source · 1 passage
Search aliases: R^d+1
Use in source · Stratified_Universality_Traditional.tex, line 5641
The same integer kernel matrix has different categorical status after changing the coefficient doctrine. Fix d≥0 and write M_d(R)=R^d+1 for the formal truncated Mahler coefficient module. Let A_M,s^(d) =[Θ_M,s(r,k)]_0≤ k,r≤ d, where the target degree k is the row index.
Exact mathematical passage
\mathcal M_d(R)=R^{d+1}
R^mUse in source · 1 passage
Search aliases: R^m
Use in source · block11_section.tex, line 71
The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.
Exact mathematical passage
\lVert\Psi(x)\rVert \le a+\sum_{m=2}^{D}b_mR^m \le R.
R^{m-1}Use in source · 1 passage
Search aliases: R^m-1
Use in source · block11_section.tex, line 43
Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.
Exact mathematical passage
\kappa_R=\varphi'(R) =\sum_{m=2}^{D}m b_mR^{m-1}.
\mathbb R^\timesUse in source · 1 passage
Search aliases: R^× mathbb times
Use in source · block12_section.tex, line 463
-1 · 0 . Then A_σσ(A_σ)=-I. The projective cocycle condition holds because -I is scalar. Its boundary is represented by -1 in R^×/N_ C/ R( C^×). Every complex norm is positive, so -1 is nontrivial. The descended central simple algebra is the Hamilton quaternion algebra. Replacing A_σ by 0 · 1
Exact mathematical passage
\mathbb R^\times/N_{\mathbb C/\mathbb R}(\mathbb C^\times).
\mathfrak R^{\simeq}Use in source · 1 passage
Search aliases: R^≃ mathfrak simeq
Use in source · Stratified_Universality_Traditional.tex, line 1911
[label=textup(roman*)] item Desc( R) exists and has the universal property of the finite pseudolimit of R. item There is a canonical equivalence Core(Desc( R)) ≃ Desc( R^≃). item Every pseudonatural transformation T: R⇒ S induces a functor Desc(T):Desc( R)→Desc( S). If every T_U is an equivalence, then Desc(T) is an equivalence. A pointwise noninvertible T remains directional. item If two global morphism expressions use only constructors in the common verified preservation profile and their equa…
Exact mathematical passage
\Core\bigl(\Desc(\mathfrak R)\bigr) \simeq \Desc\bigl(\mathfrak R^{\simeq}\bigr).
r_0Use in source · 1 passage
Search aliases: r_0
Use in source · block11_section.tex, line 91
Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum. Regrouping trees by the arity of the root gives equation textup(11.1)…
Exact mathematical passage
r_0=0
r_1Use in source · 2 passages
Search aliases: r_1
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
r_3Use in source · 1 passage
Search aliases: r_3
Use in source · Stratified_Universality_Traditional.tex, line 3372
A genuine denominator example. Over F_3, take R(z)=frac11+z. Then F(z)^2=1+z. The reduced denominator remains fixed under every Cartier section. The ambient compiler has dimension two, its reachable numerator set is {0,1,2,1+z}, and its reachable span has dimension two. Hence r_3(F)=2, κ_3(F)=4. The nonzero collapse obstruction again proves that the automatic coefficient sequence is not eventually periodic.
Exact mathematical passage
r_3(F)=2, \qquad \kappa_3(F)=4.
r_AUse in source · 2 passages
Search aliases: r_A
Use in source · block16_section_final.tex, line 185
v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, mathscr M^r_B,r_A_B,A), and composition ⋆ is isomorphic, by analysis and synthesis, to the correspon…
Exact mathematical passage
r_A
r_aUse in source · 1 passage
Search aliases: r_a
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
r_BUse in source · 1 passage
Search aliases: r_B
Use in source · block16_section_final.tex, line 187
_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, mathscr M^r_B,r_A_B,A), and composition ⋆ is isomorphic, by analysis and synthesis, to the corresponding category of lane-decorated descent objects and their underlying descend…
Exact mathematical passage
\Gamma\!\left(\mathscr G, \mathscr M^{r_B,r_A}_{B,A}\right),
r_cUse in source · 1 passage
Search aliases: r_c
Use in source · Stratified_Universality_Traditional.tex, line 537
[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.
Exact mathematical passage
\mathcal R=(r_c)
\mathfrak R_{\mathrm{DG}}Use in source · 2 passages
Search aliases: R_DG mathfrak mathrm
Use in source · Stratified_Universality_Traditional.tex, line 5284
[Effective descent on the divisor--ghost overlap] The exponent, fixed Euler-product, multiplicative ghost, and prime-power seed charts form a finite normalized atlas R_DG. Every bridge is reversible on the declared overlap, every multiple-overlap comparison is the unique identity over the represented exponent function, and Desc( R_DG) ≃ EulMult_ Z. Descent is effective for objects and morphisms.
Exact mathematical passage
\mathfrak R_{\mathrm{DG}}.
r_eUse in source · 5 passages
Search aliases: r_e
Use in source · block15_section.tex, line 239
Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of Φ_κ,τ, so the fold is split lossless.
Exact mathematical passage
r_e:C\longrightarrow B_e, \qquad r_ej_{\kappa(e)}\tau_e=1_{B_e},
R_E^{u,r}Use in source · 1 passage
Search aliases: R_E^u,r
Use in source · block15_section.tex, line 164
item Selectors and named residuals. For E⊆ I_Y,u× I_X,r, put C_E^u,r(f) :=∑_(a,i)∈ Ep_Y,a^ufp_X,i^r, R_E^u,r(f):=f-C_E^u,r(f). As additive endomorphisms of Hom_ C(X,Y), C_E^2=C_E, C_EC_F=C_E∩ F, R_E=1-C_E, f=C_E(f)+R_E(f). Thus deletion of a lane block is never silent. Its complement is an exact, named residual.
Exact mathematical passage
C_E^{u,r}(f) :=\sum_{(a,i)\in E}p_{Y,a}^ufp_{X,i}^r, \qquad R_E^{u,r}(f):=f-C_E^{u,r}(f).
R_FUse in source · 11 passages
Search aliases: R_F
Use in source · Stratified_Universality_Traditional.tex, line 690
[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .
Exact mathematical passage
f_E=C_E(f),\quad r_f=R_E(f), \qquad g_F=C_F(g),\quad r_g=R_F(g).
r_fUse in source · 7 passages
Search aliases: r_f
Use in source · Stratified_Universality_Traditional.tex, line 690
[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .
Exact mathematical passage
f_E=C_E(f),\quad r_f=R_E(f), \qquad g_F=C_F(g),\quad r_g=R_F(g).
R_F^{-1}Use in source · 1 passage
Search aliases: R_F^-1
Use in source · Stratified_Universality_Traditional.tex, line 2896
In characteristic p, Frobenius gives F(z^p)=F(z)^p. Substitution into [source reference] gives [source reference]. Thus R_F is rational exactly when F^p-1=R_F^-1 is rational. Writing the reduced rational mask as A/B gives [source reference] and [source reference].
Exact mathematical passage
F^{p-1}=R_F^{-1}
r_gUse in source · 7 passages
Search aliases: r_g
Use in source · Stratified_Universality_Traditional.tex, line 690
[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .
Exact mathematical passage
f_E=C_E(f),\quad r_f=R_E(f), \qquad g_F=C_F(g),\quad r_g=R_F(g).
r_hUse in source · 2 passages
Search aliases: r_h
Use in source · block11_section.tex, line 92
Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum. Regrouping trees by the arity of the root gives equation textup(11.1)…
Exact mathematical passage
r_{h+1}=a+\varphi(r_h),
r_{h+1}Use in source · 1 passage
Search aliases: r_h+1
Use in source · block11_section.tex, line 92
Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum. Regrouping trees by the arity of the root gives equation textup(11.1)…
Exact mathematical passage
r_{h+1}=a+\varphi(r_h),
r_jUse in source · 2 passages
Search aliases: r_j
Use in source · block09_section.tex, line 58
Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
Exact mathematical passage
R_a = S^s+\sum_{j=0}^{s-1}r_j(n)S^j.
r_nUse in source · 4 passages
Search aliases: r_n
Use in source · Stratified_Universality_Traditional.tex, line 4831
Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, the…
Exact mathematical passage
A(z)\prod_{m<n}(1-z^m)^{b_m} \equiv 1+r_nz^n\pmod{z^{n+1}}.
R_{N,j}Use in source · 1 passage
Search aliases: R_N,j
Use in source · block09_section.tex, line 782
[Effective marked binomial--Fuchsian atlas] The five normalized presentations above determine one another on the declared marked family. With equality morphisms they form equivalent discrete groupoids, every normalized fiber is contractible, and objects and morphisms descend effectively. The package J_rm BF = ( α,λ,N,j, R_N,j,L_N,j, A_N,j(λ),B_N,j, mathscr E(G), r_rm orb,r_D,A_N,C_N,j ) is complete on this marked category.
Exact mathematical passage
\boxed{ \mathcal J_{\rm BF} = \left( \alpha,\lambda,N,j, R_{N,j},L_{N,j}, A_{N,j}(\lambda),B_{N,j}, \mathscr E(G), r_{\rm orb},r_D,A_N,C_{N,j} \right)}
R_{p^a,s}Use in source · 2 passages
Search aliases: R_p^a,s
Use in source · block15_section.tex, line 907
Piecewise synthesis on the disjoint clopen cosets s+p^a Z_p is its inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix composition law. Let M:= M_p^-1:C( Z_p, Z_p) xrightarrow∼c_0( Z_p) be Mahler analysis from [source reference], and write R_p^a,sf(q):=f(p^aq+s). Conjugation gives the exact coefficient operator T_p^a,s := M R_p^a,s M^-1: c_0( Z_p)⟶ c_0( Z_p), whose affine kernel is computed in [source reference]. Its mixed-radix composition i…
Exact mathematical passage
R_{p^a,s}f(q):=f(p^aq+s)
r_qUse in source · 2 passages
Search aliases: r_q
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
r_{\rm orb}Use in source · 7 passages
Search aliases: r_rm orb rm
Use in source · block09_section.tex, line 575
Let ω_N be primitive and consider the rotated orbit O_N(F_λ,α) = {(1-αω_N^rt)^-λ:0≤ r<N}. Let r_rm orb be its dimension over the constant field.
Exact mathematical passage
r_{\rm orb}
r_{\rm tail}Use in source · 1 passage
Search aliases: r_rm tail rm
Use in source · block09_examples_table.tex, line 8
[htbp] centering small captionRepresentative exact order profiles in the binomial--Fuchsian packet family. textwidth@L0.17textwidthL0.13textwidthcccccY@ Source · Packet · r_rm tail · r_D · r_rm orb · Alias · Cancel · Interpretation
Exact mathematical passage
r_{\rm tail}
R_sUse in source · 1 passage
Search aliases: R_s
Use in source · block19_admissibility_lifting_final.tex, line 74
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
Q_s=R_s^*R_s
R_s^*Use in source · 1 passage
Search aliases: R_s^*
Use in source · block19_admissibility_lifting_final.tex, line 74
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
Q_s=R_s^*R_s
R_s^{-1}Use in source · 1 passage
Search aliases: R_s^-1
Use in source · block19_admissibility_lifting_final.tex, line 75
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
B_Q(T):=R_tTR_s^{-1}.
R_tUse in source · 2 passages
Search aliases: R_t
Use in source · block19_admissibility_lifting_final.tex, line 74
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
Q_t=R_t^*R_t
R_t^*Use in source · 1 passage
Search aliases: R_t^*
Use in source · block19_admissibility_lifting_final.tex, line 74
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
Q_t=R_t^*R_t
\operatorname{rank}Use in source · 1 passage
Search aliases: rank operatorname
Use in source · block12_section.tex, line 264
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
d_j=\operatorname{rank}D_j
\rankUse in source · 7 passages
Search aliases: rank operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1504
; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies; item for Δ_S:F_o⟶⊕_e∉ SF_o, Δ_Sx=((H_e-I)x)_e∉ S, one has C_o=kerΔ_S, dim C_o=dim F_o-rankΔ_S; item the full fibers glue to one global frame exactly when every fundamental holonomy is the identity; item a gauge change conjugate…
Exact mathematical passage
C_o=\ker\Delta_S, \qquad \dim C_o=\dim F_o-\rank\Delta_S;
\rank_{\mathbb F_p}Use in source · 1 passage
Search aliases: rank_ F_p rank mathbb operatorname mathbb
Use in source · Stratified_Universality_Traditional.tex, line 5666
e det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac dβ_prfloor+1. A local channel loses residual rank exactly at primes dividing M. item A complete neighboring packet can recover the source…
Exact mathematical passage
\boxed{ \rank_{\mathbb F_p}\overline A_{M,s}^{(d)} =\left\lfloor\frac d{\beta_p}\right\rfloor+1.}
\rank_KUse in source · 2 passages
Search aliases: rank_K rank operatorname
Use in source · Stratified_Universality_Traditional.tex, line 3664
[Regular rank theorem] Let a be k-regular over K. Then r_k(a) = rank_K H_a = min{dim_K W:W supports a digit-linear realization of a}. Moreover: item there exist finite word sets U,V⊂Σ_k^*, each of cardinality r_k(a), for which the minor H_a[U,V] is nonsingular; item every reachable-observable realization of dimension r_k(a) is isomorphic to C_k(a); item any two minimal reachable-observable realizations are related by a unique similarity intertwining all digit operators, the initial vector, and t…
Exact mathematical passage
\boxed{ r_k(a) = \rank_K H_a = \min\{\dim_K W:W\text{ supports a digit-linear realization of }a\}. } \label{eq:block05-rank-identity}
\RatUse in source · 8 passages
Search aliases: Rat mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2056
An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.
Exact mathematical passage
\Rat_K^{\mathrm{red}}
\mathrm{rat}Use in source · 2 passages
Search aliases: rat mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3203
Reduced mask chart. An object of Mask_p^rat is a reduced rational function R(z)=fracA(z)B(z), A(0)=B(0)=1. Morphisms preserve the rational function. The semantics is R↦ M_p[R]. The inverse chart map is the canonical ratio F↦ F/F(z^p).
Exact mathematical passage
\Mask_p^{\mathrm{rat}}
\Rat_K^{\mathrm{red}}Use in source · 3 passages
Search aliases: Rat_K^red Rat mathrm mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2056
An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.
Exact mathematical passage
\Rat_K^{\mathrm{red}}
\Rat_p^{\times,\mathrm{per}}Use in source · 4 passages
Search aliases: Rat_p^×,per Rat times mathrm mathbf times mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2309
An object of Rat_p^×,per is a reduced fraction F(z)=fracP(z)Q(z)∈ F_p(z), P(0)=Q(0)=1, whose coefficient sequence is purely periodic. Equivalently, Qmid 1-z^T for some T and the reduced coefficient sequence has no transient prefix. Morphisms preserve the rational function. Reducedness and the constant-term normalization make this chart rigid.
Exact mathematical passage
\Rat_p^{\times,\mathrm{per}}
\mathrm{raw}Use in source · 3 passages
Search aliases: raw mathrm
Use in source · Stratified_Universality_Traditional.tex, line 4523
Let O_k(a)={Λ_w a:w∈Σ_k^*}⊆ M_k(a) be the actual section orbit, and let O^raw_k,p^r(a) be its image in M_k(a)/p^rM_k(a).
Exact mathematical passage
\mathcal O^{\mathrm{raw}}_{k,p^r}(a)
\rceilUse in source · 2 passages
Search aliases: rceil
Use in source · Stratified_Universality_Traditional.tex, line 2570
Let R(z)=fracA(z)B(z), A(0)=B(0)=1, with a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source reference]. It computes the coefficient sequence exactly.
Exact mathematical passage
\label{eq:cartier-degree-bound-block03} D=\max\left\{b,\left\lceil\frac{c}{p-1}\right\rceil\right\}.
\RecUse in source · 6 passages
Search aliases: Rec mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2039
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\Rec_K^{\mathrm{min}}
\Rec_K^{\mathrm{min}}Use in source · 3 passages
Search aliases: Rec_K^min Rec mathrm mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2039
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\Rec_K^{\mathrm{min}}
\Rec_p^{\times,\mathrm{per}}Use in source · 2 passages
Search aliases: Rec_p^×,per Rec times mathrm mathbf times mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2300
An object of Rec_p^×,per is the normalized minimal recurrence polynomial m(x)=x^d+c_1x^d-1+⋯+c_d, c_d≠ 0, together with the initial vector (a_0,…,a_d-1), subject to a_0=1 and the requirement that the generated sequence is purely periodic. Minimality means that no monic recurrence of lower degree annihilates the sequence. Morphisms are recurrence-presentation isomorphisms preserving the coefficient sequence. Because the minimal annihilator and initial data are unique, every endomorphism is the id…
Exact mathematical passage
\Rec_p^{\times,\mathrm{per}}
\mathrm{red}Use in source · 3 passages
Search aliases: red mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2056
An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.
Exact mathematical passage
\Rat_K^{\mathrm{red}}
\mathbf{Reg}_pUse in source · 2 passages
Search aliases: Reg_p mathbf
Use in source · block13_section.tex, line 176
Fix the prime p and let O_DRLM =Reg_p( F_p) be the discrete semantic category of p-regular sequences over F_p. By the finite-field fixed-radix theorem, every object is also p-automatic. Its globally defined normalized charts are the canonical section module, the strict shortlex pointed digit-linear record, and the minimal least-significant-digit-first p-DFAO.
Exact mathematical passage
\mathcal O_{\mathrm{DRLM}} =\mathbf{Reg}_p(\mathbb F_p)
\RelUse in source · 3 passages
Search aliases: Rel
Use in source · block18_carrier_observer_change_final.tex, line 274
[Observer base-change exact sequence] The evaluation map π:K^(E)⟶Span_K(E), ∑_e∈ Ea_e[e] ⟼ ∑_e∈ Ea_e e is surjective and fits into the exact sequence 0⟶Rel_K(E) ⟶ K^(E) xrightarrowπSpan_K(E) ⟶0, where Rel_K(E):=kerπ is the module of linear relations among distinct discrete residual states. Hence Set minimization commutes with free linearization exactly when E is linearly independent.
Exact mathematical passage
0\longrightarrow\Rel_K(E) \longrightarrow K^{(E)} \xrightarrow{\pi}\Span_K(E) \longrightarrow0,
\Rel_KUse in source · 3 passages
Search aliases: Rel_K Rel
Use in source · block18_carrier_observer_change_final.tex, line 274
[Observer base-change exact sequence] The evaluation map π:K^(E)⟶Span_K(E), ∑_e∈ Ea_e[e] ⟼ ∑_e∈ Ea_e e is surjective and fits into the exact sequence 0⟶Rel_K(E) ⟶ K^(E) xrightarrowπSpan_K(E) ⟶0, where Rel_K(E):=kerπ is the module of linear relations among distinct discrete residual states. Hence Set minimization commutes with free linearization exactly when E is linearly independent.
Exact mathematical passage
0\longrightarrow\Rel_K(E) \longrightarrow K^{(E)} \xrightarrow{\pi}\Span_K(E) \longrightarrow0,
\mathfrak{Rep}Use in source · 1 passage
Search aliases: Rep mathfrak
Use in source · Stratified_Universality_Traditional.tex, line 1129
For every stratum, the invertible part is the maximal subgroupoid Rep_α( K) = Core( M_α). This core records changes of presentation that preserve all structure required by the overlap. Quotients, truncations, completions, specializations, differential annihilator maps, and compression maps generally remain outside it.
Exact mathematical passage
\mathfrak{Rep}_\alpha(\bbK) = \Core(\mathcal M_\alpha).
\operatorname{Rep}Use in source · 3 passages
Search aliases: Rep operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1757
[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].
Exact mathematical passage
\mathcal T\simeq\operatorname{Rep}_k(\mathcal G)
\operatorname{Rep}_kUse in source · 3 passages
Search aliases: Rep_k operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1757
[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].
Exact mathematical passage
\mathcal T\simeq\operatorname{Rep}_k(\mathcal G)
\mathfrak{Rep}_\alphaUse in source · 1 passage
Search aliases: Rep_α mathfrak alpha
Use in source · Stratified_Universality_Traditional.tex, line 1129
For every stratum, the invertible part is the maximal subgroupoid Rep_α( K) = Core( M_α). This core records changes of presentation that preserve all structure required by the overlap. Quotients, truncations, completions, specializations, differential annihilator maps, and compression maps generally remain outside it.
Exact mathematical passage
\mathfrak{Rep}_\alpha(\bbK) = \Core(\mathcal M_\alpha).
\operatorname{Res}Use in source · 2 passages
Search aliases: Res operatorname
Use in source · block10_section.tex, line 210
Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.
Exact mathematical passage
\operatorname{Res}_{E_N/R}\operatorname{Spec}B_{N,F}.
\operatorname{Res}_{E_N/R}Use in source · 1 passage
Search aliases: Res_E_N/R operatorname
Use in source · block10_section.tex, line 210
Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.
Exact mathematical passage
\operatorname{Res}_{E_N/R}\operatorname{Spec}B_{N,F}.
\operatorname{Res}_{H\cap x^{-1}Kx}^HUse in source · 1 passage
Search aliases: Res_H∩ x^-1Kx^H operatorname cap
Use in source · block12_section.tex, line 180
[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.
Exact mathematical passage
\boxed{ \operatorname{Res}_K^G\operatorname{Ind}_H^GW \cong \bigoplus_{x\in K\backslash G/H} \operatorname{Ind}_{K\cap xHx^{-1}}^K \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).} \tag{12.8}
\operatorname{Res}_K^GUse in source · 1 passage
Search aliases: Res_K^G operatorname
Use in source · block12_section.tex, line 180
[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.
Exact mathematical passage
\boxed{ \operatorname{Res}_K^G\operatorname{Ind}_H^GW \cong \bigoplus_{x\in K\backslash G/H} \operatorname{Ind}_{K\cap xHx^{-1}}^K \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).} \tag{12.8}
\rfloorUse in source · 4 passages
Search aliases: rfloor
Use in source · Stratified_Universality_Traditional.tex, line 3051
If deg P≤ D, then deg(PH) ≤ D+a+(p-2)b. Cartier selection divides the retained degrees by p. The definition of D gives deg T_d(P) ≤ lfloorfracD+a+(p-2)bprfloor ≤ D. This proves (i).
Exact mathematical passage
\deg T_d(P) \leq \left\lfloor\frac{D+a+(p-2)b}{p}\right\rfloor \leq D.
\RMahUse in source · 14 passages
Search aliases: RMah RMah mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2828
Fix a prime p. Let RMah_p^× = { F(z)∈ 1+z F_p[[z]]: fracF(z)F(z^p)∈ F_p(z) } be the discrete category of unit formal series with rational canonical functional-Mahler mask. This category contains the periodic objects of [source reference], but for odd p it also contains genuinely nonrational and non-C-finite automatic series.
Exact mathematical passage
\RMah_p^{\times} = \left\{ F(z)\in 1+z\mathbb F_p[[z]]: \frac{F(z)}{F(z^p)}\in\mathbb F_p(z) \right\}
\RMah_{p,\mathrm{col}}^{\times}Use in source · 2 passages
Search aliases: RMah_p,col^× RMah mathrm times mathbf mathrm times
Use in source · Stratified_Universality_Traditional.tex, line 3328
The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside RMah_p,col^×.
Exact mathematical passage
\RMah_p^{\times} = \RMah_{p,\mathrm{col}}^{\times} \sqcup \RMah_{p,\mathrm{ncol}}^{\times}.
\RMah_{p,\mathrm{ncol}}^{\times}Use in source · 1 passage
Search aliases: RMah_p,ncol^× RMah mathrm times mathbf mathrm times
Use in source · Stratified_Universality_Traditional.tex, line 3328
The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside RMah_p,col^×.
Exact mathematical passage
\RMah_p^{\times} = \RMah_{p,\mathrm{col}}^{\times} \sqcup \RMah_{p,\mathrm{ncol}}^{\times}.
\RMah_p^{\times}Use in source · 13 passages
Search aliases: RMah_p^× RMah times RMah_p^× RMah times mathbf times
Use in source · Stratified_Universality_Traditional.tex, line 2828
Fix a prime p. Let RMah_p^× = { F(z)∈ 1+z F_p[[z]]: fracF(z)F(z^p)∈ F_p(z) } be the discrete category of unit formal series with rational canonical functional-Mahler mask. This category contains the periodic objects of [source reference], but for odd p it also contains genuinely nonrational and non-C-finite automatic series.
Exact mathematical passage
\RMah_p^{\times} = \left\{ F(z)\in 1+z\mathbb F_p[[z]]: \frac{F(z)}{F(z^p)}\in\mathbb F_p(z) \right\}
\rVertUse in source · 3 passages
Search aliases: rVert
Use in source · block11_section.tex, line 35
Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.
Exact mathematical passage
a=\lVert s\rVert, \qquad b_m=\lVert A_m\rVert, \qquad \varphi(t)=\sum_{m=2}^{D}b_mt^m.
\mathfrak SUse in source · 2 passages
Search aliases: S mathfrak
Use in source · Stratified_Universality_Traditional.tex, line 1916
[label=textup(roman*)] item Desc( R) exists and has the universal property of the finite pseudolimit of R. item There is a canonical equivalence Core(Desc( R)) ≃ Desc( R^≃). item Every pseudonatural transformation T: R⇒ S induces a functor Desc(T):Desc( R)→Desc( S). If every T_U is an equivalence, then Desc(T) is an equivalence. A pointwise noninvertible T remains directional. item If two global morphism expressions use only constructors in the common verified preservation profile and their equa…
Exact mathematical passage
T:\mathfrak R\Rightarrow\mathfrak S
S^{-1}Use in source · 2 passages
Search aliases: S^-1
Use in source · Stratified_Universality_Traditional.tex, line 2096
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
C'=SCS^{-1}, \qquad v'=Sv, \qquad \lambda'=\lambda S^{-1}.
s^{-1}Use in source · 1 passage
Search aliases: s^-1
Use in source · block19_admissibility_lifting_final.tex, line 245
c · b, the equation H^*QH=Q forces a=0, contradicting positive definiteness. For Q_s, the metric-normalized matrix is s · 0 0 · 1 H s^-1 · 0 0 · 1 = 1 · s 0 · 1, whose singular-value condition number tends to one. The conditioning formula is immediate.
Exact mathematical passage
\begin{pmatrix}s&0\\0&1\end{pmatrix} H \begin{pmatrix}s^{-1}&0\\0&1\end{pmatrix} = \begin{pmatrix}1&s\\0&1\end{pmatrix},
s^{-2}Use in source · 1 passage
Search aliases: s^-2
Use in source · block19_admissibility_lifting_final.tex, line 237
0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.
Exact mathematical passage
\kappa_2(Q_s)=s^{-2}\longrightarrow\infty.
s^2Use in source · 4 passages
Search aliases: s^2
Use in source · block10_section.tex, line 433
[Quadratic-radical parity packets] The packets satisfy E^2+xO^2=1+px, 2EO=-s. Their eliminants are E^4-(1+px)E^2+fracs^2x4=0 and xO^4-(1+px)O^2+fracs^24=0. If s≠0, α^2≠β^2, then both selected packets have degree four over K(x).
Exact mathematical passage
\boxed{ E^4-(1+px)E^2+\frac{s^2x}{4}=0}
S^jUse in source · 1 passage
Search aliases: S^j
Use in source · block09_section.tex, line 58
Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
Exact mathematical passage
R_a = S^s+\sum_{j=0}^{s-1}r_j(n)S^j.
\mathfrak s^{\rm Gal}_{2,0}Use in source · 1 passage
Search aliases: s^rm Gal_2,0 mathfrak rm
Use in source · block10_section.tex, line 486
[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.
Exact mathematical passage
\mathfrak c^{\rm tuple}_{2,0}=1, \qquad \mathfrak s^{\rm Gal}_{2,0}=1.
\mathfrak s^{\rm Gal}_{2,1}Use in source · 1 passage
Search aliases: s^rm Gal_2,1 mathfrak rm
Use in source · block10_section.tex, line 498
r:block10-antisymmetric-collision If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.
Exact mathematical passage
\mathfrak c^{\rm tuple}_{2,1}=1, \qquad \mathfrak s^{\rm Gal}_{2,1}=2.
S^sUse in source · 1 passage
Search aliases: S^s
Use in source · block09_section.tex, line 58
Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
Exact mathematical passage
R_a = S^s+\sum_{j=0}^{s-1}r_j(n)S^j.
S_*Use in source · 1 passage
Search aliases: S_*
Use in source · Stratified_Universality_Traditional.tex, line 1997
Addendum 04 proves that the prefix S_*={0,1,…,L_ Z(m,d)-1} with L_ Z(m,d)=min(d+1,B(m)), B(m)=max({1}∪{p^v_p(m):pmid m}), always admits an integer decoder and is cardinality-minimal [source reference].
Exact mathematical passage
S_*=\{0,1,\dots,L_{\mathbb Z}(m,d)-1\}
S_0Use in source · 4 passages
Search aliases: S_0
Use in source · block15_section.tex, line 309
[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1
Exact mathematical passage
S_0:=\{e\in E:b_e\ne0\}, \qquad A_0:=\{e\in S_0:\tau_e(b_e)=0\}, \qquad S_1:=S_0\setminus A_0,
S_1Use in source · 7 passages
Search aliases: S_1
Use in source · block15_section.tex, line 309
[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1
Exact mathematical passage
S_0:=\{e\in E:b_e\ne0\}, \qquad A_0:=\{e\in S_0:\tau_e(b_e)=0\}, \qquad S_1:=S_0\setminus A_0,
s_2Use in source · 2 passages
Search aliases: s_2
Use in source · Stratified_Universality_Traditional.tex, line 3422
2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, contradicting pure periodicity.
Exact mathematical passage
m-\nu-s_2(q-1)
S_3Use in source · 5 passages
Search aliases: S_3
Use in source · block12_section.tex, line 165
[The S_3 nonnormal atlas] Let G=S_3 and let H be generated by one transposition. There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts. Their subgroup core is trivial, while their normal closure is all of S_3. By contrast, the order-three subgroup A_3 is normal and its fixed chart carries the quotient symmetry S_3/A_3≅ C_2.
Exact mathematical passage
S_3
\mathcal S_aUse in source · 1 passage
Search aliases: S_a mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5493
[Complete clopen packet equivalence] The analysis map B_a: C_p ⟶ ∏_s=0^p^a-1 C_p is an isometric isomorphism. Its inverse is the piecewise synthesis ( S_a g)(x) =g_s (fracx-sp^a) for x∈ s+p^a Z_p. Thus the clopen residue cover of Z_p has effective descent for both objects and equality morphisms.
Exact mathematical passage
\boxed{ (\mathcal S_a g)(x) =g_s\!\left(\frac{x-s}{p^a}\right) \quad\text{for }x\in s+p^a\mathbb Z_p.}
S_dUse in source · 2 passages
Search aliases: S_d
Use in source · block15_section.tex, line 821
[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
\Lambda_d\left(\sum_{n\ge0}a_nz^n\right) :=\sum_{n\ge0}a_{kn+d}z^n, \qquad S_d(G):=z^dG(z^k), \quad 0\le d<k.
S_eUse in source · 1 passage
Search aliases: S_e
Use in source · block15_section.tex, line 829
[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
\Lambda_dS_e=\delta_{de}1, \qquad \sum_{d=0}^{k-1}S_d\Lambda_d=1.
s_iUse in source · 2 passages
Search aliases: s_i
Use in source · block11_section.tex, line 129
For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
u_i=P_iu, \qquad s_i=P_is.
s_kUse in source · 25 passages
Search aliases: s_k
Use in source · Stratified_Universality_Traditional.tex, line 3940
[Digit-sum regularity] Let s_k(n) be the sum of the base-k digits of n, regarded as a sequence over a characteristic-zero field. Then r_k(s_k)=2, and K_k(s_k) = {s_k+c:c∈ Z_≥0}. At exact depth ℓ, the number of distinct sections is (k-1)ℓ+1. Thus s_k is k-regular but not k-automatic over a characteristic-zero field.
Exact mathematical passage
s_k(n)
s_{\max}Use in source · 1 passage
Search aliases: s_max max
Use in source · Stratified_Universality_Traditional.tex, line 3986
First, it exhaustively enumerates bounded deterministic automata in three configurations: (k,q,s_max)=(2,2,3), (2,3,3), (3,2,2). Every candidate is restricted to reachable states, minimized by behavioral equivalence, and filtered by the terminal-high-zero consistency required by integer semantics. For every resulting normalized automaton the verifier checks: item reachability and pairwise behavioral distinguishability; item construction of the canonical linear quotient from continuation-output r…
Exact mathematical passage
(k,q,s_{\max})=(2,2,3),\qquad(2,3,3),\qquad(3,2,2).
S_UUse in source · 2 passages
Search aliases: S_U
Use in source · block16_section_final.tex, line 387
item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.
Exact mathematical passage
Q_U=\sum_{i\in S_U}p^r_{A,U,i}
S_{U_0}Use in source · 2 passages
Search aliases: S_U_0
Use in source · block16_section_final.tex, line 391
item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.
Exact mathematical passage
S_{U_0}
S_VUse in source · 1 passage
Search aliases: S_V
Use in source · block16_section_final.tex, line 390
item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.
Exact mathematical passage
S_V=\lambda^A_{\gamma,r}(S_U)
S_zUse in source · 4 passages
Search aliases: S_z
Use in source · block18_carrier_observer_change_final.tex, line 293
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
S_z
S_z^kUse in source · 1 passage
Search aliases: S_z^k
Use in source · block18_carrier_observer_change_final.tex, line 318
Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving the determinant formula.
Exact mathematical passage
e_0^{\mathsf T}S_z^k
S_z^NUse in source · 1 passage
Search aliases: S_z^N
Use in source · block18_carrier_observer_change_final.tex, line 299
[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
S_z^N=zI
\operatorname{Sat}Use in source · 1 passage
Search aliases: Sat operatorname
Use in source · block17_semantic_quotients_final.tex, line 223
[Task--congruence Galois connection] For every query family Q and relation R, Q⊆Def(R) Longleftrightarrow R⊆Ind( Q). Consequently, Cl( Q):=Def(Ind( Q)) is a closure operator on query families, and Sat(R):=Ind(Def(R)) is the observational saturation of R relative to the admitted query universe.
Exact mathematical passage
\operatorname{Sat}(R):=\Ind(\Def(R))
\operatorname{SCC}Use in source · 1 passage
Search aliases: SCC operatorname
Use in source · block11_section.tex, line 276
For the packet system textup(11.4), define the directed dependency graph G_Φ on I by placing an edge j⟶ i when some active block contributing to the equation for u_i contains u_j as an input. Let SCC(G_Φ) be its strongly connected components. A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.
Exact mathematical passage
\operatorname{SCC}(G_\Phi)
\mathbf{Sec}Use in source · 2 passages
Search aliases: Sec mathbf
Use in source · Stratified_Universality_Traditional.tex, line 3782
[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence. Then Reg_k(K) ≃ Sec_k^can(K) ≃ Lin_k,K^sl. Every normalized fiber over a fixed sequence is contractible. Consequently the associated finite presenta…
Exact mathematical passage
\mathbf{Sec}_k^{\mathrm{can}}(K)
\mathbf{Sec}_k^{\mathrm{can}}Use in source · 2 passages
Search aliases: Sec_k^can mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3782
[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence. Then Reg_k(K) ≃ Sec_k^can(K) ≃ Lin_k,K^sl. Every normalized fiber over a fixed sequence is contractible. Consequently the associated finite presenta…
Exact mathematical passage
\mathbf{Sec}_k^{\mathrm{can}}(K)
\operatorname{Semilin}Use in source · 3 passages
Search aliases: Semilin operatorname
Use in source · block12_section.tex, line 342
Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.
Exact mathematical passage
L\otimes_K-:\operatorname{Vect}_K\longrightarrow\operatorname{Semilin}_G(L),
\operatorname{Semilin}_GUse in source · 3 passages
Search aliases: Semilin_G operatorname
Use in source · block12_section.tex, line 342
Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.
Exact mathematical passage
L\otimes_K-:\operatorname{Vect}_K\longrightarrow\operatorname{Semilin}_G(L),
\SeqUse in source · 3 passages
Search aliases: Seq mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2035
Let Seq_K^pers be the discrete groupoid of persistent C-finite sequences over K.
Exact mathematical passage
\Seq_K^{\mathrm{pers}}
\Seq_K^{\mathrm{pers}}Use in source · 3 passages
Search aliases: Seq_K^pers Seq mathrm mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2035
Let Seq_K^pers be the discrete groupoid of persistent C-finite sequences over K.
Exact mathematical passage
\Seq_K^{\mathrm{pers}}
\mathrm{sl}Use in source · 3 passages
Search aliases: sl mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3782
[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence. Then Reg_k(K) ≃ Sec_k^can(K) ≃ Lin_k,K^sl. Every normalized fiber over a fixed sequence is contractible. Consequently the associated finite presenta…
Exact mathematical passage
\mathbf{Lin}_{k,K}^{\mathrm{sl}}
\mathsf{SL}_{p,\mathbb F_p}Use in source · 2 passages
Search aliases: SL_p, F_p mathsf mathbb
Use in source · block13_section.tex, line 189
For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise contains the primitive period word, exact period, minimal recurrence, reduced OGF, and periodic packet profile.
Exact mathematical passage
\mathcal J_{\mathrm{DRLM},p}(a) = \left( \mathsf{SL}_{p,\mathbb F_p}(a), \mathcal P_{\mathrm{rm}}(a), \mathcal P_{\mathrm{per}}(a) \right),
\operatorname{span}_{\mathbb F_p}Use in source · 2 passages
Search aliases: span_ F_p operatorname mathbb
Use in source · Stratified_Universality_Traditional.tex, line 3069
For a digit word w=d_0⋯ d_ℓ-1, read least significant digit first and write Λ_w = Λ_d_ℓ-1⋯Λ_d_0. Define the Cartier kernel and its linear span by K_p(F) = {Λ_wF:w∈{0,…,p-1}^*}, V_p(F) = span_ F_p K_p(F).
Exact mathematical passage
\label{eq:block04-kernel-span} \mathcal K_p(F) = \{\Lambda_wF:w\in\{0,\ldots,p-1\}^{*}\}, \qquad \mathcal V_p(F) = \operatorname{span}_{\mathbb F_p}\mathcal K_p(F).
\operatorname{span}_{\mathbb Q}Use in source · 1 passage
Search aliases: span_ Q operatorname mathbb
Use in source · Stratified_Universality_Traditional.tex, line 4025
Throughout this section, k≥2 is fixed and a: N⟶ Z is an integer-valued sequence whose rational section space V_ Q(a) = span_ Q K_k(a) ⊆ Q^ N has finite dimension ρ_k(a)=dim_ QV_ Q(a). Thus a is k-regular after extension of scalars to Q.
Exact mathematical passage
V_{\mathbb Q}(a) = \operatorname{span}_{\mathbb Q}\mathcal K_k(a) \subseteq \mathbb Q^{\mathbb N}
\operatorname{span}_{K(z)}Use in source · 2 passages
Search aliases: span_K(z) operatorname
Use in source · block09_section.tex, line 74
[Derivative-module rank] For a nonzero D-finite series F, r_D(F) = dim_K(z)span_K(z) {F,DF,D^2F,…}.
Exact mathematical passage
\boxed{ r_D(F) = \dim_{K(z)}\operatorname{span}_{K(z)} \{F,DF,D^2F,\ldots\}.}
\operatorname{Spec}Use in source · 1 passage
Search aliases: Spec operatorname
Use in source · block10_section.tex, line 210
Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.
Exact mathematical passage
\operatorname{Res}_{E_N/R}\operatorname{Spec}B_{N,F}.
\operatorname{spec}Use in source · 1 passage
Search aliases: spec operatorname
Use in source · block19_admissibility_lifting_final.tex, line 103
[Coordinate covariance and composition] Simultaneous coordinate transport of T,Q_s,Q_t leaves D_Q(T) invariant. For composable invertible maps, D_Q(T_2T_1) ≤ D_Q(T_2)D_Q(T_1), so Δ_Q(T_2T_1) ≤ Δ_Q(T_2)+Δ_Q(T_1). For a closed word H, D_Q(H) ≥ fracmax_λ∈spec(H)|λ| min_λ∈spec(H)|λ|.
Exact mathematical passage
D_Q(H) \geq \frac{\max_{\lambda\in\operatorname{spec}(H)}|\lambda|} {\min_{\lambda\in\operatorname{spec}(H)}|\lambda|}.
\sqcupUse in source · 4 passages
Search aliases: sqcup
Use in source · Stratified_Universality_Traditional.tex, line 3328
The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside RMah_p,col^×.
Exact mathematical passage
\RMah_p^{\times} = \RMah_{p,\mathrm{col}}^{\times} \sqcup \RMah_{p,\mathrm{ncol}}^{\times}.
\StateUse in source · 3 passages
Search aliases: State mathbf
Use in source · Stratified_Universality_Traditional.tex, line 2081
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
\State_K^{\mathrm{min}}
\State_K^{\mathrm{min}}Use in source · 2 passages
Search aliases: State_K^min State mathrm mathbf mathrm
Use in source · Stratified_Universality_Traditional.tex, line 2081
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
\State_K^{\mathrm{min}}
\State_{\min}Use in source · 1 passage
Search aliases: State_min State min mathbf min
Use in source · Stratified_Universality_Traditional.tex, line 2698
For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)| ≤ T(a)(ν_p(T(a))+ord_M(p)), R_a(z) · =fracF_a(z)F_a(z^p), T(a)=p^ν_p(T(a))M, gcd(M,p)=1.
Exact mathematical passage
\label{eq:periodic-invariant-identities} \begin{aligned} \deg m_a &=\deg Q_a =\rank H(a) =\dim \State_{\min}(a),\\ T(a) &=\min\{N:m_a\mid x^N-1\} =\min\{N:Q_a\mid 1-z^N\},\\ |\mathcal A_p(a)| &=|\mathcal K_p(a)| \leq T(a)\bigl(\nu_p(T(a))+\operatorname{ord}_{M}(p)\bigr),\\ R_a(z) &=\frac{F_a(z)}{F_a(z^p)}, \qquad T(a)=p^{\nu_p(T(a))}M, \quad \gcd(M,p)=1. \end{aligned}
\mathrm{strong}Use in source · 1 passage
Search aliases: strong mathrm
Use in source · block19_admissibility_lifting_final.tex, line 307
Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders and should not be merged into one scalar notion of a stronger contract.
Exact mathematical passage
N_{\mathrm{strong}}\subseteq N_{\mathrm{weak}}, \qquad Q_{\mathrm{strong}}\twoheadrightarrow Q_{\mathrm{weak}}.
\subsubsectionUse in source · 1 passage
Search aliases: subsubsection
Use in source · Stratified_Universality_Traditional.tex, line 2538
[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart [4pt] Reduced OGF ↔ rational mask · equivalence on the unit-periodic essential image · F(0)=1, periodic rational F, reduced regular mask R · exact se…
Exact mathematical passage
4pt] Reduced OGF \(\leftrightarrow\) minimal DFAO & equivalence on the periodic essential image & unit purely periodic series and reachable minimal LSD-first \(p\)-DFAOs & coefficient semantics, minimal period, canonical \(p\)-kernel, output behavior & equality of represented sequences & a nonminimal residue-pair automaton is quotiented before entering the normalized chart\\[4pt] Reduced OGF \(\leftrightarrow\) rational mask & equivalence on the unit-periodic essential image & \(F(0)=1\), periodic rational \(F\), reduced regular mask \(R\) & exact series, functional equation, reduced rational mask, minimal period & equality of series and masks & undefined for nonunit series; unrestricted rational masks may have nonperiodic products\\[4pt] Mask \(\to\) Cartier numerator representation & constructor & reduced \(A/B\) over \(\mathbb F_p\) & exact coefficient semantics, digit matrices, theorem degree bound & not generally reflective & introduces basis and denominator coordinates and may contain unreachable ambient states\\[4pt] Ambient Cartier compiler \(\to\) minimal DFAO & reachable restriction & full finite numerator-state compiler & every reachable coefficient section and every future digit behavior & equality of the represented \(p\)-kernel states & removes only ambient states outside the orbit of the initial numerator\\[4pt] Unrestricted automatic \(\dashrightarrow\) rational mask & partial bridge only & automatic sequences whose unit inverse mask is rational and whose product lies in the declared periodic class & coefficient semantics on the domain of definition & no unrestricted reflection theorem & no claim that every automatic sequence has a rational periodic mask\\ \end{longtable} \end{landscape} \subsubsection{The rational-to-digital construction} For a periodic word \(w\) of length \(T\), construct the finite pair automaton from \cref{eq:pair-transition}, then quotient by equality of the induced subsequences. This gives the canonical minimal DFAO directly. Conversely, a minimal DFAO in the selected chart is promised to produce a purely periodic sequence. Its minimal period can be recovered exactly by enumerating \(N=1,2,\ldots\), constructing the finite-state transducer for addition by \(N\), and testing equivalence between the original output and its \(N\)-shift. Equivalence of finite deterministic output automata is a finite product-graph calculation. The promise of pure periodicity guarantees termination. Once \(T\) is known, \cref{eq:periodic-ogf-unreduced} gives the rational series, which is then reduced. This inverse is effective but not claimed to be complexity optimal. The point is effectivity, not a sharp automata algorithm. \subsubsection{The rational-mask compiler lane} Let \[ R(z)=\frac{A(z)}{B(z)}, \qquad A(0)=B(0)=1,
\supUse in source · 1 passage
Search aliases: sup
Use in source · block12_section.tex, line 315
For each v∈ H, strong continuity gives sup_u∈ U_n U_uv-v ⟶0. Averaging this estimate over U_n proves the first statement. Apply it to the quotient action on the N-fixed packet for the second.
Exact mathematical passage
\sup_{u\in U_n}\|U_uv-v\|\longrightarrow0.
\sup_{u\in U_n}Use in source · 1 passage
Search aliases: sup_u∈ U_n sup in
Use in source · block12_section.tex, line 315
For each v∈ H, strong continuity gives sup_u∈ U_n U_uv-v ⟶0. Averaging this estimate over U_n proves the first statement. Apply it to the quotient action on the N-fixed packet for the second.
Exact mathematical passage
\sup_{u\in U_n}\|U_uv-v\|\longrightarrow0.
\operatorname{Supp}Use in source · 1 passage
Search aliases: Supp operatorname
Use in source · block11_section.tex, line 364
N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10
Exact mathematical passage
\boxed{ \#\operatorname{Supp}_N(u)=\frac{N}{g_N}.} \tag{11.10}
\operatorname{supp}Use in source · 4 passages
Search aliases: supp operatorname
Use in source · block15_section.tex, line 328
kappa(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈Γ:overline b_t=0}. Then |S_0|-|supp(overline b)| = |A_0| +(|S_1|-|Γ|) +|K_0|. The three terms record, respectively, annihilation by the local maps τ_e, collision of surviving lane addresses under κ, and cancellation among folded values at a common address. If every τ_e is injective, then A_0=∅. If globally typed selectors satisfy the strict separation law of (L8), then K_0=∅.
Exact mathematical passage
\operatorname{supp}(\overline b) :=\{t\in T:\overline b_t\ne0\}, \qquad K_0:=\{t\in\Gamma:\overline b_t=0\}.
\operatorname{Supp}_NUse in source · 1 passage
Search aliases: Supp_N operatorname
Use in source · block11_section.tex, line 364
N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10
Exact mathematical passage
\boxed{ \#\operatorname{Supp}_N(u)=\frac{N}{g_N}.} \tag{11.10}
\operatorname{Sw}Use in source · 1 passage
Search aliases: Sw operatorname
Use in source · block12_section.tex, line 259
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
\boxed{ \operatorname{Sw}(V)=\sum_{i\geq1}\frac{|G_i|}{|G_0|}\operatorname{Tr}(I-P_i).} \tag{12.12}
\operatorname{Sym}Use in source · 1 passage
Search aliases: Sym operatorname
Use in source · block16_section_final.tex, line 344
Fix a base chart U_0. Every loop γ∈Aut_mathscr G(U_0) acts on an address set by i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the direct product of the two separate monodromy groups.
Exact mathematical passage
H_A^r := \operatorname{im}\!\left( \Aut_{\mathscr G}(U_0) \longrightarrow \operatorname{Sym}(I_{A,U_0,r}) \right)
\Syn_{\mathrm{ak}}Use in source · 5 passages
Search aliases: Syn_ak Syn mathrm operatorname mathrm
Use in source · Stratified_Universality_Traditional.tex, line 884
[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right side is the category of typed realizations satisfying all declared relations and carrying every declared packet idempotent to a split idempotent.
Exact mathematical passage
\boxed{ \Syn_{\mathrm{ak}}(\mathbb K_{\rm lin}) := \Kar\!\bigl(\Add(\kk Q/I)\bigr)}
\Synthesis_X^rUse in source · 2 passages
Search aliases: Synthesis_X^r Synthesis
Use in source · block15_section.tex, line 78
At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.
Exact mathematical passage
\Analysis_X^r :=(\pi_{X,i}^r)_i: X\longrightarrow\bigoplus_iX_i^r, \qquad \Synthesis_X^r :=(\iota_{X,i}^r)_i: \bigoplus_iX_i^r\longrightarrow X.
\mathfrak TUse in source · 1 passage
Search aliases: T mathfrak
Use in source · block11_section.tex, line 385
[Finite bottom-up residue automaton] Label every leaf by 1∈ Z/N Z and every m-ary vertex by the sum of its child labels. The reachable state set is exactly {j:j≡1pmodg_N}. Thus the finite pair T_N( A) = (d_ A,g_N,N/g_N) tag11.11 classifies packet support for the marked simply generated family.
Exact mathematical passage
\boxed{ \mathfrak T_N(\mathcal A) = (d_{\mathcal A},g_N,N/g_N)} \tag{11.11}
\mathsf TUse in source · 10 passages
Search aliases: T mathsf
Use in source · Stratified_Universality_Traditional.tex, line 5798
Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^{\mathsf T}C^nv_0,
T^2Use in source · 1 passage
Search aliases: T^2
Use in source · block10_section.tex, line 383
For the even packet, put T=E^2. Then 4xT^2-T+1=0. Its discriminant 1-16x is not a square in K(x), so [K(x)(T):K(x)]=2. The norm of T from this quadratic field is 1/(4x), which is not a square in K(x). Therefore T is not a square in its quadratic field and E has degree four.
Exact mathematical passage
4xT^2-T+1=0.
t^jUse in source · 2 passages
Search aliases: t^j
Use in source · block09_section.tex, line 260
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\boxed{ t^j\mathfrak M_{N,j}F(t^N) = \frac1N \sum_{q=0}^{N-1} \omega_N^{-jq}F(\omega_N^qt).}
T^mUse in source · 1 passage
Search aliases: T^m
Use in source · Stratified_Universality_Traditional.tex, line 5066
Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).
Exact mathematical passage
F_{f,p}(T) = \prod_{m\geq1}(1-T^m)^{-\beta_{p,m}(f)}.
t^mUse in source · 1 passage
Search aliases: t^m
Use in source · block11_section.tex, line 35
Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.
Exact mathematical passage
a=\lVert s\rVert, \qquad b_m=\lVert A_m\rVert, \qquad \varphi(t)=\sum_{m=2}^{D}b_mt^m.
t^NUse in source · 5 passages
Search aliases: t^N
Use in source · block09_section.tex, line 259
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
x=t^N
t^nUse in source · 1 passage
Search aliases: t^n
Use in source · block09_section.tex, line 595
If λ=-M, expand (1-αω_N^rt)^M = ∑_n=0^M binom Mn(-α)^nω_N^rnt^n. The rotation matrix is a truncated discrete Fourier matrix. Its rank is the number of available residue classes, namely min(N,M+1).
Exact mathematical passage
(1-\alpha\omega_N^rt)^M = \sum_{n=0}^{M} \binom Mn(-\alpha)^n\omega_N^{rn}t^n.
\mathcal T_{\mathcal A}Use in source · 1 passage
Search aliases: T_ A mathcal mathcal
Use in source · block11_section.tex, line 57
[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applied to the ordered values of its children.
Exact mathematical passage
\boxed{ u=\sum_{T\in\mathcal T_{\mathcal A}}\operatorname{ev}(T),} \tag{11.2}
T_0Use in source · 4 passages
Search aliases: T_0
Use in source · block15_section.tex, line 614
Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expansions gives an exact lane expansion of P.
Exact mathematical passage
P(x)=T_0+T_1(x)+T_2(x,x)+\cdots+T_D(x,\ldots,x),
T_1Use in source · 3 passages
Search aliases: T_1
Use in source · block15_section.tex, line 614
Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expansions gives an exact lane expansion of P.
Exact mathematical passage
P(x)=T_0+T_1(x)+T_2(x,x)+\cdots+T_D(x,\ldots,x),
T_2Use in source · 3 passages
Search aliases: T_2
Use in source · block15_section.tex, line 614
Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expansions gives an exact lane expansion of P.
Exact mathematical passage
P(x)=T_0+T_1(x)+T_2(x,x)+\cdots+T_D(x,\ldots,x),
t_{2^k+n}Use in source · 1 passage
Search aliases: t_2^k+n
Use in source · Stratified_Universality_Traditional.tex, line 3416
Negative control: automatic does not imply rational mask. Let t_n∈ F_2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, contradi…
Exact mathematical passage
t_{2^k+n}=1+t_n \qquad (0\leq n<2^k)
t_{2^m}Use in source · 1 passage
Search aliases: t_2^m
Use in source · Stratified_Universality_Traditional.tex, line 3426
(u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, contradicting pure periodicity.
Exact mathematical passage
t_{2^m-T}\neq t_{2^m},
t_{2^m-T}Use in source · 1 passage
Search aliases: t_2^m-T
Use in source · Stratified_Universality_Traditional.tex, line 3426
(u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, contradicting pure periodicity.
Exact mathematical passage
t_{2^m-T}\neq t_{2^m},
\mathcal T_aUse in source · 1 passage
Search aliases: T_a mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5587
[Complete coefficient-packet equivalence] For M=p^a, the map T_a:c_0( Z_p) ⟶ ∏_s=0^p^a-1c_0( Z_p), c⟼( T_p^a,sc)_s, is a topological isomorphism. It is the coefficient conjugate of complete clopen analysis.
Exact mathematical passage
\mathcal T_a:c_0(\mathbb Z_p) \longrightarrow \prod_{s=0}^{p^a-1}c_0(\mathbb Z_p), \qquad c\longmapsto(\mathcal T_{p^a,s}c)_s,
T_{a;i_1,\ldots,i_d}Use in source · 2 passages
Search aliases: T_a;i_1,…,i_d ldots
Use in source · block15_section.tex, line 594
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T_{a;i_1,\ldots,i_d} := \pi_{Y,a}T (\iota_{1,i_1}-,\ldots,\iota_{d,i_d}-).
T_{\bar e}Use in source · 1 passage
Search aliases: T_bar e bar
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
T_{\bar e}=T_e^{-1}
T_DUse in source · 1 passage
Search aliases: T_D
Use in source · block15_section.tex, line 614
Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expansions gives an exact lane expansion of P.
Exact mathematical passage
P(x)=T_0+T_1(x)+T_2(x,x)+\cdots+T_D(x,\ldots,x),
T_dUse in source · 5 passages
Search aliases: T_d
Use in source · Stratified_Universality_Traditional.tex, line 2999
Write a=deg A, b=deg B, H(z)=A(z)B(z)^p-2, and define D(A,B) = max{ b, lceilfraca+(p-2)bp-1rceil }. Let P_D= F_p[z]_≤ D. For 0≤ d<p, define T_d(P)=Λ_d(PH), where Λ_d selects the coefficients in degrees congruent to d modulo p.
Exact mathematical passage
\label{eq:block04-numerator-transition} T_d(P)=\Lambda_d(PH),
T_{d_0}Use in source · 1 passage
Search aliases: T_d_0
Use in source · Stratified_Universality_Traditional.tex, line 3030
Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.
Exact mathematical passage
\label{eq:block04-compiler-output} [z^n]F = \epsilon_0\!\left( \Theta_{A,B} \bigl(T_{d_{\ell-1}}\cdots T_{d_0}B\bigr) \right),
T_{d_{\ell-1}}Use in source · 1 passage
Search aliases: T_d_ℓ-1 ell
Use in source · Stratified_Universality_Traditional.tex, line 3030
Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.
Exact mathematical passage
\label{eq:block04-compiler-output} [z^n]F = \epsilon_0\!\left( \Theta_{A,B} \bigl(T_{d_{\ell-1}}\cdots T_{d_0}B\bigr) \right),
T_eUse in source · 6 passages
Search aliases: T_e
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
T_e:F_u\to F_v
T_e^*Use in source · 2 passages
Search aliases: T_e^*
Use in source · block19_admissibility_lifting_final.tex, line 126
raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge scales satisfying T_e^*Q_t(e)T_e=λ_e^2Q_s(e) exist if and only if the determinant-normalized holonomy representation preserves a positive-definite metr…
Exact mathematical passage
T_e^*Q_{t(e)}T_e=\lambda_e^2Q_{s(e)}
T_e^{-1}Use in source · 2 passages
Search aliases: T_e^-1
Use in source · Stratified_Universality_Traditional.tex, line 1474
Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
T_{\bar e}=T_e^{-1}
\mathcal T_{MN,Mt+s}Use in source · 1 passage
Search aliases: T_MN,Mt+s mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5608
[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.
Exact mathematical passage
\boxed{ \mathcal T_{N,t}\mathcal T_{M,s} = \mathcal T_{MN,Mt+s}.}
T_nUse in source · 2 passages
Search aliases: T_n
Use in source · block15_section.tex, line 618
Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expansions gives an exact lane expansion of P.
Exact mathematical passage
T_0\in Y, \qquad T_n:X^n\longrightarrow Y\quad(1\le n\le D)
\mathfrak T_NUse in source · 1 passage
Search aliases: T_N mathfrak
Use in source · block11_section.tex, line 385
[Finite bottom-up residue automaton] Label every leaf by 1∈ Z/N Z and every m-ary vertex by the sum of its child labels. The reachable state set is exactly {j:j≡1pmodg_N}. Thus the finite pair T_N( A) = (d_ A,g_N,N/g_N) tag11.11 classifies packet support for the marked simply generated family.
Exact mathematical passage
\boxed{ \mathfrak T_N(\mathcal A) = (d_{\mathcal A},g_N,N/g_N)} \tag{11.11}
t_nUse in source · 3 passages
Search aliases: t_n
Use in source · Stratified_Universality_Traditional.tex, line 3411
Negative control: automatic does not imply rational mask. Let t_n∈ F_2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, contradi…
Exact mathematical passage
t_n\in\mathbb F_2
\mathcal T_{N,t}Use in source · 1 passage
Search aliases: T_N,t mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5608
[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.
Exact mathematical passage
\boxed{ \mathcal T_{N,t}\mathcal T_{M,s} = \mathcal T_{MN,Mt+s}.}
\mathcal T_{p^a,s}Use in source · 2 passages
Search aliases: T_p^a,s mathcal
Use in source · Stratified_Universality_Traditional.tex, line 5587
[Complete coefficient-packet equivalence] For M=p^a, the map T_a:c_0( Z_p) ⟶ ∏_s=0^p^a-1c_0( Z_p), c⟼( T_p^a,sc)_s, is a topological isomorphism. It is the coefficient conjugate of complete clopen analysis.
Exact mathematical passage
\mathcal T_a:c_0(\mathbb Z_p) \longrightarrow \prod_{s=0}^{p^a-1}c_0(\mathbb Z_p), \qquad c\longmapsto(\mathcal T_{p^a,s}c)_s,
T_UUse in source · 1 passage
Search aliases: T_U
Use in source · Stratified_Universality_Traditional.tex, line 1920
Desc( R) exists and has the universal property of the finite pseudolimit of R. item There is a canonical equivalence Core(Desc( R)) ≃ Desc( R^≃). item Every pseudonatural transformation T: R⇒ S induces a functor Desc(T):Desc( R)→Desc( S). If every T_U is an equivalence, then Desc(T) is an equivalence. A pointwise noninvertible T remains directional. item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolv…
Exact mathematical passage
T_U
T_wUse in source · 1 passage
Search aliases: T_w
Use in source · Stratified_Universality_Traditional.tex, line 3150
For every word w, repeated use of [source reference] gives Θ_A,B(T_wB)=Λ_wF. Thus Θ_A,B maps the reachable numerator set onto the Cartier kernel. Injectivity of Θ_A,B makes the map bijective, and taking linear spans gives [source reference].
Exact mathematical passage
\Theta_{A,B}(T_wB)=\Lambda_wF.
\operatorname{Tor}Use in source · 2 passages
Search aliases: Tor operatorname
Use in source · Stratified_Universality_Traditional.tex, line 4231
Both M_k(a) and L_k(a) are free abelian. The long exact Tor sequence therefore begins 0⟶ Tor_1^ Z (C_k(a), Z/m Z) ⟶ M_k(a)⊗ Z/m Z ⟶ L_k(a)⊗ Z/m Z. For any abelian group C, Tor_1^ Z(C, Z/m Z) ≅ C[m], C⊗ Z/m Z ≅ C/mC. The digit-section maps preserve the original short exact sequence, so naturality of Tor makes the five-term sequence digit-equivariant.
Exact mathematical passage
0\longrightarrow \operatorname{Tor}_1^{\mathbb Z} (C_k(a),\mathbb Z/m\mathbb Z) \longrightarrow M_k(a)\otimes\mathbb Z/m\mathbb Z \longrightarrow L_k(a)\otimes\mathbb Z/m\mathbb Z.
\operatorname{Tor}_1^{\mathbb Z}Use in source · 2 passages
Search aliases: Tor_1^ Z operatorname mathbb
Use in source · Stratified_Universality_Traditional.tex, line 4231
Both M_k(a) and L_k(a) are free abelian. The long exact Tor sequence therefore begins 0⟶ Tor_1^ Z (C_k(a), Z/m Z) ⟶ M_k(a)⊗ Z/m Z ⟶ L_k(a)⊗ Z/m Z. For any abelian group C, Tor_1^ Z(C, Z/m Z) ≅ C[m], C⊗ Z/m Z ≅ C/mC. The digit-section maps preserve the original short exact sequence, so naturality of Tor makes the five-term sequence digit-equivariant.
Exact mathematical passage
0\longrightarrow \operatorname{Tor}_1^{\mathbb Z} (C_k(a),\mathbb Z/m\mathbb Z) \longrightarrow M_k(a)\otimes\mathbb Z/m\mathbb Z \longrightarrow L_k(a)\otimes\mathbb Z/m\mathbb Z.
\operatorname{Tr}Use in source · 3 passages
Search aliases: Tr operatorname
Use in source · block12_section.tex, line 254
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
\boxed{ a(V)=\sum_{i\geq0}\frac{|G_i|}{|G_0|}\operatorname{Tr}(I-P_i),} \tag{12.11}
\mathbf UUse in source · 4 passages
Search aliases: U mathbf
Use in source · block10_section.tex, line 149
Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.
Exact mathematical passage
\mathbf U=(U_0,\ldots,U_{N-1})
U'Use in source · 1 passage
Search aliases: U'
Use in source · Stratified_Universality_Traditional.tex, line 1554
rem[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V downarrow a · · downarrow b U' · xrightarrowD' · V', bD=D'a, there is a unique induced map overline b:coker D⟶coker D' with overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \downarrow a & & \downarrow b\\ U' & \xrightarrow{D'} & V', \end{array} \qquad bD=D'a,
u^{(0)}Use in source · 1 passage
Search aliases: u^(0)
Use in source · block11_section.tex, line 91
Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum. Regrouping trees by the arity of the root gives equation textup(11.1)…
Exact mathematical passage
u^{(0)}=0
u^{(h)}Use in source · 2 passages
Search aliases: u^(h)
Use in source · block11_section.tex, line 91
Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum. Regrouping trees by the arity of the root gives equation textup(11.1)…
Exact mathematical passage
u^{(h+1)}=\Psi(u^{(h)})
u^{(h+1)}Use in source · 1 passage
Search aliases: u^(h+1)
Use in source · block11_section.tex, line 91
Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum. Regrouping trees by the arity of the root gives equation textup(11.1)…
Exact mathematical passage
u^{(h+1)}=\Psi(u^{(h)})
u^pUse in source · 1 passage
Search aliases: u^p
Use in source · Stratified_Universality_Traditional.tex, line 5931
For a principal unit u=1+p, u^x=∑_r≥0p^rbinom xr. The branch x=pq has coefficients (u^p-1)^k, whose valuation is at least 2k. The branch therefore improves the original slope-one certificate.
Exact mathematical passage
\bigl(u^p-1\bigr)^k,
U^qUse in source · 5 passages
Search aliases: U^q
Use in source · block13_section.tex, line 264
Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.
Exact mathematical passage
U=z+\alpha U^q. \label{eq:block13-fuss-tree}
u^qUse in source · 3 passages
Search aliases: u^q
Use in source · block11_section.tex, line 465
Take u=z+u^q, q≥2. tag11.14 Lagrange inversion gives the exact coefficient formula [z^n]u = displaystyle frac1(q-1)m+1binomqmm, · n=1+(q-1)m,
Exact mathematical passage
\boxed{u=z+u^q,\qquad q\ge2.} \tag{11.14}
u^{q-1}Use in source · 1 passage
Search aliases: u^q-1
Use in source · block11_section.tex, line 534
Formula textup(11.15) follows from Lagrange inversion applied to z=u(1-u^q-1). The support and active-address statements are [source reference] with d_ A=q-1. Expanding u^q in complete packets and collecting the exponent carry r_1+⋯+r_q=j+Nc gives textup(11.16). The critical equations qτ^q-1=1, ρ=τ-τ^q give the printed values, and [source reference] gives the orbit count.
Exact mathematical passage
z=u(1-u^{q-1}).
u^xUse in source · 1 passage
Search aliases: u^x
Use in source · Stratified_Universality_Traditional.tex, line 5927
For a principal unit u=1+p, u^x=∑_r≥0p^rbinom xr. The branch x=pq has coefficients (u^p-1)^k, whose valuation is at least 2k. The branch therefore improves the original slope-one certificate.
Exact mathematical passage
u^x=\sum_{r\geq0}p^r\binom xr.
\mathcal U_{\mathbb Z}Use in source · 2 passages
Search aliases: U_ Z mathcal mathbb
Use in source · Stratified_Universality_Traditional.tex, line 4788
Let U_ Z=1+z Z[[z]] be the multiplicative group of integral unit series, and let E_ Z= Z^ N_≥1 be the additive group of integer exponent sequences.
Exact mathematical passage
\mathcal U_{\mathbb Z}=1+z\mathbb Z[[z]]
U_0Use in source · 22 passages
Search aliases: U_0
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
U_1Use in source · 2 passages
Search aliases: U_1
Use in source · block12_section.tex, line 302
[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.
Exact mathematical passage
U_1\supseteq U_2\supseteq\cdots
U_2Use in source · 1 passage
Search aliases: U_2
Use in source · block12_section.tex, line 302
[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.
Exact mathematical passage
U_1\supseteq U_2\supseteq\cdots
U_cUse in source · 1 passage
Search aliases: U_c
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
U_fUse in source · 2 passages
Search aliases: U_f
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
U_gUse in source · 3 passages
Search aliases: U_g
Use in source · block12_section.tex, line 116
[Normality and quotient compression] For a subgroup H≤ G, the following conditions are equivalent: item Htriangleleft G; item e_H is central in the group algebra; item P_HU_g=U_gP_H for every g∈ G; item the fixed packet V^H is G-invariant; item the fixed field L^H is Galois over K. When they hold, the compressed action on V^H factors through G/H, and Gal(L^H/K)≅ G/H. tag12.5
Exact mathematical passage
P_HU_g=U_gP_H
U_g^{-1}Use in source · 1 passage
Search aliases: U_g^-1
Use in source · block12_section.tex, line 129
Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.
Exact mathematical passage
U_gP_HU_g^{-1}=P_{gHg^{-1}}
U_hUse in source · 3 passages
Search aliases: U_h
Use in source · block12_section.tex, line 16
Let L/K be a finite Galois extension with group G=Gal(L/K). Let V be a finite-dimensional representation of G over a characteristic-zero field, or a unitary complex realization obtained after scalar extension. For every subgroup H≤ G, define P_H=frac1|H|∑_h∈ HU_h. tag12.1
Exact mathematical passage
\boxed{ P_H=\frac1{|H|}\sum_{h\in H}U_h.} \tag{12.1}
u_jUse in source · 1 passage
Search aliases: u_j
Use in source · block11_section.tex, line 275
For the packet system textup(11.4), define the directed dependency graph G_Φ on I by placing an edge j⟶ i when some active block contributing to the equation for u_i contains u_j as an input. Let SCC(G_Φ) be its strongly connected components. A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.
Exact mathematical passage
u_j
u_{j_1}Use in source · 1 passage
Search aliases: u_j_1
Use in source · block11_section.tex, line 135
I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
\boxed{ u_i = s_i+ \sum_{m=2}^{D} \sum_{j_1,\ldots,j_m} A_{m;i\mid j_1,\ldots,j_m} [u_{j_1},\ldots,u_{j_m}].} \tag{11.4}
u_{j_m}Use in source · 1 passage
Search aliases: u_j_m
Use in source · block11_section.tex, line 135
I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
\boxed{ u_i = s_i+ \sum_{m=2}^{D} \sum_{j_1,\ldots,j_m} A_{m;i\mid j_1,\ldots,j_m} [u_{j_1},\ldots,u_{j_m}].} \tag{11.4}
u_mUse in source · 1 passage
Search aliases: u_m
Use in source · block09_section.tex, line 489
B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ \prod_{r=0}^{N-1}(Nm+j+1+r)u_{m+1} = \alpha^N \prod_{r=0}^{N-1}(Nm+j+\lambda+r)u_m.}
u_{m+1}Use in source · 1 passage
Search aliases: u_m+1
Use in source · block09_section.tex, line 489
B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ \prod_{r=0}^{N-1}(Nm+j+1+r)u_{m+1} = \alpha^N \prod_{r=0}^{N-1}(Nm+j+\lambda+r)u_m.}
U_nUse in source · 5 passages
Search aliases: U_n
Use in source · block12_section.tex, line 306
[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.
Exact mathematical passage
\boxed{P_{U_n}\longrightarrow I} \tag{12.15}
u_nUse in source · 3 passages
Search aliases: u_n
Use in source · Stratified_Universality_Traditional.tex, line 3412
Negative control: automatic does not imply rational mask. Let t_n∈ F_2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, contradi…
Exact mathematical passage
u_n=1+t_n.
U_{N-1}Use in source · 4 passages
Search aliases: U_N-1
Use in source · block10_section.tex, line 149
Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.
Exact mathematical passage
\mathbf U=(U_0,\ldots,U_{N-1})
U_pUse in source · 5 passages
Search aliases: U_p
Use in source · Stratified_Universality_Traditional.tex, line 2357
[Finite periodic kernel and exact state model] Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives by equality of the sequences in [source reference] gives the canonical minimal least-significant-digit-first p-DFAO for a.
Exact mathematical passage
\label{eq:periodic-kernel-state} b_{u,r}(n)=w_{un+r\bmod T}, \qquad u\in U_p(T),\quad r\in\mathbb Z/T\mathbb Z,
U_qUse in source · 1 passage
Search aliases: U_q
Use in source · block15_section.tex, line 955
t [z^1+(q-1)n]U =frac1(q-1)n+1binomqnnα^n ≠0 (n≥0), and every other coefficient is zero. Thus its support is exactly the set of degrees congruent to 1pmodq-1. For an integer packet width N≥1, set g=gcd(N,q-1). Exactly N/g residue addresses are active, namely the congruence class 1pmod g. Packetizing the specified multilinear map (U_1,…,U_q)↦ U_1⋯ U_q gives the exact finite carry-convolution system supplied by [source reference]. On a cyclic orbit of order q-1, the power map used for packet synth…
Exact mathematical passage
(U_1,\ldots,U_q)\mapsto U_1\cdots U_q
U_{q,\alpha}Use in source · 3 passages
Search aliases: U_q,α alpha
Use in source · block13_section.tex, line 260
Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.
Exact mathematical passage
U_{q,\alpha}(z)\in zK[[z]]
U_{r_a}Use in source · 1 passage
Search aliases: U_r_a
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
U_uUse in source · 1 passage
Search aliases: U_u
Use in source · block12_section.tex, line 315
For each v∈ H, strong continuity gives sup_u∈ U_n U_uv-v ⟶0. Averaging this estimate over U_n proves the first statement. Apply it to the quotient action on the N-fixed packet for the second.
Exact mathematical passage
\sup_{u\in U_n}\|U_uv-v\|\longrightarrow0.
\underline jUse in source · 3 passages
Search aliases: underline j underline
Use in source · block09_section.tex, line 119
[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.
Exact mathematical passage
x^{\underline j}=x(x-1)\cdots(x-j+1).
\underline{j}Use in source · 1 passage
Search aliases: underlinej underline
Use in source · block09_section.tex, line 108
[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.
Exact mathematical passage
\boxed{ [z^n]LF = \sum_{j=0}^{r} \sum_{m=0}^{\min(d_j,n)} q_{j,m} (n-m+j)^{\underline{j}} a_{n-m+j},}
\mathfrak VUse in source · 3 passages
Search aliases: V mathfrak
Use in source · Stratified_Universality_Traditional.tex, line 1922
n equivalence. A pointwise noninvertible T remains directional. item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in Desc( R). item A coherent pseudonatural invariant family I: R⇒ V induces Desc(I):Desc( R)→Desc( V). item If every local invariant is complete on its normalized chart and the group…
Exact mathematical passage
I:\mathfrak R\Rightarrow\mathfrak V
V'Use in source · 1 passage
Search aliases: V'
Use in source · Stratified_Universality_Traditional.tex, line 1554
rem[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V downarrow a · · downarrow b U' · xrightarrowD' · V', bD=D'a, there is a unique induced map overline b:coker D⟶coker D' with overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \downarrow a & & \downarrow b\\ U' & \xrightarrow{D'} & V', \end{array} \qquad bD=D'a,
v'Use in source · 2 passages
Search aliases: v'
Use in source · Stratified_Universality_Traditional.tex, line 2092
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
S:(C,v,\lambda)\longrightarrow(C',v',\lambda')
V^GUse in source · 1 passage
Search aliases: V^G
Use in source · block12_section.tex, line 359
[Effective finite Galois vector descent] The two functors above are quasi-inverse equivalences: Vect_K ≃ Semilin_G(L). tag12.16 For every semilinear object V, the fixed packet W=V^G satisfies W⊗_KL≅ V. tag12.17 The equivalence is effective on both objects and morphisms.
Exact mathematical passage
W=V^G
V^{G_0}Use in source · 1 passage
Search aliases: V^G_0
Use in source · block12_section.tex, line 239
[Orthogonal ramification details] The operators D_i are mutually orthogonal projectors, and V=V^G_0⊕⊕_i D_iV. tag12.10 Every detail packet is stable under the quotient action permitted by the corresponding normal subgroup.
Exact mathematical passage
\boxed{ V=V^{G_0}\oplus\bigoplus_i D_iV.} \tag{12.10}
V^{G_i}Use in source · 3 passages
Search aliases: V^G_i
Use in source · block12_section.tex, line 274
The classical conductor formula uses codimV^G_i. Since P_i projects onto V^G_i, this codimension is Tr(I-P_i). Also codimV^G_i=∑_j≥ id_j. Reversing the finite sums gives the detail formula.
Exact mathematical passage
\operatorname{codim}V^{G_i}
V^HUse in source · 7 passages
Search aliases: V^H
Use in source · block12_section.tex, line 25
[Reynolds fixed packet and reverse order] The operator P_H is the projector onto the fixed packet V^H={v:U_hv=v for all h∈ H}. If H≤ J, then P_HP_J=P_JP_H=P_J. tag12.2 Consequently, subgroup inclusion reverses the fixed-projector order.
Exact mathematical passage
V^H=\{v:U_hv=v\text{ for all }h\in H\}.
V^JUse in source · 2 passages
Search aliases: V^J
Use in source · block12_section.tex, line 37
The average is invariant under left and right multiplication by elements of H. Therefore P_H^2=P_H, and its range is exactly the fixed subspace. If H≤ J, then V^J⊆ V^H. The two projections are nested and multiply to the projector with smaller range.
Exact mathematical passage
V^J\subseteq V^H
V^{\langle H,J\rangle}Use in source · 1 passage
Search aliases: V^⟨ H,J⟩ langle rangle
Use in source · block12_section.tex, line 85
For two subgroups H,J≤ G, the common fixed packet is always V^H∩ V^J=V^⟨ H,J⟩. The two averaging projectors need not commute.
Exact mathematical passage
V^H\cap V^J=V^{\langle H,J\rangle}.
V_0Use in source · 3 passages
Search aliases: V_0
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
v_0Use in source · 2 passages
Search aliases: v_0
Use in source · Stratified_Universality_Traditional.tex, line 5798
Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^{\mathsf T}C^nv_0,
\mathfrak V_{13}Use in source · 1 passage
Search aliases: V_13 mathfrak
Use in source · block13_section.tex, line 152
nvariant-descent Let A_13 be the atlas of [source reference]. Suppose that, in every connected component: item the local invariant is complete on every normalized chart; item the bridge comparisons are coherent; item invariant-compatible local isomorphisms satisfy effective descent. Then the local invariants induce a functor I_13: Desc( A_13) ⟶ Desc( V_13) that reflects isomorphisms. If each semantic component is discrete, equality of descended invariants is equivalent to equality of the represe…
Exact mathematical passage
\mathcal I_{13}: \Desc(\mathfrak A_{13}) \longrightarrow \Desc(\mathfrak V_{13})
V_4Use in source · 1 passage
Search aliases: V_4
Use in source · block12_section.tex, line 589
The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra. It audits cyclic groups, V_4, S_3, D_4, and A_4; every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commuting and noncommuting subgroup pairs; double-coset partitions and the Mackey dimension law; cyclic ramification-style filtrations; explicit Hilbert 90 co…
Exact mathematical passage
V_4
V_cUse in source · 2 passages
Search aliases: V_c
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
\mathcal V_{\mathrm{cert}}Use in source · 1 passage
Search aliases: V_cert mathcal mathrm
Use in source · block14_section.tex, line 19
whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object. item The semantic normalizers induce the comparison 2-cells on all nonempty triple overlaps and force the quadruple-overlap coherence equation. item The compatible native invariant family descends to a tagged global invariant J_cert: Desc( A_cert) ⟶ Desc( V_cert). item The descended invariant is complete on the declared normaliz…
Exact mathematical passage
\mathcal J_{\mathrm{cert}}: \Desc(\mathfrak A_{\mathrm{cert}}) \longrightarrow \Desc(\mathcal V_{\mathrm{cert}}).
V_fUse in source · 3 passages
Search aliases: V_f
Use in source · Stratified_Universality_Traditional.tex, line 1607
[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\ &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\ 0&\to&V_0&\to&V_f&\to&V_c&\to&0 \end{array}
v_iUse in source · 1 passage
Search aliases: v_i
Use in source · Stratified_Universality_Traditional.tex, line 3742
[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).
Exact mathematical passage
\Lambda_db_j = \sum_{i=1}^{r_k(a)}(M_d(a))_{ij}b_i, \qquad a=\sum_i v_i(a)b_i, \qquad \ell_i(a)=b_i(0).
v_{\mathrm{in}}Use in source · 2 passages
Search aliases: v_in mathrm
Use in source · Stratified_Universality_Traditional.tex, line 3688
For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.
Exact mathematical passage
(W,A_d,v_{\mathrm{in}},\lambda)
v_jUse in source · 3 passages
Search aliases: v_j
Use in source · block09_section.tex, line 395
Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
v_j=v_\alpha(q_j), \qquad \mu=\min_j(v_j-j).
v_nUse in source · 4 passages
Search aliases: v_n
Use in source · Stratified_Universality_Traditional.tex, line 5798
Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^{\mathsf T}C^nv_0,
v_{n+1}Use in source · 3 passages
Search aliases: v_n+1
Use in source · Stratified_Universality_Traditional.tex, line 5798
Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^{\mathsf T}C^nv_0,
v_{Nm+j}Use in source · 1 passage
Search aliases: v_Nm+j
Use in source · block15_section.tex, line 867
n Let K be a characteristic-zero field and let the K-valued sequence (u_n) satisfy a linear recurrence with polynomial coefficients. Beyond a finite singular prefix, write the recurrence as a finite-dimensional rational nonautonomous state model v_n+1=A(n)v_n, A(n)∈Mat_d(K(n)). For a fixed integer N≥1 and residue 0≤ j<N, the lane state w_m=v_Nm+j evolves by the exact N-step rational transfer w_m+1 = A(Nm+j+N-1)⋯ A(Nm+j)w_m whenever the displayed factors are defined. Equivalently, residue multise…
Exact mathematical passage
w_m=v_{Nm+j}
v_\alphaUse in source · 1 passage
Search aliases: v_α alpha
Use in source · block09_section.tex, line 395
Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
v_j=v_\alpha(q_j), \qquad \mu=\min_j(v_j-j).
\mathcal V_{\lambda,a}Use in source · 1 passage
Search aliases: V_λ,a mathcal lambda
Use in source · block13_section.tex, line 127
For every chart C_λ,a, let I_λ,a: C_λ,a⟶ V_λ,a be its native normalized invariant. A bridge comparison is a natural isomorphism κ_ab: J_abI_λ,a Longrightarrow I_λ,bΦ_ab, where J_ab transports invariant coordinates between their chart-specific target categories.
Exact mathematical passage
I_{\lambda,a}:\mathcal C_{\lambda,a}\longrightarrow\mathcal V_{\lambda,a}
v_{\pi}Use in source · 1 passage
Search aliases: v_π pi
Use in source · Stratified_Universality_Traditional.tex, line 2920
Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.
Exact mathematical passage
\omega_{\pi}(R) = \bigl(v_{\pi}(B)-v_{\pi}(A)\bigr)\bmod(p-1).
\varprojlimUse in source · 5 passages
Search aliases: varprojlim
Use in source · Stratified_Universality_Traditional.tex, line 4481
[Raw and semantic inverse limits] There are canonical digit-equivariant identifications varprojlim_r M_k(a)/p^rM_k(a) ≅ M_k,p(a) and varprojlim_r overline M_k,p^r(a) ≅ M_k,p(a) ⊆ L_k,p(a). Furthermore, varprojlim_r G_k,p^r(a)=0, varprojlim_r Q_k,p^r(a) ≅ C_k(a)_(p).
Exact mathematical passage
\varprojlim_r M_k(a)/p^rM_k(a) \cong M_{k,p}(a)
\varprojlim_mUse in source · 1 passage
Search aliases: varprojlim_m varprojlim
Use in source · Stratified_Universality_Traditional.tex, line 4622
Let widehatM_k(a) = varprojlim_m M_k(a)/mM_k(a) be the profinite completion over divisibility of moduli.
Exact mathematical passage
\widehat{M_k(a)} = \varprojlim_m M_k(a)/mM_k(a)
\varprojlim_rUse in source · 4 passages
Search aliases: varprojlim_r varprojlim
Use in source · Stratified_Universality_Traditional.tex, line 4481
[Raw and semantic inverse limits] There are canonical digit-equivariant identifications varprojlim_r M_k(a)/p^rM_k(a) ≅ M_k,p(a) and varprojlim_r overline M_k,p^r(a) ≅ M_k,p(a) ⊆ L_k,p(a). Furthermore, varprojlim_r G_k,p^r(a)=0, varprojlim_r Q_k,p^r(a) ≅ C_k(a)_(p).
Exact mathematical passage
\varprojlim_r M_k(a)/p^rM_k(a) \cong M_{k,p}(a)
\operatorname{Vect}Use in source · 6 passages
Search aliases: Vect operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1749
[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].
Exact mathematical passage
\omega:\mathcal T\longrightarrow\operatorname{Vect}_k
\operatorname{Vect}_KUse in source · 3 passages
Search aliases: Vect_K operatorname
Use in source · block12_section.tex, line 342
Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.
Exact mathematical passage
L\otimes_K-:\operatorname{Vect}_K\longrightarrow\operatorname{Semilin}_G(L),
\operatorname{Vect}_kUse in source · 3 passages
Search aliases: Vect_k operatorname
Use in source · Stratified_Universality_Traditional.tex, line 1749
[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].
Exact mathematical passage
\omega:\mathcal T\longrightarrow\operatorname{Vect}_k
\veeUse in source · 2 passages
Search aliases: vee
Use in source · block17_semantic_quotients_final.tex, line 282
[Combined and shared semantics] For semantic quotients with kernels N and M: [label=textup(roman*)] item the least common refinement has kernel N∩ M; item the greatest common coarsening has kernel the generated congruence join Nvee M; item in a module category, Nvee M=N+M.
Exact mathematical passage
N\vee M
\mathcal WUse in source · 2 passages
Search aliases: W mathcal
Use in source · Stratified_Universality_Traditional.tex, line 1275
[Composition law for bridge loss] Let UxrightarrowB VxrightarrowC W be certified bridges. [label=textup(roman*)] item Every constructor or universal property preserved by both B and C is preserved by CB. item MorKer_B(X,Y) ⊆ MorKer_CB(X,Y). item Coll_B ⊆ Coll_CB. item EssIm(CB) ⊆ EssIm(C). item If both bridges are conservative, then their composite is conservative. If both are classification-faithful, then their composite is classification-faithful. If both are equivalences, then their composite…
Exact mathematical passage
\mathcal U\xrightarrow{B}\mathcal V\xrightarrow{C}\mathcal W
\mathfrak WUse in source · 1 passage
Search aliases: W mathfrak
Use in source · Stratified_Universality_Traditional.tex, line 5753
Define the min-plus certificate transform ( W_M,sw)(k) = min_k≤ r≤ Mk+s (w(r)+v_p(Θ_M,s(r,k))). This transform is a certified lower bound. It need not be the exact output valuation, because cancellation can increase the valuation beyond the termwise minimum.
Exact mathematical passage
\boxed{ (\mathfrak W_{M,s}w)(k) = \min_{k\leq r\leq Mk+s} \left(w(r)+v_p(\Theta_{M,s}(r,k))\right).}
w^{\mathsf T}Use in source · 2 passages
Search aliases: w^ T mathsf
Use in source · Stratified_Universality_Traditional.tex, line 5798
Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^{\mathsf T}C^nv_0,
w^nUse in source · 1 passage
Search aliases: w^n
Use in source · block15_section.tex, line 791
[Typed polyphase realization] Let K be a field, let N∈ Z with N≥1 be invertible in K, and suppose K contains a primitive N-th root of unity. For F(z)=∑_n≥0a_nz^n∈ K[[z]], set F_j(w):=∑_n≥0a_Nn+jw^n. Then F(z)=∑_j=0^N-1z^jF_j(z^N) is exact analysis and synthesis. The Fourier residue projectors and the polyphase components satisfy the explicit coordinate relation P_N,jF(z)=z^jF_j(z^N). For every integer M≥1, refinement from N to NM is the projector-sum law of [source reference].
Exact mathematical passage
F_j(w):=\sum_{n\ge0}a_{Nn+j}w^n.
w_{(pu)n+(r+ud)}Use in source · 1 passage
Search aliases: w_(pu)n+(r+ud)
Use in source · Stratified_Universality_Traditional.tex, line 2377
After ℓ digits, the resulting subsequence is n↦ a_p^ℓ n+r, which depends only on (p^ℓbmod T,rbmod T). Applying Λ_d gives Λ_db_u,r(n) =b_u,r(pn+d) =w_(pu)n+(r+ud), proving [source reference]. The pair set is finite, so the kernel is finite. Every kernel state is reachable by construction. If two kernel sequences differ, they differ at some n; reading the base-p digits of that n distinguishes the states by their outputs. Hence the kernel automaton is minimal. Any other reachable minimal automaton …
Exact mathematical passage
\Lambda_db_{u,r}(n) =b_{u,r}(pn+d) =w_{(pu)n+(r+ud)},
w_mUse in source · 7 passages
Search aliases: w_m
Use in source · block09_section.tex, line 290
[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of order at most s.
Exact mathematical passage
\boxed{ w_{m+1}=J_{N,j}(m)w_m,}
w_{m+1}Use in source · 2 passages
Search aliases: w_m+1
Use in source · block09_section.tex, line 290
[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of order at most s.
Exact mathematical passage
\boxed{ w_{m+1}=J_{N,j}(m)w_m,}
\mathfrak W_{M,s}Use in source · 1 passage
Search aliases: W_M,s mathfrak
Use in source · Stratified_Universality_Traditional.tex, line 5753
Define the min-plus certificate transform ( W_M,sw)(k) = min_k≤ r≤ Mk+s (w(r)+v_p(Θ_M,s(r,k))). This transform is a certified lower bound. It need not be the exact output valuation, because cancellation can increase the valuation beyond the termwise minimum.
Exact mathematical passage
\boxed{ (\mathfrak W_{M,s}w)(k) = \min_{k\leq r\leq Mk+s} \left(w(r)+v_p(\Theta_{M,s}(r,k))\right).}
w_rUse in source · 1 passage
Search aliases: w_r
Use in source · Stratified_Universality_Traditional.tex, line 2372
od word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives by equality of the sequences in [source reference] gives the canonical minimal least-significant-digit-first p-DFAO for a.
Exact mathematical passage
w_r
w_{un+r\bmod T}Use in source · 1 passage
Search aliases: w_un+rbmod T bmod
Use in source · Stratified_Universality_Traditional.tex, line 2357
[Finite periodic kernel and exact state model] Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives by equality of the sequences in [source reference] gives the canonical minimal least-significant-digit-first p-DFAO for a.
Exact mathematical passage
\label{eq:periodic-kernel-state} b_{u,r}(n)=w_{un+r\bmod T}, \qquad u\in U_p(T),\quad r\in\mathbb Z/T\mathbb Z,
\mathrm{weak}Use in source · 1 passage
Search aliases: weak mathrm
Use in source · block19_admissibility_lifting_final.tex, line 307
Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders and should not be merged into one scalar notion of a stronger contract.
Exact mathematical passage
N_{\mathrm{strong}}\subseteq N_{\mathrm{weak}}, \qquad Q_{\mathrm{strong}}\twoheadrightarrow Q_{\mathrm{weak}}.
\widehat HUse in source · 2 passages
Search aliases: widehat H widehat
Use in source · block19_admissibility_lifting_final.tex, line 121
[Conformal metric-completion criterion] Let a connected finite graph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge scales satisfying T_e^*Q_t(e)T_e=λ_e^2Q_s(e) exist if and only if the determinant-norma…
Exact mathematical passage
\widehat H:=|\det H|^{-1/n}H.
\widehat{\mathbb Z}Use in source · 1 passage
Search aliases: widehat Z widehat mathbb
Use in source · Stratified_Universality_Traditional.tex, line 4662
Finite free abelian groups have profinite completion equal to the product of their p-adic completions. In Smith coordinates, the inclusion is multiplication by d_i in each profinite coordinate. Its cokernel is widehat Z/d_iwidehat Z ≅ Z/d_i Z. Taking the direct sum over i gives C_k(a). The final assertion is the definition of belonging to the diagonal image of the integral lattice inside its profinite completion.
Exact mathematical passage
\widehat{\mathbb Z}/d_i\widehat{\mathbb Z} \cong \mathbb Z/d_i\mathbb Z.
\widehat{L_k(a)}Use in source · 2 passages
Search aliases: widehatL_k(a) widehat
Use in source · Stratified_Universality_Traditional.tex, line 4632
[Profinite saturation sequence] There are canonical digit-equivariant identifications widehatM_k(a) ≅ ∏_p M_k,p(a), widehatL_k(a) ≅ ∏_p L_k,p(a), and a short exact sequence 0 ⟶ widehatM_k(a) ⟶ widehatL_k(a) ⟶ C_k(a) ⟶ 0. A compatible all-moduli raw tower has an ordinary integral lift exactly when its class in widehatM_k(a)/M_k(a) vanishes.
Exact mathematical passage
\widehat{M_k(a)} \cong \prod_p M_{k,p}(a), \qquad \widehat{L_k(a)} \cong \prod_p L_{k,p}(a),
\widehat{M_k(a)}Use in source · 5 passages
Search aliases: widehatM_k(a) widehat
Use in source · Stratified_Universality_Traditional.tex, line 4622
Let widehatM_k(a) = varprojlim_m M_k(a)/mM_k(a) be the profinite completion over divisibility of moduli.
Exact mathematical passage
\widehat{M_k(a)} = \varprojlim_m M_k(a)/mM_k(a)
\widehat{x}Use in source · 1 passage
Search aliases: widehatx widehat
Use in source · Stratified_Universality_Traditional.tex, line 4673
[Local-global separation] Every compatible prime-power tower is effective in a p-adic completion. A compatible family over all primes is effective in the profinite completion. Neither statement alone implies an ordinary integral lift. Integral effectivity is the separate diagonal condition widehatx∈ M_k(a)⊆widehatM_k(a).
Exact mathematical passage
\widehat{x}\in M_k(a)\subseteq\widehat{M_k(a)}.
\widetilde BUse in source · 4 passages
Search aliases: widetilde B widetilde
Use in source · block19_admissibility_lifting_final.tex, line 268
[Forgetful descent and enhancement lifting] A rich bridge widetilde B:widetilde C→widetilde D automatically determines an underlying bridge Vwidetilde B:widetilde C→ D. Given only a bare bridge B: C→ D, constructing widetilde B together with a comparison Vwidetilde B≅ BU is a lifting problem. Its existence, uniqueness, and residual automorphisms are additional passport fields and may carry typed obstruction objects.
Exact mathematical passage
\widetilde B:\widetilde{\mathcal C}\to\widetilde{\mathcal D}
\widetilde{\mathcal C}Use in source · 4 passages
Search aliases: widetilde C widetilde mathcal
Use in source · block19_admissibility_lifting_final.tex, line 258
Suppose richer categories widetilde C and widetilde D carry forgetful functors U:widetilde C→ C, V:widetilde D→ D.
Exact mathematical passage
\widetilde{\mathcal C}
\widetilde{\mathcal D}Use in source · 3 passages
Search aliases: widetilde D widetilde mathcal
Use in source · block19_admissibility_lifting_final.tex, line 258
Suppose richer categories widetilde C and widetilde D carry forgetful functors U:widetilde C→ C, V:widetilde D→ D.
Exact mathematical passage
\widetilde{\mathcal D}
\widetilde LUse in source · 1 passage
Search aliases: widetilde L widetilde
Use in source · block09_section.tex, line 381
Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of this operator coordinate until an explicit desingularization has been performed.
Exact mathematical passage
\widetilde L_F = \sum_{j=0}^{r}q_j(z)D^j.
\widetilde L_FUse in source · 1 passage
Search aliases: widetilde L_F widetilde
Use in source · block09_section.tex, line 381
Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of this operator coordinate until an explicit desingularization has been performed.
Exact mathematical passage
\widetilde L_F = \sum_{j=0}^{r}q_j(z)D^j.
X^{(1)}Use in source · 2 passages
Search aliases: X^(1)
Use in source · block15_section.tex, line 586
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
X^{(1)},\ldots,X^{(d)},Y
X^{(d)}Use in source · 2 passages
Search aliases: X^(d)
Use in source · block15_section.tex, line 586
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
X^{(1)},\ldots,X^{(d)},Y
x^{(r_1+\cdots+r_q-j)/N}Use in source · 1 passage
Search aliases: x^(r_1+⋯+r_q-j)/N cdots
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
x^{-1}Use in source · 3 passages
Search aliases: x^-1
Use in source · block12_section.tex, line 180
[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.
Exact mathematical passage
\boxed{ \operatorname{Res}_K^G\operatorname{Ind}_H^GW \cong \bigoplus_{x\in K\backslash G/H} \operatorname{Ind}_{K\cap xHx^{-1}}^K \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).} \tag{12.8}
x^{1/N}Use in source · 1 passage
Search aliases: x^1/N
Use in source · block09_section.tex, line 427
Away from Σ^[N] and zero, choose a local analytic branch t=x^1/N. Every rotated argument ω_N^qt avoids Σ, so the root-filter expression is analytic. A singularity can therefore occur only when one rotated preimage meets Σ.
Exact mathematical passage
t=x^{1/N}
x^2Use in source · 10 passages
Search aliases: x^2
Use in source · Stratified_Universality_Traditional.tex, line 2187
Fibonacci F_n · x^2-x-1 · z/(1-z-z^2) · 2 · The root ratio is not torsion. Every arithmetic subsequence retains rank 2.
Exact mathematical passage
x^2-x-1
x^3Use in source · 1 passage
Search aliases: x^3
Use in source · block10_section.tex, line 369
[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.
Exact mathematical passage
\boxed{4x^3O^4-8x^2O^3+5xO^2-O+1=0.}
x^dUse in source · 3 passages
Search aliases: x^d
Use in source · Stratified_Universality_Traditional.tex, line 2044
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\chi(x)=x^d-c_{d-1}x^{d-1}-\cdots-c_1x-c_0, \qquad c_0\ne0,
x^{d-1}Use in source · 3 passages
Search aliases: x^d-1
Use in source · Stratified_Universality_Traditional.tex, line 2044
An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus Rec_K^min is a discrete groupoid.
Exact mathematical passage
\chi(x)=x^d-c_{d-1}x^{d-1}-\cdots-c_1x-c_0, \qquad c_0\ne0,
X^mUse in source · 1 passage
Search aliases: X^m
Use in source · block11_section.tex, line 20
Let A_m:X^m⟶ X, 2≤ m≤ D, be bounded symmetric multilinear maps, and let s∈ X. Consider the marked equation u=s+Φ(u), Φ(u)=∑_m=2^DA_m[u,…,u]. tag11.1 The word marked means that the source term, every multilinear operation, the packet system, and the selected convergence or filtration certificate are retained as part of the presentation. The fixed point u alone is not assumed to reconstruct those data.
Exact mathematical passage
A_m:X^m\longrightarrow X, \qquad 2\le m\le D,
x^mUse in source · 5 passages
Search aliases: x^m
Use in source · block09_section.tex, line 254
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\mathfrak M_{N,j}F(x) = \sum_{m\geq0}a_{Nm+j}x^m.
x^{m-r}Use in source · 1 passage
Search aliases: x^m-r
Use in source · block11_section.tex, line 77
The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.
Exact mathematical passage
A_m[x^m]-A_m[y^m] = \sum_{r=1}^{m} A_m[y^{r-1},x-y,x^{m-r}],
X^nUse in source · 1 passage
Search aliases: X^n
Use in source · block15_section.tex, line 618
Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expansions gives an exact lane expansion of P.
Exact mathematical passage
T_0\in Y, \qquad T_n:X^n\longrightarrow Y\quad(1\le n\le D)
x^NUse in source · 4 passages
Search aliases: x^N
Use in source · Stratified_Universality_Traditional.tex, line 2261
umerate[label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0. If m_a(x) is the monic minimal recurrence polynomial and T=T(a), then m_a(x)mid x^T-1, T=min{N≥ 1:m_a(x)mid x^N-1}.
Exact mathematical passage
\label{eq:minimal-period-annihilator} m_a(x)\mid x^T-1, \qquad T=\min\{N\geq 1:m_a(x)\mid x^N-1\}.
x^nUse in source · 3 passages
Search aliases: x^n
Use in source · block11_section.tex, line 235
[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.
Exact mathematical passage
\boxed{ C_{n,M} =[x^n]\prod_{r\ge1}(1-x^r)^{-M}.} \tag{11.6}
X^{p^a}Use in source · 1 passage
Search aliases: X^p^a
Use in source · Stratified_Universality_Traditional.tex, line 5842
[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.
Exact mathematical passage
\boxed{X^{p^a}=I+p^{L+a}B_a.}
x^rUse in source · 3 passages
Search aliases: x^r
Use in source · block11_section.tex, line 235
[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.
Exact mathematical passage
\boxed{ C_{n,M} =[x^n]\prod_{r\ge1}(1-x^r)^{-M}.} \tag{11.6}
x^TUse in source · 3 passages
Search aliases: x^T
Use in source · Stratified_Universality_Traditional.tex, line 2261
umerate[label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0. If m_a(x) is the monic minimal recurrence polynomial and T=T(a), then m_a(x)mid x^T-1, T=min{N≥ 1:m_a(x)mid x^N-1}.
Exact mathematical passage
\label{eq:minimal-period-annihilator} m_a(x)\mid x^T-1, \qquad T=\min\{N\geq 1:m_a(x)\mid x^N-1\}.
x^{\underline j}Use in source · 1 passage
Search aliases: x^underline j underline
Use in source · block09_section.tex, line 119
[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.
Exact mathematical passage
x^{\underline j}=x(x-1)\cdots(x-j+1).
X_0Use in source · 2 passages
Search aliases: X_0
Use in source · Stratified_Universality_Traditional.tex, line 559
[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.
Exact mathematical passage
X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m
x_0Use in source · 3 passages
Search aliases: x_0
Use in source · block18_carrier_observer_change_final.tex, line 244
[Haar cancellation] For the normalized Haar bank, ℓ=fracx_0+x_1sqrt2, d=fracx_0-x_1sqrt2, and x_0=fracℓ+dsqrt2, x_1=fracℓ-dsqrt2. At x=(1,0), the two route contributions to x_0 are 1/2,1/2, while those to x_1 are 1/2,-1/2. Additive reconstruction returns (1,0). Replacing route addition by maximum returns (1/2,1/2). The convex/max carrier is therefore not an exact replacement for the signed additive carrier.
Exact mathematical passage
\ell=\frac{x_0+x_1}{\sqrt2}, \qquad d=\frac{x_0-x_1}{\sqrt2},
X_1Use in source · 1 passage
Search aliases: X_1
Use in source · Stratified_Universality_Traditional.tex, line 559
[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.
Exact mathematical passage
X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m
x_1Use in source · 5 passages
Search aliases: x_1
Use in source · block15_section.tex, line 601
inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T(x_1,\ldots,x_d) = \sum_{a,i_1,\ldots,i_d} \iota_{Y,a} T_{a;i_1,\ldots,i_d} (\pi_{1,i_1}x_1,\ldots,\pi_{d,i_d}x_d).
x_2Use in source · 1 passage
Search aliases: x_2
Use in source · block19_admissibility_lifting_final.tex, line 183
The two images are A_1(x_1,x_2)=(x_1,x_1), A_2(x_1,x_2)=(x_2,-x_2), so the first identity is immediate.
Exact mathematical passage
A_1(x_1,x_2)=(x_1,x_1), \qquad A_2(x_1,x_2)=(x_2,-x_2),
X_aUse in source · 1 passage
Search aliases: X_a
Use in source · Stratified_Universality_Traditional.tex, line 5853
Induct on a. If X_a=I+p^L+aB_a, the binomial expansion of X_a^p has linear term divisible by p^L+a+1, and every higher term has at least the same valuation. Apply [source reference] at the refined jump.
Exact mathematical passage
X_a=I+p^{L+a}B_a
X_a^pUse in source · 1 passage
Search aliases: X_a^p
Use in source · Stratified_Universality_Traditional.tex, line 5853
Induct on a. If X_a=I+p^L+aB_a, the binomial expansion of X_a^p has linear term divisible by p^L+a+1, and every higher term has at least the same valuation. Apply [source reference] at the refined jump.
Exact mathematical passage
X_a^p
x_dUse in source · 1 passage
Search aliases: x_d
Use in source · block15_section.tex, line 601
inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T(x_1,\ldots,x_d) = \sum_{a,i_1,\ldots,i_d} \iota_{Y,a} T_{a;i_1,\ldots,i_d} (\pi_{1,i_1}x_1,\ldots,\pi_{d,i_d}x_d).
x_iUse in source · 1 passage
Search aliases: x_i
Use in source · block18_carrier_observer_change_final.tex, line 197
Expand the raw history update and group all terms by their predecessor state and new branch. Associativity and commutativity permit the regrouping. The identity L_b (⊕_i x_i) = ⊕_i L_b(x_i) then moves aggregation through the branch action. The displayed formula is the resulting statewise sum.
Exact mathematical passage
L_b\!\left(\bigoplus_i x_i\right) = \bigoplus_i L_b(x_i)
X_jUse in source · 1 passage
Search aliases: X_j
Use in source · Stratified_Universality_Traditional.tex, line 562
[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.
Exact mathematical passage
X_j
X_j^sUse in source · 1 passage
Search aliases: X_j^s
Use in source · block15_section.tex, line 376
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathsf B^{v,s}(X,Y) := \bigoplus_{b\in I_{Y,v}} \bigoplus_{j\in I_{X,s}} \Hom_{\mathcal C}(X_j^s,Y_b^v),
X_mUse in source · 2 passages
Search aliases: X_m
Use in source · Stratified_Universality_Traditional.tex, line 559
[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.
Exact mathematical passage
X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m
x_rUse in source · 3 passages
Search aliases: x_r
Use in source · Stratified_Universality_Traditional.tex, line 4547
A section word at level r is lifted by the same word at level r+1, proving surjectivity. Every p-adic limit of orbit points gives a compatible tower. Conversely, choose for each compatible residue state an orbit representative x_r. Compatibility gives x_r+1-x_r∈ p^rM_k(a), so (x_r) is Cauchy in M_k,p(a). Its limit lies in the orbit closure and maps to the prescribed tower.
Exact mathematical passage
x_r
x_{r+1}Use in source · 1 passage
Search aliases: x_r+1
Use in source · Stratified_Universality_Traditional.tex, line 4548
A section word at level r is lifted by the same word at level r+1, proving surjectivity. Every p-adic limit of orbit points gives a compatible tower. Conversely, choose for each compatible residue state an orbit representative x_r. Compatibility gives x_r+1-x_r∈ p^rM_k(a), so (x_r) is Cauchy in M_k,p(a). Its limit lies in the orbit closure and maps to the prescribed tower.
Exact mathematical passage
x_{r+1}-x_r\in p^rM_k(a),
X_UUse in source · 1 passage
Search aliases: X_U
Use in source · block15_section.tex, line 1086
[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
X=(X_U,\theta^X_{UV}), \qquad Y=(Y_U,\theta^Y_{UV}), \qquad Z=(Z_U,\theta^Z_{UV})
x_vUse in source · 1 passage
Search aliases: x_v
Use in source · Stratified_Universality_Traditional.tex, line 1515
For x∈ C_o, set x_v=G_vx. Tree edges satisfy the parallel relation by construction, and a non-tree edge satisfies it because H_ex=x. Conversely, any global section is determined along the tree by its root value; every remaining edge forces that value to be fixed by the corresponding H_e. This proves the section classification. Closed paths are words in fundamental cycles, which gives path-flatness and tree independence. Any path-flat family must be fixed by every fundamental cycle and is therefo…
Exact mathematical passage
x_v=G_vx
x_\ellUse in source · 1 passage
Search aliases: x_ℓ ell
Use in source · block15_section.tex, line 632
Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement is then a finite sum of the specified homogeneous pieces.
Exact mathematical passage
x_\ell=\sum_{i_\ell}p_{\ell,i_\ell}x_\ell
Y^2Use in source · 1 passage
Search aliases: Y^2
Use in source · block10_section.tex, line 510
If α=β, the quadratic equation Y^2-(1-α z)^2=0 is not the normalized source polynomial. The selected branch is rational: F_α,α(z)=1-α z, so the correct source degree is one. This is a negative control against using a nonminimal defining equation in the degree ledger.
Exact mathematical passage
Y^2-(1-\alpha z)^2=0
y^mUse in source · 2 passages
Search aliases: y^m
Use in source · block11_section.tex, line 77
The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.
Exact mathematical passage
A_m[x^m]-A_m[y^m] = \sum_{r=1}^{m} A_m[y^{r-1},x-y,x^{m-r}],
Y^qUse in source · 1 passage
Search aliases: Y^q
Use in source · block13_section.tex, line 269
Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.
Exact mathematical passage
P_{q,\alpha}(z,Y)=Y-z-\alpha Y^q
y^{r-1}Use in source · 1 passage
Search aliases: y^r-1
Use in source · block11_section.tex, line 77
The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.
Exact mathematical passage
A_m[x^m]-A_m[y^m] = \sum_{r=1}^{m} A_m[y^{r-1},x-y,x^{m-r}],
y_0Use in source · 1 passage
Search aliases: y_0
Use in source · block09_section.tex, line 203
[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.
Exact mathematical passage
(y_0,\ldots,y_{r-1})\in K^r
Y_a^uUse in source · 3 passages
Search aliases: Y_a^u
Use in source · block15_section.tex, line 89
At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.
Exact mathematical passage
[f]^{u,r}_{a i} := \pi_{Y,a}^u f\iota_{X,i}^r: X_i^r\longrightarrow Y_a^u.
Y_b^vUse in source · 1 passage
Search aliases: Y_b^v
Use in source · block15_section.tex, line 376
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathsf B^{v,s}(X,Y) := \bigoplus_{b\in I_{Y,v}} \bigoplus_{j\in I_{X,s}} \Hom_{\mathcal C}(X_j^s,Y_b^v),
y_jUse in source · 1 passage
Search aliases: y_j
Use in source · block09_section.tex, line 207
[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.
Exact mathematical passage
LF=0, \qquad F^{(j)}(0)=y_j.
y_{r-1}Use in source · 1 passage
Search aliases: y_r-1
Use in source · block09_section.tex, line 203
[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.
Exact mathematical passage
(y_0,\ldots,y_{r-1})\in K^r
Y_UUse in source · 1 passage
Search aliases: Y_U
Use in source · block15_section.tex, line 1086
[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
X=(X_U,\theta^X_{UV}), \qquad Y=(Y_U,\theta^Y_{UV}), \qquad Z=(Z_U,\theta^Z_{UV})
\mathbb Z^{\mathbb N_{\geq1}}Use in source · 1 passage
Search aliases: Z^ N_≥1 mathbb mathbb geq
Use in source · Stratified_Universality_Traditional.tex, line 4792
Let U_ Z=1+z Z[[z]] be the multiplicative group of integral unit series, and let E_ Z= Z^ N_≥1 be the additive group of integer exponent sequences.
Exact mathematical passage
\mathcal E_{\mathbb Z}=\mathbb Z^{\mathbb N_{\geq1}}
z^{-1}Use in source · 2 passages
Search aliases: z^-1
Use in source · Stratified_Universality_Traditional.tex, line 2073
The recurrence-to-OGF bridge is explicit. From χ(x)=x^d-c_d-1x^d-1-⋯-c_0 set Q(z)=z^dχ(z^-1) =1-c_d-1z-⋯-c_0z^d. The numerator is the degree-<d truncation of Q(z)∑_n≥0a_nz^n. Conversely, clearing Q recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.
Exact mathematical passage
Q(z)=z^d\chi(z^{-1}) =1-c_{d-1}z-\cdots-c_0z^d.
z^0Use in source · 1 passage
Search aliases: z^0
Use in source · Stratified_Universality_Traditional.tex, line 3039
rtwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.
Exact mathematical passage
\epsilon_0(G)=[z^0]G
z^{1+(q-1)n}Use in source · 1 passage
Search aliases: z^1+(q-1)n
Use in source · block15_section.tex, line 944
[Typed Fuss--Catalan realization] Let K be a characteristic-zero field, let q∈ Z with q≥2, and let α∈ K^×. The uniquely marked formal solution of U=z+α U^q, U(0)=0, has support [z^1+(q-1)n]U =frac1(q-1)n+1binomqnnα^n ≠0 (n≥0), and every other coefficient is zero. Thus its support is exactly the set of degrees congruent to 1pmodq-1. For an integer packet width N≥1, set g=gcd(N,q-1). Exactly N/g residue addresses are active, namely the congruence class 1pmod g. Packetizing the specified multilinea…
Exact mathematical passage
[z^{1+(q-1)n}]U =\frac{1}{(q-1)n+1}\binom{qn}{n}\alpha^n \ne0 \qquad(n\ge0),
z^2Use in source · 26 passages
Search aliases: z^2
Use in source · Stratified_Universality_Traditional.tex, line 2187
Fibonacci F_n · x^2-x-1 · z/(1-z-z^2) · 2 · The root ratio is not torsion. Every arithmetic subsequence retains rank 2.
Exact mathematical passage
z/(1-z-z^2)
z^{2m}Use in source · 1 passage
Search aliases: z^2m
Use in source · Stratified_Universality_Traditional.tex, line 4959
[Distinct parts and normalization] The product ∏_m≥1(1+z^m) = ∏_m≥1frac1-z^2m1-z^m has normalized Euler exponent b_n= 1, · n odd,
Exact mathematical passage
\prod_{m\geq1}(1+z^m) = \prod_{m\geq1}\frac{1-z^{2m}}{1-z^m}
z^3Use in source · 2 passages
Search aliases: z^3
Use in source · Stratified_Universality_Traditional.tex, line 2755
Alternating unit over F_3. For the word 12, T=2, m(x)=x+1, F(z)=frac11+z, R(z)=frac1+z^31+z=1+2z+z^2. The recurrence order is one although the minimal period is two. The 3-kernel consists of the sequence and its negative, giving two minimal digital states. The compiler bound is nine.
Exact mathematical passage
T=2, \qquad m(x)=x+1, \qquad F(z)=\frac{1}{1+z}, \qquad R(z)=\frac{1+z^3}{1+z}=1+2z+z^2.
z^{3^j}Use in source · 1 passage
Search aliases: z^3^j
Use in source · Stratified_Universality_Traditional.tex, line 3347
Cantor-digit indicator. Over F_3, let R(z)=1+z. Then F(z)=∏_j≥0(1+z^3^j) has coefficient one exactly when the ternary expansion of the index uses only the digits zero and one. The functional equation is F(z)=(1+z)F(z^3), F(z)^2=frac11+z. The factor 1+z has odd divisor multiplicity, so the collapse obstruction is nonzero. Thus the sequence is automatic but not C-finite or eventually periodic. The canonical automaton has two states, while the Cartier rank is one.
Exact mathematical passage
F(z)=\prod_{j\geq0}(1+z^{3^j})
z^dUse in source · 3 passages
Search aliases: z^d
Use in source · Stratified_Universality_Traditional.tex, line 2073
The recurrence-to-OGF bridge is explicit. From χ(x)=x^d-c_d-1x^d-1-⋯-c_0 set Q(z)=z^dχ(z^-1) =1-c_d-1z-⋯-c_0z^d. The numerator is the degree-<d truncation of Q(z)∑_n≥0a_nz^n. Conversely, clearing Q recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.
Exact mathematical passage
Q(z)=z^d\chi(z^{-1}) =1-c_{d-1}z-\cdots-c_0z^d.
\mathbb Z^{d+1}Use in source · 2 passages
Search aliases: Z^d+1 mathbb
Use in source · Stratified_Universality_Traditional.tex, line 1979
Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.
Exact mathematical passage
A=P_{m,S}^{(d)}:\mathbb Z^{d+1}\to\mathbb Z^{|S|(d+1)}
z^{d_{\mathcal A}}Use in source · 2 passages
Search aliases: z^d_ A mathcal
Use in source · block11_section.tex, line 436
By [source reference], the coefficient support is contained in 1+d_ A N and the reduced exponent set is aperiodic by maximality of the gcd. Hence u(z)=zV(z^d_ A) with V aperiodic. The smooth implicit schema gives one positive dominant singularity for V, and pulling back through z^d_ A gives textup(11.12). The map z↦ z^N sends μ_d_ A onto a root group of cardinality d_ A/gcd(N,d_ A). No packet can introduce a new dominant image outside this set. Conversely, if every packet were analytic at one im…
Exact mathematical passage
u(z)=zV(z^{d_{\mathcal A}})
z^eUse in source · 1 passage
Search aliases: z^e
Use in source · block13_section.tex, line 391
Let A_b(z)=∏_m≥1(1-z^m)^-b_m be an integral unit series in the fixed Euler basis, and let c_n=∑_dmid nd b_d be its divisor-ghost sequence. Define the prime-uniform multiplicative function f_b(n)=∏_p^eparallel n[z^e]A_b(z). Every prime has the same local factor A_b(T). By [source reference], the coefficient, exponent, ghost, and prime-uniform multiplicative charts are reversible on this declared locus.
Exact mathematical passage
f_b(n)=\prod_{p^e\parallel n}[z^e]A_b(z).
z^jUse in source · 9 passages
Search aliases: z^j
Use in source · Stratified_Universality_Traditional.tex, line 4930
The logarithmic-derivative identity is equivalent to zA'(z)=(∑_j≥1c_jz^j)A(z). Comparing coefficients gives the recurrence. Conversely, the recurrence uniquely determines the coefficient sequence over Q. Since c∈ G_ Z reconstructs an integral exponent sequence, the resulting series is the Euler product from [source reference] and therefore has integral coefficients.
Exact mathematical passage
zA'(z)=\left(\sum_{j\geq1}c_jz^j\right)A(z).
z^kUse in source · 2 passages
Search aliases: z^k
Use in source · block09_section.tex, line 167
The operator p_s(θ-s) multiplies the coefficient of z^k by p_s(k-s). After multiplication by z^q-s, the coefficient at z^n+q is p_s(n)a_n+s. The recurrence kills every coefficient from degree N+q onward. Differentiating more times than the residual degree gives a homogeneous annihilator.
Exact mathematical passage
z^k
z^{mr}Use in source · 1 passage
Search aliases: z^mr
Use in source · Stratified_Universality_Traditional.tex, line 4900
Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.
Exact mathematical passage
\frac{zA'}A = \sum_{m\geq1}b_m\frac{mz^m}{1-z^m} = \sum_{m\geq1}\sum_{r\geq1}m b_m z^{mr}.
z^{N+1}Use in source · 2 passages
Search aliases: z^N+1
Use in source · Stratified_Universality_Traditional.tex, line 4776
[Filtered product completion] Every filtered factor family has a unique coefficientwise product in 1+zR[[z]]. For every N, the truncation modulo z^N+1 is equal to the product of the finitely many factors with w(m)≤ N.
Exact mathematical passage
z^{N+1}
z^{n+1}Use in source · 3 passages
Search aliases: z^n+1
Use in source · Stratified_Universality_Traditional.tex, line 4831
Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, the…
Exact mathematical passage
A(z)\prod_{m<n}(1-z^m)^{b_m} \equiv 1+r_nz^n\pmod{z^{n+1}}.
z^{n+q}Use in source · 2 passages
Search aliases: z^n+q
Use in source · block09_section.tex, line 154
[Recurrence residual as an Euler-operator polynomial] Define Q = ∑_s=0^q z^q-sp_s(θ-s). Then [z^n+q]QF = ∑_s=0^qp_s(n)a_n+s. Hence QF is a polynomial of degree less than N+q, and D^N+qQF=0.
Exact mathematical passage
\boxed{ [z^{n+q}]QF = \sum_{s=0}^{q}p_s(n)a_{n+s}.}
z^{N-1}Use in source · 2 passages
Search aliases: z^N-1
Use in source · block18_carrier_observer_change_final.tex, line 311
olyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
\det H=\pm z^{N-1}.
z^{n-m}Use in source · 1 passage
Search aliases: z^n-m
Use in source · block09_section.tex, line 126
The coefficient of z^n-m in D^jF is (n-m+j)^underline ja_n-m+j. Multiplication by q_j,mz^m shifts that contribution to degree n. Summing over j and m gives the formula.
Exact mathematical passage
z^{n-m}
z^{Nn+j}Use in source · 1 passage
Search aliases: z^Nn+j
Use in source · block15_section.tex, line 711
uppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
E_{N,j}(z) :=\mathcal P_{N,j}e^z =\sum_{n\ge0}\frac{z^{Nn+j}}{(Nn+j)!}.
z^{Nr+j}Use in source · 1 passage
Search aliases: z^Nr+j
Use in source · block11_section.tex, line 351
[Finite tree-address signature] Fix a packet width N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10
Exact mathematical passage
U_j(x)=\sum_{r\ge0}[z^{Nr+j}]u(z)\,x^r,
z^pUse in source · 18 passages
Search aliases: z^p
Use in source · Stratified_Universality_Traditional.tex, line 2242
[Why the unit condition is retained] The functional-Mahler bridge uses the formal quotient R_a(z)=fracF_a(z)F_a(z^p). It is regular at the origin exactly when F_a(0)≠ 0. The normalization a_0=1 removes scalar ambiguity and gives R_a(0)=1. Purely periodic sequences with a_0=0 remain valid C-finite and automatic objects, but they need not lie in this regular unit-mask overlap. They are not discarded; they lie outside the selected heterogeneous atlas.
Exact mathematical passage
R_a(z)=\frac{F_a(z)}{F_a(z^p)}.
z^{p^a}Use in source · 1 passage
Search aliases: z^p^a
Use in source · Stratified_Universality_Traditional.tex, line 5788
In characteristic p, (1+z)^p^a-1≡ z^p^a. Substitute this into the generating definition of Θ_p^a,s and compare coefficients.
Exact mathematical passage
(1+z)^{p^a}-1\equiv z^{p^a}.
z^{p^j}Use in source · 4 passages
Search aliases: z^p^j
Use in source · Stratified_Universality_Traditional.tex, line 2432
For a unit formal series F(0)=1, define its local functional-Mahler mask by M_p(F)(z) = fracF(z)F(z^p). Then F(z)= M_p(F)(z)F(z^p). Iterating [source reference] gives the coefficientwise product F(z) = ∏_j≥0 M_p(F)(z^p^j). Only finitely many factors contribute to each fixed coefficient.
Exact mathematical passage
\label{eq:mask-product} F(z) = \prod_{j\geq0}\mathcal M_p(F)(z^{p^j}).
z^{p^{j+1}}Use in source · 1 passage
Search aliases: z^p^j+1
Use in source · Stratified_Universality_Traditional.tex, line 2477
Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.
Exact mathematical passage
\prod_{j=0}^{N}\frac{F(z^{p^j})}{F(z^{p^{j+1}})} = \frac{F(z)}{F(z^{p^{N+1}})}.
z^{p^{N+1}}Use in source · 1 passage
Search aliases: z^p^N+1
Use in source · Stratified_Universality_Traditional.tex, line 2477
Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.
Exact mathematical passage
\prod_{j=0}^{N}\frac{F(z^{p^j})}{F(z^{p^{j+1}})} = \frac{F(z)}{F(z^{p^{N+1}})}.
z^qUse in source · 1 passage
Search aliases: z^q
Use in source · block13_section.tex, line 375
The examples show that the finite invariant retains the weight parameter as well as the arity. Support period alone recovers only q-1; the coefficient [z^q]U=α supplies the second coordinate.
Exact mathematical passage
[z^q]U=\alpha
z^{q-1}Use in source · 3 passages
Search aliases: z^q-1
Use in source · block13_section.tex, line 275
Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value
Exact mathematical passage
x=z^{q-1}, \qquad U_{q,\alpha}(z)=zC_{q,\alpha}(x).
z^{q-s}Use in source · 2 passages
Search aliases: z^q-s
Use in source · block09_section.tex, line 146
[Recurrence residual as an Euler-operator polynomial] Define Q = ∑_s=0^q z^q-sp_s(θ-s). Then [z^n+q]QF = ∑_s=0^qp_s(n)a_n+s. Hence QF is a polynomial of degree less than N+q, and D^N+qQF=0.
Exact mathematical passage
\boxed{ Q = \sum_{s=0}^{q} z^{q-s}p_s(\theta-s).}
z^TUse in source · 4 passages
Search aliases: z^T
Use in source · Stratified_Universality_Traditional.tex, line 2286
Let F_a(z)=fracP_a(z)Q_a(z) be reduced with P_a(0)=Q_a(0)=1. If d=deg m_a, then Q_a(z)=z^dm_a(z^-1), with the evident coefficient normalization. Taking reciprocal polynomials in [source reference] gives Q_a(z)mid 1-z^T, T=min{N≥ 1:Q_a(z)mid 1-z^N}. Thus pure periodicity is visible both in the linear annihilator and in the reduced rational denominator.
Exact mathematical passage
\label{eq:minimal-period-denominator} Q_a(z)\mid 1-z^T, \qquad T=\min\{N\geq 1:Q_a(z)\mid 1-z^N\}.
z^{T(a)}Use in source · 1 passage
Search aliases: z^T(a)
Use in source · Stratified_Universality_Traditional.tex, line 2229
screte category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3, the period word 12 has frac1+2z…
Exact mathematical passage
\label{eq:periodic-ogf-unreduced} F_a(z)=\sum_{n\geq 0}a_nz^n = \frac{W_a(z)}{1-z^{T(a)}}.
\mathbb Z^{|S|(d+1)}Use in source · 2 passages
Search aliases: Z^|S|(d+1) mathbb
Use in source · Stratified_Universality_Traditional.tex, line 1979
Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.
Exact mathematical passage
A=P_{m,S}^{(d)}:\mathbb Z^{d+1}\to\mathbb Z^{|S|(d+1)}
\mathbb Z^\rhoUse in source · 2 passages
Search aliases: Z^ρ mathbb rho
Use in source · Stratified_Universality_Traditional.tex, line 4087
Choose a coordinate-detection set n by [source reference]. Evaluation embeds L_k(a) into Z^ρ, so L_k(a) is a finitely generated free abelian group of rank at most ρ. The raw lattice M_k(a) is contained in L_k(a), and its rational span is all of V_ Q(a). Hence M_k(a) has rank ρ, and therefore L_k(a) also has rank ρ. A subgroup of a finitely generated free abelian group is finitely generated and free, so M_k(a) is free of rank ρ. Equal-rank lattice inclusion has finite index.
Exact mathematical passage
\mathbb Z^\rho
\mathbb Z_{>0}Use in source · 1 passage
Search aliases: Z_>0 mathbb
Use in source · block09_section.tex, line 530
item If λ∉ Z, then r_D(N,j;λ)=N. item If λ=L∈ Z_>0, then every packet is a nonzero rational function with exact denominator (1-α^Nx)^L, and r_D(N,j;L)=1. item If λ=-M∈ Z_≤0, then G_N,j^-M,α is zero when j>M. Otherwise it is a nonzero polynomial of degree lfloorfracM-jNrfloor and r_D(N,j;-M) = 1, · j≤ M,
Exact mathematical passage
\lambda=L\in\mathbb Z_{>0}
Z_UUse in source · 1 passage
Search aliases: Z_U
Use in source · block15_section.tex, line 1086
[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
X=(X_U,\theta^X_{UV}), \qquad Y=(Y_U,\theta^Y_{UV}), \qquad Z=(Z_U,\theta^Z_{UV})
\mathbb Z_{\leq0}Use in source · 2 passages
Search aliases: Z_≤0 mathbb leq
Use in source · block09_section.tex, line 538
item If λ∉ Z, then r_D(N,j;λ)=N. item If λ=L∈ Z_>0, then every packet is a nonzero rational function with exact denominator (1-α^Nx)^L, and r_D(N,j;L)=1. item If λ=-M∈ Z_≤0, then G_N,j^-M,α is zero when j>M. Otherwise it is a nonzero polynomial of degree lfloorfracM-jNrfloor and r_D(N,j;-M) = 1, · j≤ M,
Exact mathematical passage
\lambda=-M\in\mathbb Z_{\leq0}
\mathbb Z_{\ge1}Use in source · 1 passage
Search aliases: Z_≥1 mathbb ge
Use in source · block15_section.tex, line 993
[Typed cyclic-radical realization] Let K be a characteristic-zero field containing μ_d and a primitive N-th root of unity ω, let d,m,N∈ Z_≥1, and choose the formal branch F(z)=(1-α z^m)^-1/d, α∈ K^×. With g=gcd(N,m), its nonzero coefficients occupy exactly the subgroup of Z/N Z generated by m, which has N/g addresses. The rotated radicands 1-αω^jmz^m, 0≤ j<N, also have exactly N/g distinct values.
Exact mathematical passage
d,m,N\in\mathbb Z_{\ge1}
|^{-1/n}Use in source · 1 passage
Search aliases: |^-1/n
Use in source · block19_admissibility_lifting_final.tex, line 121
[Conformal metric-completion criterion] Let a connected finite graph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge scales satisfying T_e^*Q_t(e)T_e=λ_e^2Q_s(e) exist if and only if the determinant-norma…
Exact mathematical passage
\widehat H:=|\det H|^{-1/n}H.
|^{1/n}Use in source · 2 passages
Search aliases: |^1/n
Use in source · block19_admissibility_lifting_final.tex, line 135
A compatible family makes every closed root holonomy H conformal for the root metric Q_o. Taking determinants fixes its positive scale as |det H|^1/n, so widehat H preserves Q_o.
Exact mathematical passage
|\det H|^{1/n}
|^{2/n}Use in source · 2 passages
Search aliases: |^2/n
Use in source · block19_admissibility_lifting_final.tex, line 143
Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible family directly. A group preserving a positive-definite metric is conjugate into a compact orthogonal or unitary group. Conversely, averaging an inner p…
Exact mathematical passage
H_e^*Q_oH_e=|\det H_e|^{2/n}Q_o.
|_{\mathcal N_\alpha}Use in source · 1 passage
Search aliases: |_ N_α mathcal alpha
Use in source · Stratified_Universality_Traditional.tex, line 1076
[Semantic conservativity of a normalized chart] If N_α is classification-normalizing, then Sem_α|_ N_α reflects isomorphisms. Consequently, a complete invariant on N_α can be transported through a reversible overlap only after the overlap is shown to preserve the normalization and reflect the relevant isomorphisms.
Exact mathematical passage
\Sem_\alpha|_{\mathcal N_\alpha}
|_{\im A}Use in source · 1 passage
Search aliases: |_im A im operatorname
Use in source · Stratified_Universality_Traditional.tex, line 2013
The identities LA=I and Π=AL show that A identifies the source with im A, with inverse L|_im A. For any packet vector y, y=ALy+(y-ALy), where ALy∈im A and y-ALy∈ker L. The intersection is zero because Ax∈ker L implies x=LAx=0. Tensoring LA=I with any coefficient ring preserves the split identity. Intertwining maps preserve the image and kernel decompositions.
Exact mathematical passage
L|_{\im A}
\pmUse in source · 2 passages
Search aliases: ± pm
Use in source · block18_carrier_observer_change_final.tex, line 311
olyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f_N(n+k) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is N.
Exact mathematical passage
\det H=\pm z^{N-1}.
\times_CUse in source · 1 passage
Search aliases: ×_C times
Use in source · block17_semantic_quotients_final.tex, line 318
S @>>> C is a pullback in R-Mod. Equivalently, J≅ S×_C T. There is also a canonical short exact sequence 0⟶ X/(N∩ M) xrightarrowα X/N⊕ X/M xrightarrowβ X/(N+M) ⟶0, where α(x+N∩ M)=(x+N,x+M) and β(a+N,b+M)=a-b+(N+M).
Exact mathematical passage
J\cong S\times_C T.
\alpha^{-N}Use in source · 4 passages
Search aliases: α^-N alpha
Use in source · block09_section.tex, line 677
For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.
Exact mathematical passage
\boxed{ 0, \qquad \alpha^{-N}, \qquad \infty.}
\alpha^2Use in source · 4 passages
Search aliases: α^2 alpha
Use in source · block10_section.tex, line 443
[Quadratic-radical parity packets] The packets satisfy E^2+xO^2=1+px, 2EO=-s. Their eliminants are E^4-(1+px)E^2+fracs^2x4=0 and xO^4-(1+px)O^2+fracs^24=0. If s≠0, α^2≠β^2, then both selected packets have degree four over K(x).
Exact mathematical passage
s\neq0, \qquad \alpha^2\neq\beta^2,
\alpha^jUse in source · 1 passage
Search aliases: α^j alpha
Use in source · block09_section.tex, line 477
[Exact hypergeometric packet form] If (λ)_j≠0, then G_N,j^λ,α(x) = frac(λ)_jα^jj! _NF_N-1 ( A_N,j(λ) B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ G_{N,j}^{\lambda,\alpha}(x) = \frac{(\lambda)_j\alpha^j}{j!} {}_NF_{N-1} \!\left( \begin{matrix}A_{N,j}(\lambda)\\ B_{N,j}\end{matrix} ;\alpha^Nx \right).}
\alpha^NUse in source · 8 passages
Search aliases: α^N alpha
Use in source · block09_section.tex, line 417
[Singularity orbit under multisection] Assume F is an analytic germ at zero with analytic continuation away from a finite set Σ. Then every finite nonzero analytic singularity of M_N,jF lies in Σ^[N] = {α^N:α∈Σ}. Distinct singularities in one fiber of α↦α^N are aliased by the multisection map. A character projection may remove some aliased local contributions.
Exact mathematical passage
\boxed{ \Sigma^{[N]} = \{\alpha^N:\alpha\in\Sigma\}.}
\alpha^nUse in source · 3 passages
Search aliases: α^n alpha
Use in source · block09_section.tex, line 445
Its coefficients are a_n = frac(λ)_nn!α^n. For 0≤ j<N, put G_N,j^λ,α(x) = ∑_m≥0 frac(λ)_Nm+j(Nm+j)! α^Nm+jx^m. Define the upper and lower parameter packets A_N,j(λ) = { fracλ+j+rN:0≤ r<N }, B_N,j = { fracj+1+rN:0≤ r<N }∖{1}.
Exact mathematical passage
a_n = \frac{(\lambda)_n}{n!}\alpha^n.
\alpha^{Nm+j}Use in source · 2 passages
Search aliases: α^Nm+j alpha
Use in source · block09_section.tex, line 451
Its coefficients are a_n = frac(λ)_nn!α^n. For 0≤ j<N, put G_N,j^λ,α(x) = ∑_m≥0 frac(λ)_Nm+j(Nm+j)! α^Nm+jx^m. Define the upper and lower parameter packets A_N,j(λ) = { fracλ+j+rN:0≤ r<N }, B_N,j = { fracj+1+rN:0≤ r<N }∖{1}.
Exact mathematical passage
G_{N,j}^{\lambda,\alpha}(x) = \sum_{m\geq0} \frac{(\lambda)_{Nm+j}}{(Nm+j)!} \alpha^{Nm+j}x^m.
\beta^2Use in source · 2 passages
Search aliases: β^2 beta
Use in source · block10_section.tex, line 443
[Quadratic-radical parity packets] The packets satisfy E^2+xO^2=1+px, 2EO=-s. Their eliminants are E^4-(1+px)E^2+fracs^2x4=0 and xO^4-(1+px)O^2+fracs^24=0. If s≠0, α^2≠β^2, then both selected packets have degree four over K(x).
Exact mathematical passage
s\neq0, \qquad \alpha^2\neq\beta^2,
\beta_{1}Use in source · 5 passages
Search aliases: β_1 beta
Use in source · block07_named_local_table.tex, line 10
Constant-one function 1 · β_1=1 · 1,1,1,1,1,1,… · 1,1,1,1,1,1,…
Exact mathematical passage
\beta_{1}=1
\beta_{2}Use in source · 2 passages
Search aliases: β_2 beta
Use in source · block07_named_local_table.tex, line 12
Liouville λ · β_1=-1, β_2=1 · 1,-1,1,-1,1,-1,… · -1,1,-1,1,-1,1,…
Exact mathematical passage
\beta_{2}=1
\beta_pUse in source · 3 passages
Search aliases: β_p beta
Use in source · Stratified_Universality_Traditional.tex, line 5665
upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d) =lfloorfrac dβ_prfloor+1. A local channel loses residual rank exactly at primes dividing M. item A …
Exact mathematical passage
\beta_p=p^{v_p(M)}
\beta_{p,d}Use in source · 1 passage
Search aliases: β_p,d beta
Use in source · Stratified_Universality_Traditional.tex, line 5072
Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).
Exact mathematical passage
\gamma_{p,e}(f) = \sum_{d\mid e}d\,\beta_{p,d}(f).
\beta_{p,m}Use in source · 3 passages
Search aliases: β_p,m beta
Use in source · Stratified_Universality_Traditional.tex, line 5066
Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).
Exact mathematical passage
F_{f,p}(T) = \prod_{m\geq1}(1-T^m)^{-\beta_{p,m}(f)}.
\Gamma'Use in source · 3 passages
Search aliases: Γ' Gamma
Use in source · block19_admissibility_lifting_final.tex, line 24
[Admissibility monotonicity] If Γ⊆Γ', then δ_Γ'(X)≤δ_Γ(X). Thus an obstruction in a restricted realization class need not remain an obstruction after the class is enlarged.
Exact mathematical passage
\Gamma\subseteq\Gamma'
\Gamma^{(u,r)}_{(v,s)}Use in source · 2 passages
Search aliases: Γ^(u,r)_(v,s) Gamma
Use in source · block15_section.tex, line 396
[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contribute. The inverse is the corresponding fine analysis after coarse synthesis. Consequently, changing a complete lane resolution loses no information. Lo…
Exact mathematical passage
\Gamma^{(u,r)}_{(v,s)} := \mathfrak a^{u,r}_{X,Y} \bigl(\mathfrak a^{v,s}_{X,Y}\bigr)^{-1}.
\Gamma_{f,p}Use in source · 4 passages
Search aliases: Γ_f,p Gamma
Use in source · Stratified_Universality_Traditional.tex, line 5113
[Valuation logarithmic derivative] Fix a prime p. Let Γ_f,p be the arithmetic function supported on powers of p with Γ_f,p(1)=0, Γ_f,p(p^e)=γ_p,e(f) (e≥1). Then (v_p· f)(n) = (Γ_f,pmathbin∗_Df)(n) (n≥1). Equivalently, fracT F_f,p'(T)F_f,p(T) = ∑_e≥1γ_p,e(f)T^e.
Exact mathematical passage
\Gamma_{f,p}
\gamma_{p,d}Use in source · 1 passage
Search aliases: γ_p,d gamma
Use in source · Stratified_Universality_Traditional.tex, line 5102
l(β_p,m(f))_p,m, (γ_p,e(f))_p,e. Dirichlet convolution is linearized by the last two charts: β_p,m(fmathbin∗_Dg) = β_p,m(f)+β_p,m(g), γ_p,e(fmathbin∗_Dg) = γ_p,e(f)+γ_p,e(g). The ghost chart consists exactly of the prime-indexed sequences satisfying the local M obius divisibility conditions frac1e∑_dmid eμ (frac ed)γ_p,d∈ Z.
Exact mathematical passage
\frac1e\sum_{d\mid e}\mu\!\left(\frac ed\right)\gamma_{p,d}\in\mathbb Z.
\gamma_{p,e}Use in source · 5 passages
Search aliases: γ_p,e gamma
Use in source · Stratified_Universality_Traditional.tex, line 5072
Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).
Exact mathematical passage
\gamma_{p,e}(f) = \sum_{d\mid e}d\,\beta_{p,d}(f).
\gamma_{p,j}Use in source · 1 passage
Search aliases: γ_p,j gamma
Use in source · Stratified_Universality_Traditional.tex, line 5139
The local logarithmic-derivative recurrence gives e f(p^e) = ∑_j=1^eγ_p,j(f)f(p^e-j). Write n=p^e u with (u,p)=1 and multiply by f(u). Since Γ_f,p is supported on p-powers, the resulting sum is exactly the Dirichlet convolution at n.
Exact mathematical passage
e f(p^e) = \sum_{j=1}^{e}\gamma_{p,j}(f)f(p^{e-j}).
\Gamma_{tr}Use in source · 1 passage
Search aliases: Γ_tr Gamma
Use in source · block15_section.tex, line 548
X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences are compatible with every certified change of resolution.
Exact mathematical passage
\Gamma_{tr}=\Gamma_{sr}\Gamma_{ts}.
\Gamma_{ts}Use in source · 1 passage
Search aliases: Γ_ts Gamma
Use in source · block15_section.tex, line 548
X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences are compatible with every certified change of resolution.
Exact mathematical passage
\Gamma_{tr}=\Gamma_{sr}\Gamma_{ts}.
\DeltaUse in source · 13 passages
Search aliases: Δ Delta
Use in source · Stratified_Universality_Traditional.tex, line 1498
extup(roman*)] item evaluation at the root identifies the space of global parallel sections with C_o; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies; item for Δ_S:F_o⟶⊕_e∉ SF_o, Δ_Sx=((H_e-I)x)_e∉ S, one has C_o=kerΔ_S, dim C_o=dim F_o-rankΔ_S; item the full fibers glue to one…
Exact mathematical passage
\Delta_S:F_o\longrightarrow\bigoplus_{e\notin S}F_o, \qquad \Delta_Sx=((H_e-I)x)_{e\notin S},
\Delta^rUse in source · 2 passages
Search aliases: Δ^r Delta
Use in source · Stratified_Universality_Traditional.tex, line 5437
[Mahler coordinate theorem] The synthesis map M_p:c_0( Z_p)⟶ C_p, c⟼ (x⟼∑_r≥0c_rbinom xr), is an isometric isomorphism of Z_p-modules. Its inverse is c_r=Δ^r f(0). In particular, the Newton--Mahler coefficient sequence is a canonical complete coordinate for a continuous Z_p-valued function.
Exact mathematical passage
\boxed{c_r=\Delta^r f(0).}
\delta^{\rm cl}_{N,j}Use in source · 1 passage
Search aliases: δ^rm cl_N,j delta rm
Use in source · block09_section.tex, line 368
Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j measures cancellation relative to that construction. It is not intrinsic until the construction has been fixed.
Exact mathematical passage
\delta^{\rm cl}_{N,j} = \operatorname{ord}Q^{\rm cl}_{N,j} - \operatorname{ord}L_{N,j}
\delta_*Use in source · 1 passage
Search aliases: δ_* delta
Use in source · block16_section_final.tex, line 239
For (H1), additivity and invertibility follow from the corresponding properties of F_γ and the splitting comparisons β. The address and splitting cocycle laws give (δγ)_*=δ_*γ_* after the declared transition comparisons.
Exact mathematical passage
(\delta\gamma)_*=\delta_*\gamma_*
\Delta_{12\cdot2^a}^{\,r}Use in source · 1 passage
Search aliases: Δ_12·2^a^ r Delta cdot
Use in source · Stratified_Universality_Traditional.tex, line 5867
18 · 11. Hence Δ_12·2^a^ rF_n ≡0pmod2^(a+3)r.
Exact mathematical passage
\boxed{ \Delta_{12\cdot2^a}^{\,r}F_n \equiv0\pmod{2^{(a+3)r}}.}
\delta_dUse in source · 1 passage
Search aliases: δ_d delta
Use in source · Stratified_Universality_Traditional.tex, line 2338
The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.
Exact mathematical passage
\delta_d(b)=\Lambda_db,
\delta_{de}Use in source · 1 passage
Search aliases: δ_de delta
Use in source · block15_section.tex, line 829
[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
\Lambda_dS_e=\delta_{de}1, \qquad \sum_{d=0}^{k-1}S_d\Lambda_d=1.
\delta_{e,e'}Use in source · 2 passages
Search aliases: δ_e,e' delta
Use in source · block15_section.tex, line 233
Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of Φ_κ,τ, so the fold is split lossless.
Exact mathematical passage
Q_ej_{\kappa(e')}\tau_{e'}(b_{e'}) = \delta_{e,e'}j_{\kappa(e')}\tau_{e'}(b_{e'}).
\Delta_H^rUse in source · 1 passage
Search aliases: Δ_H^r Delta
Use in source · Stratified_Universality_Traditional.tex, line 5826
[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.
Exact mathematical passage
\boxed{\Delta_H^r a_n\equiv0\pmod{p^{Lr}}.}
\delta_{j,1}Use in source · 1 passage
Search aliases: δ_j,1 delta
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
\Delta_QUse in source · 2 passages
Search aliases: Δ_Q Delta
Use in source · block19_admissibility_lifting_final.tex, line 79
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
D_Q(T) := \frac{\sigma_{\max}(B_Q(T))}{\sigma_{\min}(B_Q(T))} \geq1, \qquad \Delta_Q(T):=\log D_Q(T).
\delta_{q,q'}Use in source · 1 passage
Search aliases: δ_q,q' delta
Use in source · block16_section_final.tex, line 617
are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
E_qE_{q'}=\delta_{q,q'}E_q, \qquad \sum_{q\mid N}E_q=1.
\Delta_SUse in source · 3 passages
Search aliases: Δ_S Delta
Use in source · Stratified_Universality_Traditional.tex, line 1498
extup(roman*)] item evaluation at the root identifies the space of global parallel sections with C_o; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies; item for Δ_S:F_o⟶⊕_e∉ SF_o, Δ_Sx=((H_e-I)x)_e∉ S, one has C_o=kerΔ_S, dim C_o=dim F_o-rankΔ_S; item the full fibers glue to one…
Exact mathematical passage
\Delta_S:F_o\longrightarrow\bigoplus_{e\notin S}F_o, \qquad \Delta_Sx=((H_e-I)x)_{e\notin S},
\delta_{\Gamma'}Use in source · 2 passages
Search aliases: δ_Γ' delta Gamma
Use in source · block19_admissibility_lifting_final.tex, line 25
[Admissibility monotonicity] If Γ⊆Γ', then δ_Γ'(X)≤δ_Γ(X). Thus an obstruction in a restricted realization class need not remain an obstruction after the class is enlarged.
Exact mathematical passage
\delta_{\Gamma'}(X)\leq\delta_\Gamma(X).
\varepsilonUse in source · 4 passages
Search aliases: ε varepsilon
Use in source · Stratified_Universality_Traditional.tex, line 3503
Fix an integer radix k≥2 and a field K. Let Σ_k={0,1,…,k-1}. Words are read from least significant digit to most significant digit. For w=d_0d_1⋯ d_ℓ-1∈Σ_k^*, define |w|=ℓ, [w]_k=∑_j=0^ℓ-1d_jk^j. The empty word ε has length zero and value zero.
Exact mathematical passage
\varepsilon
\zetaUse in source · 6 passages
Search aliases: ζ zeta
Use in source · block16_section_final.tex, line 591
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, whe…
Exact mathematical passage
K=\mathbb Q(\zeta_N)
\zeta_NUse in source · 2 passages
Search aliases: ζ_N zeta
Use in source · block16_section_final.tex, line 591
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, whe…
Exact mathematical passage
K=\mathbb Q(\zeta_N)
\zeta_N^{-j\ell}Use in source · 1 passage
Search aliases: ζ_N^-jℓ zeta ell
Use in source · block16_section_final.tex, line 592
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, whe…
Exact mathematical passage
e_j := \frac1N\sum_{\ell=0}^{N-1} \zeta_N^{-j\ell}g^\ell, \qquad j\in\mathbb Z/N\mathbb Z.
\zeta_N^aUse in source · 1 passage
Search aliases: ζ_N^a zeta
Use in source · block16_section_final.tex, line 601
ional descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c…
Exact mathematical passage
\sigma_a(\zeta_N)=\zeta_N^a
\zeta_qUse in source · 2 passages
Search aliases: ζ_q zeta
Use in source · block16_section_final.tex, line 642
gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projectors (E_q)_qmid N form the finest Q-rational coarsening obtainable by summing the split Fourier idempotents, and Q[C_N]E_q≅ Q(ζ_q).
Exact mathematical passage
\mathbb Q[C_N]E_q\cong\mathbb Q(\zeta_q).
\zeta_q^{-a\ell}Use in source · 1 passage
Search aliases: ζ_q^-aℓ zeta ell
Use in source · block16_section_final.tex, line 658
Write every j∈ O_q uniquely as j=(N/q)a with a∈( Z/q Z)^×. The coefficient of g^ℓ in E_q is then frac1N ∑_a∈( Z/q Z)^× ζ_q^-aℓ = fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rational sum of split primitive idempotents must be Galois invariant, so its support is a union of these orbits; this proves fineness. Finally, Q[C_N] ≅ Q…
Exact mathematical passage
\frac1N \sum_{a\in(\mathbb Z/q\mathbb Z)^\times} \zeta_q^{-a\ell} = \frac{c_q(\ell)}{N},
\eta_{abc}Use in source · 1 passage
Search aliases: η_abc eta
Use in source · block13_section.tex, line 101
[Normalizer-induced coherence] Assume that every normalized fiber of N_λ,a is contractible. Then the comparison isomorphism η_abc: Φ_bcΦ_ab Longrightarrow Φ_ac is unique. Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the normalized chart.
Exact mathematical passage
\eta_{abc}: \Phi_{bc}\Phi_{ab} \Longrightarrow \Phi_{ac}
\eta_{UV}Use in source · 2 passages
Search aliases: η_UV eta
Use in source · block15_section.tex, line 1169
et objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K_U)xrightarrow∼K_V compatible with the kernel inclusions and satisfying the triple-overlap cocycle. Then the global…
Exact mathematical passage
\eta_{UV}:F_{UV}(K_U)\xrightarrow{\sim}K_V
\theta^A_{UV}Use in source · 1 passage
Search aliases: θ^A_UV theta
Use in source · block15_section.tex, line 1137
there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the verified transition profiles preserve the chosen splittings, finite sums, composition, and refinement equations, with coherent splitting comparisons on …
Exact mathematical passage
\theta^A_{UV} F_{UV}(p_{A_U,i}^r) = p_{A_V,\lambda_{UV}^{A,r}(i)}^r \theta^A_{UV},
\theta^X_{UV}Use in source · 1 passage
Search aliases: θ^X_UV theta
Use in source · block15_section.tex, line 1086
[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
X=(X_U,\theta^X_{UV}), \qquad Y=(Y_U,\theta^Y_{UV}), \qquad Z=(Z_U,\theta^Z_{UV})
\theta^X_\gammaUse in source · 2 passages
Search aliases: θ^X_γ theta gamma
Use in source · block16_section_final.tex, line 44
A descent object A=(A_U,θ^A_γ) consists of objects A_U∈ C_U and isomorphisms θ^A_γ:F_γ(A_U)xrightarrow∼A_V satisfying the identity and cocycle equations. A descended morphism f:X→ Y is a compatible family (f_U)_U satisfying θ^Y_γ F_γ(f_U)=f_Vθ^X_γ.
Exact mathematical passage
\theta^Y_\gamma F_\gamma(f_U)=f_V\theta^X_\gamma.
\theta^Y_{UV}Use in source · 1 passage
Search aliases: θ^Y_UV theta
Use in source · block15_section.tex, line 1086
[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
X=(X_U,\theta^X_{UV}), \qquad Y=(Y_U,\theta^Y_{UV}), \qquad Z=(Z_U,\theta^Z_{UV})
\theta^Y_\gammaUse in source · 2 passages
Search aliases: θ^Y_γ theta gamma
Use in source · block16_section_final.tex, line 44
A descent object A=(A_U,θ^A_γ) consists of objects A_U∈ C_U and isomorphisms θ^A_γ:F_γ(A_U)xrightarrow∼A_V satisfying the identity and cocycle equations. A descended morphism f:X→ Y is a compatible family (f_U)_U satisfying θ^Y_γ F_γ(f_U)=f_Vθ^X_γ.
Exact mathematical passage
\theta^Y_\gamma F_\gamma(f_U)=f_V\theta^X_\gamma.
\theta^Z_{UV}Use in source · 1 passage
Search aliases: θ^Z_UV theta
Use in source · block15_section.tex, line 1086
[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z be compatible descended morphisms.
Exact mathematical passage
X=(X_U,\theta^X_{UV}), \qquad Y=(Y_U,\theta^Y_{UV}), \qquad Z=(Z_U,\theta^Z_{UV})
\Theta_{A,B}Use in source · 9 passages
Search aliases: Θ_A,B Theta
Use in source · Stratified_Universality_Traditional.tex, line 3011
[Fixed-denominator Cartier realization] Let F= M_p[A/B] and D=D(A,B). Then: [label=textup(roman*)] item every T_d preserves P_D; item the linear map Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.
Exact mathematical passage
\label{eq:block04-theta-map} \Theta_{A,B}:\mathcal P_D\longrightarrow\mathbb F_p[[z]], \qquad \Theta_{A,B}(P)=\frac{P}{B}F,
\Theta_{M',\mathcal O'}Use in source · 1 passage
Search aliases: Θ_M', O' Theta mathcal
Use in source · block17_semantic_quotients_final.tex, line 241
[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.
Exact mathematical passage
M\subseteq M',\qquad \mathcal O\subseteq\mathcal O' \quad\Longrightarrow\quad \Theta_{M',\mathcal O'}\subseteq\Theta_{M,\mathcal O},
\Theta_{M,\mathcal O}Use in source · 1 passage
Search aliases: Θ_M, O Theta mathcal
Use in source · block17_semantic_quotients_final.tex, line 241
[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.
Exact mathematical passage
M\subseteq M',\qquad \mathcal O\subseteq\mathcal O' \quad\Longrightarrow\quad \Theta_{M',\mathcal O'}\subseteq\Theta_{M,\mathcal O},
\Theta_{M,s}Use in source · 10 passages
Search aliases: Θ_M,s Theta
Use in source · Stratified_Universality_Traditional.tex, line 5535
For M≥1, s≥0, and r,k≥0, define Φ_M(z)=(1+z)^M-1 and Θ_M,s(r,k) =[z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. The support is finite: Θ_M,s(r,k)=0 unless k≤ r≤ Mk+s.
Exact mathematical passage
\boxed{ \Theta_{M,s}(r,k) =[z^r](1+z)^s\Phi_M(z)^k.}
\Theta_{MN,Mt+s}Use in source · 1 passage
Search aliases: Θ_MN,Mt+s Theta
Use in source · Stratified_Universality_Traditional.tex, line 5600
[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.
Exact mathematical passage
\boxed{ \sum_{k\geq0} \Theta_{M,s}(r,k)\Theta_{N,t}(k,\ell) = \Theta_{MN,Mt+s}(r,\ell).}
\Theta_{N,t}Use in source · 1 passage
Search aliases: Θ_N,t Theta
Use in source · Stratified_Universality_Traditional.tex, line 5600
[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.
Exact mathematical passage
\boxed{ \sum_{k\geq0} \Theta_{M,s}(r,k)\Theta_{N,t}(k,\ell) = \Theta_{MN,Mt+s}(r,\ell).}
\Theta_{p^a,s}Use in source · 3 passages
Search aliases: Θ_p^a,s Theta
Use in source · Stratified_Universality_Traditional.tex, line 5765
[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{ \Theta_{p^a,s}(r,k) \equiv \binom{s}{r-p^ak} \pmod p,}
\iota^r_{X,U,i}Use in source · 1 passage
Search aliases: ι^r_X,U,i iota
Use in source · block16_section_final.tex, line 244
For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.
Exact mathematical passage
\begin{aligned} &\beta^Y_{\gamma,u,a} F_\gamma\!\left( \pi^u_{Y,U,a}f_U\iota^r_{X,U,i} \right) \left(\beta^X_{\gamma,r,i}\right)^{-1}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} \theta^Y_\gamma F_\gamma(f_U)(\theta^X_\gamma)^{-1} \iota^r_{X,V,\lambda^X(i)}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} f_V \iota^r_{X,V,\lambda^X(i)}, \end{aligned}
\iota^r_{X,V,\lambda^X(i)}Use in source · 1 passage
Search aliases: ι^r_X,V,λ^X(i) iota lambda
Use in source · block16_section_final.tex, line 244
For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.
Exact mathematical passage
\begin{aligned} &\beta^Y_{\gamma,u,a} F_\gamma\!\left( \pi^u_{Y,U,a}f_U\iota^r_{X,U,i} \right) \left(\beta^X_{\gamma,r,i}\right)^{-1}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} \theta^Y_\gamma F_\gamma(f_U)(\theta^X_\gamma)^{-1} \iota^r_{X,V,\lambda^X(i)}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} f_V \iota^r_{X,V,\lambda^X(i)}, \end{aligned}
\iota^u_{Y,U,a}Use in source · 2 passages
Search aliases: ι^u_Y,U,a iota
Use in source · block16_section_final.tex, line 151
item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix record. It changes the global type of that record from a constantly labeled matrix to a compatible section of a matrix local system.
Exact mathematical passage
(h_U)_U\longmapsto \left( \sum_{a,i} \iota^u_{Y,U,a}h_{U,ai}\pi^r_{X,U,i} \right)_U.
\iota_{1,i_1}Use in source · 1 passage
Search aliases: ι_1,i_1 iota
Use in source · block15_section.tex, line 594
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T_{a;i_1,\ldots,i_d} := \pi_{Y,a}T (\iota_{1,i_1}-,\ldots,\iota_{d,i_d}-).
\iota_{d,i_d}Use in source · 1 passage
Search aliases: ι_d,i_d iota
Use in source · block15_section.tex, line 594
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T_{a;i_1,\ldots,i_d} := \pi_{Y,a}T (\iota_{1,i_1}-,\ldots,\iota_{d,i_d}-).
\iota_jUse in source · 2 passages
Search aliases: ι_j iota
Use in source · Stratified_Universality_Traditional.tex, line 428
Assume now that C is additive and idempotent complete. Choose a splitting X_ixrightarrowι_iXxrightarrowπ_iX_i, ι_iπ_i=p_i, π_iι_i=1_X_i. The packet equations imply π_iι_j=0 (i≠ j).
Exact mathematical passage
\pi_i\iota_j=0\quad(i\ne j).
\iota_{X,j}^rUse in source · 1 passage
Search aliases: ι_X,j^r iota
Use in source · block15_section.tex, line 259
For (L1), the first composite is ∑_iι_X,i^rπ_X,i^r =∑_i p_X,i^r=1_X. The (i,j)-entry of the other composite is π_X,i^rι_X,j^r. It is the identity when i=j, and it is zero otherwise because the corresponding idempotents annihilate one another.
Exact mathematical passage
\pi_{X,i}^r\iota_{X,j}^r
\iota_{Y,a}Use in source · 2 passages
Search aliases: ι_Y,a iota
Use in source · block15_section.tex, line 601
inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T(x_1,\ldots,x_d) = \sum_{a,i_1,\ldots,i_d} \iota_{Y,a} T_{a;i_1,\ldots,i_d} (\pi_{1,i_1}x_1,\ldots,\pi_{d,i_d}x_d).
\iota_{Y,a}^rUse in source · 1 passage
Search aliases: ι_Y,a^r iota
Use in source · block15_section.tex, line 540
[Fixed-resolution matrix-category equivalence] The functor M_r: L_ R⟶ Mat_r( L_ R), X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences are compatible with every certified change of resolution.
Exact mathematical passage
\sum_{a,i}\iota_{Y,a}^r h_{a i}\pi_{X,i}^r.
\iota_{Y,a}^uUse in source · 3 passages
Search aliases: ι_Y,a^u iota
Use in source · block15_section.tex, line 96
ion r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.
Exact mathematical passage
p_{Y,a}^u f p_{X,i}^r = \iota_{Y,a}^u[f]^{u,r}_{a i}\pi_{X,i}^r.
\iota_{Y,b}^vUse in source · 2 passages
Search aliases: ι_Y,b^v iota
Use in source · block15_section.tex, line 403
[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contribute. The inverse is the corresponding fine analysis after coarse synthesis. Consequently, changing a complete lane resolution loses no information. Lo…
Exact mathematical passage
\bigl(\Gamma^{(u,r)}_{(v,s)}h\bigr)_{a i} = \sum_{b,j} \bigl(\pi_{Y,a}^u\iota_{Y,b}^v\bigr) h_{bj} \bigl(\pi_{X,j}^s\iota_{X,i}^r\bigr).
\kappa_2Use in source · 1 passage
Search aliases: κ_2 kappa
Use in source · block19_admissibility_lifting_final.tex, line 237
0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.
Exact mathematical passage
\kappa_2(Q_s)=s^{-2}\longrightarrow\infty.
\kappa_3Use in source · 1 passage
Search aliases: κ_3 kappa
Use in source · Stratified_Universality_Traditional.tex, line 3372
A genuine denominator example. Over F_3, take R(z)=frac11+z. Then F(z)^2=1+z. The reduced denominator remains fixed under every Cartier section. The ambient compiler has dimension two, its reachable numerator set is {0,1,2,1+z}, and its reachable span has dimension two. Hence r_3(F)=2, κ_3(F)=4. The nonzero collapse obstruction again proves that the automatic coefficient sequence is not eventually periodic.
Exact mathematical passage
r_3(F)=2, \qquad \kappa_3(F)=4.
\kappa_pUse in source · 5 passages
Search aliases: κ_p kappa
Use in source · Stratified_Universality_Traditional.tex, line 3161
Let κ_p(F)=| K_p(F)|.
Exact mathematical passage
\label{eq:block04-kernel-size} \kappa_p(F)=|\mathcal K_p(F)|.
\kappa_RUse in source · 3 passages
Search aliases: κ_R kappa
Use in source · block11_section.tex, line 43
Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.
Exact mathematical passage
\kappa_R=\varphi'(R) =\sum_{m=2}^{D}m b_mR^{m-1}.
\lambda'Use in source · 2 passages
Search aliases: λ' lambda
Use in source · Stratified_Universality_Traditional.tex, line 2092
An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.
Exact mathematical passage
S:(C,v,\lambda)\longrightarrow(C',v',\lambda')
\lambda^nUse in source · 1 passage
Search aliases: λ^n lambda
Use in source · Stratified_Universality_Traditional.tex, line 2161
Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [source reference].
Exact mathematical passage
a_n=\sum_{\lambda\in\Lambda}P_\lambda(n)\lambda^n.
\lambda^XUse in source · 1 passage
Search aliases: λ^X lambda
Use in source · block16_section_final.tex, line 244
For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.
Exact mathematical passage
\begin{aligned} &\beta^Y_{\gamma,u,a} F_\gamma\!\left( \pi^u_{Y,U,a}f_U\iota^r_{X,U,i} \right) \left(\beta^X_{\gamma,r,i}\right)^{-1}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} \theta^Y_\gamma F_\gamma(f_U)(\theta^X_\gamma)^{-1} \iota^r_{X,V,\lambda^X(i)}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} f_V \iota^r_{X,V,\lambda^X(i)}, \end{aligned}
\lambda^YUse in source · 1 passage
Search aliases: λ^Y lambda
Use in source · block16_section_final.tex, line 244
For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.
Exact mathematical passage
\begin{aligned} &\beta^Y_{\gamma,u,a} F_\gamma\!\left( \pi^u_{Y,U,a}f_U\iota^r_{X,U,i} \right) \left(\beta^X_{\gamma,r,i}\right)^{-1}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} \theta^Y_\gamma F_\gamma(f_U)(\theta^X_\gamma)^{-1} \iota^r_{X,V,\lambda^X(i)}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} f_V \iota^r_{X,V,\lambda^X(i)}, \end{aligned}
\Lambda_{\mathbb K}Use in source · 3 passages
Search aliases: Λ_ K Lambda mathbb
Use in source · Stratified_Universality_Traditional.tex, line 371
[Typed realization and module equivalence, imported] Let Λ_ K:=𝕜 Q/I. In 𝕜-modules, the category of typed realizations of the linear grammar (Q,I) is equivalent to the category of unital left Λ_ K-modules. Under the equivalence, the vertex idempotent e_X acts as the projector onto the component of sort X, and each arrow acts in its declared source-target block.
Exact mathematical passage
\Lambda_{\mathbb K}:=\kk Q/I.
\Lambda_0Use in source · 2 passages
Search aliases: Λ_0 Lambda
Use in source · Stratified_Universality_Traditional.tex, line 2335
el is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.
Exact mathematical passage
(\Lambda_0b)_0=b_0
\Lambda_{13}Use in source · 1 passage
Search aliases: Λ_13 Lambda
Use in source · block13_section.tex, line 450
Let Λ_13 = {DRLM,Loc,Fuss,EM,Gal} be the component-tag set. Define J_13(X) = (λ(X), J_λ(X)(X)).
Exact mathematical passage
\Lambda_{13} = \{\mathrm{DRLM},\mathrm{Loc},\mathrm{Fuss},\mathrm{EM},\mathrm{Gal}\}
\Lambda_{d_0}Use in source · 5 passages
Search aliases: Λ_d_0 Lambda
Use in source · Stratified_Universality_Traditional.tex, line 2326
For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.
Exact mathematical passage
\label{eq:p-kernel} \mathcal K_p(a) = \left\{ \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a: \ell\geq 0,\ 0\leq d_i<p \right\}.
\Lambda_{d_1}Use in source · 2 passages
Search aliases: Λ_d_1 Lambda
Use in source · Stratified_Universality_Traditional.tex, line 3511
For a sequence a=(a_n)_n≥0∈ K^ N, define the digit-section operators (Λ_d a)(n)=a(kn+d), 0≤ d<k. For a word w=d_0⋯ d_ℓ-1, put Λ_w = Λ_d_ℓ-1⋯Λ_d_1Λ_d_0. Then (Λ_w a)(n) = a(k^|w|n+[w]_k).
Exact mathematical passage
\Lambda_w = \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_1}\Lambda_{d_0}.
\Lambda_{d_{a-1}}Use in source · 1 passage
Search aliases: Λ_d_a-1 Lambda
Use in source · block15_section.tex, line 835
eld and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
\Lambda_{d_{a-1}}\cdots\Lambda_{d_1}\Lambda_{d_0}F.
\Lambda_{d_{\ell-1}}Use in source · 4 passages
Search aliases: Λ_d_ℓ-1 Lambda ell
Use in source · Stratified_Universality_Traditional.tex, line 2326
For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.
Exact mathematical passage
\label{eq:p-kernel} \mathcal K_p(a) = \left\{ \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a: \ell\geq 0,\ 0\leq d_i<p \right\}.
\lambda_eUse in source · 1 passage
Search aliases: λ_e lambda
Use in source · block19_admissibility_lifting_final.tex, line 147
Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible family directly. A group preserving a positive-definite metric is conjugate into a compact orthogonal or unitary group. Conversely, averaging an inner p…
Exact mathematical passage
T_e^*Q_vT_e=|\det H_e|^{2/n}Q_u, \qquad \lambda_e=|\det H_e|^{1/n}>0.
\lambda_e^2Use in source · 1 passage
Search aliases: λ_e^2 lambda
Use in source · block19_admissibility_lifting_final.tex, line 126
raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge scales satisfying T_e^*Q_t(e)T_e=λ_e^2Q_s(e) exist if and only if the determinant-normalized holonomy representation preserves a positive-definite metr…
Exact mathematical passage
T_e^*Q_{t(e)}T_e=\lambda_e^2Q_{s(e)}
\Lambda_{k-1}Use in source · 1 passage
Search aliases: Λ_k-1 Lambda
Use in source · Stratified_Universality_Traditional.tex, line 3572
[Section-module realization theorem] For a sequence a∈ K^ N, the following are equivalent: item a is k-regular over K; item a admits a finite-dimensional digit-linear realization. When these conditions hold, the tuple C_k(a) = ( V_k(a), Λ_0,…,Λ_k-1, a, ϵ_0 ), ϵ_0(f)=f(0), is a canonical realization of dimension r_k(a).
Exact mathematical passage
\mathsf C_k(a) = \bigl( \mathcal V_k(a), \Lambda_0,\ldots,\Lambda_{k-1}, a, \epsilon_0 \bigr), \qquad \epsilon_0(f)=f(0), \label{eq:block05-canonical-section-module}
\lambda_{UT}^{A,r}Use in source · 1 passage
Search aliases: λ_UT^A,r lambda
Use in source · block15_section.tex, line 1111
Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global re…
Exact mathematical passage
\lambda_{UU}^{A,r}=1, \qquad \lambda_{UT}^{A,r} =\lambda_{VT}^{A,r}\lambda_{UV}^{A,r},
\lambda_{UU}^{A,r}Use in source · 1 passage
Search aliases: λ_UU^A,r lambda
Use in source · block15_section.tex, line 1111
Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global re…
Exact mathematical passage
\lambda_{UU}^{A,r}=1, \qquad \lambda_{UT}^{A,r} =\lambda_{VT}^{A,r}\lambda_{UV}^{A,r},
\lambda_{UV}^{A,r}Use in source · 6 passages
Search aliases: λ_UV^A,r lambda
Use in source · block15_section.tex, line 1106
Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global re…
Exact mathematical passage
\lambda_{UV}^{A,r}: I_{A_U,r}\xrightarrow{\sim}I_{A_V,r}
\lambda_{UV}^{A,s}Use in source · 1 passage
Search aliases: λ_UV^A,s lambda
Use in source · block15_section.tex, line 1118
carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r …
Exact mathematical passage
\rho_{sr}^{A_V}\lambda_{UV}^{A,s} = \lambda_{UV}^{A,r}\rho_{sr}^{A_U}.
\lambda_{VT}^{A,r}Use in source · 1 passage
Search aliases: λ_VT^A,r lambda
Use in source · block15_section.tex, line 1111
Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global re…
Exact mathematical passage
\lambda_{UU}^{A,r}=1, \qquad \lambda_{UT}^{A,r} =\lambda_{VT}^{A,r}\lambda_{UV}^{A,r},
\mu^2Use in source · 1 passage
Search aliases: μ^2 mu
Use in source · block07_named_local_table.tex, line 13
Squarefree indicator μ^2 · β_1=1, β_2=-1 · 1,1,0,0,0,0,… · 1,-1,1,-1,1,-1,…
Exact mathematical passage
\mu^2
\mu_1Use in source · 1 passage
Search aliases: μ_1 mu
Use in source · Stratified_Universality_Traditional.tex, line 947
[One operator and primary packets] Take one sort, one endomorphism T, and one relation μ(T)=0. The linear grammar algebra is 𝕜[x]/(μ). When μ=μ_1⋯μ_r is a product of pairwise comaximal factors, the Chinese remainder idempotents define packet objects for the primary summands. The envelope splits those idempotents formally. Over a realization where the same factorization remains valid, the packet objects become the corresponding invariant primary sectors.
Exact mathematical passage
\mu=\mu_1\cdots\mu_r
\mu_dUse in source · 2 passages
Search aliases: μ_d mu
Use in source · block11_section.tex, line 423
[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arity signature determines both the number of active packet addresses and the size of the complete dominant-singularity orbit.
Exact mathematical passage
\mu_d
\mu_{d_{\mathcal A}}Use in source · 2 passages
Search aliases: μ_d_ A mu mathcal
Use in source · block11_section.tex, line 418
[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arity signature determines both the number of active packet addresses and the size of the complete dominant-singularity orbit.
Exact mathematical passage
\boxed{ \rho\,\mu_{d_{\mathcal A}},} \tag{11.12}
\mu_HUse in source · 1 passage
Search aliases: μ_H mu
Use in source · block12_section.tex, line 292
Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.
Exact mathematical passage
\boxed{ P_H=\int_HU_h\,d\mu_H(h).} \tag{12.14}
\mu_{h_N}Use in source · 1 passage
Search aliases: μ_h_N mu
Use in source · block11_section.tex, line 424
[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arity signature determines both the number of active packet addresses and the size of the complete dominant-singularity orbit.
Exact mathematical passage
\boxed{ \rho^N\,\mu_{h_N}, \qquad h_N=\frac{d_{\mathcal A}}{\gcd(N,d_{\mathcal A})}.} \tag{11.13}
\mu_{p^r}Use in source · 1 passage
Search aliases: μ_p^r mu
Use in source · Stratified_Universality_Traditional.tex, line 4356
[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.
Exact mathematical passage
\mu_{p^r}(a) = \#\{i:\nu_i<r\}.
\mu_{q-1}Use in source · 1 passage
Search aliases: μ_q-1 mu
Use in source · block15_section.tex, line 970
power map used for packet synthesis has fracq-1gcd(N,q-1) distinct images. If a fixed embedding K↪ C sends α to a positive real number and the marked branch is interpreted in the smooth implicit analytic schema, its critical points and values satisfy qατ^q-1=1, ρ=τ-ατ^q=fracq-1qτ. The resulting orbit ρμ_q-1 is precisely the dominant singularity orbit, and packetization leaves (q-1)/g distinct images.
Exact mathematical passage
\rho\mu_{q-1}
\mu_rUse in source · 1 passage
Search aliases: μ_r mu
Use in source · Stratified_Universality_Traditional.tex, line 947
[One operator and primary packets] Take one sort, one endomorphism T, and one relation μ(T)=0. The linear grammar algebra is 𝕜[x]/(μ). When μ=μ_1⋯μ_r is a product of pairwise comaximal factors, the Chinese remainder idempotents define packet objects for the primary summands. The envelope splits those idempotents formally. Over a realization where the same factorization remains valid, the packet objects become the corresponding invariant primary sectors.
Exact mathematical passage
\mu=\mu_1\cdots\mu_r
\mu_{r,\gamma}Use in source · 7 passages
Search aliases: μ_r,γ mu gamma
Use in source · block11_section.tex, line 191
Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.
Exact mathematical passage
\mu=(\mu_{r,\gamma})_{r\ge1,\,\gamma\in\Gamma}
\mu_U^{A,r}Use in source · 6 passages
Search aliases: μ_U^A,r mu
Use in source · block15_section.tex, line 1126
UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_…
Exact mathematical passage
\mu_U^{A,r}:I_{A_U,r}\xrightarrow{\sim}I_{A,r}
\mu_U^{A,s}Use in source · 1 passage
Search aliases: μ_U^A,s mu
Use in source · block15_section.tex, line 1130
iple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the verified transition …
Exact mathematical passage
\mu_V^{A,r}\lambda_{UV}^{A,r}=\mu_U^{A,r}, \qquad \mu_U^{A,r}\rho_{sr}^{A_U} =\rho_{sr}^{A}\mu_U^{A,s}.
\mu_V^{A,r}Use in source · 1 passage
Search aliases: μ_V^A,r mu
Use in source · block15_section.tex, line 1130
iple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the verified transition …
Exact mathematical passage
\mu_V^{A,r}\lambda_{UV}^{A,r}=\mu_U^{A,r}, \qquad \mu_U^{A,r}\rho_{sr}^{A_U} =\rho_{sr}^{A}\mu_U^{A,s}.
\nu_aUse in source · 2 passages
Search aliases: ν_a nu
Use in source · block09_section.tex, line 63
Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
Exact mathematical passage
\nu_a
\nu_iUse in source · 13 passages
Search aliases: ν_i nu
Use in source · Stratified_Universality_Traditional.tex, line 4334
Fix a prime p and write ν_i=ν_p,i(a)=v_p(d_i(a)).
Exact mathematical passage
\nu_i=\nu_{p,i}(a)=v_p(d_i(a)).
\nu_pUse in source · 1 passage
Search aliases: ν_p nu
Use in source · Stratified_Universality_Traditional.tex, line 2698
For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)| ≤ T(a)(ν_p(T(a))+ord_M(p)), R_a(z) · =fracF_a(z)F_a(z^p), T(a)=p^ν_p(T(a))M, gcd(M,p)=1.
Exact mathematical passage
\label{eq:periodic-invariant-identities} \begin{aligned} \deg m_a &=\deg Q_a =\rank H(a) =\dim \State_{\min}(a),\\ T(a) &=\min\{N:m_a\mid x^N-1\} =\min\{N:Q_a\mid 1-z^N\},\\ |\mathcal A_p(a)| &=|\mathcal K_p(a)| \leq T(a)\bigl(\nu_p(T(a))+\operatorname{ord}_{M}(p)\bigr),\\ R_a(z) &=\frac{F_a(z)}{F_a(z^p)}, \qquad T(a)=p^{\nu_p(T(a))}M, \quad \gcd(M,p)=1. \end{aligned}
\nu_{p,i}Use in source · 1 passage
Search aliases: ν_p,i nu
Use in source · Stratified_Universality_Traditional.tex, line 4334
Fix a prime p and write ν_i=ν_p,i(a)=v_p(d_i(a)).
Exact mathematical passage
\nu_i=\nu_{p,i}(a)=v_p(d_i(a)).
\xiUse in source · 1 passage
Search aliases: ξ xi
Use in source · block09_section.tex, line 760
ther with its finite initial record; item the monic minimal differential operator from [source reference], together with the distinguished power-series solution selected by the marked coefficient record; item the hypergeometric parameter packet (A_N,j(λ),B_N,j;α^N) and leading coefficient; item the primitive minimal-operator singular and indicial record mathscr E(G) = ( Σ_rm op(G), (I_ξ,L_G)_ξ∈Σ_rm op(G) ), together with the orbit-rank and defect data (r_rm orb,r_D,A_N,C_N,j). In the nonintegral…
Exact mathematical passage
\mathscr E(G) = \left( \Sigma_{\rm op}(G), \bigl(I_{\xi,L_G}\bigr)_{\xi\in\Sigma_{\rm op}(G)} \right),
\Pi^2Use in source · 1 passage
Search aliases: Π^2 Pi
Use in source · Stratified_Universality_Traditional.tex, line 1983
Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.
Exact mathematical passage
LA=I, \qquad \Pi=AL, \qquad \Pi^2=\Pi,
\pi^r_{X,U,i}Use in source · 2 passages
Search aliases: π^r_X,U,i pi
Use in source · block16_section_final.tex, line 151
item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix record. It changes the global type of that record from a constantly labeled matrix to a compatible section of a matrix local system.
Exact mathematical passage
(h_U)_U\longmapsto \left( \sum_{a,i} \iota^u_{Y,U,a}h_{U,ai}\pi^r_{X,U,i} \right)_U.
\pi^u_{Y,U,a}Use in source · 1 passage
Search aliases: π^u_Y,U,a pi
Use in source · block16_section_final.tex, line 244
For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.
Exact mathematical passage
\begin{aligned} &\beta^Y_{\gamma,u,a} F_\gamma\!\left( \pi^u_{Y,U,a}f_U\iota^r_{X,U,i} \right) \left(\beta^X_{\gamma,r,i}\right)^{-1}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} \theta^Y_\gamma F_\gamma(f_U)(\theta^X_\gamma)^{-1} \iota^r_{X,V,\lambda^X(i)}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} f_V \iota^r_{X,V,\lambda^X(i)}, \end{aligned}
\pi^u_{Y,V,\lambda^Y(a)}Use in source · 1 passage
Search aliases: π^u_Y,V,λ^Y(a) pi lambda
Use in source · block16_section_final.tex, line 244
For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.
Exact mathematical passage
\begin{aligned} &\beta^Y_{\gamma,u,a} F_\gamma\!\left( \pi^u_{Y,U,a}f_U\iota^r_{X,U,i} \right) \left(\beta^X_{\gamma,r,i}\right)^{-1}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} \theta^Y_\gamma F_\gamma(f_U)(\theta^X_\gamma)^{-1} \iota^r_{X,V,\lambda^X(i)}\\ &\quad= \pi^u_{Y,V,\lambda^Y(a)} f_V \iota^r_{X,V,\lambda^X(i)}, \end{aligned}
\pi_1Use in source · 1 passage
Search aliases: π_1 pi
Use in source · Stratified_Universality_Traditional.tex, line 1781
Finite transport holonomy · connected graph, invertible edge transports, chosen structure group · fundamental cycle holonomies or the conjugacy class of π_1(Γ)→ G · one global frame on the full fibers; on a fixed subspace, path-independent transport · path dependence; possibly a smaller nonzero fixed core
Exact mathematical passage
\pi_1(\Gamma)\to G
\pi_{1,i_1}Use in source · 1 passage
Search aliases: π_1,i_1 pi
Use in source · block15_section.tex, line 601
inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T(x_1,\ldots,x_d) = \sum_{a,i_1,\ldots,i_d} \iota_{Y,a} T_{a;i_1,\ldots,i_d} (\pi_{1,i_1}x_1,\ldots,\pi_{d,i_d}x_d).
\pi_1^{\acute{e}t}Use in source · 1 passage
Search aliases: π_1^acuteet pi acute
Use in source · block12_section.tex, line 483
For a connected scheme X with geometric point bar x, the fiber functor on finite acuteetale covers identifies that category with finite continuous π_1^acuteet(X,bar x)-sets. Every finite fiber S produces a permutation resolution on ℓ^2(S): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms give equivariant packet transfers. The profinite fundamental group is the automorphism group of this fiber functor.
Exact mathematical passage
\pi_1^{\acute{e}t}(X,\bar x)
\pi_{d,i_d}Use in source · 1 passage
Search aliases: π_d,i_d pi
Use in source · block15_section.tex, line 601
inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T(x_1,\ldots,x_d) = \sum_{a,i_1,\ldots,i_d} \iota_{Y,a} T_{a;i_1,\ldots,i_d} (\pi_{1,i_1}x_1,\ldots,\pi_{d,i_d}x_d).
\pi_jUse in source · 2 passages
Search aliases: π_j pi
Use in source · block10_section.tex, line 279
Let Ω be a normal splitting field of the generic packet scheme over K(x), and let Γ=Gal(Ω/K(x)). After splitting E_N, the geometric packet branches form a set B_N of cardinality d^N. Let π_j: B_N→Ω be the jth packet coordinate, and set C_N,j=π_j( B_N). For the selected semantic packet U_j, separability gives [K(x)(U_j):K(x)] = |Γ· U_j|.
Exact mathematical passage
\pi_j:\mathcal B_N\to\Omega
\Pi_{\rm common}Use in source · 1 passage
Search aliases: Π_rm common Pi rm
Use in source · Stratified_Universality_Traditional.tex, line 770
[Common profile and composition] Let F: C→ D and G: D→ E. Every constructor preserved by both functors, with compatible chosen comparison data, is preserved by GF. For a finite diagram of realizations and bridges, the intersection Π_rm common:=⋂_FΠ_F is therefore a valid common profile for every proof transported through the whole diagram.
Exact mathematical passage
\Pi_{\rm common}:=\bigcap_F\Pi_F
\pi_{X,j}^sUse in source · 1 passage
Search aliases: π_X,j^s pi
Use in source · block15_section.tex, line 403
[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contribute. The inverse is the corresponding fine analysis after coarse synthesis. Consequently, changing a complete lane resolution loses no information. Lo…
Exact mathematical passage
\bigl(\Gamma^{(u,r)}_{(v,s)}h\bigr)_{a i} = \sum_{b,j} \bigl(\pi_{Y,a}^u\iota_{Y,b}^v\bigr) h_{bj} \bigl(\pi_{X,j}^s\iota_{X,i}^r\bigr).
\pi_{Y,a}Use in source · 2 passages
Search aliases: π_Y,a pi
Use in source · block15_section.tex, line 594
[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T_{a;i_1,\ldots,i_d} := \pi_{Y,a}T (\iota_{1,i_1}-,\ldots,\iota_{d,i_d}-).
\pi_{Y,a}^uUse in source · 5 passages
Search aliases: π_Y,a^u pi
Use in source · block15_section.tex, line 89
At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.
Exact mathematical passage
[f]^{u,r}_{a i} := \pi_{Y,a}^u f\iota_{X,i}^r: X_i^r\longrightarrow Y_a^u.
\rho^A_{sr}Use in source · 2 passages
Search aliases: ρ^A_sr rho
Use in source · block16_section_final.tex, line 408
item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.
Exact mathematical passage
P^r_{A,\mathcal O} = \sum_{\substack{\mathcal Q\text{ a fine orbit}\\ \mathcal Q\subseteq(\rho^A_{sr})^{-1}(\mathcal O)}} P^s_{A,\mathcal Q}.
\rho^NUse in source · 1 passage
Search aliases: ρ^N rho
Use in source · block11_section.tex, line 424
[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arity signature determines both the number of active packet addresses and the size of the complete dominant-singularity orbit.
Exact mathematical passage
\boxed{ \rho^N\,\mu_{h_N}, \qquad h_N=\frac{d_{\mathcal A}}{\gcd(N,d_{\mathcal A})}.} \tag{11.13}
\rho^{\underline j}Use in source · 1 passage
Search aliases: ρ^underline j rho underline
Use in source · block09_section.tex, line 401
Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
\boxed{ I_{\alpha,L}(\rho) = \sum_{v_j-j=\mu} q_{j,v_j}\rho^{\underline j}.}
\rho^Y_{vu}Use in source · 1 passage
Search aliases: ρ^Y_vu rho
Use in source · block15_section.tex, line 152
item Projector-sum refinement. If ssucceq r on X and vsucceq u on Y, then p_Y,a^u f p_X,i^r = ∑_ρ^Y_vu(b)=a ∑_ρ^X_sr(j)=i p_Y,b^v f p_X,j^s. Coarsening a fine composite gives the same ambient morphism as composing at the coarse level.
Exact mathematical passage
p_{Y,a}^u f p_{X,i}^r = \sum_{\rho^Y_{vu}(b)=a} \sum_{\rho^X_{sr}(j)=i} p_{Y,b}^v f p_{X,j}^s.
\rho_kUse in source · 2 passages
Search aliases: ρ_k rho
Use in source · Stratified_Universality_Traditional.tex, line 4033
Throughout this section, k≥2 is fixed and a: N⟶ Z is an integer-valued sequence whose rational section space V_ Q(a) = span_ Q K_k(a) ⊆ Q^ N has finite dimension ρ_k(a)=dim_ QV_ Q(a). Thus a is k-regular after extension of scalars to Q.
Exact mathematical passage
\rho_k(a)=\dim_{\mathbb Q}V_{\mathbb Q}(a).
\rho_{sr}^{A}Use in source · 2 passages
Search aliases: ρ_sr^A rho
Use in source · block15_section.tex, line 1125
I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U…
Exact mathematical passage
\rho_{sr}^{A}:I_{A,s}\twoheadrightarrow I_{A,r}
\rho_{sr}^{A_U}Use in source · 2 passages
Search aliases: ρ_sr^A_U rho
Use in source · block15_section.tex, line 1118
carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r …
Exact mathematical passage
\rho_{sr}^{A_V}\lambda_{UV}^{A,s} = \lambda_{UV}^{A,r}\rho_{sr}^{A_U}.
\rho_{sr}^{A_V}Use in source · 1 passage
Search aliases: ρ_sr^A_V rho
Use in source · block15_section.tex, line 1118
carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r …
Exact mathematical passage
\rho_{sr}^{A_V}\lambda_{UV}^{A,s} = \lambda_{UV}^{A,r}\rho_{sr}^{A_U}.
\rho_{tr}Use in source · 1 passage
Search aliases: ρ_tr rho
Use in source · Stratified_Universality_Traditional.tex, line 603
[Aggregation and transitivity] A finite aggregation of pairwise annihilating packets is again a packet system. If tsucceq ssucceq r, then refinement is transitive with address map ρ_tr=ρ_sr∘ρ_ts.
Exact mathematical passage
\rho_{tr}=\rho_{sr}\circ\rho_{ts}.
\rho_{ts}Use in source · 1 passage
Search aliases: ρ_ts rho
Use in source · Stratified_Universality_Traditional.tex, line 603
[Aggregation and transitivity] A finite aggregation of pairwise annihilating packets is again a packet system. If tsucceq ssucceq r, then refinement is transitive with address map ρ_tr=ρ_sr∘ρ_ts.
Exact mathematical passage
\rho_{tr}=\rho_{sr}\circ\rho_{ts}.
\rho_\sigmaUse in source · 1 passage
Search aliases: ρ_σ rho sigma
Use in source · block12_section.tex, line 336
Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.
Exact mathematical passage
\rho_\sigma:V\to V, \qquad \rho_\sigma(\lambda v)=\sigma(\lambda)\rho_\sigma(v),
\Sigma^{[N]}Use in source · 2 passages
Search aliases: Σ^[N] Sigma
Use in source · block09_section.tex, line 417
[Singularity orbit under multisection] Assume F is an analytic germ at zero with analytic continuation away from a finite set Σ. Then every finite nonzero analytic singularity of M_N,jF lies in Σ^[N] = {α^N:α∈Σ}. Distinct singularities in one fiber of α↦α^N are aliased by the multisection map. A character projection may remove some aliased local contributions.
Exact mathematical passage
\boxed{ \Sigma^{[N]} = \{\alpha^N:\alpha\in\Sigma\}.}
\sigma_1Use in source · 1 passage
Search aliases: σ_1 sigma
Use in source · Stratified_Universality_Traditional.tex, line 4945
[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plane-partition series. More generally, the power family converts the exponent rule n^r into the divisor-sum ghost σ_r+1.
Exact mathematical passage
\begin{array}{c|c|c} b_n & A_b(z) & c_n\\ \hline 1 & \displaystyle\prod_{n\geq1}(1-z^n)^{-1} & \sigma_1(n)\\[4pt] n & \displaystyle\prod_{n\geq1}(1-z^n)^{-n} & \sigma_2(n)\\[4pt] n^r & \displaystyle\prod_{n\geq1}(1-z^n)^{-n^r} & \sigma_{r+1}(n). \end{array}
\sigma_2Use in source · 1 passage
Search aliases: σ_2 sigma
Use in source · Stratified_Universality_Traditional.tex, line 4945
[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plane-partition series. More generally, the power family converts the exponent rule n^r into the divisor-sum ghost σ_r+1.
Exact mathematical passage
\begin{array}{c|c|c} b_n & A_b(z) & c_n\\ \hline 1 & \displaystyle\prod_{n\geq1}(1-z^n)^{-1} & \sigma_1(n)\\[4pt] n & \displaystyle\prod_{n\geq1}(1-z^n)^{-n} & \sigma_2(n)\\[4pt] n^r & \displaystyle\prod_{n\geq1}(1-z^n)^{-n^r} & \sigma_{r+1}(n). \end{array}
\sigma_aUse in source · 5 passages
Search aliases: σ_a sigma
Use in source · block16_section_final.tex, line 600
[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, whe…
Exact mathematical passage
\sigma_a\in\Gal(K/\mathbb Q)
\Sigma_kUse in source · 1 passage
Search aliases: Σ_k Sigma
Use in source · Stratified_Universality_Traditional.tex, line 3490
Fix an integer radix k≥2 and a field K. Let Σ_k={0,1,…,k-1}. Words are read from least significant digit to most significant digit. For w=d_0d_1⋯ d_ℓ-1∈Σ_k^*, define |w|=ℓ, [w]_k=∑_j=0^ℓ-1d_jk^j. The empty word ε has length zero and value zero.
Exact mathematical passage
\Sigma_k=\{0,1,\ldots,k-1\}.
\sigma_{\max}Use in source · 1 passage
Search aliases: σ_max sigma max
Use in source · block19_admissibility_lifting_final.tex, line 79
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
D_Q(T) := \frac{\sigma_{\max}(B_Q(T))}{\sigma_{\min}(B_Q(T))} \geq1, \qquad \Delta_Q(T):=\log D_Q(T).
\sigma_{\min}Use in source · 1 passage
Search aliases: σ_min sigma min
Use in source · block19_admissibility_lifting_final.tex, line 79
Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).
Exact mathematical passage
D_Q(T) := \frac{\sigma_{\max}(B_Q(T))}{\sigma_{\min}(B_Q(T))} \geq1, \qquad \Delta_Q(T):=\log D_Q(T).
\sigma_{r+1}Use in source · 4 passages
Search aliases: σ_r+1 sigma
Use in source · Stratified_Universality_Traditional.tex, line 4945
[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plane-partition series. More generally, the power family converts the exponent rule n^r into the divisor-sum ghost σ_r+1.
Exact mathematical passage
\begin{array}{c|c|c} b_n & A_b(z) & c_n\\ \hline 1 & \displaystyle\prod_{n\geq1}(1-z^n)^{-1} & \sigma_1(n)\\[4pt] n & \displaystyle\prod_{n\geq1}(1-z^n)^{-n} & \sigma_2(n)\\[4pt] n^r & \displaystyle\prod_{n\geq1}(1-z^n)^{-n^r} & \sigma_{r+1}(n). \end{array}
\Sigma_{\rm op}Use in source · 2 passages
Search aliases: Σ_rm op Sigma rm
Use in source · block09_section.tex, line 387
Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of this operator coordinate until an explicit desingularization has been performed.
Exact mathematical passage
\Sigma_{\rm op}(F) = \{q_r(z)=0\}\cup\{\infty\}.
\sigma_{vu}Use in source · 2 passages
Search aliases: σ_vu sigma
Use in source · Stratified_Universality_Traditional.tex, line 614
[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.
Exact mathematical passage
q^u_a=\sum_{\sigma_{vu}(b)=a}q^v_b
\tau^qUse in source · 2 passages
Search aliases: τ^q tau
Use in source · block11_section.tex, line 542
Formula textup(11.15) follows from Lagrange inversion applied to z=u(1-u^q-1). The support and active-address statements are [source reference] with d_ A=q-1. Expanding u^q in complete packets and collecting the exponent carry r_1+⋯+r_q=j+Nc gives textup(11.16). The critical equations qτ^q-1=1, ρ=τ-τ^q give the printed values, and [source reference] gives the orbit count.
Exact mathematical passage
q\tau^{q-1}=1, \qquad \rho=\tau-\tau^q
\tau^{q-1}Use in source · 2 passages
Search aliases: τ^q-1 tau
Use in source · block11_section.tex, line 542
Formula textup(11.15) follows from Lagrange inversion applied to z=u(1-u^q-1). The support and active-address statements are [source reference] with d_ A=q-1. Expanding u^q in complete packets and collecting the exponent carry r_1+⋯+r_q=j+Nc gives textup(11.16). The critical equations qτ^q-1=1, ρ=τ-τ^q give the printed values, and [source reference] gives the orbit count.
Exact mathematical passage
q\tau^{q-1}=1, \qquad \rho=\tau-\tau^q
\tau_3Use in source · 1 passage
Search aliases: τ_3 tau
Use in source · block07_named_local_table.tex, line 14
Three-fold divisor function τ_3 · β_1=3 · 1,3,6,10,15,21,… · 3,3,3,3,3,3,…
Exact mathematical passage
\tau_3
\tau_eUse in source · 18 passages
Search aliases: τ_e tau
Use in source · block15_section.tex, line 216
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
\tau_e:B_e\to C_{\kappa(e)}
\tau_{e'}Use in source · 3 passages
Search aliases: τ_e' tau
Use in source · block15_section.tex, line 233
Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of Φ_κ,τ, so the fold is split lossless.
Exact mathematical passage
Q_ej_{\kappa(e')}\tau_{e'}(b_{e'}) = \delta_{e,e'}j_{\kappa(e')}\tau_{e'}(b_{e'}).
\tau_UUse in source · 3 passages
Search aliases: τ_U tau
Use in source · block16_section_final.tex, line 316
Evaluation at U_0 induces an isomorphism Γ (mathscr G,mathscr M^u,r_Y,X) xrightarrow∼ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0), where the right-hand side is the subgroup fixed by every loop-transport map. Consequently, Hom_Desc(mathscr G)(X,Y) ≅ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0). After choosing one arrow τ_U:U_0→ U for each chart, the inverse sends a fixed base matrix h_0 to the section h_U=(τ_U)_*h_0. The result is independent of the chosen arrows because two choices differ by loop tra…
Exact mathematical passage
\tau_U:U_0\to U
\tau_VUse in source · 1 passage
Search aliases: τ_V tau
Use in source · block16_section_final.tex, line 325
A compatible section evaluates to a loop-fixed element at U_0. Conversely, for an arrow γ:U→ V, the two arrows γτ_U and τ_V differ by a loop at U_0; loop invariance therefore gives γ_*h_U=h_V. Evaluation and this reconstruction are mutually inverse. Compose with the analysis isomorphism in (H2).
Exact mathematical passage
\tau_V
\varphiUse in source · 15 passages
Search aliases: φ varphi
Use in source · Stratified_Universality_Traditional.tex, line 1552
[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V
Exact mathematical passage
\varphi:V\to W
\Phi'Use in source · 1 passage
Search aliases: Φ' Phi
Use in source · block11_section.tex, line 406
Assume now that the coefficients w_m are positive real numbers. The classical smooth implicit-function schema for simply generated trees applies when there is a positive solution τ of Φ'(τ)=1 and ρ=τ-Φ(τ)>0. Under its standard nondegeneracy hypotheses, the reduced aperiodic series has a square-root singularity at its positive radius [source reference]. The arity period restores the complete source orbit.
Exact mathematical passage
\Phi'(\tau)=1
\varphi'Use in source · 1 passage
Search aliases: φ' varphi
Use in source · block11_section.tex, line 43
Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.
Exact mathematical passage
\kappa_R=\varphi'(R) =\sum_{m=2}^{D}m b_mR^{m-1}.
\Phi_{ab}Use in source · 4 passages
Search aliases: Φ_ab Phi
Use in source · block13_section.tex, line 94
has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.
Exact mathematical passage
\Phi_{ab}=Q_{\lambda,b}N_{\lambda,a}.
\Phi_{ac}Use in source · 2 passages
Search aliases: Φ_ac Phi
Use in source · block13_section.tex, line 101
[Normalizer-induced coherence] Assume that every normalized fiber of N_λ,a is contractible. Then the comparison isomorphism η_abc: Φ_bcΦ_ab Longrightarrow Φ_ac is unique. Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the normalized chart.
Exact mathematical passage
\eta_{abc}: \Phi_{bc}\Phi_{ab} \Longrightarrow \Phi_{ac}
\Phi_{bc}Use in source · 2 passages
Search aliases: Φ_bc Phi
Use in source · block13_section.tex, line 101
[Normalizer-induced coherence] Assume that every normalized fiber of N_λ,a is contractible. Then the comparison isomorphism η_abc: Φ_bcΦ_ab Longrightarrow Φ_ac is unique. Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the normalized chart.
Exact mathematical passage
\eta_{abc}: \Phi_{bc}\Phi_{ab} \Longrightarrow \Phi_{ac}
\Phi_MUse in source · 3 passages
Search aliases: Φ_M Phi
Use in source · Stratified_Universality_Traditional.tex, line 5531
For M≥1, s≥0, and r,k≥0, define Φ_M(z)=(1+z)^M-1 and Θ_M,s(r,k) =[z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. The support is finite: Θ_M,s(r,k)=0 unless k≤ r≤ Mk+s.
Exact mathematical passage
\Phi_M(z)=(1+z)^M-1
\Phi_{N,r}Use in source · 4 passages
Search aliases: Φ_N,r Phi
Use in source · block10_section.tex, line 157
Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.
Exact mathematical passage
P\!\left(z,\sum_{j=0}^{N-1}z^jU_j\right) = \sum_{r=0}^{N-1}z^r\Phi_{N,r}(x,\mathbf U).
\Phi_qUse in source · 1 passage
Search aliases: Φ_q Phi
Use in source · block16_section_final.tex, line 670
= fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rational sum of split primitive idempotents must be Galois invariant, so its support is a union of these orbits; this proves fineness. Finally, Q[C_N] ≅ Q[x]/(x^N-1) ≅ ∏_qmid N Q[x]/(Φ_q(x)) ≅ ∏_qmid N Q(ζ_q), and E_q is the identity projector of the q-th factor.
Exact mathematical passage
\mathbb Q[C_N] \cong \mathbb Q[x]/(x^N-1) \cong \prod_{q\mid N}\mathbb Q[x]/(\Phi_q(x)) \cong \prod_{q\mid N}\mathbb Q(\zeta_q),
\Phi_UUse in source · 2 passages
Search aliases: Φ_U Phi
Use in source · block15_section.tex, line 1167
quare remains a projector-sum refinement square in the descent category; item every fold whose source objects, target objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K…
Exact mathematical passage
K_U=\ker(\Phi_U)
\Phi_{\alpha\gamma}Use in source · 1 passage
Search aliases: Φ_αγ Phi alpha gamma
Use in source · Stratified_Universality_Traditional.tex, line 1380
On triple overlaps, pairwise equivalences are not enough. One needs comparison cells η_αβγ: Φ_βγΦ_αβ Longrightarrow Φ_αγ. For four charts, there are two composites from Φ_γδΦ_βγΦ_αβ to Φ_αδ. Their ratio is the higher coherence defect.
Exact mathematical passage
\eta_{\alpha\beta\gamma}: \Phi_{\beta\gamma}\Phi_{\alpha\beta} \Longrightarrow \Phi_{\alpha\gamma}.
\Phi_{\beta\gamma}Use in source · 2 passages
Search aliases: Φ_βγ Phi beta gamma
Use in source · Stratified_Universality_Traditional.tex, line 1380
On triple overlaps, pairwise equivalences are not enough. One needs comparison cells η_αβγ: Φ_βγΦ_αβ Longrightarrow Φ_αγ. For four charts, there are two composites from Φ_γδΦ_βγΦ_αβ to Φ_αδ. Their ratio is the higher coherence defect.
Exact mathematical passage
\eta_{\alpha\beta\gamma}: \Phi_{\beta\gamma}\Phi_{\alpha\beta} \Longrightarrow \Phi_{\alpha\gamma}.
\Phi_{\gamma\delta}Use in source · 1 passage
Search aliases: Φ_γδ Phi gamma delta
Use in source · Stratified_Universality_Traditional.tex, line 1387
On triple overlaps, pairwise equivalences are not enough. One needs comparison cells η_αβγ: Φ_βγΦ_αβ Longrightarrow Φ_αγ. For four charts, there are two composites from Φ_γδΦ_βγΦ_αβ to Φ_αδ. Their ratio is the higher coherence defect.
Exact mathematical passage
\Phi_{\gamma\delta}\Phi_{\beta\gamma}\Phi_{\alpha\beta}
\Phi_{\kappa,\tau}Use in source · 5 passages
Search aliases: Φ_κ,τ Phi kappa tau
Use in source · block15_section.tex, line 219
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
\Phi_{\kappa,\tau}: \bigoplus_{e\in E}B_e\longrightarrow C, \qquad (b_e)_e\longmapsto \sum_{e\in E}j_{\kappa(e)}\tau_e(b_e)
\PsiUse in source · 4 passages
Search aliases: Ψ Psi
Use in source · block11_section.tex, line 67
The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.
Exact mathematical passage
\Psi(x)=s+\Phi(x)
\omegaUse in source · 27 passages
Search aliases: ω omega
Use in source · Stratified_Universality_Traditional.tex, line 1171
Reconstructed symmetry · Aut^⊗(ω) · tensor automorphisms of a fiber functor and the corresponding affine group scheme · faithful exact tensor fiber functor
Exact mathematical passage
\Aut^{\otimes}(\omega)
\omega^{-j\ell}Use in source · 1 passage
Search aliases: ω^-jℓ omega ell
Use in source · block15_section.tex, line 703
Suppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
\mathcal P_{N,j}F(z) = \frac1N\sum_{\ell=0}^{N-1} \omega^{-j\ell}F(\omega^\ell z).
\omega^{jm}Use in source · 1 passage
Search aliases: ω^jm omega
Use in source · block15_section.tex, line 1002
characteristic-zero field containing μ_d and a primitive N-th root of unity ω, let d,m,N∈ Z_≥1, and choose the formal branch F(z)=(1-α z^m)^-1/d, α∈ K^×. With g=gcd(N,m), its nonzero coefficients occupy exactly the subgroup of Z/N Z generated by m, which has N/g addresses. The rotated radicands 1-αω^jmz^m, 0≤ j<N, also have exactly N/g distinct values.
Exact mathematical passage
1-\alpha\omega^{jm}z^m, \qquad 0\le j<N,
\omega^\ellUse in source · 1 passage
Search aliases: ω^ℓ omega ell
Use in source · block15_section.tex, line 703
Suppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
\mathcal P_{N,j}F(z) = \frac1N\sum_{\ell=0}^{N-1} \omega^{-j\ell}F(\omega^\ell z).
\omega_NUse in source · 3 passages
Search aliases: ω_N omega
Use in source · block09_section.tex, line 259
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\omega_N
\omega_N^{-jq}Use in source · 2 passages
Search aliases: ω_N^-jq omega
Use in source · block09_section.tex, line 260
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\boxed{ t^j\mathfrak M_{N,j}F(t^N) = \frac1N \sum_{q=0}^{N-1} \omega_N^{-jq}F(\omega_N^qt).}
\omega_N^qUse in source · 3 passages
Search aliases: ω_N^q omega
Use in source · block09_section.tex, line 260
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\boxed{ t^j\mathfrak M_{N,j}F(t^N) = \frac1N \sum_{q=0}^{N-1} \omega_N^{-jq}F(\omega_N^qt).}
\omega_N^rUse in source · 2 passages
Search aliases: ω_N^r omega
Use in source · block09_section.tex, line 569
Let ω_N be primitive and consider the rotated orbit O_N(F_λ,α) = {(1-αω_N^rt)^-λ:0≤ r<N}. Let r_rm orb be its dimension over the constant field.
Exact mathematical passage
\mathcal O_N(F_{\lambda,\alpha}) = \left \{(1-\alpha\omega_N^rt)^{-\lambda}:0\leq r<N\right\}.
\omega_N^{rn}Use in source · 1 passage
Search aliases: ω_N^rn omega
Use in source · block09_section.tex, line 595
If λ=-M, expand (1-αω_N^rt)^M = ∑_n=0^M binom Mn(-α)^nω_N^rnt^n. The rotation matrix is a truncated discrete Fourier matrix. Its rank is the number of available residue classes, namely min(N,M+1).
Exact mathematical passage
(1-\alpha\omega_N^rt)^M = \sum_{n=0}^{M} \binom Mn(-\alpha)^n\omega_N^{rn}t^n.
\omega_{p-1}Use in source · 5 passages
Search aliases: ω_p-1 omega
Use in source · Stratified_Universality_Traditional.tex, line 2932
Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.
Exact mathematical passage
\label{eq:block04-collapse-obstruction} \omega_{p-1}(R) = \bigl(\omega_v(R)\bigr)_v
\omega_vUse in source · 1 passage
Search aliases: ω_v omega
Use in source · Stratified_Universality_Traditional.tex, line 2932
Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.
Exact mathematical passage
\label{eq:block04-collapse-obstruction} \omega_{p-1}(R) = \bigl(\omega_v(R)\bigr)_v
\omega_{\pi}Use in source · 1 passage
Search aliases: ω_π omega pi
Use in source · Stratified_Universality_Traditional.tex, line 2920
Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.
Exact mathematical passage
\omega_{\pi}(R) = \bigl(v_{\pi}(B)-v_{\pi}(A)\bigr)\bmod(p-1).
\omega_{\infty}Use in source · 1 passage
Search aliases: ω_∞ omega infty
Use in source · Stratified_Universality_Traditional.tex, line 2926
Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.
Exact mathematical passage
\omega_{\infty}(R) = \bigl(\deg A-\deg B\bigr)\bmod(p-1).
\ell^2Use in source · 1 passage
Search aliases: ℓ^2 ell
Use in source · block12_section.tex, line 483
For a connected scheme X with geometric point bar x, the fiber functor on finite acuteetale covers identifies that category with finite continuous π_1^acuteet(X,bar x)-sets. Every finite fiber S produces a permutation resolution on ℓ^2(S): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms give equivariant packet transfers. The profinite fundamental group is the automorphism group of this fiber functor.
Exact mathematical passage
\ell^2(S)
\ell_iUse in source · 1 passage
Search aliases: ℓ_i ell
Use in source · Stratified_Universality_Traditional.tex, line 3742
[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).
Exact mathematical passage
\Lambda_db_j = \sum_{i=1}^{r_k(a)}(M_d(a))_{ij}b_i, \qquad a=\sum_i v_i(a)b_i, \qquad \ell_i(a)=b_i(0).
\leftrightarrowUse in source · 22 passages
Search aliases: ↔ leftrightarrow
Use in source · Stratified_Universality_Traditional.tex, line 1845
item constant recurrence ↔ rational ordinary generating function ↔ finite linear state; item P-recursive ↔ D-finite coefficient translation; item finite digit realization ↔ finite k-kernel; item rational functional-Mahler mask → Cartier representation → finite automaton; item finite-mask Cartier transitions ↔ carry matrices; item selected algebraic branch → packet polynomial system; item invertible transform ↔ transported packet resolution; item Euler-product exponent coordinates ↔ logarithmic-d…
Exact mathematical passage
\leftrightarrow
\mapstoUse in source · 17 passages
Search aliases: ↦ mapsto
Use in source · Stratified_Universality_Traditional.tex, line 1599
The invariance of im D makes [v]↦[Q_av] well defined. Idempotence, annihilation, and reconstruction descend to the quotient. In the block-diagonal case the image of D is the direct sum of the images of the restrictions.
Exact mathematical passage
[v]\mapsto[Q_av]
\hookrightarrowUse in source · 3 passages
Search aliases: ↪ hookrightarrow
Use in source · Stratified_Universality_Traditional.tex, line 4277
Choose Smith bases in which the inclusion M_k(a)↪ L_k(a) is the diagonal matrix D=diag(d_1,…,d_ρ). Modulo m, the kernel and cokernel of multiplication by d_i are both cyclic of order gcd(m,d_i), and the image has order m/gcd(m,d_i). The map is invertible in every coordinate exactly when every d_i is a unit modulo m, which is equivalent to gcd(m,d_ρ)=1.
Exact mathematical passage
M_k(a)\hookrightarrow L_k(a)
\partialUse in source · 4 passages
Search aliases: ∂ partial
Use in source · block10_section.tex, line 72
[Simple-branch normalization] If P_F(0,a_0)=0, ∂_Y P_F(0,a_0)≠0, then a_0 alone selects a unique formal branch. Equivalently, σ(P_F,F)≤1.
Exact mathematical passage
P_F(0,a_0)=0, \qquad \partial_Y P_F(0,a_0)\neq0,
\partial_YUse in source · 4 passages
Search aliases: ∂_Y partial
Use in source · block10_section.tex, line 72
[Simple-branch normalization] If P_F(0,a_0)=0, ∂_Y P_F(0,a_0)≠0, then a_0 alone selects a unique formal branch. Equivalently, σ(P_F,F)≤1.
Exact mathematical passage
P_F(0,a_0)=0, \qquad \partial_Y P_F(0,a_0)\neq0,
\partial_zUse in source · 1 passage
Search aliases: ∂_z partial
Use in source · block10_section.tex, line 534
[Algebraic derivative closure] Let F have separable minimal polynomial P_F of degree d. Then F' = -frac∂_zP_F(z,F)∂_YP_F(z,F) ∈ K(z,F). Every higher derivative lies in the same degree-d function field. Consequently, F has a differential annihilator of order at most d.
Exact mathematical passage
F' = -\frac{\partial_zP_F(z,F)}{\partial_YP_F(z,F)} \in K(z,F).
\varnothingUse in source · 6 passages
Search aliases: ∅ varnothing
Use in source · block13_section.tex, line 12
Write E, FF, D, P, ∅ for an equivalence on a named normalized overlap, a fully faithful inclusion, an information-losing directional functor, a partial or hypothesis-dependent bridge, and the absence of a certified broad bridge, respectively. The matrix below records only relations already established in the preceding sections or in the audited source concordance. A symbol carrying a subscript refers to the named overlap rather than to the whole broad stratum.
Exact mathematical passage
\mathsf E, \quad \mathsf{FF}, \quad \mathsf D, \quad \mathsf P, \quad \varnothing
\notinUse in source · 8 passages
Search aliases: ∉ notin
Use in source · Stratified_Universality_Traditional.tex, line 721
[Hilbert--Schmidt optimality, imported] Assume X,Y are Hilbert spaces, the packet idempotents are orthogonal projectors, and f:X→ Y is Hilbert--Schmidt. Then C_E is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and R_E(f) _HS^2 = ∑_(a,i)∉ E q_afp_i _HS^2. Consequently, C_E(f) is the unique best Hilbert--Schmidt approximation to f among operators supported on E.
Exact mathematical passage
\|R_E(f)\|_{\mathrm{HS}}^2 = \sum_{(a,i)\notin E}\|q_afp_i\|_{\mathrm{HS}}^2.
\prod_{a=1}^{q}Use in source · 1 passage
Search aliases: ∏_a=1^q prod
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
\prod_{a\in A_{N,j}(\lambda)}Use in source · 1 passage
Search aliases: ∏_a∈ A_N,j(λ) prod in lambda
Use in source · block09_section.tex, line 497
B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ \mathcal H_{N,j}^{\lambda,\alpha} = \theta \prod_{b\in B_{N,j}}(\theta+b-1) - \alpha^Nx \prod_{a\in A_{N,j}(\lambda)}(\theta+a),}
\prod_{b\in B_{N,j}}Use in source · 1 passage
Search aliases: ∏_b∈ B_N,j prod in
Use in source · block09_section.tex, line 497
B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ \mathcal H_{N,j}^{\lambda,\alpha} = \theta \prod_{b\in B_{N,j}}(\theta+b-1) - \alpha^Nx \prod_{a\in A_{N,j}(\lambda)}(\theta+a),}
\prod_{g\in G}Use in source · 1 passage
Search aliases: ∏_g∈ G prod in
Use in source · block12_section.tex, line 145
[Canonical envelopes of a nonnormal Galois atlas] Let M=L^H. Then L^Core_G(H) =∏_g∈ Gg(M) tag12.6 is the smallest Galois extension inside L containing every conjugate chart, while L^Ncl_G(H) =⋂_g∈ Gg(M) tag12.7 is the largest Galois subfield contained in every chart. The core and normal-closure projectors are the two canonical global envelopes of the conjugate atlas.
Exact mathematical passage
\boxed{ L^{\operatorname{Core}_G(H)} =\prod_{g\in G}g(M)} \tag{12.6}
\prod_iUse in source · 1 passage
Search aliases: ∏_i prod
Use in source · Stratified_Universality_Traditional.tex, line 502
[Universal mapping property of packet coordinates] For every object Z, restriction to the packet objects gives natural bijections Hom(X,Z) ≅ ∏_iHom(X_i,Z), Hom(Z,X) ≅ ∏_iHom(Z,X_i), where the products are finite. Packetwise maps therefore assemble uniquely into a global map.
Exact mathematical passage
\operatorname{Hom}(X,Z) \cong \prod_i\operatorname{Hom}(X_i,Z), \qquad \operatorname{Hom}(Z,X) \cong \prod_i\operatorname{Hom}(Z,X_i),
\prod_{i=1}^{\rho}Use in source · 1 passage
Search aliases: ∏_i=1^ρ prod rho
Use in source · Stratified_Universality_Traditional.tex, line 4263
[Exact Smith formulas] Let g_i(m)=gcd(m,d_i(a)). Then G_k,m(a) ≅ Q_k,m(a) ≅ ⊕_i=1^ρ Z/g_i(m) Z, and |overline M_k,m(a)| = ∏_i=1^ρ fracmg_i(m). The reduction map is an isomorphism if and only if gcd(m,d_ρ(a))=1. Equivalently, every prime dividing a reduction defect divides the saturation index.
Exact mathematical passage
\left|\overline M_{k,m}(a)\right| = \prod_{i=1}^{\rho} \frac{m}{g_i(m)}.
\prod_{j=0}^{N}Use in source · 1 passage
Search aliases: ∏_j=0^N prod
Use in source · Stratified_Universality_Traditional.tex, line 2477
Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.
Exact mathematical passage
\prod_{j=0}^{N}\frac{F(z^{p^j})}{F(z^{p^{j+1}})} = \frac{F(z)}{F(z^{p^{N+1}})}.
\prod_{j\geq 0}Use in source · 1 passage
Search aliases: ∏_j≥ 0 prod geq
Use in source · Stratified_Universality_Traditional.tex, line 2848
For a unit formal series, define its canonical local mask by R_F(z) = fracF(z)F(z^p). The coefficientwise infinite product M_p[R](z) = ∏_j≥ 0R (z^p^j) is well defined for every R∈1+z F_p[[z]], because only finitely many factors contribute to a fixed coefficient.
Exact mathematical passage
\label{eq:block04-mask-product} \mathcal M_p[R](z) = \prod_{j\geq 0}R\!\left(z^{p^j}\right)
\prod_{j\geq0}Use in source · 3 passages
Search aliases: ∏_j≥0 prod geq
Use in source · Stratified_Universality_Traditional.tex, line 2432
For a unit formal series F(0)=1, define its local functional-Mahler mask by M_p(F)(z) = fracF(z)F(z^p). Then F(z)= M_p(F)(z)F(z^p). Iterating [source reference] gives the coefficientwise product F(z) = ∏_j≥0 M_p(F)(z^p^j). Only finitely many factors contribute to each fixed coefficient.
Exact mathematical passage
\label{eq:mask-product} F(z) = \prod_{j\geq0}\mathcal M_p(F)(z^{p^j}).
\prod_{k=0}^{n-1}Use in source · 1 passage
Search aliases: ∏_k=0^n-1 prod
Use in source · block15_section.tex, line 659
For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0
Exact mathematical passage
(a)_{n;N}:=\prod_{k=0}^{n-1}(a+Nk), \qquad (a)_{0;N}:=1.
\prod_{m<n}Use in source · 1 passage
Search aliases: ∏_m<n prod
Use in source · Stratified_Universality_Traditional.tex, line 4831
Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, the…
Exact mathematical passage
A(z)\prod_{m<n}(1-z^m)^{b_m} \equiv 1+r_nz^n\pmod{z^{n+1}}.
\prod_{n\geq1}Use in source · 2 passages
Search aliases: ∏_n≥1 prod geq
Use in source · Stratified_Universality_Traditional.tex, line 4945
[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plane-partition series. More generally, the power family converts the exponent rule n^r into the divisor-sum ghost σ_r+1.
Exact mathematical passage
\begin{array}{c|c|c} b_n & A_b(z) & c_n\\ \hline 1 & \displaystyle\prod_{n\geq1}(1-z^n)^{-1} & \sigma_1(n)\\[4pt] n & \displaystyle\prod_{n\geq1}(1-z^n)^{-n} & \sigma_2(n)\\[4pt] n^r & \displaystyle\prod_{n\geq1}(1-z^n)^{-n^r} & \sigma_{r+1}(n). \end{array}
\prod_pUse in source · 1 passage
Search aliases: ∏_p prod
Use in source · Stratified_Universality_Traditional.tex, line 4632
[Profinite saturation sequence] There are canonical digit-equivariant identifications widehatM_k(a) ≅ ∏_p M_k,p(a), widehatL_k(a) ≅ ∏_p L_k,p(a), and a short exact sequence 0 ⟶ widehatM_k(a) ⟶ widehatL_k(a) ⟶ C_k(a) ⟶ 0. A compatible all-moduli raw tower has an ordinary integral lift exactly when its class in widehatM_k(a)/M_k(a) vanishes.
Exact mathematical passage
\widehat{M_k(a)} \cong \prod_p M_{k,p}(a), \qquad \widehat{L_k(a)} \cong \prod_p L_{k,p}(a),
\prod_{p\text{ prime}}Use in source · 1 passage
Search aliases: ∏_p prime prod text
Use in source · Stratified_Universality_Traditional.tex, line 5032
[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).
Exact mathematical passage
\boxed{ (\mathbf{Mult}_{\mathbb Z},\mathbin{\ast_D}) \simeq \prod_{p\text{ prime}} (1+T\mathbb Z[[T]],\cdot). }
\prod_{p^e\parallel n}Use in source · 2 passages
Search aliases: ∏_p^eparallel n prod parallel
Use in source · Stratified_Universality_Traditional.tex, line 5041
[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).
Exact mathematical passage
\boxed{ f(n)=\prod_{p^e\parallel n}f(p^e).}
\prod_{q\mid N}Use in source · 1 passage
Search aliases: ∏_qmid N prod mid
Use in source · block16_section_final.tex, line 670
= fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rational sum of split primitive idempotents must be Galois invariant, so its support is a union of these orbits; this proves fineness. Finally, Q[C_N] ≅ Q[x]/(x^N-1) ≅ ∏_qmid N Q[x]/(Φ_q(x)) ≅ ∏_qmid N Q(ζ_q), and E_q is the identity projector of the q-th factor.
Exact mathematical passage
\mathbb Q[C_N] \cong \mathbb Q[x]/(x^N-1) \cong \prod_{q\mid N}\mathbb Q[x]/(\Phi_q(x)) \cong \prod_{q\mid N}\mathbb Q(\zeta_q),
\prod_{r,\gamma}Use in source · 2 passages
Search aliases: ∏_r,γ prod gamma
Use in source · block11_section.tex, line 206
[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.
Exact mathematical passage
\boxed{ \begin{aligned} &P_\delta D^n(F\circ G)(x)[h,\ldots,h]\\ &=\sum_{\mu} \frac{n!} {\displaystyle\prod_{r,\gamma} (r!)^{\mu_{r,\gamma}}\mu_{r,\gamma}!} \,P_\delta D^{k(\mu)}F(Gx) \bigl[ (g_{r,\gamma})^{\mu_{r,\gamma}} \bigr]_{r,\gamma}. \end{aligned}} \tag{11.5}
\prod_{r=0}^{N-1}Use in source · 1 passage
Search aliases: ∏_r=0^N-1 prod
Use in source · block09_section.tex, line 489
B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.
Exact mathematical passage
\boxed{ \prod_{r=0}^{N-1}(Nm+j+1+r)u_{m+1} = \alpha^N \prod_{r=0}^{N-1}(Nm+j+\lambda+r)u_m.}
\prod_{r=1}^{N}Use in source · 1 passage
Search aliases: ∏_r=1^N prod
Use in source · block15_section.tex, line 752
The projector laws follow coefficient by coefficient. The residue classes modulo NM lying over jpmod N are precisely j,j+N,…,j+(M-1)N, which proves refinement. Character orthogonality gives the Fourier formula. Finally, (Nn+j)! = j! N^Nn ∏_r=1^N(fracj+rN)_n. Exactly one factor in the product is (1)_n=n!. Removing it leaves the parameter set b_N,j and gives the hypergeometric series.
Exact mathematical passage
(Nn+j)! = j!\,N^{Nn} \prod_{r=1}^{N}\left(\frac{j+r}{N}\right)_n.
\prod_{r=1}^{q}Use in source · 1 passage
Search aliases: ∏_r=1^q prod
Use in source · block13_section.tex, line 300
x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value c_0=1.
Exact mathematical passage
B_q(n) =\prod_{r=1}^{q}(qn+r).
\prod_{r=2}^{q}Use in source · 1 passage
Search aliases: ∏_r=2^q prod
Use in source · block13_section.tex, line 296
Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence with initial value c_0=1.
Exact mathematical passage
A_q(n) =(n+1)\prod_{r=2}^{q}\bigl((q-1)n+r\bigr),
\prod_{r=c+1}^{c+N}Use in source · 1 passage
Search aliases: ∏_r=c+1^c+N prod
Use in source · block15_section.tex, line 676
[Exact factorial--residue realization] Let N,c,n∈ Z with N≥1, c≥0, and n≥0. Then frac(Nn+c)!c! = ∏_r=c+1^c+N(r)_n;N. Each integer factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_N,j = ∑_ℓ=0^M-1 P_NM,j+ℓ N.
Exact mathematical passage
\boxed{ \frac{(Nn+c)!}{c!} = \prod_{r=c+1}^{c+N}(r)_{n;N}.}
\prod_{r\ge1}Use in source · 1 passage
Search aliases: ∏_r≥1 prod ge
Use in source · block11_section.tex, line 235
[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.
Exact mathematical passage
\boxed{ C_{n,M} =[x^n]\prod_{r\ge1}(1-x^r)^{-M}.} \tag{11.6}
\prod_{s=0}^{p^a-1}Use in source · 2 passages
Search aliases: ∏_s=0^p^a-1 prod
Use in source · Stratified_Universality_Traditional.tex, line 5487
[Complete clopen packet equivalence] The analysis map B_a: C_p ⟶ ∏_s=0^p^a-1 C_p is an isometric isomorphism. Its inverse is the piecewise synthesis ( S_a g)(x) =g_s (fracx-sp^a) for x∈ s+p^a Z_p. Thus the clopen residue cover of Z_p has effective descent for both objects and equality morphisms.
Exact mathematical passage
\mathcal B_a:\mathcal C_p \longrightarrow \prod_{s=0}^{p^a-1}\mathcal C_p
\sum_{\mathcal R\in H\backslash(K\times I)}Use in source · 1 passage
Search aliases: ∑_ R∈ Hbackslash(K× I) sum mathcal in backslash times
Use in source · block16_section_final.tex, line 507
Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.
Exact mathematical passage
B_{\mathcal Q}B_{\mathcal P} = \sum_{\mathcal R\in H\backslash(K\times I)} c^{\mathcal R}_{\mathcal Q,\mathcal P}B_{\mathcal R}.
\sum_{(a,i)\notin E}Use in source · 1 passage
Search aliases: ∑_(a,i)∉ E sum notin
Use in source · Stratified_Universality_Traditional.tex, line 721
[Hilbert--Schmidt optimality, imported] Assume X,Y are Hilbert spaces, the packet idempotents are orthogonal projectors, and f:X→ Y is Hilbert--Schmidt. Then C_E is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and R_E(f) _HS^2 = ∑_(a,i)∉ E q_afp_i _HS^2. Consequently, C_E(f) is the unique best Hilbert--Schmidt approximation to f among operators supported on E.
Exact mathematical passage
\|R_E(f)\|_{\mathrm{HS}}^2 = \sum_{(a,i)\notin E}\|q_afp_i\|_{\mathrm{HS}}^2.
\sum_aUse in source · 8 passages
Search aliases: ∑_a sum
Use in source · Stratified_Universality_Traditional.tex, line 538
[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.
Exact mathematical passage
\boxed{ (gf)_{ci} = \sum_a g_{ca}f_{ai}.}
\sum_{a,i}Use in source · 5 passages
Search aliases: ∑_a,i sum
Use in source · Stratified_Universality_Traditional.tex, line 529
[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.
Exact mathematical passage
\boxed{ f = \sum_{a,i}q_afp_i = \sum_{a,i}\jmath_a f_{ai}\pi_i.}
\sum_{a,i_1,\ldots,i_d}Use in source · 1 passage
Search aliases: ∑_a,i_1,…,i_d sum ldots
Use in source · block15_section.tex, line 601
inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps is governed by finite contraction over every intermediate lane address.
Exact mathematical passage
T(x_1,\ldots,x_d) = \sum_{a,i_1,\ldots,i_d} \iota_{Y,a} T_{a;i_1,\ldots,i_d} (\pi_{1,i_1}x_1,\ldots,\pi_{d,i_d}x_d).
\sum_{a,i\in\{1,2\}}Use in source · 1 passage
Search aliases: ∑_a,i∈{1,2} sum in
Use in source · block15_section.tex, line 1270
The displayed matrix multiplication gives the two values immediately. The full refinement identity recovers f=∑_a,i∈{1,2}p_afp_i, whose only nonzero term here is p_2fp_2.
Exact mathematical passage
f=\sum_{a,i\in\{1,2\}}p_afp_i
\sum_{a\in I_{Y,u}}Use in source · 1 passage
Search aliases: ∑_a∈ I_Y,u sum in
Use in source · block15_section.tex, line 142
item Composition and path law. If f:X→ Y and g:Y→ Z, with the same resolution u used on the intermediate object, then [gf]^v,r_c i = ∑_a∈ I_Y,u[g]^v,u_c a[f]^u,r_a i. More generally, a block of a composable string is the finite sum of the products along all compatible intermediate address paths.
Exact mathematical passage
[gf]^{v,r}_{c i} = \sum_{a\in I_{Y,u}}[g]^{v,u}_{c a}[f]^{u,r}_{a i}.
\sum_{a\in I_{Y,U,u}}Use in source · 1 passage
Search aliases: ∑_a∈ I_Y,U,u sum in
Use in source · block16_section_final.tex, line 170
item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, maths…
Exact mathematical passage
(k\star h)_{U,ci} := \sum_{a\in I_{Y,U,u}}k_{U,ca}h_{U,ai}.
\sum_{a\in(\mathbb Z/q\mathbb Z)^\times}Use in source · 1 passage
Search aliases: ∑_a∈( Z/q Z)^× sum in mathbb mathbb times
Use in source · block16_section_final.tex, line 658
Write every j∈ O_q uniquely as j=(N/q)a with a∈( Z/q Z)^×. The coefficient of g^ℓ in E_q is then frac1N ∑_a∈( Z/q Z)^× ζ_q^-aℓ = fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rational sum of split primitive idempotents must be Galois invariant, so its support is a union of these orbits; this proves fineness. Finally, Q[C_N] ≅ Q…
Exact mathematical passage
\frac1N \sum_{a\in(\mathbb Z/q\mathbb Z)^\times} \zeta_q^{-a\ell} = \frac{c_q(\ell)}{N},
\sum_{b,j}Use in source · 1 passage
Search aliases: ∑_b,j sum
Use in source · block15_section.tex, line 403
[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contribute. The inverse is the corresponding fine analysis after coarse synthesis. Consequently, changing a complete lane resolution loses no information. Lo…
Exact mathematical passage
\bigl(\Gamma^{(u,r)}_{(v,s)}h\bigr)_{a i} = \sum_{b,j} \bigl(\pi_{Y,a}^u\iota_{Y,b}^v\bigr) h_{bj} \bigl(\pi_{X,j}^s\iota_{X,i}^r\bigr).
\sum_{d=0}^{k-1}Use in source · 1 passage
Search aliases: ∑_d=0^k-1 sum
Use in source · block15_section.tex, line 829
[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path law.
Exact mathematical passage
\Lambda_dS_e=\delta_{de}1, \qquad \sum_{d=0}^{k-1}S_d\Lambda_d=1.
\sum_{d\mid e}Use in source · 2 passages
Search aliases: ∑_dmid e sum mid
Use in source · Stratified_Universality_Traditional.tex, line 5072
Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).
Exact mathematical passage
\gamma_{p,e}(f) = \sum_{d\mid e}d\,\beta_{p,d}(f).
\sum_{d\mid\gcd(q,\ell)}Use in source · 1 passage
Search aliases: ∑_dmidgcd(q,ℓ) sum mid gcd ell
Use in source · block16_section_final.tex, line 630
Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projectors (E_q)_qmid N form the finest Q-rational coarsening obtainable by summing the split Fourier idempotents, and Q[C_N]E_q≅ Q(ζ_q).
Exact mathematical passage
c_q(\ell) := \sum_{\substack{1\le a\le q\\\gcd(a,q)=1}} e^{2\pi i a\ell/q} = \sum_{d\mid\gcd(q,\ell)}d\,\mu(q/d) \in\mathbb Z
\sum_{e\in E}Use in source · 3 passages
Search aliases: ∑_e∈ E sum in
Use in source · block15_section.tex, line 219
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
\Phi_{\kappa,\tau}: \bigoplus_{e\in E}B_e\longrightarrow C, \qquad (b_e)_e\longmapsto \sum_{e\in E}j_{\kappa(e)}\tau_e(b_e)
\sum_{e\geq1}Use in source · 1 passage
Search aliases: ∑_e≥1 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 5130
[Valuation logarithmic derivative] Fix a prime p. Let Γ_f,p be the arithmetic function supported on powers of p with Γ_f,p(1)=0, Γ_f,p(p^e)=γ_p,e(f) (e≥1). Then (v_p· f)(n) = (Γ_f,pmathbin∗_Df)(n) (n≥1). Equivalently, fracT F_f,p'(T)F_f,p(T) = ∑_e≥1γ_p,e(f)T^e.
Exact mathematical passage
\frac{T F_{f,p}'(T)}{F_{f,p}(T)} = \sum_{e\geq1}\gamma_{p,e}(f)T^e.
\sum_{g\in G}Use in source · 1 passage
Search aliases: ∑_g∈ G sum in
Use in source · block15_section.tex, line 1035
[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map is block diagonal in these isotypic lanes, while an arbitrary K-linear map has the full lane matrix of [source reference].
Exact mathematical passage
e_\chi = \frac{\chi(1)}{|G|} \sum_{g\in G}\chi(g^{-1})g \in K[G]
\sum_{g\in G_i}Use in source · 1 passage
Search aliases: ∑_g∈ G_i sum in
Use in source · block12_section.tex, line 224
Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.
Exact mathematical passage
P_i=\frac1{|G_i|}\sum_{g\in G_i}U_g.
\sum_{h\in H}Use in source · 3 passages
Search aliases: ∑_h∈ H sum in
Use in source · block12_section.tex, line 16
Let L/K be a finite Galois extension with group G=Gal(L/K). Let V be a finite-dimensional representation of G over a characteristic-zero field, or a unitary complex realization obtained after scalar extension. For every subgroup H≤ G, define P_H=frac1|H|∑_h∈ HU_h. tag12.1
Exact mathematical passage
\boxed{ P_H=\frac1{|H|}\sum_{h\in H}U_h.} \tag{12.1}
\sum_{i,j_1,\ldots,j_m}Use in source · 1 passage
Search aliases: ∑_i,j_1,…,j_m sum ldots
Use in source · block11_section.tex, line 120
For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
\boxed{ A_m = \sum_{i,j_1,\ldots,j_m} A_{m;i\mid j_1,\ldots,j_m}.} \tag{11.3}
\sum_{i=0}^{j}Use in source · 1 passage
Search aliases: ∑_i=0^j sum
Use in source · block12_section.tex, line 265
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
\boxed{ a(V)=\sum_jd_j\sum_{i=0}^{j}\frac{|G_i|}{|G_0|}.} \tag{12.13}
\sum_{i=1}^{r_k(a)}Use in source · 1 passage
Search aliases: ∑_i=1^r_k(a) sum
Use in source · Stratified_Universality_Traditional.tex, line 3742
[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).
Exact mathematical passage
\Lambda_db_j = \sum_{i=1}^{r_k(a)}(M_d(a))_{ij}b_i, \qquad a=\sum_i v_i(a)b_i, \qquad \ell_i(a)=b_i(0).
\sum_{i=1}^{\rho}Use in source · 1 passage
Search aliases: ∑_i=1^ρ sum rho
Use in source · Stratified_Universality_Traditional.tex, line 4350
[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.
Exact mathematical passage
\left|\overline M_{k,p^r}(a)\right| = p^{\sum_{i=1}^{\rho}(r-\min(r,\nu_i))}.
\sum_{i_\ell}Use in source · 1 passage
Search aliases: ∑_i_ℓ sum ell
Use in source · block15_section.tex, line 632
Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement is then a finite sum of the specified homogeneous pieces.
Exact mathematical passage
x_\ell=\sum_{i_\ell}p_{\ell,i_\ell}x_\ell
\sum_{i\in\mathcal O_U}Use in source · 1 passage
Search aliases: ∑_i∈ O_U sum in mathcal
Use in source · block16_section_final.tex, line 375
[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.
Exact mathematical passage
P^r_{A,U,\mathcal O} := \sum_{i\in\mathcal O_U}p^r_{A,U,i}
\sum_{i\in S_U}Use in source · 1 passage
Search aliases: ∑_i∈ S_U sum in
Use in source · block16_section_final.tex, line 387
item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.
Exact mathematical passage
Q_U=\sum_{i\in S_U}p^r_{A,U,i}
\sum_{i\geq0}Use in source · 1 passage
Search aliases: ∑_i≥0 sum geq
Use in source · block12_section.tex, line 254
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
\boxed{ a(V)=\sum_{i\geq0}\frac{|G_i|}{|G_0|}\operatorname{Tr}(I-P_i),} \tag{12.11}
\sum_{i\geq1}Use in source · 1 passage
Search aliases: ∑_i≥1 sum geq
Use in source · block12_section.tex, line 259
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
\boxed{ \operatorname{Sw}(V)=\sum_{i\geq1}\frac{|G_i|}{|G_0|}\operatorname{Tr}(I-P_i).} \tag{12.12}
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · block12_section.tex, line 265
[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
Exact mathematical passage
\boxed{ a(V)=\sum_jd_j\sum_{i=0}^{j}\frac{|G_i|}{|G_0|}.} \tag{12.13}
\sum_{j=0}^{e}Use in source · 1 passage
Search aliases: ∑_j=0^e sum
Use in source · Stratified_Universality_Traditional.tex, line 5240
[Multiplicativity is preserved and reflected] Assume b(1)=1 and c= G(b). Then b is multiplicative Longleftrightarrow c is multiplicative. On this locus, both b and c are reconstructed from prime-power seeds. For every prime p and e≥1, c(p^e)=∑_j=0^ep^j b(p^j), b(p^e)=fracc(p^e)-c(p^e-1)p^e.
Exact mathematical passage
\boxed{ c(p^e)=\sum_{j=0}^{e}p^j b(p^j),}
\sum_{j=0}^{k}Use in source · 1 passage
Search aliases: ∑_j=0^k sum
Use in source · Stratified_Universality_Traditional.tex, line 5541
For M≥1, s≥0, and r,k≥0, define Φ_M(z)=(1+z)^M-1 and Θ_M,s(r,k) =[z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. The support is finite: Θ_M,s(r,k)=0 unless k≤ r≤ Mk+s.
Exact mathematical passage
\boxed{ \Theta_{M,s}(r,k) =\sum_{j=0}^{k}(-1)^{k-j}\binom kj\binom{Mj+s}{r}.}
\sum_{j=0}^{N}Use in source · 1 passage
Search aliases: ∑_j=0^N sum
Use in source · Stratified_Universality_Traditional.tex, line 4598
[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
Exact mathematical passage
c_N=\sum_{j=0}^{N}p^{2j}
\sum_{j=0}^{N-1}Use in source · 7 passages
Search aliases: ∑_j=0^N-1 sum
Use in source · block10_section.tex, line 141
Fix N≥1. For F(z)=∑_n≥0a_nz^n, define the arithmetic packets U_j(x) = ∑_m≥0a_Nm+jx^m, 0≤ j<N. They satisfy the exact synthesis identity F(z) = ∑_j=0^N-1z^jU_j(z^N).
Exact mathematical passage
\boxed{ F(z) = \sum_{j=0}^{N-1}z^jU_j(z^N).}
\sum_{j=0}^{r}Use in source · 3 passages
Search aliases: ∑_j=0^r sum
Use in source · block09_section.tex, line 97
Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.
Exact mathematical passage
L = \sum_{j=0}^{r}q_j(z)D^j, \qquad q_j(z)=\sum_{m=0}^{d_j}q_{j,m}z^m.
\sum_{j=0}^{s-1}Use in source · 1 passage
Search aliases: ∑_j=0^s-1 sum
Use in source · block09_section.tex, line 58
Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
Exact mathematical passage
R_a = S^s+\sum_{j=0}^{s-1}r_j(n)S^j.
\sum_{j=0}^{\ell-1}Use in source · 2 passages
Search aliases: ∑_j=0^ℓ-1 sum ell
Use in source · Stratified_Universality_Traditional.tex, line 2346
The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.
Exact mathematical passage
\Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a = \bigl(n\longmapsto a_{p^\ell n+r}\bigr), \qquad r=\sum_{j=0}^{\ell-1}d_jp^j.
\sum_{j=1}^{e}Use in source · 1 passage
Search aliases: ∑_j=1^e sum
Use in source · Stratified_Universality_Traditional.tex, line 5139
The local logarithmic-derivative recurrence gives e f(p^e) = ∑_j=1^eγ_p,j(f)f(p^e-j). Write n=p^e u with (u,p)=1 and multiply by f(u). Since Γ_f,p is supported on p-powers, the resulting sum is exactly the Dirichlet convolution at n.
Exact mathematical passage
e f(p^e) = \sum_{j=1}^{e}\gamma_{p,j}(f)f(p^{e-j}).
\sum_{j=1}^{n}Use in source · 1 passage
Search aliases: ∑_j=1^n sum
Use in source · Stratified_Universality_Traditional.tex, line 4917
[Exact coefficient recurrence] Let A(z)=1+∑_n≥1a_nz^n and let c_n be its Euler ghost coordinates. Then na_n = ∑_j=1^nc_j a_n-j (n≥1), with a_0=1. Conversely, any c∈ G_ Z and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.
Exact mathematical passage
\boxed{ na_n = \sum_{j=1}^{n}c_j a_{n-j} \qquad(n\geq1), }
\sum_{j_1,\ldots,j_m}Use in source · 1 passage
Search aliases: ∑_j_1,…,j_m sum ldots
Use in source · block11_section.tex, line 135
I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [u_j_1,…,u_j_m]. tag11.4
Exact mathematical passage
\boxed{ u_i = s_i+ \sum_{m=2}^{D} \sum_{j_1,\ldots,j_m} A_{m;i\mid j_1,\ldots,j_m} [u_{j_1},\ldots,u_{j_m}].} \tag{11.4}
\sum_{j\in\mathcal O_q}Use in source · 1 passage
Search aliases: ∑_j∈ O_q sum in mathcal
Use in source · block16_section_final.tex, line 613
ll=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
E_q:=\sum_{j\in\mathcal O_q}e_j
\sum_{j\geq i}Use in source · 1 passage
Search aliases: ∑_j≥ i sum geq
Use in source · block12_section.tex, line 275
The classical conductor formula uses codimV^G_i. Since P_i projects onto V^G_i, this codimension is Tr(I-P_i). Also codimV^G_i=∑_j≥ id_j. Reversing the finite sums gives the detail formula.
Exact mathematical passage
\operatorname{codim}V^{G_i}=\sum_{j\geq i}d_j.
\sum_{j\geq1}Use in source · 1 passage
Search aliases: ∑_j≥1 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 4930
The logarithmic-derivative identity is equivalent to zA'(z)=(∑_j≥1c_jz^j)A(z). Comparing coefficients gives the recurrence. Conversely, the recurrence uniquely determines the coefficient sequence over Q. Since c∈ G_ Z reconstructs an integral exponent sequence, the resulting series is the Euler product from [source reference] and therefore has integral coefficients.
Exact mathematical passage
zA'(z)=\left(\sum_{j\geq1}c_jz^j\right)A(z).
\sum_{k=1}^{n}Use in source · 1 passage
Search aliases: ∑_k=1^n sum
Use in source · block11_section.tex, line 242
[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.
Exact mathematical passage
N_{n,M}=\sum_{k=1}^{n}S(n,k)M^k.
\sum_{k\geq0}Use in source · 3 passages
Search aliases: ∑_k≥0 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 5560
[Affine Newton--Mahler transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion and put g_s(q)=f(Mq+s). Then g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k =∑_r=k^Mk+sc_rΘ_M,s(r,k). Every output coefficient depends on a finite source window.
Exact mathematical passage
g_s(q)=\sum_{k\geq0}d_{s,k}\binom qk,
\sum_{m=0}^{d_j}Use in source · 1 passage
Search aliases: ∑_m=0^d_j sum
Use in source · block09_section.tex, line 97
Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.
Exact mathematical passage
L = \sum_{j=0}^{r}q_j(z)D^j, \qquad q_j(z)=\sum_{m=0}^{d_j}q_{j,m}z^m.
\sum_{m=0}^{\min(d_j,n)}Use in source · 1 passage
Search aliases: ∑_m=0^min(d_j,n) sum min
Use in source · block09_section.tex, line 108
[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.
Exact mathematical passage
\boxed{ [z^n]LF = \sum_{j=0}^{r} \sum_{m=0}^{\min(d_j,n)} q_{j,m} (n-m+j)^{\underline{j}} a_{n-m+j},}
\sum_{m=2}^{D}Use in source · 5 passages
Search aliases: ∑_m=2^D sum
Use in source · block11_section.tex, line 25
Let A_m:X^m⟶ X, 2≤ m≤ D, be bounded symmetric multilinear maps, and let s∈ X. Consider the marked equation u=s+Φ(u), Φ(u)=∑_m=2^DA_m[u,…,u]. tag11.1 The word marked means that the source term, every multilinear operation, the packet system, and the selected convergence or filtration certificate are retained as part of the presentation. The fixed point u alone is not assumed to reconstruct those data.
Exact mathematical passage
\boxed{ u=s+\Phi(u), \qquad \Phi(u)=\sum_{m=2}^{D}A_m[u,\ldots,u].} \tag{11.1}
\sum_{m\mid n}Use in source · 1 passage
Search aliases: ∑_mmid n sum mid
Use in source · Stratified_Universality_Traditional.tex, line 4907
Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.
Exact mathematical passage
\sum_{m\mid n}m b_m=c_n
\sum_{m\in\mathcal A}Use in source · 1 passage
Search aliases: ∑_m∈ A sum in mathcal
Use in source · block11_section.tex, line 322
Consider the scalar marked equation u(z)=z+Φ(u(z)), Φ(y)=∑_m∈ Aw_my^m, tag11.8 where A⊆{2,3,…} is finite and every w_m is a positive integer. Its unique formal solution belongs to z Z[[z]] and counts plane rooted trees whose internal arities lie in A, weighted by the w_m.
Exact mathematical passage
\boxed{ u(z)=z+\Phi(u(z)), \qquad \Phi(y)=\sum_{m\in\mathcal A}w_my^m,} \tag{11.8}
\sum_{m\ge0}Use in source · 1 passage
Search aliases: ∑_m≥0 sum ge
Use in source · block15_section.tex, line 664
For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0
Exact mathematical passage
F(z)=\sum_{m\ge0}a_mz^m\in A[[z]]
\sum_{m\geq0}Use in source · 3 passages
Search aliases: ∑_m≥0 sum geq
Use in source · block09_section.tex, line 254
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\mathfrak M_{N,j}F(x) = \sum_{m\geq0}a_{Nm+j}x^m.
\sum_{m\geq1}Use in source · 1 passage
Search aliases: ∑_m≥1 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 4900
Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.
Exact mathematical passage
\frac{zA'}A = \sum_{m\geq1}b_m\frac{mz^m}{1-z^m} = \sum_{m\geq1}\sum_{r\geq1}m b_m z^{mr}.
\sum_{n=0}^{M}Use in source · 1 passage
Search aliases: ∑_n=0^M sum
Use in source · block09_section.tex, line 595
If λ=-M, expand (1-αω_N^rt)^M = ∑_n=0^M binom Mn(-α)^nω_N^rnt^n. The rotation matrix is a truncated discrete Fourier matrix. Its rank is the number of available residue classes, namely min(N,M+1).
Exact mathematical passage
(1-\alpha\omega_N^rt)^M = \sum_{n=0}^{M} \binom Mn(-\alpha)^n\omega_N^{rn}t^n.
\sum_{n\geq 0}Use in source · 1 passage
Search aliases: ∑_n≥ 0 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 2229
screte category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3, the period word 12 has frac1+2z…
Exact mathematical passage
\label{eq:periodic-ogf-unreduced} F_a(z)=\sum_{n\geq 0}a_nz^n = \frac{W_a(z)}{1-z^{T(a)}}.
\sum_{n\ge0}Use in source · 5 passages
Search aliases: ∑_n≥0 sum ge
Use in source · Stratified_Universality_Traditional.tex, line 2077
The recurrence-to-OGF bridge is explicit. From χ(x)=x^d-c_d-1x^d-1-⋯-c_0 set Q(z)=z^dχ(z^-1) =1-c_d-1z-⋯-c_0z^d. The numerator is the degree-<d truncation of Q(z)∑_n≥0a_nz^n. Conversely, clearing Q recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.
Exact mathematical passage
Q(z)\sum_{n\ge0}a_nz^n
\sum_{n\geq1}Use in source · 4 passages
Search aliases: ∑_n≥1 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 4845
[Finite-prefix triangularity] For each N, the coefficient prefix (a_1,…,a_N) of A(z)=1+∑_n≥1a_nz^n and the exponent prefix (b_1,…,b_N) determine one another by integral triangular recursions. No exponent b_m with m>N affects the coefficient prefix through degree N.
Exact mathematical passage
(a_1,\ldots,a_N) \quad\text{of}\quad A(z)=1+\sum_{n\geq1}a_nz^n
\sum_{q=0}^{N-1}Use in source · 2 passages
Search aliases: ∑_q=0^N-1 sum
Use in source · block09_section.tex, line 260
For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.
Exact mathematical passage
\boxed{ t^j\mathfrak M_{N,j}F(t^N) = \frac1N \sum_{q=0}^{N-1} \omega_N^{-jq}F(\omega_N^qt).}
\sum_{q\mid N}Use in source · 1 passage
Search aliases: ∑_qmid N sum mid
Use in source · block16_section_final.tex, line 617
are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q
Exact mathematical passage
E_qE_{q'}=\delta_{q,q'}E_q, \qquad \sum_{q\mid N}E_q=1.
\sum_rUse in source · 1 passage
Search aliases: ∑_r sum
Use in source · Stratified_Universality_Traditional.tex, line 2491
[Functional Mahler is not Newton-Mahler] The mask equation in this section is F(z)=R(z)F(z^p). It is a digit-section and automaton construction. It is not the Newton-Mahler interpolation expansion f(x)=∑_rc_rbinomxr. Both occur elsewhere in the Library, but this atlas uses only the functional-Mahler meaning.
Exact mathematical passage
f(x)=\sum_rc_r\binom{x}{r}.
\sum_{r,\gamma}Use in source · 2 passages
Search aliases: ∑_r,γ sum gamma
Use in source · block11_section.tex, line 195
G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.
Exact mathematical passage
\sum_{r,\gamma}r\mu_{r,\gamma}=n.
\sum_{r=0}^{e}Use in source · 1 passage
Search aliases: ∑_r=0^e sum
Use in source · Stratified_Universality_Traditional.tex, line 5055
Unique factorization writes every positive integer as a finite product of pairwise coprime prime powers, so multiplicativity gives the reconstruction formula. Conversely, an arbitrary family of local unit series defines a multiplicative function by the same finite product. For n=p^e, Dirichlet convolution becomes (fmathbin∗_Dg)(p^e) = ∑_r=0^ef(p^r)g(p^e-r), which is exactly the coefficient of T^e in the local Cauchy product.
Exact mathematical passage
(f\mathbin{\ast_D}g)(p^e) = \sum_{r=0}^{e}f(p^r)g(p^{e-r}),
\sum_{r=0}^{\lfloor n/N\rfloor}Use in source · 1 passage
Search aliases: ∑_r=0^lfloor n/Nrfloor sum lfloor rfloor
Use in source · block18_carrier_observer_change_final.tex, line 321
For positive z, f_N(n)=∑_r=0^lfloor n/Nrfloorbinom nNrz^r. The sequence is strictly increasing from index N-1 onward. Prefixes below N are distinguished by choosing the first suffix that crosses index N, and later prefixes differ already at suffix zero. Thus all residual functions are distinct.
Exact mathematical passage
f_N(n)=\sum_{r=0}^{\lfloor n/N\rfloor}\binom n{Nr}z^r.
\sum_{r=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_r=0^N-1 sum
Use in source · block10_section.tex, line 157
Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.
Exact mathematical passage
P\!\left(z,\sum_{j=0}^{N-1}z^jU_j\right) = \sum_{r=0}^{N-1}z^r\Phi_{N,r}(x,\mathbf U).
\sum_{r=1}^{m}Use in source · 1 passage
Search aliases: ∑_r=1^m sum
Use in source · block11_section.tex, line 77
The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.
Exact mathematical passage
A_m[x^m]-A_m[y^m] = \sum_{r=1}^{m} A_m[y^{r-1},x-y,x^{m-r}],
\sum_{r=k}^{Mk+s}Use in source · 1 passage
Search aliases: ∑_r=k^Mk+s sum
Use in source · Stratified_Universality_Traditional.tex, line 5564
[Affine Newton--Mahler transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion and put g_s(q)=f(Mq+s). Then g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k =∑_r=k^Mk+sc_rΘ_M,s(r,k). Every output coefficient depends on a finite source window.
Exact mathematical passage
\boxed{ d_{s,k} =\sum_{r=k}^{Mk+s}c_r\Theta_{M,s}(r,k).}
\sum_{rs\equiv k\, (N)}Use in source · 1 passage
Search aliases: ∑_rs≡ k (N) sum equiv
Use in source · Stratified_Universality_Traditional.tex, line 5149
[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.
Exact mathematical passage
\boxed{ P_{N,k}(f\mathbin{\ast_D}g) = \sum_{rs\equiv k\, (N)} (P_{N,r}f)\mathbin{\ast_D}(P_{N,s}g). }
\sum_{r\ge0}Use in source · 1 passage
Search aliases: ∑_r≥0 sum ge
Use in source · block11_section.tex, line 351
[Finite tree-address signature] Fix a packet width N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10
Exact mathematical passage
U_j(x)=\sum_{r\ge0}[z^{Nr+j}]u(z)\,x^r,
\sum_{r\geq0}Use in source · 6 passages
Search aliases: ∑_r≥0 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 2911
[Functional Mahler versus Newton-Mahler] The word ``Mahler'' continues to name two distinct constructions. This section concerns the functional equation F(z)=R(z)F(z^p). It does not identify rational masks with Newton-Mahler expansions f(x)=∑_r≥0c_rbinom xr or with p-adic analytic interpolation. Any bridge to that local analytic chart still requires its own passport and proof.
Exact mathematical passage
f(x)=\sum_{r\geq0}c_r\binom xr
\sum_{r\geq1}Use in source · 1 passage
Search aliases: ∑_r≥1 sum geq
Use in source · Stratified_Universality_Traditional.tex, line 4900
Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.
Exact mathematical passage
\frac{zA'}A = \sum_{m\geq1}b_m\frac{mz^m}{1-z^m} = \sum_{m\geq1}\sum_{r\geq1}m b_m z^{mr}.
\sum_{s=0}^{q}Use in source · 3 passages
Search aliases: ∑_s=0^q sum
Use in source · block09_section.tex, line 134
Conversely, suppose ∑_s=0^qp_s(n)a_n+s=0 (n≥ N), where p_s∈ K[n], and put θ=zD.
Exact mathematical passage
\sum_{s=0}^{q}p_s(n)a_{n+s}=0 \qquad(n\geq N),
\sum_{\substack{\mathcal Q\text{ a fine orbit}\\
\mathcal Q\subseteq(\rho^A_{sr})^{-1}(\mathcal O)}}Use in source · 1 passage
Search aliases: ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) sum substack mathcal text mathcal subseteq rho mathcal
Use in source · block16_section_final.tex, line 408
item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.
Exact mathematical passage
P^r_{A,\mathcal O} = \sum_{\substack{\mathcal Q\text{ a fine orbit}\\ \mathcal Q\subseteq(\rho^A_{sr})^{-1}(\mathcal O)}} P^s_{A,\mathcal Q}.
\sum_{\substack{0\le r_1,\ldots,r_q<N\\
r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}}Use in source · 1 passage
Search aliases: ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) sum substack le ldots cdots equiv mathrm
Use in source · block11_section.tex, line 505
[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,…
Exact mathematical passage
\boxed{ U_j(x) = \delta_{j,1} + \sum_{\substack{0\le r_1,\ldots,r_q<N\\ r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}} x^{(r_1+\cdots+r_q-j)/N} \prod_{a=1}^{q}U_{r_a}(x);} \tag{11.16}
\sum_{\substack{1\le a\le q\\\gcd(a,q)=1}}Use in source · 1 passage
Search aliases: ∑_substack1≤ a≤ q gcd(a,q)=1 sum substack le le gcd
Use in source · block16_section_final.tex, line 630
Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projectors (E_q)_qmid N form the finest Q-rational coarsening obtainable by summing the split Fourier idempotents, and Q[C_N]E_q≅ Q(ζ_q).
Exact mathematical passage
c_q(\ell) := \sum_{\substack{1\le a\le q\\\gcd(a,q)=1}} e^{2\pi i a\ell/q} = \sum_{d\mid\gcd(q,\ell)}d\,\mu(q/d) \in\mathbb Z
\sum_{\substack{d\mid n\\d<n}}Use in source · 1 passage
Search aliases: ∑_substackdmid n d<n sum substack mid d
Use in source · Stratified_Universality_Traditional.tex, line 5007
The coefficient equation at n is triangular: g(1)=1, g(n)=-∑_substackdmid n d<nf(n/d)g(d). Hence the inverse exists integrally and uniquely. The standard coprime-divisor factorization of a divisor of mn proves closure of multiplicative functions under convolution; uniqueness of the inverse then proves multiplicativity of f^-1_D.
Exact mathematical passage
g(1)=1, \qquad g(n)=-\sum_{\substack{d\mid n\\d<n}}f(n/d)g(d).
\sum_{\substack{e\in S_1\\\kappa(e)=t}}Use in source · 1 passage
Search aliases: ∑_substacke∈ S_1 κ(e)=t sum substack in kappa
Use in source · block15_section.tex, line 321
[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1 κ(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈Γ:overline b_t=0}. Then |S_0|-|supp(overline b)| = |A_0| +(|S_1|-|Γ|) +|K_0|. The three terms record, respectively, annihilation by the local maps τ_e…
Exact mathematical passage
\overline b_t := \sum_{\substack{e\in S_1\\\kappa(e)=t}}\tau_e(b_e) \in C_t,
\sum_{\substack{m\ge0\\m\equiv j\ (\mathrm{mod}\ N)}}Use in source · 1 passage
Search aliases: ∑_substackm≥0 m≡ j (mod N) sum substack ge m equiv mathrm
Use in source · block15_section.tex, line 665
For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0 m≡ j (mod N)a_mz^m, 0≤ j<N.
Exact mathematical passage
\mathcal P_{N,j}F(z) := \sum_{\substack{m\ge0\\m\equiv j\ (\mathrm{mod}\ N)}}a_mz^m, \qquad 0\le j<N.
\sum_{T\in\mathcal T_{\mathcal A}}Use in source · 1 passage
Search aliases: ∑_T∈ T_ A sum in mathcal mathcal
Use in source · block11_section.tex, line 57
[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applied to the ordered values of its children.
Exact mathematical passage
\boxed{ u=\sum_{T\in\mathcal T_{\mathcal A}}\operatorname{ev}(T),} \tag{11.2}
\sum_{u=0}^{s}Use in source · 1 passage
Search aliases: ∑_u=0^s sum
Use in source · Stratified_Universality_Traditional.tex, line 5773
[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{ d_{s,k} \equiv \sum_{u=0}^{s}\binom su c_{p^ak+u} \pmod p.}
\sum_{v}Use in source · 1 passage
Search aliases: ∑_v sum
Use in source · block11_section.tex, line 340
Define the arity period d_ A = gcd{m-1:m∈ A}. tag11.9 Every tree with internal vertices v has leaf count ℓ(T)=1+∑_v(deg(v)-1).
Exact mathematical passage
\ell(T)=1+\sum_{v}(\deg(v)-1).
\sum_{v_j-j=\mu}Use in source · 1 passage
Search aliases: ∑_v_j-j=μ sum mu
Use in source · block09_section.tex, line 401
Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
Exact mathematical passage
\boxed{ I_{\alpha,L}(\rho) = \sum_{v_j-j=\mu} q_{j,v_j}\rho^{\underline j}.}
\sum_{x\in K\backslash G/H}Use in source · 1 passage
Search aliases: ∑_x∈ Kbackslash G/H sum in backslash
Use in source · block12_section.tex, line 197
The dimension identity for the trivial H-representation is [G:H] = ∑_x∈ Kbackslash G/H [K:K∩ xHx^-1]. tag12.9 This is the finite packet-count law audited by the Galois-Tannakian atlas verifier.
Exact mathematical passage
\boxed{ [G:H] = \sum_{x\in K\backslash G/H} [K:K\cap xHx^{-1}].} \tag{12.9}
\sum_{\kappa(e')=t}Use in source · 1 passage
Search aliases: ∑_κ(e')=t sum kappa
Use in source · block15_section.tex, line 364
Injectivity of τ_e makes τ_e(b_e)≠0 for every e∈ S_0, so there is no annihilation. For strict selectors, suppose t∈ K_0 and choose e∈ S_1 with κ(e)=t. Applying Q_e to j_toverline b_t =∑_κ(e')=tj_tτ_e'(b_e')=0 leaves j_tτ_e(b_e)=0, contrary to e∈ S_1. Hence K_0=∅.
Exact mathematical passage
j_t\overline b_t =\sum_{\kappa(e')=t}j_t\tau_{e'}(b_{e'})=0
\sum_{\lambda\in\Lambda}Use in source · 1 passage
Search aliases: ∑_λ∈Λ sum lambda in Lambda
Use in source · Stratified_Universality_Traditional.tex, line 2161
Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [source reference].
Exact mathematical passage
a_n=\sum_{\lambda\in\Lambda}P_\lambda(n)\lambda^n.
\sum_{\mu}Use in source · 1 passage
Search aliases: ∑_μ sum mu
Use in source · block11_section.tex, line 206
[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.
Exact mathematical passage
\boxed{ \begin{aligned} &P_\delta D^n(F\circ G)(x)[h,\ldots,h]\\ &=\sum_{\mu} \frac{n!} {\displaystyle\prod_{r,\gamma} (r!)^{\mu_{r,\gamma}}\mu_{r,\gamma}!} \,P_\delta D^{k(\mu)}F(Gx) \bigl[ (g_{r,\gamma})^{\mu_{r,\gamma}} \bigr]_{r,\gamma}. \end{aligned}} \tag{11.5}
\sum_{\rho^Y_{vu}(b)=a}Use in source · 1 passage
Search aliases: ∑_ρ^Y_vu(b)=a sum rho
Use in source · block15_section.tex, line 152
item Projector-sum refinement. If ssucceq r on X and vsucceq u on Y, then p_Y,a^u f p_X,i^r = ∑_ρ^Y_vu(b)=a ∑_ρ^X_sr(j)=i p_Y,b^v f p_X,j^s. Coarsening a fine composite gives the same ambient morphism as composing at the coarse level.
Exact mathematical passage
p_{Y,a}^u f p_{X,i}^r = \sum_{\rho^Y_{vu}(b)=a} \sum_{\rho^X_{sr}(j)=i} p_{Y,b}^v f p_{X,j}^s.
\sum_{\sigma_{vu}(b)=a}Use in source · 2 passages
Search aliases: ∑_σ_vu(b)=a sum sigma
Use in source · Stratified_Universality_Traditional.tex, line 614
[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.
Exact mathematical passage
q^u_a=\sum_{\sigma_{vu}(b)=a}q^v_b
\sum_{\sigma\in G}Use in source · 1 passage
Search aliases: ∑_σ∈ G sum sigma in
Use in source · block12_section.tex, line 382
A useful constructive form is obtained by choosing any matrix C∈ M_n(L) and setting T(C)=∑_σ∈ Ga_σσ(C). Then a_ττ(T(C))=T(C). Whenever T(C) is invertible, B=T(C)^-1 is a Hilbert 90 trivialization. Standard descent guarantees that such a C exists.
Exact mathematical passage
T(C)=\sum_{\sigma\in G}a_\sigma\sigma(C).
\sum_{\ell=0}^{M-1}Use in source · 1 passage
Search aliases: ∑_ℓ=0^M-1 sum ell
Use in source · block15_section.tex, line 694
factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_N,j = ∑_ℓ=0^M-1 P_NM,j+ℓ N.
Exact mathematical passage
\mathcal P_{N,j} = \sum_{\ell=0}^{M-1}\mathcal P_{NM,j+\ell N}.
\sum_{\ell=0}^{N-1}Use in source · 3 passages
Search aliases: ∑_ℓ=0^N-1 sum ell
Use in source · block15_section.tex, line 703
Suppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( -
Exact mathematical passage
\mathcal P_{N,j}F(z) = \frac1N\sum_{\ell=0}^{N-1} \omega^{-j\ell}F(\omega^\ell z).
\intUse in source · 1 passage
Search aliases: ∫ int
Use in source · block12_section.tex, line 292
Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.
Exact mathematical passage
\boxed{ P_H=\int_HU_h\,d\mu_H(h).} \tag{12.14}
\int_HUse in source · 1 passage
Search aliases: ∫_H int
Use in source · block12_section.tex, line 292
Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.
Exact mathematical passage
\boxed{ P_H=\int_HU_h\,d\mu_H(h).} \tag{12.14}
\leUse in source · 20 passages
Search aliases: ≤ le
Use in source · Stratified_Universality_Traditional.tex, line 703
[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .
Exact mathematical passage
\|gf-g_Ff_E\| \le \|r_g\|\,\|f_E\| +\|g_F\|\,\|r_f\| +\|r_g\|\,\|r_f\|.
\geUse in source · 33 passages
Search aliases: ≥ ge
Use in source · Stratified_Universality_Traditional.tex, line 1966
Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with trivial holonomy on those diamonds.
Exact mathematical passage
d\ge0
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · Stratified_Universality_Traditional.tex, line 3676
[Regular rank theorem] Let a be k-regular over K. Then r_k(a) = rank_K H_a = min{dim_K W:W supports a digit-linear realization of a}. Moreover: item there exist finite word sets U,V⊂Σ_k^*, each of cardinality r_k(a), for which the minor H_a[U,V] is nonsingular; item every reachable-observable realization of dimension r_k(a) is isomorphic to C_k(a); item any two minimal reachable-observable realizations are related by a unique similarity intertwining all digit operators, the initial vector, and t…
Exact mathematical passage
U,V\subset\Sigma_k^*
\supseteqUse in source · 3 passages
Search aliases: ⊇ supseteq
Use in source · block12_section.tex, line 60
intheorem[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P_H≥ P_J. Thus the subgroup poset, the intermediate-field poset, and the faithful fixed-projector poset form three normalized charts of one finite semi…
Exact mathematical passage
H\leq J \quad\Longleftrightarrow\quad L^H\supseteq L^J \quad\Longleftrightarrow\quad P_H\geq P_J.
\bigoplus_{\mathcal O\in H\backslash(J\times I)}Use in source · 1 passage
Search aliases: ⊕_ O∈ Hbackslash(J× I) bigoplus mathcal in backslash times
Use in source · block16_section_final.tex, line 480
ion-orbital-calculus Let kk be a commutative ring, let a finite group H act on finite sets I,J,K, and let kk[I],kk[J],kk[K] be the corresponding permutation modules. For an orbit O⊆ J× I under the diagonal action, let B_ O:kk[I]→kk[J] be the matrix whose (j,i)-entry is one when (j,i)∈ O and zero otherwise. Then Hom_kk H(kk[I],kk[J]) = ⊕_ O∈ Hbackslash(J× I) kk B_ O. In particular, an equivariant matrix is exactly a matrix constant on every pair orbit.
Exact mathematical passage
\Hom_{\kk H}(\kk[I],\kk[J]) = \bigoplus_{\mathcal O\in H\backslash(J\times I)} \kk B_{\mathcal O}.
\bigoplus_aUse in source · 2 passages
Search aliases: ⊕_a bigoplus
Use in source · Stratified_Universality_Traditional.tex, line 1584
[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summands and D is block diagonal, this becomes coker D≅⊕_acoker D_a. Off-diagonal correction blocks can couple packet defects, so the block-diagonal hypoth…
Exact mathematical passage
\coker D \cong \bigoplus_a\overline Q_a(\coker D), \qquad \overline Q_a\Ob_D(r)=[Q_ar].
\bigoplus_{b\in I_{Y,v}}Use in source · 1 passage
Search aliases: ⊕_b∈ I_Y,v bigoplus in
Use in source · block15_section.tex, line 376
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathsf B^{v,s}(X,Y) := \bigoplus_{b\in I_{Y,v}} \bigoplus_{j\in I_{X,s}} \Hom_{\mathcal C}(X_j^s,Y_b^v),
\bigoplus_eUse in source · 1 passage
Search aliases: ⊕_e bigoplus
Use in source · block15_section.tex, line 450
Suppose a finite family of lane objects (B_e)_e∈ E, target objects (C_t)_t∈ T, and morphisms τ_e:B_e→ C_κ(e) is given. Let q_e:⊕_e'∈ EB_e'→ B_e be the coordinate projection and define Φ:=∑_e∈ Ej_κ(e)τ_eq_e: ⊕_e∈ EB_e⟶ ⊕_t∈ TC_t. Then there are canonical short exact sequences 0⟶kerΦ ⟶⊕_eB_e ⟶imΦ ⟶0 and 0⟶imΦ ⟶⊕_tC_t ⟶cokerΦ ⟶0. The first sequence is the exact fold-loss certificate. Suppose every τ_e is monic and there are endomorphisms Q_e∈End(⊕_tC_t) satisfying Q_ej_κ(e')τ_e' =δ_e,e'j_κ(e')τ_e' …
Exact mathematical passage
0\longrightarrow\ker\Phi \longrightarrow\bigoplus_eB_e \longrightarrow\im\Phi \longrightarrow0
\bigoplus_{e'\in E}Use in source · 1 passage
Search aliases: ⊕_e'∈ E bigoplus in
Use in source · block15_section.tex, line 442
[Abelian fold-loss exact sequence] Let A be an abelian category. Suppose a finite family of lane objects (B_e)_e∈ E, target objects (C_t)_t∈ T, and morphisms τ_e:B_e→ C_κ(e) is given. Let q_e:⊕_e'∈ EB_e'→ B_e be the coordinate projection and define Φ:=∑_e∈ Ej_κ(e)τ_eq_e: ⊕_e∈ EB_e⟶ ⊕_t∈ TC_t. Then there are canonical short exact sequences 0⟶kerΦ ⟶⊕_eB_e ⟶imΦ ⟶0 and 0⟶imΦ ⟶⊕_tC_t ⟶cokerΦ ⟶0. The first sequence is the exact fold-loss certificate. Suppose every τ_e is monic and there are endomorphi…
Exact mathematical passage
q_e:\bigoplus_{e'\in E}B_{e'}\to B_e
\bigoplus_{e\in E}Use in source · 3 passages
Search aliases: ⊕_e∈ E bigoplus in
Use in source · block15_section.tex, line 219
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
\Phi_{\kappa,\tau}: \bigoplus_{e\in E}B_e\longrightarrow C, \qquad (b_e)_e\longmapsto \sum_{e\in E}j_{\kappa(e)}\tau_e(b_e)
\bigoplus_{e\notin S}Use in source · 1 passage
Search aliases: ⊕_e∉ S bigoplus notin
Use in source · Stratified_Universality_Traditional.tex, line 1498
extup(roman*)] item evaluation at the root identifies the space of global parallel sections with C_o; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies; item for Δ_S:F_o⟶⊕_e∉ SF_o, Δ_Sx=((H_e-I)x)_e∉ S, one has C_o=kerΔ_S, dim C_o=dim F_o-rankΔ_S; item the full fibers glue to one…
Exact mathematical passage
\Delta_S:F_o\longrightarrow\bigoplus_{e\notin S}F_o, \qquad \Delta_Sx=((H_e-I)x)_{e\notin S},
\bigoplus_iUse in source · 5 passages
Search aliases: ⊕_i bigoplus
Use in source · Stratified_Universality_Traditional.tex, line 934
[Declared packets become literal objects] Every declared idempotent p:X→ X defines an object (X,p) of Syn_ak( K_rm lin). A complete packet family (p_i) gives the canonical isomorphism (X,1_X)≅⊕_i(X,p_i).
Exact mathematical passage
(X,1_X)\cong\bigoplus_i(X,p_i).
\oplus_iUse in source · 2 passages
Search aliases: ⊕_i oplus
Use in source · Stratified_Universality_Traditional.tex, line 452
[Exact packet-coordinate isomorphism, imported] For every finite packet system in an additive idempotent-complete category, S_ P A_ P=1_X, A_ P S_ P =1_⊕_iX_i. Consequently, X≅⊕_i∈ IX_i.
Exact mathematical passage
\boxed{ \Synthesis_{\mathcal P}\Analysis_{\mathcal P}=1_X, \qquad \Analysis_{\mathcal P}\Synthesis_{\mathcal P} =1_{\oplus_iX_i}.}
\bigoplus_{i=1}^{\rho}Use in source · 4 passages
Search aliases: ⊕_i=1^ρ bigoplus rho
Use in source · Stratified_Universality_Traditional.tex, line 4114
By the Smith normal form, there are uniquely determined integers 1≤ d_1(a)mid d_2(a)mid⋯mid d_ρ(a) such that C_k(a) ≅ ⊕_i=1^ρ Z/d_i(a) Z. Factors with d_i=1 are harmless but are retained so that the list has length ρ.
Exact mathematical passage
C_k(a) \cong \bigoplus_{i=1}^{\rho}\mathbb Z/d_i(a)\mathbb Z.
\bigoplus_{i\in\mathcal O_U}Use in source · 1 passage
Search aliases: ⊕_i∈ O_U bigoplus in mathcal
Use in source · block16_section_final.tex, line 430
(U and transport O along its address bijection. Two path choices differ by a loop at U_0, which preserves O setwise. Hence O_U is path independent. Projector compatibility gives θ^A_γ F_γ(P^r_A,U, O) = P^r_A,V, Oθ^A_γ, so the orbit sums descend. Their local splittings are the finite direct sums ⊕_i∈ O_UA^r_U,i; the direct sums of the β-maps, with the induced permutation of summands, provide their descent data. This proves (O1).
Exact mathematical passage
\bigoplus_{i\in\mathcal O_U}A^r_{U,i}
\bigoplus_{j\in I_{X,s}}Use in source · 1 passage
Search aliases: ⊕_j∈ I_X,s bigoplus in
Use in source · block15_section.tex, line 376
For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.
Exact mathematical passage
\mathsf B^{v,s}(X,Y) := \bigoplus_{b\in I_{Y,v}} \bigoplus_{j\in I_{X,s}} \Hom_{\mathcal C}(X_j^s,Y_b^v),
\bigoplus_tUse in source · 5 passages
Search aliases: ⊕_t bigoplus
Use in source · block15_section.tex, line 300
[Why the fold selectors live in one group] The endomorphisms Q_e in (L8) all act on the single group C=⊕_tC_t. Writing separate selectors on unrelated codomains would make the Kronecker relation ill-typed and would not prove injectivity of the combined fold.
Exact mathematical passage
C=\bigoplus_tC_t
\bigoplus_{t\in T}Use in source · 2 passages
Search aliases: ⊕_t∈ T bigoplus in
Use in source · block15_section.tex, line 217
item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact information-loss group kerΦ_κ,τ. In particular, two retained lane matrices have the same folded record exactly when their difference belongs to this ke…
Exact mathematical passage
C=\bigoplus_{t\in T}C_t
\bigoplus_{x\in K\backslash G/H}Use in source · 1 passage
Search aliases: ⊕_x∈ Kbackslash G/H bigoplus in backslash
Use in source · block12_section.tex, line 180
[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.
Exact mathematical passage
\boxed{ \operatorname{Res}_K^G\operatorname{Ind}_H^GW \cong \bigoplus_{x\in K\backslash G/H} \operatorname{Ind}_{K\cap xHx^{-1}}^K \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).} \tag{12.8}
\bigoplus_{X\in Q_0}Use in source · 1 passage
Search aliases: ⊕_X∈ Q_0 bigoplus in
Use in source · Stratified_Universality_Traditional.tex, line 379
A typed realization defines the action of a path by composition and then extends 𝕜-linearly. Every element of I acts by zero, so the action factors through Λ_ K. Conversely, for a unital module M, the pairwise orthogonal vertex idempotents give M=⊕_X∈ Q_0e_XM, and an arrow a:X→ Y maps e_XM into e_YM. These constructions are inverse up to their canonical natural isomorphisms. This is the typed-module theorem of Paper I [source reference].
Exact mathematical passage
M=\bigoplus_{X\in Q_0}e_XM,
\bigoplus_{\delta(s,b)=t}Use in source · 1 passage
Search aliases: ⊕_δ(s,b)=t bigoplus delta
Use in source · block18_carrier_observer_change_final.tex, line 187
ed-observer Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(s,b)=tL_b(a_s). If a finite family of raw histories has already been aggregated by their state using ⊕, then this update gives exactly the same result as advancing every raw histor…
Exact mathematical passage
a'_t := \bigoplus_{\delta(s,b)=t}L_b(a_s).
\bigoplus_{\chi\in\operatorname{Irr}_K(G)}Use in source · 1 passage
Search aliases: ⊕_χ∈Irr_K(G) bigoplus chi in operatorname
Use in source · block15_section.tex, line 1045
ation] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map is block diagonal in these isotypic lanes, while an arbitrary K-linear map has the full lane matrix of [source reference].
Exact mathematical passage
V=\bigoplus_{\chi\in\operatorname{Irr}_K(G)} e_\chi V.
\otimesUse in source · 11 passages
Search aliases: ⊗ otimes
Use in source · Stratified_Universality_Traditional.tex, line 1171
Reconstructed symmetry · Aut^⊗(ω) · tensor automorphisms of a fiber functor and the corresponding affine group scheme · faithful exact tensor fiber functor
Exact mathematical passage
\Aut^{\otimes}(\omega)
\otimes_{\mathbb Z}Use in source · 2 passages
Search aliases: ⊗_ Z otimes mathbb
Use in source · Stratified_Universality_Traditional.tex, line 4438
Define M_k,p(a)=M_k(a)⊗_ Z Z_p, L_k,p(a)=L_k(a)⊗_ Z Z_p.
Exact mathematical passage
M_{k,p}(a)=M_k(a)\otimes_{\mathbb Z}\mathbb Z_p, \qquad L_{k,p}(a)=L_k(a)\otimes_{\mathbb Z}\mathbb Z_p.
\otimes_KUse in source · 3 passages
Search aliases: ⊗_K otimes
Use in source · Stratified_Universality_Traditional.tex, line 1705
vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
W\otimes_KL\xrightarrow{\sim}L^n.
\bigcap_{a\in A}Use in source · 2 passages
Search aliases: ⋂_a∈ A bigcap in
Use in source · block17_semantic_quotients_final.tex, line 147
Assume M is generated by a family A. On congruences or submodules define Φ(R) := K_0∩⋂_a∈ Aρ(a)^-1(R), with the corresponding submodule formula in the linear setting.
Exact mathematical passage
\Phi(R) := K_0\cap\bigcap_{a\in A}\rho(a)^{-1}(R),
\bigcap_{e\notin S}Use in source · 1 passage
Search aliases: ⋂_e∉ S bigcap notin
Use in source · Stratified_Universality_Traditional.tex, line 1479
a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
C_o = \bigcap_{e\notin S}\ker(H_e-I) = \Fix\langle H_e:e\notin S\rangle, \qquad C_v=G_vC_o.
\bigcap_FUse in source · 1 passage
Search aliases: ⋂_F bigcap
Use in source · Stratified_Universality_Traditional.tex, line 770
[Common profile and composition] Let F: C→ D and G: D→ E. Every constructor preserved by both functors, with compatible chosen comparison data, is preserved by GF. For a finite diagram of realizations and bridges, the intersection Π_rm common:=⋂_FΠ_F is therefore a valid common profile for every proof transported through the whole diagram.
Exact mathematical passage
\Pi_{\rm common}:=\bigcap_F\Pi_F
\bigcap_{g\in G}Use in source · 2 passages
Search aliases: ⋂_g∈ G bigcap in
Use in source · block12_section.tex, line 136
If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.
Exact mathematical passage
\operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1}, \qquad \operatorname{Ncl}_G(H)=\left\langle gHg^{-1}:g\in G\right\rangle.
\bigcap_jUse in source · 2 passages
Search aliases: ⋂_j bigcap
Use in source · block17_semantic_quotients_final.tex, line 108
Let K_0:=⋂_jkerpair(o_j) in the relational setting and N_0:=⋂_jker(o_j) in the linear setting.
Exact mathematical passage
K_0:=\bigcap_j\kerpair(o_j)
\bigcap_{m,j}Use in source · 1 passage
Search aliases: ⋂_m,j bigcap
Use in source · block17_semantic_quotients_final.tex, line 92
In the relational setting, the kernel pair of q is contained in Θ_ S, because equality under q forces equality under every factorized future observation. The universal property of the quotient by Θ_ S gives the unique map. In the module setting, ker q⊆ ⋂_m,jker(o_jρ(m))=N_ S, so the ordinary quotient universal property gives the same conclusion.
Exact mathematical passage
\ker q\subseteq \bigcap_{m,j}\ker(o_j\rho(m))=N_{\mathsf S},
\bigcap_{q\in\mathcal Q}Use in source · 1 passage
Search aliases: ⋂_q∈ Q bigcap in mathcal
Use in source · block17_semantic_quotients_final.tex, line 198
Fix a universe mathscr Q(X) of admissible queries on X. For Q⊆mathscr Q(X), define Ind( Q) := ⋂_q∈ Qkerpair(q). For an equivalence relation or congruence R, define Def(R) := {q∈mathscr Q(X):R⊆kerpair(q)}.
Exact mathematical passage
\Ind(\mathcal Q) := \bigcap_{q\in\mathcal Q}\kerpair(q).
\bigcap_{w\in\Sigma_k^*}Use in source · 1 passage
Search aliases: ⋂_w∈Σ_k^* bigcap in Sigma
Use in source · Stratified_Universality_Traditional.tex, line 3627
[Reachability and observability] The canonical section realization C_k(a) is linearly reachable and observable: span_K{Λ_wa:w∈Σ_k^*} = V_k(a), and ⋂_w∈Σ_k^* ker(ϵ_0Λ_w) = {0}.
Exact mathematical passage
\bigcap_{w\in\Sigma_k^*} \ker(\epsilon_0\Lambda_w) = \{0\}.
\bigcupUse in source · 1 passage
Search aliases: ⋃ bigcup
Use in source · block18_carrier_observer_change_final.tex, line 342
ite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.
Exact mathematical passage
C'_t = \operatorname{conv} \bigcup_{\delta(s,b)=t}A_b^{\mathsf T}C_s.
\bigcup_{\delta(s,b)=t}Use in source · 1 passage
Search aliases: ⋃_δ(s,b)=t bigcup delta
Use in source · block18_carrier_observer_change_final.tex, line 342
ite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.
Exact mathematical passage
C'_t = \operatorname{conv} \bigcup_{\delta(s,b)=t}A_b^{\mathsf T}C_s.
\starUse in source · 6 passages
Search aliases: ⋆ star
Use in source · Stratified_Universality_Traditional.tex, line 1701
[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is effective [source reference].
Exact mathematical passage
W=(L^n)^{G,\star}
\langleUse in source · 8 passages
Search aliases: ⟨ langle
Use in source · Stratified_Universality_Traditional.tex, line 1479
a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
C_o = \bigcap_{e\notin S}\ker(H_e-I) = \Fix\langle H_e:e\notin S\rangle, \qquad C_v=G_vC_o.
\rangleUse in source · 8 passages
Search aliases: ⟩ rangle
Use in source · Stratified_Universality_Traditional.tex, line 1479
a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(H_e-I) = Fix⟨ H_e:e∉ S⟩, C_v=G_vC_o.
Exact mathematical passage
C_o = \bigcap_{e\notin S}\ker(H_e-I) = \Fix\langle H_e:e\notin S\rangle, \qquad C_v=G_vC_o.
\rangle_{\mathbb N}Use in source · 1 passage
Search aliases: ⟩_ N rangle mathbb
Use in source · block11_section.tex, line 373
Replacing one leaf by an m-ary vertex increases the leaf count by m-1. Starting from the one-leaf tree, every nonnegative sum of allowed increments is realized by a sequence of leaf replacements. Positivity of the weights prevents cancellation, so the coefficient support is exactly 1+⟨ m-1:m∈ A⟩_ N. Modulo N, the monoid generated by the increments equals the subgroup they generate because every element of a finite cyclic subgroup has a nonnegative representative. That subgroup is g_N Z/N Z, whic…
Exact mathematical passage
1+\left\langle m-1:m\in\mathcal A\right\rangle_{\mathbb N}.
\mathbb{F}Use in source · 3 passages
Search aliases: 𝔽 mathbb
Use in source · block13_periodic_examples.tex, line 10
𝔽_2/ word 10 · 2 · 2 · 2 · 3 · 2
Exact mathematical passage
\mathbb {F}_2
\mathbb{F}_2Use in source · 2 passages
Search aliases: 𝔽_2 mathbb
Use in source · block13_periodic_examples.tex, line 10
𝔽_2/ word 10 · 2 · 2 · 2 · 3 · 2
Exact mathematical passage
\mathbb {F}_2
\mathbb{F}_3Use in source · 1 passage
Search aliases: 𝔽_3 mathbb
Use in source · block13_periodic_examples.tex, line 12
𝔽_3/ word 12 · 2 · 1 · 1 · 2 · 1
Exact mathematical passage
\mathbb {F}_3
Source coverage
Automatic extraction covers TeX and Markdown math delimiters and common equation environments in these selected files. It does not interpret diagrams, images, PDF text, program identifiers, or historical archives. Occurrences are navigation aids, not a proof review.
Download all entries, passages and source checksums (JSON) · Collection coverage
- research/stratified-universality/README.md · 0 mathematical passages
- research/stratified-universality/traditional/Stratified_Universality_Traditional.tex · 1818 mathematical passages
- research/stratified-universality/traditional/appendix_claim_ledger_final.tex · 13 mathematical passages
- research/stratified-universality/traditional/appendix_open_program.tex · 5 mathematical passages
- research/stratified-universality/traditional/block09_section.tex · 245 mathematical passages
- research/stratified-universality/traditional/block10_section.tex · 186 mathematical passages
- research/stratified-universality/traditional/block11_section.tex · 160 mathematical passages
- research/stratified-universality/traditional/block12_section.tex · 217 mathematical passages
- research/stratified-universality/traditional/block13_section.tex · 146 mathematical passages
- research/stratified-universality/traditional/block14_section.tex · 5 mathematical passages
- research/stratified-universality/traditional/block15_section.tex · 416 mathematical passages
- research/stratified-universality/traditional/block16_section_final.tex · 200 mathematical passages
- research/stratified-universality/traditional/block17_semantic_quotients_final.tex · 140 mathematical passages
- research/stratified-universality/traditional/block18_carrier_observer_change_final.tex · 166 mathematical passages
- research/stratified-universality/traditional/block19_admissibility_lifting_final.tex · 99 mathematical passages
- research/stratified-universality/traditional/final_generated/compact_concordance.tex · 11 mathematical passages
- research/stratified-universality/traditional/final_generated/edition_ledger.tex · 2 mathematical passages
- research/stratified-universality/traditional/generated/block03_examples_table.tex · 12 mathematical passages
- research/stratified-universality/traditional/generated/block03_validation_table.tex · 2 mathematical passages
- research/stratified-universality/traditional/generated/block04_examples_table.tex · 18 mathematical passages
- research/stratified-universality/traditional/generated/block04_validation_table.tex · 6 mathematical passages
- research/stratified-universality/traditional/generated/block05_examples_table.tex · 8 mathematical passages
- research/stratified-universality/traditional/generated/block05_validation_table.tex · 4 mathematical passages
- research/stratified-universality/traditional/generated/block06_defect_profiles_table.tex · 13 mathematical passages
- research/stratified-universality/traditional/generated/block06_validation_table.tex · 1 mathematical passage
- research/stratified-universality/traditional/generated/block07_named_local_table.tex · 22 mathematical passages
- research/stratified-universality/traditional/generated/block07_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block08_examples_table.tex · 17 mathematical passages
- research/stratified-universality/traditional/generated/block08_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block09_examples_table.tex · 13 mathematical passages
- research/stratified-universality/traditional/generated/block09_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block10_examples_table.tex · 8 mathematical passages
- research/stratified-universality/traditional/generated/block10_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block11_faa_counts_table.tex · 1 mathematical passage
- research/stratified-universality/traditional/generated/block11_qary_phase_table.tex · 13 mathematical passages
- research/stratified-universality/traditional/generated/block11_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block12_group_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block12_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block13_bridge_matrix.tex · 37 mathematical passages
- research/stratified-universality/traditional/generated/block13_combined_validation_table.tex · 1 mathematical passage
- research/stratified-universality/traditional/generated/block13_component_table.tex · 4 mathematical passages
- research/stratified-universality/traditional/generated/block13_counts.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block13_periodic_examples.tex · 3 mathematical passages
- research/stratified-universality/traditional/generated/block13_qary_examples_table.tex · 15 mathematical passages
- research/stratified-universality/traditional/generated/block13_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block14_counts.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block14_development_ledger.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block14_scope_table.tex · 3 mathematical passages
- research/stratified-universality/traditional/generated/block14_theorem_suite_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block15_counts.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block15_publication_audit_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block15_publication_counts.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block15_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block16_counts.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block16_publication_audit_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block16_publication_counts.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block16_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block17_counts.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/block17_validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/domain_summary_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/formal_foundations_v2_publication_audit_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/proof_obligations.tex · 2 mathematical passages
- research/stratified-universality/traditional/generated/status_summary_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/generated/validation_table.tex · 0 mathematical passages
- research/stratified-universality/traditional/publication_conclusion.tex · 0 mathematical passages
- research/stratified-universality/traditional/publication_scope_section.tex · 0 mathematical passages
PDF reading editions (3)
The text index does not extract PDF glyphs. Check the reading edition for notation in figures, tables, and PDF-only material.