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ell","display":"\\rangle_\\ell^{\\,p^{\\ell-1}}","kind":"use","occurrences":[9973]},{"id":"symbol-0da92c8c15988310","tex":"\\mathbb{F}","aliases":"𝔽 mathbb","display":"\\mathbb{F}","kind":"use","occurrences":[1291]}],"contexts":["\\subsection*{The basic conceptual chain}\n\\ConceptChain{\\(\\begin{gathered}\n\\text{typed grammar}\\longrightarrow\\text{realized algebra}\\longrightarrow\\text{seed hull}\\\\[.45em]\n\\longrightarrow\\text{seed rank}\\longrightarrow\\text{packet coordinates}\\\\[.45em]\n\\longrightarrow\\text{transfer blocks}\\longrightarrow\\text{functorial realization}.\n\\end{gathered}\\)}","What can a finite seed family generate? & Evaluation map \\(A^{\\oplus r}\\to M\\) & The generated hull is a quotient of a finite free module.","How many seeds are required? & \\(\\srank_A(M)\\) & For a finitely generated module, pass to \\(M/\\Jac(A)M\\). When the quotient algebra is semisimple Artinian, compute its simple multiplicities.","How are vectors and operators resolved? & Complete idempotents \\((P_\\alpha)\\) & \\(X\\cong\\bigoplus_\\alpha P_\\alpha X\\) and \\(T\\leftrightarrow(P_\\beta TP_\\alpha)\\).","\\midrule\n\\endhead\n\\(\\mathfrak G=(Q,I)\\) & Typed grammar & A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.","\\(\\Lambda_{\\mathfrak G}\\) & Grammar algebra & The quotient path algebra \\(\\bbk Q/I\\).","\\(\\A_X\\) & Realized algebra & The image of the grammar algebra in \\(\\End_{\\bbk}(|X|)\\).","\\(S\\subseteq M\\) & Seed family & The vectors from which the operator orbit is generated.","\\(\\Hull_A(S)\\) & Generated hull & The submodule \\(AS\\), or its declared topological closure.","\\(\\srank_A(M)\\) & Seed rank & The smallest cardinality of a generating family. A finite value is the minimum number of seeds.","\\(\\Pcal=(P_\\alpha)\\) & Packet system & Pairwise annihilating idempotents summing to the identity.","\\(X_\\alpha\\) & Packet space & The range \\(P_\\alpha X\\).","\\(\\mathcal A_{\\Pcal}\\) & Analysis & The map \\(x\\mapsto(P_\\alpha x)_\\alpha\\).","\\(\\mathcal S_{\\Pcal}\\) & Synthesis & The map \\((x_\\alpha)\\mapsto\\sum_\\alpha x_\\alpha\\).","\\(T_{\\beta\\alpha}\\) & Transfer block & The source and target block \\(P_\\beta TP_\\alpha:X_\\alpha\\to X_\\beta\\).","\\(\\Supp_{\\Pcal}(x)\\) & Packet support & The packet indices on which \\(x\\) has a nonzero coordinate.","\\(\\psrank\\) & Rank for seeds confined to individual packets & The minimal number of generating seeds each supported in a single packet.","\\begin{microexample}[title={R1: one operator with several primary blocks}]\nLet \\(T\\) act on a finite dimensional vector space.  Regard \\(X\\) as a \\(\\bbk[t]\\)-module by \\(t\\cdot x=Tx\\).  If \\(T\\) has three Jordan blocks for the same eigenvalue, then three seeds are necessary and sufficient for that primary component.  The largest number of blocks over any irreducible polynomial is the seed rank.\n\\end{microexample}","\\begin{microexample}[title={R2: a matrix algebra with multiplicity}]\nLet \\(A=M_2(\\bbk)\\) act diagonally on \\(M=(\\bbk^2)^{\\oplus3}\\).  One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module.  Hence\n\\[\n\\srank_A(M)=\\left\\lceil\\frac{3}{2}\\right\\rceil=2.\n\\]\nThis is the simplest nontrivial instance of the semisimple multiplicity formula.\n\\end{microexample}","\\begin{microexample}[title={R3: an exact but oblique packet split}]\nOn \\(X=\\bbk^2\\), set\n\\[\nP_1=\\begin{pmatrix}1&-1","\\begin{microexample}[title={R3: an exact but oblique packet split}]\nOn \\(X=\\bbk^2\\), set\n\\[\nP_1=\\begin{pmatrix}1&-1\\\\0&0\\end{pmatrix},\n\\qquad\nP_2=\\begin{pmatrix}0&1\\\\0&1\\end{pmatrix}.\n\\]\nThen \\(P_i^2=P_i\\), \\(P_1P_2=P_2P_1=0\\), and \\(P_1+P_2=I\\).  The ranges form a direct sum, but over \\(\\mathbb R\\) the projectors are not orthogonal in the Euclidean inner product.  Exact reconstruction and numerical conditioning are therefore separate questions.\n\\end{microexample}","0&1\\end{pmatrix}.\n\\]\nThen \\(P_i^2=P_i\\), \\(P_1P_2=P_2P_1=0\\), and \\(P_1+P_2=I\\).  The ranges form a direct sum, but over \\(\\mathbb R\\) the projectors are not orthogonal in the Euclidean inner product.  Exact reconstruction and numerical conditioning are therefore separate questions.\n\\end{microexample}","\\begin{microexample}[title={R4: a directed packet system}]\nLet \\(X=X_1\\oplus X_2\\oplus X_3\\).  Suppose the generating operators permit transfers \\(1\\to2\\) and \\(2\\to3\\), with no path entering packet \\(1\\).  Packet \\(1\\) is the unique source component.  If the corner action on \\(X_1\\) is cyclic and the propagation hypotheses of \\cref{thm:source-component-rank} hold, one seed in \\(X_1\\) can generate all three packets.\n\\end{microexample}","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$.","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finit","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partial","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the th","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder cou","begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder counts strict steps.","vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder counts strict steps.","f continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder counts strict steps.","\\begin{definition}[Canonical theorem record]\n\\label{def:theorem-record}\nA canonical theorem record is the tuple\n\\[\n(\\mathrm{ID},\\mathrm{status},\\mathrm{hypotheses},\\mathrm{statement},\n\\mathrm{dependencies},\\mathrm{proof},\\mathrm{exports},\\mathrm{boundary}).\n\\]\nA later chapter cites the stable record ID.  A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.\n\\end{definition}\n\\RecordID{PSC-I-D-THEOREM-RECORD}","\\begin{definition}[Typed operator grammar]\n\\label{def:typed-grammar}\nLet $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","ion}[Typed operator grammar]\n\\label{def:typed-grammar}\nLet $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","r]\n\\label{def:typed-grammar}\nLet $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","k G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","l on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","\\begin{definition}[Realization on a single space]\n\\label{def:collapsed-realization}\nFor a typed realization $X$, define its realization on a single space by\n\\[\n|X|:=\\bigoplus_{v\\in Q_0}X_v.\n\\]\nEvery vertex idempotent acts as a coordinate projector on $|X|$, and every path acts as a block operator from its source space to its target space.\n\\end{definition}\n\\RecordID{PSC-I-D-COLLAPSED-REALIZATION}","\\begin{theorem}[Typed realization and module equivalence]\n\\label{thm:typed-module-equivalence}\nThe category of realizations of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","egin{theorem}[Typed realization and module equivalence]\n\\label{thm:typed-module-equivalence}\nThe category of realizations of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","ealizations of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","ns of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constr","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original verte","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}",", the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","ambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","n a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","fine the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","ts arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","ment of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","\\begin{microexample}[title={A grammar with two vertices}]\nTake two vertices and one arrow \\(a:1\\to2\\).  A realization is a pair of spaces and a linear map\n\\[\nX_1\\xrightarrow{\\,X_a\\,}X_2.\n\\]\nAfter collapse, \\(|X|=X_1\\oplus X_2\\), the vertex idempotents are the two coordinate projectors, and \\(a\\) is a single block outside the diagonal.  This elementary case already contains the viewpoint of packets and transfer blocks used later.\n\\end{microexample}","\\subsection{The realized action and faithful reduction}\n\\begin{definition}[Realized operator algebra]\n\\label{def:realized-algebra}\nFor a realization $X$, the realized operator algebra is\n\\[\n\\A_X:=\\im\\!\\left(\\Lambda_{\\mathfrak G}\\longrightarrow\\End_{\\bbk}(|X|)\\right).\n\\]\nIt records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.\n\\end{definition}\n\\RecordID{PSC-I-D-REALIZED-ALGEBRA}","\\begin{proposition}[Action factorization and hull invariance]\n\\label{prop:action-factorization}\nLet $S\\subseteq |X|$.  The generated submodule under the grammar algebra equals the generated submodule under the realized operator algebra:\n\\[\n\\Lambda_{\\mathfrak G}S=\\A_XS.\n\\]\nConsequently, every statement about seed rank depends only on the realized action, not on redundant relations in a chosen presentation of the grammar.\n\\end{proposition}\n\\RecordID{PSC-I-P-ACTION-FACTORIZATION}\n\\Depends{PSC-I-D-REALIZED-ALGEBRA.}","\\begin{proof}\nThe action map $\\rho:\\Lambda_{\\mathfrak G}\\twoheadrightarrow\\A_X$ is surjective by definition and satisfies $\\lambda s=\\rho(\\lambda)s$.  Hence every grammar orbit vector lies in $\\A_XS$.  The reverse inclusion holds because every element of $\\A_X$ is $\\rho(\\lambda)$ for some $\\lambda$.\n\\end{proof}","\\begin{keyidea}\nReplacing the abstract grammar algebra by its image \\(\\A_X\\) removes syntax that acts as zero.  This is a faithful reduction of the action.  It does not change any orbit, hull, or computation of seed rank.\n\\end{keyidea}","\\subsection{Seeds, addressed hulls, and natural presentations}\n\\begin{definition}[Linear seed system]\n\\label{def:seed-system}\nA linear seed system is a tuple\n\\[\n\\mathfrak S=(X,\\A,S,\\Pcal,\\tau),\n\\]\nwhere $X$ is a unital left module over $\\A$, $\\A$ is fixed independently of the target element, $S=(s_1,\\ldots,s_r)$ is a finite distinguished seed family, $\\Pcal$ is either empty or a declared packet system, and $\\tau$ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-SYSTEM}","{Seeds, addressed hulls, and natural presentations}\n\\begin{definition}[Linear seed system]\n\\label{def:seed-system}\nA linear seed system is a tuple\n\\[\n\\mathfrak S=(X,\\A,S,\\Pcal,\\tau),\n\\]\nwhere $X$ is a unital left module over $\\A$, $\\A$ is fixed independently of the target element, $S=(s_1,\\ldots,s_r)$ is a finite distinguished seed family, $\\Pcal$ is either empty or a declared packet system, and $\\tau$ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-SYSTEM}","\\begin{definition}[Generated hull]\n\\label{def:generated-hull}\nFor $S\\subseteq X$, define\n\\[\n\\Hull_{\\A}(S):=\\sum_{s\\in S}\\A s.\n\\]\nIn the topological regime define\n\\[\n\\Hull_{\\A,\\tau}(S):=\\cl{\\Hull_{\\A}(S)}^{\\,\\tau}.\n\\]\nFor a singleton $S=\\{s\\}$, write $\\Hull_{\\A}(s)$.\n\\end{definition}\n\\RecordID{PSC-I-D-GENERATED-HULL}","\\begin{proposition}[Algebra generation and module generation are distinct]\n\\label{prop:algebra-module-generation}\nEvery nonzero unital algebra $A$, regarded as a left module over itself, has\n\\[\n\\srank_A(A)=1,\n\\]\nwith cyclic seed $1_A$.  This conclusion is independent of the number of algebra generators required to present $A$ as an algebra.\n\\end{proposition}\n\\RecordID{PSC-I-P-ALGEBRA-MODULE-GENERATION}\n\\Depends{PSC-I-D-GENERATED-HULL.}","\\begin{proof}\nFor every $a\\in A$, one has $a=a1_A$.  Hence $A1_A=A$.  The number of operations needed to generate $A$ as an algebra is a different invariant and does not enter the module calculation.\n\\end{proof}","\\begin{boundary}[Operators chosen for a target]\n\\label{bnd:target-dependent-operators}\nFor a fixed seed $s$ and target $x$, the existence of a specially defined operator $A_x$ satisfying $A_xs=x$ does not establish a natural seed calculus unless $A_x$ belongs to a grammar that is fixed independently of the target and satisfies \\cref{def:natural-presentation}.\n\\end{boundary}\n\\RecordID{PSC-I-B-TARGET-DEPENDENT-OPERATORS}","tions by seeds and relations}\n\\begin{readerroadmap}\nA seed family gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","ns}\n\\begin{readerroadmap}\nA seed family gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","A seed family gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","nerated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","ed hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","\\begin{theorem}[Finite free presentation]\n\\label{thm:free-presentation}\nFor the map in \\cref{def:seed-evaluation},\n\\[\n\\im\\varepsilon_S=\\Hull_A(S),\n\\qquad\nN_S:=\\Ker\\varepsilon_S\\leq A^{\\oplus r},\n\\]\nand the induced map is an isomorphism of left modules over $A$\n\\[\n\\boxed{A^{\\oplus r}/N_S\\cong\\Hull_A(S).}\n\\]\nConsequently, $S$ generates $M$ if and only if $\\varepsilon_S$ is surjective.\n\\end{theorem}\n\\RecordID{PSC-I-T-FREE-PRESENTATION}\n\\Depends{PSC-I-D-SEED-EVALUATION and the first isomorphism theorem.}\n\\Exports{All later free seed presentations.}","\\begin{proof}\nThe image consists exactly of finite sums $\\sum_ja_js_j$, which is $\\Hull_A(S)$.  Because $\\varepsilon_S$ is linear over $A$, its kernel is a left submodule.  The first isomorphism theorem gives\n\\[\nA^{\\oplus r}/N_S\\cong\\im\\varepsilon_S=\\Hull_A(S).\n\\]\nThe final equivalence is the equality $\\im\\varepsilon_S=M$.\n\\end{proof}","\\begin{microexample}[title={A cyclic nilpotent chain}]\nLet \\(A=\\bbk[t]\\) and \\(M=\\bbk[t]/(t^3)\\).  The residue class \\(s=[1]\\) is one seed and\n\\[\n\\varepsilon_s:A\\to M,\n\\qquad\nf(t)\\mapsto f(t)[1]\n\\]\nhas kernel \\((t^3)\\).  Hence \\(M\\cong A/(t^3)\\), and the addressed vectors \\(s,ts,t^2s\\) are the visible orbit chain.\n\\end{microexample}","\\begin{corollary}[Characterization by minimal free covers]\n\\label{cor:minimal-free-cover}\nFor every finitely generated left module $M$ over $A$,\n\\[\n\\boxed{\n\\srank_A(M)=\\min\\{r\\geq0:\\text{there exists a surjection }A^{\\oplus r}\\twoheadrightarrow M\\}.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-MINIMAL-FREE-COVER}\n\\Depends{PSC-I-T-FREE-PRESENTATION.}","\\begin{proof}\nA seed family of size $r$ gives a surjection by \\cref{thm:free-presentation}.  Conversely, the images of the standard basis vectors under any surjection $A^{\\oplus r}\\twoheadrightarrow M$ form a generating family of size at most $r$.\n\\end{proof}","\\begin{corollary}[Presentation with one seed]\n\\label{cor:one-seed-presentation}\nFor $s\\in M$, the annihilator\n\\[\n\\operatorname{Ann}_A(s):=\\{a\\in A:as=0\\}\n\\]\nis a left ideal and\n\\[\n\\boxed{A/\\operatorname{Ann}_A(s)\\cong\\Hull_A(s).}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-ONE-SEED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION.}","\\begin{proof}\nThe evaluation map for one seed is $a\\mapsto as$.  Its kernel is $\\operatorname{Ann}_A(s)$, which is therefore a left ideal, and \\cref{thm:free-presentation} gives the quotient isomorphism.\n\\end{proof}","\\subsection{One seed and the distinction between module and algebra relations}\n\\begin{proposition}[Criterion for the annihilator to be an ideal on both sides]\n\\label{prop:two-sided-annihilator}\nFor $s\\in M$, the left ideal $L=\\operatorname{Ann}_A(s)$ is an ideal on both sides if and only if\n\\[\nL\\,\\Hull_A(s)=0.\n\\]\nIf $s$ is cyclic, these conditions are equivalent to\n\\[\n\\operatorname{Ann}_A(s)=\\operatorname{Ann}_A(M).\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-ANNIHILATOR-TWO-SIDED}\n\\Depends{PSC-I-C-ONE-SEED-PRESENTATION.}","\\begin{proof}\nThe left ideal $L$ is an ideal on both sides exactly when $ab\\in L$ for every $a\\in L$ and $b\\in A$.  This is equivalent to\n\\[\n0=(ab)s=a(bs)\n\\qquad(a\\in L,\\ b\\in A).\n\\]\nBy additivity, the last identity is equivalent to $L\\Hull_A(s)=0$.","If $s$ is cyclic, then $\\Hull_A(s)=M$.  Hence the preceding condition is $L\\subseteq\\operatorname{Ann}_A(M)$.  The reverse inclusion always holds because an operator annihilating all of $M$ annihilates $s$.  Thus equality follows.  Conversely, $\\operatorname{Ann}_A(M)$ is an ideal on both sides, so equality implies that $L$ is an ideal on both sides.\n\\end{proof}","\\begin{microexample}[title={Why a seed annihilator need not be an ideal on both sides}]\nLet \\(A=M_2(\\bbk)\\) act on \\(M=\\bbk^2\\), and let \\(s=e_1\\).  Then \\(\\operatorname{Ann}_A(s)\\) consists of matrices whose first column is zero.  It is a left ideal because left multiplication preserves a zero first column.  It is not generally a right ideal.  Thus the natural quotient for one seed is a quotient of the left regular module, not automatically a quotient algebra.\n\\end{microexample}","\\subsection{Topological completion of a presentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPL","\\subsection{Topological completion of a presentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of","\\subsection{Topological completion of a presentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","sentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","$X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","ausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","ically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","ilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and c","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\wid","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  The","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every de","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to ze","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in t","f}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\wideh…","f modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal prope…","$\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly bec","orted seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was","that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","tinuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","$X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","o X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","us and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","ypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","ause $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","\\begin{definition}[Seed rank]\n\\label{def:seed-rank}\nFor a unital left module $M$ over $A$, define\n\\[\n\\srank_A(M):=\\min\\{|S|:AS=M\\}.\n\\]\nFor a topological module $(X,\\tau)$, define\n\\[\n\\srank_{\\A,\\tau}(X):=\\min\\{|S|:\\cl{\\A S}^{\\,\\tau}=X\\}.\n\\]\nSet $\\srank_A(0)=0$.  When the value is finite, it is the minimal number of module generators.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-RANK}","\\begin{proposition}[Cyclic characterization]\n\\label{prop:cyclic-characterization}\nFor $M\\neq0$,\n\\[\n\\srank_A(M)=1\n\\quad\\Longleftrightarrow\\quad\nM\\text{ contains a cyclic vector}.\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-CYCLIC-CHARACTERIZATION}\n\\Depends{PSC-I-D-SEED-RANK.}","\\begin{proof}\nBoth conditions say that there is $m\\in M$ with $Am=M$.\n\\end{proof}","\\begin{proposition}[Monotonicity under enlargement of the acting algebra]\n\\label{prop:acting-algebra-monotonicity}\nLet $B\\subseteq A\\subseteq\\End_{\\bbk}(X)$ be unital operator algebras with the same identity.  Then\n\\[\n\\boxed{\\srank_A(X)\\leq\\srank_B(X).}\n\\]\nThe same statement holds for topological seed rank when both algebras act continuously.\n\\end{proposition}\n\\RecordID{PSC-I-P-ACTING-ALGEBRA-MONOTONICITY}\n\\Depends{PSC-I-D-SEED-RANK.}","\\begin{proof}\nFor every $S\\subseteq X$, one has $BS\\subseteq AS$.  Hence every generating family over $B$ is also a generating family over $A$.  Taking closures gives the topological statement.\n\\end{proof}","\\subsection{Quotients, isomorphisms, and exact sequences}\n\\begin{theorem}[Quotient monotonicity]\n\\label{thm:quotient-monotonicity}\nLet $\\varphi:A\\to B$ be a unital ring homomorphism and let\n\\[\n\\Phi:M\\twoheadrightarrow N\n\\]\nbe a surjective additive map satisfying\n\\[\n\\Phi(am)=\\varphi(a)\\Phi(m)\n\\qquad(a\\in A,\\ m\\in M).\n\\]\nThen\n\\[\n\\boxed{\\srank_B(N)\\leq\\srank_A(M).}\n\\]\nIf $M$ and $N$ are topological modules, $\\Phi$ is continuous and surjective, and the actions are continuous, then\n\\[\n\\srank_{B,\\tau_N}(N)\\leq\\srank_{A,\\tau_M}(M).\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-MONOTONICITY}\n\\Depends{PSC-I-D-GENERATED-HULL and PSC-I-D-SEED-RANK.}\n\\Exports{All quotient lower bounds and realization inequalities.}","on{Quotients, isomorphisms, and exact sequences}\n\\begin{theorem}[Quotient monotonicity]\n\\label{thm:quotient-monotonicity}\nLet $\\varphi:A\\to B$ be a unital ring homomorphism and let\n\\[\n\\Phi:M\\twoheadrightarrow N\n\\]\nbe a surjective additive map satisfying\n\\[\n\\Phi(am)=\\varphi(a)\\Phi(m)\n\\qquad(a\\in A,\\ m\\in M).\n\\]\nThen\n\\[\n\\boxed{\\srank_B(N)\\leq\\srank_A(M).}\n\\]\nIf $M$ and $N$ are topological modules, $\\Phi$ is continuous and surjective, and the actions are continuous, then\n\\[\n\\srank_{B,\\tau_N}(N)\\leq\\srank_{A,\\tau_M}(M).\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-MONOTONICITY}\n\\Depends{PSC-I-D-GENERATED-HULL and PSC-I-D-SEED-RANK.}\n\\Exports{All quotient lower bounds and realization inequalities.}","onotonicity]\n\\label{thm:quotient-monotonicity}\nLet $\\varphi:A\\to B$ be a unital ring homomorphism and let\n\\[\n\\Phi:M\\twoheadrightarrow N\n\\]\nbe a surjective additive map satisfying\n\\[\n\\Phi(am)=\\varphi(a)\\Phi(m)\n\\qquad(a\\in A,\\ m\\in M).\n\\]\nThen\n\\[\n\\boxed{\\srank_B(N)\\leq\\srank_A(M).}\n\\]\nIf $M$ and $N$ are topological modules, $\\Phi$ is continuous and surjective, and the actions are continuous, then\n\\[\n\\srank_{B,\\tau_N}(N)\\leq\\srank_{A,\\tau_M}(M).\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-MONOTONICITY}\n\\Depends{PSC-I-D-GENERATED-HULL and PSC-I-D-SEED-RANK.}\n\\Exports{All quotient lower bounds and realization inequalities.}","\\begin{proof}\nLet $S$ generate $M$.  For every $n\\in N$, choose $m\\in M$ with $\\Phi(m)=n$ and write $m=\\sum_ia_is_i$.  Then\n\\[\nn=\\sum_i\\varphi(a_i)\\Phi(s_i),\n\\]\nso $\\Phi(S)$ generates $N$.","For the topological statement, let $S$ generate $M$ densely.  Given $n=\\Phi(m)$, choose a net $m_\\lambda\\in AS$ converging to $m$.  Continuity gives\n\\[\n\\Phi(m_\\lambda)\\longrightarrow n,\n\\]\nand every $\\Phi(m_\\lambda)$ lies in $B\\Phi(S)$.  Thus $B\\Phi(S)$ is dense in $N$.\n\\end{proof}","\\begin{microexample}[title={A quotient can strictly lower seed rank}]\nLet \\(A=\\bbk\\) act by scalars on \\(M=\\bbk^3\\).  Then \\(\\srank_A(M)=3\\).  Quotienting by a subspace of dimension one gives a scalar module of dimension two, whose seed rank is \\(2\\).  Quotient monotonicity is therefore often strict.\n\\end{microexample}","\\begin{theorem}[Short exact sequence bounds]\n\\label{thm:exact-sequence-bounds}\nLet\n\\[\n0\\longrightarrow L\\xrightarrow{\\iota}M\\xrightarrow{\\pi}N\\longrightarrow0\n\\]\nbe a short exact sequence of left modules over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(N)\\leq\\srank_A(M)\n\\leq\\srank_A(L)+\\srank_A(N).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-EXACT-SEQUENCE-BOUNDS}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nThe lower inequality is quotient monotonicity. For the upper inequality, choose generating families $(\\ell_i)_{i\\in I}$ of $L$ and $(n_j)_{j\\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","f}\nThe lower inequality is quotient monotonicity. For the upper inequality, choose generating families $(\\ell_i)_{i\\in I}$ of $L$ and $(n_j)_{j\\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","e upper inequality, choose generating families $(\\ell_i)_{i\\in I}$ of $L$ and $(n_j)_{j\\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","st possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","e a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","\\begin{hypothesisguide}\nFor a short exact sequence, a generating family for \\(M\\) always maps to a generating family for the quotient.  It need not generate the submodule \\(L\\).  For the upper bound, choose generators of \\(L\\) separately and combine them with lifts of quotient generators.\n\\end{hypothesisguide}","\\subsection{Direct sums under shared and independent actions}\n\\begin{proposition}[Direct sum bounds for a shared action]\n\\label{prop:shared-direct-sum}\nIf one algebra $A$ acts diagonally on $M\\oplus N$, then\n\\[\n\\boxed{\n\\max\\{\\srank_A(M),\\srank_A(N)\\}\n\\leq\\srank_A(M\\oplus N)\n\\leq\\srank_A(M)+\\srank_A(N).\n}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-SHARED-DIRECT-SUM}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nThe coordinate projections are intertwining quotients and give the lower bound.  For the upper bound, embed minimal generating families of $M$ as $(m_i,0)$ and of $N$ as $(0,n_j)$. Their union generates by finite sums and has cardinality at most the sum of the two seed ranks.\n\\end{proof}","\\begin{theorem}[Direct sum formula for independent actions]\n\\label{thm:independent-direct-sum}\nLet $M$ be a left module over $A$ and $N$ a left module over $B$.  Under the action of the product algebra\n\\[\n(a,b)(m,n)=(am,bn),\n\\]\none has\n\\[\n\\boxed{\n\\srank_{A\\times B}(M\\oplus N)\n=\n\\max\\{\\srank_A(M),\\srank_B(N)\\}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-INDEPENDENT-DIRECT-SUM}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nThe coordinate projections give the lower bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{pro","\\begin{proof}\nThe coordinate projections give the lower bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","dinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","inimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","ros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","\\subsection{Tensor generation and witnesses for lower bounds}\n\\begin{proposition}[Tensor product bound]\n\\label{prop:tensor-bound}\nLet $A$ and $B$ be unital algebras over $\\bbk$, let $M$ be a left module over $A$, and let $N$ be a left module over $B$.  Under the natural $A\\otimes_{\\bbk}B$ action,\n\\[\n\\boxed{\n\\srank_{A\\otimes B}(M\\otimes_{\\bbk}N)\n\\leq\n\\srank_A(M)\\srank_B(N).\n}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-TENSOR-BOUND}\n\\Depends{PSC-I-D-SEED-RANK.}","\\begin{proof}\nChoose minimal generating families $(m_i)_{i\\in I}$ for $M$ and $(n_j)_{j\\in J}$ for $N$. Each vector in either module is a finite linear combination of its generators. Bilinearity therefore expresses every simple tensor as a finite sum of terms\n\\[\n(am_i)\\otimes(bn_j)=(a\\otimes b)(m_i\\otimes n_j).\n\\]\nSimple tensors span the algebraic tensor product, so the family $(m_i\\otimes n_j)_{(i,j)\\in I\\times J}$ generates it. This family has cardinality at most $|I||J|$, which is the claimed bound. The proof also covers an empty generating family, when the corresponding module and tensor product are zero.\n\\end{proof}","M$ and $(n_j)_{j\\in J}$ for $N$. Each vector in either module is a finite linear combination of its generators. Bilinearity therefore expresses every simple tensor as a finite sum of terms\n\\[\n(am_i)\\otimes(bn_j)=(a\\otimes b)(m_i\\otimes n_j).\n\\]\nSimple tensors span the algebraic tensor product, so the family $(m_i\\otimes n_j)_{(i,j)\\in I\\times J}$ generates it. This family has cardinality at most $|I||J|$, which is the claimed bound. The proof also covers an empty generating family, when the corresponding module and tensor product are zero.\n\\end{proof}","\\begin{proposition}[Monotonicity under extension of scalars]\n\\label{prop:scalar-extension}\nLet \\(K/\\bbk\\) be a field extension, let \\(A\\) be a unital algebra over \\(\\bbk\\), and let \\(M\\) be a finitely generated left module over \\(A\\).  Set\n\\[\nA_K:=K\\otimes_{\\bbk}A,\n\\qquad\nM_K:=K\\otimes_{\\bbk}M.\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{A_K}(M_K)\\leq\\srank_A(M).\n}\n\\]\nEvery generating family \\(S\\subseteq M\\) extends to the generating family \\(1\\otimes S\\subseteq M_K\\).\n\\end{proposition}\n\\RecordID{PSC-I-P-SCALAR-EXTENSION-MONOTONICITY}\n\\Depends{PSC-I-D-SEED-RANK.}\n\\Exports{Extension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.}","ty under extension of scalars]\n\\label{prop:scalar-extension}\nLet \\(K/\\bbk\\) be a field extension, let \\(A\\) be a unital algebra over \\(\\bbk\\), and let \\(M\\) be a finitely generated left module over \\(A\\).  Set\n\\[\nA_K:=K\\otimes_{\\bbk}A,\n\\qquad\nM_K:=K\\otimes_{\\bbk}M.\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{A_K}(M_K)\\leq\\srank_A(M).\n}\n\\]\nEvery generating family \\(S\\subseteq M\\) extends to the generating family \\(1\\otimes S\\subseteq M_K\\).\n\\end{proposition}\n\\RecordID{PSC-I-P-SCALAR-EXTENSION-MONOTONICITY}\n\\Depends{PSC-I-D-SEED-RANK.}\n\\Exports{Extension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.}","\\begin{proof}\nLet \\(S=\\{s_1,\\ldots,s_r\\}\\) generate \\(M\\).  Every pure tensor \\(\\lambda\\otimes m\\in M_K\\) has an expression\n\\[\nm=\\sum_{i=1}^r a_is_i,\n\\]\nso\n\\[\n\\lambda\\otimes m\n=\n\\sum_{i=1}^r(\\lambda\\otimes a_i)(1\\otimes s_i).\n\\]\nPure tensors span \\(M_K\\), hence \\(1\\otimes S\\) generates \\(M_K\\) over \\(A_K\\).  Taking the minimum over \\(S\\) proves the inequality.\n\\end{proof}","\\subsection{Quotient witnesses for lower bounds}\n\\begin{definition}[Witness for seed rank]\n\\label{def:seed-rank-witness}\nA surjective intertwiner $\\Phi:M\\twoheadrightarrow N$ is a witness for seed rank of order $r$ when $\\srank(N)=r$.  It certifies $\\srank(M)\\geq r$.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-RANK-WITNESS}","\\begin{proposition}[Scalar quotient witness]\n\\label{prop:scalar-quotient}\nLet $M$ be a module over $A$ with coefficient field $\\bbk$.  Suppose $M$ has a quotient $N$ of dimension $r$ on which $A$ acts through a character\n\\[\n\\chi:A\\longrightarrow\\bbk.\n\\]\nThen\n\\[\n\\boxed{\\srank_A(M)\\geq r.}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-SCALAR-QUOTIENT-WITNESS}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nFor every $n\\in N$, the cyclic submodule $An$ is contained in the scalar span $\\bbk n$.  Therefore $q$ seed images span a subspace of dimension at most $q$.  Generating the quotient of dimension $r$ requires at least $r$ seeds.  Quotient monotonicity transfers the lower bound to $M$.\n\\end{proof}","\\begin{classicalinput}[Nakayama lemma]\n\\label{ci:nakayama}\nLet $A$ be a unital ring, let $J\\subseteq\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  If $M=N+JM$ for a submodule $N\\leq M$, then $M=N$.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-NAKAYAMA}","\\begin{theorem}[Generator number is detected modulo the Jacobson radical]\n\\label{thm:module-top}\nLet $A$ be a unital ring, let $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  A family $S\\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$.  When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top.  No semisimplicity of $A/J$ is required for the general equality.  Consequently,\n\\[\n\\boxed{\n\\srank_A(M)=\\srank_{A/J}(M/JM).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-MODULE-TOP}\n\\Depends{PSC-I-CI-NAKAYAMA and PSC-I-T-QUOTIENT-MONOTONICITY.}\n\\Exports{Generator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modules.}","n{theorem}[Generator number is detected modulo the Jacobson radical]\n\\label{thm:module-top}\nLet $A$ be a unital ring, let $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  A family $S\\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$.  When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top.  No semisimplicity of $A/J$ is required for the general equality.  Consequently,\n\\[\n\\boxed{\n\\srank_A(M)=\\srank_{A/J}(M/JM).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-MODULE-TOP}\n\\Depends{PSC-I-CI-NAKAYAMA and PSC-I-T-QUOTIENT-MONOTONICITY.}\n\\Exports{Generator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modules.}","radical]\n\\label{thm:module-top}\nLet $A$ be a unital ring, let $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  A family $S\\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$.  When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top.  No semisimplicity of $A/J$ is required for the general equality.  Consequently,\n\\[\n\\boxed{\n\\srank_A(M)=\\srank_{A/J}(M/JM).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-MODULE-TOP}\n\\Depends{PSC-I-CI-NAKAYAMA and PSC-I-T-QUOTIENT-MONOTONICITY.}\n\\Exports{Generator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modules.}","\\begin{proof}\nIf $S$ generates $M$, its image generates every quotient.  Conversely, suppose the image of $S$ generates $M/JM$ and put $N=AS$.  Then\n\\[\nM=N+JM.\n\\]\nNakayama's lemma gives $M=N$.  Thus the same finite families generate before and after passage to the top, and the minimal cardinalities agree.\n\\end{proof}","\\begin{interpretation}\nThe theorem says that nilpotent or radical directions can lengthen orbit chains but cannot reduce the number of independent directions in the module top that must be seeded.  Once the images of the seeds generate \\(M/JM\\), Nakayama's lemma forces the entire module to be generated.\n\\end{interpretation}","\\begin{microexample}[title={A local nilpotent algebra}]\nLet \\(A=\\bbk[\\varepsilon]/(\\varepsilon^3)\\) and \\(M=A^{\\oplus2}\\).  The radical is \\(J=(\\varepsilon)\\), and\n\\[\nM/JM\\cong\\bbk^2.\n\\]\nTherefore two seeds are necessary and sufficient.  The extra nilpotent layers \\(\\varepsilon M\\) and \\(\\varepsilon^2M\\) are generated automatically once the two top directions are seeded.\n\\end{microexample}","\\begin{corollary}[Local algebra formula]\n\\label{cor:local-algebra-formula}\nLet $A$ be a local ring with Jacobson radical $J$ and residue division ring $D=A/J$.  For every finitely generated left module $M$ over $A$,\n\\[\n\\boxed{\n\\srank_A(M)=\\dim_D(M/JM),\n}\n\\]\nwhere the dimension is that of a left vector space over $D$.\n\\end{corollary}\n\\RecordID{PSC-I-C-LOCAL-ALGEBRA-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP.}","\\begin{proof}\nThe module $M/JM$ over $A/J$ is a left vector space over $D$.  Its minimal number of generators is its dimension over $D$.\n\\end{proof}","\\subsection{Semisimple multiplicity calculus}\n\\begin{classicalinput}[Wedderburn and Artin decomposition]\n\\label{ci:wedderburn}\nEvery semisimple Artinian ring is isomorphic to a finite product\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nof full matrix rings over division rings.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-WEDDERBURN-ARTIN}","\\begin{lemma}[Multiplicities in the regular module]\n\\label{lem:regular-multiplicities}\nLet\n\\[\nA=\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nand let $S_i=D_i^{d_i}$ denote the natural simple left module for the $i$th factor, extended by zero across the other factors.  Then, as a left module over $A$,\n\\[\n\\boxed{A\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus d_i}.}\n\\]\n\\end{lemma}\n\\RecordID{PSC-I-L-REGULAR-MODULE-MULTIPLICITIES}\n\\Depends{PSC-I-CI-WEDDERBURN-ARTIN.}","\\begin{proof}\nFor one factor $M_d(D)$, let $e_{jj}$ be the diagonal matrix units.  The left regular module decomposes as\n\\[\nM_d(D)=\\bigoplus_{j=1}^{d}M_d(D)e_{jj}.\n\\]\nEach minimal left ideal $M_d(D)e_{jj}$ is isomorphic to the natural column module $D^d$.  Taking the direct product of factors gives the stated decomposition.\n\\end{proof}","\\begin{theorem}[Semisimple seed rank formula over arbitrary division rings]\n\\label{thm:semisimple-seed-rank}\nLet\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","l{thm:semisimple-seed-rank}\nLet\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","eed-rank}\nLet\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","ng\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","\\begin{proof}\nBy \\cref{cor:minimal-free-cover}, $M$ is generated by $r$ elements exactly when it is a quotient of $A^{\\oplus r}$.  By \\cref{lem:regular-multiplicities},\n\\[\nA^{\\oplus r}\\cong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","\\cref{cor:minimal-free-cover}, $M$ is generated by $r$ elements exactly when it is a quotient of $A^{\\oplus r}$.  By \\cref{lem:regular-multiplicities},\n\\[\nA^{\\oplus r}\\cong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","\\[\nA^{\\oplus r}\\cong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","ong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","\\begin{interpretation}\nFor the factor \\(M_{d_i}(D_i)\\), one free generator contributes \\(d_i\\) copies of the corresponding simple module.  If the top of \\(M\\) contains \\(m_i\\) copies, at least \\(\\lceil m_i/d_i\\rceil\\) free generators are required.  The worst simple type controls the global answer.","\\begin{microexample}[title={Multiplicity consumes seed capacity}]\nFor \\(A=M_2(\\bbk)\\) acting on \\((\\bbk^2)^{\\oplus3}\\), the regular module contributes two copies of the natural simple module per generator.  One generator cannot cover all three multiplicity directions.  Two can, giving seed rank \\(2\\).\n\\end{microexample}","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnot","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-C","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEM","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","la]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","on over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","lus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\begin{proof}\nBy \\cref{thm:module-top}, $\\srank_A(M)=\\srank_{A/J}(M/JM)$.  Apply \\cref{thm:semisimple-seed-rank} to the displayed decompositions of $A/J$ and $M/JM$, with every $D_i=\\bbk$.  This gives the stated maximum.  If $q=0$, the unital $A/J$-module $M/JM$ is zero, and \\cref{thm:module-top} gives seed rank zero.\n\\end{proof}","\\begin{corollary}[Full matrix algebra]\n\\label{cor:full-matrix}\nFor the natural module $S=D^d$ over $M_d(D)$,\n\\[\n\\srank_{M_d(D)}(S)=1.\n\\]\nMore strongly, every nonzero vector of $S$ is cyclic.\n\\end{corollary}\n\\RecordID{PSC-I-C-FULL-MATRIX}\n\\Depends{PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\begin{proof}\nThe formula gives $\\lceil1/d\\rceil=1$.  Directly, let $v\\neq0$ and choose a coordinate $v_j\\neq0$.  For any target $w=(w_i)$, the matrix whose $j$th column has entries $w_iv_j^{-1}$ and whose other columns vanish sends $v$ to $w$.\n\\end{proof}","\\begin{corollary}[Multiplicity obstruction]\n\\label{cor:multiplicity-obstruction}\nFor the diagonal action of $M_d(D)$ on $S^{\\oplus m}$,\n\\[\n\\boxed{\n\\srank_{M_d(D)}(S^{\\oplus m})\n=\n\\left\\lceil\\frac{m}{d}\\right\\rceil.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-MULTIPLICITY-OBSTRUCTION}\n\\Depends{PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\begin{proof}\nApply \\cref{thm:semisimple-seed-rank} to the single simple type with multiplicity $m$.\n\\end{proof}","\\begin{sectionrecap}\nFor finitely generated modules, the minimal number of seeds is detected on the quotient by the Jacobson radical.  In the semisimple top, each generator has a fixed multiplicity capacity \\(d_i\\) in simple type \\(i\\), and the largest capacity ratio determines the exact seed rank.\n\\end{sectionrecap}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{04_semisimple_seed_rank.pdf}\n\\caption{The semisimple multiplicity formula for one simple block. A seed can expose at most $d$ independent multiplicity directions, so $m$ copies require $\\lceil m/d\\rceil$ seeds.}\n\\label{fig:semisimple}\n\\end{figure}","\\begin{readerroadmap}\nOne operator is equivalent to a module over \\(\\bbk[t]\\).  Primary decomposition isolates the irreducible polynomial sectors, and the quotient by \\(p(T)\\) counts how many cyclic chains begin in each sector.\n\\end{readerroadmap}\n\\subsection{Primary components and generator counts}","\\begin{classicalinput}[Primary decomposition for polynomial modules]\n\\label{ci:primary-decomposition}\nEvery module $X$ over $\\bbk[t]$ with finite dimension over the coefficient field decomposes as\n\\[\nX=\\bigoplus_{p\\in\\Irr(\\bbk[t])}X_p,\n\\]\nwhere $\\Irr(\\bbk[t])$ denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero $X_p$ is a finite direct sum of modules $\\bbk[t]/(p^e)$.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-PID-PRIMARY-DECOMPOSITION}","{classicalinput}[Primary decomposition for polynomial modules]\n\\label{ci:primary-decomposition}\nEvery module $X$ over $\\bbk[t]$ with finite dimension over the coefficient field decomposes as\n\\[\nX=\\bigoplus_{p\\in\\Irr(\\bbk[t])}X_p,\n\\]\nwhere $\\Irr(\\bbk[t])$ denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero $X_p$ is a finite direct sum of modules $\\bbk[t]/(p^e)$.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-PID-PRIMARY-DECOMPOSITION}","\\begin{theorem}[Seed rank for a single operator]\n\\label{thm:single-operator}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subs","\\begin{theorem}[Seed rank for a single operator]\n\\label{thm:single-operator}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","d rank for a single operator]\n\\label{thm:single-operator}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","\\begin{proof}\nLet the minimal polynomial be\n\\[\nm_T(t)=\\prod_{p}p(t)^{e_p}.\n\\]\nThe acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the el","\\begin{proof}\nLet the minimal polynomial be\n\\[\nm_T(t)=\\prod_{p}p(t)^{e_p}.\n\\]\nThe acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","the minimal polynomial be\n\\[\nm_T(t)=\\prod_{p}p(t)^{e_p}.\n\\]\nThe acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","The acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","d annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","ent.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","\\begin{interpretation}\nFor one operator, the quotient \\(X_p/p(T)X_p\\) counts the number of independent primary chains for the irreducible polynomial \\(p\\).  Over an algebraic closure these become Jordan chains.  The seed rank is the largest chain count among the primary sectors, not the largest chain length.\n\\end{interpretation}","\\begin{microexample}[title={Block count versus block size}]\nA nilpotent operator with Jordan block sizes \\((5,1,1)\\) has seed rank \\(3\\), even though its largest block has length \\(5\\).  Three seeds are needed because there are three independent chains.  Once their initial vectors are chosen, repeated application of \\(T\\) generates every position in those chains.\n\\end{microexample}","\\subsection{Invariant factors, Jordan blocks, and cyclicity}\n\\begin{corollary}[Invariant factor count]\n\\label{cor:invariant-factor-count}\nIf $X\\neq0$ and the rational canonical decomposition is written with nonconstant invariant factors\n\\[\nX\\cong\\bigoplus_{j=1}^{r}\\bbk[t]/(f_j),\n\\qquad\nf_1\\mid f_2\\mid\\cdots\\mid f_r,\n\\]\nthen\n\\[\n\\boxed{\\srank_{\\bbk[T]}(X)=r.}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-INVARIANT-FACTOR-COUNT}\n\\Depends{PSC-I-T-SINGLE-OPERATOR.}","\\begin{proof}\nFor each irreducible $p$, the number of invariant factors divisible by $p$ equals the number of elementary divisors for $p$.  Because $f_1\\mid\\cdots\\mid f_r$ and $f_1\\neq1$ after zero summands are removed, some irreducible divisor of $f_1$ divides all $r$ invariant factors.  Hence $\\max_p\\mu_p(T)=r$.\n\\end{proof}","\\begin{corollary}[Formula for geometric multiplicity]\n\\label{cor:geometric-multiplicity}\nIf $\\bbk$ is algebraically closed, then\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)\n=\n\\max_{\\lambda\\in\\operatorname{Spec}(T)}\n\\dim_{\\bbk}\\Ker(T-\\lambda I).\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-SINGLE-OPERATOR}\n\\Depends{PSC-I-T-SINGLE-OPERATOR.}","\\begin{proof}\nThe irreducibles are $p=t-\\lambda$.  On the primary component for $\\lambda$, the quotient by $(T-\\lambda I)X_\\lambda$ has dimension equal to the number of Jordan blocks, which is also $\\dim\\Ker(T-\\lambda I)$.\n\\end{proof}","\\begin{corollary}[Criterion for a cyclic matrix]\n\\label{cor:cyclic-matrix}\nFor $X\\ne0$ of finite dimension, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $T$ has a cyclic vector.\n\\item $\\srank_{\\bbk[T]}(X)=1$.\n\\item Every nonzero primary component for $p$ is cyclic.\n\\item The rational canonical form has one invariant factor.\n\\item The minimal polynomial equals the characteristic polynomial.\n\\item After scalar extension to an algebraic closure, every eigenvalue has geometric multiplicity one.\n\\end{enumerate}\n\\end{corollary}\n\\RecordID{PSC-I-C-CYCLIC-MATRIX}\n\\Depends{PSC-I-T-SINGLE-OPERATOR, PSC-I-C-INVARIANT-FACTOR-COUNT, and PSC-I-P-CYCLIC-CHARACTERIZATION.}","\\begin{proof}\nThe equivalence of (i) and (ii) is \\cref{prop:cyclic-characterization}.  By \\cref{thm:single-operator}, (ii) holds exactly when every $\\mu_p(T)$ belonging to a nonzero primary component equals one, which is (iii).  Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv).  In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)$\\Leftrightarrow$(v).  Scalar extension preserves the equality of minimal and characteristic polynomials.  Over an algebraic closure, that equality is equivalent to one Jordan block for each eigenvalue, which is (vi).\n\\end{proof}","ii) holds exactly when every $\\mu_p(T)$ belonging to a nonzero primary component equals one, which is (iii).  Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv).  In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)$\\Leftrightarrow$(v).  Scalar extension preserves the equality of minimal and characteristic polynomials.  Over an algebraic closure, that equality is equivalent to one Jordan block for each eigenvalue, which is (vi).\n\\end{proof}","\\begin{definition}[Finite packet system]\n\\label{def:packet-system}\nA finite packet system on a vector space $X$ over $\\bbk$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\subseteq\\End_{\\bbk}(X)\n\\]\nsatisfying\n\\[\nP_\\alpha^2=P_\\alpha,\n\\qquad\nP_\\alpha P_\\beta=0\\quad(\\alpha\\neq\\beta),\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nSet $X_\\alpha=P_\\alpha X$.  Zero packet spaces may be removed without changing any identity.  No orthogonality is assumed unless stated.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-SYSTEM}","\\begin{definition}[Packet support and packet pure vectors]\n\\label{def:packet-support}\nFor a finite packet system \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(x\\in X\\), define\n\\[\n\\Supp_{\\Pcal}(x)\n:=\n\\{\\alpha\\in I:P_\\alpha x\\neq0\\}.\n\\]\nThe vector \\(x\\) is \\emph{packet pure} if \\(|\\Supp_{\\Pcal}(x)|\\leq1\\).  A finite family is packet pure if each of its members is packet pure, although different members may lie in different packets.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-SUPPORT}","\\begin{lemma}[Finite packet aggregation]\n\\label{lem:packet-aggregation}\nLet $\\sigma:I\\twoheadrightarrow J$ be a surjection and define\n\\[\nQ_j:=\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nThen $(Q_j)_{j\\in J}$ is a finite packet system and\n\\[\nQ_jX=\\bigoplus_{\\sigma(\\alpha)=j}X_\\alpha.\n\\]\n\\end{lemma}\n\\RecordID{PSC-I-L-PACKET-AGGREGATION}\n\\Depends{PSC-I-D-PACKET-SYSTEM.}","\\begin{proof}\nPairwise annihilation of the $P_\\alpha$ gives\n\\[\nQ_j^2=Q_j,\n\\qquad\nQ_jQ_k=0\\quad(j\\neq k).\n\\]\nSumming over $j$ recovers $\\sum_\\alpha P_\\alpha=I$.  The range statement follows from the directness forced by pairwise annihilation.\n\\end{proof}","\\begin{definition}[Packet analysis and synthesis]\n\\label{def:analysis-synthesis}\nFor a finite packet system, define\n\\[\n\\mathcal A_{\\Pcal}:X\\longrightarrow\\bigoplus_{\\alpha\\in I}X_\\alpha,\n\\qquad\nx\\longmapsto(P_\\alpha x)_\\alpha,\n\\]\nand\n\\[\n\\mathcal S_{\\Pcal}:\\bigoplus_{\\alpha\\in I}X_\\alpha\\longrightarrow X,\n\\qquad\n(x_\\alpha)_\\alpha\\longmapsto\\sum_\\alpha x_\\alpha.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-ANALYSIS-SYNTHESIS}","\\begin{theorem}[Exact packet coordinate isomorphism]\n\\label{thm:exact-packet-coordinates}\nFor every finite packet system,\n\\[\n\\boxed{\n\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}=I_X,\n\\qquad\n\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I_{\\oplus X_\\alpha}.\n}\n\\]\nHence\n\\[\n\\boxed{X\\cong\\bigoplus_{\\alpha\\in I}X_\\alpha.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-EXACT-PACKET-COORDINATES}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-ANALYSIS-SYNTHESIS.}\n\\Exports{Every exact packet reconstruction in the series.}","\\begin{proof}\nFor $x\\in X$,\n\\[\n\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x\n=\n\\sum_\\alpha P_\\alpha x\n=x.\n\\]\nIf $x_\\alpha\\in X_\\alpha$, then $P_\\beta x_\\alpha=\\delta_{\\alpha\\beta}x_\\alpha$, so\n\\[\n\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}(x_\\alpha)_\\alpha\n=\n\\left(P_\\beta\\sum_\\alpha x_\\alpha\\right)_\\beta\n=(x_\\beta)_\\beta.\n\\]\nThus analysis and synthesis are inverse linear maps.\n\\end{proof}","\\begin{corollary}[Packetwise equality]\n\\label{cor:packetwise-equality}\nFor $x,y\\in X$,\n\\[\nx=y\n\\quad\\Longleftrightarrow\\quad\nP_\\alpha x=P_\\alpha y\\text{ for every }\\alpha.\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKETWISE-EQUALITY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}","\\begin{proof}\nThe forward implication is immediate.  For the reverse implication, sum the packet equalities and use $\\sum_\\alpha P_\\alpha=I$.\n\\end{proof}","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of pa","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, co","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","pha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","\\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","\\begin{proof}\nFor \\(x=\\sum_\\alpha P_\\alpha x\\), any extension of the \\(f_\\alpha\\) must satisfy\n\\[\nf(x)=\\sum_\\alpha f_\\alpha(P_\\alpha x),\n\\]\nwhich proves both existence and uniqueness of the first formula.  For the second, the sum \\(g=\\sum_\\alpha g_\\alpha\\) satisfies\n\\[\nP_\\beta g\n=\\sum_\\alpha P_\\beta g_\\alpha\n=g_\\beta\n\\]\nbecause \\(g_\\alpha(Y)\\subseteq X_\\alpha\\).  Uniqueness follows from \\(\\sum_\\alpha P_\\alpha=I\\).\n\\end{proof}","\\begin{theorem}[Tensor product packet systems]\n\\label{thm:tensor-packet-system}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realization","\\begin{theorem}[Tensor product packet systems]\n\\label{thm:tensor-packet-system}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realizations.}","ct packet systems]\n\\label{thm:tensor-packet-system}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realizations.}","lpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realizations.}","\\begin{proof}\nFor every pair of packet indices,\n\\[\n(P_\\alpha\\otimes Q_\\gamma)^2\n=P_\\alpha^2\\otimes Q_\\gamma^2\n=P_\\alpha\\otimes Q_\\gamma.\n\\]\nIf \\((\\alpha,\\gamma)\\neq(\\beta,\\delta)\\), then either \\(P_\\alpha P_\\beta=0\\) or \\(Q_\\gamma Q_\\delta=0\\), so the corresponding tensor idempotents annihilate each other.  Finally,\n\\[\n\\sum_{\\alpha,\\gamma}P_\\alpha\\otimes Q_\\gamma\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\otimes\n\\left(\\sum_\\gamma Q_\\gamma\\right)\n=I_X\\otimes I_Y.\n\\]\nThe range is generated by pure tensors \\(P_\\alpha x\\otimes Q_\\gamma y\\), hence equals \\(X_\\alpha\\otimes Y_\\gamma\\).  Exact reconstruction follows from \\cref{thm:exact-packet-coordinates}.\n\\end{proof}","ma^2\n=P_\\alpha\\otimes Q_\\gamma.\n\\]\nIf \\((\\alpha,\\gamma)\\neq(\\beta,\\delta)\\), then either \\(P_\\alpha P_\\beta=0\\) or \\(Q_\\gamma Q_\\delta=0\\), so the corresponding tensor idempotents annihilate each other.  Finally,\n\\[\n\\sum_{\\alpha,\\gamma}P_\\alpha\\otimes Q_\\gamma\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\otimes\n\\left(\\sum_\\gamma Q_\\gamma\\right)\n=I_X\\otimes I_Y.\n\\]\nThe range is generated by pure tensors \\(P_\\alpha x\\otimes Q_\\gamma y\\), hence equals \\(X_\\alpha\\otimes Y_\\gamma\\).  Exact reconstruction follows from \\cref{thm:exact-packet-coordinates}.\n\\end{proof}",",\\gamma)\\neq(\\beta,\\delta)\\), then either \\(P_\\alpha P_\\beta=0\\) or \\(Q_\\gamma Q_\\delta=0\\), so the corresponding tensor idempotents annihilate each other.  Finally,\n\\[\n\\sum_{\\alpha,\\gamma}P_\\alpha\\otimes Q_\\gamma\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\otimes\n\\left(\\sum_\\gamma Q_\\gamma\\right)\n=I_X\\otimes I_Y.\n\\]\nThe range is generated by pure tensors \\(P_\\alpha x\\otimes Q_\\gamma y\\), hence equals \\(X_\\alpha\\otimes Y_\\gamma\\).  Exact reconstruction follows from \\cref{thm:exact-packet-coordinates}.\n\\end{proof}","\\begin{proposition}[Dual packet systems]\n\\label{prop:dual-packet-system}\nLet \\(X\\) be finite dimensional and let \\(\\Pcal=(P_\\alpha)\\) be a finite packet system.  The algebraic dual maps\n\\[\nP_\\alpha^*:X^*\\to X^*,\n\\qquad\nP_\\alpha^*(\\varphi)=\\varphi\\circ P_\\alpha,\n\\]\nform a finite packet system on \\(X^*\\).  Its packet with index \\(\\alpha\\) is canonically isomorphic to \\(X_\\alpha^*\\), and\n\\[\nX^*\\cong\\bigoplus_\\alpha X_\\alpha^*.\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-DUAL-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Dual observables, covectors, trace pairings, and adjoint packet calculations.}","\\begin{proof}\nBecause dualization reverses products,\n\\[\nP_\\alpha^*P_\\beta^*=(P_\\beta P_\\alpha)^*\n=\\delta_{\\alpha\\beta}P_\\alpha^*.\n\\]\nAlso \\(\\sum_\\alpha P_\\alpha^*=I_{X^*}\\).  Thus the dual maps form a packet system.  Restriction gives a map from \\(P_\\alpha^*X^*\\) to \\(X_\\alpha^*\\).  It is injective because a covector in the range of \\(P_\\alpha^*\\) vanishes on every other packet, and it is surjective because any covector on \\(X_\\alpha\\) extends to \\(X\\) through the direct sum decomposition.\n\\end{proof}","tion reverses products,\n\\[\nP_\\alpha^*P_\\beta^*=(P_\\beta P_\\alpha)^*\n=\\delta_{\\alpha\\beta}P_\\alpha^*.\n\\]\nAlso \\(\\sum_\\alpha P_\\alpha^*=I_{X^*}\\).  Thus the dual maps form a packet system.  Restriction gives a map from \\(P_\\alpha^*X^*\\) to \\(X_\\alpha^*\\).  It is injective because a covector in the range of \\(P_\\alpha^*\\) vanishes on every other packet, and it is surjective because any covector on \\(X_\\alpha\\) extends to \\(X\\) through the direct sum decomposition.\n\\end{proof}","\\[\nP_\\alpha^*P_\\beta^*=(P_\\beta P_\\alpha)^*\n=\\delta_{\\alpha\\beta}P_\\alpha^*.\n\\]\nAlso \\(\\sum_\\alpha P_\\alpha^*=I_{X^*}\\).  Thus the dual maps form a packet system.  Restriction gives a map from \\(P_\\alpha^*X^*\\) to \\(X_\\alpha^*\\).  It is injective because a covector in the range of \\(P_\\alpha^*\\) vanishes on every other packet, and it is surjective because any covector on \\(X_\\alpha\\) extends to \\(X\\) through the direct sum decomposition.\n\\end{proof}","\\begin{microexample}[title={Tensoring the oblique packet split}]\nTensor the two oblique packets in Running Example R3 with the coordinate packets of \\(\\bbk^3\\).  The result has six exact packets.  No orthogonality is created by tensoring, but the analysis and synthesis identities remain exact because they are idempotent identities.\n\\end{microexample}","\\begin{theorem}[Finite Banach packet isomorphism]\n\\label{thm:banach-packets}\nLet $X$ be Banach and let $(P_\\alpha)_{\\alpha\\in I}$ be a finite family of bounded pairwise annihilating idempotents with sum $I_X$.  Then every range $X_\\alpha=P_\\alpha X$ is closed.  For any finite product norm equivalent to the maximum norm on $\\bigoplus_\\alpha X_\\alpha$, packet analysis and synthesis are bounded inverse operators.\n\\end{theorem}\n\\RecordID{PSC-I-T-BANACH-PACKET-ISOMORPHISM}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}","\\begin{proof}\nThe range of a bounded idempotent is its space of fixed points,\n\\[\nP_\\alpha X=\\Ker(I-P_\\alpha),\n\\]\nwhich is closed.  Because $I$ is finite, analysis and synthesis are bounded for the maximum and sum norms, respectively, and therefore for every equivalent finite product norm.  Their inverse identities are algebraic and were proved in \\cref{thm:exact-packet-coordinates}.\n\\end{proof}","\\begin{definition}[Packet coordinate condition number]\n\\label{def:packet-condition-number}\nFix a finite product norm on $\\bigoplus_\\alpha X_\\alpha$ equivalent to the maximum norm, as in \\cref{thm:banach-packets}. Define\n\\[\n\\kappa(\\Pcal):=\n\\|\\mathcal A_{\\Pcal}\\|\\,\n\\|\\mathcal S_{\\Pcal}\\|.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-CONDITION-NUMBER}","\\begin{proposition}[Conditioned coordinate transport]\n\\label{prop:conditioned-coordinates}\nFor a finite Banach packet system with $X\\ne0$,\n\\[\n\\kappa(\\Pcal)\\geq1.\n\\]\nIf $T\\in\\mathcal B(X)$ and\n\\[\n[T]_{\\Pcal}:=\\mathcal A_{\\Pcal}T\\mathcal S_{\\Pcal},\n\\]\nthen\n\\[\n\\boxed{\n\\|[T]_{\\Pcal}\\|\n\\leq\\kappa(\\Pcal)\\|T\\|,\n\\qquad\n\\|T\\|\n\\leq\\kappa(\\Pcal)\\|[T]_{\\Pcal}\\|.\n}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-CONDITIONED-COORDINATES}\n\\Depends{PSC-I-T-BANACH-PACKET-ISOMORPHISM.}","\\begin{proof}\nSince $I=\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}$,\n\\[\n1=\\|I\\|\\leq\\|\\mathcal S_{\\Pcal}\\|\\|\\mathcal A_{\\Pcal}\\|.\n\\]\nThe first displayed estimate follows from submultiplicativity.  The inverse identity\n\\[\nT=\\mathcal S_{\\Pcal}[T]_{\\Pcal}\\mathcal A_{\\Pcal}\n\\]\ngives the second estimate.\n\\end{proof}","\\begin{interpretation}\nThe quantity \\(\\kappa(\\Pcal)\\) measures how strongly the chosen packet coordinates can amplify perturbations.  An orthogonal Hilbert decomposition has the best possible behavior in the natural \\(\\ell^2\\) sum norm.  An oblique packet system can reconstruct every vector exactly and still have a large condition number.\n\\end{interpretation}","\\begin{corollary}[Completion preserves a finite packet split]\n\\label{cor:completion-packet-split}\nLet $M$ be a dense normed subspace of a Banach space $X$.  Suppose every bounded $P_\\alpha$ leaves $M$ invariant and the restricted projectors form a packet system on $M$.  Then\n\\[\nP_\\alpha X=\\cl{P_\\alpha M}\n\\]\nand the packet decomposition of $M$ extends uniquely to the completion $X$.\n\\end{corollary}\n\\RecordID{PSC-I-C-COMPLETION-PACKET-SPLIT}\n\\Depends{PSC-I-T-BANACH-PACKET-ISOMORPHISM.}","\\begin{proof}\nFor $x\\in X$, choose $m_n\\in M$ with $m_n\\to x$.  Continuity gives $P_\\alpha m_n\\to P_\\alpha x$, so $P_\\alpha X\\subseteq\\cl{P_\\alpha M}$.  The reverse inclusion follows because $P_\\alpha X$ is closed and contains $P_\\alpha M$.  The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.\n\\end{proof}","\\begin{definition}[Unconditional packet sum space]\n\\label{def:unconditional-packet-space}\nLet $(X_n)_{n\\geq1}$ be closed subspaces of a real or complex Banach space $X$.  Define\n\\[\n\\bigoplus_{n\\geq1}^{u}X_n\n\\]\nto be the space of sequences $x=(x_n)$ with $x_n\\in X_n$ for which $\\sum_nx_n$ converges unconditionally in $X$.  Equip it with\n\\[\n\\|x\\|_u\n:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-UNCONDITIONAL-PACKET-SPACE}","\\begin{lemma}[Completeness of the unconditional packet sum]\n\\label{lem:unconditional-completeness}\nThe normed space $\\bigoplus_{n\\geq1}^{u}X_n$ in \\cref{def:unconditional-packet-space} is Banach.\n\\end{lemma}\n\\RecordID{PSC-I-L-UNCONDITIONAL-COMPLETENESS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE.}","\\begin{proof}\nLet $x^{(k)}=(x_n^{(k)})$ be Cauchy in $\\|\\cdot\\|_u$.  For every fixed $n$,\n\\[\n\\|x_n^{(k)}-x_n^{(\\ell)}\\|\n\\leq\n\\|x^{(k)}-x^{(\\ell)}\\|_u,\n\\]\nso $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","k)}=(x_n^{(k)})$ be Cauchy in $\\|\\cdot\\|_u$.  For every fixed $n$,\n\\[\n\\|x_n^{(k)}-x_n^{(\\ell)}\\|\n\\leq\n\\|x^{(k)}-x^{(\\ell)}\\|_u,\n\\]\nso $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","}\\|\n\\leq\n\\|x^{(k)}-x^{(\\ell)}\\|_u,\n\\]\nso $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","so $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","Choose such a $k\\geq k_0$.  Since $x^{(k)}$ is unconditionally summable, there is a finite $F_0$ such that every finite $F\\subseteq\\mathbb N\\setminus F_0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","s a finite $F_0$ such that every finite $F\\subseteq\\mathbb N\\setminus F_0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","thbb N\\setminus F_0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","\\begin{theorem}[Countable unconditional packet isomorphism]\n\\label{thm:countable-packets}\nLet $X$ be a real or complex Banach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions unde","\\begin{theorem}[Countable unconditional packet isomorphism]\n\\label{thm:countable-packets}\nLet $X$ be a real or complex Banach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions under explicit convergence control.}","sm]\n\\label{thm:countable-packets}\nLet $X$ be a real or complex Banach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions under explicit convergence control.}","ach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions under explicit convergence control.}","\\begin{proof}\nEach $P_nX=\\Ker(I-P_n)$ is closed.  For $x\\in X$,\n\\[\n\\|\\mathcal A_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end","\\begin{proof}\nEach $P_nX=\\Ker(I-P_n)$ is closed.  For $x\\in X$,\n\\[\n\\|\\mathcal A_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","roof}\nEach $P_nX=\\Ker(I-P_n)$ is closed.  For $x\\in X$,\n\\[\n\\|\\mathcal A_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","For $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","\\begin{remark}[Uniform boundedness]\n\\label{rem:uniform-boundedness}\nIf every expansion $\\sum_nP_nx$ is unconditional, the operators obtained from finite partial sums with signs form a pointwise bounded family.  The uniform boundedness principle therefore yields a finite constant $K$.  We retain this constant explicitly in \\cref{thm:countable-packets} because later estimates use it to bound the coordinate map.\n\\end{remark}\n\\RecordID{PSC-I-R-UNIFORM-BOUNDEDNESS}","\\begin{boundary}[Infinite packet sums]\n\\label{bnd:infinite-packet-topology}\nFor an infinite family, an identity such as $\\sum_\\alpha P_\\alpha=I$ requires a declared convergence mode and the corresponding uniform boundedness statement.  Pointwise, strong, norm, weak, and unconditional convergence are not interchangeable.\n\\end{boundary}\n\\RecordID{PSC-I-B-INFINITE-PACKET-TOPOLOGY}","\\begin{definition}[Transfer block]\n\\label{def:transfer-block}\nFor $T\\in\\End_{\\bbk}(X)$ and a finite packet system $\\Pcal$, define\n\\[\nT_{\\beta\\alpha}:=P_\\beta TP_\\alpha\n\\in\\Hom_{\\bbk}(X_\\alpha,X_\\beta).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-BLOCK}","\\begin{theorem}[Exact operator reconstruction]\n\\label{thm:operator-reconstruction}\nEvery $T\\in\\End_{\\bbk}(X)$ satisfies\n\\[\n\\boxed{T=\\sum_{\\alpha,\\beta\\in I}T_{\\beta\\alpha}.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-OPERATOR-RECONSTRUCTION}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}","\\begin{proof}\nInsert the finite packet identity on both sides:\n\\[\nT=\n\\left(\\sum_\\beta P_\\beta\\right)\nT\n\\left(\\sum_\\alpha P_\\alpha\\right)\n=\n\\sum_{\\alpha,\\beta}P_\\beta TP_\\alpha.\n\\]\n\\end{proof}","\\begin{interpretation}\nOperator reconstruction inserts the identity on both sides of \\(T\\).  The source index \\(\\alpha\\) records where a vector begins, and the target index \\(\\beta\\) records where its image lands.  The resulting matrix is exact, and each entry is itself a linear map.\n\\end{interpretation}","\\begin{theorem}[Packet matrix multiplication]\n\\label{thm:block-multiplication}\nFor $S,T\\in\\End_{\\bbk}(X)$,\n\\[\n\\boxed{\n(ST)_{\\gamma\\alpha}\n=\n\\sum_{\\beta\\in I}S_{\\gamma\\beta}T_{\\beta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-MULTIPLICATION}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}\n\\Exports{All packet path expansions.}","\\begin{proof}\nInsert $I=\\sum_\\beta P_\\beta$ between $S$ and $T$:\n\\[\nP_\\gamma STP_\\alpha\n=\n\\sum_\\beta\nP_\\gamma SP_\\beta TP_\\alpha.\n\\]\n\\end{proof}","\\begin{corollary}[Packet powers and paths]\n\\label{cor:block-powers}\nFor $m\\geq1$,\n\\[\n(T^m)_{\\beta_m\\beta_0}\n=\n\\sum_{\\beta_1,\\ldots,\\beta_{m-1}}\nT_{\\beta_m\\beta_{m-1}}\n\\cdots\nT_{\\beta_1\\beta_0}.\n\\]\nThe same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order.\n\\end{corollary}\n\\RecordID{PSC-I-C-BLOCK-POWERS}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION.}","\\begin{proof}\nInduct on $m$ using \\cref{thm:block-multiplication}.  For a product of different operators, use the same induction with the specified operator at each step.\n\\end{proof}","\\begin{corollary}[Adjoint reversal]\n\\label{cor:adjoint-reversal}\nIf $X$ is Hilbert, every $P_\\alpha$ is orthogonal, and $T\\in\\mathcal B(X)$, then\n\\[\n\\boxed{(T_{\\beta\\alpha})^*=(T^*)_{\\alpha\\beta}.}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-ADJOINT-REVERSAL}\n\\Depends{PSC-I-D-TRANSFER-BLOCK.}","\\begin{proof}\nBecause $P_\\alpha^*=P_\\alpha$,\n\\[\n(P_\\beta TP_\\alpha)^*=P_\\alpha T^*P_\\beta.\n\\]\n\\end{proof}","\\begin{corollary}[Tensor product transfer blocks]\n\\label{cor:tensor-transfer-blocks}\nLet \\(S\\in\\End(X)\\) and \\(T\\in\\End(Y)\\), with packet systems \\(\\Pcal\\) and \\(\\Qcal\\).  Relative to the tensor product packet system of \\cref{thm:tensor-packet-system},\n\\[\n\\boxed{\n(S\\otimes T)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n=\nS_{\\beta\\alpha}\\otimes T_{\\delta\\gamma}.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-TENSOR-TRANSFER-BLOCKS}\n\\Depends{PSC-I-T-TENSOR-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}\n\\Exports{Tensor network, quantum circuit, and multivariate packet calculations.}","\\begin{proof}\nBy definition,\n\\begin{align*}\n(S\\otimes T)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n&=(P_\\beta\\otimes Q_\\delta)(S\\otimes T)(P_\\alpha\\otimes Q_\\gamma)\\\\\n&=(P_\\beta SP_\\alpha)\\otimes(Q_\\delta TQ_\\gamma)\\\\\n&=S_{\\beta\\alpha}\\otimes T_{\\delta\\gamma}.\n\\end{align*}\n\\end{proof}","\\begin{theorem}[Blockwise intertwining equations]\n\\label{thm:block-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representat","\\begin{theorem}[Blockwise intertwining equations]\n\\label{thm:block-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","ons]\n\\label{thm:block-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","k-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","et \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","\\begin{proof}\nApply \\(Q_\\gamma\\) on the left and \\(P_\\alpha\\) on the right.  Inserting \\(I_X=\\sum_\\beta P_\\beta\\) gives\n\\[\nQ_\\gamma UTP_\\alpha\n=\n\\sum_\\beta Q_\\gamma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","begin{proof}\nApply \\(Q_\\gamma\\) on the left and \\(P_\\alpha\\) on the right.  Inserting \\(I_X=\\sum_\\beta P_\\beta\\) gives\n\\[\nQ_\\gamma UTP_\\alpha\n=\n\\sum_\\beta Q_\\gamma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","P_\\beta\\) gives\n\\[\nQ_\\gamma UTP_\\alpha\n=\n\\sum_\\beta Q_\\gamma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","amma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","\\begin{theorem}[Full block algebra representation]\n\\label{thm:block-algebra}\nThe map\n\\[\n\\Theta_{\\Pcal}:\\End_{\\bbk}(X)\n\\longrightarrow\n\\prod_{\\beta,\\alpha\\in I}\\Hom_{\\bbk}(X_\\alpha,X_\\beta),\n\\qquad\nT\\longmapsto(T_{\\beta\\alpha})_{\\beta,\\alpha},\n\\]\nis an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators.\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-ALGEBRA}\n\\Depends{\\nolinkurl{PSC-I-T-OPERATOR-RECONSTRUCTION} and\n\\nolinkurl{PSC-I-T-BLOCK-MULTIPLICATION}.}","\\begin{proof}\nInjectivity follows from exact reconstruction.  Given arbitrary linear maps $B_{\\beta\\alpha}:X_\\alpha\\to X_\\beta$, define\n\\[\nT\\left(\\sum_\\alpha x_\\alpha\\right)\n:=\n\\sum_{\\alpha,\\beta}B_{\\beta\\alpha}x_\\alpha.\n\\]\nThe packet decomposition is unique and the index set is finite, so this defines a linear operator whose blocks are the prescribed maps.  Thus $\\Theta_{\\Pcal}$ is surjective.  Compatibility with multiplication is \\cref{thm:block-multiplication}.\n\\end{proof}","\\begin{corollary}[Propagation of packet support]\n\\label{cor:packet-support-propagation}\nLet \\(T\\in\\End(X)\\) and \\(x\\in X\\).  Then\n\\[\n\\boxed{\n\\Supp_{\\Pcal}(Tx)\n\\subseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reacha","\\begin{corollary}[Propagation of packet support]\n\\label{cor:packet-support-propagation}\nLet \\(T\\in\\End(X)\\) and \\(x\\in X\\).  Then\n\\[\n\\boxed{\n\\Supp_{\\Pcal}(Tx)\n\\subseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reachability, leakage","\\begin{corollary}[Propagation of packet support]\n\\label{cor:packet-support-propagation}\nLet \\(T\\in\\End(X)\\) and \\(x\\in X\\).  Then\n\\[\n\\boxed{\n\\Supp_{\\Pcal}(Tx)\n\\subseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reachability, leakage bounds, and support pruning.}","bseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reachability, leakage bounds, and support pruning.}","\\begin{proof}\nFor every target packet \\(\\beta\\),\n\\[\nP_\\beta Tx\n=\n\\sum_{\\alpha\\in\\Supp_{\\Pcal}(x)}T_{\\beta\\alpha}P_\\alpha x.\n\\]\nIf every block in this sum is zero, then the coordinate with index \\(\\beta\\) vanishes.  This proves the inclusion.  Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right.  The final sufficient condition leaves at most one nonzero summand at each target, and kernel avoidance makes that summand nonzero.\n\\end{proof}","\\begin{corollary}[Dimension count]\n\\label{cor:dimension-count}\nIf $X$ is finite dimensional and $d_\\alpha=\\dim X_\\alpha$, then\n\\[\n\\dim\\End_{\\bbk}(X)\n=\n\\sum_{\\alpha,\\beta}d_\\alpha d_\\beta\n=\n\\left(\\sum_\\alpha d_\\alpha\\right)^2.\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-DIMENSION-COUNT}\n\\Depends{PSC-I-T-BLOCK-ALGEBRA.}","\\begin{proof}\nTake dimensions in the block algebra isomorphism and use $\\dim\\Hom(X_\\alpha,X_\\beta)=d_\\alpha d_\\beta$.\n\\end{proof}","\\begin{proposition}[Block diagonal projection and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","begin{proposition}[Block diagonal projection and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","n}[Block diagonal projection and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","bel{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","\\begin{proof}\nPairwise annihilation gives $\\Delta_{\\Pcal}^2=\\Delta_{\\Pcal}$ and identifies its range with diagonal block families.  If $T$ commutes with all $P_\\alpha$, then for $\\beta\\neq\\alpha$,\n\\[\nP_\\beta TP_\\alpha=P_\\beta P_\\alpha T=0.\n\\]\nThus (i) implies (ii).  Exact reconstruction shows (ii) implies (iii).  If $T=\\sum_\\alpha P_\\alpha TP_\\alpha$, then\n\\[\nTP_\\gamma=P_\\gamma TP_\\gamma=P_\\gamma T,\n\\]\nso (iii) implies (i).\n\\end{proof}","\\subsection{Support patterns, preorders, and triangular structure}\n\\begin{definition}[Support relation and support space]\n\\label{def:support-relation}\nFor a relation $R\\subseteq I\\times I$, with $(\\beta,\\alpha)$ interpreted as an allowed transfer $\\alpha\\to\\beta$, define\n\\[\n\\mathfrak M_R\n:=\n\\{T\\in\\End(X):T_{\\beta\\alpha}=0\\text{ whenever }(\\beta,\\alpha)\\notin R\\}.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SUPPORT-RELATION}","\\begin{theorem}[Support pattern algebra criterion]\n\\label{thm:support-pattern}\nAssume $X_\\alpha\\neq0$ for every $\\alpha\\in I$.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $I_X\\in\\mathfrak M_R$ if and only if $R$ is reflexive.\n\\item $\\mathfrak M_R\\mathfrak M_R\\subseteq\\mathfrak M_R$ if and only if $R$ is transitive.\n\\item $\\mathfrak M_R$ is a unital subalgebra if and only if $R$ is a preorder.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-SUPPORT-PATTERN-ALGEBRA}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION.}","\\begin{proof}\nThe identity has diagonal blocks $I_{X_\\alpha}$ and no blocks outside the diagonal, so it belongs to $\\mathfrak M_R$ exactly when $(\\alpha,\\alpha)\\in R$ for every $\\alpha$.","Assume $R$ is transitive.  If a summand\n\\[\nS_{\\gamma\\beta}T_{\\beta\\alpha}\n\\]\nin the $(\\gamma,\\alpha)$ block of $ST$ is nonzero, then $(\\gamma,\\beta)$ and $(\\beta,\\alpha)$ lie in $R$.  Transitivity gives $(\\gamma,\\alpha)\\in R$, so every forbidden block of $ST$ vanishes.","Conversely, suppose transitivity fails.  Choose $\\alpha,\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}",", suppose transitivity fails.  Choose $\\alpha,\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","ails.  Choose $\\alpha,\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","lpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","\\begin{corollary}[Bounded support patterns closed under $*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","corollary}[Bounded support patterns closed under $*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","support patterns closed under $*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","$*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","\\begin{proof}\nAdjoint reversal sends an allowed $(\\beta,\\alpha)$ block to an $(\\alpha,\\beta)$ block.  Symmetry is therefore sufficient.  If symmetry fails, choose a nonzero map of rank one in an allowed nonsymmetric block.  Its adjoint occupies the forbidden reverse block, proving necessity.  A reflexive, transitive, symmetric relation is an equivalence relation.\n\\end{proof}","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorp","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathf","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREO","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular …","ad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","lpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","arries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","rtial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","k M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","\\begin{proof}\nA nonzero block in $\\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","A nonzero block in $\\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","$\\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","s to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","Let $P_C=\\sum_{\\alpha\\in C}P_\\alpha$.  Define\n\\[\n\\rho:\\mathfrak M_R\\longrightarrow\\prod_C\\End(X_C),\n\\qquad\n\\rho(T)=(P_CTP_C)_C.\n\\]\nIf an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","C=\\sum_{\\alpha\\in C}P_\\alpha$.  Define\n\\[\n\\rho:\\mathfrak M_R\\longrightarrow\\prod_C\\End(X_C),\n\\qquad\n\\rho(T)=(P_CTP_C)_C.\n\\]\nIf an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","C),\n\\qquad\n\\rho(T)=(P_CTP_C)_C.\n\\]\nIf an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","rmediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","\\begin{definition}[Transfer graph of a family generating the algebra]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","efinition}[Transfer graph of a family generating the algebra]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","rating the algebra]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","l=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","t system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","\\begin{theorem}[Reachability support of the generated algebra]\n\\label{thm:algebra-reachability-support}\nFor every $A\\in\\A$,\n\\[\nP_\\beta AP_\\alpha=0\n\\qquad\\text{whenever }\\beta\\notin\\Reach(\\alpha).\n\\]\nEquivalently,\n\\[\n\\A\\subseteq\\mathfrak M_{R_\\Gamma},\n\\qquad\nR_\\Gamma:=\\{(\\beta,\\alpha):\\beta\\in\\Reach(\\alpha)\\}.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-ALGEBRA-REACHABILITY-SUPPORT}\n\\Depends{PSC-I-D-TRANSFER-GRAPH and PSC-I-C-BLOCK-POWERS.}","\\begin{proof}\nIt is enough to treat a word in the generators and packet projectors.  Insert\n\\[\nI=\\sum_{\\gamma\\in I}P_\\gamma\n\\]\nbetween consecutive nonprojector generators.  Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","oof}\nIt is enough to treat a word in the generators and packet projectors.  Insert\n\\[\nI=\\sum_{\\gamma\\in I}P_\\gamma\n\\]\nbetween consecutive nonprojector generators.  Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","ve nonprojector generators.  Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","\\begin{interpretation}\nThe graph belongs to the declared generating family within \\(\\End(X)\\).  Multiplying its generators concatenates paths, so a word in them cannot create a block outside graph reachability.  Reachability alone gives less information in the other direction.  A nonzero edge block may reach only a proper subspace of its target packet.\n\\end{interpretation}","\\subsection{Reachability obstructions for seeds}\n\\begin{proposition}[Reachability obstruction for generated hulls]\n\\label{prop:reachability-obstruction}\nLet $I_0\\subseteq I$ and let\n\\[\nS\\subseteq\\bigoplus_{\\alpha\\in I_0}X_\\alpha.\n\\]\nThen\n\\[\n\\boxed{\n\\Hull_{\\A}(S)\n\\subseteq\n\\bigoplus_{\\beta\\in\\Reach(I_0)}X_\\beta.\n}\n\\]\nIn particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component.\n\\end{proposition}\n\\RecordID{PSC-I-P-REACHABILITY-OBSTRUCTION}\n\\Depends{PSC-I-T-ALGEBRA-REACHABILITY-SUPPORT.}","\\begin{proof}\nFor $s\\in S$, write $s=\\sum_{\\alpha\\in I_0}P_\\alpha s$.  If $\\beta$ is unreachable from $I_0$, then for every $A\\in\\A$,\n\\[\nP_\\beta As\n=\n\\sum_{\\alpha\\in I_0}P_\\beta AP_\\alpha s\n=0\n\\]\nby \\cref{thm:algebra-reachability-support}.  The same holds for every finite linear combination of orbit vectors.\n\\end{proof}","\\Needspace{15\\baselineskip}\n\\begin{definition}[Packet pure seed rank]\n\\label{def:directed-seed-rank}\nA vector is packet pure when it is zero or lies in one packet space $X_\\alpha$, equivalently when its packet support has cardinality at most one.  Define\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n:=\n\\min\\{|S|:S\\text{ is packet-pure and }\\A S=X\\}.\n\\]\nFor $C\\subseteq I$, put\n\\[\nP_C:=\\sum_{\\alpha\\in C}P_\\alpha,\n\\qquad\nX_C:=P_CX,\n\\qquad\n\\A_C:=P_C\\A P_C,\n\\]\nand define the local packet pure rank\n\\[\nr_C:=\\psrank_{\\A_C,\\Pcal|_C}(X_C).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-DIRECTED-SEED-RANK}","f:directed-seed-rank}\nA vector is packet pure when it is zero or lies in one packet space $X_\\alpha$, equivalently when its packet support has cardinality at most one.  Define\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n:=\n\\min\\{|S|:S\\text{ is packet-pure and }\\A S=X\\}.\n\\]\nFor $C\\subseteq I$, put\n\\[\nP_C:=\\sum_{\\alpha\\in C}P_\\alpha,\n\\qquad\nX_C:=P_CX,\n\\qquad\n\\A_C:=P_C\\A P_C,\n\\]\nand define the local packet pure rank\n\\[\nr_C:=\\psrank_{\\A_C,\\Pcal|_C}(X_C).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-DIRECTED-SEED-RANK}","\\begin{proposition}[Basic packet pure rank bounds]\n\\label{prop:packet-pure-rank-bounds}\nFor every $A$ module resolved into packets with $P_\\alpha\\in A$,\n\\[\n\\boxed{\n\\srank_A(X)\n\\leq\n\\psrank_{A,\\Pcal}(X)\n\\leq\n\\sum_{\\alpha\\in I}\n\\srank_{P_\\alpha A P_\\alpha}(X_\\alpha).\n}\n\\]\nThe corner algebra on the right is unital on $X_\\alpha$ with identity $P_\\alpha$.\n\\end{proposition}\n\\RecordID{PSC-I-P-PACKET-PURE-RANK-BOUNDS}\n\\Depends{PSC-I-D-SEED-RANK and PSC-I-D-DIRECTED-SEED-RANK.}","\\begin{proof}\nEvery packet pure generating family is an unrestricted generating family, which gives the first inequality.  For each $\\alpha$, choose a minimal generating family $S_\\alpha\\subseteq X_\\alpha$ for the corner action $P_\\alpha A P_\\alpha$.  The union $S=\\bigcup_\\alpha S_\\alpha$ is packet pure.  Since every $x\\in X$ has the unique decomposition $x=\\sum_\\alpha x_\\alpha$ and every $x_\\alpha$ lies in $(P_\\alpha A P_\\alpha)S_\\alpha\\subseteq AS$, the union generates $X$.  Its cardinality is at most the displayed sum.\n\\end{proof}","}\nEvery packet pure generating family is an unrestricted generating family, which gives the first inequality.  For each $\\alpha$, choose a minimal generating family $S_\\alpha\\subseteq X_\\alpha$ for the corner action $P_\\alpha A P_\\alpha$.  The union $S=\\bigcup_\\alpha S_\\alpha$ is packet pure.  Since every $x\\in X$ has the unique decomposition $x=\\sum_\\alpha x_\\alpha$ and every $x_\\alpha$ lies in $(P_\\alpha A P_\\alpha)S_\\alpha\\subseteq AS$, the union generates $X$.  Its cardinality is at most the displayed sum.\n\\end{proof}","ng family, which gives the first inequality.  For each $\\alpha$, choose a minimal generating family $S_\\alpha\\subseteq X_\\alpha$ for the corner action $P_\\alpha A P_\\alpha$.  The union $S=\\bigcup_\\alpha S_\\alpha$ is packet pure.  Since every $x\\in X$ has the unique decomposition $x=\\sum_\\alpha x_\\alpha$ and every $x_\\alpha$ lies in $(P_\\alpha A P_\\alpha)S_\\alpha\\subseteq AS$, the union generates $X$.  Its cardinality is at most the displayed sum.\n\\end{proof}","\\begin{definition}[Packet system surjective along edges]\n\\label{def:edge-surjective}\nThe module resolved into packets is surjective along edges relative to $\\Gamma(\\mathcal G,\\Pcal)$ when, for every graph edge $\\alpha\\to\\beta$, there exists $B\\in\\A$ such that the pure block\n\\[\nP_\\beta BP_\\alpha:X_\\alpha\\twoheadrightarrow X_\\beta\n\\]\nis surjective.\n\\end{definition}\n\\RecordID{PSC-I-D-EDGE-SURJECTIVE}","\\begin{theorem}[Source component formula for packet pure seed rank]\n\\label{thm:source-component-rank}\nAssume $I$ is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges.  Let $\\mathfrak C_0$ be the set of source strongly connected components of the condensation graph of $\\Gamma(\\mathcal G,\\Pcal)$.  Then\n\\[\n\\boxed{\n\\psrank_{\\A,\\Pcal}(X)\n=\n\\sum_{C\\in\\mathfrak C_0}r_C.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-SOURCE-COMPONENT-SEED-RANK}\n\\Depends{PSC-I-P-REACHABILITY-OBSTRUCTION and PSC-I-D-EDGE-SURJECTIVE.}\n\\Exports{Residual transport forests and directed source rank calculations.}","\\begin{proof}\nLet $S$ be a packet pure generating family for $X$.  Discard zero seeds, which do not change the generated hull, and fix a source component $C$.  Seeds outside $C$ cannot reach $C$ by \\cref{prop:reachability-obstruction}.  Moreover, for $s\\in X_C$ and $A\\in\\A$,\n\\[\nP_CAs=(P_CAP_C)s\\in\\A_Cs.\n\\]\nThus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\\A_C$, and there are at least $r_C$ of them.  Summing over disjoint source components gives\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n\\geq\n\\sum_{C\\in\\mathfrak C_0}r_C.\n\\]","gin{proof}\nLet $S$ be a packet pure generating family for $X$.  Discard zero seeds, which do not change the generated hull, and fix a source component $C$.  Seeds outside $C$ cannot reach $C$ by \\cref{prop:reachability-obstruction}.  Moreover, for $s\\in X_C$ and $A\\in\\A$,\n\\[\nP_CAs=(P_CAP_C)s\\in\\A_Cs.\n\\]\nThus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\\A_C$, and there are at least $r_C$ of them.  Summing over disjoint source components gives\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n\\geq\n\\sum_{C\\in\\mathfrak C_0}r_C.\n\\]","Discard zero seeds, which do not change the generated hull, and fix a source component $C$.  Seeds outside $C$ cannot reach $C$ by \\cref{prop:reachability-obstruction}.  Moreover, for $s\\in X_C$ and $A\\in\\A$,\n\\[\nP_CAs=(P_CAP_C)s\\in\\A_Cs.\n\\]\nThus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\\A_C$, and there are at least $r_C$ of them.  Summing over disjoint source components gives\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n\\geq\n\\sum_{C\\in\\mathfrak C_0}r_C.\n\\]","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\ga","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, cho","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed pa","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected compon","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive sur","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subsete","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$,","e component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ily $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}",", and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","s in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ical order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ponents preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ent $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","e have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","hoose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","$\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","es a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","a}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","teq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","\\begin{definition}[Packet propagating system]\n\\label{def:packet-propagating}\nA packet system that is surjective along edges is called packet propagating when every packet space $X_\\alpha$ is cyclic under its diagonal corner algebra\n\\[\nP_\\alpha\\A P_\\alpha.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-PROPAGATING}","\\begin{corollary}[Unweighted source component count]\n\\label{cor:source-component-count}\nAssume the module resolved into packets is packet propagating and $X_\\alpha\\neq0$ for every $\\alpha\\in I$.  If $c_0$ denotes the number of source strongly connected components, then\n\\[\n\\boxed{\n\\psrank_{\\A,\\Pcal}(X)=c_0.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-SOURCE-COMPONENT-COUNT}\n\\Depends{PSC-I-T-SOURCE-COMPONENT-SEED-RANK.}","\\begin{proof}\nFix a source component $C$ and a vertex $\\alpha\\in C$.  Choose a cyclic vector $s\\in X_\\alpha$ for $P_\\alpha\\A P_\\alpha$.  It generates all of $X_\\alpha$.  Strong connectivity and surjectivity along edges then propagate the whole packet space to every vertex of $C$.  Thus $r_C=1$ for every source component.  Apply \\cref{thm:source-component-rank}.\n\\end{proof}","\\begin{boundary}[Mixed support seeds can merge source requirements]\n\\label{bnd:packet-pure-boundary}\nThe source component formulas concern packet pure seeds.  They do not give the unrestricted seed rank.  For example, let\n\\[\n\\A=\\bbk\\times\\bbk,\n\\qquad\nX=\\bbk\\oplus\\bbk\n\\]\nwith the coordinate action and the two coordinate packet projectors.  The transfer graph has two source components and\n\\[\n\\psrank_{\\A,\\Pcal}(X)=2,\n\\]\nbut the mixed vector $(1,1)$ is cyclic, so\n\\[\n\\srank_{\\A}(X)=1.\n\\]\n\\end{boundary}\n\\RecordID{PSC-I-B-PACKET-PURE-BOUNDARY}","rge source requirements]\n\\label{bnd:packet-pure-boundary}\nThe source component formulas concern packet pure seeds.  They do not give the unrestricted seed rank.  For example, let\n\\[\n\\A=\\bbk\\times\\bbk,\n\\qquad\nX=\\bbk\\oplus\\bbk\n\\]\nwith the coordinate action and the two coordinate packet projectors.  The transfer graph has two source components and\n\\[\n\\psrank_{\\A,\\Pcal}(X)=2,\n\\]\nbut the mixed vector $(1,1)$ is cyclic, so\n\\[\n\\srank_{\\A}(X)=1.\n\\]\n\\end{boundary}\n\\RecordID{PSC-I-B-PACKET-PURE-BOUNDARY}","]\n\\label{bnd:packet-pure-boundary}\nThe source component formulas concern packet pure seeds.  They do not give the unrestricted seed rank.  For example, let\n\\[\n\\A=\\bbk\\times\\bbk,\n\\qquad\nX=\\bbk\\oplus\\bbk\n\\]\nwith the coordinate action and the two coordinate packet projectors.  The transfer graph has two source components and\n\\[\n\\psrank_{\\A,\\Pcal}(X)=2,\n\\]\nbut the mixed vector $(1,1)$ is cyclic, so\n\\[\n\\srank_{\\A}(X)=1.\n\\]\n\\end{boundary}\n\\RecordID{PSC-I-B-PACKET-PURE-BOUNDARY}","\\begin{microexample}[title={Why mixed seeds change the answer}]\nLet the acting algebra be diagonal on \\(X=X_1\\oplus X_2\\), with no transfer between the packets.  Generation by packet pure seeds requires one seed in each source component, so \\(\\psrank=2\\).  A single mixed vector \\(x_1+x_2\\) may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can be \\(1\\).\n\\end{microexample}","\\begin{theorem}[Coordinate transport]\n\\label{thm:coordinate-transport}\nLet $U:X\\to Y$ be a vector space isomorphism and let $\\Pcal=(P_\\alpha)$ be a packet system on $X$.  Define\n\\[\nQ_\\alpha:=UP_\\alpha U^{-1}.\n\\]\nThen $\\Qcal=(Q_\\alpha)$ is a packet system on $Y$.  For $\\widetilde T=UTU^{-1}$,\n\\[\n\\boxed{\nQ_\\beta\\widetilde TQ_\\alpha\n=\nU(P_\\beta TP_\\alpha)U^{-1}.\n}\n\\]\nHence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport.  If a declared algebra generating family $\\mathcal G$ is transported to $U\\mathcal G U^{-1}$, then its transfer graph and reachability relation are also preserved.\n\\end{theorem}\n\\RecordID{PSC-I-T-COORDINATE-TRANSPORT}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}","sm and let $\\Pcal=(P_\\alpha)$ be a packet system on $X$.  Define\n\\[\nQ_\\alpha:=UP_\\alpha U^{-1}.\n\\]\nThen $\\Qcal=(Q_\\alpha)$ is a packet system on $Y$.  For $\\widetilde T=UTU^{-1}$,\n\\[\n\\boxed{\nQ_\\beta\\widetilde TQ_\\alpha\n=\nU(P_\\beta TP_\\alpha)U^{-1}.\n}\n\\]\nHence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport.  If a declared algebra generating family $\\mathcal G$ is transported to $U\\mathcal G U^{-1}$, then its transfer graph and reachability relation are also preserved.\n\\end{theorem}\n\\RecordID{PSC-I-T-COORDINATE-TRANSPORT}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}","e a packet system on $X$.  Define\n\\[\nQ_\\alpha:=UP_\\alpha U^{-1}.\n\\]\nThen $\\Qcal=(Q_\\alpha)$ is a packet system on $Y$.  For $\\widetilde T=UTU^{-1}$,\n\\[\n\\boxed{\nQ_\\beta\\widetilde TQ_\\alpha\n=\nU(P_\\beta TP_\\alpha)U^{-1}.\n}\n\\]\nHence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport.  If a declared algebra generating family $\\mathcal G$ is transported to $U\\mathcal G U^{-1}$, then its transfer graph and reachability relation are also preserved.\n\\end{theorem}\n\\RecordID{PSC-I-T-COORDINATE-TRANSPORT}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}","\\begin{proof}\nConjugation preserves products and sums, so\n\\[\nQ_\\alpha Q_\\beta\n=UP_\\alpha P_\\beta U^{-1}\n=\\delta_{\\alpha\\beta}Q_\\alpha,\n\\qquad\n\\sum_\\alpha Q_\\alpha=I_Y.\n\\]\nThe block identity is direct multiplication:\n\\[\nQ_\\beta\\widetilde TQ_\\alpha\n=UP_\\beta TP_\\alpha U^{-1}.\n\\]\nAll listed invariants are determined by these transported blocks.\n\\end{proof}","\\begin{definition}[Packet compatible operator morphism]\n\\label{def:packet-morphism}\nLet $(X,\\A,\\Pcal)$ and $(Y,\\B,\\Qcal)$ be packet resolved operator modules with packet index sets $I$ and $J$.  A packet compatible operator morphism is a triple\n\\[\n(\\Phi,\\varphi,\\sigma)\n\\]\nconsisting of a unital algebra homomorphism $\\varphi:\\A\\to\\B$, a linear map $\\Phi:X\\to Y$ satisfying\n\\[\n\\Phi(Ax)=\\varphi(A)\\Phi(x),\n\\]\nand a map $\\sigma:I\\to J$ satisfying\n\\[\nQ_j\\Phi\n=\n\\Phi P_{\\sigma^{-1}(j)},\n\\qquad\nP_{\\sigma^{-1}(j)}:=\n\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nAn empty fiber is interpreted as the zero projector.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-MORPHISM}","Packet compatible operator morphism]\n\\label{def:packet-morphism}\nLet $(X,\\A,\\Pcal)$ and $(Y,\\B,\\Qcal)$ be packet resolved operator modules with packet index sets $I$ and $J$.  A packet compatible operator morphism is a triple\n\\[\n(\\Phi,\\varphi,\\sigma)\n\\]\nconsisting of a unital algebra homomorphism $\\varphi:\\A\\to\\B$, a linear map $\\Phi:X\\to Y$ satisfying\n\\[\n\\Phi(Ax)=\\varphi(A)\\Phi(x),\n\\]\nand a map $\\sigma:I\\to J$ satisfying\n\\[\nQ_j\\Phi\n=\n\\Phi P_{\\sigma^{-1}(j)},\n\\qquad\nP_{\\sigma^{-1}(j)}:=\n\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nAn empty fiber is interpreted as the zero projector.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-MORPHISM}","orphism]\n\\label{def:packet-morphism}\nLet $(X,\\A,\\Pcal)$ and $(Y,\\B,\\Qcal)$ be packet resolved operator modules with packet index sets $I$ and $J$.  A packet compatible operator morphism is a triple\n\\[\n(\\Phi,\\varphi,\\sigma)\n\\]\nconsisting of a unital algebra homomorphism $\\varphi:\\A\\to\\B$, a linear map $\\Phi:X\\to Y$ satisfying\n\\[\n\\Phi(Ax)=\\varphi(A)\\Phi(x),\n\\]\nand a map $\\sigma:I\\to J$ satisfying\n\\[\nQ_j\\Phi\n=\n\\Phi P_{\\sigma^{-1}(j)},\n\\qquad\nP_{\\sigma^{-1}(j)}:=\n\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nAn empty fiber is interpreted as the zero projector.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-MORPHISM}","\\begin{definition}[Morphism of seeded systems]\n\\label{def:seed-morphism}\nA morphism\n\\[\n(X,\\A,S,\\Pcal)\\longrightarrow(Y,\\B,S',\\Qcal)\n\\]\nis a packet compatible operator morphism satisfying the seed condition\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S').\n\\]\nIn the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-MORPHISM}","\\begin{theorem}[Naturality of generated hulls and transfer blocks]\n\\label{thm:naturality}\nLet $(\\Phi,\\varphi,\\sigma)$ be a packet compatible operator morphism.  For every subset $S\\subseteq X$,\n\\[\n\\boxed{\n\\Phi(\\Hull_{\\A}(S))\n=\n\\Hull_{\\varphi(\\A)}(\\Phi(S))\n\\subseteq\n\\Hull_{\\B}(\\Phi(S)).\n}\n\\]\nFor $T\\in\\A$ and $i,j\\in J$,\n\\[\n\\boxed{\nQ_j\\varphi(T)Q_i\\Phi\n=\n\\sum_{\\substack{\\sigma(\\beta)=j","\\begin{theorem}[Naturality of generated hulls and transfer blocks]\n\\label{thm:naturality}\nLet $(\\Phi,\\varphi,\\sigma)$ be a packet compatible operator morphism.  For every subset $S\\subseteq X$,\n\\[\n\\boxed{\n\\Phi(\\Hull_{\\A}(S))\n=\n\\Hull_{\\varphi(\\A)}(\\Phi(S))\n\\subseteq\n\\Hull_{\\B}(\\Phi(S)).\n}\n\\]\nFor $T\\in\\A$ and $i,j\\in J$,\n\\[\n\\boxed{\nQ_j\\varphi(T)Q_i\\Phi\n=\n\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n}\n\\]\nIf $\\Phi$ is surjective and $S$ generates $X$, then $\\Phi(S)$ generates $Y$ under $\\varphi(\\A)$ and hence under $\\B$.  If, in addition, the morphism is seeded, then $S'$ generates $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-NATURALITY}\n\\Depends{PSC-I-D-PACKET-MORPHISM and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{The canonical naturality theorem for every later realization.}","sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n}\n\\]\nIf $\\Phi$ is surjective and $S$ generates $X$, then $\\Phi(S)$ generates $Y$ under $\\varphi(\\A)$ and hence under $\\B$.  If, in addition, the morphism is seeded, then $S'$ generates $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-NATURALITY}\n\\Depends{PSC-I-D-PACKET-MORPHISM and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{The canonical naturality theorem for every later realization.}","\\begin{proof}\nEvery element of $\\Phi(\\Hull_{\\A}(S))$ is a finite sum of vectors\n\\[\n\\Phi(As)=\\varphi(A)\\Phi(s),\n\\]\nso it belongs to $\\Hull_{\\varphi(\\A)}(\\Phi(S))$.  Conversely, every generator $\\varphi(A)\\Phi(s)$ equals $\\Phi(As)$, giving equality.","For the block formula, use intertwining and packet compatibility twice:\n\\begin{align*}\nQ_j\\varphi(T)Q_i\\Phi\n&=Q_j\\varphi(T)\\Phi P_{\\sigma^{-1}(i)}\\\\\n&=Q_j\\Phi TP_{\\sigma^{-1}(i)}\\\\\n&=\\Phi P_{\\sigma^{-1}(j)}TP_{\\sigma^{-1}(i)}\\\\\n&=\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n\\end{align*}\nIf $\\Phi$ is surjective and $\\Hull_{\\A}(S)=X$, the hull equality gives $Y=\\Hull_{\\varphi(\\A)}(\\Phi(S))$.  If $\\Phi(S)\\subseteq\\Hull_{\\B}(S')$, then $\\Hull_{\\B}(\\Phi(S))\\subseteq\\Hull_{\\B}(S')$, so $S'$ generates $Y$.\n\\end{proof}","sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n\\end{align*}\nIf $\\Phi$ is surjective and $\\Hull_{\\A}(S)=X$, the hull equality gives $Y=\\Hull_{\\varphi(\\A)}(\\Phi(S))$.  If $\\Phi(S)\\subseteq\\Hull_{\\B}(S')$, then $\\Hull_{\\B}(\\Phi(S))\\subseteq\\Hull_{\\B}(S')$, so $S'$ generates $Y$.\n\\end{proof}","\\begin{proposition}[Category of seeded packet systems]\n\\label{prop:seed-category}\nPacket compatible operator morphisms compose, seeded morphisms compose, and identity morphisms satisfy both definitions.  Thus packet resolved operator modules form a category $\\mathbf{PktMod}$ and seeded packet systems form a category $\\mathbf{SeedSys}$.\n\\end{proposition}\n\\RecordID{PSC-I-P-SEED-SYSTEM-CATEGORY}\n\\Depends{\\nolinkurl{PSC-I-D-PACKET-MORPHISM},\n\\nolinkurl{PSC-I-D-SEED-MORPHISM}, and\n\\nolinkurl{PSC-I-T-NATURALITY}.}","\\begin{proof}\nLet $(\\Phi,\\varphi,\\sigma):(X,\\A,\\Pcal)\\to(Y,\\B,\\Qcal)$ and $(\\Psi,\\psi,\\tau):(Y,\\B,\\Qcal)\\to(Z,\\Ccal,\\Rcal)$ be morphisms.  Intertwining is preserved because\n\\[\n\\Psi\\Phi(Ax)\n=\n\\Psi(\\varphi(A)\\Phi x)\n=\n(\\psi\\circ\\varphi)(A)\\Psi\\Phi x.\n\\]\nFor $k$ in the target packet set,\n\\begin{align*}\nR_k\\Psi\\Phi\n&=\\Psi Q_{\\tau^{-1}(k)}\\Phi","\\begin{proof}\nLet $(\\Phi,\\varphi,\\sigma):(X,\\A,\\Pcal)\\to(Y,\\B,\\Qcal)$ and $(\\Psi,\\psi,\\tau):(Y,\\B,\\Qcal)\\to(Z,\\Ccal,\\Rcal)$ be morphisms.  Intertwining is preserved because\n\\[\n\\Psi\\Phi(Ax)\n=\n\\Psi(\\varphi(A)\\Phi x)\n=\n(\\psi\\circ\\varphi)(A)\\Psi\\Phi x.\n\\]\nFor $k$ in the target packet set,\n\\begin{align*}\nR_k\\Psi\\Phi\n&=\\Psi Q_{\\tau^{-1}(k)}\\Phi\\\\\n&=\\Psi\\Phi\n\\sum_{\\tau(\\sigma(\\alpha))=k}P_\\alpha.\n\\end{align*}\nThus the composite packet map is $\\tau\\circ\\sigma$.  If the morphisms are seeded, then\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S'),\n\\qquad\n\\Psi(S')\\subseteq\\Hull_{\\Ccal}(S'').\n\\]\nBy \\cref{thm:naturality},\n\\[\n\\Psi\\Phi(S)\n\\subseteq\n\\Psi(\\Hull_{\\B}(S'))\n\\subseteq\n\\Hull_{\\Ccal}(\\Psi(S'))\n\\subseteq\n\\Hull_{\\Ccal}(S'').\n\\]\nIdentity maps satisfy the definitions with identity algebra and packet maps.\n\\end{proof}","&=\\Psi\\Phi\n\\sum_{\\tau(\\sigma(\\alpha))=k}P_\\alpha.\n\\end{align*}\nThus the composite packet map is $\\tau\\circ\\sigma$.  If the morphisms are seeded, then\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S'),\n\\qquad\n\\Psi(S')\\subseteq\\Hull_{\\Ccal}(S'').\n\\]\nBy \\cref{thm:naturality},\n\\[\n\\Psi\\Phi(S)\n\\subseteq\n\\Psi(\\Hull_{\\B}(S'))\n\\subseteq\n\\Hull_{\\Ccal}(\\Psi(S'))\n\\subseteq\n\\Hull_{\\Ccal}(S'').\n\\]\nIdentity maps satisfy the definitions with identity algebra and packet maps.\n\\end{proof}","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\R","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PAC","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and q","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","n invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","q X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","line P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","\\begin{proof}\nRestriction preserves idempotency, pairwise annihilation, and the identity sum on $K$.  If $x\\in K\\cap X_\\alpha$, then $x=P_\\alpha x\\in P_\\alpha K$, proving $K_\\alpha=K\\cap X_\\alpha$.","The quotient operator is well defined because replacing $x$ by $x+k$ changes $P_\\alpha x$ by $P_\\alpha k\\in K$.  The idempotent and annihilation identities descend, and the sum is the identity on $X/K$.  The map $X_\\alpha\\to X/K$ has image $(X_\\alpha+K)/K$ and kernel $X_\\alpha\\cap K$, giving the short exact sequence and the quotient isomorphisms.  Exact packet coordinates identify the original sequence with the direct sum over $\\alpha$.\n\\end{proof}","fined because replacing $x$ by $x+k$ changes $P_\\alpha x$ by $P_\\alpha k\\in K$.  The idempotent and annihilation identities descend, and the sum is the identity on $X/K$.  The map $X_\\alpha\\to X/K$ has image $(X_\\alpha+K)/K$ and kernel $X_\\alpha\\cap K$, giving the short exact sequence and the quotient isomorphisms.  Exact packet coordinates identify the original sequence with the direct sum over $\\alpha$.\n\\end{proof}","\\begin{microexample}[title={Packet descent to a quotient}]\nLet \\(X=X_1\\oplus X_2\\) and let \\(K\\subseteq X_1\\) be a subspace.  The quotient has packets \\(X_1/K\\) and \\(X_2\\).  Any operator preserving \\(K\\) descends with the same organization by source and target packets, with blocks entering or leaving the first packet replaced by their induced quotient maps.  An induced block can become zero even when its original block is nonzero, so the pattern of nonzero blocks can shrink.  Any resulting zero packet is removed from the quotient packet system.\n\\end{microexample}","\\begin{theorem}[Packet descent through an invariant quotient]\n\\label{thm:quotient-packet-descent}\nUnder the hypotheses of \\cref{thm:packet-exact-sequence}, the operators\n\\[\n\\overline P_\\alpha(x+K)=P_\\alpha x+K\n\\]\nare well defined pairwise annihilating idempotents with sum $I_{X/K}$, and the quotient map satisfies\n\\[\n\\boxed{\\pi P_\\alpha=\\overline P_\\alpha\\pi.}\n\\]\nAfter zero packets are removed, they form the canonical quotient packet system.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-PACKET-DESCENT}\n\\Depends{PSC-I-T-PACKET-EXACT-SEQUENCE.}","\\begin{proof}\nAll assertions are the quotient part of \\cref{thm:packet-exact-sequence}.  The intertwining identity follows directly from the definition:\n\\[\n\\overline P_\\alpha\\pi(x)=P_\\alpha x+K=\\pi P_\\alpha(x).\n\\]\n\\end{proof}","\\begin{corollary}[Restriction and quotient of transfer blocks]\n\\label{cor:block-descent}\nUnder the hypotheses of \\cref{thm:packet-exact-sequence}, let $T\\in\\End(X)$ preserve $K$.  Then every block $T_{\\beta\\alpha}$ maps $K_\\alpha$ into $K_\\beta$, and the induced quotient operator $\\overline T$ satisfies\n\\[\n\\boxed{\n\\overline T_{\\beta\\alpha}\\,\\pi\n=\n\\pi T_{\\beta\\alpha},\n}\n\\]\nwhere $\\pi:X\\twoheadrightarrow X/K$.\n\\end{corollary}\n\\RecordID{PSC-I-C-QUOTIENT-PACKET-DESCENT}\n\\Depends{PSC-I-T-PACKET-EXACT-SEQUENCE.}","\\begin{proof}\nIf $x\\in K_\\alpha$, then $Tx\\in K$, so\n\\[\nT_{\\beta\\alpha}x=P_\\beta Tx\\in K\\cap X_\\beta=K_\\beta.\n\\]\nOn the quotient,\n\\[\n\\overline T_{\\beta\\alpha}\\pi\n=\n\\overline P_\\beta\\overline T\\overline P_\\alpha\\pi\n=\n\\pi P_\\beta TP_\\alpha.\n\\]\n\\end{proof}","\\begin{corollary}[Topological packet exact sequence]\n\\label{cor:topological-packet-exact-sequence}\nIf $X$ is Banach, the $P_\\alpha$ are bounded, and $K$ is closed and packet invariant, then every packet sequence in \\cref{thm:packet-exact-sequence} is a short exact sequence of Banach spaces with bounded maps, and the quotient projectors are bounded.\n\\end{corollary}\n\\RecordID{PSC-I-C-TOPOLOGICAL-PACKET-EXACT-SEQUENCE}\n\\Depends{\\nolinkurl{PSC-I-T-PACKET-EXACT-SEQUENCE} and\n\\nolinkurl{PSC-I-T-BANACH-PACKET-ISOMORPHISM}.}","\\begin{proof}\nClosedness of $K$ makes $X/K$ Banach.  The restricted projectors are bounded, and the quotient projectors are bounded because\n\\[\n\\|\\overline P_\\alpha(x+K)\\|_{X/K}\n\\leq\n\\|P_\\alpha\\|\\,\\|x+K\\|_{X/K}.\n\\]\nThe packet spaces are closed ranges of bounded idempotents.\n\\end{proof}","\\begin{theorem}[Dense realization theorem]\n\\label{thm:dense-realization}\nLet $X$ and $Y$ be Hausdorff topological vector spaces, let $\\Phi:X\\to Y$ be continuous, surjective, and intertwining with respect to $\\varphi$, and let $S$ topologically generate $X$.  Then $\\Phi(S)$ topologically generates $Y$.  If finite continuous packet systems satisfy\n\\[\nQ_j\\Phi=\\Phi P_{\\sigma^{-1}(j)},\n\\]\nthen\n\\[\n\\boxed{\nQ_jY=\\Phi(P_{\\sigma^{-1}(j)}X)\n}\n\\]\nfor every $j$.\n\\end{theorem}\n\\RecordID{PSC-I-T-DENSE-REALIZATION}\n\\Depends{PSC-I-T-NATURALITY.}","thm:dense-realization}\nLet $X$ and $Y$ be Hausdorff topological vector spaces, let $\\Phi:X\\to Y$ be continuous, surjective, and intertwining with respect to $\\varphi$, and let $S$ topologically generate $X$.  Then $\\Phi(S)$ topologically generates $Y$.  If finite continuous packet systems satisfy\n\\[\nQ_j\\Phi=\\Phi P_{\\sigma^{-1}(j)},\n\\]\nthen\n\\[\n\\boxed{\nQ_jY=\\Phi(P_{\\sigma^{-1}(j)}X)\n}\n\\]\nfor every $j$.\n\\end{theorem}\n\\RecordID{PSC-I-T-DENSE-REALIZATION}\n\\Depends{PSC-I-T-NATURALITY.}","\\begin{proof}\nGiven $y=\\Phi(x)$, choose a net $x_\\lambda\\in\\Hull_{\\A}(S)$ converging to $x$.  Then\n\\[\n\\Phi(x_\\lambda)\\to y,\n\\]\nand naturality places every $\\Phi(x_\\lambda)$ in $\\Hull_{\\B}(\\Phi(S))$.  Hence the latter hull is dense.  For packets, one inclusion follows from compatibility.  Conversely, if $y=Q_jy$, choose $x$ with $\\Phi(x)=y$.  Then\n\\[\ny=Q_j\\Phi(x)=\\Phi P_{\\sigma^{-1}(j)}x.\n\\]\n\\end{proof}","\\begin{theorem}[Quotient topology and completed realization]\n\\label{thm:quotient-topology}\nLet $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","m}[Quotient topology and completed realization]\n\\label{thm:quotient-topology}\nLet $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","leted realization]\n\\label{thm:quotient-topology}\nLet $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","et $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","e a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","\\begin{proof}\nThe algebraic first isomorphism theorem gives a bijection $\\widetilde\\Phi$.  By definition of the quotient topology, a set $U\\subseteq X/K$ is open exactly when its preimage in $X$ is open.  Because $\\Phi$ is a quotient map, this is equivalent to $\\widetilde\\Phi(U)$ being open in $Y$.  Thus $\\widetilde\\Phi$ is a homeomorphism.  The second statement applies the dense completion argument of \\cref{thm:completed-presentation} to the injective map $D/(D\\cap K)\\to Y$.\n\\end{proof}","bijection $\\widetilde\\Phi$.  By definition of the quotient topology, a set $U\\subseteq X/K$ is open exactly when its preimage in $X$ is open.  Because $\\Phi$ is a quotient map, this is equivalent to $\\widetilde\\Phi(U)$ being open in $Y$.  Thus $\\widetilde\\Phi$ is a homeomorphism.  The second statement applies the dense completion argument of \\cref{thm:completed-presentation} to the injective map $D/(D\\cap K)\\to Y$.\n\\end{proof}","\\begin{boundary}[Continuous surjections and quotient topology]\n\\label{bnd:topological-quotient}\nContinuity and surjectivity transport density as in \\cref{thm:dense-realization}.  They do not by themselves identify the given topology on $Y$ with the quotient topology from $X$.  The stronger conclusion in \\cref{thm:quotient-topology} requires the quotient map hypothesis or an equivalent open mapping condition.\n\\end{boundary}\n\\RecordID{PSC-I-B-TOPOLOGICAL-QUOTIENT}","\\begin{definition}[Realization functor and preservation profile]\n\\label{def:realization-functor}\nLet $\\mathbf{SeedSys}_{\\mathrm{abs}}$ be an abstract category of seed systems and let $\\mathbf C_D$ be a domain category.  A realization functor is a functor\n\\[\n\\mathsf{Ev}_D:\\mathbf{SeedSys}_{\\mathrm{abs}}\\longrightarrow\\mathbf C_D\n\\]\nwith a declared preservation profile specifying which of the following structures are preserved:\n\\[\n\\begin{gathered}\n0,\\ I,\\ \\text{finite sums},\\ \\text{composition},\\ \\text{biproducts},","unctor and preservation profile]\n\\label{def:realization-functor}\nLet $\\mathbf{SeedSys}_{\\mathrm{abs}}$ be an abstract category of seed systems and let $\\mathbf C_D$ be a domain category.  A realization functor is a functor\n\\[\n\\mathsf{Ev}_D:\\mathbf{SeedSys}_{\\mathrm{abs}}\\longrightarrow\\mathbf C_D\n\\]\nwith a declared preservation profile specifying which of the following structures are preserved:\n\\[\n\\begin{gathered}\n0,\\ I,\\ \\text{finite sums},\\ \\text{composition},\\ \\text{biproducts},\\\\\n\\text{kernels},\\ \\text{cokernels},\\ \\text{quotients},\\ \\text{tensor products},\\\\\n\\text{packet idempotents},\\ \\text{transfer blocks},\\ \\text{topological completions}.\n\\end{gathered}\n\\]\nPreservation of a structure means preservation of its defining identity or universal property, not merely a similarity of notation.\n\\end{definition}\n\\RecordID{PSC-I-D-REALIZATION-FUNCTOR}","\\begin{theorem}[Equational realization principle]\n\\label{thm:realization-export}\nLet $F:\\mathbf C\\to\\mathbf D$ be an additive functor between preadditive categories.  Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATIO","\\begin{theorem}[Equational realization principle]\n\\label{thm:realization-export}\nLet $F:\\mathbf C\\to\\mathbf D$ be an additive functor between preadditive categories.  Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","label{thm:realization-export}\nLet $F:\\mathbf C\\to\\mathbf D$ be an additive functor between preadditive categories.  Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","rom declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","reserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","\\begin{proof}\nFunctoriality gives\n\\[\nF(I)=I,\n\\qquad\nF(g\\circ f)=F(g)\\circ F(f),\n\\]\nand additivity gives\n\\[\nF(0)=0,\n\\qquad\nF(f+g)=F(f)+F(g),\n\\qquad\nF(-f)=-F(f).\n\\]\nIf the declared enrichment is present, linearity additionally gives $F(af)=aF(f)$ for every coefficient $a$.  A structural induction on the construction of a morphism expression therefore carries every permitted equational proof to its image under $F$.  If a step invokes a universal construction, preservation means that the image diagram satisfies the same universal property.  The uniquely induced morphism in the source proof is therefore carried to the uniquely induced morphism in the target proof.  Repeating this argument through the finite proof diagram transports the conclusion.\n\\end{proof}","}\nFunctoriality gives\n\\[\nF(I)=I,\n\\qquad\nF(g\\circ f)=F(g)\\circ F(f),\n\\]\nand additivity gives\n\\[\nF(0)=0,\n\\qquad\nF(f+g)=F(f)+F(g),\n\\qquad\nF(-f)=-F(f).\n\\]\nIf the declared enrichment is present, linearity additionally gives $F(af)=aF(f)$ for every coefficient $a$.  A structural induction on the construction of a morphism expression therefore carries every permitted equational proof to its image under $F$.  If a step invokes a universal construction, preservation means that the image diagram satisfies the same universal property.  The uniquely induced morphism in the source proof is therefore carried to the uniquely induced morphism in the target proof.  Repeating this argument through the finite proof diagram transports the conclusion.\n\\end{proof}","\\begin{definition}[Chapter I export object]\n\\label{def:paper-one-export}\nThe canonical Chapter I export is\n\\[\n\\boxed{\n\\mathsf{PSC}_I=\n\\left(\n\\begin{array}{c}\n\\text{typed grammars and realized operator algebras},\\\\\n\\text{seed systems, free presentations, and seed rank},\\\\\n\\text{module-top, semisimple, and single-operator formulas},\\\\\n\\text{finite and unconditional packet coordinates},\\\\\n\\text{transfer blocks and support algebras},\\\\\n\\text{reachability and packet-pure source generation},\\\\\n\\text{morphisms, invariant exact sequences, quotients, completions},\\\\\n\\text{realization functors and equational theorem transport}\n\\end{array}\n\\right).\n}\n\\]\n\\end{definition}\n\\RecordID{PSC-I-EXPORT}","Let the quiver have vertices $U,V$ and arrows\n\\[\nR:U\\to U,\n\\qquad\nA:U\\to V,\n\\qquad\nB:V\\to U,\n\\]\nwith the relation in the path algebra\n\\[\nBA=e_U,\n\\]\nwhose realization is the operator identity $BA=I_U$.\nRealize\n\\[\nU=\\bbk^2,\n\\qquad\nV=\\bbk^3,\n\\]\nby\n\\[\nR=\n\\begin{pmatrix}0&-1","Let the quiver have vertices $U,V$ and arrows\n\\[\nR:U\\to U,\n\\qquad\nA:U\\to V,\n\\qquad\nB:V\\to U,\n\\]\nwith the relation in the path algebra\n\\[\nBA=e_U,\n\\]\nwhose realization is the operator identity $BA=I_U$.\nRealize\n\\[\nU=\\bbk^2,\n\\qquad\nV=\\bbk^3,\n\\]\nby\n\\[\nR=\n\\begin{pmatrix}0&-1\\\\1&0\\end{pmatrix},\n\\qquad\nA(u_1,u_2)=(u_1,u_2,0),\n\\qquad\nB(v_1,v_2,v_3)=(v_1,v_2).\n\\]\nStarting from the seed $e_1\\in U$, repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.","1&0\\end{pmatrix},\n\\qquad\nA(u_1,u_2)=(u_1,u_2,0),\n\\qquad\nB(v_1,v_2,v_3)=(v_1,v_2).\n\\]\nStarting from the seed $e_1\\in U$, repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.","2,0),\n\\qquad\nB(v_1,v_2,v_3)=(v_1,v_2).\n\\]\nStarting from the seed $e_1\\in U$, repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.",", repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.","\\begin{proposition}[Typed reachability certificate]\n\\label{prop:typed-reachability}\nIn the preceding quiver realization, no seed supported at $U$ can generate the full representation, whereas the two seeds\n\\[\ne_1\\in U,\n\\qquad\ne_3\\in V\n\\]\ndo generate it.\n\\end{proposition}","\\begin{proof}\nThe first claim follows because every path that starts at $U$ and ends in $V$ has final arrow $A$, so its image lies in $\\im A$. For the second, $e_1$ and $Re_1$ span $U$. Their images under $A$ span $\\im A$, and $e_3$ spans $\\ker B$, the missing complementary direction in $V$.\n\\end{proof}","This example shows why a typed grammar is more than a notational convenience. The obstruction lies in the subspace $\\ker B\\subset V$, which is unreachable from seeds supported at $U$, and is visible before any coordinate calculation.","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of de","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concr","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","c curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","mplication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","ghtarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","ities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","alize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","ctive smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","e geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","Take the seed\n\\[\ns=1\n\\]\nand let $M_x,M_y$ be multiplication by $x$ and $y$. Then\n\\[\nM_x^iM_y^js=x^iy^j,\n\\]\nso the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","The two packet spaces are\n\\[\nA_0=\\bbk[x],\n\\qquad\nA_1=y\\bbk[x].\n\\]\nLet\n\\[\nP_0(a+yb)=a,\n\\qquad\nP_1(a+yb)=yb.\n\\]\nThese idempotents are linear over $\\bbk$, annihilate one another, and satisfy $P_0+P_1=I$. They are not module maps over $A$, because multiplication by $y$ exchanges the two packets.","Identify the packet coefficients by\n\\[\nA_0\\oplus A_1\\longrightarrow\\bbk[x]\\oplus\\bbk[x],\n\\qquad\n(a,yb)\\longmapsto(a,b).\n\\]\nIn these ordered coefficient coordinates, with $M_x$ and $M_{x^3-x}$ denoting multiplication on $\\bbk[x]$,\n\\[\n{\nM_x=\n\\begin{pmatrix}\nM_x&0","Identify the packet coefficients by\n\\[\nA_0\\oplus A_1\\longrightarrow\\bbk[x]\\oplus\\bbk[x],\n\\qquad\n(a,yb)\\longmapsto(a,b).\n\\]\nIn these ordered coefficient coordinates, with $M_x$ and $M_{x^3-x}$ denoting multiplication on $\\bbk[x]$,\n\\[\n{\nM_x=\n\\begin{pmatrix}\nM_x&0\\\\\n0&M_x\n\\end{pmatrix},\n\\qquad\nM_y=\n\\begin{pmatrix}\n0&M_{x^3-x}\\\\\nI&0\n\\end{pmatrix}.\n}\n\\]\nThe two rows of the second matrix record the outgoing packet coefficients. An even input enters the odd packet without an additional factor. An odd input returns to the even packet through the factor imposed by the curve equation. Applying the matrix twice therefore preserves each packet and multiplies its coefficient by that factor.","The relation is reconstructed as an operator identity:\n\\[\nM_y^2=M_x^3-M_x.\n\\]","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{08_elliptic_packet_transfer.pdf}\n\\caption{Multiplication by $y$ exchanges the even and odd packets. The transfer from the odd packet to the even packet carries the relation weight $x^3-x$.}\n\\label{fig:elliptic-transfer}\n\\end{figure}","Write\n\\[\nf=a_0+ya_1,\n\\qquad\ng=b_0+yb_1.\n\\]\nThen\n\\[\n{\nfg\n=\n\\bigl(a_0b_0+(x^3-x)a_1b_1\\bigr)\n+\ny\\bigl(a_0b_1+a_1b_0\\bigr).\n}\n\\]\nThis is a convolution law for the two packets. It is already visible at the linear operator level because multiplication by a fixed element is a transfer operator. The even output collects the product of the two even coefficients and the weighted product of the two odd coefficients. The odd output collects the two products with one even and one odd coefficient. The quotient relation is thus retained in the weight rather than discarded during the decomposition.","\\begin{example}[Worked reduction]\nTake\n\\[\nf=x^2+y,\n\\qquad\ng=x-y.\n\\]\nThen\n\\[\n\\begin{aligned}\nfg\n&=x^3-x^2y+xy-y^2","\\begin{example}[Worked reduction]\nTake\n\\[\nf=x^2+y,\n\\qquad\ng=x-y.\n\\]\nThen\n\\[\n\\begin{aligned}\nfg\n&=x^3-x^2y+xy-y^2\\\\\n&=x^3-x^2y+xy-(x^3-x)\\\\\n&=x+(x-x^2)y.\n\\end{aligned}\n\\]\nThus\n\\[\nP_0(fg)=x,\n\\qquad\nP_1(fg)=(x-x^2)y.\n\\]\nThe symbolic certificate in the companion code performs the same quotient reduction exactly.\n\\end{example}","&=x+(x-x^2)y.\n\\end{aligned}\n\\]\nThus\n\\[\nP_0(fg)=x,\n\\qquad\nP_1(fg)=(x-x^2)y.\n\\]\nThe symbolic certificate in the companion code performs the same quotient reduction exactly.\n\\end{example}","The coordinate ring illustrates four distinct parts of the calculus:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the unit is a module seed.\n\\item $M_x$ and $M_y$ are address operators.\n\\item the curve equation is an operator relation.\n\\item parity in $y$ gives exact packet analysis, transfer, and reconstruction.\n\\end{enumerate}\nIt does not imply that every algebra can be presented as an algebra generated by one element. It shows instead that unital algebras are cyclic as modules over their full multiplication algebra.","Let $\\mathbb{F}\\in\\{\\R,\\C\\}$ and let\n\\[\nX=\\ell^2(\\Z;\\mathbb F)\n\\]\nwith a family of bounded reindexing operators indexed by the integers\n\\[\n(S_mx)[n]=x[n-m]\n\\qquad(m,n\\in\\Z).\n\\]\nThus $S_0=I$, $S_{m+r}=S_mS_r$, and $S_1$ is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of \\cite{conway1990}.\nThe impulse $\\delta_0$ satisfies\n\\[\nS_m\\delta_0=\\delta_m\n\\]\nfor every $m\\in\\Z$. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in $\\ell^2(\\Z)$.","\\in\\{\\R,\\C\\}$ and let\n\\[\nX=\\ell^2(\\Z;\\mathbb F)\n\\]\nwith a family of bounded reindexing operators indexed by the integers\n\\[\n(S_mx)[n]=x[n-m]\n\\qquad(m,n\\in\\Z).\n\\]\nThus $S_0=I$, $S_{m+r}=S_mS_r$, and $S_1$ is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of \\cite{conway1990}.\nThe impulse $\\delta_0$ satisfies\n\\[\nS_m\\delta_0=\\delta_m\n\\]\nfor every $m\\in\\Z$. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in $\\ell^2(\\Z)$.","dexed by the integers\n\\[\n(S_mx)[n]=x[n-m]\n\\qquad(m,n\\in\\Z).\n\\]\nThus $S_0=I$, $S_{m+r}=S_mS_r$, and $S_1$ is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of \\cite{conway1990}.\nThe impulse $\\delta_0$ satisfies\n\\[\nS_m\\delta_0=\\delta_m\n\\]\nfor every $m\\in\\Z$. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in $\\ell^2(\\Z)$.","\\begin{proposition}[Shift seed]\n\\label{prop:shift-seed}\nFor $n\\in\\N$, let $G(n)$ mean that some finite seed set with $n$ elements has dense hull under the declared family $\\{S_m:m\\in\\Z\\}$. Then\n\\[\n{\n1=\\min\\{n\\in\\N:G(n)\\}.\n}\n\\]\nEquivalently, one seed suffices and no empty seed family generates densely.\n\\end{proposition}","\\begin{proof}\nThe orbit of $\\delta_0$ contains every standard basis vector. Their finite linear span is dense, so $G(1)$ holds. The hull of the empty seed set is zero, while $\\ell^2(\\Z;\\mathbb F)$ is nonzero, so $G(0)$ fails. Therefore $1$ is the least admissible finite cardinality.\n\\end{proof}","For $N\\ge1$, define residue projectors\n\\[\n(P_{N,j}x)[n]\n=\n\\begin{cases}\nx[n],&n\\equiv j\\pmod N,","For $N\\ge1$, define residue projectors\n\\[\n(P_{N,j}x)[n]\n=\n\\begin{cases}\nx[n],&n\\equiv j\\pmod N,\\\\0,&\\text{otherwise.}\\end{cases}\n\\]\nThey form an orthogonal packet system and satisfy\n\\[\nS_1P_{N,j}=P_{N,j+1}S_1,\n\\]\nwith indices modulo $N$. The shift is therefore a cyclic transfer between residue packets.","0,&\\text{otherwise.}\\end{cases}\n\\]\nThey form an orthogonal packet system and satisfy\n\\[\nS_1P_{N,j}=P_{N,j+1}S_1,\n\\]\nwith indices modulo $N$. The shift is therefore a cyclic transfer between residue packets.","Let $X=\\bbk^n$ and let $\\cA=M_n(\\bbk)=\\End_{\\bbk}(X)$, with the full matrix algebra declared before a seed or target is chosen. By \\cref{cor:full-matrix}, every nonzero seed is cyclic. This is the maximal linear grammar on $X$. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.","For one operator $T$, \\cref{thm:single-operator} turns verified primary data\ninto an exact certificate for the minimum number of seeds.  The numerical companion compares:\n\\begin{enumerate}[label=(\\alph*)]\n\\item a nilpotent companion operator with one Jordan chain, for which the Krylov rank is $1,2,\\ldots,n$.\n\\item two equal Jordan blocks, for which one diagonal seed produces a chain only as long as one block and then stalls.\n\\end{enumerate}","Let $G$ be a nonempty finite graph with adjacency matrix $A_G$, and regard\n$\\bbk^{V(G)}$ as a module over $\\bbk[A_G]$.  The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.","\\begin{corollary}[Certified graph seed number]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","gin{corollary}[Certified graph seed number]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","aph seed number]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","r]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","et $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","ld and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","se\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}\\\\[2pt]\n\\bbk^{V(G)}=\\Span_{\\bbk}\n  \\{p(A_G)s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\end{array}\n\\right\\}.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nThe distinct eigenvalue labels give Lagrange polynomials that project onto\nthe individual eigenspace packets.  Padding one basis vector from every\npacket into each of $R$ seeds gives the upper bound.  Projection to the\n$m_{\\lambda}$ eigenspace coordinates is a surjective scalar quotient, so any\ngenerating family has at least $m_{\\lambda}$ members for every $\\lambda$.\nThe label $\\lambda_*$ gives the matching lower bound $R$.\n\\end{proof}","labels give Lagrange polynomials that project onto\nthe individual eigenspace packets.  Padding one basis vector from every\npacket into each of $R$ seeds gives the upper bound.  Projection to the\n$m_{\\lambda}$ eigenspace coordinates is a surjective scalar quotient, so any\ngenerating family has at least $m_{\\lambda}$ members for every $\\lambda$.\nThe label $\\lambda_*$ gives the matching lower bound $R$.\n\\end{proof}","\\begin{boundary}[Input from spectral classification]\nFor a finite undirected graph over $\\C$, the spectral theorem for real symmetric matrices\nclassically supplies a certificate of the displayed kind, and the numbers\n$m_{\\lambda}$ are the usual eigenvalue multiplicities. See\n\\cite{brouwerHaemers2012}.  The formal declaration verifies the\nformula for the number of seeds from the supplied eigenbasis and distinct eigenvalue\nlabels.  It does not derive diagonalizability from symmetry or implement an\neigensolver.\n\\end{boundary}","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex t","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for e","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path s","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","quently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","ctors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","point coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","inate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","Two adjacent vertex vectors generate the adjacency module of every simple\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\","Two adjacent vertex vectors generate the adjacency module of every simple\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for","Two adjacent vertex vectors generate the adjacency module of every simple\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","le\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","plies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","tching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","ce, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","ery $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","least finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","y seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","d directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.",", index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","from $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","The usual spectral description writes the eigenvalues as\n\\[\n\\lambda_k=2\\cos\\frac{2\\pi k}{n},\n\\qquad 0\\le k<n.\n\\]\nThe values for $k$ and $n-k$ coincide. Apart from the simple characters at $k=0$ and, for even $n$, $k=n/2$, the remaining Fourier characters occur in pairs. Therefore the maximum eigenspace multiplicity is two, although some eigenspaces have multiplicity one, and\nthe certified minimum above is two.","\\midrule\n\\endhead\nCoordinate ring & $1$ & $M_x,M_y$ & even/odd $y$ degree & quotient by $y^2-x^3+x$","Shift system & $\\delta_0$ & $(S_m)_{m\\in\\Z}$ & residue classes of $n$ & $\\ell^2$ closure","Full matrix module & $s\\ne0$ & $M_n(\\bbk)$ & chosen idempotents & finite dimensional","Single operator & $s_1,\\ldots,s_r$ & $\\bbk[T]$ & primary or spectral & finite dimensional","Graph adjacency & vertex seeds & $\\bbk[A_G]$ & eigenspaces & finite dimensional","Typed grammar & vertex seeds & path algebra $\\bbk Q/I$ & vertex idempotents & relation quotient","\\Needspace{16\\baselineskip}\nFor a finite ordered seed family $S=(s_1,\\ldots,s_r)$, let $\\varepsilon_S:\\cA^{\\oplus r}\\to X$ be the seed evaluation map of \\cref{def:seed-evaluation}.  Its kernel is the relation module.  The free module modulo that kernel presents the generated hull by \\cref{thm:free-presentation}.  The invariant content is:\n\\[\n{\n\\begin{aligned}\n&\\text{generated hull: }\\Hull_{\\cA}(S),","\\Needspace{16\\baselineskip}\nFor a finite ordered seed family $S=(s_1,\\ldots,s_r)$, let $\\varepsilon_S:\\cA^{\\oplus r}\\to X$ be the seed evaluation map of \\cref{def:seed-evaluation}.  Its kernel is the relation module.  The free module modulo that kernel presents the generated hull by \\cref{thm:free-presentation}.  The invariant content is:\n\\[\n{\n\\begin{aligned}\n&\\text{generated hull: }\\Hull_{\\cA}(S),\\\\\n&\\text{seed complexity: }\\srank_{\\cA}(X),\\\\\n&\\text{packet coordinates: }x\\leftrightarrow(P_{\\alpha}x)_{\\alpha},\\\\\n&\\text{operator coordinates: }T\\leftrightarrow(P_{\\beta}TP_{\\alpha})_{\\beta,\\alpha},\\\\\n&\\text{relation module: }N_S:=\\Ker\\varepsilon_S\\leq\\cA^{\\oplus r},\\\\\n&\\text{free presentation: }\\cA^{\\oplus r}/N_S\\cong\\Hull_{\\cA}(S),\\\\\n&\\text{coordinate invariance: }P_{\\alpha}\\mapsto UP_{\\alpha}U^{-1},\\\\\n&\\text{realization: }\\Phi A=\\varphi(A)\\Phi.\n\\end{aligned}\n}\n\\]","\\midrule\nProjector idempotence residual & $0$","Projector annihilation residual & $0$","Projector completeness residual & $0$","Operator reconstruction residual & $0$","Residual for transported projectors, maximum & $1.85\\times10^{-15}$","Companion seed rank & $1$","Two equal Jordan blocks, seed rank & $2$","$M_3(\\C)$ acting on seven copies, seed rank & $3$","Sampled path graph seed ranks, $2\\le n\\le12$ & $1$","Sampled cycle graph seed ranks, $3\\le n\\le12$ & $2$","Full $M_4(\\R)$ orbit dimensions for two sampled nonzero seeds & $4,4$","Automated tests & $33/33$ passed","The finite shift pattern recovered by the experiment is\n\\[\n\\begin{pmatrix}\n0&0&0&1\\\\\n1&0&0&0\\\\\n0&1&0&0\\\\\n0&0&1&0\n\\end{pmatrix},\n\\]\nwhich is the exact packet adjacency of a cyclic transfer among four phases.","\\begin{construction}[Closure specification for the orbit hull]\n\\label{con:orbit-hull-algorithm}\nLet $V$ have finite dimension, let $(O_i)_{i\\in I}$ be a finite operator\nfamily, and let $S\\subseteq V$.  Define\n\\[\nW_0=\\Span_{\\bbk}S,\n\\qquad\nW_{n+1}=W_n+\\sum_{i\\in I}O_i(W_n).\n\\]\nThen there is an $n\\in\\N$ such that\n\\[\nW_{n+1}=W_n\n\\qquad\\text{and}\\qquad\nW_n=\\Hull_{(O_i)}(S),\n\\]\nwhere the right side is the least subspace containing $S$ and invariant under\nevery $O_i$.  Thus the ascending subspace iteration terminates and its stable\nvalue is exactly the finite operator hull.","pecification for the orbit hull]\n\\label{con:orbit-hull-algorithm}\nLet $V$ have finite dimension, let $(O_i)_{i\\in I}$ be a finite operator\nfamily, and let $S\\subseteq V$.  Define\n\\[\nW_0=\\Span_{\\bbk}S,\n\\qquad\nW_{n+1}=W_n+\\sum_{i\\in I}O_i(W_n).\n\\]\nThen there is an $n\\in\\N$ such that\n\\[\nW_{n+1}=W_n\n\\qquad\\text{and}\\qquad\nW_n=\\Hull_{(O_i)}(S),\n\\]\nwhere the right side is the least subspace containing $S$ and invariant under\nevery $O_i$.  Thus the ascending subspace iteration terminates and its stable\nvalue is exactly the finite operator hull.","l]\n\\label{con:orbit-hull-algorithm}\nLet $V$ have finite dimension, let $(O_i)_{i\\in I}$ be a finite operator\nfamily, and let $S\\subseteq V$.  Define\n\\[\nW_0=\\Span_{\\bbk}S,\n\\qquad\nW_{n+1}=W_n+\\sum_{i\\in I}O_i(W_n).\n\\]\nThen there is an $n\\in\\N$ such that\n\\[\nW_{n+1}=W_n\n\\qquad\\text{and}\\qquad\nW_n=\\Hull_{(O_i)}(S),\n\\]\nwhere the right side is the least subspace containing $S$ and invariant under\nevery $O_i$.  Thus the ascending subspace iteration terminates and its stable\nvalue is exactly the finite operator hull.","\\begin{construction}[Table specification for transfer blocks]\n\\label{con:transfer-block-algorithm}\nGiven finite projector systems $(P_i)_{i\\in I}$ and $(Q_j)_{j\\in J}$ and a\nlinear map $T$, form the table\n\\[\nB_{ji}=Q_jTP_i.\n\\]\nIts assembly reconstructs the input map exactly,\n\\[\n\\sum_{j\\in J}\\sum_{i\\in I}B_{ji}=T,\n\\]\nand its reported support relation is exactly\n$\\{(i,j):B_{ji}\\ne0\\}$.  The latter equality is the specification of the\ntransfer graph output. It does not claim that Lean extracts a routine that decides the edge list for every coefficient field.\n\\end{construction}","\\begin{construction}[Exact graph seed rank from a certificate]\n\\label{con:exact-graph-seed-rank}\nFor a finite square matrix $A$ over a field $K$, define the noncomputable exact\nspectral quantity\n\\[\nr_{\\mathrm{spec}}(A)\n=\n\\max_{\\mu\\in\\Spec_K(A)}\n\\dim_K\\ker(A-\\mu I),\n\\]\nwith value zero for an empty spectrum.  Suppose an exact certificate supplies\na finite eigenbasis indexed by labels and multiplicities, the eigenvalue\nequations, a multiplicity bound attained by one label, projector polynomials\nthat extract the padded eigenbasis generators, coverage of every spectral\nvalue by a label, equality of the declared and actual eigenspace dimensions,\nand membership of the maximizing eigenvalue in the spectrum.  Then\n$r_{\\mathrm{spec}}(A)$ is the least natural number of seeds under polynomial orbits\nfor $A$.","an exact certificate supplies\na finite eigenbasis indexed by labels and multiplicities, the eigenvalue\nequations, a multiplicity bound attained by one label, projector polynomials\nthat extract the padded eigenbasis generators, coverage of every spectral\nvalue by a label, equality of the declared and actual eigenspace dimensions,\nand membership of the maximizing eigenvalue in the spectrum.  Then\n$r_{\\mathrm{spec}}(A)$ is the least natural number of seeds under polynomial orbits\nfor $A$.","the eigenvalue\nequations, a multiplicity bound attained by one label, projector polynomials\nthat extract the padded eigenbasis generators, coverage of every spectral\nvalue by a label, equality of the declared and actual eigenspace dimensions,\nand membership of the maximizing eigenvalue in the spectrum.  Then\n$r_{\\mathrm{spec}}(A)$ is the least natural number of seeds under polynomial orbits\nfor $A$.","\\subsection{Theorem spine at a glance}\n\\ConceptChain{\n\\(\n\\begin{array}{c}\n\\text{free seed presentation}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{minimal generator number}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{reduction modulo the Jacobson radical}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{semisimple or primary computation}\n\\end{array}\n\\qquad\\qquad\n\\begin{array}{c}\n\\text{packet idempotents}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{exact analysis/synthesis}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{transfer block algebra}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{graphs, quotients, and realization}\n\\end{array}\n\\)\n}","\\item[Block support.] The set of packet pairs \\((\\beta,\\alpha)\\) for which \\(P_\\beta TP_\\alpha\\neq0\\).\n\\item[Collapsed realization.] The direct sum of all typed vertex spaces.\n\\item[Condensation graph.] The acyclic graph obtained by collapsing strongly connected components.\n\\item[Generated hull.] The submodule generated by a seed family, or its declared closure.\n\\item[Grammar.] A typed quiver and relation ideal describing legal operator words.\n\\item[Packet.] The range of one idempotent in a complete annihilating family.\n\\item[Packet pure.] Supported in at most one packet.\n\\item[Preservation profile.] The operations a realization functor is required to preserve.\n\\item[Realized algebra.] The image of the grammar algebra in the endomorphism algebra of a r","\\item[Block support.] The set of packet pairs \\((\\beta,\\alpha)\\) for which \\(P_\\beta TP_\\alpha\\neq0\\).\n\\item[Collapsed realization.] The direct sum of all typed vertex spaces.\n\\item[Condensation graph.] The acyclic graph obtained by collapsing strongly connected components.\n\\item[Generated hull.] The submodule generated by a seed family, or its declared closure.\n\\item[Grammar.] A typed quiver and relation ideal describing legal operator words.\n\\item[Packet.] The range of one idempotent in a complete annihilating family.\n\\item[Packet pure.] Supported in at most one packet.\n\\item[Preservation profile.] The operations a realization functor is required to preserve.\n\\item[Realized algebra.] The image of the grammar algebra in the endomorphism algebra of a realization.\n\\item[Seed rank.] The sma","trary division rings, with the conventions for left and right modules kept explicit in \\cref{thm:semisimple-seed-rank}.\n\\item The single operator formula and its cyclic criteria apply over arbitrary fields in \\cref{thm:single-operator,cor:cyclic-matrix}.\n\\item The countable unconditional packet theorem is proved in \\cref{thm:countable-packets}.\n\\item The classification of support compatible with $*$ is proved in \\cref{cor:star-support}.\n\\item The weighted source component formula gives the packet pure generation requirement under the propagation hypotheses in \\cref{thm:source-component-rank}.\n\\item Seed system composition, packet exact sequences, quotient block descent, dense realization, quotient completion, and semantic theorem transport are proved in \\cref{prop:seed-category,thm:packet-exact-sequence,cor:block-descent,thm:dense-realization,thm:quotient-topology,thm:realization-export}.\n\\item Exact sequence bounds, monotonicity under enlargement of the acting algebra, coordinate stability, projection onto diagonal blocks, and preorder decomposition are established in \\cref{thm:exac…","\\nolinkurl{PSC-I-C-SUPPORT-PATTERN-STAR} & \\ref{cor:star-support} & Bounded support patterns closed under $*$","Use the same space of dimension six throughout the first comparison:\n\\[\nX=\\QQ^6,\\qquad N=J_3(0)\\oplus J_3(0),\\qquad\ns=e_0+f_0.\n\\]\nHere each $J_3(0)$ moves a basis vector forward along a chain of three basis vectors. The projectors $P_e,P_f$ select the two chains. Their formulas and the proofs of the minimum seed counts appear on \\pagerefV{3} and \\pagerefV{4}.","[4pt]\n\\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\\\[4pt]\n\\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?\n\\end{tabularx}\n\nA packet projector can be available as a way to \\emph{inspect} a vector without belonging to the \\emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.\n\n\\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}\n\n\\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \\cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}\nMost of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.\n\n{\\small\n\\begin{tabularx}{\\linewidth}{@{}>{\\raggedright\\arraybackslash}p{.15\\linewidth}Y>{\\raggedrig…","\\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \\cref{thm:single-operator}. Matrices over $\\QQ$}\nUse the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define\n\\[\nNe_0=e_1,\\quad Ne_1=e_2,\\quad Ne_2=0,\n\\qquad Nf_0=f_1,\\quad Nf_1=f_2,\\quad Nf_2=0.\n\\]\nStarting with $s=e_0+f_0$, the two chains advance together:\n\\[\ns=e_0+f_0,\\qquad Ns=e_1+f_1,\\qquad N^2s=e_2+f_2,\\qquad N^3s=0.\n\\]","\\vfig[.94]{linked_chains}\n\\atlasCaption{The vertical dotted pairings record equal coefficients, not an additional allowed operator. Every polynomial in $N$ uses the same coefficients on both chains.}","Since $N^3=0$, every polynomial orbit vector has the form\n\\[\np(N)s=a(e_0+f_0)+b(e_1+f_1)+c(e_2+f_2).\n\\]\nThe generated hull therefore has dimension three, even though the seed has a nonzero component in each packet of dimension three. The Krylov ranks are $1,2,3,3,3,3$.","\\head{What changes when a projector becomes admissible?}\nLet $P_e$ keep the coordinates along the $e$ chain and $P_f$ those along the $f$ chain. If these are merely measurement maps, the preceding hull is unchanged. If they become allowed generators, then\n\\[\nP_es=e_0,\\qquad P_fs=f_0,\n\\]\nand powers of $N$ now recover all six basis vectors. The same seed becomes cyclic for the larger algebra $\\QQ\\langle N,P_e,P_f\\rangle$.","Define a surjective linear map\n\\[\nq:X\\longrightarrow\\QQ^2,\\qquad\nq\\Bigl(\\sum a_je_j+\\sum b_jf_j\\Bigr)=(a_0,b_0).\n\\]","\\vfig[.78]{quotient_heads}\n\\atlasCaption{An actual $2\\times6$ quotient matrix. It forgets the advanced positions in both chains and retains two independent head coordinates.}","Because $qN=0$, one has, for every polynomial $p$,\n\\[\nq\\bigl(p(N)x\\bigr)=p(0)q(x).\n\\]\nThus every operator acts as a \\emph{scalar} on the quotient. The image of the orbit of one seed stays on one line in $\\QQ^2$. It cannot span this quotient of dimension two.","\\head{Lower and upper bounds meet}\nIf $r$ seeds generated $X$, their images would generate the quotient. Since scalar multiplication cannot create a new independent direction, $r\\ge2$. Conversely, the two seeds $e_0,f_0$ generate their respective chains, so\n\\[\n\\boxed{\\srank_{\\QQ[N]}(X)=2.}\n\\]\nThis proof uses the displayed operator and quotient. It does not ask a computer to search every possible seed, and it does not need an existence theorem for Jordan form.","\\head{A general pattern worth recognizing}\nA surjective intertwiner $q:X\\twoheadrightarrow Y$ turns any generating family of $X$ into a generating family of $Y$. Therefore a quotient on which the grammar loses mixing power can certify a lower bound in the original, much larger space.","\\vpage{5}{Relations identify words, not just outputs}{PRESENTATIONS / WHY A LEFT IDEAL APPEARS}{Main text: \\cref{thm:free-presentation,cor:one-seed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","uts}{PRESENTATIONS / WHY A LEFT IDEAL APPEARS}{Main text: \\cref{thm:free-presentation,cor:one-seed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","IDEAL APPEARS}{Main text: \\cref{thm:free-presentation,cor:one-seed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","eed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","rop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","et $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","\\head{The distinction is visible in two dimensions}\nUse $X=\\QQ^2$, seed $e_1=(1,0)^\\mathsf T$, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column:\n\\[\nI_{e_1}=\\left\\{\\mat{0&a","\\head{The distinction is visible in two dimensions}\nUse $X=\\QQ^2$, seed $e_1=(1,0)^\\mathsf T$, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column:\n\\[\nI_{e_1}=\\left\\{\\mat{0&a\\\\0&b}:a,b\\in\\QQ\\right\\}.\n\\]","\\vfig[.90]{left_ideal}\n\\atlasCaption{The relation $Ce_1=0$ is destroyed by multiplying $C$ on the right by $B$. The product $CB$ sends the seed to itself.}","For arbitrary $D$, $(DC)e_1=D(Ce_1)=0$. But $(CB)e_1=e_1\\ne0$. Thus $I_{e_1}$ is a left ideal, but it is not stable under multiplication on the right. The quotient retains the action on generated vectors. Calling it a ring would lose the distinction that makes the presentation correct.","\\head{Several seeds give a module of relations}\nFor $(s_1,\\ldots,s_m)$, evaluation is $(f_1,\\ldots,f_m)\\mapsto\\sum_j f_j(T)s_j$. Its kernel is a submodule of $F^m$ and can include relations \\emph{between} seeds, not only relations annihilating each seed separately.","\\vpage{6}{A reached type can still hide a direction}{TYPED MODEL / $U=\\QQ^2$ AND $V=\\QQ^3$}{Main text: \\cref{prop:typed-reachability,def:directed-seed-rank}}\nThe chapter's typed model has a rotation $R:U\\to U$, an inclusion $A:U\\to V$, and a retraction $B:V\\to U$:\n\\[\nR=\\mat{0&-1","\\vpage{6}{A reached type can still hide a direction}{TYPED MODEL / $U=\\QQ^2$ AND $V=\\QQ^3$}{Main text: \\cref{prop:typed-reachability,def:directed-seed-rank}}\nThe chapter's typed model has a rotation $R:U\\to U$, an inclusion $A:U\\to V$, and a retraction $B:V\\to U$:\n\\[\nR=\\mat{0&-1\\\\1&0},\\qquad\nA=\\mat{1&0\\\\0&1\\\\0&0},\\qquad\nB=\\mat{1&0&0\\\\0&1&0}.\n\\]\nThe typed relation is $BA=I_U$. In the opposite order,\n\\[\nAB=\\diag(1,1,0)\\ne I_V.\n\\]\nThe missing direction is $\\Ker B=\\QQ v_3$, not an entirely missing vertex.","0&1&0}.\n\\]\nThe typed relation is $BA=I_U$. In the opposite order,\n\\[\nAB=\\diag(1,1,0)\\ne I_V.\n\\]\nThe missing direction is $\\Ker B=\\QQ v_3$, not an entirely missing vertex.","\\head{One seed supported at $U$}\nThe vectors $u_1,Ru_1$ span $U$. Their images under $A$ span $\\QQ v_1+\\QQ v_2$. Every path from $U$ to $V$ ends with $A$, so it cannot supply $v_3$. The hull has dimension four. Adding the pure seed $v_3$ gives all five dimensions.","\\head{Why this does not prove an unrestricted minimum of two}\nThe bound path algebra includes the vertex idempotents $P_U,P_V$. A mixed seed\n\\[\ns=u_1+v_3\\in U\\oplus V\n\\]\nsatisfies $P_Us=u_1$ and $P_Vs=v_3$. It therefore generates the entire representation by itself. The chapter proves that no seed \\emph{supported at $U$} is enough and that two displayed pure seeds suffice. It does not claim that two arbitrary mixed seeds are necessary.","\\vpage{7}{Exact does not mean orthogonal}{PACKET COORDINATES / AN OBLIQUE SPLITTING}{Main text: \\cref{thm:exact-packet-coordinates,cor:adjoint-reversal}}\nIn $\\RR^2$, take\n\\[\nP=\\mat{1&-1","\\vpage{7}{Exact does not mean orthogonal}{PACKET COORDINATES / AN OBLIQUE SPLITTING}{Main text: \\cref{thm:exact-packet-coordinates,cor:adjoint-reversal}}\nIn $\\RR^2$, take\n\\[\nP=\\mat{1&-1\\\\0&0},\\qquad Q=\\mat{0&1\\\\0&1}.\n\\]\nThese maps satisfy $P^2=P$, $Q^2=Q$, $PQ=QP=0$ and $P+Q=I$. Their images are the horizontal line and the diagonal line, respectively. They are not self-adjoint in the usual Euclidean inner product.","0&1}.\n\\]\nThese maps satisfy $P^2=P$, $Q^2=Q$, $PQ=QP=0$ and $P+Q=I$. Their images are the horizontal line and the diagonal line, respectively. They are not self-adjoint in the usual Euclidean inner product.","\\vfig[.68]{oblique_vectors}\n\\atlasCaption{Actual Euclidean vector geometry for $x=(3,2)$. The parallelogram reconstructs $x$ from $Px=(1,0)$ and $Qx=(2,2)$. Neither component is an orthogonal projection of $x$.}","Analysis records $(Px,Qx)$ and synthesis adds the two vectors. For a packet tuple $(u,v)\\in\\im P\\times\\im Q$,\n\\[\nP(u+v)=u,\\qquad Q(u+v)=v.\n\\]\nBoth inverse identities follow from annihilation and completeness, without any angle condition.","\\head{What the algebra does not preserve}\nFor the displayed $x$,\n\\[\n\\norm{x}^2=13,\\qquad \\norm{Px}^2+\\norm{Qx}^2=1+8=9.\n\\]\nThere is no Parseval identity here. The cross term is nonzero because the packet directions are not perpendicular. Likewise, the simple formula for reversing adjoints of packet blocks requires self-adjoint projectors. It is not justified for this oblique pair.","\\vpage{8}{A transfer block has an input and an output}{OPERATOR COORDINATES / MULTIPLICATION AS A SUM}{Main text: \\cref{thm:operator-reconstruction,thm:block-multiplication,cor:block-powers}}\nUse the three coordinate packets of dimension one in $\\QQ^3$. The block $T_{ij}=P_iTP_j$ sends input packet $j$ into output packet $i$. For this example, its nonzero scalar is the $(i,j)$ matrix entry.","\\head{Reconstruction retains every block}\nThe identity $P_1+P_2+P_3=I$ gives\n\\[\nT=\\sum_{i,j=1}^3 P_iTP_j.\n\\]\nSelecting only the diagonal would change the operator. The entries $2,3,4$ carry transfers between different packets.","\\head{Composition sums over intermediate packets}\nThe $(1,2)$ entry of the square has two contributing routes:\n\\[\n(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}\n=1\\cdot2+2\\cdot1+0=4.\n\\]\nFor input $3$ and output $1$, only the route through packet $2$ survives:\n\\[\n(T^2)_{13}=T_{12}T_{23}=2\\cdot3=6.\n\\]\nFor packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.","\\vpage{9}{Two visible routes can cancel exactly}{REACHABILITY / NECESSARY IS NOT SUFFICIENT}{Main text: \\cref{cor:block-powers,prop:reachability-obstruction}}\nLet a single operator $D$ on $\\QQ^4$ act by\n\\[\nDe_1=e_2+e_3,\\qquad De_2=e_4,\\qquad De_3=-e_4,\\qquad De_4=0.\n\\]\nEvery vertex of its transfer graph for the coordinate packets is reachable from vertex $1$. Nevertheless,\n\\[\nD^2e_1=e_4-e_4=0,\\qquad D^2=0.\n\\]","\\vfig[.84]{cancelled_paths}\n\\atlasCaption{The diamond is the actual weighted action of $D$. The two paths of length two exist, but their contributions to the same transfer block sum to zero.}","\\head{What the seed generates}\nWith the fixed algebra $\\QQ[D]$ and seed $e_1$,\n\\[\n\\Hull_{\\QQ[D]}(e_1)=\\Span\\{e_1,e_2+e_3\\}.\n\\]\nIt has dimension two, not four. Both the independent difference $e_2-e_3$ and the last coordinate $e_4$ are missing.","\\head{Why projecting in the middle changes the experiment}\nIf $P_2$ becomes an admissible primitive, the word $DP_2D$ satisfies\n\\[\nDP_2De_1=DP_2(e_2+e_3)=De_2=e_4.\n\\]\nBut $P_2$ was not an allowed generator of the original $\\QQ[D]$ grammar. Using it to argue that the original seed already reached $e_4$ would silently enlarge the algebra.","\\vpage{10}{Which zero patterns survive multiplication?}{SUPPORT ALGEBRAS / A UNIVERSAL TEST}{Main text: \\cref{thm:support-pattern}. All packet spaces are nonzero}\nSuppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let $E_{ij}$ denote the matrix that sends $e_j$ to $e_i$ and kills the other basis vectors.","\\vfig[.96]{support_closure}\n\\atlasCaption{Here $1$ means ``allowed,'' not that every admissible operator has that numerical value. The middle matrix is an actual product that violates the original pattern.}","The first pattern contains the diagonal and permits $1\\to2$ and $2\\to3$, but forbids $1\\to3$. Yet\n\\[\nE_{32}E_{21}=E_{31}\\ne0.\n\\]\nBoth factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","\\head{Reflexivity and transitivity have different jobs}\nThe identity has a nonzero diagonal block on every nonzero packet, so including the identity requires every packet to be related to itself. Multiplicative closure requires every composite pair. Consequently, for arbitrary algebraic block maps on nonzero packet spaces,\n\\[\n\\boxed{\\text{unital universal support algebra}\n\\quad\\Longleftrightarrow\\quad\n\\text{reflexive, transitive support relation}.}\n\\]\nThe transitive closure in the figure is the lower triangular pattern. It does define a unital algebra.","\\vpage{11}{Move the packets with the coordinates}{CONJUGACY / STRUCTURE VERSUS RAW SPARSITY}{Main text: \\cref{thm:coordinate-transport,cor:isomorphism-invariance,thm:naturality}}\nStart with $T=\\diag(1,2)$ and the two coordinate projections. Apply the invertible shear $U$. The transformed operator is $T^U=UTU^{-1}$, and the transformed projectors are $Q_i=UP_iU^{-1}$.","Indeed,\n\\[\nQ_1T^UQ_2=Q_2T^UQ_1=0,\n\\qquad\nQ_iT^UQ_j=U(P_iTP_j)U^{-1}.\n\\]\nA change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.","\\head{When the map is not invertible}\nFor an intertwiner $\\Phi A=\\varphi(A)\\Phi$, the hull identity uses the \\emph{image algebra}:\n\\[\n\\Phi\\bigl(\\Hull_{\\AA}(S)\\bigr)=\\Hull_{\\varphi(\\AA)}(\\Phi S).\n\\]\nReplacing $\\varphi(\\AA)$ with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not automatic equality.","\\vpage{12}{Shared control and independent control differ}{DIRECT SUMS / ONE SEED CAN CARRY TWO COMPONENTS}{Main text: \\cref{prop:shared-direct-sum,thm:independent-direct-sum}}\nTake $X=\\kk\\oplus\\kk$ and $s=(1,1)$. With one shared scalar acting on both coordinates, every orbit vector is $(a,a)$. With independent scalars, the orbit contains every $(a,b)$.","\\vfig[.94]{shared_independent}\n\\atlasCaption{The geometric picture uses $\\kk=\\RR$. On the right, shading shows a finite window of the whole plane. On the left, only the diagonal line is generated. The algebraic distinction holds over any field.}","The corresponding minimum seed numbers are\n\\[\n\\srank_{\\kk I}(\\kk^2)=2,\n\\qquad\n\\srank_{\\kk\\times\\kk}(\\kk\\oplus\\kk)=1.\n\\]\nIn the independent case, the algebra contains $(1,0)$ and $(0,1)$, which recover the two coordinates of $s$ separately.","\\head{This is why the independent formula uses a maximum}\nFor independent actions of $\\AA$ on $X$ and $\\BB$ on $Y$, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving\n\\[\n\\srank_{\\AA\\times\\BB}(X\\oplus Y)\n=\\max\\{\\srank_{\\AA}(X),\\srank_{\\BB}(Y)\\}.\n\\]\nWith a shared action, this selection need not be available. The general bound is the sum.","\\head{One operator can supply independent control}\nOver $\\QQ$, $T=\\diag(1,2)$ has polynomial projectors $2I-T$ and $T-I$. Thus $\\QQ[T]$ already separates the two eigenspaces. If the eigenvalues coincide, these formulas no longer separate them. Distinct primary labels play the same role on \\pagerefV{14}.","\\vpage{13}{How much row space can one seed supply?}{SPLIT SEMISIMPLE MODEL / A CONSTRUCTIVE FORMULA}{Main text: \\cref{thm:semisimple-seed-rank,cor:algebraically-closed-formula}}\nLet $M_2(\\QQ)$ act on $X=M_{2\\times5}(\\QQ)$ by left multiplication. Think of the five columns as five copies of the same natural module of dimension two. One seed is a $2\\times5$ matrix $V$.","Every row of $AV$ is a linear combination of the two rows of $V$. Hence one seed exposes at most two independent directions in the multiplicity space of dimension five. Two seeds expose at most four. At least three are necessary.","\\vfig[.98]{row_budget}\n\\atlasCaption{The three fixed seeds cover multiplicity directions $1,2$, then $3,4$, then $5$. The exact dimensions of their cumulative matrix orbits are $4,8,10$.}","\\head{An explicit synthesis, not just dimension counting}\nFor\n\\[\nW=\\mat{1&2&3&4&5\\\\6&7&8&9&10},\n\\]\nchoose $A_1$ from columns $1,2$ of $W$, $A_2$ from columns $3,4$, and $A_3$ with first column equal to column $5$ of $W$ and second column zero. Then\n\\[\nA_1V_1+A_2V_2+A_3V_3=W.\n\\]\nThe same construction works for arbitrary entries of $W$, so the lower bound is attained:\n\\[\n\\boxed{\\srank_{M_2(\\QQ)}M_{2\\times5}(\\QQ)=\\lceil5/2\\rceil=3.}\n\\]","6&7&8&9&10},\n\\]\nchoose $A_1$ from columns $1,2$ of $W$, $A_2$ from columns $3,4$, and $A_3$ with first column equal to column $5$ of $W$ and second column zero. Then\n\\[\nA_1V_1+A_2V_2+A_3V_3=W.\n\\]\nThe same construction works for arbitrary entries of $W$, so the lower bound is attained:\n\\[\n\\boxed{\\srank_{M_2(\\QQ)}M_{2\\times5}(\\QQ)=\\lceil5/2\\rceil=3.}\n\\]","\\head{Several split simple blocks}\nIf the acting algebra and module are supplied in compatible product coordinates, the row budget is computed separately in each block and the formula for independent actions takes the maximum. For block data $(d,m)=(2,5)$ and $(3,4)$, the result is $\\max(3,2)=3$, not $3+2$.","\\vpage{14}{Distinct labels let one polynomial separate chains}{PRIMARY DATA / AN ACTUAL EXTRACTION CERTIFICATE}{Main text: \\cref{thm:single-operator,cor:invariant-factor-count}}\nKeep the model in six dimensions, but compare two different operators:\n\\[\nN=J\\oplus J,\\qquad T=J\\oplus(2I+J),\\qquad J=J_3(0).\n\\]\nFor the same mixed seed $s=e_0+f_0$, the Krylov matrices are different:","\\vfig[.89]{primary_krylov}\n\\atlasCaption{Exact rational calculations. Changing the primary label from $0$ to $2$ changes the operator, not merely its coordinates. The determinant on the right is $512\\ne0$.}","\\head{The missing separating operation is now a polynomial}\nDefine\n\\[\np_2(t)=\\frac{3t^5-15t^4+20t^3}{16},\\qquad p_0(t)=1-p_2(t).\n\\]\nThese polynomials satisfy\n\\[\np_2\\equiv0\\pmod{t^3},\\qquad p_2\\equiv1\\pmod{(t-2)^3}.\n\\]\nConsequently $p_0(T)=P_e$ and $p_2(T)=P_f$. Applying them to $s$ extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on \\pagerefV{3}, the separating maps already belong to the polynomial grammar.","\\vpage{15}{Walk counts make an endpoint cyclic}{GRAPH MODEL / THE PATH CERTIFICATE}{Main text: Worked graph model. Endpoint Krylov proof for every path size}\nLet $A$ be the adjacency matrix of the path with five vertices. The $(i,j)$ entry of\n\\[\nK=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1]\n\\]\ncounts walks of length $j-1$ from vertex $1$ to vertex $i$.","A walk cannot reach vertex $i$ in fewer than $i-1$ steps. Therefore entries below the diagonal vanish. There is exactly one walk of length $i-1$ from the first vertex to vertex $i$: the shortest path. The diagonal entries are all one, so $\\det K=1$.","\\head{The same reason works at every size}\nFor a path with any $n\\ge1$, the endpoint Krylov matrix is upper triangular with unit diagonal. This is a combinatorial proof of cyclicity over \\emph{every field}. It does not rely on numerical eigenvalues. The appendix checks the finite cases $1\\le n\\le12$ as implementation examples, separately from the chapter's proof for every path size.","\\vpage{16}{Closing a path changes the seed obstruction}{GRAPH MODEL / A CYCLE WITH SIX VERTICES}{Main text: Worked graph model. Proof of the minimum seed count for complex cycles}\nFor the adjacency matrix $A$ of $C_6$, the distinct eigenvalues are $2,1,-1,-2$, with multiplicities $1,2,2,1$. A single vertex seed spans only four directions. Two adjacent vertex seeds span all six.","\\head{A lower bound without a numerical eigensolver}\nThe two rows of the following matrix are independent and obey $qA=q$:\n\\[\nq=\\mat{2&1&-1&-2&-1&1","\\head{A lower bound without a numerical eigensolver}\nThe two rows of the following matrix are independent and obey $qA=q$:\n\\[\nq=\\mat{2&1&-1&-2&-1&1\\\\0&1&1&0&-1&-1}.\n\\]\nThus $q:\\QQ^6\\twoheadrightarrow\\QQ^2$ is a scalar quotient for the eigenvalue $1$. Its extension over $\\CC$ gives the same obstruction of dimension two. The orbit of any single seed projects to a line, so no single seed can generate the full module.","0&1&1&0&-1&-1}.\n\\]\nThus $q:\\QQ^6\\twoheadrightarrow\\QQ^2$ is a scalar quotient for the eigenvalue $1$. Its extension over $\\CC$ gives the same obstruction of dimension two. The orbit of any single seed projects to a line, so no single seed can generate the full module.","\\head{The matching construction}\nFrom adjacent vertex vectors $e_0,e_1$, the identity\n\\[\ne_{j+1}=Ae_j-e_{j-1}\n\\]\nrecovers successive vertices around the cycle. The lower and upper bounds therefore give two seeds for this example. The chapter proves the corresponding minimum for every complex simple cycle of size at least three.","\\vpage{17}{A relation becomes a packet product rule}{COORDINATE ALGEBRA / NORMAL FORM RATHER THAN A CURVE SKETCH}{Main text: Worked coordinate ring model. Normal form and exact multiplication}\nWork over $\\QQ$ in\n\\[\nA=\\QQ[x,y]/(y^2-x^3+x).\n\\]\nBecause the relation is monic of degree two in $y$, each element has one unique form $a(x)+yb(x)$. The two packet coefficients are polynomials, not scalars.","\\vfig[.97]{normal_form}\n\\atlasCaption{A finite window of the two infinite coefficient lanes. Higher powers of $y$ are rewritten using the relation rather than discarded. No geometric genus or smoothness claim is encoded by this picture.}","\\head{Multiplication by $y$ transfers between the lanes}\nIn the ordered coefficient coordinates $(a,b)$,\n\\[\nM_y=\\mat{0&x^3-x","\\head{Multiplication by $y$ transfers between the lanes}\nIn the ordered coefficient coordinates $(a,b)$,\n\\[\nM_y=\\mat{0&x^3-x\\\\1&0},\\qquad M_y^2=(x^3-x)I.\n\\]\nThe entries denote multiplication maps on $\\QQ[x]$. For $f=a_0+ya_1$ and $g=b_0+yb_1$,\n\\[\nfg=\\bigl(a_0b_0+(x^3-x)a_1b_1\\bigr)\n+y\\bigl(a_0b_1+a_1b_0\\bigr).\n\\]\nIn particular,\n\\[\n(x^2+y)(x-y)=x+(x-x^2)y.\n\\]","1&0},\\qquad M_y^2=(x^3-x)I.\n\\]\nThe entries denote multiplication maps on $\\QQ[x]$. For $f=a_0+ya_1$ and $g=b_0+yb_1$,\n\\[\nfg=\\bigl(a_0b_0+(x^3-x)a_1b_1\\bigr)\n+y\\bigl(a_0b_1+a_1b_0\\bigr).\n\\]\nIn particular,\n\\[\n(x^2+y)(x-y)=x+(x-x^2)y.\n\\]","\\head{One module seed does not mean one algebra generator}\nStarting from $1$, the fixed multiplication operators $M_x,M_y$ generate every monomial in normal form. Thus the algebra is cyclic as a module over its multiplication algebra. This does not remove either address operator, and does not say that the algebra is generated by one algebra element.","\\takeaway{The relation controls an exact weighted transfer between coefficient lanes. Packet reconstruction is $a+yb$, while packet multiplication retains the weight $x^3-x$.}","\\vpage{18}{Density is not finite generation of each vector}{INFINITE MODEL / THE BILATERAL SHIFT}{Main text: \\cref{prop:shift-seed,thm:completed-presentation}}\nOn $\\ell^2(\\ZZ)$, declare every integer shift\n\\[\n(S_mx)[n]=x[n-m].\n\\]\nThe seed $\\delta_0$ generates $\\delta_m=S_m\\delta_0$. Finite linear combinations give precisely the sequences with finite support. Their norm closure is all of $\\ell^2(\\ZZ)$.","\\vfig[.93]{shift_tail}\n\\atlasCaption{A finite window of the actual sequence $x[n]=2^{-|n|}$. Outside every finite window, coefficients remain nonzero. The shaded window illustrates the truncation $x^{(3)}$, not a periodic wraparound.}","This $x$ cannot be a finite linear combination of shifted impulses: it has infinite support. But if $x^{(N)}$ agrees with $x$ on $|n|\\le N$ and is zero elsewhere, then\n\\[\n\\norm{x-x^{(N)}}_2^2\n=2\\sum_{n>N}4^{-n}\n=\\frac{2}{3}\\,4^{-N}\\longrightarrow0.\n\\]\nThe exact squared errors for $N=1,3,5$ are $1/6$, $1/96$, and $1/1536$. These are identities from a geometric series, not floating residuals.","\\vpage{19}{Residue packets retain an address carry}{INFINITE MODEL / THREE EXACT SUBSEQUENCES}{Main text: \\cref{thm:banach-packets}. Residue packets}\nFor $j=0,1,2$, let $P_j$ keep the entries with $n\\equiv j\\pmod3$. These are bounded orthogonal projectors whose sum is the identity. The packet coordinates can be written as\n\\[\nx_j[k]=x[3k+j],\\qquad k\\in\\ZZ.\n\\]","\\head{What one shift does in these coordinates}\nLet $S=S_1$ and $z=Sx$. Substituting $3k+j$ into the shift definition gives\n\\[\nz_1[k]=x_0[k],\\qquad z_2[k]=x_1[k],\\qquad z_0[k]=x_2[k-1].\n\\]\nThe passage from phase $2$ to phase $0$ carries a shift of the coarse address. If $Lh[k]=h[k-1]$, the exact packet matrix is\n\\[\nS_{\\mathrm{packets}}=\\mat{0&0&L","\\head{What one shift does in these coordinates}\nLet $S=S_1$ and $z=Sx$. Substituting $3k+j$ into the shift definition gives\n\\[\nz_1[k]=x_0[k],\\qquad z_2[k]=x_1[k],\\qquad z_0[k]=x_2[k-1].\n\\]\nThe passage from phase $2$ to phase $0$ carries a shift of the coarse address. If $Lh[k]=h[k-1]$, the exact packet matrix is\n\\[\nS_{\\mathrm{packets}}=\\mat{0&0&L\\\\I&0&0\\\\0&I&0}.\n\\]\nReplacing $L$ by a scalar $1$ would preserve the phase graph but would lose the operator's action on the infinite sequences.","0&I&0}.\n\\]\nReplacing $L$ by a scalar $1$ would preserve the phase graph but would lose the operator's action on the infinite sequences.","\\head{Synthesis keeps the positions}\nTo reconstruct $x$, place $x_j[k]$ at its original address $3k+j$. In the ambient space formulation, this is the finite sum $x=P_0x+P_1x+P_2x$. Boundedness makes the same formulas valid after taking the $\\ell^2$ completion.","\\vpage{20}{Stop when the subspace stops growing}{EXACT COMPUTATION / AN INVARIANCE CERTIFICATE}{Main text: \\cref{con:orbit-hull-algorithm}}\nFor a fixed finite operator family $(O_i)$, start from $W_0=\\Span S$ and iterate\n\\[\nW_{n+1}=W_n+\\sum_i O_iW_n.\n\\]\nThis is an ascending chain. When two consecutive subspaces agree, every generator maps the stable subspace into itself. The stable value is therefore the entire hull of operator words, not just the span of the words tested so far.","\\vfig[.58]{hull_iteration}\n\\atlasCaption{Exact dimensions for the same mixed seed in Model A. The dashed curve allows only $N$. The solid curve also admits the two chain projectors. Plateau values are repeated to show the certified stop.}","\\head{Why the process stops in finite dimensions}\nIf an inclusion is strict, the dimension rises by at least one. It cannot rise beyond $\\dim X$. At equality, invariance and containment of the seeds prove both inclusions between the stable space and the least invariant hull.","\\vpage{21}{Tensor products require every cross term}{OPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONAL}{Main text: \\cref{prop:tensor-bound}. Algebraic tensor products}\nIf $(x_i)$ generates $X$ under $\\AA$ and $(y_j)$ generates $Y$ under $\\BB$, then the candidate tensor seeds are \\emph{all} $x_i\\otimes y_j$. For finite expansions,\n\\[\nx=\\sum_i A_ix_i,\\qquad y=\\sum_j B_jy_j,\n\\]\nbilinearity gives the required double sum:\n\\[\nx\\otimes y=\\sum_i\\sum_j(A_i\\otimes B_j)(x_i\\otimes y_j).\n\\]","require every cross term}{OPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONAL}{Main text: \\cref{prop:tensor-bound}. Algebraic tensor products}\nIf $(x_i)$ generates $X$ under $\\AA$ and $(y_j)$ generates $Y$ under $\\BB$, then the candidate tensor seeds are \\emph{all} $x_i\\otimes y_j$. For finite expansions,\n\\[\nx=\\sum_i A_ix_i,\\qquad y=\\sum_j B_jy_j,\n\\]\nbilinearity gives the required double sum:\n\\[\nx\\otimes y=\\sum_i\\sum_j(A_i\\otimes B_j)(x_i\\otimes y_j).\n\\]","\\vfig[.95]{tensor_rectangle}\n\\atlasCaption{The exact expansion of $(2e_1+3e_2)\\otimes(5f_1+7f_2)$. The terms away from the diagonal $14$ and $15$ cannot be omitted.}","Keeping only the tensors with matching indices $e_1\\otimes f_1$ and $e_2\\otimes f_2$ spans a subspace of dimension two in a tensor product of dimension four. It misses both cross directions. Under scalar actions on $\\QQ^2$ and $\\QQ^2$, all four basis tensors are necessary and sufficient.","\\head{The resulting upper bound}\nSince simple tensors span the algebraic tensor product, the complete rectangular family proves\n\\[\n\\boxed{\\srank_{\\AA\\otimes\\BB}(X\\otimes Y)\n\\le\\srank_{\\AA}(X)\\,\\srank_{\\BB}(Y).}\n\\]\nThis is not a universal equality claim. Changing the algebra changes the answer: the full independent matrix action can make a single nonzero pure tensor cyclic. Nor does the algebraic argument itself establish a theorem for completed topological tensor products.","The second part studies inversion in a complete filtered algebra.  Suppose an operator has the form\n\\[\nT=D+N,\n\\]\nwhere $D$ is invertible and every packet transfer in $N$ raises the filtration by a positive amount.  The ordinary Neumann series can then be expanded into weighted paths.  The weight is not the number of edges.  It is the total filtration gain along the path.  At precision $L$, every path of weight at least $L$ vanishes.  A finite transfer graph also gives a rational generating function for the number of paths of each total weight.  The relevant generating function facts are classical and are discussed in \\cite{Stanley2012}.","inversion in a complete filtered algebra.  Suppose an operator has the form\n\\[\nT=D+N,\n\\]\nwhere $D$ is invertible and every packet transfer in $N$ raises the filtration by a positive amount.  The ordinary Neumann series can then be expanded into weighted paths.  The weight is not the number of edges.  It is the total filtration gain along the path.  At precision $L$, every path of weight at least $L$ vanishes.  A finite transfer graph also gives a rational generating function for the number of paths of each total weight.  The relevant generating function facts are classical and are discussed in \\cite{Stanley2012}.","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation als","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves tha","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation we","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on l","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matri","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matrix polynomials may be found in \\cite{Serre1979,","$\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matrix polynomials may be found in \\cite{Serre1979,Newman1972,GohbergLancasterRodman2009}.","ambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matrix polynomials may be found in \\cite{Serre1979,Newman1972,GohbergLancasterRodman2009}.","The last part records how a defect between fine and coarse operators propagates.  If\n\\[\n\\Delta=CT_f-T_cC,\n\\]\nthen exact formulas describe the corresponding defect for powers, resolvents, holomorphic functions, exponentials, and a chain of several refinement levels.  These formulas are elementary, but collecting them in one place makes the later analytic uses transparent.  The spectral and resolvent background follows the usual operator theory in \\cite{Kato1995}.","The second is filtration depth. A transfer in $F^r\\Bcal$ vanishes modulo $F^r\\Bcal$. This membership gives a lower bound on its depth. A nonzero transfer may first become visible only at a higher precision. Two paths with the same number of edges can have different filtration depths. A longer path can appear earlier when its edges have smaller weights.","The third is separation depth.  A rational packet projector may require multiplication by $\\pi^s$ before it preserves the chosen lattice.  This number depends on both the rational decomposition and the lattice.  It can be zero even when the corresponding eigenvalues are close, and it can be positive even when the rational projector is simple.","The fourth notion is the compatibility defect. A fine operator and a separately chosen coarse operator need not commute with coarsening. Their discrepancy is\n\\[\n\\Delta=CT_f-T_cC.\n\\]\nThis operator records the failure of one model to represent another exactly. A chosen norm measures its size, while a chosen filtration can assign a depth. No integer compatibility depth is assigned without that additional structure.","A simple two packet model already contains all four notions.  Let\n\\[\nX=X_0\\oplus X_1,\n\\qquad\nD=\\begin{pmatrix}D_0&0\\\\0&D_1\\end{pmatrix},\n\\qquad\nN=\\begin{pmatrix}0&N_{01}\\\\N_{10}&0\\end{pmatrix}.\n\\]\nA finer family may split $X_0$ and $X_1$ into smaller summands.  The blocks $N_{01}$ and $N_{10}$ may enter at different filtration weights.  Over a fraction field, the rational packet spaces may fail to split the chosen lattice.  A coarse operator on one selected summand may also fail to intertwine with the full matrix.  The later sections isolate these four effects one at a time.","N_{10}&0\\end{pmatrix}.\n\\]\nA finer family may split $X_0$ and $X_1$ into smaller summands.  The blocks $N_{01}$ and $N_{10}$ may enter at different filtration weights.  Over a fraction field, the rational packet spaces may fail to split the chosen lattice.  A coarse operator on one selected summand may also fail to intertwine with the full matrix.  The later sections isolate these four effects one at a time.","\\begin{definition}[Packet system]\\label{def:packet_system}\nLet $X$ be a module over a unital ring.  A finite packet system on $X$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\n\\subseteq\\End(X)\n\\]\nof nonzero idempotents such that\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{when }\\alpha\\ne\\beta,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nWrite\n\\[\nX_\\alpha=P_\\alpha X,\n\\qquad\nX^{\\Pcal}=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nThe analysis and synthesis maps are\n\\[\n\\Acal_{\\Pcal}x=(P_\\alpha x)_\\alpha,\n\\qquad\n\\Scal_{\\Pcal}(x_\\alpha)_\\alpha=\\sum_\\alpha x_\\alpha.\n\\]\nThey are inverse isomorphisms.\n\\end{definition}","ket_system}\nLet $X$ be a module over a unital ring.  A finite packet system on $X$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\n\\subseteq\\End(X)\n\\]\nof nonzero idempotents such that\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{when }\\alpha\\ne\\beta,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nWrite\n\\[\nX_\\alpha=P_\\alpha X,\n\\qquad\nX^{\\Pcal}=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nThe analysis and synthesis maps are\n\\[\n\\Acal_{\\Pcal}x=(P_\\alpha x)_\\alpha,\n\\qquad\n\\Scal_{\\Pcal}(x_\\alpha)_\\alpha=\\sum_\\alpha x_\\alpha.\n\\]\nThey are inverse isomorphisms.\n\\end{definition}","\\begin{definition}[Refinement]\\label{def:refinement}\\label{def:refinement-map}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on the same module.  We say that $\\Qcal$ refines $\\Pcal$ when there is a surjection\n\\[\n\\rho:J\\longrightarrow I\n\\]\nsuch that\n\\[\nP_\\alpha=\\sum_{\\rho(\\beta)=\\alpha}Q_\\beta\n\\]\nfor every $\\alpha\\in I$.  The index $\\beta$ is then called a child of $\\rho(\\beta)$.\n\\end{definition}","ition}[Refinement]\\label{def:refinement}\\label{def:refinement-map}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on the same module.  We say that $\\Qcal$ refines $\\Pcal$ when there is a surjection\n\\[\n\\rho:J\\longrightarrow I\n\\]\nsuch that\n\\[\nP_\\alpha=\\sum_{\\rho(\\beta)=\\alpha}Q_\\beta\n\\]\nfor every $\\alpha\\in I$.  The index $\\beta$ is then called a child of $\\rho(\\beta)$.\n\\end{definition}","\\begin{theorem}[Children, parents, and transitivity]\\label{thm:children_transitivity}\\label{thm:nested-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nP_\\alpha Q_\\beta\n=\nQ_\\beta P_\\alpha\n=\n\\begin{cases}\nQ_\\beta & \\rho(\\beta)=\\alpha,","\\begin{theorem}[Children, parents, and transitivity]\\label{thm:children_transitivity}\\label{thm:nested-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nP_\\alpha Q_\\beta\n=\nQ_\\beta P_\\alpha\n=\n\\begin{cases}\nQ_\\beta & \\rho(\\beta)=\\alpha,\\\\\n0 & \\rho(\\beta)\\ne\\alpha.\n\\end{cases}\n\\]\nHence $Q_\\beta X\\subseteq P_{\\rho(\\beta)}X$.  If a third packet system $\\Rcal$ refines $\\Qcal$, then $\\Rcal$ refines $\\Pcal$ through the composite parent map.\n\\end{theorem}","0 & \\rho(\\beta)\\ne\\alpha.\n\\end{cases}\n\\]\nHence $Q_\\beta X\\subseteq P_{\\rho(\\beta)}X$.  If a third packet system $\\Rcal$ refines $\\Qcal$, then $\\Rcal$ refines $\\Pcal$ through the composite parent map.\n\\end{theorem}","\\begin{proof}\nMultiply the identity\n\\[\nP_\\alpha=\\sum_{\\rho(\\gamma)=\\alpha}Q_\\gamma\n\\]\nby $Q_\\beta$ on either side.  Pairwise annihilation in the fine family leaves $Q_\\beta$ when $\\rho(\\beta)=\\alpha$ and leaves zero otherwise.  For transitivity, substitute the child expansion of each $Q_\\beta$ into the expansion of $P_\\alpha$ and group the resulting terms by the composite parent.\n\\end{proof}","A refinement tower is a sequence $(\\Pcal_\\ell)_{\\ell\\geq0}$ with parent maps\n\\[\n\\rho_{m,\\ell}:I_m\\longrightarrow I_\\ell\n\\qquad(m\\geq\\ell)\n\\]\nsuch that $\\Pcal_m$ refines $\\Pcal_\\ell$ through $\\rho_{m,\\ell}$ for every $m\\geq\\ell$, and\n\\[\n\\rho_{\\ell,\\ell}=\\id_{I_\\ell},\n\\qquad\n\\rho_{n,\\ell}=\\rho_{m,\\ell}\\circ\\rho_{n,m}\n\\qquad(n\\geq m\\geq\\ell).\n\\]\nThus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","\\begin{definition}[Coordinate change across levels]\\label{def:coordinate_change}\\label{def:coordinate-refinement}\nFor two packet systems $\\Pcal$ and $\\Qcal$ on $X$, define\n\\[\nJ_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Qcal}\\Scal_{\\Pcal}:X^{\\Pcal}\\longrightarrow X^{\\Qcal}\n\\]\nand\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}:X^{\\Qcal}\\longrightarrow X^{\\Pcal}.\n\\]\nWhen $\\Qcal$ refines $\\Pcal$, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.\n\\end{definition}","\\begin{theorem}[Exact coordinate change and coherence]\\label{thm:coordinate_coherence}\\label{prop:perfect-reconstruction}\nFor any two packet systems on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","herence]\\label{thm:coordinate_coherence}\\label{prop:perfect-reconstruction}\nFor any two packet systems on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","wo packet systems on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","s on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","w\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","\\begin{proof}\nThe packet analysis and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","n{proof}\nThe packet analysis and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","ket analysis and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","rse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","arrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","\\begin{remark}[Refinement and reduction]\\label{rem:refinement_reduction}\nThe maps in \\cref{thm:coordinate_coherence} are isomorphisms because both coordinate spaces represent all of $X$.  They do not reduce dimension and they do not discard packets.  Approximation begins only after one chooses a truncation or compression rule.\n\\end{remark}","\\begin{corollary}[Orthogonal Hilbert packets]\\label{cor:orthogonal_isometry}\nIf $X$ is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product $\\langle(x_\\alpha),(y_\\alpha)\\rangle=\\sum_\\alpha\\langle x_\\alpha,y_\\alpha\\rangle$. Then analysis and synthesis are unitary. Every coordinate change in \\cref{thm:coordinate_coherence} is unitary.\n\\end{corollary}","\\begin{proof}\nFor an orthogonal packet system,\n\\[\n\\|x\\|^2=\\sum_\\alpha\\|P_\\alpha x\\|^2.\n\\]\nThus analysis is an isometry onto the full packet direct sum.  Synthesis is its inverse.  A coordinate change is a composition of two unitary maps.\n\\end{proof}","Every coarse packet is also uniquely reconstructed from its fine children.  If vectors $x_\\beta\\in Q_\\beta X$ sum to a coarse packet $x$, applying $Q_\\gamma$ recovers $x_\\gamma=Q_\\gamma x$.  This is the uniqueness clause of the canonical nested-refinement statement.","\\subsection{Typed coordinate naturality}\nThese coordinate changes are coherent through every finite tower. If three\ncomplete packet systems are denoted by $a,b,c$, then\n\\[\nJ_{c,b}J_{b,a}=J_{c,a},\n\\qquad J_{a,a}=I,\n\\qquad J_{b,a}^{-1}=J_{a,b}.\n\\]\nThese laws hold for arbitrary exact packet systems representing the same\nambient module. The parent maps supply the component formulas linking\nchildren to their parents. Coordinate coherence itself does not require them.","The same coherence applies to operators. Let $P_a,P_b$ be packet systems on $X$, let $Q_a,Q_b$ be packet\nsystems on $Y$, where $X$ and $Y$ are modules over the same scalar ring, and let $T:X\\to Y$ be a linear map over that ring. Its exact coordinate\nrepresentation\n\\[\n\\widehat T_{Q,P}=\\cA_QT\\cS_P\n\\]\npreserves identities and typed composition, and the naturality square is\n\\[\n\\boxed{\nJ^Y_{Q_b,Q_a}\\,\\widehat T_{Q_a,P_a}\n=\n\\widehat T_{Q_b,P_b}\\,J^X_{P_b,P_a}.\n}\n\\]\nThe identity, cocycle, composition, inverse, and naturality laws are all proved\nin Lean. The separate refinement hypotheses supply the local component\nformulas linking each child to its parent.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{01_nested_dyadic_refinement.pdf}\n\\caption{A nested residue hierarchy. Each residue modulo $2$ is the union of two residues modulo $4$, and each residue modulo $4$ is the union of two residues modulo $8$.}\n\\label{fig:refinement-tree}\n\\end{figure}\n\\subsection{Operator blocks across refinement}","For $T\\in\\End(X)$ and a packet system $\\Pcal$, write\n\\[\nT^{\\Pcal}_{\\beta\\alpha}=P_\\beta T P_\\alpha.\n\\]","\\begin{theorem}[Refinement of operator blocks]\\label{thm:block_refinement}\\label{thm:transfer-block-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nT^{\\Pcal}_{ba}\n=\n\\sum_{\\substack{\\rho(\\beta)=b","\\begin{theorem}[Refinement of operator blocks]\\label{thm:block_refinement}\\label{thm:transfer-block-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nT^{\\Pcal}_{ba}\n=\n\\sum_{\\substack{\\rho(\\beta)=b\\\\\\rho(\\alpha)=a}}\nT^{\\Qcal}_{\\beta\\alpha}.\n\\]\nThe matrix of $T$ in fine coordinates is related to the matrix in coarse coordinates by\n\\[\n[T]_{\\Pcal}\n=\nC_{\\Pcal\\leftarrow\\Qcal}[T]_{\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}.\n\\]\n\\end{theorem}","rho(\\alpha)=a}}\nT^{\\Qcal}_{\\beta\\alpha}.\n\\]\nThe matrix of $T$ in fine coordinates is related to the matrix in coarse coordinates by\n\\[\n[T]_{\\Pcal}\n=\nC_{\\Pcal\\leftarrow\\Qcal}[T]_{\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}.\n\\]\n\\end{theorem}","\\begin{proof}\nExpand the two coarse projectors into their children:\n\\[\nP_bTP_a\n=\n\\left(\\sum_{\\rho(\\beta)=b}Q_\\beta\\right)\nT\n\\left(\\sum_{\\rho(\\alpha)=a}Q_\\alpha\\right).\n\\]\nDistributing the finite sums gives the first identity.  For the coordinate identity, use\n\\[\n[T]_{\\Pcal}=\\Acal_{\\Pcal}T\\Scal_{\\Pcal}\n\\]\nand insert\n\\[\nI_X=\\Scal_{\\Qcal}\\Acal_{\\Qcal}.\n\\]\n\\end{proof}","\\begin{corollary}[Support under refinement]\\label{cor:block_support_refinement}\nUnder the hypotheses of \\cref{thm:block_refinement}, a coarse block $T^{\\Pcal}_{ba}$ is zero if and only if every fine descendant block\n\\[\nT^{\\Qcal}_{\\beta\\alpha}\n\\qquad\n\\bigl(\\rho(\\beta)=b,\\ \\rho(\\alpha)=a\\bigr)\n\\]\nis zero.\n\\end{corollary}","\\begin{proof}\nIf every fine block is zero, the coarse block is zero by \\cref{thm:block_refinement}.  Conversely, suppose $P_bTP_a=0$.  Multiply on the left by $Q_\\beta$ and on the right by $Q_\\alpha$.  The child identities in \\cref{thm:children_transitivity} give\n\\[\nQ_\\beta TQ_\\alpha=0.\n\\]\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{03_refinement_incidence.pdf}\n\\caption{The incidence matrix $M_{r,s}=1$ when the fine residue $s$ belongs to the coarse residue $r$. Its rows implement exact aggregation from fine packets to coarse packets.}\n\\label{fig:incidence}\n\\end{figure}\n\\subsection{Common refinements}","\\begin{theorem}[Common refinement of commuting packet systems]\\label{thm:common_refinement}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on $X$.  Assume\n\\[\nP_\\alpha Q_\\beta=Q_\\beta P_\\alpha\n\\]\nfor all $\\alpha$ and $\\beta$.  Remove the zero operators from the family\n\\[\nR_{\\alpha\\beta}=P_\\alpha Q_\\beta.\n\\]\nThe remaining operators form a packet system that refines both $\\Pcal$ and $\\Qcal$.  The same statement holds for any finite collection of pairwise commuting packet systems.\n\\end{theorem}","[Common refinement of commuting packet systems]\\label{thm:common_refinement}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on $X$.  Assume\n\\[\nP_\\alpha Q_\\beta=Q_\\beta P_\\alpha\n\\]\nfor all $\\alpha$ and $\\beta$.  Remove the zero operators from the family\n\\[\nR_{\\alpha\\beta}=P_\\alpha Q_\\beta.\n\\]\nThe remaining operators form a packet system that refines both $\\Pcal$ and $\\Qcal$.  The same statement holds for any finite collection of pairwise commuting packet systems.\n\\end{theorem}","nement of commuting packet systems]\\label{thm:common_refinement}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on $X$.  Assume\n\\[\nP_\\alpha Q_\\beta=Q_\\beta P_\\alpha\n\\]\nfor all $\\alpha$ and $\\beta$.  Remove the zero operators from the family\n\\[\nR_{\\alpha\\beta}=P_\\alpha Q_\\beta.\n\\]\nThe remaining operators form a packet system that refines both $\\Pcal$ and $\\Qcal$.  The same statement holds for any finite collection of pairwise commuting packet systems.\n\\end{theorem}","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, u","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family wh","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","egin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","nd the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","\\begin{proposition}[Coarsest common refinement]\\label{prop:coarsest_common_refinement}\nUnder the hypotheses of \\cref{thm:common_refinement}, let $\\Rcal$ be a packet system that refines both $\\Pcal$ and $\\Qcal$.  Then $\\Rcal$ refines the nonzero product family $(P_\\alpha Q_\\beta)$.  Hence that product family is the coarsest packet system that refines both $\\Pcal$ and $\\Qcal$.\n\\end{proposition}","\\begin{proof}\nLet $R_\\gamma$ be a packet in $\\Rcal$.  Because $\\Rcal$ refines $\\Pcal$, there is a unique $\\alpha$ with\n\\[\nP_\\alpha R_\\gamma=R_\\gamma.\n\\]\nLikewise there is a unique $\\beta$ with\n\\[\nQ_\\beta R_\\gamma=R_\\gamma.\n\\]\nTherefore\n\\[\nP_\\alpha Q_\\beta R_\\gamma=R_\\gamma.\n\\]\nEvery packet of $\\Rcal$ lies in one product packet. Multiplying $\\sum_\\gamma R_\\gamma=I_X$ by $P_\\alpha Q_\\beta$ and using the parent identities gives\n\\[\nP_\\alpha Q_\\beta\n=\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma","a packet in $\\Rcal$.  Because $\\Rcal$ refines $\\Pcal$, there is a unique $\\alpha$ with\n\\[\nP_\\alpha R_\\gamma=R_\\gamma.\n\\]\nLikewise there is a unique $\\beta$ with\n\\[\nQ_\\beta R_\\gamma=R_\\gamma.\n\\]\nTherefore\n\\[\nP_\\alpha Q_\\beta R_\\gamma=R_\\gamma.\n\\]\nEvery packet of $\\Rcal$ lies in one product packet. Multiplying $\\sum_\\gamma R_\\gamma=I_X$ by $P_\\alpha Q_\\beta$ and using the parent identities gives\n\\[\nP_\\alpha Q_\\beta\n=\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma\\\\Q_\\beta R_\\gamma=R_\\gamma}}R_\\gamma.\n\\]\nThus the packets with the same pair of parents reconstruct the product packet. If that product packet is nonzero, the sum has a child, so the parent map onto the supported product family is surjective.\n\\end{proof}","\\begin{example}[A two level split]\\label{ex:two_level_split}\nLet\n\\[\nX=X_{00}\\oplus X_{01}\\oplus X_{10}\\oplus X_{11}.\n\\]\nAssume that all four summands are nonzero. The four coordinate projections form a fine packet system.  The two sums\n\\[\nP_0=Q_{00}+Q_{01},\n\\qquad\nP_1=Q_{10}+Q_{11}\n\\]\nform a coarse packet system.  If $x=(x_{00},x_{01},x_{10},x_{11})$, the coarse coordinates are\n\\[\n(x_{00}+x_{01},x_{10}+x_{11})\n\\]\nwhen the sums are interpreted inside the two coarse direct summands.  Fine analysis recovers the four entries exactly.  Nothing has been approximated.\n\\end{example}","\\begin{definition}[Residue selector]\\label{def:residue_selector}\nLet $R$ be a commutative ring and let\n\\[\nF(z)=\\sum_{n\\geq0}a_nz^n\\in R[[z]].\n\\]\nFor $N\\geq1$ and $0\\leq j<N$, define\n\\[\n(P_{N,j}F)(z)\n=\n\\sum_{n\\equiv j\\pmod N}a_nz^n.\n\\]\n\\end{definition}","\\begin{proposition}[Residue packet algebra]\\label{prop:residue_algebra}\\label{prop:residue-algebra}\nFor every commutative ring $R$, the operators $(P_{N,j})_{0\\leq j<N}$ satisfy the identities below.  If $R$ is nonzero, they form a packet system on $R[[z]]$:\n\\[\nP_{N,i}P_{N,j}=\\delta_{ij}P_{N,j},\n\\qquad\n\\sum_{j=0}^{N-1}P_{N,j}=I.\n\\]\n\\end{proposition}","\\begin{proof}\nEvery nonnegative integer belongs to one residue class modulo $N$.  A coefficient survives both $P_{N,i}$ and $P_{N,j}$ exactly when the two labels are equal to its residue.  Summing all selectors preserves every coefficient once.\n\\end{proof}","The coefficient-mask definition of $P_{N,j}$ also makes sense over a commutative semiring: it only retains or sets coefficients to zero. We use that same definition in the following semiring statement.","Let $U$ act on coefficient sequences by\n\\[\n(UF)(z)=F(\\omega z).\n\\]\nIf $\\omega^N=1$, then $U^N=I$.","\\begin{theorem}[Root-of-unity realization]\\label{thm:root-of-unity}\nLet $R$ be a commutative semiring, let $N>0$, and suppose\n$\\omega,\\omega^\\vee,\\eta\\in R$ satisfy $\\omega^N=1$ and the finite\n$N\\times N$ Fourier orthogonality table\n\\[\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}\\omega^{rk}\n=\n\\begin{cases}\n1,&r=j,","\\begin{theorem}[Root-of-unity realization]\\label{thm:root-of-unity}\nLet $R$ be a commutative semiring, let $N>0$, and suppose\n$\\omega,\\omega^\\vee,\\eta\\in R$ satisfy $\\omega^N=1$ and the finite\n$N\\times N$ Fourier orthogonality table\n\\[\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}\\omega^{rk}\n=\n\\begin{cases}\n1,&r=j,\\\\\n0,&r\\ne j,\n\\end{cases}\n\\qquad 0\\le j,r<N.\n\\]\nThen\n\\[\n\\boxed{\nP_{N,j}\n=\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}U^k.\n}\n\\]\nIn particular, if $R$ is a field, $N>0$, and $\\omega$ is a primitive\n$N$th root of unity, the certificate holds with\n$\\omega^\\vee=\\omega^{-1}$ and $\\eta=N^{-1}$.\n\\end{theorem}","0,&r\\ne j,\n\\end{cases}\n\\qquad 0\\le j,r<N.\n\\]\nThen\n\\[\n\\boxed{\nP_{N,j}\n=\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}U^k.\n}\n\\]\nIn particular, if $R$ is a field, $N>0$, and $\\omega$ is a primitive\n$N$th root of unity, the certificate holds with\n$\\omega^\\vee=\\omega^{-1}$ and $\\eta=N^{-1}$.\n\\end{theorem}","\\begin{proof}\nOn the monomial $z^n$, the right-hand side acts by\n\\[\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}\\omega^{nk}.\n\\]\nThe identity $\\omega^N=1$ reduces this expression to the table entry indexed\nby $r=n\\bmod N$. The finite certificate therefore gives exactly the\ncoefficient mask defining $P_{N,j}$.\nOver a field, the primitive-root geometric sum supplies the certificate.\n\\end{proof}","The field or orthogonality condition is essential. A primitive element of\norder $N$ in a ring with zero divisors does not by itself force the geometric\nsums away from the diagonal to vanish. For example, in $\\mathbb F_3\\times\\mathbb F_3$ the\nelement $(-1,1)$ has order two, but the second component of its nontrivial\ncharacter sum is nonzero. The coefficient selector uses no division and remains valid\nwithout any Fourier certificate. This distinction becomes decisive at the\ndyadic prime.","\\subsection{The field specialization}\n\\begin{theorem}[Character realization]\\label{thm:character_realization}\nLet $\\bbk$ be a field, assume that $N$ is nonzero in $\\bbk$, and suppose that $\\bbk$ contains a primitive $N$th root of unity $\\omega$.  Define\n\\[\n(UF)(z)=F(\\omega z).\n\\]\nThen\n\\[\nP_{N,j}\n=\n\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k.\n\\]\n\\end{theorem}","\\begin{proof}\nOn $z^n$, the right side acts by the scalar\n\\[\n\\frac1N\\sum_{k=0}^{N-1}\\omega^{k(n-j)}.\n\\]\nThis is one when $n\\equiv j\\pmod N$ and zero otherwise by the geometric sum identity in the field $\\bbk$.\n\\end{proof}","\\begin{remark}[Two distinct constructions]\\label{rem:selector_character}\nThe coefficient selector of \\cref{def:residue_selector} uses no division.  The character formula in \\cref{thm:character_realization} divides by $N$.  At a prime that divides $N$, the coefficient selector may remain integral even when the character average does not.\n\\end{remark}","\\begin{theorem}[Nested residue refinement]\\label{thm:residue_refinement}\\label{thm:polyphase-refinement}\nLet $N,M$ be positive integers with $N$ dividing $M$, and let $0\\leq j<N$.  Then\n\\[\nP_{N,j}\n=\n\\sum_{\\substack{0\\leq s<M","\\begin{theorem}[Nested residue refinement]\\label{thm:residue_refinement}\\label{thm:polyphase-refinement}\nLet $N,M$ be positive integers with $N$ dividing $M$, and let $0\\leq j<N$.  Then\n\\[\nP_{N,j}\n=\n\\sum_{\\substack{0\\leq s<M\\\\s\\equiv j\\pmod N}}P_{M,s}.\n\\]\nWhen $R$ is nonzero, the packet system modulo $M$ refines the packet system modulo $N$.\n\\end{theorem}","s\\equiv j\\pmod N}}P_{M,s}.\n\\]\nWhen $R$ is nonzero, the packet system modulo $M$ refines the packet system modulo $N$.\n\\end{theorem}","\\begin{proof}\nAn exponent $n$ satisfies $n\\equiv j\\pmod N$ exactly when its unique residue $s$ modulo $M$ satisfies $s\\equiv j\\pmod N$.\n\\end{proof}","\\begin{example}[The shift graph of residue packets]\\label{ex:residue_shift}\nAssume that $R$ is nonzero and $N>0$. Multiplication by $z$ sends the residue packet $P_{N,j}R[[z]]$ into $P_{N,j+1}R[[z]]$, where the index is read modulo $N$.  Its packet graph is the directed cycle\n\\[\n0\\longrightarrow1\\longrightarrow\\cdots\\longrightarrow N-1\\longrightarrow0.\n\\]\nMultiplication by a polynomial supported on a set of residues gives the corresponding finite set of phase shifts.  This is the simplest transfer graph behind a polyphase representation.\n\\end{example}","\\begin{theorem}[Coprime factorization]\\label{thm:crt_factorization}\nLet $A$ and $B$ be coprime positive integers.  For each $j$ modulo $AB$, write\n\\[\nj_A=j\\pmod A,\n\\qquad\nj_B=j\\pmod B.\n\\]\nThen\n\\[\nP_{AB,j}=P_{A,j_A}P_{B,j_B}.\n\\]\nEvery pair $(j_A,j_B)$ occurs for one residue $j$ modulo $AB$.\n\\end{theorem}","\\begin{proof}\nA coefficient survives the product on the right exactly when its exponent satisfies both congruences.  The Chinese remainder theorem gives one residue class modulo $AB$ with those two coordinates.  The needed form of the theorem is standard and may be found in \\cite{Lang2002}.\n\\end{proof}","\\begin{corollary}[Dyadic and odd coordinates]\\label{cor:dyadic_odd_coordinates}\\label{thm:dyadic-odd}\nWrite $N=2^aM$ with $M$ odd.  Every residue packet modulo $N$ is the product of one packet modulo $2^a$ and one packet modulo $M$.\n\\end{corollary}","\\begin{proof}\nApply \\cref{thm:crt_factorization} to the coprime moduli $2^a$ and $M$.\n\\end{proof}","\\begin{example}[Residues modulo twelve]\\label{ex:mod_twelve}\nThe pair\n\\[\n(j\\pmod4,j\\pmod3)=(3,1)\n\\]\ncorresponds to $j=7$ modulo $12$.  Hence\n\\[\nP_{12,7}=P_{4,3}P_{3,1}.\n\\]\nThis factorization concerns addresses.  It does not say that the two factors have the same integral spectral behavior.\n\\end{example}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{02_crt_dyadic_odd_grid.pdf}\n\\caption{A composite residue packet is addressed by its dyadic and odd coordinates. The displayed mod-$24$ system is the product of mod $8$ and mod $3$.}\n\\label{fig:crt-grid}\n\\end{figure}","\\begin{definition}[Complete multiplicative filtration]\\label{def:filtered_algebra}\nA complete filtered algebra is a unital algebra $\\Bcal$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","ef:filtered_algebra}\nA complete filtered algebra is a unital algebra $\\Bcal$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","bra is a unital algebra $\\Bcal$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","In the filtration topology, reduction to $\\Bcal/F^L\\Bcal$ is continuous when the quotient has the discrete topology. Its kernel $F^L\\Bcal$ is therefore closed. In particular, a convergent sum of terms in $F^L\\Bcal$ remains in that ideal.","If $\\OO$ is a complete discrete valuation ring with uniformizer $\\pi$ and $\\Bcal$ is a separated and complete $\\OO$ algebra for the filtration\n\\[\nF^L\\Bcal=\\pi^L\\Bcal,\n\\]\nthen this is a standard example. Finite matrix algebras over $\\OO$ satisfy these conditions. Completeness of $\\OO$ alone does not imply completeness of every $\\OO$ algebra. Basic facts about completeness in this setting are reviewed in \\cite{Serre1979}.","\\begin{theorem}[Weighted inverse]\\label{thm:weighted_inverse}\nLet $\\Bcal$ be a complete filtered algebra.  Let $D\\in\\Bcal$ be invertible with\n\\[\nD,D^{-1}\\in F^0\\Bcal.\n\\]\nLet $E$ be a finite set and suppose\n\\[\nT=D+\\sum_{e\\in E}N_e,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","red algebra.  Let $D\\in\\Bcal$ be invertible with\n\\[\nD,D^{-1}\\in F^0\\Bcal.\n\\]\nLet $E$ be a finite set and suppose\n\\[\nT=D+\\sum_{e\\in E}N_e,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","\\Bcal$ be invertible with\n\\[\nD,D^{-1}\\in F^0\\Bcal.\n\\]\nLet $E$ be a finite set and suppose\n\\[\nT=D+\\sum_{e\\in E}N_e,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","N_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","l,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","\\begin{proof}\nPut\n\\[\nA=\\sum_{e\\in E}A_e.\n\\]\nSince left multiplication by $D^{-1}$ preserves the filtration, $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}","nce left multiplication by $D^{-1}$ preserves the filtration, $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}","e filtration, $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}",", $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}","The preceding formula does not yet use packet blocks. When $D$ is packet diagonal and the terms $N_e$ are individual transfers, only words with compatible packet endpoints can contribute. These words become directed paths.","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\be","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\a","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ket system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ery packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","pha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","e, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","eta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","uad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","\\begin{proof}\nThe diagonal inverse is\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha.\n\\]\nThe block $P_\\beta D^{-1}NP_\\alpha$ is nonzero only when there is a declared edge from $\\alpha$ to $\\beta$.  Repeated block multiplication shows that the $r$th power of $D^{-1}N$ is the sum over all directed paths of length $r$, with the operator factors ordered along the path.  Apply \\cref{thm:weighted_inverse} and multiply on the right by the packet diagonal inverse.\n\\end{proof}","\\begin{corollary}[Minimum path weight]\\label{cor:minimum_path_weight}\nFor packets $\\alpha$ and $\\beta$, let $d(\\alpha,\\beta)$ be the least total weight of a directed path from $\\alpha$ to $\\beta$, with $d=\\infty$ when no path exists.  Use the convention $F^\\infty\\Bcal=\\bigcap_{L\\geq0}F^L\\Bcal=0$.  If $\\alpha\\ne\\beta$, then\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n\\in\nF^{d(\\alpha,\\beta)}\\Bcal.\n\\]\nIf there is no path from $\\alpha$ to $\\beta$, the block is zero.\n\\end{corollary}","ary}[Minimum path weight]\\label{cor:minimum_path_weight}\nFor packets $\\alpha$ and $\\beta$, let $d(\\alpha,\\beta)$ be the least total weight of a directed path from $\\alpha$ to $\\beta$, with $d=\\infty$ when no path exists.  Use the convention $F^\\infty\\Bcal=\\bigcap_{L\\geq0}F^L\\Bcal=0$.  If $\\alpha\\ne\\beta$, then\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n\\in\nF^{d(\\alpha,\\beta)}\\Bcal.\n\\]\nIf there is no path from $\\alpha$ to $\\beta$, the block is zero.\n\\end{corollary}","path weight]\\label{cor:minimum_path_weight}\nFor packets $\\alpha$ and $\\beta$, let $d(\\alpha,\\beta)$ be the least total weight of a directed path from $\\alpha$ to $\\beta$, with $d=\\infty$ when no path exists.  Use the convention $F^\\infty\\Bcal=\\bigcap_{L\\geq0}F^L\\Bcal=0$.  If $\\alpha\\ne\\beta$, then\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n\\in\nF^{d(\\alpha,\\beta)}\\Bcal.\n\\]\nIf there is no path from $\\alpha$ to $\\beta$, the block is zero.\n\\end{corollary}","\\begin{proof}\nEvery term in the block path formula is indexed by a path from $\\alpha$ to $\\beta$ and lies in the filtration level given by its total weight.  If no such path exists, the sum is empty.\n\\end{proof}","\\begin{example}[A longer path that appears earlier]\\label{ex:weighted_route}\nConsider three packets $\\alpha$, $\\beta$, and $\\gamma$.  Suppose there is a direct edge from $\\alpha$ to $\\gamma$ of weight three, together with edges\n\\[\n\\alpha\\longrightarrow\\beta,\n\\qquad\n\\beta\\longrightarrow\\gamma\n\\]\nof weight one each.  The two edge route has total weight two, so it contributes to the inverse before the direct edge does.  This is why the truncation rule uses total weight rather than path length. The lower bound in \\cref{cor:minimum_path_weight} need not be an equality because several paths of the same weight may cancel.\n\\end{example}","\\begin{corollary}[Acyclic transfer graph]\\label{cor:acyclic_inverse}\nIf the packet transfer graph is acyclic, then $D^{-1}N$ is nilpotent.  The inverse in \\cref{thm:block_path_formula} is a finite sum and does not require completeness of the filtration.\n\\end{corollary}","\\begin{corollary}[Stability of inversion at finite precision]\\label{cor:inverse_stability}\nLet $T$ and $\\widetilde T$ be invertible elements of $F^0\\Bcal$.  If\n\\[\nT-\\widetilde T\\in F^L\\Bcal,\n\\]\nthen\n\\[\nT^{-1}-\\widetilde T^{-1}\\in F^L\\Bcal.\n\\]\n\\end{corollary}","\\begin{proof}\nUse the resolvent identity at zero in the form\n\\[\nT^{-1}-\\widetilde T^{-1}\n=\nT^{-1}(\\widetilde T-T)\\widetilde T^{-1}.\n\\]\nBoth inverse factors belong to $F^0\\Bcal$, so the product lies in $F^L\\Bcal$.\n\\end{proof}","\\begin{corollary}[Exact finite precision truncation]\\label{cor:finite_precision}\nUnder the hypotheses of \\cref{thm:weighted_inverse}, define\n\\[\nR_L\n=\n\\sum_{W(\\gamma)<L}(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThen\n\\[\nT^{-1}-R_L\\in F^L\\Bcal,\n\\]\n\\[\nTR_L-I\\in F^L\\Bcal,\n\\qquad\nR_LT-I\\in F^L\\Bcal.\n\\]\nThus $R_L$ is the exact inverse in $\\Bcal/F^L\\Bcal$.\n\\end{corollary}","\\begin{proof}\nEvery omitted word has weight at least $L$, so the difference from the full inverse lies in $F^L\\Bcal$.  Multiplication by $T$ or by its inverse preserves membership in $F^L\\Bcal$ under the stated filtration assumptions.  The two residual statements follow.\n\\end{proof}","\\Needspace{4\\baselineskip}\n\\begin{example}[A triangular transfer]\\label{ex:triangular_transfer}\nLet $\\OO$ be complete with uniformizer $\\pi$, and consider\n\\[\nT=\n\\begin{pmatrix}\n1&\\pi^a","\\Needspace{4\\baselineskip}\n\\begin{example}[A triangular transfer]\\label{ex:triangular_transfer}\nLet $\\OO$ be complete with uniformizer $\\pi$, and consider\n\\[\nT=\n\\begin{pmatrix}\n1&\\pi^a\\\\\n0&1\n\\end{pmatrix}\n\\]\nwith $a\\geq1$.  The only nontrivial transfer has weight $a$, and its square is zero.  Hence\n\\[\nT^{-1}\n=\n\\begin{pmatrix}\n1&-\\pi^a","0&1\n\\end{pmatrix}\n\\]\nwith $a\\geq1$.  The only nontrivial transfer has weight $a$, and its square is zero.  Hence\n\\[\nT^{-1}\n=\n\\begin{pmatrix}\n1&-\\pi^a","0&1\n\\end{pmatrix}\n\\]\nwith $a\\geq1$.  The only nontrivial transfer has weight $a$, and its square is zero.  Hence\n\\[\nT^{-1}\n=\n\\begin{pmatrix}\n1&-\\pi^a\\\\\n0&1\n\\end{pmatrix}.\n\\]\nAt precision $L\\leq a$, the transfer vanishes and the inverse is the identity modulo $\\pi^L$.  At depth $a+1$, the transfer between the two packets first becomes visible.\n\\end{example}","0&1\n\\end{pmatrix}.\n\\]\nAt precision $L\\leq a$, the transfer vanishes and the inverse is the identity modulo $\\pi^L$.  At depth $a+1$, the transfer between the two packets first becomes visible.\n\\end{example}","Let $\\OO$ be a complete discrete valuation ring with uniformizer $\\pi$, fraction field $K$, and filtration\n\\[\nF^m\\cB=\\pi^m\\cB\n\\]\non an $\\OO$-algebra $\\cB$. The finite quotient statements below do not require $\\cB$ itself to be complete. Suppose\n\\[\nT=D+H,\n\\]\nwhere $D$ is invertible and the perturbation $H$ raises filtration. The symbol\n$K$ is reserved for the fraction field.","A packet decomposition resolves $H=\\sum_{e\\in E}H_e$ and assigns a\ncertified filtration gain\n\\[\nD^{-1}H_e\\in F^{w(e)}\\cB,\n\\qquad w(e)\\in\\Z_{>0}.\n\\]\nEach $e$ is a directed transfer edge. For an ordered transfer word\n\\[\n\\gamma=(e_1,\\ldots,e_r),\n\\]\ndefine\n\\[\nW(\\gamma)=\\sum_{j=1}^rw(e_j),\n\\qquad\nA_\\gamma=A_{e_1}\\cdots A_{e_r},\n\\qquad A_e=D^{-1}H_e.\n\\]","\\Needspace{8\\baselineskip}\n\\begin{theorem}[Certified weighted inverse]\\label{thm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","lineskip}\n\\begin{theorem}[Certified weighted inverse]\\label{thm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","[Certified weighted inverse]\\label{thm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","hm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFactor\n\\[\nT=D(I+A),\n\\qquad A=D^{-1}H.\n\\]\nFor a finite cutoff $r$, multiplication on either side gives the\nnoncommutative telescoping identity\n\\[\n(I+A)\\sum_{k=0}^{r-1}(-A)^k\n=\n\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)(I+A)\n=1-(-A)^r.\n\\]\nThis proves (a). In (b), the strict norm bound makes the geometric series\nsummable. Continuity of multiplication passes the same identities to the\nlimit. Multiplication by $D^{-1}$ gives the displayed inverse in both orders.\n\\end{proof}","tor\n\\[\nT=D(I+A),\n\\qquad A=D^{-1}H.\n\\]\nFor a finite cutoff $r$, multiplication on either side gives the\nnoncommutative telescoping identity\n\\[\n(I+A)\\sum_{k=0}^{r-1}(-A)^k\n=\n\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)(I+A)\n=1-(-A)^r.\n\\]\nThis proves (a). In (b), the strict norm bound makes the geometric series\nsummable. Continuity of multiplication passes the same identities to the\nlimit. Multiplication by $D^{-1}$ gives the displayed inverse in both orders.\n\\end{proof}","\\Needspace{4\\baselineskip}\n\\begin{corollary}[Finite precision]\\label{cor:finite-precision}\nLet $F^\\bullet R$ be a multiplicative decreasing filtration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","egin{corollary}[Finite precision]\\label{cor:finite-precision}\nLet $F^\\bullet R$ be a multiplicative decreasing filtration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","llet R$ be a multiplicative decreasing filtration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","tration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","et\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n\\\\W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le …","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{coroll","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","s directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","ly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","$q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","wo-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","d q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","Here $A_\\gamma=(D^{-1}H_{e_1})\\cdots(D^{-1}H_{e_r})$ in the written\norder. Multiplicativity places it in $F^{W(\\gamma)}R$, so the quotient kills\nit when $W(\\gamma)\\ge L$. Positive weights also imply\n$|\\gamma|\\le W(\\gamma)$, making the retained search finite.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{06_filtered_inverse_depth.pdf}\n\\caption{At precision $L$, weighted inversion requires only transfer words whose cumulative weight is below $L$.}\n\\label{fig:inverse-depth}\n\\end{figure}","\\begin{definition}[Weighted transfer series]\\label{def:transfer_series}\nLet $G$ be a finite directed multigraph.  Give every edge $e$ a positive integer weight $w(e)$ and a scalar multiplicity $m(e)$ in a field $\\bbk$.  Define the polynomial adjacency matrix\n\\[\nA(t)_{ij}\n=\n\\sum_{e:i\\to j}m(e)t^{w(e)}.\n\\]\nFor column vectors $u$ and $v$, define\n\\[\nZ_{u,v}(t)\n=\n\\sum_{n\\geq0}a_nt^n,\n\\]\nwhere\n\\[\na_n=\\sum_{i,j}\\ \\sum_{\\substack{\\gamma:i\\leadsto j","\\begin{definition}[Weighted transfer series]\\label{def:transfer_series}\nLet $G$ be a finite directed multigraph.  Give every edge $e$ a positive integer weight $w(e)$ and a scalar multiplicity $m(e)$ in a field $\\bbk$.  Define the polynomial adjacency matrix\n\\[\nA(t)_{ij}\n=\n\\sum_{e:i\\to j}m(e)t^{w(e)}.\n\\]\nFor column vectors $u$ and $v$, define\n\\[\nZ_{u,v}(t)\n=\n\\sum_{n\\geq0}a_nt^n,\n\\]\nwhere\n\\[\na_n=\\sum_{i,j}\\ \\sum_{\\substack{\\gamma:i\\leadsto j\\\\W(\\gamma)=n}}\n u_i v_j\\prod_{\\nu=1}^{|\\gamma|}m(e_\\nu).\n\\]\nHere $\\gamma=(e_1,\\ldots,e_{|\\gamma|})$ is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.\n\\end{definition}","W(\\gamma)=n}}\n u_i v_j\\prod_{\\nu=1}^{|\\gamma|}m(e_\\nu).\n\\]\nHere $\\gamma=(e_1,\\ldots,e_{|\\gamma|})$ is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.\n\\end{definition}","\\begin{theorem}[Rationality of the transfer series]\\label{thm:transfer_rationality}\nThe weighted transfer series satisfies\n\\[\nZ_{u,v}(t)\n=\nu^{\\mathsf T}(I-A(t))^{-1}v.\n\\]\nIt is a rational function whose denominator divides\n\\[\n\\det(I-A(t)).\n\\]\n\\end{theorem}","\\begin{proof}\nSince every entry of $A(t)$ has zero constant term, the inverse exists in the formal power series ring and\n\\[\n(I-A(t))^{-1}=\\sum_{r\\geq0}A(t)^r.\n\\]\nThe $(i,j)$ entry of $A(t)^r$ is the sum of the monomials attached to all paths of length $r$ from $i$ to $j$.  Summing over $r$ records paths of every length.  Multiplication by the endpoint vectors gives the stated series.  The adjugate formula gives\n\\[\n(I-A(t))^{-1}\n=\n\\frac{\\operatorname{adj}(I-A(t))}{\\det(I-A(t))},\n\\]\nso every entry is rational.\n\\end{proof}","ce every entry of $A(t)$ has zero constant term, the inverse exists in the formal power series ring and\n\\[\n(I-A(t))^{-1}=\\sum_{r\\geq0}A(t)^r.\n\\]\nThe $(i,j)$ entry of $A(t)^r$ is the sum of the monomials attached to all paths of length $r$ from $i$ to $j$.  Summing over $r$ records paths of every length.  Multiplication by the endpoint vectors gives the stated series.  The adjugate formula gives\n\\[\n(I-A(t))^{-1}\n=\n\\frac{\\operatorname{adj}(I-A(t))}{\\det(I-A(t))},\n\\]\nso every entry is rational.\n\\end{proof}","\\begin{corollary}[Recurrence for path counts]\\label{cor:path_recurrence}\nWrite\n\\[\nZ_{u,v}(t)=\\frac{P(t)}{Q(t)},\n\\qquad\nQ(t)=1+q_1t+\\cdots+q_dt^d.\n\\]\nUse the convention $\\deg 0=-\\infty$. Then, for every nonnegative integer $n>\\deg P$,\n\\[\na_n+q_1a_{n-1}+\\cdots+q_da_{n-d}=0.\n\\]\n\\end{corollary}","\\begin{proof}\nCompare coefficients in the identity\n\\[\nQ(t)Z_{u,v}(t)=P(t).\n\\]\nBeyond the degree of $P$, the coefficient on the right is zero.\n\\end{proof}","In the recurrence, set $a_k=0$ when $k<0$. This is the coefficient convention for the formal power series $Z_{u,v}(t)$ and makes the first recurrence equations meaningful even when $n<d$.","\\begin{corollary}[Several independent weights]\\label{cor:multigraded_series}\nSuppose every edge has a weight vector\n\\[\nw(e)=(w_1(e),\\ldots,w_r(e))\\in\\mathbb N^r\\setminus\\{0\\}.\n\\]\nReplace $t^{w(e)}$ by\n\\[\nt_1^{w_1(e)}\\cdots t_r^{w_r(e)}.\n\\]\nThe resulting multigraded transfer series is rational in $t_1,\\ldots,t_r$.\n\\end{corollary}","\\begin{example}[A transfer series with one loop]\\label{ex:simple_transfer_series}\nTake vertices $0$, $1$, and $2$.  Give the edge $0\\to1$ weight one, the loop $1\\to1$ weight two, and the edge $1\\to2$ weight one.  Every path from $0$ to $2$ uses the first and last edges and may traverse the loop any number of times.  Hence\n\\[\nZ_{0,2}(t)\n=\nt^2(1+t^2+t^4+\\cdots)\n=\n\\frac{t^2}{1-t^2}.\n\\]\nThere is one path at every positive even weight and no path at an odd weight.  The denominator records the increase of two units produced by the loop.\n\\end{example}","Let $G$ be a finite directed graph with positive integer edge weights. Define the polynomial adjacency matrix\n\\[\nA(t)_{ij}\n=\n\\sum_{e:i\\to j}t^{w(e)},\n\\]\nwhere parallel edges are distinct elements of the finite edge type and are\ntherefore counted with their full multiplicity.","For fixed start and target vertices $s,t$, define\n\\[\nZ_{s,t}(t)\n=\n\\sum_{n\\ge0}a_n(s,t)t^n,\n\\]\nwhere $a_n(s,t)$ counts directed walks from $s$ to $t$ with total weight $n$.\nVertices and edges may repeat. Thus ``path zeta'' is retained as the name of\nthe series and Lean API, but the walks are not required to visit each vertex only once.","\\begin{theorem}[Automatic finite-graph transfer-zeta rationality]\\label{thm:transfer-zeta}\nAssume the vertex and edge sets are finite and every edge has positive integer\nweight. For every $n$, exact enumeration of edge tuples of lengths at most\n$n$, followed by endpoint and total-weight filtering, returns $a_n(s,t)$.\nThese coefficients obey the recurrence obtained by removing the first edge\n\\[\na_n(s,t)\n=\n\\mathbf 1_{s=t,\\,n=0}\n+\\sum_{\\substack{e:\\,\\operatorname{source}(e)=s","\\begin{theorem}[Automatic finite-graph transfer-zeta rationality]\\label{thm:transfer-zeta}\nAssume the vertex and edge sets are finite and every edge has positive integer\nweight. For every $n$, exact enumeration of edge tuples of lengths at most\n$n$, followed by endpoint and total-weight filtering, returns $a_n(s,t)$.\nThese coefficients obey the recurrence obtained by removing the first edge\n\\[\na_n(s,t)\n=\n\\mathbf 1_{s=t,\\,n=0}\n+\\sum_{\\substack{e:\\,\\operatorname{source}(e)=s\\\\w(e)\\le n}}\na_{n-w(e)}(\\operatorname{target}(e),t).\n\\]\nLet\n\\[\nT(t)=I-A(t),\\qquad\nQ(t)=\\det T(t),\\qquad\nP_{s,t}(t)=\\operatorname{adj}(T(t))_{s,t}.\n\\]\nThen $Q(0)=1$, hence $Q\\ne0$ and $Q$ is a unit in $\\Z[[t]]$, and the\nindependently enumerated walk series satisfies\n\\[\n\\boxed{\nZ_{s,t}(t)=Q(t)^{-1}P_{s,t}(t),\n\\qquad\nQ(t)Z_{s,t}(t)=P_{s,t}(t)\n}\n\\quad\\text{in }\\Z[[t]].\n\\]\n\\end{theorem}","w(e)\\le n}}\na_{n-w(e)}(\\operatorname{target}(e),t).\n\\]\nLet\n\\[\nT(t)=I-A(t),\\qquad\nQ(t)=\\det T(t),\\qquad\nP_{s,t}(t)=\\operatorname{adj}(T(t))_{s,t}.\n\\]\nThen $Q(0)=1$, hence $Q\\ne0$ and $Q$ is a unit in $\\Z[[t]]$, and the\nindependently enumerated walk series satisfies\n\\[\n\\boxed{\nZ_{s,t}(t)=Q(t)^{-1}P_{s,t}(t),\n\\qquad\nQ(t)Z_{s,t}(t)=P_{s,t}(t)\n}\n\\quad\\text{in }\\Z[[t]].\n\\]\n\\end{theorem}","\\begin{proof}\nEvery edge has weight at least one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that con","\\begin{proof}\nEvery edge has weight at least one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the inde","\\begin{proof}\nEvery edge has weight at least one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","t one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ord of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","$n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","refore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ning predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","tive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","This is the finite transfer-matrix method of \\cite{Stanley2012}, but here its\ndeterminant, adjugate, normalization, recurrence from the first edge, and identification\nwith the enumerated series are all formalized. The older API accepting an\nindependently supplied normalized coefficient identity $QZ=P$ is retained for\nexternal certificates. It is no longer needed to obtain rationality for the\nfinite graph defined above. The Python companion performs an exact symbolic\nconstruction as an independent executable check.","\\begin{corollary}[Linear recurrence]\\label{cor:zeta-recurrence}\nWith the canonical polynomials from \\cref{thm:transfer-zeta}, write\n\\[\nQ(t)=\\sum_kq_kt^k,\n\\qquad\nZ_{s,t}(t)=\\sum_{n\\ge0}a_n(s,t)t^n.\n\\]\nThen the walk counts satisfy the recurrence with constant coefficients determined by\n$Q$ after the finite initial range contributed by $P_{s,t}$, with $\\deg 0=-\\infty$:\n\\[\n\\boxed{\n\\sum_{k=0}^{n}q_k a_{n-k}(s,t)=0\n\\qquad\\bigl(n>\\deg P_{s,t}\\bigr).\n}\n\\]\n\\end{corollary}","\\begin{example}[A computed transfer zeta]\nFor the graph used in the numerical companion, the walk series from packet $0$ to packet $2$ is\n\\[\nZ_{0,2}(t)\n=\n\\frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.\n\\]\nThe first nonzero coefficients are\n\\[\n1,1,1,3,1,5,4,10,7,19,17,37,\\ldots\n\\]\nat their corresponding weights. Direct enumeration through weight $30$ agrees coefficient by coefficient with the rational expansion.\n\\end{example}","Let $\\OO$ be a discrete valuation ring with fraction field $K$ and uniformizer $\\pi$.  Let $\\Lambda$ be a finite free $\\OO$ module and put\n\\[\nV=K\\otimes_{\\OO}\\Lambda.\n\\]\nSuppose $A\\in\\End_{\\OO}(\\Lambda)$.","\\begin{theorem}[Bézout spectral projector]\\label{thm:bezout_projector}\nAssume that\n\\[\n\\chi_A(x)=f(x)g(x)\n\\]\nwith monic polynomials $f,g\\in\\OO[x]$ that are coprime in $K[x]$.  Choose $u,v\\in\\OO[x]$ and a nonzero $r\\in\\OO$ such that\n\\[\nu(x)f(x)+v(x)g(x)=r.\n\\]\nThen\n\\[\nE_f=r^{-1}v(A)g(A)\n\\]\nis the projector of $V$ onto $\\Ker f(A)$ along $\\Ker g(A)$.  In particular,\n\\[\nE_f^2=E_f,\n\\qquad\nrE_f\\in\\End_{\\OO}(\\Lambda).\n\\]\n\\end{theorem}","\\begin{proof}\nCayley Hamilton gives\n\\[\nf(A)g(A)=0.\n\\]\nIf $x\\in\\Ker f(A)$, then the Bézout identity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","(A)=0.\n\\]\nIf $x\\in\\Ker f(A)$, then the Bézout identity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","$, then the Bézout identity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","tity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","\\begin{theorem}[Several coprime spectral packets]\\label{thm:several_projectors}\nAssume\n\\[\n\\chi_A(x)=\\prod_{i=1}^rf_i(x),\n\\]\nwhere the monic factors $f_i\\in\\OO[x]$ are pairwise coprime in $K[x]$.  Put\n\\[\ng_i=\\prod_{j\\ne i}f_j.\n\\]\nThere are complementary projectors $E_i$ with\n\\[\n\\im E_i=\\Ker f_i(A),\n\\qquad\n\\sum_iE_i=I,\n\\qquad\nE_iE_j=0\n\\quad(i\\ne j).\n\\]\nOne may choose $E_i$ so that\n\\[\n\\Res(f_i,g_i)E_i\\in\\End_{\\OO}(\\Lambda)\n\\]\nup to multiplication of the resultant by a unit.\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:bezout_projector} to the factorization $\\chi_A=f_ig_i$.  The resulting projector is one on $\\Ker f_i(A)$ and zero on every $\\Ker f_j(A)$ with $j\\ne i$.  The direct sum decomposition follows from the polynomial Chinese remainder theorem.  The orthogonality and sum identities follow by evaluating the corresponding idempotents in $K[x]/(\\chi_A)$.  The Sylvester Bézout relation for $(f_i,g_i)$ gives the stated denominator bound.\n\\end{proof}","}\nApply \\cref{thm:bezout_projector} to the factorization $\\chi_A=f_ig_i$.  The resulting projector is one on $\\Ker f_i(A)$ and zero on every $\\Ker f_j(A)$ with $j\\ne i$.  The direct sum decomposition follows from the polynomial Chinese remainder theorem.  The orthogonality and sum identities follow by evaluating the corresponding idempotents in $K[x]/(\\chi_A)$.  The Sylvester Bézout relation for $(f_i,g_i)$ gives the stated denominator bound.\n\\end{proof}","\\begin{corollary}[Resultant denominator bound]\\label{cor:resultant_bound}\nUnder the hypotheses of \\cref{thm:bezout_projector}, one may take\n\\[\nr=\\Res(f,g)\n\\]\nup to multiplication by a unit.  Therefore\n\\[\n\\pi^{\\val_\\pi(\\Res(f,g))}E_f\\Lambda\\subseteq\\Lambda.\n\\]\n\\end{corollary}","\\begin{example}[A resultant bound that is not exact]\\label{ex:loose_resultant}\nLet\n\\[\nA=\\diag(0,\\pi^s)\n\\]\non the standard lattice $\\Lambda=\\OO^2$.  The factors $x$ and $x-\\pi^s$ have resultant $-\\pi^s$, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors.  Their exact separation weight is zero.  The resultant sees the closeness of the two eigenvalues.  It does not see that the chosen lattice already splits along their eigenspaces.\n\\end{example}","Let $\\OO$ be a DVR, $K=\\Frac(\\OO)$, and $A\\in\\End_\\OO(\\Lambda)$ for a finite free lattice $\\Lambda$. Suppose the characteristic polynomial factors over $\\OO$ as\n\\[\n\\chi_A(x)=f(x)g(x),\n\\]\nwhere $f$ and $g$ are coprime over $K[x]$.","\\begin{theorem}[Evaluated B\\'ezout spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","begin{theorem}[Evaluated B\\'ezout spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","}[Evaluated B\\'ezout spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","out spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","\\begin{proof}\nThe B\\'ezout identity gives $N=rI-UF$. The two stated commutation relations\nand $FG=0$ give\n\\[\nN(UF)=VGUF=VUGF=VUFG=0.\n\\]\nTherefore\n\\[\nN^2=N(rI-UF)=rN.\n\\]\nSince $r\\ne0$, $E^2=r^{-2}N^2=E$. If $Fx=0$, the evaluated B\\'ezout\nidentity gives $VGx=rx$, hence $Ex=x$. If $Gx=0$, then $Ex=0$.\nThe lattice assertion follows directly from $rE=VG$.\n\\end{proof}","The equality $N^2=rN$ before division uses neither $r\\ne0$ nor the other\npairwise commutation relations that hold in the polynomial application.\nNormalizing to $E$ uses $r\\ne0$. Its action on $\\ker F$ uses only the B\\'ezout\nidentity and $r\\ne0$, while vanishing on $\\ker G$ follows immediately from $N=VG$.\nThe bundled Lean certificate records all pairwise commutations as convenient\nevaluated polynomial data, but the minimal theorems expose only the hypotheses\nthey consume.","For the polynomial application, one supplies $F=f(A)$, $G=g(A)$,\n$U=u(A)$, and $V=v(A)$. A B\\'ezout or resultant identity\n$uf+vg=r$ evaluates to the first certificate, while\n$(fg)(A)=0$ supplies the second. When $fg=\\chi_A$, this last equality is\nCayley Hamilton. Standard B\\'ezout, resultant, and DVR facts are recalled\nfrom \\cite{Lang2002,Serre1979}. The inherited certificate theorem begins with the evaluated\ncertificate. It does not infer exact action merely from nilpotence of\n$f(A)$ or $g(A)$, which would be insufficient.","A)$. A B\\'ezout or resultant identity\n$uf+vg=r$ evaluates to the first certificate, while\n$(fg)(A)=0$ supplies the second. When $fg=\\chi_A$, this last equality is\nCayley Hamilton. Standard B\\'ezout, resultant, and DVR facts are recalled\nfrom \\cite{Lang2002,Serre1979}. The inherited certificate theorem begins with the evaluated\ncertificate. It does not infer exact action merely from nilpotence of\n$f(A)$ or $g(A)$, which would be insufficient.","ezout or resultant identity\n$uf+vg=r$ evaluates to the first certificate, while\n$(fg)(A)=0$ supplies the second. When $fg=\\chi_A$, this last equality is\nCayley Hamilton. Standard B\\'ezout, resultant, and DVR facts are recalled\nfrom \\cite{Lang2002,Serre1979}. The inherited certificate theorem begins with the evaluated\ncertificate. It does not infer exact action merely from nilpotence of\n$f(A)$ or $g(A)$, which would be insufficient.","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothes","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nt","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is ob","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","ight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","rojector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","ice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","\\begin{theorem}[Projector precision delay]\\label{thm:precision-delay}\nLet $E$ be a rational projector with $\\pi^sE(\\Lambda)\\subseteq\\Lambda$. If\n\\[\nx\\equiv y\\pmod{\\pi^{m+s}\\Lambda},\n\\]\nthen\n\\[\nEx\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $m+s$ input digits suffice to determine $m$ output digits of the projected packet.\n\\end{theorem}","\\begin{proof}\nWrite $x-y=\\pi^{m+s}z$. Since $\\pi^sE(z)\\in\\Lambda$,\n\\[\nE(x-y)=\\pi^m\\bigl(\\pi^sE(z)\\bigr)\\in\\pi^m\\Lambda.\n\\]\n\\end{proof}","The theorem gives a precision budget for delayed separation. A packet with weight $s$ can be reconstructed at the requested output precision by retaining $s$ additional input digits.","\\Needspace{8\\baselineskip}\n\\begin{definition}[Lattice packet data]\\label{def:lattice_packet}\nLet\n\\[\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha\n\\]\nbe a finite direct sum over $K$, with packet projectors $E_\\alpha$.  Let $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","ttice_packet}\nLet\n\\[\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha\n\\]\nbe a finite direct sum over $K$, with packet projectors $E_\\alpha$.  Let $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","et\n\\[\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha\n\\]\nbe a finite direct sum over $K$, with packet projectors $E_\\alpha$.  Let $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","et $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","it}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","s_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","mbda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","e gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","e\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","The gluing module $G_{\\Lambda,\\Pcal}$ is a finitely generated torsion $\\OO$ module, hence has finite length, because $\\Lambda_{\\mathrm{split}}$ and $\\Lambda$ are full lattices in the same vector space. To see that the split lattice is full, clear denominators of bases of the rational summands. Each $\\Lambda\\cap V_\\alpha$ then spans $V_\\alpha$, so their direct sum spans $V$.","\\begin{theorem}[Gluing module and exact separation weight]\\label{thm:gluing_weight}\nWith the notation of \\cref{def:lattice_packet}, the map\n\\[\n\\Phi:G_{\\Lambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","ght]\\label{thm:gluing_weight}\nWith the notation of \\cref{def:lattice_packet}, the map\n\\[\n\\Phi:G_{\\Lambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","_packet}, the map\n\\[\n\\Phi:G_{\\Lambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","ambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","\\begin{proof}\nIf $x\\in\\Lambda$ maps to zero, then $E_\\alpha x\\in\\Lambda_\\alpha$ for every $\\alpha$.  Since\n\\[\nx=\\sum_\\alpha E_\\alpha x,\n\\]\nwe have $x\\in\\Lambda_{\\mathrm{split}}$.  Thus $\\Phi$ is injective.  The coordinate map onto $E_\\alpha\\Lambda/\\Lambda_\\alpha$ is surjective by the definition of $E_\\alpha\\Lambda$.","Let $s=s_\\Lambda(\\Pcal)$.  For every $x\\in\\Lambda$ and every $\\alpha$,\n\\[\n\\pi^sE_\\alpha x\\in\\Lambda_\\alpha.\n\\]\nHence\n\\[\n\\pi^sx=\\sum_\\alpha\\pi^sE_\\alpha x\\in\\Lambda_{\\mathrm{split}},\n\\]\nso $\\pi^sG_{\\Lambda,\\Pcal}=0$.  This proves\n\\[\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\\leq s.\n\\]\nIf $d<s$, then for some $\\alpha$ there is an $x\\in\\Lambda$ with\n\\[\n\\pi^dE_\\alpha x\\notin\\Lambda_\\alpha.\n\\]\nThe class $\\pi^d[x]$ is therefore nonzero in the gluing module.  Thus $\\pi^d$ does not annihilate the gluing module, and equality follows.  The final formula is the usual description of the exponent through Smith normal form.\n\\end{proof}","$ and every $\\alpha$,\n\\[\n\\pi^sE_\\alpha x\\in\\Lambda_\\alpha.\n\\]\nHence\n\\[\n\\pi^sx=\\sum_\\alpha\\pi^sE_\\alpha x\\in\\Lambda_{\\mathrm{split}},\n\\]\nso $\\pi^sG_{\\Lambda,\\Pcal}=0$.  This proves\n\\[\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\\leq s.\n\\]\nIf $d<s$, then for some $\\alpha$ there is an $x\\in\\Lambda$ with\n\\[\n\\pi^dE_\\alpha x\\notin\\Lambda_\\alpha.\n\\]\nThe class $\\pi^d[x]$ is therefore nonzero in the gluing module.  Thus $\\pi^d$ does not annihilate the gluing module, and equality follows.  The final formula is the usual description of the exponent through Smith normal form.\n\\end{proof}","\\begin{corollary}[Total gluing and worst precision]\\label{cor:length_and_weight}\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO,\n\\]\nthen\n\\[\n\\length_{\\OO}(G_{\\Lambda,\\Pcal})\n=\n\\sum_{j=1}^ra_j,\n\\qquad\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\nThus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.\n\\end{corollary}","\\begin{corollary}[Two complementary packets]\\label{cor:binary_gluing}\nSuppose\n\\[\nV=V_1\\oplus V_2\n\\]\nwith projector $E$ onto $V_1$ and complementary projector $I-E$.  Then\n\\[\n\\Lambda/(\\Lambda_1\\oplus\\Lambda_2)\n\\cong\nE\\Lambda/\\Lambda_1\n\\cong\n(I-E)\\Lambda/\\Lambda_2.\n\\]\nConsequently\n\\[\ns_\\Lambda(E)=s_\\Lambda(I-E)\n=\n\\expn_\\pi\\bigl(\\Lambda/(\\Lambda_1\\oplus\\Lambda_2)\\bigr).\n\\]\n\\end{corollary}","\\begin{proof}\nMap $x\\in\\Lambda$ to $Ex+\\Lambda_1$.  Its kernel consists of the $x$ for which $Ex\\in\\Lambda_1$.  In that case\n\\[\n(I-E)x=x-Ex\\in\\Lambda\\cap V_2=\\Lambda_2,\n\\]\nso the kernel is $\\Lambda_1\\oplus\\Lambda_2$.  The map is surjective by definition.  The complementary map is handled in the same way.  The equality of weights follows from \\cref{thm:gluing_weight}.\n\\end{proof}","\\begin{corollary}[Integral splitting criterion]\\label{cor:integral_split}\nEvery packet projector preserves $\\Lambda$ if and only if\n\\[\n\\Lambda=\\bigoplus_\\alpha(\\Lambda\\cap V_\\alpha).\n\\]\nEquivalently,\n\\[\ns_\\Lambda(\\Pcal)=0\n\\quad\\Longleftrightarrow\\quad\nG_{\\Lambda,\\Pcal}=0.\n\\]\n\\end{corollary}","\\begin{proof}\nIf every projector preserves $\\Lambda$, then\n\\[\nx=\\sum_\\alpha E_\\alpha x\n\\]\nexpresses every $x\\in\\Lambda$ as a sum of vectors in $\\Lambda_\\alpha$.  The converse is immediate because each projector is the identity on one direct summand and zero on the others.  The final equivalence follows from \\cref{thm:gluing_weight}.\n\\end{proof}","\\begin{theorem}[Gluing exact sequence under refinement]\\label{thm:gluing_refinement}\nLet $\\Qcal=(Q_\\beta)$ refine $\\Pcal=(P_\\alpha)$ on $V$, and let $\\Lambda$ be a lattice.  Define\n\\[\n\\Lambda_{\\Pcal}\n=\n\\bigoplus_\\alpha(\\Lambda\\cap P_\\alpha V),\n\\qquad\n\\Lambda_{\\Qcal}\n=\n\\bigoplus_\\beta(\\Lambda\\cap Q_\\beta V).\n\\]\nThen\n\\[\n\\Lambda_{\\Qcal}\\subseteq\\Lambda_{\\Pcal}\\subseteq\\Lambda\n\\]\nand there is a short exact sequence\n\\[\n0\n\\longrightarrow\n\\Lambda_{\\Pcal}/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Pcal}\n\\longrightarrow0.\n\\]\nIn particular,\n\\[\n\\boxed{\ns_\\Lambda(\\Pcal)\\leq s_\\Lambda(\\Qcal).\n}\n\\]\n\\end{theorem}","luing exact sequence under refinement]\\label{thm:gluing_refinement}\nLet $\\Qcal=(Q_\\beta)$ refine $\\Pcal=(P_\\alpha)$ on $V$, and let $\\Lambda$ be a lattice.  Define\n\\[\n\\Lambda_{\\Pcal}\n=\n\\bigoplus_\\alpha(\\Lambda\\cap P_\\alpha V),\n\\qquad\n\\Lambda_{\\Qcal}\n=\n\\bigoplus_\\beta(\\Lambda\\cap Q_\\beta V).\n\\]\nThen\n\\[\n\\Lambda_{\\Qcal}\\subseteq\\Lambda_{\\Pcal}\\subseteq\\Lambda\n\\]\nand there is a short exact sequence\n\\[\n0\n\\longrightarrow\n\\Lambda_{\\Pcal}/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Pcal}\n\\longrightarrow0.\n\\]\nIn particular,\n\\[\n\\boxed{\ns_\\Lambda(\\Pcal)\\leq s_\\Lambda(\\Qcal).\n}\n\\]\n\\end{theorem}","ambda_{\\Pcal}\n=\n\\bigoplus_\\alpha(\\Lambda\\cap P_\\alpha V),\n\\qquad\n\\Lambda_{\\Qcal}\n=\n\\bigoplus_\\beta(\\Lambda\\cap Q_\\beta V).\n\\]\nThen\n\\[\n\\Lambda_{\\Qcal}\\subseteq\\Lambda_{\\Pcal}\\subseteq\\Lambda\n\\]\nand there is a short exact sequence\n\\[\n0\n\\longrightarrow\n\\Lambda_{\\Pcal}/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Pcal}\n\\longrightarrow0.\n\\]\nIn particular,\n\\[\n\\boxed{\ns_\\Lambda(\\Pcal)\\leq s_\\Lambda(\\Qcal).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nEvery fine packet space lies in its parent packet space, so every element of $\\Lambda_{\\Qcal}$ belongs to $\\Lambda_{\\Pcal}$.  The displayed exact sequence is the standard quotient sequence for nested submodules.  The exponent of a quotient cannot exceed the exponent of the module from which it is obtained.  Apply \\cref{thm:gluing_weight} to the two gluing modules.\n\\end{proof}","\\begin{example}[Strict growth under refinement]\\label{ex:strict_weight_growth}\nTake the standard lattice $\\mathbb Z_2^2$ and the involution from \\cref{prop:sign_projectors}.  The one packet system $(I)$ has weight zero.  Refining it into the two rational sign packets produces weight one.  The coarse lattice is already split because there is only one summand.  The fine eigenspaces are glued together by one dyadic digit.\n\\end{example}","% Keep the final proof marker with its displayed conclusion.\n\\enlargethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","Keep the final proof marker with its displayed conclusion.\n\\enlargethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","splayed conclusion.\n\\enlargethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","largethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","skip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","gin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","ary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","n delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","\\begin{proof}\nIf\n\\[\nx-y=\\pi^{m+s}z,\n\\]\nthen\n\\[\nE(x-y)=\\pi^m(\\pi^sEz)\\in\\pi^m\\Lambda.\n\\]\nThis proves that the rule is well defined.  Now take $d<s$.  By the minimality of $s$, there is a $z\\in\\Lambda$ with\n\\[\n\\pi^dEz\\notin\\Lambda.\n\\]\nSet\n\\[\nx=\\pi^{m+d}z,\n\\qquad\ny=0.\n\\]\nThen $x\\equiv y$ modulo $\\pi^{m+d}\\Lambda$, while\n\\[\nE(x-y)=\\pi^m(\\pi^dEz)\\notin\\pi^m\\Lambda.\n\\]\n\\end{proof}","\\begin{example}[The same rational split with two lattices]\\label{ex:lattice_dependence}\nLet\n\\[\nV=Ke_1\\oplus Ke_2\n\\]\nand let $E$ project onto $Ke_1$.  On the split lattice\n\\[\n\\Lambda_0=\\OO e_1\\oplus\\OO e_2,\n\\]\nthe weight is zero.  Fix $s\\geq1$ and consider\n\\[\n\\Lambda_s\n=\n\\OO e_1+\n\\OO(\\pi^{-s}e_1+e_2).\n\\]\nThen\n\\[\n\\Lambda_s\\cap Ke_1=\\OO e_1,\n\\qquad\n\\Lambda_s\\cap Ke_2=\\pi^s\\OO e_2.\n\\]\nThe gluing module is isomorphic to $\\OO/\\pi^s\\OO$, so\n\\[\ns_{\\Lambda_s}(E)=s.\n\\]\nThe rational decomposition has not changed.  Only the lattice has changed.\n\\end{example}","same rational split with two lattices]\\label{ex:lattice_dependence}\nLet\n\\[\nV=Ke_1\\oplus Ke_2\n\\]\nand let $E$ project onto $Ke_1$.  On the split lattice\n\\[\n\\Lambda_0=\\OO e_1\\oplus\\OO e_2,\n\\]\nthe weight is zero.  Fix $s\\geq1$ and consider\n\\[\n\\Lambda_s\n=\n\\OO e_1+\n\\OO(\\pi^{-s}e_1+e_2).\n\\]\nThen\n\\[\n\\Lambda_s\\cap Ke_1=\\OO e_1,\n\\qquad\n\\Lambda_s\\cap Ke_2=\\pi^s\\OO e_2.\n\\]\nThe gluing module is isomorphic to $\\OO/\\pi^s\\OO$, so\n\\[\ns_{\\Lambda_s}(E)=s.\n\\]\nThe rational decomposition has not changed.  Only the lattice has changed.\n\\end{example}","with two lattices]\\label{ex:lattice_dependence}\nLet\n\\[\nV=Ke_1\\oplus Ke_2\n\\]\nand let $E$ project onto $Ke_1$.  On the split lattice\n\\[\n\\Lambda_0=\\OO e_1\\oplus\\OO e_2,\n\\]\nthe weight is zero.  Fix $s\\geq1$ and consider\n\\[\n\\Lambda_s\n=\n\\OO e_1+\n\\OO(\\pi^{-s}e_1+e_2).\n\\]\nThen\n\\[\n\\Lambda_s\\cap Ke_1=\\OO e_1,\n\\qquad\n\\Lambda_s\\cap Ke_2=\\pi^s\\OO e_2.\n\\]\nThe gluing module is isomorphic to $\\OO/\\pi^s\\OO$, so\n\\[\ns_{\\Lambda_s}(E)=s.\n\\]\nThe rational decomposition has not changed.  Only the lattice has changed.\n\\end{example}","The example can be realized by an integral operator.  Define\n\\[\nAe_1=0,\n\\qquad\nAe_2=\\pi^se_2.\n\\]\nIn the basis\n\\[\ne_1,\n\\quad\n\\pi^{-s}e_1+e_2,\n\\]\nthe matrix of $A$ is\n\\[\n\\begin{pmatrix}\n0&-1","The example can be realized by an integral operator.  Define\n\\[\nAe_1=0,\n\\qquad\nAe_2=\\pi^se_2.\n\\]\nIn the basis\n\\[\ne_1,\n\\quad\n\\pi^{-s}e_1+e_2,\n\\]\nthe matrix of $A$ is\n\\[\n\\begin{pmatrix}\n0&-1\\\\\n0&\\pi^s\n\\end{pmatrix}.\n\\]\nIts eigenvalue factors have resultant equal to $\\pi^s$ up to a unit, and the exact projector weight on $\\Lambda_s$ is also $s$.","0&\\pi^s\n\\end{pmatrix}.\n\\]\nIts eigenvalue factors have resultant equal to $\\pi^s$ up to a unit, and the exact projector weight on $\\Lambda_s$ is also $s$.","\\begin{proposition}[Sign projectors over a lattice]\\label{prop:sign_projectors}\nLet\n\\[\nA=\n\\begin{pmatrix}\n1&1\\\\\n0&-1\n\\end{pmatrix}\n\\]\nact on $\\mathbb Z_p^2$.  Over $\\mathbb Q_p$, the sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2","0&-1\n\\end{pmatrix}\n\\]\nact on $\\mathbb Z_p^2$.  Over $\\mathbb Q_p$, the sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2","0&-1\n\\end{pmatrix}\n\\]\nact on $\\mathbb Z_p^2$.  Over $\\mathbb Q_p$, the sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2\\\\\n0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2","0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2\\\\\n0&1\n\\end{pmatrix}.\n\\]\nFor every odd prime $p$, both projectors preserve $\\mathbb Z_p^2$.  At $p=2$,\n\\[\ns_2(E_+)=s_2(E_-)=1.\n\\]\nThe factors $x-1$ and $x+1$ have resultant $2$.\n\\end{proposition}","0&1\n\\end{pmatrix}.\n\\]\nFor every odd prime $p$, both projectors preserve $\\mathbb Z_p^2$.  At $p=2$,\n\\[\ns_2(E_+)=s_2(E_-)=1.\n\\]\nThe factors $x-1$ and $x+1$ have resultant $2$.\n\\end{proposition}","\\begin{proof}\nThe displayed matrices are complementary idempotents.  The scalar $2$ is a unit in $\\mathbb Z_p$ for odd $p$, so both matrices are integral there.  Over $\\mathbb Z_2$, multiplying by $2$ clears the denominators, while neither projector preserves the standard lattice.  Thus both weights are one.  The resultant of the two linear factors is their root difference, which is $2$.\n\\end{proof}","Modulo $2$, the two eigenvalues become equal.  The rational sign packets remain distinct over $\\mathbb Q_2$, but the standard lattice does not split into their intersections until one more dyadic digit is retained.","\\Needspace{4\\baselineskip}\n\\begin{proposition}[Integral character averages away from the group order]\\label{prop:character_integrality}\nLet $\\OO$ be a discrete valuation ring, let $N$ be a unit in $\\OO$, and suppose $\\OO$ contains a primitive $N$th root of unity $\\omega$.  If an integral operator $U$ satisfies $U^N=I$, then\n\\[\nE_j=\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k\n\\]\npreserves every $U$ stable $\\OO$ lattice.  At primes dividing $N$, this formula alone gives no integral projector.\n\\end{proposition}","edspace{4\\baselineskip}\n\\begin{proposition}[Integral character averages away from the group order]\\label{prop:character_integrality}\nLet $\\OO$ be a discrete valuation ring, let $N$ be a unit in $\\OO$, and suppose $\\OO$ contains a primitive $N$th root of unity $\\omega$.  If an integral operator $U$ satisfies $U^N=I$, then\n\\[\nE_j=\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k\n\\]\npreserves every $U$ stable $\\OO$ lattice.  At primes dividing $N$, this formula alone gives no integral projector.\n\\end{proposition}","osition}[Integral character averages away from the group order]\\label{prop:character_integrality}\nLet $\\OO$ be a discrete valuation ring, let $N$ be a unit in $\\OO$, and suppose $\\OO$ contains a primitive $N$th root of unity $\\omega$.  If an integral operator $U$ satisfies $U^N=I$, then\n\\[\nE_j=\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k\n\\]\npreserves every $U$ stable $\\OO$ lattice.  At primes dividing $N$, this formula alone gives no integral projector.\n\\end{proposition}","\\begin{proof}\nEvery scalar coefficient in the displayed sum belongs to $\\OO$ because $N$ is a unit and $\\omega$ is integral.  Each power of $U$ preserves the lattice, so their $\\OO$ linear combination does as well.  When $N$ is not a unit, division by $N$ is not an operation in $\\OO$, so the same argument is unavailable.\n\\end{proof}","At a prime dividing $N$, the factor $1/N$ no longer guarantees that the character average preserves the lattice. Cancellation may still make a particular projector integral. Coefficient residue selectors remain exact because they never divide by the group order.","Consider the integral involution\n\\[\nA=\n\\begin{pmatrix}\n1&1\\\\\n0&-1\n\\end{pmatrix},\n\\qquad\nA^2=I.\n\\]\nOver a field of characteristic different from two, its sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2","0&-1\n\\end{pmatrix},\n\\qquad\nA^2=I.\n\\]\nOver a field of characteristic different from two, its sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2\\\\\n0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2","0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2\\\\\n0&1\n\\end{pmatrix}.\n\\]\nThey are orthogonal idempotents and sum to $I$.","0&1\n\\end{pmatrix}.\n\\]\nThey are orthogonal idempotents and sum to $I$.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{07_dyadic_delayed_separation.pdf}\n\\caption{The rational sign projectors for an integral involution. Their denominator has $2$-adic weight one.}\n\\label{fig:dyadic-projectors}\n\\end{figure}","\\begin{proposition}[Odd-adic versus dyadic behavior]\\label{prop:dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, matching both local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","sus dyadic behavior]\\label{prop:dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, matching both local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","behavior]\\label{prop:dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, matching both local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}",":dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, matching both local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","ign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, matching both local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","d each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, matching both local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","or weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, matching both local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","\\begin{proof}\nThe reduced denominators of the displayed entries are $1$ or $2$, which proves\nthe local integrality and minimality statements directly. The resultant is\n\\[\n\\Res(x-1,x+1)=2.\n\\]\nThus $v_2(2)=1$ and $v_p(2)=0$ for primes $p\\ne2$.\n\\end{proof}","The two characteristic factors $x-1$ and $x+1$ have the same reduction in\n$\\F_2[x]$. The rational matrices $E_+$ and $E_-$ themselves cannot be reduced\nmodulo $2$, because their entries contain $1/2$. Consequently the integral\nspecial fiber does not inherit their separate idempotent splitting, even\nthough the two projectors remain distinct over $\\Q_2$. This is the simplest\nmodel of delayed dyadic separation.","The failure of the character formula\n\\[\n\\frac12(I\\pm A)\n\\]\nto be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished:\n\\[\n\\boxed{\n\\text{integral address selector}\n\\quad\\neq\\quad\n\\text{character-average spectral projector}\n}\n\\]\nwhen the group order is not a unit. The first can remain exact over $\\Z_2$. The second can acquire a denominator weight.","\\begin{standardfact}[Smith normal form over a discrete valuation ring]\\label{fact:smith}\nLet $M\\in M_{m,n}(\\OO)$.  There are invertible matrices\n\\[\nU\\in\\operatorname{GL}_m(\\OO),\n\\qquad\nV\\in\\operatorname{GL}_n(\\OO)\n\\]\nsuch that\n\\[\nUMV\n=\n\\diag(\\pi^{s_1},\\ldots,\\pi^{s_r},0,\\ldots,0),\n\\]\nwhere\n\\[\n0\\leq s_1\\leq\\cdots\\leq s_r\n\\]\nand $r$ is the rank over $K$.  The exponents are uniquely determined.  See \\cite{Newman1972}.\n\\end{standardfact}","\\begin{proposition}[Computing a packet weight from an inclusion matrix]\\label{prop:weight_from_inclusion}\nChoose an $\\OO$ basis of $\\Lambda$ and an $\\OO$ basis of $\\Lambda_{\\mathrm{split}}$.  Let $B$ be the matrix of the inclusion\n\\[\n\\Lambda_{\\mathrm{split}}\\hookrightarrow\\Lambda.\n\\]\nIf the Smith form of $B$ is\n\\[\n\\diag(\\pi^{a_1},\\ldots,\\pi^{a_n}),\n\\qquad\n0\\leq a_1\\leq\\cdots\\leq a_n,\n\\]\nthen\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^n\\OO/\\pi^{a_j}\\OO,\n\\qquad\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq n\\}\\bigr).\n\\]\n\\end{proposition}","\\begin{proof}\nThe gluing module is the cokernel of the inclusion matrix $B$.  Smith normal form gives its invariant factors.  The weight formula follows from \\cref{thm:gluing_weight}.\n\\end{proof}","\\begin{theorem}[Smith weighted congruence solve]\\label{thm:smith_congruence}\nLet $M\\in M_{m,n}(\\OO)$ have Smith exponents $s_1,\\ldots,s_r$.  Assume the residue field is finite of cardinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","bel{thm:smith_congruence}\nLet $M\\in M_{m,n}(\\OO)$ have Smith exponents $s_1,\\ldots,s_r$.  Assume the residue field is finite of cardinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","e field is finite of cardinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","rdinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nSet $x=Vy$.  Multiplying the congruence by $U$ reduces it to independent equations\n\\[\n\\pi^{s_i}y_i\\equiv\\widetilde b_i\\pmod{\\pi^L}\n\\qquad(1\\leq i\\leq r)\n\\]\nand zero equations in the remaining rows.  Multiplication by $\\pi^{s_i}$ on $\\OO/\\pi^L\\OO$ has image equal to the elements divisible by $\\pi^{\\min(s_i,L)}$ and kernel of size\n\\[\nq^{\\min(s_i,L)}.\n\\]\nEach of the $n-r$ unused variables is free and contributes $q^L$ choices.  Multiplying the independent counts gives the formula.\n\\end{proof}","ultiplying the congruence by $U$ reduces it to independent equations\n\\[\n\\pi^{s_i}y_i\\equiv\\widetilde b_i\\pmod{\\pi^L}\n\\qquad(1\\leq i\\leq r)\n\\]\nand zero equations in the remaining rows.  Multiplication by $\\pi^{s_i}$ on $\\OO/\\pi^L\\OO$ has image equal to the elements divisible by $\\pi^{\\min(s_i,L)}$ and kernel of size\n\\[\nq^{\\min(s_i,L)}.\n\\]\nEach of the $n-r$ unused variables is free and contributes $q^L$ choices.  Multiplying the independent counts gives the formula.\n\\end{proof}","\\begin{corollary}[Obstruction and branching data]\\label{cor:smith_obstruction}\nThe transformed entries of $\\widetilde b$ that fail the stated divisibility conditions are the complete finite congruence obstructions.  When all obstructions vanish, the Smith exponents determine the number of solution branches at depth $L$.\n\\end{corollary}","\\begin{example}[Three weighted directions]\\label{ex:smith_three}\nTake $p=2$, $L=6$, and the diagonal weights\n\\[\n(0,1,3).\n\\]\nThe right side\n\\[\n(1,2,8)\n\\]\nis admissible.  The three coordinate kernels have sizes\n\\[\n1,\n\\quad2,\n\\quad8,\n\\]\nso there are $16$ solutions.  Replacing the final entry $8$ by $4$ violates the required divisibility by $2^3$ and makes the system unsolvable.\n\\end{example}","Let $M$ be a matrix over a DVR. Smith normal form gives invertible matrices $U,V$ such that\n\\[\nUMV\n=\n\\diag(\\pi^{s_1}u_1,\\ldots,\\pi^{s_r}u_r,0,\\ldots,0),\n\\]\nwhere each $u_i$ is a unit and\n\\[\n0\\le s_1\\le\\cdots\\le s_r.\n\\]\nThe exponents $s_i$ are directional lifting weights.","\\begin{theorem}[Concrete Smith-weighted congruence solve]\\label{thm:smith-solve}\nLet $p>1$ and consider the diagonal system over\n$(\\Z/(p^L)\\Z)^d$\n\\[\n\\diag(p^{s_1},\\ldots,p^{s_d})x=b\n\\pmod{p^L}.\n\\]\nThe system is solvable if and only if\n\\[\np^{\\min(s_i,L)}\\mid (b_i)_{\\mathrm{val}}\n\\]\nfor every $i$, where $(b_i)_{\\mathrm{val}}$ is the canonical natural\nrepresentative. Conditional on this admissibility condition, the number of\nsolutions is\n\\[\n\\boxed{\np^{\\sum_{i=1}^d\\min(s_i,L)}.\n}\n\\]\nIf any coordinate is inadmissible, the solution set is empty and its\ncardinality is zero.\n\\end{theorem}","ongruence solve]\\label{thm:smith-solve}\nLet $p>1$ and consider the diagonal system over\n$(\\Z/(p^L)\\Z)^d$\n\\[\n\\diag(p^{s_1},\\ldots,p^{s_d})x=b\n\\pmod{p^L}.\n\\]\nThe system is solvable if and only if\n\\[\np^{\\min(s_i,L)}\\mid (b_i)_{\\mathrm{val}}\n\\]\nfor every $i$, where $(b_i)_{\\mathrm{val}}$ is the canonical natural\nrepresentative. Conditional on this admissibility condition, the number of\nsolutions is\n\\[\n\\boxed{\np^{\\sum_{i=1}^d\\min(s_i,L)}.\n}\n\\]\nIf any coordinate is inadmissible, the solution set is empty and its\ncardinality is zero.\n\\end{theorem}","\\begin{proof}\nLean proves the one-coordinate arithmetic directly over\n$\\mathbb Z/p^L\\mathbb Z$ for every $p>1$:\n\\[\n\\bigl(\\exists x\\in\\mathbb Z/p^L\\mathbb Z:\\quad p^s x=b\\pmod{p^L}\\bigr)\n\\quad\\Longleftrightarrow\\quad\np^{\\min(s,L)}\\mid b_{\\mathrm{val}},\n\\]\nwhere $b_{\\mathrm{val}}$ is the canonical natural representative, and every\nadmissible fiber has exactly $p^{\\min(s,L)}$ elements. The coordinates are\nindependent, so the product equivalence gives the displayed global count. If\none coordinate fiber is empty, the global fiber is empty. Lean proves both the\nglobal solvability equivalence and the conditional count, including\ncardinality zero on the inadmissible branch.\n\\end{proof}","For an arbitrary DVR $\\OO$ with residue field cardinality $q$, the analogous\nformula\n\\[\nq^{\\sum_i\\min(s_i,L)}\n\\]\nfollows from the solvability and fiber count for each coordinate. The formal\ncorrespondence records direct proofs of this DVR coordinate arithmetic and\nthe rectangular transport as separate targets.\nThe inherited abstract product interface remains available when the coordinate\nfacts are supplied as certificates. A Smith form and its changes of basis are\ninputs to the transport statement. It does not provide an extracted\ndiagonalization routine. See \\cite{Serre1979,Lang2002}.","If a Smith form is rank deficient, a literal zero diagonal factor requires\nthe corresponding target coordinate to vanish modulo $p^L$ and then\ncontributes all $p^L$ possible coordinate values. In the concrete quotient it\ncan equivalently be represented by any weight $s_i\\ge L$, since\n$p^{s_i}=0$ in $\\Z/(p^L)\\Z$. The theorem then gives exactly that condition and\nfactor. The numbered theorem takes the diagonal data as its input.","\\begin{example}\nAt $p=2$, depth $L=6$, and weights $(0,1,3)$, the right side $(1,2,8)$ is admissible. The coordinate kernels have sizes\n\\[\n1,2,8,\n\\]\nso the system has $16$ solutions. Replacing $8$ by $4$ in the final coordinate violates the divisibility condition and makes the system unsolvable.\n\\end{example}","\\begin{interpretation}\nThe gluing calculation takes the largest Smith weight because every packet must be separated to the requested precision. The congruence calculation retains the individual directional weights. A coordinate with weight $s_i$ contributes a solvability threshold and $\\min(s_i,L)$ to the branching exponent. The independent coordinate counts multiply, so these truncated weights add in the exponent.\n\\end{interpretation}","Let $X_f$ and $X_c$ be vector spaces or Banach spaces over a common field, and let\n\\[\nC:X_f\\longrightarrow X_c\n\\]\nbe a linear map,\nand let\n\\[\nT_f\\in\\End(X_f),\n\\qquad\nT_c\\in\\End(X_c).\n\\]\nDefine\n\\[\n\\Delta=CT_f-T_cC.\n\\]","\\begin{theorem}[Power defect identity]\\label{thm:power_defect}\\label{thm:power-defect}\nFor every integer $n\\geq1$,\n\\[\nCT_f^n-T_c^nC\n=\n\\sum_{k=0}^{n-1}T_c^{n-1-k}\\Delta T_f^k.\n\\]\n\\end{theorem}","\\begin{proof}\nThe statement is clear for $n=1$.  For the induction step, write\n\\[\nCT_f^{n+1}-T_c^{n+1}C\n=\n(CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC)\n\\]\nand apply the induction hypothesis.\n\\end{proof}","\\begin{corollary}[Power norm bound]\\label{cor:power_norm}\nFor bounded operators between Banach spaces,\n\\[\n\\|CT_f^n-T_c^nC\\|\n\\leq\n\\|\\Delta\\|\n\\sum_{k=0}^{n-1}\n\\|T_c\\|^{n-1-k}\\|T_f\\|^k.\n\\]\n\\end{corollary}","\\Needspace{4\\baselineskip}\n\\begin{corollary}[Norm bound]\\label{cor:defect-norm}\nFor a submultiplicative operator norm,\n\\[\n\\|CT_f^n-T_c^nC\\|\n\\le\n\\|\\Delta\\|\n\\sum_{k=0}^{n-1}\n\\|T_c\\|^{n-1-k}\\|T_f\\|^k.\n\\]\nIf $\\|T_f\\|,\\|T_c\\|\\le\\rho$, then\n\\[\n\\boxed{\n\\|CT_f^n-T_c^nC\\|\n\\le n\\rho^{n-1}\\|\\Delta\\|.\n}\n\\]\n\\end{corollary}","\\begin{corollary}[Polynomial defect]\\label{cor:polynomial_defect}\nFor\n\\[\np(z)=\\sum_{n=0}^Na_nz^n,\n\\]\nwe have\n\\[\nCp(T_f)-p(T_c)C\n=\n\\sum_{n=1}^Na_n\n\\sum_{k=0}^{n-1}T_c^{n-1-k}\\Delta T_f^k.\n\\]\n\\end{corollary}","\\begin{proof}\nThe constant terms cancel because $CI=IC$.  Apply \\cref{thm:power_defect} to every positive power and sum.\n\\end{proof}","\\begin{theorem}[Resolvent defect identity]\\label{thm:resolvent_defect}\\label{thm:resolvent-defect}\nIf $z-T_f$ and $z-T_c$ are invertible, then\n\\[\nC(z-T_f)^{-1}-(z-T_c)^{-1}C\n=\n(z-T_c)^{-1}\\Delta(z-T_f)^{-1}.\n\\]\n\\end{theorem}","\\begin{proof}\nMultiply the claimed identity on the left by $z-T_c$ and on the right by $z-T_f$.  The left side becomes\n\\[\n(z-T_c)C-C(z-T_f)\n=\nCT_f-T_cC\n=\n\\Delta.\n\\]\nMultiplying back by the two resolvents proves the formula.\n\\end{proof}","\\subsection{Inverse witnesses from one side and norm control}\nFor continuous linear maps, the same identity gives the certified norm bound\n\\[\n\\boxed{\n\\|C R_f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the m","ntinuous linear maps, the same identity gives the certified norm bound\n\\[\n\\boxed{\n\\|C R_f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the more general theory of refinement defects reserved for Chapter III.","d norm bound\n\\[\n\\boxed{\n\\|C R_f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the more general theory of refinement defects reserved for Chapter III.","f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the more general theory of refinement defects reserved for Chapter III.","\\subsection{Analytic functional calculus}\n\\begin{theorem}[Holomorphic functional calculus defect]\\label{thm:holomorphic_defect}\nLet $X_f$ and $X_c$ be complex Banach spaces, and let $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded complex-linear maps.  Let $h$ be holomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|","lculus defect]\\label{thm:holomorphic_defect}\nLet $X_f$ and $X_c$ be complex Banach spaces, and let $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded complex-linear maps.  Let $h$ be holomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|","$X_c$ be complex Banach spaces, and let $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded complex-linear maps.  Let $h$ be holomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|","olomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|\\\\\n&\\quad\\times\n\\sup_{z\\in\\Gamma}\\|(z-T_c)^{-1}\\|\n\\sup_{z\\in\\Gamma}\\|(z-T_f)^{-1}\\|.\n\\end{align*}\n\\end{theorem}","\\begin{proof}\nThe assertion is immediate if either space is zero. Otherwise, use the rectifiable-cycle form of the holomorphic functional calculus in \\cite[Chapter~VII, \\S4, Theorem~4.1, formula~(4.5), and Proposition~4.6]{Conway1990}. For each operator $T$, compare $\\Gamma$ with a positive simple system around its spectrum. Their difference has zero winding outside $U\\setminus\\sigma(T)$, so Theorem~4.1 applied to the resolvent-valued integrand identifies the two integrals. Consequently,\n\\[\nh(T)\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma h(z)(z-T)^{-1}\\,dz.\n\\]\nSubtract the two formulas after composing with $C$, then use \\cref{thm:resolvent_defect} inside the integral.  The norm estimate follows from the standard contour bound.\n\\end{proof}","e-cycle form of the holomorphic functional calculus in \\cite[Chapter~VII, \\S4, Theorem~4.1, formula~(4.5), and Proposition~4.6]{Conway1990}. For each operator $T$, compare $\\Gamma$ with a positive simple system around its spectrum. Their difference has zero winding outside $U\\setminus\\sigma(T)$, so Theorem~4.1 applied to the resolvent-valued integrand identifies the two integrals. Consequently,\n\\[\nh(T)\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma h(z)(z-T)^{-1}\\,dz.\n\\]\nSubtract the two formulas after composing with $C$, then use \\cref{thm:resolvent_defect} inside the integral.  The norm estimate follows from the standard contour bound.\n\\end{proof}","nd Proposition~4.6]{Conway1990}. For each operator $T$, compare $\\Gamma$ with a positive simple system around its spectrum. Their difference has zero winding outside $U\\setminus\\sigma(T)$, so Theorem~4.1 applied to the resolvent-valued integrand identifies the two integrals. Consequently,\n\\[\nh(T)\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma h(z)(z-T)^{-1}\\,dz.\n\\]\nSubtract the two formulas after composing with $C$, then use \\cref{thm:resolvent_defect} inside the integral.  The norm estimate follows from the standard contour bound.\n\\end{proof}","\\begin{theorem}[Exponential defect identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","onential defect identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","l defect identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","ntial_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","t}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","\\begin{proof}\nFor real $t$, differentiate\n\\[\nF(s)=e^{(t-s)T_c}Ce^{sT_f}.\n\\]\nWe obtain\n\\[\nF'(s)=e^{(t-s)T_c}\\Delta e^{sT_f}.\n\\]\nIntegration from $0$ to $t$ gives the identity.  The complex statement follows by the same calculation along the straight segment.  The norm bound follows from the integral formula.\n\\end{proof}","Let\n\\[\nC_\\ell:X_{\\ell+1}\\longrightarrow X_\\ell\n\\]\nbe linear maps for $0\\leq\\ell<L$, and let $T_\\ell\\in\\End(X_\\ell)$.  Put\n\\[\n\\Delta_\\ell=C_\\ell T_{\\ell+1}-T_\\ell C_\\ell.\n\\]\nFor $a<b$, write\n\\[\nC_{a,b}=C_aC_{a+1}\\cdots C_{b-1},\n\\]\nwith $C_{a,a}=I$.","\\begin{theorem}[Multilevel defect decomposition]\\label{thm:multilevel_defect}\nThe total defect from level $L$ to level zero is\n\\[\nC_{0,L}T_L-T_0C_{0,L}\n=\n\\sum_{\\ell=0}^{L-1}\nC_{0,\\ell}\\Delta_\\ell C_{\\ell+1,L}.\n\\]\nFor bounded maps,\n\\[\n\\|C_{0,L}T_L-T_0C_{0,L}\\|\n\\leq\n\\sum_{\\ell=0}^{L-1}\n\\|C_{0,\\ell}\\|\n\\|\\Delta_\\ell\\|\n\\|C_{\\ell+1,L}\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nInsert the intermediate terms\n\\[\nC_{0,\\ell}T_\\ell C_{\\ell,L}\n\\]\nfor $0\\leq\\ell\\leq L$.  Consecutive differences telescope.  The difference between the terms indexed by $\\ell+1$ and $\\ell$ is\n\\[\nC_{0,\\ell}(C_\\ell T_{\\ell+1}-T_\\ell C_\\ell)C_{\\ell+1,L}.\n\\]\nSumming gives the identity.  The norm estimate follows from the triangle inequality.\n\\end{proof}","\\begin{example}[A visible one step defect]\\label{ex:defect_example}\nLet $X_f=\\bbk^2$, let $X_c=\\bbk$, and let\n\\[\nC(x_1,x_2)=x_1.\n\\]\nTake\n\\[\nT_f=\n\\begin{pmatrix}\na&b","\\begin{example}[A visible one step defect]\\label{ex:defect_example}\nLet $X_f=\\bbk^2$, let $X_c=\\bbk$, and let\n\\[\nC(x_1,x_2)=x_1.\n\\]\nTake\n\\[\nT_f=\n\\begin{pmatrix}\na&b\\\\\nc&d\n\\end{pmatrix},\n\\qquad\nT_c=(a).\n\\]\nThen\n\\[\n\\Delta=CT_f-T_cC=(0,b).\n\\]\nThe coarse operator is exactly compatible with the fine operator precisely when the discarded coordinate does not feed into the retained coordinate.  The entry $c$ does not appear in the one step defect because it moves information in the opposite direction.  It can still affect later powers through the terms in \\cref{thm:power_defect}.\n\\end{example}","c&d\n\\end{pmatrix},\n\\qquad\nT_c=(a).\n\\]\nThen\n\\[\n\\Delta=CT_f-T_cC=(0,b).\n\\]\nThe coarse operator is exactly compatible with the fine operator precisely when the discarded coordinate does not feed into the retained coordinate.  The entry $c$ does not appear in the one step defect because it moves information in the opposite direction.  It can still affect later powers through the terms in \\cref{thm:power_defect}.\n\\end{example}","\\begin{corollary}[Exact compatibility]\\label{cor:exact_compatibility}\nIf every one step defect $\\Delta_\\ell$ vanishes, then\n\\[\nC_{0,L}p(T_L)=p(T_0)C_{0,L}\n\\]\nfor every polynomial $p$.  The analogous resolvent, holomorphic, and exponential identities also hold whenever the relevant expressions are defined.\n\\end{corollary}","Let $x=(x_n)_{n\\in\\Z}\\in c_{00}(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","\\in\\Z}\\in c_{00}(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","in c_{00}(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","A banded operator need not remain banded after residue reordering, but its\nnonzero couplings become a structured packet graph. With the convention\n$(S_rx)_n=x_{n-r}$, the only active transfer is\n\\[\nP_{N,j+r}S_rP_{N,j},\n\\]\nwith indices taken modulo $N$. More general finite stencils produce a finite set of active phase differences.","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges n","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpe","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these v","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sample","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two v","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Highe","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compa","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","t $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","isk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","ma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","y any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{11_smith_polyphase_orbit.pdf}\n\\caption{Nested phase orbits at powers of $2$ on the Smith reflection disk. The full treatment of the Smith chart is deferred to the signal chapter.}\n\\label{fig:smith-orbit}\n\\end{figure}","The symmetric weight\n\\[\nw_{\\mathrm{Smith}}(\\Gamma)\n=\n\\frac1{|1-\\Gamma|^2}\n+\n\\frac1{|1+\\Gamma|^2}\n\\]\nindicates where small errors in reflection coordinates produce large impedance or admittance errors.","\\midrule\nPacket synthesis residuals for moduli $2,3,4,6,8,12$ & $0$","Nested refinement residuals for $2\\to4$, $2\\to8$, $3\\to6$, $3\\to12$, $4\\to8$, $4\\to12$, $6\\to12$ & $0$","CRT residuals for dyadic and odd moduli for $4\\times3$, $8\\times3$, $4\\times5$, $8\\times5$ & $0$","Maximum finite Fourier projector residual & $6.60\\times10^{-15}$","Weighted inverse residual modulo $2^{10}$ & $0$","Weighted inverse residual modulo $3^8$ & $0$","Weighted inverse residual modulo $5^7$ & $0$","Difference between filtered and direct modular inverses & $0$","Transfer zeta coefficients versus direct walk enumeration through weight $30$ & exact match","Dyadic sign projector resultant weight & $v_2(2)=1$","Sign projector resultant weight at the odd prime $3$ & $v_3(2)=0$","Power defect telescoping identity residual & at most $6.94\\times10^{-18}$","Resolvent defect identity residual & $1.09\\times10^{-17}$","Smith $N=2$ phase versus $z\\mapsto1/z$ & residual below $1.4\\times10^{-16}$","\\begin{certificate}[Weighted inverse]\nFor the tested diagonal-plus-transfer matrices, the filtered inverse agrees exactly with direct modular inversion at all tested precisions:\n\\[\n2^{10},\\qquad3^8,\\qquad5^7.\n\\]\nThe maximum modular residual is zero in every case.\n\\end{certificate}","\\begin{certificate}[Transfer zeta]\nThe rational function\n\\[\n\\frac{-t^3}{t^9-t^7-t^6-t^5+2t^4+t^3+t^2-1}\n\\]\nwas expanded through degree $30$. Every coefficient agrees with direct weighted-walk enumeration.\n\\end{certificate}","\\begin{certificate}[Dyadic separation]\nThe code verifies exactly that the two rational sign projectors are idempotent, orthogonal, and complementary, that scaling by $2$ makes both integral, and that the same projectors are already integral modulo every tested odd prime.\n\\end{certificate}","\\midrule\n\\endhead\n$P_{\\ell,\\alpha}$ & packet projector at refinement level $\\ell$","$I_\\ell$ & packet address set at level $\\ell$","$\\beta\\succ\\alpha$ & fine packet $\\beta$ is a child of coarse packet $\\alpha$","$\\cA_\\ell,\\cS_\\ell$ & packet analysis and synthesis maps","$J_{\\ell+1,\\ell}$ & exact coordinate refinement map","$C_{\\ell,\\ell+1}$ & exact coordinate regrouping from fine to coarse","$T^{(\\ell)}_{\\beta\\alpha}$ & transfer block $P_{\\ell,\\beta}TP_{\\ell,\\alpha}$","$P_{N,j}$ & residue or phase selector for address $j\\pmod N$","$w(e)$ & positive filtration weight of a transfer edge","$W(\\gamma)$ & cumulative weight of a transfer word or graph walk","$A(t)$ & polynomial weighted adjacency matrix","$Z_{s,t}(t)$ & weighted transfer-zeta function from vertex $s$ to vertex $t$","$s_{\\pi,\\Lambda}(E)$ & separation weight of a rational projector on the chosen lattice, when a clearing exponent exists","$\\Delta$ & linear refinement defect for one step $CT_f-T_cC$","The verified contract is\n$\\operatorname{synthesize}(\\operatorname{analyze}(x,N))=x$. The Lean\ncounterpart uses packet coordinates restricted to projector ranges and proves\nboth inverse laws. The finite Python routine checks the concrete array\nrealization.","Lean proves that the mapped pruned sum is exactly the mapped finite geometric\ninverse. The lower bound $m$ gives $m|\\gamma|\\le W(\\gamma)$, so\n$L\\le mN$ derives $q_L(A^N)=0$ from the filtration certificate and makes the\nreturned value an inverse on both sides of $q_L(D+H)$. The Lean lemma\n$L\\le m\\lceil L/m\\rceil$ justifies the displayed executable cutoff for\n$m>0$. Taking $m=1$ and $N=L$ gives\nthe unconditional cutoff for positive weights. The Python implementation checks the\nDVR matrix specialization against direct modular inversion.","Positive weights make the coefficient loop finite because every contributing\nword has length at most $n$. Lean proves the enumeration correct, derives its\nrecurrence from the first edge, proves $(I-A)Z=I$, constructs the\ndeterminant/adjugate inverse, and identifies the two matrices by inverse\nuniqueness. Thus the normalized $QZ=P$ identity and its induced recurrence\nfollow from finite graph data. The Python companion repeats the exact symbolic\nconstruction as an executable check. The Lean proof does not use it as an\noracle.","The proof order is:\n\\[\n\\begin{aligned}\n&\\text{complete packet axioms}\n  \\Rightarrow\\text{finite-packet uniqueness and coordinate coherence}\\\\\n&\\qquad\\Rightarrow\\text{coordinate naturality and refined transfer blocks},\\\\\n&\\text{residue selectors and a finite Fourier table}\n  \\Rightarrow\\text{Fourier formulas},\\\\\n&\\text{coprime moduli}\\Rightarrow\\text{dyadic--odd formulas},\\\\\n&\\text{quotient, filtration, and edge-weight certificates}\n  \\Rightarrow\\text{explicit pruned inverse},\\\\\n&\\text{finite positive-weight graph}\n  \\Rightarrow\\text{semantic enumeration and first-edge recurrence}\\\\\n&\\qquad\\Rightarrow\\det/\\operatorname{adj}\\text{ rationality and recurrence},\\\\\n&\\text{minimal evaluated B\\'ezout data and inverse-unit lattice action}\\\\\n&\\qquad\\Rightarrow\\text{separation bound, upward closure, and precision delay},\\\\\n&\\text{concrete }\\mathbb Z/p^L\\mathbb Z\\text{ coordinate fibres}\n  \\Rightarrow\\text{finite Smith product count},\\\\\n&\\text{one-step typed defect}\n  \\Rightarrow\\text{powers, polynomial actions, and witnessed resolvents}.\n\\end{aligned}\n\\]","The walk branch for transfer zeta is logically independent of operator\ninversion. Its rationality conclusion follows from finite graph data without\na supplied $QZ=P$ premise. Likewise, CRT factorization follows directly from\nresidue selectors and coprimality. It does not require the abstract refinement\ntheorem, although it supplies a concrete refinement family. General Smith\ndiagonalization and the construction of polynomial B\\'ezout/resultant data\nremain inputs to the formalized branches that use those certificates.","\\head{Address resolution}\nA coefficient such as $x_6$ can sit in the even packet, then the residue-$2$ packet modulo $4$, then the residue-$6$ packet modulo $8$. Its value does not become more accurate. Its \\emph{address becomes more specific}, while the full object remains exactly reconstructible.","\\head{Arithmetic precision}\nAn operator word with total filtration weight at least $L$ vanishes in the specified depth-$L$ quotient. Increasing $L$ can make additional words relevant. This is a change in the precision of the ring in which the answer is required, not a lossless rearrangement of one fixed list.","\\head{Separation precision}\nFor the integral involution used later, $E_+=(I+A)/2$ is an exact rational projector but not an integral one at $2$. One extra input digit suffices to recover the projected difference to the requested precision. The packet is present over the fraction field even when its integral separation is delayed.","midrule\n$P_{N,j}$ & Retain coefficients with address $j$ modulo $N$. & Even/odd masks, then mod-$4$ and mod-$8$ masks.","$\\mathcal A_N,\\mathcal S_N$ & Split into range-restricted packet vectors, then sum them. & A permutation/interleaving of eight coordinates.","$J,C$ & Exact refinement and regrouping. In the defect section, $C$ may instead be any supplied coarsening map. & A permutation in Plate II.5 and a shear in Plate II.22.","$T_{ba}$ & Input packet $a$, output packet $b$: $P_bTP_a$. & Blocks of a finite cyclic filter.","$D,H,A$ & $T=D+H=D(I+A)$, with $A=D^{-1}H$. & Explicit integral $2\\times2$ matrices.","$w(e),W(\\gamma)$ & Positive certified edge gain and its additive word weight. & $w(a)=1$, $w(b)=2$.","$L$ & Precision depth in a quotient such as $\\ZZ/2^L\\ZZ$. & $L=5$ gives arithmetic modulo $32$.","$A(t),Z(t)$ & Polynomial graph adjacency and unsigned walk-count series. & Two parallel edges remain distinct.","$E,\\Lambda,s$ & A rational projector, a specified lattice, and a clearing exponent. & $E_+$ has local weight $1$ at $2$.","$(s_i)$ & Directional weights of a supplied Smith diagonal. & $(1,2)$ modulo $8$ or $(0,1,3)$ across depths.","$\\Delta$ & The failure of an independently chosen operator to intertwine: $CT_f-T_cC$. & One nonzero entry $1/16$.","\\vpage{3}{Follow one coefficient through the tower}{MODEL A / LOSSLESS ADDRESS REFINEMENT}{Chapter II, mathematical text: Theorem 3.3, Theorem 4.6}\nTake a vector $x=(x_0,\\ldots,x_{15})$. A packet is not a single summary number: it retains every coefficient at its selected addresses. For example,\n\\[\nP_{2,0}x=(x_0,0,x_2,0,\\ldots,x_{14},0).\n\\]\nThe rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on Plate II.5 instead uses increasing residue order.","The address $6$ is even, is $2$ modulo $4$, and is $6$ modulo $8$. Its coefficient therefore lies in the successive ranges\n\\[\nP_{8,6}X\\subseteq P_{4,2}X\\subseteq P_{2,0}X.\n\\]\nThe corresponding identities hold as operators:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2},\\qquad\nP_{4,2}=P_{8,2}+P_{8,6}.\n\\]\nApplying either composite parent selector to $P_{8,6}x$ leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.","For $x=(1,2,3,4)$, the fine packets are the four coordinate-supported vectors. Regrouping gives $(1,0,3,0)$ and $(0,2,0,4)$. Every original coordinate remains available in one of these two vectors.","\\vfig[.98]{regrouping_not_averaging}\n\\platecaption{The left and middle panels store the same four independent coordinates. The right panel instead applies a $2\\times4$ numerical incidence matrix to four scalar entries. That is not the range-restricted exact coordinate map.}","Define\n\\[\nX_N^{\\rm pkt}=\\bigoplus_j P_{N,j}X,\n\\qquad\n\\mathcal A_Nx=(P_{N,j}x)_j,\n\\qquad\n\\mathcal S_N(x_j)_j=\\sum_jx_j.\n\\]\nCompleteness and mutual annihilation give both inverse laws. Consequently,\n\\[\nJ=\\mathcal A_4\\mathcal S_2,\\qquad C=\\mathcal A_2\\mathcal S_4,\n\\qquad CJ=I,\\quad JC=I.\n\\]\nThe larger list of fine packets has smaller individual packet spaces.","\\head{A kernel witnesses genuine information loss}\nFor the scalar incidence matrix $B$ drawn on the right,\n\\[\nB(1,0,-1,0)^\\mathsf T=0.\n\\]\nThe zero vector and $(1,0,-1,0)$ therefore have the same two scalar summaries. No inverse can recover the original four scalars from that output alone.","\\vpage{5}{The operator changes coordinates too}{MODEL A / COHERENCE AND NATURALITY}{Chapter II, mathematical text: Theorem 3.5, Theorem 3.8}\nUse a finite periodic carrier $X=\\QQ^8$ for this example. Let $Se_j=e_{j+1\\bmod8}$ and let\n\\[\nT=I+2S+S^2.\n\\]\nThis is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. Write $R_N$ for the permutation that groups coordinates in increasing residue order modulo $N$.","NCE AND NATURALITY}{Chapter II, mathematical text: Theorem 3.5, Theorem 3.8}\nUse a finite periodic carrier $X=\\QQ^8$ for this example. Let $Se_j=e_{j+1\\bmod8}$ and let\n\\[\nT=I+2S+S^2.\n\\]\nThis is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. Write $R_N$ for the permutation that groups coordinates in increasing residue order modulo $N$.","\\vfig[.99]{operator_naturality}\n\\platecaption{Actual $8\\times8$ matrices. Dots are zero entries. The other entries are exact integers. Heavy lines mark the packet blocks: size four at level two and size two at level four.}","The reordered operators and change of coordinates are\n\\[\n\\widehat T_N=R_NTR_N^{-1},\\qquad J_{4,2}=R_4R_2^{-1}.\n\\]\nMultiplication cancels the intermediate coordinate maps:\n\\[\nJ_{4,2}\\widehat T_2=\\widehat T_4J_{4,2},\\qquad\nJ_{8,4}J_{4,2}=J_{8,2}.\n\\]\nThus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.","\\head{Refined blocks still reconstruct the same map}\nA coarse block expands as\n\\[\nP_bTP_a=\\sum_{\\beta\\succ b,\\,\\alpha\\succ a}P_\\beta TP_\\alpha.\n\\]\nThe terms are maps embedded in the original source and target spaces. Adding their norms, or adding their matrix entries after forgetting the embeddings, is not this operator identity.","\\vpage{6}{Selectors survive without division}{MODEL A / RESIDUE MASKS IN CHARACTERISTIC TWO}{Chapter II, mathematical text: Proposition 4.2}\nWork first in $\\mathbb F_2$, where $1+1=0$. The eight coefficients below are just a finite display of a coefficient sequence. The even and odd projectors are diagonal masks with entries zero and one.","\\vfig[.95]{division_free_masks}\n\\platecaption{Each nonzero input coefficient is retained by exactly one mask. Reconstruction is exact over $\\mathbb F_2$. No average and no root of unity is used.}","For a general coefficient ring and a positive integer $N$, the definition is\n\\[\n(P_{N,j}x)_n=\\begin{cases}x_n,&n\\equiv j\\pmod N,","For a general coefficient ring and a positive integer $N$, the definition is\n\\[\n(P_{N,j}x)_n=\\begin{cases}x_n,&n\\equiv j\\pmod N,\\\\0,&\\text{otherwise}.\\end{cases}\n\\]\nAn address belongs to one residue class, so applying two masks either keeps that same coefficient or kills it. This proves\n\\[\nP_{N,i}P_{N,j}=\\delta_{ij}P_{N,j},\\qquad \\sum_jP_{N,j}=I.\n\\]\nThe argument is coordinatewise and does not require the coefficients to form a field.","0,&\\text{otherwise}.\\end{cases}\n\\]\nAn address belongs to one residue class, so applying two masks either keeps that same coefficient or kills it. This proves\n\\[\nP_{N,i}P_{N,j}=\\delta_{ij}P_{N,j},\\qquad \\sum_jP_{N,j}=I.\n\\]\nThe argument is coordinatewise and does not require the coefficients to form a field.","\\head{Why the usual even/odd average is a different formula}\nOver a field of characteristic not two, parity extraction can be represented using $F(z)$ and $F(-z)$ and division by $2$. In characteristic two, $-z=z$ and $2=0$, so that formula cannot be used. The coefficient mask, however, still distinguishes even from odd \\emph{exponents}.","This distinction also explains the later involution example. Failure to reduce a rational spectral projector containing $1/2$ does not make every integral coefficient-address selector disappear.","\\vpage{7}{A Fourier certificate you can inspect}{MODEL A / AN EXACT NINE-ENTRY TABLE}{Chapter II, mathematical text: Theorem 4.3}\nTake $N=3$ and work in $\\mathbb F_7$. Set\n\\[\n\\omega=2,\\qquad \\omega^{-1}=4,\\qquad \\eta=3^{-1}=5.\n\\]\nHere $2^3=1$ and neither $2$ nor $2^2$ is one. The unnormalized geometric sums are the entries in the first matrix.","\\vfig[.97]{finite_fourier_table}\n\\platecaption{All entries are exact residues modulo seven. Multiplying the diagonal value $3$ by $5$ produces $1$. The lower calculation is a genuine failed Fourier certificate in a different coefficient ring.}","On the monomial $z^n$, define $UF(z)=F(2z)$. The proposed projector acts by\n\\[\n5\\sum_{k=0}^2 4^{jk}2^{nk}.\n\\]\nThe identity $2^3=1$ makes this depend only on $n\\bmod3$. The nine-entry table therefore establishes the formula for every exponent once periodicity is used, rather than requiring a separate test at every coefficient.","\\head{Order alone is not sufficient over every ring}\nIn $\\mathbb F_3\\times\\mathbb F_3$, the element $(-1,1)$ has order two, but one component is a trivial character. Its nontrivial Fourier sum does not vanish in that component. Thus an element of order $N$ in a ring with zero divisors is not, by itself, the full orthogonality certificate.","\\vpage{8}{One address, two independent residues}{MODEL A / THE DYADIC AND ODD RECTANGLE}{Chapter II, mathematical text: Corollary 4.9}\nBecause $4$ and $3$ are coprime, a residue modulo $12$ is specified by a column $a=n\\bmod4$ and a row $b=n\\bmod3$. Each cell contains the unique combined address.","\\vfig[.96]{crt_address_grid}\n\\platecaption{The twelve cells are a permutation of $0,\\ldots,11$, not twelve extra degrees of freedom. The selected cell contains every integer congruent to $7$ modulo $12$.}","An explicit reconstruction formula is\n\\[\nn\\equiv9a+4b\\pmod{12},\n\\]\nbecause $9$ is one modulo four and zero modulo three, while $4$ has the opposite pair of residues. For $(a,b)=(3,1)$ this gives $31\\equiv7$.","Applying the two masks in either order gives\n\\[\nP_{12,7}=P_{4,3}P_{3,1}=P_{3,1}P_{4,3}.\n\\]\nMore generally, an address survives exactly when it satisfies both congruences. The coprime reconstruction identifies this intersection with one combined residue class.","\\head{The hypothesis can be seen failing}\nReplace $4$ and $3$ by $4$ and $6$. The requirements $n\\equiv0\\pmod4$ and $n\\equiv1\\pmod6$ are incompatible: the first makes $n$ even and the second makes it odd. A full rectangular address product no longer exists. Compatible pairs must agree modulo the greatest common divisor.","\\vpage{9}{Which words can survive precision five?}{MODEL B / ORDERED TRANSFERS IN A FILTERED RING}{Chapter II, mathematical text: Theorem 5.10, Corollary 5.11}\nLet $E_{ij}$ send the $j$th basis vector to the $i$th. In integral $2\\times2$ matrices set\n\\[\na=2E_{01},\\qquad b=4E_{10},\\qquad A=a+b.\n\\]\nTheir certified dyadic weights are $1$ and $2$. At precision $L=5$, every word of weight at least five vanishes modulo $32$.","rds can survive precision five?}{MODEL B / ORDERED TRANSFERS IN A FILTERED RING}{Chapter II, mathematical text: Theorem 5.10, Corollary 5.11}\nLet $E_{ij}$ send the $j$th basis vector to the $i$th. In integral $2\\times2$ matrices set\n\\[\na=2E_{01},\\qquad b=4E_{10},\\qquad A=a+b.\n\\]\nTheir certified dyadic weights are $1$ and $2$. At precision $L=5$, every word of weight at least five vanishes modulo $32$.","\\vfig[.97]{weighted_word_cutoff}\n\\platecaption{All nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes $I$. Other words of weight below five can still be zero because $a^2=b^2=0$.}","The words are \\emph{written matrix products}: in $ab$, the map $b$ acts first. Therefore\n\\[\nab=8E_{00},\\qquad ba=8E_{11},\\qquad aba=16E_{01}.\n\\]\nThe two products $ab$ and $ba$ cannot be identified merely because they have equal weight. Both their order and their source/target information matter.","\\vpage{10}{An inverse computed in both orders}{MODEL B / A COMPLETE MODULO-32 CALCULATION}{Chapter II, mathematical text: Theorem 5.10, Corollary 5.11}\nKeep the matrices from Plate II.9 and choose\n\\[\nD=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix},\\quad\nD^{-1}=\\begin{pmatrix}1&-1\\\\0&1\\end{pmatrix},\\quad\nT=D(I+A)=\\begin{pmatrix}5&3\\\\4&1\\end{pmatrix}.\n\\]\nThe six retained contributions sum to\n\\[\nS=I-a-b+ab+ba-aba\n=\\begin{pmatrix}9&-18","4&1\\end{pmatrix}.\n\\]\nThe six retained contributions sum to\n\\[\nS=I-a-b+ab+ba-aba\n=\\begin{pmatrix}9&-18\\\\-4&9\\end{pmatrix}.\n\\]\nThe required inverse is $K=SD^{-1}$ in the quotient, not $D^{-1}S$.","-4&9\\end{pmatrix}.\n\\]\nThe required inverse is $K=SD^{-1}$ in the quotient, not $D^{-1}S$.","\\vfig[.98]{modular_inverse}\n\\platecaption{The displayed $K$ uses canonical residues from $0$ to $31$. Both left and right products are the identity modulo $32$. These are exact modular equalities, not floating-point residuals.}","One can verify the claim without trusting the word expansion:\n\\[\nTK=\\begin{pmatrix}129&64\\\\64&33\\end{pmatrix}\\equiv I,\n\\qquad\nKT=\\begin{pmatrix}65&32\\\\192&97\\end{pmatrix}\\equiv I\\pmod{32}.\n\\]\nThe cancellation of the exterior factor $D$ explains the order. Since $T=D(I+A)$, multiply its inverse by $D^{-1}$ on the \\emph{right} of the inverse of $I+A$.","192&97\\end{pmatrix}\\equiv I\\pmod{32}.\n\\]\nThe cancellation of the exterior factor $D$ explains the order. Since $T=D(I+A)$, multiply its inverse by $D^{-1}$ on the \\emph{right} of the inverse of $I+A$.","\\vpage{11}{The residual tells you when to stop}{MODEL B / A CERTIFIED CUTOFF, NOT A HEURISTIC}{Chapter II, mathematical text: Theorem 5.10, Corollary 5.11, Section B.2}\nFor $S_N=\\sum_{k=0}^{N-1}(-A)^k$, telescoping gives\n\\[\n(I+A)S_N=S_N(I+A)=I-(-A)^N.\n\\]\nThus the exact residual is known. To certify precision $L$, it is enough to show that every residual entry is divisible by $2^L$.","\\vfig[.94]{inverse_residual_depth}\n\\platecaption{The plotted values are the minimum dyadic valuations of the nonzero entries of $(-A)^N$. The dashed line is the requested cutoff $L=5$. This is an exact valuation profile, not a measured numerical convergence curve.}","In this particular model, $A^2=8I$. Therefore\n\\[\nv_2(A^{2k})=3k,\\qquad v_2(A^{2k+1})=3k+1,\n\\]\nwhere the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo $32$. Four terms leave valuation six and do suffice.","The generic positive-weight estimate is more conservative. If every edge weight is at least $m>0$, a word of length $N$ has weight at least $mN$. Hence the cutoff\n\\[\nN=\\left\\lceil\\frac Lm\\right\\rceil\n\\]\nis sufficient. Here the certified minimum weight is $m=1$, so the generic choice is five. The extra identity $A^2=8I$ justifies stopping at four in this example.","\\vpage{12}{Walks are not simple paths}{MODEL C / A GRAPH WITH PARALLEL EDGES}{Chapter II, mathematical text: Theorem 6.6}\nConsider the directed graph below. The loop $\\ell$ has weight one, the two distinct edges $a,b$ from $0$ to $1$ each have weight two, and the return edge $r$ has weight one.","A walk may revisit a vertex and reuse an edge. At total weight five there are six walks from $0$ to $1$. Two use three loops followed by either $a$ or $b$. Four travel to $1$, return to $0$, and travel to $1$ again, with two independent choices of parallel edge.","Their multiplicities enter the polynomial adjacency matrix:\n\\[\nA(t)=\\begin{pmatrix}t&2t^2\\\\t&0\\end{pmatrix}.\n\\]\nThe coefficient $2$ remembers the two distinct edges. Replacing it by $1$ would count a different graph.","t&0\\end{pmatrix}.\n\\]\nThe coefficient $2$ remembers the two distinct edges. Replacing it by $1$ would count a different graph.","\\head{The first-edge decomposition}\nLet $a_n(s,t)$ count walks of total weight $n$ from $s$ to $t$. Removing the first edge gives\n\\[\na_n(s,t)=\\mathbf1_{s=t,\\,n=0}\n +\\sum_{\\substack{e:\\,\\mathrm{source}(e)=s","\\head{The first-edge decomposition}\nLet $a_n(s,t)$ count walks of total weight $n$ from $s$ to $t$. Removing the first edge gives\n\\[\na_n(s,t)=\\mathbf1_{s=t,\\,n=0}\n +\\sum_{\\substack{e:\\,\\mathrm{source}(e)=s\\\\w(e)\\le n}}\n a_{n-w(e)}(\\mathrm{target}(e),t).\n\\]\nThe empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most $n$, so enumeration at each fixed weight is finite.","w(e)\\le n}}\n a_{n-w(e)}(\\mathrm{target}(e),t).\n\\]\nThe empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most $n$, so enumeration at each fixed weight is finite.","\\vpage{13}{A finite graph produces its denominator}{MODEL C / AUTOMATIC RATIONALITY AND RECURRENCE}{Chapter II, mathematical text: Theorem 6.6, Corollary 6.7}\nFor the graph on Plate II.12, compute the determinant and adjugate rather than guessing a recurrence:\n\\[\nI-A(t)=\\begin{pmatrix}1-t&-2t^2\\\\-t&1\\end{pmatrix},\n\\quad Q(t)=1-t-2t^3,\n\\quad \\operatorname{adj}(I-A)_{0,1}=2t^2.\n\\]\nSince $Q(0)=1$, its inverse exists as a formal power series with integer coefficients.","-t&1\\end{pmatrix},\n\\quad Q(t)=1-t-2t^3,\n\\quad \\operatorname{adj}(I-A)_{0,1}=2t^2.\n\\]\nSince $Q(0)=1$, its inverse exists as a formal power series with integer coefficients.","\\vfig[.99]{zeta_recurrence}\n\\platecaption{The coefficients are independently computed by the first-edge recurrence and checked against the rational series through weight $24$. Literal edge-walk enumeration supplies a third check through weight $10$.}","From $QZ_{0,1}=2t^2$ one obtains\n\\[\na_0=a_1=0,\\quad a_2=2,\\qquad a_n=a_{n-1}+2a_{n-3}\\quad(n\\ge3).\n\\]\nThe numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","\\head{Three boundaries of this calculation}\nThis is a \\emph{formal} series identity. No analytic convergence at a chosen real or complex value of $t$ is needed. The canonical determinant need not be the minimal denominator after cancellation. Finally, positive weights prevent infinitely many walks at a fixed weight: a single weight-zero loop would already give the infinitely many distinct walks $1,\\ell,\\ell^2,\\ldots$ at weight zero.","\\takeaway{Finite positive-weight graph data determine a rational series and an eventual recurrence. The proof does not begin by assuming a certificate $QZ=P$ that already contains that conclusion.}","\\vpage{14}{Separate before dividing}{MODEL D / AN EVALUATED B\\'EZOUT CERTIFICATE}{Chapter II, mathematical text: Theorem 7.5, Definition 7.6}\nUse the integral involution\n\\[\nA=\\begin{pmatrix}1&1\\\\0&-1\\end{pmatrix},\\qquad A^2=I.\n\\]\nWith $F=A-I$ and $G=A+I$, the identities\n\\[\n-F+G=2I,\\qquad FG=0\n\\]\nare an explicit evaluated B\\'ezout certificate. Here all the required commutations follow because these operators are polynomials in the same matrix.","0&-1\\end{pmatrix},\\qquad A^2=I.\n\\]\nWith $F=A-I$ and $G=A+I$, the identities\n\\[\n-F+G=2I,\\qquad FG=0\n\\]\nare an explicit evaluated B\\'ezout certificate. Here all the required commutations follow because these operators are polynomials in the same matrix.","\\vfig[.96]{bezout_normalization}\n\\platecaption{The denominator-free numerator is quasi-idempotent: $N^2=2N$. Division by the nonzero scalar $2$ gives an idempotent over the fraction field. It does not yet make the projector integral on the selected lattice.}","The numerator $N=G$ acts by $2$ on $\\ker(A-I)$ and by zero on $\\ker(A+I)$. Thus\n\\[\nE_+=\\frac12N\n=\\begin{pmatrix}1&1/2","The numerator $N=G$ acts by $2$ on $\\ker(A-I)$ and by zero on $\\ker(A+I)$. Thus\n\\[\nE_+=\\frac12N\n=\\begin{pmatrix}1&1/2\\\\0&0\\end{pmatrix}\n\\]\nis the required separator. The complementary projector is $E_-=I-E_+$.","0&0\\end{pmatrix}\n\\]\nis the required separator. The complementary projector is $E_-=I-E_+$.","\\head{A denominator bound need not be minimal}\nThe same projector can be presented with the inflated certificate $-2F+2G=4I$. Its numerator is $2G$ and its normalization is still $E_+$. This supplies a two-digit clearing bound at $2$, although one digit is enough. Thus the valuation of a supplied certificate gives an upper bound, not automatically the least possible weight.","For the original sign factors the concrete resultant is $\\operatorname{Res}(x-1,x+1)=2$ and the bound is sharp. That equality is special to this displayed calculation, not a universal minimality theorem for every resultant certificate.","\\vpage{15}{Where the dyadic digit is lost}{MODEL D / THE EIGENLATTICE HAS INDEX TWO}{Chapter II, mathematical text: Proposition 9.3}\nThe eigenspaces are spanned by $(1,0)$ and $(-1,2)$. Their integral eigenvectors do not form an integral basis of $\\ZZ^2$. The change-of-basis determinant is two.","\\vfig[.94]{dyadic_eigenlattice}\n\\platecaption{A picture of the ordinary integer lattice, not a drawing of the topology of $\\QQ_2$. Filled dots have even second coordinate. Unfilled dots lie in the other coset. The hollow point at $(1/2,0)$ records the image of an integral vector under $E_+$.}","Write\n\\[\nB=\\begin{pmatrix}1&-1\\\\0&2\\end{pmatrix},\\qquad\nAB=B\\diag(1,-1),\\qquad\nE_+=B\\diag(1,0)B^{-1}.\n\\]\nThe index-two eigenlattice explains the denominator in $B^{-1}$ and in the projector. Over the localized lattice $\\Lambda_p=\\ZZ_{(p)}^2$, the number $2$ is a unit for every odd prime $p$. Hence the local denominator weight is zero there and one at $p=2$.","0&2\\end{pmatrix},\\qquad\nAB=B\\diag(1,-1),\\qquad\nE_+=B\\diag(1,0)B^{-1}.\n\\]\nThe index-two eigenlattice explains the denominator in $B^{-1}$ and in the projector. Over the localized lattice $\\Lambda_p=\\ZZ_{(p)}^2$, the number $2$ is a unit for every odd prime $p$. Hence the local denominator weight is zero there and one at $p=2$.","\\head{The special fiber does not retain the rational splitting}\nModulo two, $x-1=x+1$ and $A$ becomes a nontrivial unipotent matrix. The entries $1/2$ in $E_\\pm$ cannot be reduced modulo two. By contrast, the coefficient selectors on Plate II.6 use only zero and one and remain available.","\\vpage{16}{One extra digit is necessary}{MODEL D / A SHARP EXAMPLE OF PRECISION DELAY}{Chapter II, mathematical text: Theorem 7.7, Proposition 9.3}\nThe projector acts by\n\\[\nE_+(x,y)=(x+y/2,0).\n\\]\nIts factor $1/2$ can consume one dyadic digit. The claim concerns differences: if the inputs agree modulo $2^{m+1}\\Lambda$, their projected differences lie in $2^m\\Lambda$.","\\vfig[.97]{precision_delay}\n\\platecaption{For $m=3$, inputs differing by $(0,8)$ agree modulo eight, but their projected outputs differ by $(4,0)$ and do not. Raising the input agreement to modulo sixteen makes the projected outputs agree modulo eight.}","The example works at every positive depth:\n\\[\nE_+(0,2^m)=(2^{m-1},0)\\notin2^m\\Lambda,\n\\qquad\nE_+(0,2^{m+1})=(2^m,0)\\in2^m\\Lambda.\n\\]\nThus a zero-delay promise fails, while one extra input digit is sufficient. This witnesses sharpness for the specified projector and lattice.","\\head{The general calculation is just as short}\nIf $\\pi^sE\\Lambda\\subseteq\\Lambda$ and $x-y=\\pi^{m+s}v$, then\n\\[\nE(x-y)=\\pi^m(\\pi^sEv)\\in\\pi^m\\Lambda.\n\\]\nNo rounding procedure is hiding in this identity. Nor does it authorize reducing $E$ modulo $\\pi$ when its entries are nonintegral. Individual projected points may lie in a fractional lattice. The theorem controls their difference at the stated integral scale.","\\vpage{17}{See every Smith fiber modulo eight}{MODEL E / TWO DIRECTIONAL WEIGHTS}{Chapter II, mathematical text: Theorem 10.6}\nConsider the actual map of finite sets\n\\[\n(\\ZZ/8\\ZZ)^2\\longrightarrow(\\ZZ/8\\ZZ)^2,\n\\qquad (x,y)\\longmapsto(2x,4y).\n\\]\nThe first coordinate of the output must be even. The second must be zero or four. Those are precisely the attainable targets in the right panel.","For a specific attainable target $(2,4)$,\n\\[\n2x\\equiv2\\pmod8\\iff x\\in\\{1,5\\},\\qquad\n4y\\equiv4\\pmod8\\iff y\\in\\{1,3,5,7\\}.\n\\]\nThe coordinate choices are independent, giving $2\\cdot4=8$ solutions. The entire carrier is partitioned consistently:\n\\[\n8\\text{ attainable targets}\\ \\times\\ 8\\text{ inputs per target}=64.\n\\]\nFor an inadmissible target, such as $(1,4)$ or $(2,2)$, the fiber is empty. The same nonzero cardinality formula must not be applied to it.","\\head{The arithmetic behind the picture}\nMultiplication by $p^s$ modulo $p^L$ has kernel size $p^{\\min(s,L)}$. Its image consists exactly of residues whose canonical representatives are divisible by that same power. The displayed model has $p=2$, $L=3$ and weights $(1,2)$.","\\vpage{18}{Transport the whole solution fiber}{MODEL E / EXPLICIT SMITH COORDINATES}{Chapter II, mathematical text: Theorem 10.6}\nThe preceding diagonal system is connected to a nondiagonal one by the unimodular change of variables\n\\[\nM=\\begin{pmatrix}2&2\\\\0&4\\end{pmatrix},\\quad\nV=\\begin{pmatrix}1&-1\\\\0&1\\end{pmatrix},\\quad\nMV=\\diag(2,4)=D.\n\\]\nSince $\\det V=1$, $x=Vy$ is bijective modulo eight. Thus $Mx=b$ is equivalent to $Dy=b$.","0&1\\end{pmatrix},\\quad\nMV=\\diag(2,4)=D.\n\\]\nSince $\\det V=1$, $x=Vy$ is bijective modulo eight. Thus $Mx=b$ is equivalent to $Dy=b$.","\\vfig[.95]{smith_coordinate_transport}\n\\platecaption{The eight solutions for the target $(2,4)$, before and after the coordinate shear. Coordinates are residues in $\\{0,\\ldots,7\\}$. No interpolation on this modular grid is intended.}","For example, the diagonal solution $y=(1,3)$ corresponds to\n\\[\nx=Vy=(-2,3)\\equiv(6,3)\\pmod8,\n\\]\nand $M(6,3)^\\mathsf T=(18,12)^\\mathsf T\\equiv(2,4)^\\mathsf T$. Applying the inverse shear recovers the original solution.","\\vpage{19}{Precision exposes directional freedom}{MODEL E / A FIBER BUDGET AT EACH DEPTH}{Chapter II, mathematical text: Theorem 10.6}\nTake $p=2$ and the supplied diagonal weights $(s_0,s_1,s_2)=(0,1,3)$. At depth $L$, an admissible target has\n\\[\n2^{W(L)}\\text{ solutions},\\qquad\nW(L)=\\min(0,L)+\\min(1,L)+\\min(3,L).\n\\]\nEach direction contributes only the amount visible at the current depth.","\\vfig[.93]{smith_depth_budget}\n\\platecaption{The stacked height records the exponent $W(L)$. The label above each bar records the actual number of solutions. The zero-weight direction contributes no branching. This is a finite cardinality calculation, not an information-theoretic assumption about random inputs.}","The values are\n\\[\n\\begin{array}{c|rrrrrr}\nL&1&2&3&4&5&6\\\\\\hline\nW(L)&2&3&4&4&4&4\\\\\n\\#\\text{ solutions}&4&8&16&16&16&16.\n\\end{array}\n\\]\nOnce $L\\ge3$, the nonzero weights have been fully exposed and the admissible fiber cardinality stabilizes at sixteen. The input space still grows as $2^{3L}$. The number of attainable targets grows accordingly.","\\#\\text{ solutions}&4&8&16&16&16&16.\n\\end{array}\n\\]\nOnce $L\\ge3$, the nonzero weights have been fully exposed and the admissible fiber cardinality stabilizes at sixteen. The input space still grows as $2^{3L}$. The number of attainable targets grows accordingly.","At depth six, $(1,2,8)$ is admissible and gives sixteen solutions. Replacing the last coordinate by four makes the target inadmissible and gives zero solutions, even though the weights and depth are unchanged.","\\head{What changes with a zero diagonal factor?}\nA literal zero factor contributes $L$, not a fixed eventual weight, and requires a zero target coordinate. Its fiber therefore keeps growing with the modulus. That case must not be mistaken for a finite nonzero Smith weight.","\\vpage{20}{Filter first or split first: same output}{MODEL F / A FINITE POLYPHASE REALIZATION}{Chapter II, mathematical text: Section B.1}\nUse the finitely supported signal $x=(1,2,3,4,5,6)$ and the filter $h=(1,2,1)$. Outside the displayed support, the input is zero. Let $e_m=x_{2m}$ and $o_m=x_{2m+1}$.","If $w$ denotes one delay on a phase sequence, the filter becomes\n\\[\n\\begin{pmatrix}y_{\\rm even}","If $w$ denotes one delay on a phase sequence, the filter becomes\n\\[\n\\begin{pmatrix}y_{\\rm even}\\\\y_{\\rm odd}\\end{pmatrix}\n=\n\\underbrace{\\begin{pmatrix}1+w&2w\\\\2&1+w\\end{pmatrix}}_{H_{\\rm phase}(w)}\n\\begin{pmatrix}e\\\\o\\end{pmatrix}.\n\\]\nThe cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","Here $e=1+3w+5w^2$ and $o=2+4w+6w^2$, so direct polynomial multiplication gives\n\\[\ny_{\\rm even}=1+8w+16w^2+17w^3,\n\\qquad\ny_{\\rm odd}=4+12w+20w^2+6w^3.\n\\]\nThis is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.","\\vpage{21}{Two Smith constructions, different roles}{MODEL G / A NONLINEAR PHASE ORBIT}{Chapter II, mathematical text: Section 12.3, Smith-phase identity (p. 43)}\nSmith \\emph{normal form} concerns linear congruences. The Smith \\emph{impedance chart} uses the nonlinear coordinate change\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThey share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points $\\Gamma_k=\\tfrac12 e^{ik\\pi/4}$.","A NONLINEAR PHASE ORBIT}{Chapter II, mathematical text: Section 12.3, Smith-phase identity (p. 43)}\nSmith \\emph{normal form} concerns linear congruences. The Smith \\emph{impedance chart} uses the nonlinear coordinate change\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThey share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points $\\Gamma_k=\\tfrac12 e^{ik\\pi/4}$.","\\vfig[.96]{smith_chart_orbit}\n\\platecaption{Matching labels identify the same point in the two coordinates. All eight points are computed algebraically. Only their placement on the page is rendered numerically. The outer left circle is $|\\Gamma|=1$, not part of the sampled radius-$1/2$ orbit.}","The half-turn $\\Gamma\\mapsto-\\Gamma$ gives\n\\[\nz\\longmapsto\\frac1z,\\qquad z\\ne0,-1.\n\\]\nThus the labels $0$ and $4$ are $3$ and $1/3$. Label $2$ is $(3+4i)/5$, and the opposite point is its reciprocal. The domain restrictions come from the forward and inverse coordinate denominators.","\\takeaway{The chart is a geometric realization of phase action. Its nonlinear coordinate sensitivity near $\\Gamma=\\pm1$ is not the Smith congruence-fiber count and not a new linear projector theorem.}","\\vpage{22}{A small defect can grow under powers}{MODEL H / EXACT TELESCOPING, EXPLICIT GROWTH}{Chapter II, mathematical text: Theorem 11.1, Corollary 11.3}\nChoose an invertible coordinate shear but independently keep the same numerical operator at both levels:\n\\[\nC=\\begin{pmatrix}1&\\varepsilon\\\\0&1\\end{pmatrix},\\quad\nT_f=T_c=\\diag(1,2),\\quad \\varepsilon=\\frac1{16}.\n\\]\nThen $\\Delta=CT_f-T_cC=\\varepsilon E_{01}$. Exact coordinate invertibility has not forced these independently selected operators to intertwine.","0&1\\end{pmatrix},\\quad\nT_f=T_c=\\diag(1,2),\\quad \\varepsilon=\\frac1{16}.\n\\]\nThen $\\Delta=CT_f-T_cC=\\varepsilon E_{01}$. Exact coordinate invertibility has not forced these independently selected operators to intertwine.","\\vfig[.95]{power_defect_growth}\n\\platecaption{Actual defects and certified upper bounds in the induced infinity norm, both divided by $\\varepsilon$. All heights are exact integers. The upper estimate is valid but need not be tight.}","Direct multiplication gives\n\\[\nCT_f^n-T_c^nC=\\varepsilon(2^n-1)E_{01}.\n\\]\nThe general telescoping formula explains every contribution:\n\\[\nCT_f^n-T_c^nC\n=\\sum_{k=0}^{n-1}T_c^{n-1-k}\\Delta T_f^k.\n\\]\nSince both operator norms are two, the source theorem bounds the error by $n2^{n-1}\\varepsilon$. It does not promise a small error uniformly in $n$ merely because $\\varepsilon$ is small.","Had we chosen the transported operator $T_c'=CT_fC^{-1}$, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.","\\vpage{23}{Resolvents amplify errors near poles}{MODEL H / A DOMAIN-SENSITIVE COMPARISON}{Chapter II, mathematical text: Theorem 11.5}\nFor the same model, take $z\\notin\\{1,2\\}$ and form the actual inverse matrices\n\\[\nR_f=(zI-T_f)^{-1},\\qquad R_c=(zI-T_c)^{-1}.\n\\]\nTheir off-diagonal intertwining error is\n\\[\nCR_f-R_cC=R_c\\Delta R_f\n=\\frac{\\varepsilon}{(z-1)(z-2)}E_{01}.\n\\]\nBoth inverse hypotheses matter. Neither resolvent is asserted at a pole.","\\vfig[.94]{resolvent_amplification}\n\\platecaption{The bars show exact amplification factors for five rational choices $z>2$. Rows are separate samples, not equally spaced positions on the $z$ axis. The coefficient $\\varepsilon=1/16$ has been divided out.}","At $z=3$ the amplification factor is only $1/2$. At $z=201/100$, it is $10000/101$, so a small one-step defect can create a much larger resolvent defect. This follows from the exact formula rather than from unstable floating-point subtraction of two nearly singular inverses.","The certified general norm estimate is\n\\[\n\\|CR_f-R_cC\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|.\n\\]\nThe chapter also proves a version using one-sided witnesses: a left inverse for the coarse shift and a right inverse for the fine shift suffice for the algebraic identity. That does not turn arbitrary one-sided witnesses into a claim of two-sided resolvent existence.","Weighted inversion & A two-sided inverse of $D$, a multiplicative filtration, and positive certified gains. & Convergence in every possible complete filtered algebra.","Smith congruences & Concrete diagonal data modulo $p^L$, with $p>1$. & An automatically constructed general-DVR Smith normal form.","Let $V$ be a finite packet type, $E$ a finite edge type, and $R$ a semiring.\nA directed multigraph is a pair of endpoint functions\n\\[\n  \\src,\\tgt:E\\longrightarrow V.\n\\]\nParallel edges remain distinct elements of $E$.  Rows are targets and columns\nare sources.  Thus an edge $e:i\\to j$ is represented by a matrix supported at\nrow $j$, column $i$.","\\begin{definition}[Packet graph and scalar edge system]\n\\label{def:edge-system}\nFor $a,b\\in V$ and $r\\in R$, define\n\\[\n  \\operatorname{packetMatrix}(a,b;r)_{yx}\n  =\n  \\begin{cases}\n  r,&y=b\\text{ and }x=a,","\\begin{definition}[Packet graph and scalar edge system]\n\\label{def:edge-system}\nFor $a,b\\in V$ and $r\\in R$, define\n\\[\n  \\operatorname{packetMatrix}(a,b;r)_{yx}\n  =\n  \\begin{cases}\n  r,&y=b\\text{ and }x=a,\\\\\n  0,&\\text{otherwise}.\n  \\end{cases}\n\\]\nAn edge system is a coefficient function $c:E\\to R$.  Its elementary edge\noperator and total transfer operator are\n\\[\n  L_e=\\operatorname{packetMatrix}(\\src(e),\\tgt(e);c(e)),\n  \\qquad\n  T=\\sum_{e\\in E}L_e.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp}","0,&\\text{otherwise}.\n  \\end{cases}\n\\]\nAn edge system is a coefficient function $c:E\\to R$.  Its elementary edge\noperator and total transfer operator are\n\\[\n  L_e=\\operatorname{packetMatrix}(\\src(e),\\tgt(e);c(e)),\n  \\qquad\n  T=\\sum_{e\\in E}L_e.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp}","Words are written in execution order. In $L_fL_e$, the rightmost operator\n$L_e$ acts first. For a written word $(e_1,\\ldots,e_n)$, $e_1$ acts first\nand the operator product is $L_{e_n}\\cdots L_{e_1}$.","\\begin{lemma}[Declared support]\n\\label{lem:edge-support}\nFor every edge $e$, $(L_e)_{yx}=0$ unless\n$y=\\tgt(e)$ and $x=\\src(e)$.\n\\end{lemma}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp_eq_zero_off_support}","\\begin{theorem}[Typed interface annihilation]\n\\label{thm:edge-annihilation}\nIf $\\tgt(e)\\ne\\src(f)$, then\n\\[\n  L_fL_e=0.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp_mul_edgeOp_eq_zero}","\\begin{proof}\nFor a matrix entry $(y,x)$,\n\\[\n  (L_fL_e)_{yx}=\\sum_{z\\in V}(L_f)_{yz}(L_e)_{zx}.\n\\]\nA potentially nonzero summand would require simultaneously\n$z=\\src(f)$ and $z=\\tgt(e)$, contrary to the hypothesis.\n\\end{proof}","\\begin{theorem}[Collapse of a matched edge pair]\n\\label{thm:edge-composable}\nIf $\\tgt(e)=\\src(f)$, then\n\\[\nL_fL_e\n=\n\\operatorname{packetMatrix}\n\\bigl(\\src(e),\\tgt(f);c(f)c(e)\\bigr).\n\\]\nTogether with \\cref{thm:edge-annihilation}, this gives both cases for\nthe composition of two edge operators.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp_mul_edgeOp_of_composable}","\\begin{proof}\nThe sum defining the matrix product has exactly one potentially nonzero intermediate\nindex, namely $\\tgt(e)=\\src(f)$.  At that index the coefficients occur in\noperator order as $c(f)c(e)$, and the outer source and target are\n$\\src(e)$ and $\\tgt(f)$.\n\\end{proof}","\\begin{boundary}[Current fiber scope]\nThe compiled Chapter III model above has one scalar coordinate per packet.\nFor a family of modules $(X_i)_{i\\in V}$, an intrinsic edge map would instead\nhave type $\\ell_e:X_{\\src(e)}\\to X_{\\tgt(e)}$, and its totalization on\n$\\bigoplus_iX_i$ would be\n$\\iota_{\\tgt(e)}\\ell_e\\pi_{\\src(e)}$.  The same support proof applies, but\nthis generalization with dependent fibers is not attributed to the released\nChapter III modules for scalar edge systems.\n\\end{boundary}","\\begin{definition}[Composable word and execution operator]\n\\label{def:word-op}\nA finite word $w=(e_1,\\ldots,e_n)$ is composable when\n\\[\n  \\tgt(e_r)=\\src(e_{r+1})\n  \\qquad (1\\le r<n).\n\\]\nSet\n\\[\n  L_w=L_{e_n}\\cdots L_{e_1},\n  \\qquad L_{\\varnothing}=I.\n\\]\nThis definition records the written order of execution even though matrix\ncomposition is read right to left.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp}","\\begin{theorem}[First broken interface]\n\\label{thm:first-break}\nIf a word begins $(e,f,\\ldots)$ and $\\tgt(e)\\ne\\src(f)$, then its word\noperator is zero.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp_eq_zero_of_first_break}","\\begin{theorem}[Noncomposable words vanish]\n\\label{thm:nonwalk-vanish}\nIf $w$ is not composable, then $L_w=0$.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp_eq_zero_of_not_composable}","\\begin{definition}[Recursive word expansion]\n\\label{def:recursive-word-expansion}\nDefine\n\\[\n  \\cW_0=I,\n  \\qquad\n  \\cW_{n+1}=T\\cW_n.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion}","\\begin{theorem}[Transfer power equals recursive word expansion]\n\\label{thm:power-recursive-expansion}\nFor every $n\\in\\N$,\n\\[\n  T^n=\\cW_n.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_wordExpansion}","\\begin{proof}\nThe zero case is the identity matrix.  If $T^n=\\cW_n$, then the\nleft-successor power convention gives\n$T^{n+1}=TT^n=T\\cW_n=\\cW_{n+1}$.\n\\end{proof}","\\begin{remark}\nThe formal module also defines an explicit sum over tuples\n$w:\\operatorname{Fin}(n)\\to E$ and a second sum in which noncomposable tuples\nare replaced by zero.  It proves that the recursive matrix, the explicit tuple sum, and the sum\nfiltered by composability agree. The separate declarations make the finite\nenumeration claim directly checkable.\n\\end{remark}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.walkExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion_eq_explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.explicitWordExpansion_eq_walkExpansion}","For a nonempty composable word $w=(e_1,\\ldots,e_n)$, define its reverse\ncoefficient product\n\\[\n  c^\\leftarrow(w)=c(e_n)c(e_{n-1})\\cdots c(e_1).\n\\]\nThe reversal is essential over a noncommutative semiring.","\\begin{lemma}[One word is one endpoint block]\n\\label{lem:word-endpoint}\nIf $w=(e_1,\\ldots,e_n)$ is nonempty and composable, then\n\\[\n  L_w\n  =\n  \\operatorname{packetMatrix}\n  \\bigl(\\src(e_1),\\tgt(e_n);c^\\leftarrow(w)\\bigr).\n\\]\n\\end{lemma}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.reverseCoefficient; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.terminalTarget; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp_eq_packetMatrix_of_composable}","\\begin{proof}\nInduct on $n$.  The case $n=1$ is the definition of $L_e$.  Suppose the\nformula holds for the prefix $(e_1,\\ldots,e_n)$ and append $f$.  Composability\ngives $\\tgt(e_n)=\\src(f)$.  Multiplying the two packet matrices leaves one\nintermediate index and changes the coefficient to\n$c(f)c(e_n)\\cdots c(e_1)$, with source $\\src(e_1)$ and target $\\tgt(f)$.\n\\end{proof}","Write $\\Walk_n(i,j)$ for the finite set of composable edge tuples of length $n$\nwhose first source is $i$ and whose final target is $j$. For $n=0$, it has\none empty walk when $i=j$ and is empty otherwise. Assign the empty walk the\ncoefficient $1$.","\\begin{theorem}[Finite operator expansion from words to walks]\n\\label{thm:word-to-walk}\nFor every $n\\in\\N$,\n\\[\n  \\boxed{\n  T^n\n  =\n  \\sum_{w\\in E^n}L_w\n  =\n  \\sum_{w\\in E^n}\n  \\begin{cases}\n  L_w,&w\\text{ composable},","\\begin{theorem}[Finite operator expansion from words to walks]\n\\label{thm:word-to-walk}\nFor every $n\\in\\N$,\n\\[\n  \\boxed{\n  T^n\n  =\n  \\sum_{w\\in E^n}L_w\n  =\n  \\sum_{w\\in E^n}\n  \\begin{cases}\n  L_w,&w\\text{ composable},\\\\\n  0,&w\\text{ not composable}.\n  \\end{cases}\n  }\n\\]\nParallel edges occur as distinct summands.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_walkExpansion}","\\begin{proof}\nInduction and finite distributivity give\n\\[\n\\left(\\sum_{e\\in E}L_e\\right)^n\n=\n\\sum_{(e_1,\\ldots,e_n)\\in E^n}L_{e_n}\\cdots L_{e_1}.\n\\]\nEvery noncomposable summand is zero by\n\\cref{thm:nonwalk-vanish}, so inserting the composability filter does not\nchange the sum.  In Lean, tuple concatenation and the successor sum are\nhandled by \\leanname{tupleWordOp_snoc} and\n\\leanname{explicitWordExpansion_succ}. The two bridge equalities then yield\nthe displayed theorem.\n\\end{proof}","\\begin{corollary}[Endpoint coefficient walk sum]\n\\label{cor:word-to-walk-entry}\nFor all $n\\in\\N$ and $i,j\\in V$,\n\\[\n  \\boxed{\n  (T^n)_{ji}\n  =\n  \\sum_{w\\in\\Walk_n(i,j)}c^\\leftarrow(w).\n  }\n\\]\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Connects; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.endpointWalkCoefficientSum; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_apply_eq_sum_walks}","\\begin{proof}\nTake the $(j,i)$ entry in \\cref{thm:word-to-walk}.  By\n\\cref{lem:word-endpoint}, a surviving word contributes\n$c^\\leftarrow(w)$ exactly when its endpoints are $i$ and $j$, and contributes\nzero otherwise.  At $n=0$, the identity matrix gives the unique empty\ndiagonal walk.\n\\end{proof}","\\begin{corollary}[Reachability obstruction at fixed length]\n\\label{cor:fixed-reachability}\nIf $\\Walk_n(i,j)=\\varnothing$, then $(T^n)_{ji}=0$.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_apply_eq_zero_of_no_walk}","\\begin{corollary}[Orbit support obstruction]\n\\label{cor:orbit-reachability}\nLet $S\\subseteq V$, and let $x\\in R^V$ be supported on $S$.  If $j$ is not\nreachable by a directed path from any vertex of $S$, then\n\\[\n  (T^nx)_j=0\\qquad\\text{for every }n\\in\\N.\n\\]\nHence the orbit under $T$ of $x$ cannot generate a vector with a nonzero\ncoordinate at $j$.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Reachable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_mulVec_apply_eq_zero_of_unreachable_support}\n\\statusleanprior{Komorebi.PacketNaturalityCompletion.Reachability.algebraicOrbitHull_le_reachable_support; Komorebi.PacketNaturalityCompletion.Reachability.unreachable_nonzero_packet_obstruction}","\\begin{proof}\nExpand $(T^nx)_j=\\sum_i(T^n)_{ji}x_i$.  Terms with $i\\notin S$ vanish by\nthe support of $x$, and terms with $i\\in S$ vanish by\n\\cref{cor:fixed-reachability}.  The final claim follows because the\n$j$-coordinate also vanishes on every finite linear combination of orbit\nvectors.\n\\end{proof}","Let every chart $a$ carry a linear frame equivalence\n$F_a:M\\xrightarrow{\\sim}M$.  The formal Chapter III module defines transport\nfrom $a$ to $b$ by\n\\[\nJ_{b\\leftarrow a}=F_b^{-1}F_a.\n\\]\nThis frame formula means \\emph{synthesize in chart \\(a\\), then\nanalyze in chart \\(b\\)}.","\\begin{theorem}[Exact frame transport is coherent]\n\\label{thm:exact-frame-transport}\nFor every $a,b,c$,\n\\[\nJ_{a\\leftarrow a}=\\id,\\qquad\nJ_{a\\leftarrow b}J_{b\\leftarrow a}=\\id,\\qquad\nJ_{c\\leftarrow b}J_{b\\leftarrow a}=J_{c\\leftarrow a},\n\\]\nand therefore\n\\[\nJ_{a\\leftarrow c}J_{c\\leftarrow b}J_{b\\leftarrow a}=\\id.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTransport_triangleLoop}","\\begin{proof}\nEvery identity is cancellation of adjacent $F_rF_r^{-1}$ factors.  The Lean\ndefinitions use \\leanname{LinearEquiv.trans}. The displayed formulas use the\nequivalent right-to-left notation for function composition fixed in this chapter.\n\\end{proof}","\\begin{theorem}[Every finite canonical path telescopes]\n\\label{thm:exact-path-transport}\nLet $p=(a_0,a_1,\\ldots,a_n)$ be any finite vertex list and put\n\\[\nJ_p\n=J_{a_n\\leftarrow a_{n-1}}\\cdots J_{a_1\\leftarrow a_0},\n\\qquad\nJ_{(a_0)}=\\id.\n\\]\nThen\n\\[\nJ_p=J_{a_n\\leftarrow a_0}.\n\\]\nIn particular, if $a_n=a_0$, then $J_p=\\id$.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTransport_eq_endpoint; Komorebi.PaperIII.Transport.exactPathTransport_loop}","The graph model above resolves a fixed scalar matrix.  Coordinate transport\nis a separate construction.  Let $X$ be an $R$-module, and let\n\\[\n\\cP^a=(P^a_i)_{i\\in I_a}\n\\]\nbe a finite complete family of pairwise orthogonal linear idempotents:\n\\[\nP^a_iP^a_j=\\delta_{ij}P^a_i,\n\\qquad\n\\sum_{i\\in I_a}P^a_i=I_X.\n\\]\nDefine the exact packet coordinate module\n\\[\n\\Coord(\\cP^a)\n=\n\\left\\{(x_i)_{i\\in I_a}:P^a_ix_i=x_i\\right\\}.\n\\]\nIts analysis and synthesis maps are\n\\[\n\\cA_a(x)=(P^a_ix)_i,\n\\qquad\n\\cS_a((x_i)_i)=\\sum_ix_i.\n\\]\nCompleteness and orthogonality give\n\\[\n\\cS_a\\cA_a=\\id_X,\n\\qquad\n\\cA_a\\cS_a=\\id_{\\Coord(\\cP^a)}.\n\\]","ordinate transport\nis a separate construction.  Let $X$ be an $R$-module, and let\n\\[\n\\cP^a=(P^a_i)_{i\\in I_a}\n\\]\nbe a finite complete family of pairwise orthogonal linear idempotents:\n\\[\nP^a_iP^a_j=\\delta_{ij}P^a_i,\n\\qquad\n\\sum_{i\\in I_a}P^a_i=I_X.\n\\]\nDefine the exact packet coordinate module\n\\[\n\\Coord(\\cP^a)\n=\n\\left\\{(x_i)_{i\\in I_a}:P^a_ix_i=x_i\\right\\}.\n\\]\nIts analysis and synthesis maps are\n\\[\n\\cA_a(x)=(P^a_ix)_i,\n\\qquad\n\\cS_a((x_i)_i)=\\sum_ix_i.\n\\]\nCompleteness and orthogonality give\n\\[\n\\cS_a\\cA_a=\\id_X,\n\\qquad\n\\cA_a\\cS_a=\\id_{\\Coord(\\cP^a)}.\n\\]","i)_{i\\in I_a}\n\\]\nbe a finite complete family of pairwise orthogonal linear idempotents:\n\\[\nP^a_iP^a_j=\\delta_{ij}P^a_i,\n\\qquad\n\\sum_{i\\in I_a}P^a_i=I_X.\n\\]\nDefine the exact packet coordinate module\n\\[\n\\Coord(\\cP^a)\n=\n\\left\\{(x_i)_{i\\in I_a}:P^a_ix_i=x_i\\right\\}.\n\\]\nIts analysis and synthesis maps are\n\\[\n\\cA_a(x)=(P^a_ix)_i,\n\\qquad\n\\cS_a((x_i)_i)=\\sum_ix_i.\n\\]\nCompleteness and orthogonality give\n\\[\n\\cS_a\\cA_a=\\id_X,\n\\qquad\n\\cA_a\\cS_a=\\id_{\\Coord(\\cP^a)}.\n\\]","\\begin{definition}[Canonical exact coordinate transport]\n\\label{def:canonical-J}\nFor two exact packet systems on the same ambient module, define\n\\[\n  J^X_{b\\leftarrow a}\n  =\n  \\cA_b\\cS_a:\n  \\Coord(\\cP^a)\\longrightarrow\\Coord(\\cP^b).\n\\]\nThe arrow points from the source coordinate system $a$ to the target\ncoordinate system $b$.\n\\end{definition}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv}","\\begin{theorem}[Identity, inverse, and cocycle for canonical transport]\n\\label{thm:J-cocycle}\nFor any exact packet systems $a,b,c$ on $X$,\n\\[\nJ^X_{a\\leftarrow a}=\\id,\n\\qquad\n\\left(J^X_{b\\leftarrow a}\\right)^{-1}=J^X_{a\\leftarrow b},\n\\qquad\nJ^X_{c\\leftarrow b}J^X_{b\\leftarrow a}=J^X_{c\\leftarrow a}.\n\\]\n\\end{theorem}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_coarsenCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv_trans}","\\begin{proof}\nInsert $\\cS_b\\cA_b=\\id_X$:\n\\[\nJ^X_{c\\leftarrow b}J^X_{b\\leftarrow a}\n=\\cA_c\\cS_b\\cA_b\\cS_a\n=\\cA_c\\cS_a\n=J^X_{c\\leftarrow a}.\n\\]\nThe identity statement is $\\cA_a\\cS_a=\\id$.  Setting $c=a$ in the cocycle\ngives $J^X_{a\\leftarrow b}J^X_{b\\leftarrow a}=\\id$. Interchanging $a$ and $b$\ngives the other composition, so the two transports are inverses.\n\\end{proof}","If $F:X\\to Y$ is linear and $\\cP^a,\\cP^b$ are source packet systems while\n$\\cQ^r,\\cQ^s$ are target packet systems, put\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_rF\\cS^X_a.\n\\]","\\begin{theorem}[Canonical coordinate naturality]\n\\label{thm:canonical-naturality}\nThe exact coordinate representation preserves typed composition and satisfies\n\\[\nJ^Y_{s\\leftarrow r}\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}J^X_{b\\leftarrow a}.\n\\]\n\\end{theorem}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality}","\\begin{proof}\nBoth sides reduce to $\\cA^Y_sF\\cS^X_a$ after inserting the appropriate\nsynthesis--analysis identity.  The composition law is the same calculation\nwith the middle ambient module inserted.\n\\end{proof}","\\begin{corollary}[Canonical triangle holonomy is trivial]\n\\label{cor:J-flat}\nFor every triple $a,b,c$,\n\\[\nJ^X_{a\\leftarrow c}J^X_{c\\leftarrow b}J^X_{b\\leftarrow a}=\\id.\n\\]\n\\end{corollary}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.refineCoordinates_self}","\\begin{boundary}[Refinement is not reduction]\nAlthough Chapter II uses the name \\emph{refineCoordinates}, $J$ is an\nequivalence between two complete coordinate presentations.  It neither\ntruncates packets nor projects to a smaller state space.  Any lossy\nrestriction, interpolation, or learned identification is extra data and\ncannot inherit \\cref{thm:J-cocycle} without proof.\n\\end{boundary}","The compiled Chapter III API assumes only normalization at each chart.  All\nchart transports act on one module $M$, so chart labels distinguish routes\nbut not Lean types.","\\begin{definition}[Noncanonical connection normalized at each chart]\n\\label{def:minimal-connection}\nA minimal connection is a family\n\\[\n\\Gamma_{b\\leftarrow a}\\in\\operatorname{Aut}_R(M)\n\\qquad(a,b\\in B)\n\\]\nwith only\n\\[\n\\Gamma_{a\\leftarrow a}=\\id_M.\n\\]\nNeither inversion on reversed edges nor the cocycle law is a field of the\nstructure.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection}","At a triple $(a,b,c)$, the compiled cocycle predicate is\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\\Gamma_{c\\leftarrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.C","At a triple $(a,b,c)$, the compiled cocycle predicate is\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\\Gamma_{c\\leftarrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.Connection.triangleHolonomy}","b\\leftarrow a}\n=\\Gamma_{c\\leftarrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.Connection.triangleHolonomy}","rrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.Connection.triangleHolonomy}","\\begin{theorem}[Checked flatness consequences]\n\\label{thm:minimal-flatness}\nFor the minimal connection API, flatness implies\n\\[\nD^\\times_{c,b,a}=\\id_M,\n\\qquad\nH_{c,b,a}=\\id_M\n\\]\nfor every $a,b,c$.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.cocycleDefect_eq_refl_of_flat; Komorebi.PaperIII.Transport.Connection.triangleHolonomy_eq_refl_of_flat}","\\begin{proof}\nFlatness replaces the two-edge route in $D^\\times$ by the direct route,\nwhich cancels its inverse.  For $H$, flatness first collapses $a\\to b\\to c$\nto $a\\to c$, then collapses $a\\to c\\to a$ to the normalized transport\n$a\\to a$.\n\\end{proof}","\\begin{proposition}[The minimal API admits nontrivial holonomy]\n\\label{prop:twisted-loop}\nThere is a connection on three points over $\\mathbb Q$ for which transport\n$0\\to1$ is multiplication by $2$, every other transport between distinct\nendpoints is the identity, and\n\\[\nH_{2,1,0}(1)=2.\n\\]\nIn particular $H_{2,1,0}\\ne\\id$.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.twistedThreePointConnection; Komorebi.PaperIII.Transport.twisted_triangleHolonomy_apply_one; Komorebi.PaperIII.Transport.twisted_triangleHolonomy_ne_refl}","\\begin{boundary}[Why the minimal defect and loop are distinct]\nWithout reverse-edge inversion,\n$\\Gamma_{a\\leftarrow c}$ need not equal\n$\\Gamma_{c\\leftarrow a}^{-1}$.  Therefore the multiplicative relative defect\n$D^\\times_{c,b,a}$ and triangle holonomy $H_{c,b,a}$ are different\ncomparisons.  The formal module proves that global flatness trivializes both.\nIt does not identify them for an arbitrary connection.\n\\end{boundary}","Let $R$ be a commutative ring, let $B$ be a type of charts or refinement\nstates, and let each $C_a$ be an $R$-module.  The chart type is not assumed\nfinite.  The following definition is deliberately not derived from analysis\nand synthesis.","\\begin{definition}[Normalized reversible connection]\n\\label{def:connection}\nA normalized reversible connection is a family of linear equivalences\n\\[\n\\Gamma_{b\\leftarrow a}:C_a\\xrightarrow{\\;\\sim\\;}C_b\n\\qquad(a,b\\in B)\n\\]\nsuch that\n\\[\n\\Gamma_{a\\leftarrow a}=\\id,\n\\qquad\n\\Gamma_{a\\leftarrow b}=\\Gamma_{b\\leftarrow a}^{-1}.\n\\]\nNo cocycle equation is included in the structure.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection}","\\begin{definition}[Calibration and flatness]\n\\label{def:calibration-flatness}\nWhen the $C_a$ are exact packet coordinates of one ambient module, the\nconnection is \\emph{canonically calibrated} if\n\\[\n\\Gamma_{b\\leftarrow a}=J_{b\\leftarrow a}\n\\quad\\text{for every }a,b.\n\\]\nIt is \\emph{flat} if\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\n\\Gamma_{c\\leftarrow a}\n\\quad\\text{for every }a,b,c.\n\\]\nThe canonical connection is flat.  A general flat connection is proved in\n\\cref{thm:flat-gauge-canonical} to be related to it by a gauge change.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalConnection; Komorebi.PaperIII.Transport.Rev","\\begin{definition}[Calibration and flatness]\n\\label{def:calibration-flatness}\nWhen the $C_a$ are exact packet coordinates of one ambient module, the\nconnection is \\emph{canonically calibrated} if\n\\[\n\\Gamma_{b\\leftarrow a}=J_{b\\leftarrow a}\n\\quad\\text{for every }a,b.\n\\]\nIt is \\emph{flat} if\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\n\\Gamma_{c\\leftarrow a}\n\\quad\\text{for every }a,b,c.\n\\]\nThe canonical connection is flat.  A general flat connection is proved in\n\\cref{thm:flat-gauge-canonical} to be related to it by a gauge change.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalConnection; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalConnection_flat}","\\begin{remark}[A connection on a chart graph]\nOne may specify $\\Gamma_e$ only on the oriented edges of a chart graph,\ntogether with inverse maps on reversed edges.  Path transport and loop\nholonomy below still make sense.  A direct cocycle defect for $(a,b,c)$,\nhowever, requires a declared direct transport from $a$ to $c$.  The pairwise\nAPI in \\cref{def:connection} keeps this comparison total.\n\\end{remark}","\\begin{definition}[Cocycle defect]\n\\label{def:cocycle-defect}\nFor $a,b,c\\in B$, define the additive cocycle defect\n\\[\n\\Delta_{c,b,a}\n=\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n-\\Gamma_{c\\leftarrow a}\n\\in\\Hom_R(C_a,C_c).\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect}","\\begin{definition}[Triangle holonomy]\n\\label{def:triangle-holonomy}\nThe triangle holonomy based at $a$ is\n\\[\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}\n\\in\\operatorname{Aut}_R(C_a).\n\\]\nIt traverses $a\\to b\\to c\\to a$.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction}","Holonomy is attached to the chosen connection $\\Gamma$, never to an\nunspecified coordinate change.","\\begin{theorem}[Identity between defect and holonomy]\n\\label{thm:defect-holonomy}\nFor every triple $(a,b,c)$,\n\\[\n\\boxed{\nH_{c,b,a}-\\id_{C_a}\n=\n\\Gamma_{a\\leftarrow c}\\Delta_{c,b,a}.\n}\n\\]\nConsequently,\n\\[\n\\Delta_{c,b,a}=0\n\\quad\\Longleftrightarrow\\quad\nH_{c,b,a}=\\id_{C_a}.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy_sub_id; Komorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect_eq_zero_iff_triangleHolonomy_eq_refl}","\\begin{proof}\nUsing reversibility,\n\\[\n\\begin{aligned}\n\\Gamma_{a\\leftarrow c}\\Delta_{c,b,a}\n&=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}\n-\n\\Gamma_{a\\leftarrow c}\\Gamma_{c\\leftarrow a}\\\\\n&=H_{c,b,a}-\\id_{C_a}.\n\\end{aligned}\n\\]\nLeft composition by the equivalence $\\Gamma_{a\\leftarrow c}$ is injective,\nwhich gives the equivalence of the two vanishing statements.\n\\end{proof}","&=H_{c,b,a}-\\id_{C_a}.\n\\end{aligned}\n\\]\nLeft composition by the equivalence $\\Gamma_{a\\leftarrow c}$ is injective,\nwhich gives the equivalence of the two vanishing statements.\n\\end{proof}","\\begin{corollary}[Canonical transport has zero connection defect]\n\\label{cor:canonical-zero-defect}\nIf $\\Gamma_{b\\leftarrow a}=J_{b\\leftarrow a}$, then\n\\[\n\\Delta_{c,b,a}=0,\n\\qquad\nH_{c,b,a}=\\id\n\\]\nfor all triples.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.canonicalTransport_zero_connectionDefect; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalTransport_triangleHolonomy_eq_refl}","\\begin{definition}[Connection transport along a path]\n\\label{def:path-transport}\nFor a vertex sequence\n$p=(a_0,a_1,\\ldots,a_n)$, define\n\\[\n\\Gamma_p\n=\n\\Gamma_{a_n\\leftarrow a_{n-1}}\\cdots\n\\Gamma_{a_1\\leftarrow a_0},\n\\qquad\n\\Gamma_{(a_0)}=\\id_{C_{a_0}}.\n\\]\nIf $a_n=a_0$, then $\\Gamma_p$ is the holonomy of the loop $p$.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport}","\\begin{proposition}[Flat pair transport collapses to endpoints]\n\\label{prop:flat-endpoints}\nIf $\\Gamma$ is flat, then every path satisfies\n\\[\n\\Gamma_p=\\Gamma_{a_n\\leftarrow a_0}.\n\\]\nIn particular, every loop has identity holonomy.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.pathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedLoopTransport_eq_refl_of_flat}","\\begin{proof}\nInduct on the path length.  Appending $a_{n+1}$ left-multiplies the\ninductive expression by $\\Gamma_{a_{n+1}\\leftarrow a_n}$, and flatness\nreplaces the two factors by\n$\\Gamma_{a_{n+1}\\leftarrow a_0}$.  For a loop, normalization gives\n$\\Gamma_{a_0\\leftarrow a_0}=\\id$.\n\\end{proof}","\\begin{definition}[Gauge change]\n\\label{def:gauge-change}\nA gauge is a family of automorphisms $G_a\\in\\operatorname{Aut}_R(C_a)$.\nIt acts on a connection by\n\\[\n\\Gamma^G_{b\\leftarrow a}\n=\nG_b\\Gamma_{b\\leftarrow a}G_a^{-1}.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge}","\\begin{theorem}[Gauge covariance]\n\\label{thm:gauge-covariance}\nThe defect and triangle holonomy transform as\n\\[\n\\Delta^G_{c,b,a}\n=G_c\\Delta_{c,b,a}G_a^{-1},\n\\qquad\nH^G_{c,b,a}\n=G_aH_{c,b,a}G_a^{-1}.\n\\]\nThus flatness and triviality of triangle holonomy are gauge invariant.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_flat_iff; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy_eq_refl_iff}","\\begin{proof}\nSubstitute the definition of $\\Gamma^G$.  Adjacent factors\n$G_b^{-1}G_b$ and $G_c^{-1}G_c$ cancel.  In the defect, both the composite\nand direct terms acquire the same outer factors $G_c$ and $G_a^{-1}$.  In\nthe loop, all inner factors cancel and only conjugation at the base remains.\n\\end{proof}","\\begin{theorem}[Flat connections are gauges of the canonical pair transport]\n\\label{thm:flat-gauge-canonical}\nSuppose the $C_a$ are exact packet coordinate modules for one ambient module,\nso the canonical $J_{b\\leftarrow a}$ is available.  If $\\Gamma$ is a flat\nnormalized reversible pair connection, choose $o\\in B$ and set\n\\[\nG_a=\\Gamma_{a\\leftarrow o}J_{o\\leftarrow a}.\n\\]\nThen every pair transport satisfies\n\\[\n\\boxed{\n\\Gamma_{b\\leftarrow a}\n=G_bJ_{b\\leftarrow a}G_a^{-1}.\n}\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.flatGauge; Komorebi.PaperIII.Transport.ReversibleConnection.flat_isGaugeOfCanonical}","}[Flat connections are gauges of the canonical pair transport]\n\\label{thm:flat-gauge-canonical}\nSuppose the $C_a$ are exact packet coordinate modules for one ambient module,\nso the canonical $J_{b\\leftarrow a}$ is available.  If $\\Gamma$ is a flat\nnormalized reversible pair connection, choose $o\\in B$ and set\n\\[\nG_a=\\Gamma_{a\\leftarrow o}J_{o\\leftarrow a}.\n\\]\nThen every pair transport satisfies\n\\[\n\\boxed{\n\\Gamma_{b\\leftarrow a}\n=G_bJ_{b\\leftarrow a}G_a^{-1}.\n}\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.flatGauge; Komorebi.PaperIII.Transport.ReversibleConnection.flat_isGaugeOfCanonical}","\\begin{proof}\nReversibility gives\n$G_a^{-1}=J_{a\\leftarrow o}\\Gamma_{o\\leftarrow a}$.  Therefore\n\\[\n\\begin{aligned}\nG_bJ_{b\\leftarrow a}G_a^{-1}\n&=\n\\Gamma_{b\\leftarrow o}\nJ_{o\\leftarrow b}J_{b\\leftarrow a}\nJ_{a\\leftarrow o}\n\\Gamma_{o\\leftarrow a}","\\begin{proof}\nReversibility gives\n$G_a^{-1}=J_{a\\leftarrow o}\\Gamma_{o\\leftarrow a}$.  Therefore\n\\[\n\\begin{aligned}\nG_bJ_{b\\leftarrow a}G_a^{-1}\n&=\n\\Gamma_{b\\leftarrow o}\nJ_{o\\leftarrow b}J_{b\\leftarrow a}\nJ_{a\\leftarrow o}\n\\Gamma_{o\\leftarrow a}\\\\\n&=\n\\Gamma_{b\\leftarrow o}\\Gamma_{o\\leftarrow a}\\\\\n&=\n\\Gamma_{b\\leftarrow a}.\n\\end{aligned}\n\\]\nThe first reduction uses the canonical cocycle twice. The second uses\nflatness of $\\Gamma$.\n\\end{proof}","&=\n\\Gamma_{b\\leftarrow a}.\n\\end{aligned}\n\\]\nThe first reduction uses the canonical cocycle twice. The second uses\nflatness of $\\Gamma$.\n\\end{proof}","\\begin{boundary}[Flat does not mean calibrated]\nThe theorem produces a gauge relating $\\Gamma$ to $J$.  It does not prove\n$\\Gamma=J$ before that gauge is chosen.  Conversely, nontrivial triangle\nholonomy cannot be removed by a gauge, because conjugation cannot turn a\nnonidentity automorphism into the identity.\n\\end{boundary}","Let $\\mathsf{Candidate}$, $\\mathsf{Packet}$, and $\\mathsf{Residual}$ be\narbitrary types, with a distinguished\n$0\\in\\mathsf{Residual}$, and let\n\\[\nr:\\mathsf{Candidate}\\longrightarrow\n\\mathsf{Packet}\\longrightarrow\\mathsf{Residual}\n\\]\nbe a residual function.","\\begin{definition}[Solution and obstruction packet for a candidate]\n\\label{def:generic-obstruction}\nA candidate $x$ is a solution when\n\\[\n\\operatorname{IsSolution}_r(x)\n\\quad\\Longleftrightarrow\\quad\n\\forall p,\\ r(x,p)=0.\n\\]\nAn obstruction packet for $x$ is a pair consisting of a named packet $p$\nand a certificate $r(x,p)\\ne0$.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket}","\\begin{theorem}[Exact logical force of a local obstruction]\n\\label{thm:generic-obstruction}\nEvery obstruction packet for $x$ proves that $x$ is not a solution, and\n\\[\n\\neg\\operatorname{IsSolution}_r(x)\n\\quad\\Longleftrightarrow\\quad\n\\text{there exists an obstruction packet for }x.\n\\]\nThe reverse implication from failure to a packet witness is classical and\ndoes not require the packet type to be finite.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket.not_isSolution; Komorebi.PaperIII.NonlinearObstructions.obstructionPacket_of_not_isSolution; Komorebi.PaperIII.NonlinearObstructions.not_isSolution_iff_nonempty_obstructionPacket}","\\begin{boundary}[Candidate-specific means candidate-specific]\nThe theorem rules out the named $x$.  It neither constructs a solution nor\nshows that every candidate is obstructed.  It provides no minimality,\nstability, computability, or classification theorem for the packet witness.\n\\end{boundary}","For a reversible connection, specialize the generic residual at a fixed\ntriangle by taking a packet label to be a pair $(\\mu,\\nu)$ and the residual\nto be the corresponding block of $H-\\id$.  A nonzero block, together with its\nlabel and a proof of nonvanishing, gives an obstruction packet certificate\nin the sense of \\cref{def:generic-obstruction}.  The reconstruction theorem\nbelow supplies the further step: vanishing of all these local residuals is\nequivalent to the cocycle equation at that triangle.","Fix a triangle $(a,b,c)$ and resolve the base fiber $C_a$ by a finite complete\nfamily of idempotents $(\\Pi^a_\\mu)_{\\mu\\in I_a}$.  Orthogonality may also hold,\nas it does for the exact range coordinate projectors below, but the compiled\nreconstruction theorem does not require it.  In exact range coordinates,\n$\\Pi^a_\\mu$ is simply the projector that retains component $\\mu$ and zeros\nthe other components.","\\begin{definition}[Holonomy obstruction and packet residuals]\n\\label{def:obstruction-packet}\nDefine\n\\[\n\\Ob_{c,b,a}=H_{c,b,a}-\\id_{C_a},\n\\qquad\n\\Ob^{\\mu\\nu}_{c,b,a}\n=\n\\Pi^a_\\mu\\Ob_{c,b,a}\\Pi^a_\\nu.\n\\]\nThe map $\\Ob^{\\mu\\nu}_{c,b,a}$ is the obstruction block, or packet residual,\nfrom input packet $\\nu$ to output packet $\\mu$.  If it is nonzero, then\n$(\\mu,\\nu)$ and that nonvanishing proof form an obstruction packet\ncertificate.  Diagonal blocks detect holonomy within one packet.  Off-diagonal\nblocks detect packet mixing induced by holonomy.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi.PaperIII.HolonomyBlocks.PacketResolution.block}","\\begin{theorem}[Exact reconstruction and detection by obstruction blocks]\n\\label{thm:obstruction-detection}\nThe total obstruction has the exact finite resolution\n\\[\n\\boxed{\n\\Ob_{c,b,a}\n=\n\\sum_{\\mu,\\nu\\in I_a}\n\\Ob^{\\mu\\nu}_{c,b,a}.\n}\n\\]\nMoreover, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Delta_{c,b,a}=0$;\n\\item $H_{c,b,a}=\\id$;\n\\item $\\Ob_{c,b,a}=0$;\n\\item $\\Ob^{\\mu\\nu}_{c,b,a}=0$ for every $\\mu,\\nu$.\n\\end{enumerate}\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.obstruction_sum_blocks; Komorebi.PaperIII.HolonomyBlocks.obstructionBlocks_zero_iff; Komorebi.PaperIII.HolonomyBlocks.obstructionDetection_equivalences}","onstruction and detection by obstruction blocks]\n\\label{thm:obstruction-detection}\nThe total obstruction has the exact finite resolution\n\\[\n\\boxed{\n\\Ob_{c,b,a}\n=\n\\sum_{\\mu,\\nu\\in I_a}\n\\Ob^{\\mu\\nu}_{c,b,a}.\n}\n\\]\nMoreover, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Delta_{c,b,a}=0$;\n\\item $H_{c,b,a}=\\id$;\n\\item $\\Ob_{c,b,a}=0$;\n\\item $\\Ob^{\\mu\\nu}_{c,b,a}=0$ for every $\\mu,\\nu$.\n\\end{enumerate}\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.obstruction_sum_blocks; Komorebi.PaperIII.HolonomyBlocks.obstructionBlocks_zero_iff; Komorebi.PaperIII.HolonomyBlocks.obstructionDetection_equivalences}","\\begin{proof}\nInsert completeness on both sides:\n\\[\n\\Ob\n=\n\\left(\\sum_\\mu\\Pi^a_\\mu\\right)\n\\Ob\n\\left(\\sum_\\nu\\Pi^a_\\nu\\right)\n=\n\\sum_{\\mu,\\nu}\\Pi^a_\\mu\\Ob\\Pi^a_\\nu.\n\\]\nThus (iv) implies (iii), and (iii) implies (iv) by composition with the\nprojectors.  Conditions (ii) and (iii) are equivalent by the definition of\n$\\Ob$ and additive cancellation.\nConditions (i) and (ii) are equivalent by\n\\cref{thm:defect-holonomy}.\n\\end{proof}","\\begin{corollary}[Witness supported in one packet]\n\\label{cor:packet-witness}\nIf $\\Ob^{\\mu\\nu}_{c,b,a}\\ne0$, then there is a vector $x$ supported in packet\n$\\nu$ such that\n\\[\n\\Pi^a_\\mu\\Ob_{c,b,a}x\\ne0.\n\\]\nSuch an $x$ is an exact finite witness of nonflatness.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.exists_packetWitness_of_obstructionBlock_ne_zero}","\\begin{proof}\nClassically choose $y$ on which the nonzero linear map\n$\\Pi^a_\\mu\\Ob\\Pi^a_\\nu$ is nonzero and set $x=\\Pi^a_\\nu y$.\nIdempotence gives $\\Pi^a_\\nu x=x$, and the required output is the chosen\nnonzero value.\n\\end{proof}","Vanishing or nonvanishing of the total triangle obstruction is gauge invariant\nby \\cref{thm:gauge-covariance}, but a fixed packet pattern need not be.  A\ngeneral $G_a$ can mix the projectors $\\Pi^a_\\mu$.","\\begin{proposition}[Gauge covariance when packet projectors are preserved]\n\\label{prop:packet-gauge}\nIf $G_a\\Pi^a_\\mu=\\Pi^a_\\mu G_a$ for every $\\mu$, then\n\\[\n\\left(\\Ob^G\\right)^{\\mu\\nu}\n=\nG_a\\Ob^{\\mu\\nu}G_a^{-1}.\n\\]\nConsequently the zero/nonzero status of each obstruction block is preserved\nby gauges that preserve the packet projectors.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_eq_zero_iff}","\\begin{proof}\nUse $\\Ob^G=G_a\\Ob G_a^{-1}$ and commute $G_a$ and $G_a^{-1}$ through the\ntwo packet projectors.  Composition by an equivalence is injective in both\ndirections.\n\\end{proof}","The passage from words to walks also applies to products of different matrices.\nLet $V$\nbe one finite packet type, let $R$ be a semiring, and let\n$B_1,\\ldots,B_m\\in R^{V\\times V}$ be a finite sequence of possibly different\npacket matrices.  Rows are targets and columns are sources as before.","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.he","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPa","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpan","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komore","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.het","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousPr","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","pansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","rices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","\\begin{boundary}[Changing packet types along a chain]\nThe compiled theorems\n\\leanname{heterogeneousPacketPathExpansion_two} and\n\\leanname{heterogeneousPacketPathExpansion_three} allow different packet\ntypes at the successive stages of two- and three-factor products.  The\narbitrary-length theorem above permits different factors but uses one common\nfinite packet type $V$.  A fully dependent arbitrary-length chain of changing\nmodules and packet types remains outside the finite algebraic contract.\n\\end{boundary}","\\begin{remark}[Checked specialization]\nWhen every $B_r$ is one repeated endomorphism $T$, the recursive\nspecialization to packet powers is inherited from Chapter I.\n\\end{remark}\n\\statusleanprior{Komorebi.PacketNaturalityCompletion.Packets.transferBlock_pow_eq_packetPathSum}","For a loop represented on one common finite scalar packet coordinate type,\nthe proposition resolves its successive transport matrices into packet\nwalks.  If $R$ is a ring, as in the reversible connection API, then\n$(H-I)_{ji}=H_{ji}-\\delta_{ji}$.  In particular, off-diagonal entries are\nunchanged.  Thus a nonzero off-diagonal holonomy obstruction in this model\nwith a common index type requires a transport word along the closed chart\nloop whose packet path runs from its input packet to its output packet.  This\nconclusion requires declared block supports for the connection matrices.\nAn arbitrary connection need not respect the transfer graph from\n\\cref{sec:edge-operators}.","\\begin{example}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet","\\begin{example}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{examp","\\begin{example}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","le}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","amma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","se scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","\\begin{example}[A transfer walk with two steps]\nLet a packet graph have edges $e:0\\to1$ and $f:1\\to2$, with no other edges.\nThen\n\\[\nT=L_e+L_f,\\qquad\n(T^2)_{20}=c(f)c(e).\n\\]\nThe products $L_eL_e$, $L_eL_f$, and $L_fL_f$ vanish by typed mismatch.\nOnly $L_fL_e$, corresponding to the walk $0\\to1\\to2$, contributes to the\n$(2,0)$ entry.  This example also fixes the noncommutative coefficient order.\n\\end{example}","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exac","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representati","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreser","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarr","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\ci","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{pro","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthre","begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komor…","n}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Tra…",":X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komor…","th inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","tarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","fferentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","tarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","\\begin{proof}\nBoth sides of the naturality equation reduce to\n$\\cA^Y_s\\circ F\\circ\\cS^X_a$ using\n$\\cS\\circ\\cA=\\id$ in $X$ and $Y$.  For composition, the middle\n$\\cS^Y\\circ\\cA^Y$ cancels in the same way.\n\\end{proof}","\\begin{boundary}[Where nonlinear defects can enter]\nA nonlinear \\emph{coordinate} defect cannot be blamed on the exact canonical\n$J$ of \\cref{def:canonical-J}. By \\cref{prop:set-naturality}, that defect is\nidentically zero.  Nonzero defects require extra structure such as a\nnoncanonical connection, truncation, approximation, restricted domains, or\nfailure of an operation to respect a declared packet model.\n\\end{boundary}","\\begin{definition}[Finite grammar for operation trees]\n\\label{def:compiled-operation-tree}\nFor types of input labels $I$, constants $K$, unary symbols $U$, and binary\nsymbols $B$, the finite formal grammar has four constructors:\n\\[\n\\begin{aligned}\nt ::={}&\\operatorname{input}(i)\n\\mid\\operatorname{constant}(k)\n\\mid\\operatorname{unary}(u,t)","\\begin{definition}[Finite grammar for operation trees]\n\\label{def:compiled-operation-tree}\nFor types of input labels $I$, constants $K$, unary symbols $U$, and binary\nsymbols $B$, the finite formal grammar has four constructors:\n\\[\n\\begin{aligned}\nt ::={}&\\operatorname{input}(i)\n\\mid\\operatorname{constant}(k)\n\\mid\\operatorname{unary}(u,t)\\\\\n&\\mid\\operatorname{binary}(b,t_1,t_2).\n\\end{aligned}\n\\]\nGiven an arbitrary value type $Y$, evaluation takes completely explicit maps\n\\[\nI\\to Y,\\qquad K\\to Y,\\qquad U\\to Y\\to Y,\\qquad\nB\\to Y\\to Y\\to Y.\n\\]\nNo algebraic, continuity, or differentiability law is imposed on these\nsemantics.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.OperationTree; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval}","&\\mid\\operatorname{binary}(b,t_1,t_2).\n\\end{aligned}\n\\]\nGiven an arbitrary value type $Y$, evaluation takes completely explicit maps\n\\[\nI\\to Y,\\qquad K\\to Y,\\qquad U\\to Y\\to Y,\\qquad\nB\\to Y\\to Y\\to Y.\n\\]\nNo algebraic, continuity, or differentiability law is imposed on these\nsemantics.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.OperationTree; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval}","When $I$ has decidable equality, let $\\operatorname{inputs}(t)$ be the finite\nset of input labels actually occurring in $t$.","\\begin{theorem}[Input locality for operation trees]\n\\label{thm:tree-locality}\nLet $\\eta_1,\\eta_2:I\\to Y$ be two input environments.  If\n\\[\n\\eta_1(i)=\\eta_2(i)\n\\qquad\\text{for every }i\\in\\operatorname{inputs}(t),\n\\]\nthen the two evaluations of $t$ are equal under the same constant, unary, and\nbinary semantics.  In particular, if\n$\\operatorname{inputs}(t)=\\varnothing$, evaluation is independent of the\ninput environment.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval_congr_on_inputs; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval_eq_of_inputs_eq_empty}","\\begin{proof}\nStructural induction on $t$.  The input case is the agreement hypothesis.\nThe constant case is reflexive. The unary case applies the induction\nhypothesis to its child. The binary case applies it independently to the\nleft and right subtrees, whose input sets form a union.\n\\end{proof}","\\begin{definition}[Finitary colored operation signature]\n\\label{def:signature}\nA finitary colored signature consists of:\n\\begin{itemize}\n\\item a finite type $V$ of colors;\n\\item a type $\\Sigma$ of operation symbols;\n\\item an arity $r(\\sigma)\\in\\N$ for every $\\sigma\\in\\Sigma$;\n\\item input colors\n$\\operatorname{in}_\\sigma:\\operatorname{Fin}(r(\\sigma))\\to V$;\n\\item an output color $\\operatorname{out}(\\sigma)\\in V$.\n\\end{itemize}\nA realization assigns a space $X_v$ to each color and a map\n\\[\n\\Phi_\\sigma:\n\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}X_{\\operatorname{in}_\\sigma(q)}\n\\longrightarrow X_{\\operatorname{out}(\\sigma)}.\n\\]\n\\end{definition}\n\\statusroadmap{The intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.}","el{def:signature}\nA finitary colored signature consists of:\n\\begin{itemize}\n\\item a finite type $V$ of colors;\n\\item a type $\\Sigma$ of operation symbols;\n\\item an arity $r(\\sigma)\\in\\N$ for every $\\sigma\\in\\Sigma$;\n\\item input colors\n$\\operatorname{in}_\\sigma:\\operatorname{Fin}(r(\\sigma))\\to V$;\n\\item an output color $\\operatorname{out}(\\sigma)\\in V$.\n\\end{itemize}\nA realization assigns a space $X_v$ to each color and a map\n\\[\n\\Phi_\\sigma:\n\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}X_{\\operatorname{in}_\\sigma(q)}\n\\longrightarrow X_{\\operatorname{out}(\\sigma)}.\n\\]\n\\end{definition}\n\\statusroadmap{The intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.}","lored signature consists of:\n\\begin{itemize}\n\\item a finite type $V$ of colors;\n\\item a type $\\Sigma$ of operation symbols;\n\\item an arity $r(\\sigma)\\in\\N$ for every $\\sigma\\in\\Sigma$;\n\\item input colors\n$\\operatorname{in}_\\sigma:\\operatorname{Fin}(r(\\sigma))\\to V$;\n\\item an output color $\\operatorname{out}(\\sigma)\\in V$.\n\\end{itemize}\nA realization assigns a space $X_v$ to each color and a map\n\\[\n\\Phi_\\sigma:\n\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}X_{\\operatorname{in}_\\sigma(q)}\n\\longrightarrow X_{\\operatorname{out}(\\sigma)}.\n\\]\n\\end{definition}\n\\statusroadmap{The intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.}","\\begin{definition}[Typed operation tree and evaluation]\n\\label{def:operation-tree}\nA typed operation tree is recursively either a leaf of declared color or a\nnode labeled by $\\sigma$, with one correctly colored child for each input\nslot of $\\sigma$.  Evaluation is recursive:\n\\[\n\\ev(\\operatorname{node}(\\sigma;t_1,\\ldots,t_{r(\\sigma)}))\n=\n\\Phi_\\sigma(\\ev(t_1),\\ldots,\\ev(t_{r(\\sigma)})).\n\\]\n\\end{definition}\n\\statusroadmap{Candidate API: \\leanname{Komorebi.PaperIII.NonlinearObstructions.ColoredOperationTree} and \\leanname{ColoredOperationTree.eval}.  This is stronger than the compiled uncolored unary--binary grammar.}","Suppose every $X_v$ has a finite complete packet family\n$(P^v_i)_{i\\in I_v}$.  A packet labeling assigns an admissible packet index\nto each edge or node of a typed tree.  For multilinear $\\Phi_\\sigma$, inserting\n\\[\n\\id_{X_v}=\\sum_{i\\in I_v}P^v_i\n\\]\nat every input and output yields a finite sum over trees with packet labels.\nThis is the branching analogue of inserting intermediate projectors in\n\\cref{prop:heterogeneous-path}.","For coordinate models of the source and target colors, an additive target\ncoordinate space, and a chosen noncanonical connection, a pointwise node\ndefect can be defined by\n\\[\n\\mathfrak D^\\Gamma_{b,a}(\\Phi)\n=\n\\Gamma^Y_{b\\leftarrow a}\\circ\\widehat\\Phi_a\n-\n\\widehat\\Phi_b\\circ\n\\left(\\prod_q\\Gamma^{X_q}_{b\\leftarrow a}\\right).\n\\]\nThis expression compares two \\emph{chosen} routes.  With canonical exact\ntransport in every slot it vanishes by \\cref{prop:set-naturality}.","\\addlinespace\nTwisted three-point loop\n& Lean-backed III\n& \\leanname{twistedThreePointConnection}; \\leanname{twisted_triangleHolonomy_ne_refl}\n& Concrete rational connection. The selected loop sends \\(1\\) to \\(2\\).","\\medskip\\noindent The notational change from \\(\\cA,\\cS\\) to \\(\\Acal,\\Scal\\) is typographical: both denote analysis and synthesis for the declared packet system. A connection remains additional data. Smooth canonical transport is constructed from a moving projector family, whereas the algebraic exact coordinate transport in Part~I compares fixed exact descriptions of one ambient object.\n\\section{Introduction}\\label{sec:introduction}","A finite family of pairwise annihilating idempotents gives exact coordinates.  If\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{for }\\alpha\\ne\\beta,\n\\qquad\n\\sum_\\alpha P_\\alpha=I,\n\\]\nthen every vector is the sum of its packet components.  Linear operators become block matrices in these coordinates.  A nonlinear map cannot be reconstructed from its diagonal restrictions, since mixed interactions between packet components are usually present.  It can still be represented exactly if its input is the complete packet tuple.","A linear correction map\n\\[\nD:U\\longrightarrow V\n\\]\nplaces the obstruction to solving $Du=-r$ in the canonical quotient $\\coker D$.  This quotient has a universal property and behaves naturally under commuting squares.  Mapping cones extend the same construction to cochain complexes.  In a Hilbert space with closed range, the quotient is represented by the orthogonal projection onto $\\Ker D^*$.  The Moore--Penrose inverse then gives the least norm correction.  Relevant background appears in \\cite{Weibel1994,Penrose1955}.","The first question is representation.  Given a nonlinear map $F$, can its packet form be written without approximation?  The answer is yes.  Analysis and synthesis are inverse coordinate maps, so conjugation preserves the full equation.","The third question is correction.  Suppose an approximate solution leaves a defect $r$.  A declared linearized correction map $D$ removes the part of $r$ lying in $\\im D$.  The quotient class\n\\[\n[r]\\in\\coker D\n\\]\nis the part that no allowed first-order correction can remove.  This quotient class is the obstruction.  Its vanishing must be supplemented by a nonlinear theorem.","These four layers fit into the sequence\n\\[\n\\begin{gathered}\n\\text{exact coordinates}\n\\longrightarrow\n\\text{packet interactions}\n\\longrightarrow\n\\text{correction classes}\n\\\\\n\\longrightarrow\n\\text{global gluing}.\n\\end{gathered}\n\\]\nEach arrow needs hypotheses.  Exact coordinates need a complete packet system.  Differential interactions need regularity.  A cokernel obstruction needs a specified correction map.  Holonomy needs invertible local transports.","\\subsection*{Standing conventions}\nAll packet systems are finite unless a statement says otherwise.  In a normed setting every packet projector and every coarsening operator is bounded.  Differentiability of operator-valued maps is taken in operator norm.  A Banach space isomorphism is a bounded linear bijection with bounded inverse.  For a decreasing filtration $(F^mX)$, completeness means that the canonical map\n\\[\nX\\longrightarrow\\varprojlim_m X/F^mX\n\\]\nis an isomorphism and separatedness means $\\bigcap_mF^mX=0$.  Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filtration congruences at the stated depth.  These assumptions are repeated where they are essential to a proof.","ng conventions}\nAll packet systems are finite unless a statement says otherwise.  In a normed setting every packet projector and every coarsening operator is bounded.  Differentiability of operator-valued maps is taken in operator norm.  A Banach space isomorphism is a bounded linear bijection with bounded inverse.  For a decreasing filtration $(F^mX)$, completeness means that the canonical map\n\\[\nX\\longrightarrow\\varprojlim_m X/F^mX\n\\]\nis an isomorphism and separatedness means $\\bigcap_mF^mX=0$.  Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filtration congruences at the stated depth.  These assumptions are repeated where they are essential to a proof.","a normed setting every packet projector and every coarsening operator is bounded.  Differentiability of operator-valued maps is taken in operator norm.  A Banach space isomorphism is a bounded linear bijection with bounded inverse.  For a decreasing filtration $(F^mX)$, completeness means that the canonical map\n\\[\nX\\longrightarrow\\varprojlim_m X/F^mX\n\\]\nis an isomorphism and separatedness means $\\bigcap_mF^mX=0$.  Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filtration congruences at the stated depth.  These assumptions are repeated where they are essential to a proof.","Let $X$ be a module over a unital ring.  A finite packet system on $X$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\subseteq\\End(X)\n\\]\nof nonzero idempotents such that\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{when }\\alpha\\ne\\beta,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nWrite\n\\[\nX_\\alpha=P_\\alpha X,\n\\qquad\nX^{\\Pcal}=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nThe analysis and synthesis maps are\n\\[\n\\Acal_{\\Pcal}x=(P_\\alpha x)_\\alpha,\n\\qquad\n\\Scal_{\\Pcal}(x_\\alpha)_\\alpha=\\sum_\\alpha x_\\alpha.\n\\]\nThey are inverse isomorphisms.","\\begin{definition}[Nonlinear packet coordinate map]\\label{def:nonlinear_packet_map}\nLet $\\Pcal$ be a packet system on $X$ and let $\\Qcal$ be a packet system on $Y$.  For any map $F:X\\to Y$, define\n\\[\nF_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Qcal}F\\Scal_{\\Pcal}.\n\\]\nThus\n\\[\nF_{\\Qcal\\leftarrow\\Pcal}((x_\\alpha)_\\alpha)\n=\n\\left(\nQ_\\gamma F\\left(\\sum_\\alpha x_\\alpha\\right)\n\\right)_\\gamma.\n\\]\n\\end{definition}","\\begin{theorem}[Exact nonlinear coordinate theorem]\\label{thm:exact_nonlinear_coordinates}\nFor every map $F:X\\to Y$,\n\\[\n\\Scal_{\\Qcal}F_{\\Qcal\\leftarrow\\Pcal}\\Acal_{\\Pcal}=F.\n\\]\nIf $G:Y\\to Z$ and $\\Rcal$ is a packet system on $Z$, then\n\\[\n(G\\circ F)_{\\Rcal\\leftarrow\\Pcal}\n=\nG_{\\Rcal\\leftarrow\\Qcal}\n\\circ\nF_{\\Qcal\\leftarrow\\Pcal}.\n\\]\nNo linearity, continuity, or differentiability assumption is needed.\n\\end{theorem}","\\begin{proof}\nThe identities $\\Scal_{\\Pcal}\\Acal_{\\Pcal}=I_X$ and $\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_Y$ give\n\\[\n\\Scal_{\\Qcal}F_{\\Qcal\\leftarrow\\Pcal}\\Acal_{\\Pcal}\n=\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}F\\Scal_{\\Pcal}\\Acal_{\\Pcal}\n=F.\n\\]\nFor composition,\n\\[\nG_{\\Rcal\\leftarrow\\Qcal}F_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Rcal}G\\Scal_{\\Qcal}\\Acal_{\\Qcal}F\\Scal_{\\Pcal}\n=\n\\Acal_{\\Rcal}(G\\circ F)\\Scal_{\\Pcal}.\n\\]\n\\end{proof}","\\begin{corollary}[Dynamical equivalence]\\label{cor:dynamical_equivalence}\nSuppose $X=Y$ and the same packet system is used on both sides.  Packet coordinates preserve fixed points, periodic points, injectivity, surjectivity, and invertibility.  In particular,\n\\[\nF(x)=x\n\\quad\\Longleftrightarrow\\quad\nF_{\\Pcal\\leftarrow\\Pcal}(\\Acal_{\\Pcal}x)=\\Acal_{\\Pcal}x.\n\\]\n\\end{corollary}","\\begin{proof}\nThe packet map is conjugate to $F$ through the isomorphism $\\Acal_{\\Pcal}$.  Every listed property is invariant under conjugacy.\n\\end{proof}","\\begin{proposition}[Change of packet coordinates]\\label{prop:change_packet_coordinates}\nLet $\\Pcal$ and $\\Pcal'$ be packet systems on $X$.  Let $\\Qcal$ and $\\Qcal'$ be packet systems on $Y$.  For any two packet systems $\\Ucal$ and $\\Vcal$ on the same space, write\n\\[\nJ_{\\Ucal\\leftarrow\\Vcal}\n=\n\\Acal_{\\Ucal}\\Scal_{\\Vcal}.\n\\]\nThen\n\\[\nF_{\\Qcal'\\leftarrow\\Pcal'}\n=\nJ_{\\Qcal'\\leftarrow\\Qcal}\nF_{\\Qcal\\leftarrow\\Pcal}\nJ_{\\Pcal\\leftarrow\\Pcal'}.\n\\]\n\\end{proposition}","\\begin{proof}\nInsert $\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_Y$ and $\\Scal_{\\Pcal}\\Acal_{\\Pcal}=I_X$ into the definition of $F_{\\Qcal'\\leftarrow\\Pcal'}$.\n\\end{proof}","\\begin{example}[A mixed quadratic map]\\label{ex:mixed_quadratic}\nLet $X=Y=\\bbk^2$ with coordinate packets.  For\n\\[\nF(x_1,x_2)=(x_1x_2,x_1^2+x_2),\n\\]\nthe first output packet depends on both input packets.  The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second.  Packet coordinates record these interactions exactly.  Replacing $F$ by the two diagonal restrictions would lose the product $x_1x_2$.\n\\end{example}","egin{example}[A mixed quadratic map]\\label{ex:mixed_quadratic}\nLet $X=Y=\\bbk^2$ with coordinate packets.  For\n\\[\nF(x_1,x_2)=(x_1x_2,x_1^2+x_2),\n\\]\nthe first output packet depends on both input packets.  The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second.  Packet coordinates record these interactions exactly.  Replacing $F$ by the two diagonal restrictions would lose the product $x_1x_2$.\n\\end{example}","c}\nLet $X=Y=\\bbk^2$ with coordinate packets.  For\n\\[\nF(x_1,x_2)=(x_1x_2,x_1^2+x_2),\n\\]\nthe first output packet depends on both input packets.  The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second.  Packet coordinates record these interactions exactly.  Replacing $F$ by the two diagonal restrictions would lose the product $x_1x_2$.\n\\end{example}","Let $X$ and $Y$ be Banach spaces over the same field $\\mathbb R$ or $\\mathbb C$ and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr\\'echet derivatives and the Taylor integral below has a real parameter. Assume that $F:X\\to Y$ is $m$ times continuously Fr\\'echet differentiable. The same local statements hold on open domains containing the points and segments in question. For $1\\le r\\le m$, the derivative\n\\[\nD^rF(x):X^r\\longrightarrow Y\n\\]\nis a continuous symmetric multilinear map.","s over the same field $\\mathbb R$ or $\\mathbb C$ and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr\\'echet derivatives and the Taylor integral below has a real parameter. Assume that $F:X\\to Y$ is $m$ times continuously Fr\\'echet differentiable. The same local statements hold on open domains containing the points and segments in question. For $1\\le r\\le m$, the derivative\n\\[\nD^rF(x):X^r\\longrightarrow Y\n\\]\nis a continuous symmetric multilinear map.","R$ or $\\mathbb C$ and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr\\'echet derivatives and the Taylor integral below has a real parameter. Assume that $F:X\\to Y$ is $m$ times continuously Fr\\'echet differentiable. The same local statements hold on open domains containing the points and segments in question. For $1\\le r\\le m$, the derivative\n\\[\nD^rF(x):X^r\\longrightarrow Y\n\\]\nis a continuous symmetric multilinear map.","The range $X_\\alpha$ is closed because\n$X_\\alpha=\\ker(I-P_\\alpha)$. It is therefore Banach with its inherited norm. The same holds for every $Y_\\gamma$. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from \\cref{sec:nonlinear_coordinates} make them bounded linear isomorphisms. For a nonempty input family,\n$\\|\\Acal_{\\Pcal}\\|\\le\\max_\\alpha\\|P_\\alpha\\|$ and\n$\\|\\Scal_{\\Pcal}\\|\\le |I|$. An empty complete family can occur only on the zero space. Its coordinate product is the zero vector space with norm zero, and the coordinate identities hold directly.","It is therefore Banach with its inherited norm. The same holds for every $Y_\\gamma$. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from \\cref{sec:nonlinear_coordinates} make them bounded linear isomorphisms. For a nonempty input family,\n$\\|\\Acal_{\\Pcal}\\|\\le\\max_\\alpha\\|P_\\alpha\\|$ and\n$\\|\\Scal_{\\Pcal}\\|\\le |I|$. An empty complete family can occur only on the zero space. Its coordinate product is the zero vector space with norm zero, and the coordinate identities hold directly.","e same holds for every $Y_\\gamma$. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from \\cref{sec:nonlinear_coordinates} make them bounded linear isomorphisms. For a nonempty input family,\n$\\|\\Acal_{\\Pcal}\\|\\le\\max_\\alpha\\|P_\\alpha\\|$ and\n$\\|\\Scal_{\\Pcal}\\|\\le |I|$. An empty complete family can occur only on the zero space. Its coordinate product is the zero vector space with norm zero, and the coordinate identities hold directly.","\\begin{definition}[Packet jet]\\label{def:packet_jet}\nFor $1\\le r\\le m$, define\n\\[\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n=\nQ_\\gamma D^rF(x)\n\\big|_{X_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}}.\n\\]\nThis is a continuous multilinear map\n\\[\nX_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}\n\\longrightarrow\nY_\\gamma.\n\\]\n\\end{definition}","\\begin{theorem}[Exact reconstruction of derivatives]\\label{thm:jet_reconstruction}\nFor $h_1,\\ldots,h_r\\in X$,\n\\[\nD^rF(x)[h_1,\\ldots,h_r]\n=\n\\sum_{\\gamma}\n\\sum_{\\alpha_1,\\ldots,\\alpha_r}\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n[P_{\\alpha_1}h_1,\\ldots,P_{\\alpha_r}h_r].\n\\]\n\\end{theorem}","\\begin{proof}\nInsert $I_Y=\\sum_\\gamma Q_\\gamma$ on the output and $I_X=\\sum_\\alpha P_\\alpha$ in every input slot.  Multilinearity expands the finite sums.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{packet_jet_worked_example.pdf}\n\\caption{Second packet jets for $F(x,y)=(x+xy,y+x^2)$ at the origin. Coordinate axis projectors give $DF(0)=I$. The first output has mixed second derivatives equal to one, while the second has a repeated first input derivative equal to two. Each grid records the ordered pair of input packets. The mixed entries retain an interaction that diagonal restrictions would lose.}\n\\label{fig:packet-jet-example}\n\\end{figure}\n\\clearpage","\\begin{proposition}[Jet norm bound]\\label{prop:jet_norm_bound}\nGive every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate\n\\[\n\\left\\|J^r_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le \\|Q_\\gamma\\|\\,\\|D^rF(x)\\|.\n\\]\nIts ambient extension inserts $P_{\\alpha_j}$ in the $j$th slot and has norm at most\n$\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\\prod_j\\|P_{\\alpha_j}\\|$.\nConsequently the operator norm of the restricted packet jet also satisfies\n\\[\n\\left\\|J^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le\n\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\n\\prod_{j=1}^r\\|P_{\\alpha_j}\\|.\n\\]\nWhen all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound reduces to the norm of the full derivative.\n\\end{proposition}","cket range its inherited subspace norm. The restricted jet satisfies the sharper estimate\n\\[\n\\left\\|J^r_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le \\|Q_\\gamma\\|\\,\\|D^rF(x)\\|.\n\\]\nIts ambient extension inserts $P_{\\alpha_j}$ in the $j$th slot and has norm at most\n$\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\\prod_j\\|P_{\\alpha_j}\\|$.\nConsequently the operator norm of the restricted packet jet also satisfies\n\\[\n\\left\\|J^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le\n\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\n\\prod_{j=1}^r\\|P_{\\alpha_j}\\|.\n\\]\nWhen all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound reduces to the norm of the full derivative.\n\\end{proposition}","\\begin{proof}\nFor $h_j\\in X_{\\alpha_j}$, the inherited norm gives\n\\[\n\\|Q_\\gamma D^rF(x)[h_1,\\ldots,h_r]\\|\n\\le \\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\\prod_j\\|h_j\\|.\n\\]\nTaking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace $h_j$ by $P_{\\alpha_j}h_j$ and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its range. The displayed product bound therefore also follows for the restricted jet. If a zero packet is retained as a redundant index, multilinearity makes any jet with that input or output packet zero, so both estimates remain valid.\n\\end{proof}","\\begin{corollary}[Packet Taylor formula]\\label{cor:packet_taylor}\nAssume $F$ is $m+1$ times continuously Fr\\'echet differentiable on an open set containing the segment $x+th$ for $0\\le t\\le1$.  Then\n\\[\n\\begin{aligned}\nQ_\\gamma F(x+h)\n={}&Q_\\gamma F(x)\n+\n\\sum_{r=1}^{m}\\frac1{r!}\n\\sum_{\\alpha_1,\\ldots,\\alpha_r}\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n[P_{\\alpha_1}h,\\ldots,P_{\\alpha_r}h]","\\begin{corollary}[Packet Taylor formula]\\label{cor:packet_taylor}\nAssume $F$ is $m+1$ times continuously Fr\\'echet differentiable on an open set containing the segment $x+th$ for $0\\le t\\le1$.  Then\n\\[\n\\begin{aligned}\nQ_\\gamma F(x+h)\n={}&Q_\\gamma F(x)\n+\n\\sum_{r=1}^{m}\\frac1{r!}\n\\sum_{\\alpha_1,\\ldots,\\alpha_r}\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n[P_{\\alpha_1}h,\\ldots,P_{\\alpha_r}h]\\\\\n&+Q_\\gamma R_{m+1}(x,h),\n\\end{aligned}\n\\]\nwhere the Bochner integral remainder is\n\\[\nR_{m+1}(x,h)\n=\n\\frac1{m!}\n\\int_0^1(1-t)^m\nD^{m+1}F(x+th)[h,\\ldots,h]\\,dt.\n\\]\n\\end{corollary}","&+Q_\\gamma R_{m+1}(x,h),\n\\end{aligned}\n\\]\nwhere the Bochner integral remainder is\n\\[\nR_{m+1}(x,h)\n=\n\\frac1{m!}\n\\int_0^1(1-t)^m\nD^{m+1}F(x+th)[h,\\ldots,h]\\,dt.\n\\]\n\\end{corollary}","\\begin{proof}\nSet $g(t)=F(x+th)$ on the open preimage of the domain under $t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","$g(t)=F(x+th)$ on the open preimage of the domain under $t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","open preimage of the domain under $t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","Apply $Q_\\gamma$ to the identity and use \\cref{thm:jet_reconstruction}. Pairwise annihilation and idempotence retain exactly the output packet $\\gamma$. Boundedness of $Q_\\gamma$ allows it to pass through the Bochner integral. In particular, if\n$\\|D^{m+1}F(x+th)\\|\\le M$ along the segment, then\n\\[\n\\|Q_\\gamma R_{m+1}(x,h)\\|\n\\le \\frac{\\|Q_\\gamma\\|M\\|h\\|^{m+1}}{(m+1)!}.\n\\]\nFor $m=0$ the Taylor sum is empty. For $h=0$, multilinearity makes every positive-order term and the remainder zero.\n\\end{proof}","mma$ to the identity and use \\cref{thm:jet_reconstruction}. Pairwise annihilation and idempotence retain exactly the output packet $\\gamma$. Boundedness of $Q_\\gamma$ allows it to pass through the Bochner integral. In particular, if\n$\\|D^{m+1}F(x+th)\\|\\le M$ along the segment, then\n\\[\n\\|Q_\\gamma R_{m+1}(x,h)\\|\n\\le \\frac{\\|Q_\\gamma\\|M\\|h\\|^{m+1}}{(m+1)!}.\n\\]\nFor $m=0$ the Taylor sum is empty. For $h=0$, multilinearity makes every positive-order term and the remainder zero.\n\\end{proof}","The relation to derivatives of the complete coordinate representation is also literal. For an open domain $U\\subseteq X$, put\n$\\widehat F=\\Acal_{\\Qcal}\\circ F\\circ\\Scal_{\\Pcal}$ on the open coordinate set $\\Acal_{\\Pcal}(U)$. Boundedness and linearity of analysis and synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","on the open coordinate set $\\Acal_{\\Pcal}(U)$. Boundedness and linearity of analysis and synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","(U)$. Boundedness and linearity of analysis and synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","d synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","is imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","At a fixed base point, the nonzero packet jets form a directed hypergraph.  An order $r$ interaction has $r$ input packet labels and one output packet label.  The support therefore records interactions relative to the chosen packet system.","For a block $B=\\{i_1<\\cdots<i_s\\}$ and an intermediate packet $\\gamma$, write\n\\[\nG_{\\gamma,B}(x\\mid h)\n=\nJ^s_{\\gamma\\mid\\alpha_{i_1},\\ldots,\\alpha_{i_s}}G(x)\n[h_{i_1},\\ldots,h_{i_s}].\n\\]","\\begin{theorem}[Packet Fa\\`a di Bruno formula]\\label{thm:packet_faa}\nLet $n\\ge1$, let $X,Y,Z$ be Banach spaces over the same real or complex field, and let $G:X\\to Y$ and $F:Y\\to Z$ be $n$ times continuously Fr\\'echet differentiable. Let $\\Pcal$, $\\Qcal$, and $\\Rcal$ be packet systems on $X$, $Y$, and $Z$.  Fix input packets $\\alpha_1,\\ldots,\\alpha_n$, an output packet $\\delta$, and directions $h_i\\in X_{\\alpha_i}$.  Then\n\\[\n\\begin{aligned}\n&J^{n}_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_n}(F\\circ G)(x)\n[h_1,\\ldots,h_n]","di Bruno formula]\\label{thm:packet_faa}\nLet $n\\ge1$, let $X,Y,Z$ be Banach spaces over the same real or complex field, and let $G:X\\to Y$ and $F:Y\\to Z$ be $n$ times continuously Fr\\'echet differentiable. Let $\\Pcal$, $\\Qcal$, and $\\Rcal$ be packet systems on $X$, $Y$, and $Z$.  Fix input packets $\\alpha_1,\\ldots,\\alpha_n$, an output packet $\\delta$, and directions $h_i\\in X_{\\alpha_i}$.  Then\n\\[\n\\begin{aligned}\n&J^{n}_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_n}(F\\circ G)(x)\n[h_1,\\ldots,h_n]\\\\\n&=\n\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}\n\\sum_{\\gamma_1,\\ldots,\\gamma_k}\nJ^{k}_{\\delta\\mid\\gamma_1,\\ldots,\\gamma_k}F(Gx)\n\\big[\nG_{\\gamma_1,B_1}(x\\mid h),\\ldots,\nG_{\\gamma_k,B_k}(x\\mid h)\n\\big].\n\\end{aligned}\n\\]\nThe first sum runs over all set partitions of $\\{1,\\ldots,n\\}$.\n\\end{theorem}","&=\n\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}\n\\sum_{\\gamma_1,\\ldots,\\gamma_k}\nJ^{k}_{\\delta\\mid\\gamma_1,\\ldots,\\gamma_k}F(Gx)\n\\big[\nG_{\\gamma_1,B_1}(x\\mid h),\\ldots,\nG_{\\gamma_k,B_k}(x\\mid h)\n\\big].\n\\end{aligned}\n\\]\nThe first sum runs over all set partitions of $\\{1,\\ldots,n\\}$.\n\\end{theorem}","\\begin{proof}\nFirst derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","begin{proof}\nFirst derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","artition formula. At order one it is the ordinary chain rule. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","le. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","Insert $I_Y=\\sum_\\gamma Q_\\gamma$ in each argument of the outer derivative and expand the finite sums by multilinearity. Apply $R_\\delta$ to the output. The directions already lie in the declared input ranges, and each intermediate projection lands in its literal output range. The resulting summands are exactly the displayed packet jets. List directions within a block in increasing index order and blocks in increasing least-element order. Symmetry of the higher derivative makes the final value independent of the block ordering.\n\\end{proof}","\\begin{proposition}[Number of labeled composition terms]\\label{prop:faa_term_count}\nFor $n\\ge1$, if $Y$ has $M$ packets, then the number of packet labeled terms at order $n$ before zero jets are removed is\n\\[\nN_{n,M}\n=\n\\sum_{k=1}^{n}S(n,k)M^k,\n\\]\nwhere $S(n,k)$ is a Stirling number of the second kind.  Thus $N_{n,M}$ is the $n$th Touchard polynomial evaluated at $M$.\n\\end{proposition}","\\begin{proof}\nA set partition with $k$ blocks receives one independent intermediate packet label on each block.  It therefore gives $M^k$ labeled terms.  There are $S(n,k)$ such partitions.\n\\end{proof}","\\begin{corollary}[Structural pruning]\\label{cor:structural_pruning}\nFor a map $H$ with fixed input and output packet systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","pruning]\\label{cor:structural_pruning}\nFor a map $H$ with fixed input and output packet systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","with fixed input and output packet systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","t systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","ernal profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","\\begin{example}[The second-order composition law]\\label{ex:second_order_composition}\nAt order two, the partition formula has two shapes.  For packet directions $h_1$ and $h_2$,\n\\[\n\\begin{aligned}\nR_\\delta D^2(F\\circ G)(x)[h_1,h_2]\n={}&\n\\sum_{\\gamma_1,\\gamma_2}\nJ^2_{\\delta\\mid\\gamma_1,\\gamma_2}F(Gx)\n[J^1_{\\gamma_1\\mid\\alpha_1}G(x)h_1,J^1_{\\gamma_2\\mid\\alpha_2}G(x)h_2]","\\begin{example}[The second-order composition law]\\label{ex:second_order_composition}\nAt order two, the partition formula has two shapes.  For packet directions $h_1$ and $h_2$,\n\\[\n\\begin{aligned}\nR_\\delta D^2(F\\circ G)(x)[h_1,h_2]\n={}&\n\\sum_{\\gamma_1,\\gamma_2}\nJ^2_{\\delta\\mid\\gamma_1,\\gamma_2}F(Gx)\n[J^1_{\\gamma_1\\mid\\alpha_1}G(x)h_1,J^1_{\\gamma_2\\mid\\alpha_2}G(x)h_2]\\\\\n&+\n\\sum_\\gamma\nJ^1_{\\delta\\mid\\gamma}F(Gx)\n[J^2_{\\gamma\\mid\\alpha_1,\\alpha_2}G(x)[h_1,h_2]].\n\\end{aligned}\n\\]\nThe first line records two first-order changes entering a quadratic interaction of $F$.  The second records a genuine second-order interaction created inside $G$ before $F$ is applied.\n\\end{example}","&+\n\\sum_\\gamma\nJ^1_{\\delta\\mid\\gamma}F(Gx)\n[J^2_{\\gamma\\mid\\alpha_1,\\alpha_2}G(x)[h_1,h_2]].\n\\end{aligned}\n\\]\nThe first line records two first-order changes entering a quadratic interaction of $F$.  The second records a genuine second-order interaction created inside $G$ before $F$ is applied.\n\\end{example}","Let $X$ be a Banach space.  Fix $s\\in X$ and continuous multilinear maps\n\\[\nB_r:X^r\\longrightarrow X,\n\\qquad\n2\\le r\\le d.\n\\]\nSet\n\\[\na=\\|s\\|,\n\\qquad\nb_r=\\|B_r\\|,\n\\qquad\n\\phi(t)=a+\\sum_{r=2}^{d}b_rt^r.\n\\]\nAn ordered rooted tree has leaves labeled by $s$ and every internal vertex of arity $r$ labeled by $B_r$.  Its value is obtained by evaluating from the leaves toward the root.","\\begin{theorem}[Polynomial tree expansion]\\label{thm:polynomial_tree_expansion}\nSuppose there is $R>0$ such that\n\\[\n\\phi(R)\\le R\n\\]\nand\n\\[\nq=\n\\sum_{r=2}^{d}r b_rR^{r-1}<1.\n\\]\nThen the map\n\\[\n\\Phi(x)=s+\\sum_{r=2}^{d}B_r(x,\\ldots,x)\n\\]\nhas a unique fixed point in the closed ball $\\overline B_R(0)$.  The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.\n\\end{theorem}","\\begin{proof}\nFor $x,y\\in\\overline B_R(0)$, multilinearity and telescoping in the $r$ input positions give\n\\[\n\\|B_r(x,\\ldots,x)-B_r(y,\\ldots,y)\\|\n\\le\nr b_rR^{r-1}\\|x-y\\|.\n\\]\nHence $\\Phi$ is $q$-Lipschitz on the ball.  The inequality $\\phi(R)\\le R$ shows that $\\Phi$ preserves the ball.  The contraction mapping theorem gives a unique fixed point $u$ in that ball.","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  T","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative c","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bou","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\p","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tr","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolu","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","eaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","red subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","terate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","u\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","alar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","ight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","\\begin{corollary}[Quadratic Catalan expansion]\\label{cor:quadratic_tree_expansion}\nLet $B:X\\times X\\to X$ be bounded and consider\n\\[\nu=s+B(u,u).\n\\]\nPut $a=\\|s\\|$ and $b=\\|B\\|$.  If $4ab<1$, then\n\\[\nu=\n\\sum_{n\\ge1}\n\\sum_{\\substack{\\tau\\text{ full ordered binary tree}","\\begin{corollary}[Quadratic Catalan expansion]\\label{cor:quadratic_tree_expansion}\nLet $B:X\\times X\\to X$ be bounded and consider\n\\[\nu=s+B(u,u).\n\\]\nPut $a=\\|s\\|$ and $b=\\|B\\|$.  If $4ab<1$, then\n\\[\nu=\n\\sum_{n\\ge1}\n\\sum_{\\substack{\\tau\\text{ full ordered binary tree}\\\\\\text{with }n\\text{ leaves}}}\nB_\\tau(s)\n\\]\nconverges absolutely.  It also satisfies\n\\[\n\\|u\\|\n\\le\n\\frac{1-\\sqrt{1-4ab}}{2b}\n\\]\nwhen $b>0$.\n\\end{corollary}","text{with }n\\text{ leaves}}}\nB_\\tau(s)\n\\]\nconverges absolutely.  It also satisfies\n\\[\n\\|u\\|\n\\le\n\\frac{1-\\sqrt{1-4ab}}{2b}\n\\]\nwhen $b>0$.\n\\end{corollary}","\\begin{proof}\nIf $b=0$, the equation is $u=s$ and the tree series contains only its one leaf.  If $a=0$, every tree value is zero.  Choose any $R$ with $0<R<1/(2b)$ and apply \\cref{thm:polynomial_tree_expansion} to obtain uniqueness of the zero fixed point in that ball.","Assume now that $a>0$ and $b>0$.  Take\n\\[\nR=\n\\frac{1-\\sqrt{1-4ab}}{2b}.\n\\]\nThen $a+bR^2=R$ and $2bR<1$.  Apply \\cref{thm:polynomial_tree_expansion}.  The number of full ordered binary trees with $n$ leaves is the Catalan number $C_{n-1}$, which gives the displayed majorant.\n\\end{proof}","\\begin{theorem}[Packet labeled tree reconstruction]\\label{thm:packet_tree_reconstruction}\nLet $X$ carry a packet system $\\Pcal$.  At each internal vertex of a tree, insert the output identity and one input identity for every child.  Summing over all internal packet labels reconstructs the original tree value exactly.\n\\end{theorem}","\\begin{proof}\nAt a vertex labeled by an $r$ multilinear map $B_r$, insert\n\\[\nI_X=\\sum_\\beta P_\\beta\n\\]\non the output and in every input slot.  Multilinearity gives an exact finite sum of packet blocks at that vertex.  Apply this identity successively from the leaves toward the root.  Since the tree and the packet set are finite, all sums are finite and their total reconstructs the original tree value.\n\\end{proof}","The tree theorem separates combinatorial growth from analytic convergence.  The number of trees may grow rapidly, while multilinear bounds control their values. The scalar map $\\phi$ controls the total weight of the expansion.  Packet pruning changes the combinatorics but not the proof.  It removes trees whose values are already zero before any norm estimate is used.","\\begin{example}[Catalan parity packets]\\label{ex:catalan_parity}\nOver a field of characteristic zero, the formal equation\n\\[\nu=z+u^2\n\\]\nhas the branch\n\\[\nu(z)=\\frac{1-\\sqrt{1-4z}}2.\n\\]\nWrite $u=u_0+u_1$, where $u_0$ contains the even powers of $z$ and $u_1$ contains the odd powers.  Exact packet projection gives\n\\[\nu_0=u_0^2+u_1^2,\n\\qquad\nu_1=z+2u_0u_1.\n\\]\nThe two equations remain coupled.  Their sum is the original fixed point equation.\n\\end{example}","Let $G$ be a finite abelian group and let\n\\[\nA=\\bigoplus_{g\\in G}A_g\n\\]\nbe a $G$ graded module.","\\begin{definition}[Grade respecting operation]\\label{def:grade_respecting}\nAn $r$ multilinear map $\\mu:A^r\\to A$ respects the grading when\n\\[\n\\mu(A_{g_1},\\ldots,A_{g_r})\n\\subseteq\nA_{g_1+\\cdots+g_r}.\n\\]\n\\end{definition}","\\begin{theorem}[Graded packet convolution]\\label{thm:graded_convolution}\nIf $a_j=\\sum_{g\\in G}a_{j,g}$ and $\\mu$ respects the grading, then\n\\[\nP_k\\mu(a_1,\\ldots,a_r)\n=\n\\sum_{g_1+\\cdots+g_r=k}\n\\mu(a_{1,g_1},\\ldots,a_{r,g_r}).\n\\]\n\\end{theorem}","\\begin{proof}\nExpand every input into its homogeneous components.  Each term lands in the component whose label is the sum of its input labels.  Projection onto $A_k$ retains exactly the displayed terms.\n\\end{proof}","\\begin{corollary}[Cyclic product formula]\\label{cor:cyclic_product}\nFor a cyclically graded algebra\n\\[\nA=\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}A_j,\n\\qquad\nA_iA_j\\subseteq A_{i+j},\n\\]\none has\n\\[\nP_k(ab)=\n\\sum_{i+j=k}a_ib_j.\n\\]\n\\end{corollary}","\\begin{theorem}[Packet Maurer Cartan equations]\\label{thm:packet_mc}\nLet\n\\[\n\\Lcal=\\bigoplus_{g\\in G}\\Lcal_g\n\\]\nbe a differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\nd\\Lcal_g\\subseteq\\Lcal_g,\n\\qquad\n[\\Lcal_g,\\Lcal_h]\\subseteq\\Lcal_{g+h}.\n\\]\nFor a degree one element $\\eta=\\sum_g\\eta_g$, the Maurer Cartan equation\n\\[\nd\\eta+\\frac12[\\eta,\\eta]=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\frac12\n\\sum_{g+h=k}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k\\in G$.\n\\end{theorem}","t_mc}\nLet\n\\[\n\\Lcal=\\bigoplus_{g\\in G}\\Lcal_g\n\\]\nbe a differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\nd\\Lcal_g\\subseteq\\Lcal_g,\n\\qquad\n[\\Lcal_g,\\Lcal_h]\\subseteq\\Lcal_{g+h}.\n\\]\nFor a degree one element $\\eta=\\sum_g\\eta_g$, the Maurer Cartan equation\n\\[\nd\\eta+\\frac12[\\eta,\\eta]=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\frac12\n\\sum_{g+h=k}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k\\in G$.\n\\end{theorem}","\\begin{proof}\nThe differential preserves the group label and the bracket adds labels.  Apply the projector onto $\\Lcal_k$ and use exact reconstruction.\n\\end{proof}","\\begin{proposition}[Packet equations for $L_\\infty$ operations]\\label{prop:packet_l_infinity}\nLet $\\Lcal$ be complete and carry operations $\\ell_r$ that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges.  Assume the Maurer Cartan series\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\\ell_r(\\eta,\\ldots,\\eta)\n\\]\nconverges and the required factorials are invertible.  Then its packet of label $k$ is\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\n\\sum_{g_1+\\cdots+g_r=k}\n\\ell_r(\\eta_{g_1},\\ldots,\\eta_{g_r}).\n\\]\nThe full Maurer Cartan equation holds if and only if every packet equation vanishes.\n\\end{proposition}","ions]\\label{prop:packet_l_infinity}\nLet $\\Lcal$ be complete and carry operations $\\ell_r$ that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges.  Assume the Maurer Cartan series\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\\ell_r(\\eta,\\ldots,\\eta)\n\\]\nconverges and the required factorials are invertible.  Then its packet of label $k$ is\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\n\\sum_{g_1+\\cdots+g_r=k}\n\\ell_r(\\eta_{g_1},\\ldots,\\eta_{g_r}).\n\\]\nThe full Maurer Cartan equation holds if and only if every packet equation vanishes.\n\\end{proposition}","bel{prop:packet_l_infinity}\nLet $\\Lcal$ be complete and carry operations $\\ell_r$ that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges.  Assume the Maurer Cartan series\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\\ell_r(\\eta,\\ldots,\\eta)\n\\]\nconverges and the required factorials are invertible.  Then its packet of label $k$ is\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\n\\sum_{g_1+\\cdots+g_r=k}\n\\ell_r(\\eta_{g_1},\\ldots,\\eta_{g_r}).\n\\]\nThe full Maurer Cartan equation holds if and only if every packet equation vanishes.\n\\end{proposition}","\\begin{proof}\nApply \\cref{thm:graded_convolution} to each operation $\\ell_r$.  Continuity of the packet projectors permits the convergent sums to be projected term by term.\n\\end{proof}","\\begin{definition}[Divided square]\\label{def:divided_square}\nA divided square on degree one elements is a map $q$ satisfying\n\\[\nq(x+y)=q(x)+q(y)+[x,y],\n\\qquad\nq(ax)=a^2q(x),\n\\]\nwhenever the expressions are defined.\n\\end{definition}","\\begin{proposition}[Divided square packet equations]\\label{prop:divided_square_packets}\nAssume $q(\\Lcal_g)\\subseteq\\Lcal_{2g}$.  Choose an ordering of the finite label set.  The equation\n\\[\nd\\eta+q(\\eta)=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\sum_{2g=k}q(\\eta_g)\n+\n\\sum_{\\substack{g<h","\\begin{proposition}[Divided square packet equations]\\label{prop:divided_square_packets}\nAssume $q(\\Lcal_g)\\subseteq\\Lcal_{2g}$.  Choose an ordering of the finite label set.  The equation\n\\[\nd\\eta+q(\\eta)=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\sum_{2g=k}q(\\eta_g)\n+\n\\sum_{\\substack{g<h\\\\g+h=k}}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k$.\n\\end{proposition}","g+h=k}}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k$.\n\\end{proposition}","\\begin{proof}\nRepeated use of the polarization identity for $q$ gives\n\\[\nq\\left(\\sum_g\\eta_g\\right)\n=\n\\sum_gq(\\eta_g)\n+\n\\sum_{g<h}[\\eta_g,\\eta_h].\n\\]\nProject onto the component of label $k$.\n\\end{proof}","\\begin{example}[Matrix difference grading]\\label{ex:matrix_difference_grading}\nLet $A=M_N(\\bbk)$ and assign the matrix unit $E_{rs}$ the label $r-s$ modulo $N$.  Since\n\\[\nE_{rs}E_{uv}=\\delta_{su}E_{rv},\n\\]\nthe output label is\n\\[\n(r-s)+(u-v)=r-v\n\\]\nwhenever the product is nonzero.  Matrix multiplication therefore obeys the cyclic convolution law exactly.  The same grading resolves commutators and polynomial matrix equations.\n\\end{example}","Let\n\\[\nF_f:X_f\\longrightarrow Y_f,\n\\qquad\nF_c:X_c\\longrightarrow Y_c\n\\]\nbe fine and coarse maps.  Let\n\\[\nC_X:X_f\\longrightarrow X_c,\n\\qquad\nC_Y:Y_f\\longrightarrow Y_c\n\\]\nbe linear coarsening maps.  Whenever a norm, a Lipschitz constant, or a Fr\\'echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","\\begin{definition}[Nonlinear refinement defect]\\label{def:nonlinear_defect}\nThe pointwise defect of $F$ is\n\\[\n\\Delta_F(x)\n=\nC_YF_f(x)-F_c(C_Xx).\n\\]\nThe two models are exactly compatible when $\\Delta_F$ vanishes everywhere.\n\\end{definition}","The defect is a vector in $Y_c$.  It determines an obstruction class after a correction map into $Y_c$ has been chosen and the defect has been passed to the corresponding cokernel.  Norm estimates measure the size of the defect.  The quotient determines whether an allowed linear correction can remove it.","\\begin{theorem}[Exact defect under composition]\\label{thm:defect_composition_identity}\nLet $G_f:Y_f\\to Z_f$ and $G_c:Y_c\\to Z_c$, with coarsening $C_Z:Z_f\\to Z_c$.  Then\n\\[\n\\Delta_{G\\circ F}(x)\n=\n\\Delta_G(F_f(x))\n+\nG_c\\big(F_c(C_Xx)+\\Delta_F(x)\\big)\n-\nG_c\\big(F_c(C_Xx)\\big).\n\\]\nIf $G_c$ is $L$-Lipschitz on the relevant set, then\n\\[\n\\|\\Delta_{G\\circ F}(x)\\|\n\\le\n\\|\\Delta_G(F_f(x))\\|\n+\nL\\|\\Delta_F(x)\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nAdd and subtract $G_c(C_YF_f(x))$.  Use\n\\[\nC_YF_f(x)=F_c(C_Xx)+\\Delta_F(x).\n\\]\nThe norm estimate follows from the Lipschitz bound.\n\\end{proof}","\\begin{corollary}[Linear product rule]\\label{cor:linear_defect_product}\nIf all maps in \\cref{thm:defect_composition_identity} are linear, then\n\\[\n\\Delta_{GF}\n=\n\\Delta_G F_f+G_c\\Delta_F.\n\\]\n\\end{corollary}","\\begin{proof}\nLinearity turns the last difference in \\cref{thm:defect_composition_identity} into $G_c\\Delta_F(x)$.\n\\end{proof}","\\begin{theorem}[Iterated orbit defect]\\label{thm:iterated_orbit_defect}\nLet $F_f:X_f\\to X_f$ and $F_c:X_c\\to X_c$, with coarsening $C:X_f\\to X_c$.  Suppose $F_c$ is $q$-Lipschitz.  For a fine orbit starting at $x$, assume\n\\[\n\\|\\Delta_F(F_f^j x)\\|\\le\\varepsilon_j.\n\\]\nThen\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\sum_{j=0}^{n-1}q^{n-1-j}\\varepsilon_j.\n\\]\nIf $0\\le q<1$ and every $\\varepsilon_j\\le\\varepsilon$, then\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\varepsilon\\frac{1-q^n}{1-q}.\n\\]\n\\end{theorem}","rated orbit defect]\\label{thm:iterated_orbit_defect}\nLet $F_f:X_f\\to X_f$ and $F_c:X_c\\to X_c$, with coarsening $C:X_f\\to X_c$.  Suppose $F_c$ is $q$-Lipschitz.  For a fine orbit starting at $x$, assume\n\\[\n\\|\\Delta_F(F_f^j x)\\|\\le\\varepsilon_j.\n\\]\nThen\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\sum_{j=0}^{n-1}q^{n-1-j}\\varepsilon_j.\n\\]\nIf $0\\le q<1$ and every $\\varepsilon_j\\le\\varepsilon$, then\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\varepsilon\\frac{1-q^n}{1-q}.\n\\]\n\\end{theorem}","\\begin{proof}\nLet\n\\[\ne_n=\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|.\n\\]\nThe definition of the defect and the Lipschitz property give\n\\[\ne_{n+1}\\le\\varepsilon_n+qe_n.\n\\]\nIterating this recurrence proves the first estimate.  The second is the geometric sum.\n\\end{proof}","\\begin{theorem}[Fixed point stability]\\label{thm:fixed_point_stability}\nAssume $F_c$ is a contraction with constant $q<1$.  Let $x_f$ and $x_c$ be fixed points of $F_f$ and $F_c$.  If\n\\[\n\\|\\Delta_F(x_f)\\|\\le\\varepsilon,\n\\]\nthen\n\\[\n\\|Cx_f-x_c\\|\n\\le\n\\frac{\\varepsilon}{1-q}.\n\\]\nWhen the defect at $x_f$ is zero, the fixed points are compatible.\n\\end{theorem}","\\begin{proof}\nUsing the fixed-point equations,\n\\[\n\\begin{aligned}\n\\|Cx_f-x_c\\|\n&\\le\n\\|CF_f(x_f)-F_c(Cx_f)\\|\n+\n\\|F_c(Cx_f)-F_c(x_c)\\|\\\\\n&\\le\n\\varepsilon+q\\|Cx_f-x_c\\|.\n\\end{aligned}\n\\]\nRearrange.\n\\end{proof}","\\begin{corollary}[Several refinement levels]\\label{cor:multilevel_fixed_points}\nLet $C_\\ell:X_{\\ell+1}\\to X_\\ell$ for $0\\le\\ell<m$.  Suppose $x_\\ell$ is a fixed point at level $\\ell$, the coarse map at level $\\ell$ has contraction constant $q_\\ell<1$, and the adjacent defect at $x_{\\ell+1}$ is at most $\\varepsilon_\\ell$.  Write\n\\[\nC_{a,b}=C_aC_{a+1}\\cdots C_{b-1}\n\\quad\\text{for }a<b,\n\\qquad\nC_{a,a}=I.\n\\]\nThen\n\\[\n\\|C_{0,m}x_m-x_0\\|\n\\le\n\\sum_{\\ell=0}^{m-1}\n\\|C_{0,\\ell}\\|\n\\frac{\\varepsilon_\\ell}{1-q_\\ell}.\n\\]\n\\end{corollary}","y}[Several refinement levels]\\label{cor:multilevel_fixed_points}\nLet $C_\\ell:X_{\\ell+1}\\to X_\\ell$ for $0\\le\\ell<m$.  Suppose $x_\\ell$ is a fixed point at level $\\ell$, the coarse map at level $\\ell$ has contraction constant $q_\\ell<1$, and the adjacent defect at $x_{\\ell+1}$ is at most $\\varepsilon_\\ell$.  Write\n\\[\nC_{a,b}=C_aC_{a+1}\\cdots C_{b-1}\n\\quad\\text{for }a<b,\n\\qquad\nC_{a,a}=I.\n\\]\nThen\n\\[\n\\|C_{0,m}x_m-x_0\\|\n\\le\n\\sum_{\\ell=0}^{m-1}\n\\|C_{0,\\ell}\\|\n\\frac{\\varepsilon_\\ell}{1-q_\\ell}.\n\\]\n\\end{corollary}","\\begin{proof}\nInsert the intermediate vectors $C_{0,\\ell}x_\\ell$ and telescope.  Apply \\cref{thm:fixed_point_stability} to every adjacent pair.\n\\end{proof}","\\begin{proposition}[Derivative defects]\\label{prop:derivative_defects}\nAssume $F_f$ and $F_c$ are $r$ times Fr\\'echet differentiable and the coarsening maps are bounded and linear.  Then\n\\[\nD\\Delta_F(x)\n=\nC_YDF_f(x)-DF_c(C_Xx)C_X.\n\\]\nFor $r\\ge2$,\n\\[\nD^r\\Delta_F(x)[h_1,\\ldots,h_r]\n=\nC_YD^rF_f(x)[h_1,\\ldots,h_r]\n-\nD^rF_c(C_Xx)[C_Xh_1,\\ldots,C_Xh_r].\n\\]\n\\end{proposition}","\\begin{proof}\nDifferentiate the defining identity for $\\Delta_F$.  The coarsening maps are linear, so their higher derivatives vanish.\n\\end{proof}","\\begin{example}[Sharp affine bound]\\label{ex:sharp_affine_bound}\nLet $F_c(y)=qy+b$ with $0<q<1$.  Choose a fine fixed point whose coarsening has constant defect $\\varepsilon$ in the direction of the scalar state.  The fixed point mismatch is exactly $\\varepsilon/(1-q)$.  Thus the constant in \\cref{thm:fixed_point_stability} cannot be improved without more information.\n\\end{example}","\\begin{definition}[Linear correction datum]\\label{def:linear_correction_datum}\nA linear correction datum consists of a homomorphism\n\\[\nD:U\\longrightarrow V\n\\]\nand a defect $r\\in V$.  A correction is an element $u\\in U$ satisfying\n\\[\nDu=-r.\n\\]\nThe obstruction class is\n\\[\n\\Ob_D(r)=[r]\\in\\coker D=V/\\im D.\n\\]\n\\end{definition}","\\begin{theorem}[Cokernel obstruction criterion]\\label{thm:cokernel_obstruction}\nA linear correction exists if and only if\n\\[\n\\Ob_D(r)=0.\n\\]\nThe quotient map\n\\[\nq_D:V\\longrightarrow\\coker D\n\\]\nhas the following universal property.  If $\\varphi:V\\to W$ is linear and $\\varphi D=0$, then there is a unique linear map\n\\[\n\\overline\\varphi:\\coker D\\longrightarrow W\n\\]\nwith $\\varphi=\\overline\\varphi q_D$.\n\\end{theorem}","\\begin{proof}\nThe class of $r$ is zero exactly when $r\\in\\im D$, which is equivalent to solvability of $Du=-r$.  If $\\varphi D=0$, then $\\varphi$ vanishes on $\\im D$.  It therefore descends uniquely to the quotient.\n\\end{proof}","The universal property explains why the cokernel is the canonical obstruction object relative to $D$.  Any linear obstruction test that vanishes on correctable defects factors through it.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{correction_cokernel_plane.pdf}\n\\caption{A specified correction range in $\\mathbb R^2$. For $L(a,b)=(a,0)$, the image is the horizontal axis. The defect $(2,3)$ splits into $(2,0)$ in that image and the remaining representative $(0,3)$. Subtracting the image component leaves the same cokernel class. The functional $(x,y)\\mapsto y$ vanishes on the correction range and detects the value three. The correction equation $Lu=-(2,3)$ therefore has no solution.}\n\\label{fig:correction-cokernel-example}\n\\end{figure}\n\\clearpage","_plane.pdf}\n\\caption{A specified correction range in $\\mathbb R^2$. For $L(a,b)=(a,0)$, the image is the horizontal axis. The defect $(2,3)$ splits into $(2,0)$ in that image and the remaining representative $(0,3)$. Subtracting the image component leaves the same cokernel class. The functional $(x,y)\\mapsto y$ vanishes on the correction range and detects the value three. The correction equation $Lu=-(2,3)$ therefore has no solution.}\n\\label{fig:correction-cokernel-example}\n\\end{figure}\n\\clearpage","\\begin{theorem}[Naturality of obstruction classes]\\label{thm:obstruction_naturality}\nSuppose\n\\[\n\\begin{array}{ccc}\nU & \\xrightarrow{D} & V\\\\\n\\mathllap{a}\\downarrow & & \\downarrow\\mathrlap{b}\\\\\nU' & \\xrightarrow{D'} & V'\n\\end{array}\n\\]\nis a commuting square.  Then there is a unique induced map\n\\[\n\\overline b:\\coker D\\longrightarrow\\coker D'\n\\]\nsatisfying\n\\[\n\\overline b\\,\\Ob_D(r)=\\Ob_{D'}(br).\n\\]\n\\end{theorem}","U' & \\xrightarrow{D'} & V'\n\\end{array}\n\\]\nis a commuting square.  Then there is a unique induced map\n\\[\n\\overline b:\\coker D\\longrightarrow\\coker D'\n\\]\nsatisfying\n\\[\n\\overline b\\,\\Ob_D(r)=\\Ob_{D'}(br).\n\\]\n\\end{theorem}","\\begin{proof}\nThe relation $bD=D'a$ shows that the quotient map $q_{D'}b$ vanishes on $\\im D$.  Apply the universal property in \\cref{thm:cokernel_obstruction}.\n\\end{proof}","\\begin{theorem}[Packet decomposition of an obstruction]\\label{thm:packet_obstruction_decomposition}\nSuppose\n\\[\nU=\\bigoplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","of an obstruction]\\label{thm:packet_obstruction_decomposition}\nSuppose\n\\[\nU=\\bigoplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","truction_decomposition}\nSuppose\n\\[\nU=\\bigoplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","goplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","\\begin{proof}\nThe grading assumption gives\n\\[\n\\im D=\\bigoplus_\\alpha\\im D_\\alpha.\n\\]\nTaking the quotient of the direct sum by this direct sum of images gives the displayed isomorphism.\n\\end{proof}","\\begin{proposition}[Projectors on the obstruction quotient]\\label{prop:obstruction_quotient_projectors}\nLet $(Q_\\alpha)_{\\alpha\\in I}$ be a complete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","obstruction quotient]\\label{prop:obstruction_quotient_projectors}\nLet $(Q_\\alpha)_{\\alpha\\in I}$ be a complete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","n quotient]\\label{prop:obstruction_quotient_projectors}\nLet $(Q_\\alpha)_{\\alpha\\in I}$ be a complete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","lete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","\\begin{proof}\nIf $v-v'\\in\\im D$, then\n\\[\nQ_\\alpha(v-v')\\in\\im D.\n\\]\nHence $[v]\\mapsto[Q_\\alpha v]$ is well defined on the quotient.  The idempotent, annihilation, and reconstruction identities descend from $V$ to $V/\\im D$.  The formula for an obstruction class follows from the definition.\n\\end{proof}","\\begin{theorem}[Relative obstruction exact sequence]\\label{thm:relative_obstruction_sequence}\nSuppose the following diagram of modules commutes and has exact rows:\n\\[\n\\begin{array}{ccccccccc}\n0&\\longrightarrow&U_0&\\xrightarrow{i_U}&U_f&\\xrightarrow{p_U}&U_c&\\longrightarrow&0\\\\[4pt]\n&&\\mathllap{D_0}\\downarrow&&\\mathllap{D_f}\\downarrow&&\\mathllap{D_c}\\downarrow&&\\\\[4pt]\n0&\\longrightarrow&V_0&\\xrightarrow{i_V}&V_f&\\xrightarrow{p_V}&V_c&\\longrightarrow&0.\n\\end{array}\n\\]\nThere is a natural exact sequence\n\\[\n\\begin{aligned}\n0&\\longrightarrow\\Ker D_0\n\\longrightarrow\\Ker D_f\n\\longrightarrow\\Ker D_c\n\\xrightarrow{\\delta}\\coker D_0","[4pt]\n0&\\longrightarrow&V_0&\\xrightarrow{i_V}&V_f&\\xrightarrow{p_V}&V_c&\\longrightarrow&0.\n\\end{array}\n\\]\nThere is a natural exact sequence\n\\[\n\\begin{aligned}\n0&\\longrightarrow\\Ker D_0\n\\longrightarrow\\Ker D_f\n\\longrightarrow\\Ker D_c\n\\xrightarrow{\\delta}\\coker D_0\\\\\n&\\longrightarrow\\coker D_f\n\\longrightarrow\\coker D_c\n\\longrightarrow0.\n\\end{aligned}\n\\]\nFor $u_c\\in\\Ker D_c$, choose $u_f\\in U_f$ with $p_Uu_f=u_c$.  There is a unique $v_0\\in V_0$ such that\n\\[\nD_fu_f=i_Vv_0,\n\\]\nand the connecting map is\n\\[\n\\delta(u_c)=[v_0].\n\\]\n\\end{theorem}","&\\longrightarrow\\coker D_f\n\\longrightarrow\\coker D_c\n\\longrightarrow0.\n\\end{aligned}\n\\]\nFor $u_c\\in\\Ker D_c$, choose $u_f\\in U_f$ with $p_Uu_f=u_c$.  There is a unique $v_0\\in V_0$ such that\n\\[\nD_fu_f=i_Vv_0,\n\\]\nand the connecting map is\n\\[\n\\delta(u_c)=[v_0].\n\\]\n\\end{theorem}","\\begin{proof}\nSince $D_cu_c=0$, commutativity gives\n\\[\np_VD_fu_f=D_cp_Uu_f=0.\n\\]\nExactness of the lower row gives a unique $v_0$ with $D_fu_f=i_Vv_0$.  A different lift has the form $u_f+i_Uu_0$.  The corresponding element of $V_0$ changes by $D_0u_0$, so $[v_0]$ is independent of the lift.","The map $\\Ker D_0\\to\\Ker D_f$ is injective because $i_U$ is injective.  If $u_f\\in\\Ker D_f$ maps to zero in $U_c$, then $u_f=i_Uu_0$ for a unique $u_0$.  The identity\n\\[\ni_VD_0u_0=D_fi_Uu_0=0\n\\]\nand injectivity of $i_V$ show that $u_0\\in\\Ker D_0$.  This proves exactness at $\\Ker D_f$.","If $u_c$ comes from an element of $\\Ker D_f$, one may choose that element as its lift and obtain $\\delta(u_c)=0$.  Conversely, if $\\delta(u_c)=0$, then $v_0=D_0u_0$ for some $u_0$.  Replacing $u_f$ by $u_f-i_Uu_0$ gives an element of $\\Ker D_f$ that maps to $u_c$.  This proves exactness at $\\Ker D_c$.","The class $[v_0]\\in\\coker D_0$ maps to zero in $\\coker D_f$ whenever it equals $\\delta(u_c)$, since the chosen lift satisfies $i_Vv_0=D_fu_f$.  Conversely, if $i_Vv_0=D_fu_f$, then $u_c=p_Uu_f$ lies in $\\Ker D_c$ and $\\delta(u_c)=[v_0]$.  This proves exactness at $\\coker D_0$.","If a class in $\\coker D_f$ comes from $\\coker D_0$, its image in $\\coker D_c$ is zero.  Conversely, suppose $[v_f]$ maps to zero in $\\coker D_c$.  Choose $u_c$ with $D_cu_c=p_Vv_f$ and lift it to $u_f\\in U_f$.  Then\n\\[\np_V(v_f-D_fu_f)=0,\n\\]\nso $v_f-D_fu_f=i_Vv_0$ for some $v_0\\in V_0$.  Thus $[v_f]$ comes from $[v_0]$.  Surjectivity at the last term follows from surjectivity of $p_V$.\n\\end{proof}","\\begin{corollary}[Coarse vanishing and fine vanishing]\\label{cor:coarse_fine_obstruction}\nLet $[r_f]\\in\\coker D_f$.  If its image in $\\coker D_c$ is zero, then $[r_f]$ comes from $\\coker D_0$.  Coarse vanishing therefore means that the fine obstruction is supported in information removed by coarsening.  It does not imply $[r_f]=0$.\n\\end{corollary}","\\begin{proof}\nThis is exactness of the sequence in \\cref{thm:relative_obstruction_sequence} at $\\coker D_f$.\n\\end{proof}","\\begin{example}[A fine obstruction erased by coarsening]\\label{ex:coarse_cancellation}\nTake all three correction spaces to be zero.  Let\n\\[\nV_f=\\bbk^2,\n\\qquad\nV_c=\\bbk,\n\\qquad\np_V(x,y)=x+y,\n\\]\nand let\n\\[\nV_0=\\Ker p_V=\\Span\\{(1,-1)\\}.\n\\]\nThe fine defect $(1,-1)$ is a nonzero class in $\\coker D_f=V_f$, while its coarse image is zero.  The class comes from $\\coker D_0=V_0$, exactly as \\cref{cor:coarse_fine_obstruction} predicts.\n\\end{example}","\\begin{definition}[Mapping cone]\\label{def:mapping_cone}\nLet $D:C^\\bullet\\to E^\\bullet$ be a cochain map.  Its mapping cone is\n\\[\n\\Cone(D)^n=E^n\\oplus C^{n+1}\n\\]\nwith differential\n\\[\nd_{\\Cone}(e,c)=(d_Ee+Dc,-d_Cc).\n\\]\n\\end{definition}","\\begin{theorem}[Cohomological obstruction through the cone]\\label{thm:cone_obstruction}\nLet $e\\in E^n$ satisfy $d_Ee=0$.  The class of $(e,0)$ in\n\\[\nH^n(\\Cone(D))\n\\]\nvanishes if and only if the class $[e]\\in H^n(E)$ lies in the image of\n\\[\nH^n(D):H^n(C)\\longrightarrow H^n(E).\n\\]\nWhen $C$ and $E$ are concentrated in degree zero,\n\\[\nH^0(\\Cone(D))\\cong\\coker D.\n\\]\n\\end{theorem}","\\begin{proof}\nThe class of $(e,0)$ vanishes exactly when there are $e'\\in E^{n-1}$ and $c\\in C^n$ with\n\\[\n(e,0)=d_{\\Cone}(e',c).\n\\]\nThis means\n\\[\ne=d_Ee'+Dc,\n\\qquad\nd_Cc=0.\n\\]\nEquivalently, $[e]=H^n(D)[c]$.  The degree zero statement follows directly from the definition of the cone differential.\n\\end{proof}","\\begin{corollary}[Packet obstruction complexes]\\label{cor:packet_obstruction_complexes}\nIf $C^\\bullet$ and $E^\\bullet$ split into finite packet subcomplexes and $D$ preserves those packets, then $\\Cone(D)$ splits into packet subcomplexes and\n\\[\nH^n(\\Cone(D))\n\\cong\n\\bigoplus_\\alpha H^n(\\Cone(D_\\alpha)).\n\\]\n\\end{corollary}","\\begin{example}[A two component obstruction]\\label{ex:two_component_obstruction}\nLet\n\\[\nD:\\bbk\\longrightarrow\\bbk^2,\n\\qquad\nD(t)=(t,t).\n\\]\nThe cokernel is one dimensional.  The defect $(a,b)$ is correctable exactly when $a=b$.  The obstruction can be represented by $a-b$, but this scalar representative depends on a choice of isomorphism from the cokernel to $\\bbk$.  The quotient class itself is canonical.\n\\end{example}","\\begin{theorem}[Orthogonal obstruction projector]\\label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore--Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","onal obstruction projector]\\label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore--Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","ction projector]\\label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore--Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore--Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore--Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","$D^\\dagger$ be its Moore--Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","gger$ be its Moore--Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nClosed range gives the orthogonal decomposition\n\\[\nH_2=\\im D\\oplus\\Ker D^*.\n\\]\nThe Moore--Penrose identities show that $DD^\\dagger$ is the orthogonal projector onto $\\im D$.  The stated equivalences and distance formula follow from this decomposition.  The least norm property is the standard characterization of $D^\\dagger r$.\n\\end{proof}","\\begin{theorem}[Right inverse correction theorem]\\label{thm:right_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$-Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","begin{theorem}[Right inverse correction theorem]\\label{thm:right_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$-Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","em}[Right inverse correction theorem]\\label{thm:right_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$-Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$-Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$-Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","\\begin{proof}\nThe space $X_R$ is closed.  Indeed, if $Rz_n\\to x$, then $z_n=DRz_n\\to Dx$, so $x=R(Dx)$.  Define\n\\[\n\\Psi(x)=R(y-N(x)).\n\\]\nThis map takes $X_R$ to itself.  For $x\\in\\overline B_\\rho(0)\\cap X_R$,\n\\[\n\\begin{aligned}\n\\|\\Psi(x)\\|\n&\\le\n\\|R(y-N(0))\\|\n+\n\\|R\\|\\,\\|N(x)-N(0)\\|","\\begin{proof}\nThe space $X_R$ is closed.  Indeed, if $Rz_n\\to x$, then $z_n=DRz_n\\to Dx$, so $x=R(Dx)$.  Define\n\\[\n\\Psi(x)=R(y-N(x)).\n\\]\nThis map takes $X_R$ to itself.  For $x\\in\\overline B_\\rho(0)\\cap X_R$,\n\\[\n\\begin{aligned}\n\\|\\Psi(x)\\|\n&\\le\n\\|R(y-N(0))\\|\n+\n\\|R\\|\\,\\|N(x)-N(0)\\|\\\\\n&\\le\n(1-q)\\rho+q\\rho\n=\\rho.\n\\end{aligned}\n\\]\nAlso,\n\\[\n\\|\\Psi(x)-\\Psi(x')\\|\n\\le q\\|x-x'\\|.\n\\]\nThe contraction mapping theorem gives a unique fixed point in the stated set.  Applying $D$ to the fixed point identity gives the equation.  Conversely, a solution in $X_R$ equals $R(Dx)$ and is therefore a fixed point of $\\Psi$.\n\\end{proof}","&\\le\n(1-q)\\rho+q\\rho\n=\\rho.\n\\end{aligned}\n\\]\nAlso,\n\\[\n\\|\\Psi(x)-\\Psi(x')\\|\n\\le q\\|x-x'\\|.\n\\]\nThe contraction mapping theorem gives a unique fixed point in the stated set.  Applying $D$ to the fixed point identity gives the equation.  Conversely, a solution in $X_R$ equals $R(Dx)$ and is therefore a fixed point of $\\Psi$.\n\\end{proof}","\\begin{corollary}[Stability with respect to the target]\\label{cor:correction_stability}\nUnder the hypotheses of \\cref{thm:right_inverse_correction}, let $x$ and $x'$ be the fixed points associated with targets $y$ and $y'$.  If both fixed point maps preserve the same ball, then\n\\[\n\\|x-x'\\|\n\\le\n\\frac{\\|R\\|}{1-q}\\|y-y'\\|.\n\\]\n\\end{corollary}","\\begin{proof}\nSubtract the two fixed-point equations and use the Lipschitz bound for $N$:\n\\[\n\\|x-x'\\|\n\\le\n\\|R\\|\\|y-y'\\|+q\\|x-x'\\|.\n\\]\n\\end{proof}","\\begin{theorem}[Local Lyapunov--Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points","\\begin{theorem}[Local Lyapunov--Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[","\\begin{theorem}[Local Lyapunov--Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\","\\begin{theorem}[Local Lyapunov--Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the b","\\begin{theorem}[Local Lyapunov--Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,","\\begin{theorem}[Local Lyapunov--Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,","\\begin{theorem}[Local Lyapunov--Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","hmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","ion]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","et $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","en neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","ro and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","en Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","me there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","\\begin{proof}\nDefine\n\\[\nG(k,x_1)=\\pi_RF(k+x_1).\n\\]\nThe derivative of $G$ in the $X_1$ variable at $(0,0)$ is $L_1$, which is invertible.  The implicit function theorem gives the unique map $\\psi$.  Every nearby $x$ has a unique decomposition $x=k+x_1$.  The range equation forces $x_1=\\psi(k)$, and the remaining equation is precisely $\\kappa(k)=0$.","Differentiate $G(k,\\psi(k))=0$ at zero.  Since $L$ vanishes on $K$, one obtains $L_1D\\psi(0)=0$, hence $D\\psi(0)=0$.  Finally,\n\\[\nD\\kappa(0)=\\pi_OL|_K=0.\n\\]\n\\end{proof}","\\begin{corollary}[Quadratic reduced obstruction]\\label{cor:quadratic_reduced_obstruction}\nAssume the hypotheses of \\cref{thm:lyapunov_schmidt} with $r\\ge2$.  For $k_1,k_2\\in K$,\n\\[\nD^2\\kappa(0)[k_1,k_2]\n=\n\\pi_O D^2F(0)[k_1,k_2].\n\\]\n\\end{corollary}","\\begin{proof}\nDifferentiate $\\kappa(k)=\\pi_OF(k+\\psi(k))$ twice.  The terms containing $D\\psi(0)$ vanish.  The term containing $D^2\\psi(0)$ is multiplied by $\\pi_OL$, which is zero because $L(X)=R$.\n\\end{proof}","\\begin{example}[A purely quadratic obstruction]\\label{ex:pure_quadratic_obstruction}\nLet\n\\[\nF(x,y)=(y,x^2)\n\\]\non $\\mathbb R^2$.  At the origin,\n\\[\nL(h,k)=(k,0).\n\\]\nThe kernel is the $x$ axis, the range is the first output axis, and the cokernel is the second output axis.  The range equation forces $y=0$.  The reduced obstruction map is\n\\[\n\\kappa(x)=x^2.\n\\]\nThe linear obstruction vanishes for every kernel direction, but the nonlinear equation still has only the solution $x=0$ near the origin.\n\\end{example}","truction}\nLet\n\\[\nF(x,y)=(y,x^2)\n\\]\non $\\mathbb R^2$.  At the origin,\n\\[\nL(h,k)=(k,0).\n\\]\nThe kernel is the $x$ axis, the range is the first output axis, and the cokernel is the second output axis.  The range equation forces $y=0$.  The reduced obstruction map is\n\\[\n\\kappa(x)=x^2.\n\\]\nThe linear obstruction vanishes for every kernel direction, but the nonlinear equation still has only the solution $x=0$ near the origin.\n\\end{example}","\\begin{remark}\\label{rem:higher_obstruction}\nThe reduced map $\\kappa$ begins at quadratic order.  This is why vanishing of the first-order cokernel class need not settle a nonlinear problem.  Higher terms can produce a genuine obstruction on the kernel directions.\n\\end{remark}","Let $X$ and $Y$ be modules with decreasing filtrations\n\\[\nX=F^0X\\supseteq F^1X\\supseteq\\cdots,\n\\qquad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","g filtrations\n\\[\nX=F^0X\\supseteq F^1X\\supseteq\\cdots,\n\\qquad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","\\cdots,\n\\qquad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","uad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","Suppose $x_m$ satisfies\n\\[\n\\Phi(x_m)\\in F^mY.\n\\]\nAssume that corrections $h\\in F^mX$ obey a first-order congruence\n\\[\n\\Phi(x_m+h)\n\\equiv\n\\Phi(x_m)+L_mh\n\\pmod{F^{m+1}Y},\n\\]\nwhere $L_m:F^mX\\to F^mY$ is a module homomorphism satisfying\n\\[\nL_m(F^{m+1}X)\\subseteq F^{m+1}Y\n\\]\nand therefore induces\n\\[\n\\overline L_m:\nF^mX/F^{m+1}X\n\\longrightarrow\nF^mY/F^{m+1}Y.\n\\]","At depth $m$, only the associated graded correction is visible. The cokernel below has finite length whenever the target graded piece has finite length, even if the completed problem is infinite. Vanishing of the depth obstruction decides whether the current approximation can be improved by one level. When the graded pieces are finite, the branching count below applies. Finite length alone does not imply finite cardinality.","\\begin{definition}[Depth obstruction]\\label{def:depth_obstruction}\nThe depth $m$ obstruction of $x_m$ is\n\\[\n\\Ob_m(x_m)\n=\n[\\Phi(x_m)]\n\\in\n\\coker\\overline L_m,\n\\]\nwhere $\\Phi(x_m)$ is first viewed in $F^mY/F^{m+1}Y$.\n\\end{definition}","\\begin{theorem}[One-step lifting criterion]\\label{thm:one_step_lifting}\nThe approximation $x_m$ has a correction $h_m\\in F^mX$ such that\n\\[\n\\Phi(x_m+h_m)\\in F^{m+1}Y\n\\]\nif and only if\n\\[\n\\Ob_m(x_m)=0.\n\\]\nWhen a correction exists, the set of its classes in\n\\[\nF^mX/F^{m+1}X\n\\]\nis a coset of $\\Ker\\overline L_m$.  If the graded pieces are vector spaces, this coset is an affine space.\n\\end{theorem}","\\begin{proof}\nThe next depth condition is equivalent to\n\\[\n\\overline L_m[h_m]\n=-[\\Phi(x_m)]\n\\]\nin $F^mY/F^{m+1}Y$.  This is solvable exactly when the displayed defect has zero cokernel class.  If $h_m$ and $h_m'$ are two solutions, their difference lies in $\\Ker\\overline L_m$.  Adding an element of the kernel to one solution gives another, so the solution set is a coset of the kernel.\n\\end{proof}","\\begin{corollary}[Finite branching count]\\label{cor:finite_branch_count}\nSuppose the graded pieces are finite vector spaces over a finite field.  If the depth obstruction vanishes, then the number of correction classes at depth $m$ is\n\\[\n|\\Ker\\overline L_m|.\n\\]\nIf the obstruction does not vanish, the number is zero.\n\\end{corollary}","\\begin{lemma}[Well-defined lift classes]\\label{lem:well_defined_lift_classes}\nUnder the filtration congruence assumption, the condition\n\\[\n\\Phi(x)\\in F^mY\n\\]\ndepends only on the class of $x$ in $X/F^mX$.  Fix a representative $x_m$ of a depth $m$ class.  The map\n\\[\n[h]\\longmapsto[x_m+h]_{m+1}\n\\]\nidentifies correction classes $[h]\\in F^mX/F^{m+1}X$ with the classes at depth $m+1$ that reduce to $[x_m]_m$.  Under this identification, the solutions of the associated graded correction equation in \\cref{thm:one_step_lifting} are exactly the children whose defect lies in $F^{m+1}Y$.  A different representative of the same parent gives a translated parametrization of the same set of children.\n\\end{lemma}","the class of $x$ in $X/F^mX$.  Fix a representative $x_m$ of a depth $m$ class.  The map\n\\[\n[h]\\longmapsto[x_m+h]_{m+1}\n\\]\nidentifies correction classes $[h]\\in F^mX/F^{m+1}X$ with the classes at depth $m+1$ that reduce to $[x_m]_m$.  Under this identification, the solutions of the associated graded correction equation in \\cref{thm:one_step_lifting} are exactly the children whose defect lies in $F^{m+1}Y$.  A different representative of the same parent gives a translated parametrization of the same set of children.\n\\end{lemma}","\\begin{proof}\nIf $x'-x\\in F^mX$, then filtration congruence preservation gives\n\\[\n\\Phi(x')-\\Phi(x)\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first-order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselv","\\begin{proof}\nIf $x'-x\\in F^mX$, then filtration congruence preservation gives\n\\[\n\\Phi(x')-\\Phi(x)\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first-order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","'-x\\in F^mX$, then filtration congruence preservation gives\n\\[\n\\Phi(x')-\\Phi(x)\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first-order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}",")\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first-order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first-order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","$m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first-order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","\\begin{definition}[Lift tree]\\label{def:lift_tree}\nFix an initial approximation.  A vertex at depth $m$ is a class $[x]_m\\in X/F^mX$ with $\\Phi(x)\\in F^mY$ that reduces to the fixed initial class.  A depth $m+1$ vertex is joined to its reduction in $X/F^mX$.  By \\cref{lem:well_defined_lift_classes}, this rooted graph is independent of the chosen representatives.  It is called the lift tree.\n\\end{definition}","\\begin{theorem}[Finitely branching lift tree]\\label{thm:finitely_branching_lift_tree}\nAssume every level of the lift tree is nonempty and every vertex has only finitely many children.  Then the tree contains an infinite branch.  If $X$ and $Y$ are separated and complete and $\\Phi$ is continuous, that branch determines an exact solution of $\\Phi(x)=0$.\n\\end{theorem}","Choose representatives recursively.  Once $x_m$ represents the selected depth $m$ vertex, choose a representative $x_{m+1}$ of its selected child with\n\\[\nx_{m+1}-x_m\\in F^mX.\n\\]\nThe resulting sequence is Cauchy for the filtration topology.  Each representative satisfies $\\Phi(x_m)\\in F^mY$.  The conclusion now follows from \\cref{thm:complete_filtered_lifting}.\n\\end{proof}","\\begin{corollary}[Unique lift branch]\\label{cor:unique_lift_branch}\nSuppose an initial approximation is fixed, every depth obstruction vanishes along its descendants, and every map $\\overline L_m$ has zero kernel.  Then at most one compatible lift branch extends the initial approximation.\n\\end{corollary}","\\begin{proof}\nBy \\cref{thm:one_step_lifting}, the correction classes at each depth form a coset of $\\Ker\\overline L_m$.  A zero kernel leaves at most one correction class, so induction gives at most one branch.\n\\end{proof}","\\begin{theorem}[Complete filtered lifting]\\label{thm:complete_filtered_lifting}\nAssume there is a compatible sequence\n\\[\nx_{m+1}\\equiv x_m\\pmod{F^mX}\n\\]\nwith\n\\[\n\\Phi(x_m)\\in F^mY\n\\]\nfor every $m$.  Then $x_m$ converges to an element $x_\\infty\\in X$ and\n\\[\n\\Phi(x_\\infty)=0.\n\\]\nIn particular, if the obstruction vanishes at every depth and compatible corrections are chosen, an exact solution exists.\n\\end{theorem}","\\begin{proof}\nFor every fixed $r$, the congruences imply\n\\[\nx_n-x_m\\in F^rX\n\\]\nwhenever $n\\ge m\\ge r$.  Thus $(x_m)$ is Cauchy for the filtration topology.  Completeness gives a limit $x_\\infty$.","Fix $r$.  For every $m\\ge r$, one has $\\Phi(x_m)\\in F^mY\\subseteq F^rY$.  The submodule $F^rY$ is open and therefore closed in the filtration topology.  Continuity of $\\Phi$ gives $\\Phi(x_m)\\to\\Phi(x_\\infty)$, so $\\Phi(x_\\infty)\\in F^rY$.  Since this holds for every $r$ and $Y$ is separated,\n\\[\n\\Phi(x_\\infty)\\in\\bigcap_rF^rY=0.\n\\]\n\\end{proof}","\\begin{theorem}[Naturality of filtered obstructions]\\label{thm:filtered_obstruction_naturality}\nSuppose a morphism of filtered lifting problems sends $x_m$ to $x_m'$, intertwines $L_m$ with $L_m'$, and sends $\\Phi(x_m)$ to $\\Phi'(x_m')$ on the associated graded pieces.  Then the induced cokernel map sends\n\\[\n\\Ob_m(x_m)\n\\quad\\text{to}\\quad\n\\Ob_m'(x_m').\n\\]\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:obstruction_naturality} to the induced commuting square on the depth $m$ graded pieces.\n\\end{proof}","\\begin{example}[Three lifting outcomes]\\label{ex:three_lifting_outcomes}\nThe scalar equation $x^2-2=0$ at $x=3$ modulo $7$ has invertible derivative.  It has one correction modulo $49$ and gives $x=10$ modulo $49$.","For $x^2-3=0$ at $x=0$ modulo $3$, the derivative is zero and the defect class is nonzero.  No correction exists.","For $x^2=0$ at $x=0$ modulo $3$, the derivative and defect both vanish.  Every correction digit lies in the kernel, so the lift branches.\n\\end{example}","\\begin{proposition}[Maurer Cartan lifting obstruction]\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","tion}[Maurer Cartan lifting obstruction]\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","lifting obstruction]\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","e.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","uad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","\\begin{proof}\nDirect expansion gives\n\\[\n\\Omega(\\eta+h)\n=\n\\Omega(\\eta)+dh+[\\eta,h]+\\frac12[h,h].\n\\]\nThe filtration rule places $[h,h]$ in $F^{2m}\\Lcal^2$, which is contained in $F^{m+1}\\Lcal^2$ because $m\\ge1$.  The congruence follows.  The last statement is \\cref{thm:one_step_lifting} applied to the induced graded correction map.\n\\end{proof}","\\begin{standardfact}[Spectral sequence of a finite filtered complex]\\label{fact:filtered_spectral_sequence}\nLet $C^\\bullet$ be a cochain complex with a finite decreasing filtration by subcomplexes\n\\[\nC^\\bullet=F^0C^\\bullet\\supseteq F^1C^\\bullet\\supseteq\\cdots\\supseteq F^{N+1}C^\\bullet=0.\n\\]\nThere is a natural spectral sequence with\n\\[\nE_0^{p,q}=\\gr^pC^{p+q},\n\\qquad\nE_1^{p,q}=H^{p+q}(\\gr^pC^\\bullet),\n\\]\nwhich converges to the associated graded object of $H^{p+q}(C^\\bullet)$ with its induced filtration.  A filtered cochain map induces a morphism of spectral sequences.\n\\end{standardfact}","d complex]\\label{fact:filtered_spectral_sequence}\nLet $C^\\bullet$ be a cochain complex with a finite decreasing filtration by subcomplexes\n\\[\nC^\\bullet=F^0C^\\bullet\\supseteq F^1C^\\bullet\\supseteq\\cdots\\supseteq F^{N+1}C^\\bullet=0.\n\\]\nThere is a natural spectral sequence with\n\\[\nE_0^{p,q}=\\gr^pC^{p+q},\n\\qquad\nE_1^{p,q}=H^{p+q}(\\gr^pC^\\bullet),\n\\]\nwhich converges to the associated graded object of $H^{p+q}(C^\\bullet)$ with its induced filtration.  A filtered cochain map induces a morphism of spectral sequences.\n\\end{standardfact}","This is the standard construction for a filtered complex, in the form given in \\cite{Weibel1994,McCleary2001}. When $C^\\bullet$ is a correction or deformation complex, the groups on its pages can be read as graded obstruction candidates. For an arbitrary filtered complex, they are cohomology classes of the associated graded problem.","\\begin{proposition}[Fine obstruction survival]\\label{prop:fine_obstruction_survival}\nA class on the first page contributes to the total obstruction only if it is represented by a permanent cycle, so it is neither killed nor identified as a boundary on any later page.  If the spectral sequence degenerates at the first page, then $H^n(C^\\bullet)=0$ exactly when $E_1^{p,n-p}=0$ for every $p$.\n\\end{proposition}","\\begin{proof}\nConvergence identifies the last page with the associated graded object of the total cohomology.  A first page class contributes to that object exactly when it determines a nonzero class on the last page.  If the sequence degenerates at the first page, the associated graded pieces of $H^n(C^\\bullet)$ are the groups $E_1^{p,n-p}$.  A finite filtered module is zero exactly when all its associated graded pieces are zero.\n\\end{proof}","\\begin{corollary}[Direct packet complexes]\\label{cor:direct_packet_no_cancellation}\nSuppose\n\\[\nC^\\bullet=\\bigoplus_{\\alpha\\in I}C_\\alpha^\\bullet\n\\]\nis a finite direct sum of subcomplexes.  Then\n\\[\nH^n(C^\\bullet)\n\\cong\n\\bigoplus_{\\alpha\\in I}H^n(C_\\alpha^\\bullet).\n\\]\nA total cohomology class vanishes if and only if every packet class vanishes.\n\\end{corollary}","\\begin{example}[A fine class killed at the next page]\\label{ex:spectral_cancellation}\nLet $C^0=\\bbk x$ and $C^1=\\bbk y$, with differential $d(x)=y$.  Give $C^0$ the filtration\n\\[\nF^0C^0=\\bbk x,\n\\qquad\nF^1C^0=0,\n\\]\nand give $C^1$ the filtration\n\\[\nF^0C^1=F^1C^1=\\bbk y,\n\\qquad\nF^2C^1=0.\n\\]\nThe associated graded differential is zero in the relevant positions, so both $x$ and $y$ give nonzero classes on the next page.  The next differential sends the class of $x$ to the class of $y$.  The total complex is acyclic.  Thus nonzero fine obstruction classes can cancel through a later refinement differential.\n\\end{example}","x:spectral_cancellation}\nLet $C^0=\\bbk x$ and $C^1=\\bbk y$, with differential $d(x)=y$.  Give $C^0$ the filtration\n\\[\nF^0C^0=\\bbk x,\n\\qquad\nF^1C^0=0,\n\\]\nand give $C^1$ the filtration\n\\[\nF^0C^1=F^1C^1=\\bbk y,\n\\qquad\nF^2C^1=0.\n\\]\nThe associated graded differential is zero in the relevant positions, so both $x$ and $y$ give nonzero classes on the next page.  The next differential sends the class of $x$ to the class of $y$.  The total complex is acyclic.  Thus nonzero fine obstruction classes can cancel through a later refinement differential.\n\\end{example}","ion}\nLet $C^0=\\bbk x$ and $C^1=\\bbk y$, with differential $d(x)=y$.  Give $C^0$ the filtration\n\\[\nF^0C^0=\\bbk x,\n\\qquad\nF^1C^0=0,\n\\]\nand give $C^1$ the filtration\n\\[\nF^0C^1=F^1C^1=\\bbk y,\n\\qquad\nF^2C^1=0.\n\\]\nThe associated graded differential is zero in the relevant positions, so both $x$ and $y$ give nonzero classes on the next page.  The next differential sends the class of $x$ to the class of $y$.  The total complex is acyclic.  Thus nonzero fine obstruction classes can cancel through a later refinement differential.\n\\end{example}","Let $U,V,Y_P,Y_Q$ be Banach spaces. On an open neighborhood of $(u_0,v_0)\\in U\\times V$, split the variables and equations as\n\\[\nx=(u,v),\n\\qquad\nF(u,v)=\n\\begin{pmatrix}\nF_P(u,v)","Let $U,V,Y_P,Y_Q$ be Banach spaces. On an open neighborhood of $(u_0,v_0)\\in U\\times V$, split the variables and equations as\n\\[\nx=(u,v),\n\\qquad\nF(u,v)=\n\\begin{pmatrix}\nF_P(u,v)\\\\\nF_Q(u,v)\n\\end{pmatrix}.\n\\]\nAssume $F$ takes values in $Y_P\\times Y_Q$, is twice continuously Fr\\'echet differentiable near $(u_0,v_0)$, satisfies $F_Q(u_0,v_0)=0$, and\n\\[\nA=D_vF_Q(u_0,v_0)\n\\]\nis a Banach space isomorphism from $V$ to $Y_Q$. The implicit function theorem gives a local map $v=\\varphi(u)$, with $\\varphi(u_0)=v_0$, satisfying\n\\[\nF_Q(u,\\varphi(u))=0.\n\\]\nDefine the reduced equation\n\\[\nR(u)=F_P(u,\\varphi(u)).\n\\]","F_Q(u,v)\n\\end{pmatrix}.\n\\]\nAssume $F$ takes values in $Y_P\\times Y_Q$, is twice continuously Fr\\'echet differentiable near $(u_0,v_0)$, satisfies $F_Q(u_0,v_0)=0$, and\n\\[\nA=D_vF_Q(u_0,v_0)\n\\]\nis a Banach space isomorphism from $V$ to $Y_Q$. The implicit function theorem gives a local map $v=\\varphi(u)$, with $\\varphi(u_0)=v_0$, satisfying\n\\[\nF_Q(u,\\varphi(u))=0.\n\\]\nDefine the reduced equation\n\\[\nR(u)=F_P(u,\\varphi(u)).\n\\]","\\begin{theorem}[Exact local elimination]\\label{thm:exact_local_elimination}\nAfter the neighborhoods are chosen small enough, the equation\n\\[\nF(u,v)=0\n\\]\nis equivalent to\n\\[\nv=\\varphi(u)\n\\qquad\\text{and}\\qquad\nR(u)=0.\n\\]\nThus eliminating the hidden variable loses no local solutions.\n\\end{theorem}","\\begin{proof}\nThe hidden equation $F_Q(u,v)=0$ has the unique local solution $v=\\varphi(u)$ by the implicit function theorem.  After this identity is imposed, the remaining component of $F$ is exactly $R(u)$.\n\\end{proof}","\\begin{theorem}[Nonlinear Schur derivative]\\label{thm:schur_derivative}\nAt $(u_0,v_0)$,\n\\[\nDR\n=\nD_uF_P\n-\nD_vF_P A^{-1}D_uF_Q.\n\\]\n\\end{theorem}","\\begin{proof}\nDifferentiate $F_Q(u,\\varphi(u))=0$:\n\\[\nD_uF_Q+A D\\varphi=0.\n\\]\nThus\n\\[\nD\\varphi=-A^{-1}D_uF_Q.\n\\]\nSubstitute this into\n\\[\nDR=D_uF_P+D_vF_PD\\varphi.\n\\]\n\\end{proof}","\\begin{theorem}[Second Schur derivative]\\label{thm:schur_second_derivative}\nFor retained directions $h,k$, set\n\\[\n\\widehat h=(h,D\\varphi h),\n\\qquad\n\\widehat k=(k,D\\varphi k).\n\\]\nThen\n\\[\nD^2\\varphi[h,k]\n=\n-A^{-1}D^2F_Q[\\widehat h,\\widehat k]\n\\]\nand\n\\[\nD^2R[h,k]\n=\nD^2F_P[\\widehat h,\\widehat k]\n-\nD_vF_P A^{-1}D^2F_Q[\\widehat h,\\widehat k].\n\\]\nAll derivatives on the right are evaluated at $(u_0,v_0)$.\n\\end{theorem}","\\begin{proof}\nDifferentiate the identity $F_Q(u,\\varphi(u))=0$ twice.  The chain rule gives\n\\[\nD^2F_Q[\\widehat h,\\widehat k]\n+A D^2\\varphi[h,k]\n=0.\n\\]\nThis proves the first formula.  Differentiate $R(u)=F_P(u,\\varphi(u))$ twice and substitute the first formula.\n\\end{proof}","\\begin{corollary}[Packet jets of the reduced equation]\\label{cor:reduced_packet_jets}\nIf the retained and hidden spaces carry packet systems, then the first and second packet jets of $R$ are obtained by projecting the formulas in \\cref{thm:schur_derivative,thm:schur_second_derivative}.  Hidden packet feedback is carried by the factors involving $A^{-1}$.\n\\end{corollary}","\\begin{example}[Two retained variables and one hidden variable]\\label{ex:finite_schur}\nSuppose the derivative of $F$ at a solution has block form\n\\[\nDF=\n\\begin{pmatrix}\nA&B","\\begin{example}[Two retained variables and one hidden variable]\\label{ex:finite_schur}\nSuppose the derivative of $F$ at a solution has block form\n\\[\nDF=\n\\begin{pmatrix}\nA&B\\\\\nC&D\n\\end{pmatrix},\n\\]\nwhere $D$ is invertible.  Eliminating the hidden variable replaces the retained derivative $A$ by\n\\[\nA-BD^{-1}C.\n\\]\nThe feedback term is exact.  Keeping only $A$ would ignore the path from retained variables into the hidden variable and back.\n\\end{example}","C&D\n\\end{pmatrix},\n\\]\nwhere $D$ is invertible.  Eliminating the hidden variable replaces the retained derivative $A$ by\n\\[\nA-BD^{-1}C.\n\\]\nThe feedback term is exact.  Keeping only $A$ would ignore the path from retained variables into the hidden variable and back.\n\\end{example}","Let $\\Gamma=(V,E)$ be a finite connected graph and fix a vector space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph t","Let $\\Gamma=(V,E)$ be a finite connected graph and fix a vector space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","e connected graph and fix a vector space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","or space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","nsport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","\\begin{definition}[Gauge change]\\label{def:gauge_change}\nA gauge is a family $G_v\\in\\operatorname{GL}(X)$.  It changes the edge transports to\n\\[\nT'_{vu}=G_vT_{vu}G_u^{-1}.\n\\]\n\\end{definition}","\\begin{proposition}[Gauge covariance]\\label{prop:gauge_covariance}\nFor a loop $\\gamma$ based at $v_0$,\n\\[\n\\Hol'(\\gamma)\n=\nG_{v_0}\\Hol(\\gamma)G_{v_0}^{-1}.\n\\]\nThus the conjugacy class of each loop holonomy is gauge invariant.\n\\end{proposition}","\\begin{theorem}[Spanning tree normalization]\\label{thm:tree_normalization}\nChoose a root and a spanning tree $\\Tcal$ of $\\Gamma$.  There is a gauge in which every tree edge transport is the identity.  In that gauge, the transport on each edge outside the tree is the holonomy of its fundamental cycle, with the orientation used to define that cycle.\n\\end{theorem}","\\begin{corollary}[Number of independent cycle tests]\\label{cor:cycle_number}\nFor a connected graph with $|V|$ vertices and $|E|$ unoriented edges, a spanning tree leaves\n\\[\n|E|-|V|+1\n\\]\nfundamental cycle holonomies.\n\\end{corollary}","\\begin{proof}\nA spanning tree has $|V|-1$ edges.  Every remaining edge creates one fundamental cycle.\n\\end{proof}","Let $A$ be a transport system on copies of a fixed vector space $V_0$, and let $B$ be a transport system on copies of a fixed vector space $W_0$.  Fix a root $r$ and an isomorphism\n\\[\nU_r:V_0\\longrightarrow W_0.\n\\]","\\begin{theorem}[Fundamental holonomy intertwining criterion]\\label{thm:holonomy_intertwining}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item There is a family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","omy_intertwining}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item There is a family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","begin{enumerate}\n\\item There is a family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","Conversely, choose a spanning tree.  Starting from $U_r$, propagate $U_v$ along the unique tree path by the rule\n\\[\nU_v=B_{vu}U_uA_{vu}^{-1}\n\\]\non every tree edge.  This is the only possible extension satisfying the tree edge equations.  An edge outside the tree closes one fundamental loop.  Its intertwining equation holds exactly when $U_r$ intertwines the two corresponding fundamental holonomies.  The second condition therefore supplies every remaining edge equation.  Uniqueness follows from propagation along the tree.\n\\end{proof}","\\begin{theorem}[Gauge classification of transports]\\label{thm:gauge_classification}\nLet $G\\subseteq\\operatorname{GL}(X)$ be the chosen structure group and fix a root $r$.  Restrict both edge transports and gauge values to $G$.  Gauge equivalence classes of $G$ valued transport systems are naturally identified with\n\\[\n\\Hom(\\pi_1(\\Gamma,r),G)/G,\n\\]\nwhere $G$ acts by conjugation.  After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of\n\\[\nb_1=|E|-|V|+1\n\\]\ntuples in $G$.\n\\end{theorem}","$ be the chosen structure group and fix a root $r$.  Restrict both edge transports and gauge values to $G$.  Gauge equivalence classes of $G$ valued transport systems are naturally identified with\n\\[\n\\Hom(\\pi_1(\\Gamma,r),G)/G,\n\\]\nwhere $G$ acts by conjugation.  After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of\n\\[\nb_1=|E|-|V|+1\n\\]\ntuples in $G$.\n\\end{theorem}","up and fix a root $r$.  Restrict both edge transports and gauge values to $G$.  Gauge equivalence classes of $G$ valued transport systems are naturally identified with\n\\[\n\\Hom(\\pi_1(\\Gamma,r),G)/G,\n\\]\nwhere $G$ acts by conjugation.  After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of\n\\[\nb_1=|E|-|V|+1\n\\]\ntuples in $G$.\n\\end{theorem}","\\begin{proof}\nA transport system assigns a group element to every oriented edge, with inversion under reversal.  The product along a based loop gives a homomorphism from the free group $\\pi_1(\\Gamma,r)$ to $G$.  A gauge change conjugates this homomorphism by the gauge value at the root.  Conversely, normalize a transport system on a spanning tree.  The remaining edge values prescribe the images of a free basis of $\\pi_1(\\Gamma,r)$, and every homomorphism arises in this way.  A further gauge preserving the tree normalization is constant at all vertices and acts by simultaneous conjugation.  The number of non-tree edges is $b_1$.\n\\end{proof}","ansport system assigns a group element to every oriented edge, with inversion under reversal.  The product along a based loop gives a homomorphism from the free group $\\pi_1(\\Gamma,r)$ to $G$.  A gauge change conjugates this homomorphism by the gauge value at the root.  Conversely, normalize a transport system on a spanning tree.  The remaining edge values prescribe the images of a free basis of $\\pi_1(\\Gamma,r)$, and every homomorphism arises in this way.  A further gauge preserving the tree normalization is constant at all vertices and acts by simultaneous conjugation.  The number of non-tree edges is $b_1$.\n\\end{proof}","jugates this homomorphism by the gauge value at the root.  Conversely, normalize a transport system on a spanning tree.  The remaining edge values prescribe the images of a free basis of $\\pi_1(\\Gamma,r)$, and every homomorphism arises in this way.  A further gauge preserving the tree normalization is constant at all vertices and acts by simultaneous conjugation.  The number of non-tree edges is $b_1$.\n\\end{proof}","\\begin{corollary}[Abelian cohomology class]\\label{cor:abelian_holonomy_class}\nWhen the structure group $A$ is abelian, gauge classes of transports are naturally\n\\[\nH^1(\\Gamma,A)\n\\cong\n\\Hom(H_1(\\Gamma,\\mathbb Z),A).\n\\]\nAfter a basis of $H_1(\\Gamma,\\mathbb Z)$ is chosen, this group is identified with $A^{b_1}$.\n\\end{corollary}","\\begin{proof}\nAn edge assignment is an $A$ valued one cochain.  A gauge change multiplies it by a coboundary.  A graph has no two cells, so every one cochain is a cocycle.  The usual computation of the first cohomology of a connected graph gives $A^{b_1}$.\n\\end{proof}","Finite graph transport has a continuous counterpart.  Let $H$ be a Hilbert space and let\n\\[\n\\Pcal(t)=(P_\\alpha(t))_{\\alpha\\in I}\n\\]\nbe a complete orthogonal packet system with finitely many labels. Assume that it is continuously differentiable in operator norm for $t\\in[0,T]$, where $T\\ge0$. Here every $P_\\alpha(t)$ is a bounded self-adjoint projection, so $P_\\alpha(t)^*=P_\\alpha(t)=P_\\alpha(t)^2$.","Zero projections are permitted in this section and contribute zero to all generator sums. When $T=0$, every partition product is the empty product $I$. Completeness and orthogonality give\n\\[\nP_\\alpha(t)P_\\beta(t)=0\n\\quad\\text{for }\\alpha\\ne\\beta,\n\\qquad\n\\sum_\\alpha P_\\alpha(t)=I.\n\\]\nDefine the canonical generator\n\\[\nK(t)=\\sum_\\alpha \\dot P_\\alpha(t)P_\\alpha(t).\n\\]","\\begingroup\n\\setlength{\\abovedisplayskip}{6pt plus 2pt minus 2pt}\n\\setlength{\\belowdisplayskip}{6pt plus 2pt minus 2pt}\n\\begin{lemma}[Complete packet commutator identity]\\label{lem:packet_commutator}\nFor every packet label $\\beta$,\n\\[\n\\dot P_\\beta(t)=[K(t),P_\\beta(t)].\n\\]\nThe generator is also skew-adjoint:\n\\[\nK(t)^*=-K(t).\n\\]\n\\end{lemma}","\\begin{proof}\nDifferentiating $P_\\alpha^2=P_\\alpha$ gives\n\\[\nP_\\alpha\\dot P_\\alpha P_\\alpha=0.\n\\]\nDifferentiating $P_\\beta P_\\alpha=0$ for $\\alpha\\ne\\beta$ gives\n\\[\nP_\\beta\\dot P_\\alpha=-\\dot P_\\beta P_\\alpha.\n\\]\nHence\n\\[\nK P_\\beta=\\dot P_\\beta P_\\beta\n\\]\nand\n\\[\nP_\\beta K=-\\dot P_\\beta(I-P_\\beta).\n\\]\nTheir difference is $\\dot P_\\beta$.  Also,\n\\[\nK+K^*\n=\n\\sum_\\alpha(\\dot P_\\alpha P_\\alpha+P_\\alpha\\dot P_\\alpha)\n=\n\\sum_\\alpha\\dot P_\\alpha\n=0.\n\\]\n\\end{proof}\n\\endgroup","\\begin{theorem}[Canonical packet transport]\\label{thm:kato_transport}\nLet $U(t)$ solve\n\\[\n\\dot U(t)=K(t)U(t),\n\\qquad\nU(0)=I.\n\\]\nThen $U(t)$ is unitary and transports every packet exactly:\n\\[\nP_\\alpha(t)U(t)=U(t)P_\\alpha(0)\n\\]\nfor all $\\alpha$.\n\\end{theorem}","\\begin{proof}\nThe continuous bounded generator has an evolution inverse. Indeed, solve $\\dot V=-VK$ with $V(0)=I$. Differentiation gives $(VU)^{\\prime}=0$, hence $VU=I$. The product $W=UV$ satisfies $\\dot W=KW-WK$ and $W(0)=I$. Uniqueness of this bounded linear equation gives $UV=I$. Thus $U$ is invertible.","Skew-adjointness of $K(t)$ gives\n\\[\n\\frac{d}{dt}(U(t)^*U(t))=0,\n\\]\nso $U(t)$ is unitary.  By \\cref{lem:packet_commutator},\n\\[\n\\frac{d}{dt}\\big(U(t)^{-1}P_\\alpha(t)U(t)\\big)\n=\nU(t)^{-1}\\big(\\dot P_\\alpha-[K,P_\\alpha]\\big)U(t)\n=0.\n\\]\nThe expression equals $P_\\alpha(0)$ at time zero.\n\\end{proof}","\\begin{corollary}[One moving subspace]\\label{cor:one_moving_subspace}\nFor one orthogonal projection $P(t)$ and its complement, the canonical generator is\n\\[\nK(t)=[\\dot P(t),P(t)].\n\\]\nThe transport in \\cref{thm:kato_transport} maps $\\im P(0)$ unitarily onto $\\im P(t)$.\n\\end{corollary}","\\begin{proof}\nApply the complete packet formula to $P(t)$ and $I-P(t)$, then simplify the sum.\n\\end{proof}","\\begin{definition}[Direct rotation of nearby packet systems]\\label{def:packet_direct_rotation}\nLet $(P_\\alpha)_{\\alpha\\in I}$ and $(Q_\\alpha)_{\\alpha\\in I}$ be complete orthogonal packet systems on the same Hilbert space.  Put\n\\[\nS(\\Qcal,\\Pcal)=\\sum_{\\alpha\\in I}Q_\\alpha P_\\alpha.\n\\]\nWhen $S(\\Qcal,\\Pcal)$ is invertible, define\n\\[\n\\Ucal(\\Qcal,\\Pcal)\n=\nS(\\Qcal,\\Pcal)\n\\big(S(\\Qcal,\\Pcal)^*S(\\Qcal,\\Pcal)\\big)^{-1/2}.\n\\]\n\\end{definition}","\\begin{proposition}[Exact direct packet rotation]\\label{prop:packet_direct_rotation}\nIf\n\\[\n\\max\\bigl(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\}\\bigr)<1,\n\\]\nthen $S(\\Qcal,\\Pcal)$ is invertible.  The operator $\\Ucal(\\Qcal,\\Pcal)$ is unitary and satisfies\n\\[\nQ_\\alpha\\Ucal(\\Qcal,\\Pcal)\n=\n\\Ucal(\\Qcal,\\Pcal)P_\\alpha\n\\]\nfor every $\\alpha$.\n\\end{proposition}","\\begin{proof}\nThe identities\n\\[\nQ_\\alpha S(\\Qcal,\\Pcal)\n=\nS(\\Qcal,\\Pcal)P_\\alpha\n=\nQ_\\alpha P_\\alpha\n\\]\nfollow from pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}",")\n=\nS(\\Qcal,\\Pcal)P_\\alpha\n=\nQ_\\alpha P_\\alpha\n\\]\nfollow from pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","a P_\\alpha\n\\]\nfollow from pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","m pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","a P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","$\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","a\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","\\begin{lemma}[One-step direct rotation estimate]\\label{lem:one_step_direct_rotation}\nAssume $\\Pcal(t)$ is twice continuously differentiable in operator norm on $[0,T]$.  There are constants $h_0>0$ and $C>0$ such that, whenever $t,t+h\\in[0,T]$ and $|h|\\le h_0$,\n\\[\n\\left\\|\n\\Ucal(\\Pcal(t+h),\\Pcal(t))\n-\nI\n-\nhK(t)\n\\right\\|\n\\le\nCh^2.\n\\]\nThe constants can be chosen uniformly in $t$.\n\\end{lemma}","\\begin{proof}\nTaylor's theorem in the Banach space of bounded operators gives\n\\[\nP_\\alpha(t+h)\n=\nP_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","proof}\nTaylor's theorem in the Banach space of bounded operators gives\n\\[\nP_\\alpha(t+h)\n=\nP_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","bounded operators gives\n\\[\nP_\\alpha(t+h)\n=\nP_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","P_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","\\begin{theorem}[Direct rotations converge to packet transport]\\label{thm:direct_rotation_convergence}\nAssume the complete packet family $\\Pcal(t)$ is twice continuously differentiable in operator norm on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T\n\\]\nwith sufficiently small mesh, define\n\\[\nV_\\pi\n=\n\\Ucal(\\Pcal(t_N),\\Pcal(t_{N-1}))\n\\cdots\n\\Ucal(\\Pcal(t_1),\\Pcal(t_0)).\n\\]\nThen\n\\[\nV_\\pi\\longrightarrow U(T)\n\\]\nin operator norm, where $U$ is the canonical transport from \\cref{thm:kato_transport}.  On a fixed compact interval the error is bounded by a constant times the mesh.\n\\end{theorem}","to packet transport]\\label{thm:direct_rotation_convergence}\nAssume the complete packet family $\\Pcal(t)$ is twice continuously differentiable in operator norm on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T\n\\]\nwith sufficiently small mesh, define\n\\[\nV_\\pi\n=\n\\Ucal(\\Pcal(t_N),\\Pcal(t_{N-1}))\n\\cdots\n\\Ucal(\\Pcal(t_1),\\Pcal(t_0)).\n\\]\nThen\n\\[\nV_\\pi\\longrightarrow U(T)\n\\]\nin operator norm, where $U$ is the canonical transport from \\cref{thm:kato_transport}.  On a fixed compact interval the error is bounded by a constant times the mesh.\n\\end{theorem}","\\begin{proof}\nLet $U(t,s)$ be the exact propagator generated by $K$.  Since $K$ is continuously differentiable, the integral equation for $U(t+h,t)$ gives a uniform estimate\n\\[\n\\|U(t+h,t)-I-hK(t)\\|\n\\le\nC_1h^2.\n\\]\nBy \\cref{lem:one_step_direct_rotation}, the direct rotation $V_j$ from $t_j$ to $t_{j+1}$ therefore satisfies\n\\[\n\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2h_j^2,\n\\qquad\nh_j=t_{j+1}-t_j.\n\\]\nBoth $V_j$ and the exact propagator are unitary.  A telescoping product comparison gives\n\\[\n\\|V_\\pi-U(T,0)\\|\n\\le\n\\sum_j\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2\\sum_jh_j^2\n\\le\nC_2T|\\pi|.\n\\]\nThis proves norm convergence and the stated first-order global bound.\n\\end{proof}","s continuously differentiable, the integral equation for $U(t+h,t)$ gives a uniform estimate\n\\[\n\\|U(t+h,t)-I-hK(t)\\|\n\\le\nC_1h^2.\n\\]\nBy \\cref{lem:one_step_direct_rotation}, the direct rotation $V_j$ from $t_j$ to $t_{j+1}$ therefore satisfies\n\\[\n\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2h_j^2,\n\\qquad\nh_j=t_{j+1}-t_j.\n\\]\nBoth $V_j$ and the exact propagator are unitary.  A telescoping product comparison gives\n\\[\n\\|V_\\pi-U(T,0)\\|\n\\le\n\\sum_j\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2\\sum_jh_j^2\n\\le\nC_2T|\\pi|.\n\\]\nThis proves norm convergence and the stated first-order global bound.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{rotating_projector_transport.pdf}\n\\caption{A rotating rank-one packet and its orthogonal complement. With $u(\\theta)=(\\cos\\theta,\\sin\\theta)$ and $P(\\theta)=u(\\theta)u(\\theta)^{\\mathsf T}$, the canonical generator is $[P',P]=\\bigl(\\begin{smallmatrix}0&-1","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{rotating_projector_transport.pdf}\n\\caption{A rotating rank-one packet and its orthogonal complement. With $u(\\theta)=(\\cos\\theta,\\sin\\theta)$ and $P(\\theta)=u(\\theta)u(\\theta)^{\\mathsf T}$, the canonical generator is $[P',P]=\\bigl(\\begin{smallmatrix}0&-1\\\\1&0\\end{smallmatrix}\\bigr)$. For a positive step $\\delta<\\pi/2$, the polar unitary is the usual rotation $R(\\delta)$. On each displayed mesh, the direct rotation product equals $R(\\pi/3)$ exactly. This special planar example illustrates the construction. The general theorem supplies convergence for families whose step products need not already equal the continuous transport.}\n\\label{fig:rotating-projector-example}\n\\end{figure}\n\\clearpage","1&0\\end{smallmatrix}\\bigr)$. For a positive step $\\delta<\\pi/2$, the polar unitary is the usual rotation $R(\\delta)$. On each displayed mesh, the direct rotation product equals $R(\\pi/3)$ exactly. This special planar example illustrates the construction. The general theorem supplies convergence for families whose step products need not already equal the continuous transport.}\n\\label{fig:rotating-projector-example}\n\\end{figure}\n\\clearpage","\\begin{theorem}[Convergence of sampled transports]\\label{thm:sampled_transport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","egin{theorem}[Convergence of sampled transports]\\label{thm:sampled_transport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","of sampled transports]\\label{thm:sampled_transport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","nsport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","\\begin{proof}\nOn each interval, compare the exact propagator with the frozen exponential.  Variation of constants and the Lipschitz bound give a local error bounded by a constant times the square of the interval length.  Every factor is unitary, so multiplication does not amplify norms.  Summing the local errors gives a bound by a constant times\n\\[\n\\sum_j(t_{j+1}-t_j)^2\n\\le\nT|\\pi|.\n\\]\n\\end{proof}","\\Needspace{10\\baselineskip}\n\\begin{corollary}[Continuous holonomy from graph products]\\label{cor:continuous_holonomy}\nAssume the complete packet family is twice continuously differentiable in operator norm on $[0,T]$. If $P_\\alpha(0)=P_\\alpha(T)$ for every $\\alpha$, then $U(T)$ preserves every initial packet.  Its restriction to each packet is the holonomy of the closed packet path.  The products in \\cref{thm:direct_rotation_convergence,thm:sampled_transport_convergence} converge to these holonomies.  The direct rotation product has the added property that every factor maps each sampled packet exactly to its successor.\n\\end{corollary}","\\begin{example}[A rotating line]\\label{ex:rotating_line}\nIn $\\mathbb R^2$, let\n\\[\nv(t)=(\\cos t,\\sin t)\n\\]\nand let $P(t)$ be the orthogonal projection onto the line spanned by $v(t)$.  The canonical transport is the ordinary rotation matrix\n\\[\nU(t)=\n\\begin{pmatrix}\n\\cos t&-\\sin t","\\begin{example}[A rotating line]\\label{ex:rotating_line}\nIn $\\mathbb R^2$, let\n\\[\nv(t)=(\\cos t,\\sin t)\n\\]\nand let $P(t)$ be the orthogonal projection onto the line spanned by $v(t)$.  The canonical transport is the ordinary rotation matrix\n\\[\nU(t)=\n\\begin{pmatrix}\n\\cos t&-\\sin t\\\\\n\\sin t&\\cos t\n\\end{pmatrix}.\n\\]\nIt carries the initial line and its orthogonal complement to the two moving packets.  Products of short rotations converge to the same transport.\n\\end{example}","\\begin{standardfact}[Curvature of a projected connection]\\label{fact:projected_curvature}\nLet $P$ be a smooth orthogonal projector-valued map on a parameter manifold.  The connection\n\\[\n\\nabla=P\\,d\n\\]\non the range bundle has curvature\n\\[\n\\Fcal=P(dP\\wedge dP)P.\n\\]\n\\end{standardfact}","\\begin{proof}\nFor a section $s$ satisfying $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\\qedhere\n\\]\n\\end{proof}","\\vpage{2}{An atlas of the objects}{ORIENTATION / KEEP THE TYPES VISIBLE}{Chapter III, mathematical text: Definition 2.1, Definition 5.5, Definition 7.4}\nRows are targets and columns are sources throughout. If $E_{ji}$ is a matrix unit, then $E_{ji}e_i=e_j$. The notation $e_i$ means a basis vector. A named graph edge is written $a,b,c,d$ or $e,f$ when its role is explicit.","midrule\n$\\src,\\tgt:E\\to V$ & The ordered endpoints of each distinct edge, including parallel edges. & \\pagerefV{3} to \\pagerefV{4}","$L_e=c(e)E_{\\tgt(e),\\src(e)}$ & One operator supported on a packet. & \\pagerefV{3}","$L_w=L_{e_n}\\cdots L_{e_1}$ & Execute the written word from left to right. & \\pagerefV{4} to \\pagerefV{5}","$T=\\sum_eL_e$ & Add all edge operators before taking powers. & \\pagerefV{4} to \\pagerefV{7}","$F_a$ & A synthesis frame from chart $a$ into the ambient space. & \\pagerefV{9}","$\\mathcal A_a,\\mathcal S_a$ & Inverse analysis and synthesis maps, which may be maps of sets. & \\pagerefV{10}","$J_{b\\leftarrow a}$ & Canonical coordinate transport, forced by the exact charts. & \\pagerefV{9} to \\pagerefV{10}","$\\Gamma_{b\\leftarrow a}$ & Additional connection data, with reversibility stated explicitly. & \\pagerefV{11} to \\pagerefV{12}","$\\Delta_{c,b,a}$ & The difference between transport through two steps and direct transport, a map from $C_a$ to $C_c$. & \\pagerefV{12}","$H_{c,b,a}$ & Loop transport based at $a$, an automorphism of $C_a$. & \\pagerefV{13}","$G_a$ & An invertible gauge change at each chart. & \\pagerefV{14} to \\pagerefV{15}","$\\Pi_\\mu\\Ob\\Pi_\\nu$ & Block from input to output of $\\Ob=H-I$. & \\pagerefV{16} to \\pagerefV{18}","$r(x,p)$ & A residual at packet $p$ for the named candidate $x$. & \\pagerefV{19}","$t,\\operatorname{inputs}(t)$ & A finite operation tree and its syntactically used input labels. & \\pagerefV{20}","\\takeaway{A $C_a\\to C_c$ defect and a $C_a\\to C_a$ holonomy are not the same typed map. Returning the defect to the base fiber requires a specified inverse transport.}","\\vpage{3}{Read the product in execution order}{MODEL A / ONE EDGE IS ONE MATRIX ENTRY}{Chapter III, mathematical text: Lemma 2.2, Theorem 2.3, Theorem 2.4}\nWork over $\\ZZ$ with packets $0,1,2$. Let $e:0\\to1$ have coefficient $2$, and let $f:1\\to2$ have coefficient $3$. The elementary operators are\n\\[\nL_e=2E_{10},\\qquad L_f=3E_{21}.\n\\]\nThe location of the nonzero entry records the type of the transfer, not an arbitrary matrix convention.","\\vfig[.94]{edge_orientation}\n\\platecaption{Each grid is an actual $3\\times3$ matrix. The rightmost matrix factor acts first. In $L_fL_e$, packet $1$ is the shared intermediate index. In $L_eL_f$ the interface is broken.}","On a vector $(x_0,x_1,x_2)^\\mathsf T$, the first step produces $(0,2x_0,0)^\\mathsf T$, and the second produces $(0,0,6x_0)^\\mathsf T$. Thus\n\\[\nL_fL_e=6E_{20},\\qquad L_eL_f=0.\n\\]\nFor general edges, a summand in the entrywise product can survive only when the same intermediate packet is both $\\tgt(e)$ and $\\src(f)$:\n\\[\n(L_fL_e)_{yx}=\\sum_z(L_f)_{yz}(L_e)_{zx}.\n\\]\nThis is why a broken interface annihilates the entire product, even if both edge coefficients are nonzero.","\\takeaway{Write the traversed edges first. Then reverse their order when writing the operator product: $e$ followed by $f$ is $L_fL_e$.}","\\vpage{4}{All sixteen words, only five walks}{MODEL A / PARALLEL EDGES REMAIN DISTINCT}{Chapter III, mathematical text: Theorem 3.8, Corollary 3.9}\nAdd four edges on three packets: $a:0\\to1$ with coefficient $2$, $b:0\\to1$ with coefficient $-1$, $c:1\\to2$ with coefficient $3$, and $d:2\\to0$ with coefficient $1$. The parallel edges $a$ and $b$ are separate labels even though their source and target agree.","The surviving words are $ac,bc,cd,da,db$. Their coefficients are $6,-3,3,2,-1$, respectively. Their endpoints determine where these numbers enter the matrix:\n\\[\nT=\\mat{0&0&1","The surviving words are $ac,bc,cd,da,db$. Their coefficients are $6,-3,3,2,-1$, respectively. Their endpoints determine where these numbers enter the matrix:\n\\[\nT=\\mat{0&0&1\\\\1&0&0\\\\0&3&0},\\qquad\nT^2=\\mat{0&3&0\\\\0&0&1\\\\3&0&0}.\n\\]\nAt row $2$, column $0$, the two parallel choices give $6-3=3$. Deduplicating the edge labels before summing would alter the operator. On the other hand, adding the parallel coefficients in the already assembled matrix is legitimate: both act in the same supported matrix position.","3&0&0}.\n\\]\nAt row $2$, column $0$, the two parallel choices give $6-3=3$. Deduplicating the edge labels before summing would alter the operator. On the other hand, adding the parallel coefficients in the already assembled matrix is legitimate: both act in the same supported matrix position.","\\vpage{5}{Coefficients need not commute}{MODEL A / A MATRIX RING AS THE COEFFICIENT SEMIRING}{Chapter III, mathematical text: Lemma 3.7, Section 9}\nThe edge theorem allows a noncommutative coefficient semiring. Use $R=M_2(\\QQ)$ and let the two coefficients themselves be matrices:\n\\[\nA=\\mat{1&1","\\vpage{5}{Coefficients need not commute}{MODEL A / A MATRIX RING AS THE COEFFICIENT SEMIRING}{Chapter III, mathematical text: Lemma 3.7, Section 9}\nThe edge theorem allows a noncommutative coefficient semiring. Use $R=M_2(\\QQ)$ and let the two coefficients themselves be matrices:\n\\[\nA=\\mat{1&1\\\\0&1},\\qquad B=\\mat{1&0\\\\1&1}.\n\\]\nAttach $A$ to $0\\to1$ and $B$ to $1\\to2$. The single surviving endpoint coefficient is $BA$, not $AB$.","1&1}.\n\\]\nAttach $A$ to $0\\to1$ and $B$ to $1\\to2$. The single surviving endpoint coefficient is $BA$, not $AB$.","\\vfig[.96]{ordered_matrix_coefficients}\n\\platecaption{All entries are exact integers. The two products differ on the diagonal and act differently on the same input vector. This example is within the scalar model with semiring coefficients by taking the scalar ring to be $M_2(\\QQ)$.}","The distinction is visible before any abstraction:\n\\[\nBA\\mat{1\\\\0}=\\mat{1\\\\1},\\qquad\nAB\\mat{1\\\\0}=\\mat{2\\\\1}.\n\\]\nFor a written walk $(e_1,\\ldots,e_n)$, its coefficient must therefore be\n\\[\nc^\\leftarrow(w)=c(e_n)c(e_{n-1})\\cdots c(e_1).\n\\]\nA commutative example can hide an orientation mistake because reversing the coefficients changes nothing. This one cannot.","1}.\n\\]\nFor a written walk $(e_1,\\ldots,e_n)$, its coefficient must therefore be\n\\[\nc^\\leftarrow(w)=c(e_n)c(e_{n-1})\\cdots c(e_1).\n\\]\nA commutative example can hide an orientation mistake because reversing the coefficients changes nothing. This one cannot.","\\head{One ring is not a dependent family of fibers}\nThe $3\\times3$ matrix over $M_2(\\QQ)$ can be displayed as an ordinary $6\\times6$ matrix with $2\\times2$ blocks. This is a concrete representation of the example over one ring. It does not establish an arbitrary family of differently typed edge fibers, which is outside the finite formal contract's scalar interface for edge systems.","\\vpage{6}{A reachable packet can receive zero}{MODEL A / CANCELLATION IS NOT A BROKEN INTERFACE}{Chapter III, mathematical text: Corollary 3.9, Corollary 3.10}\nConsider a diamond: two routes of length two from packet $0$ to packet $3$. Three edge coefficients are $1$ and the lower final coefficient is $-1$. Both routes are correctly typed. Neither is annihilated by an interface mismatch.","\\vfig[.96]{walk_cancellation}\n\\platecaption{The dashed edge carries the coefficient $-1$. It is an actual edge, not an uncertain or missing one. Both paths exist. The unsigned count is two, while their exact operator contributions sum to zero.}","The endpoint identity gives\n\\[\n(T^2)_{30}=(1)(1)+(-1)(1)=0.\n\\]\nThus the chapter's reachability implication has a definite direction:\n\\[\n\\text{no compatible walk}\\ \\Longrightarrow\\ (T^n)_{ji}=0.\n\\]\nIts converse is false. A zero entry may come from absent paths, zero coefficients, zero divisors in the coefficient ring, or cancellation among nonzero terms. The diamond exhibits the last possibility over $\\ZZ$, where no zero divisor is needed.","endpoint identity gives\n\\[\n(T^2)_{30}=(1)(1)+(-1)(1)=0.\n\\]\nThus the chapter's reachability implication has a definite direction:\n\\[\n\\text{no compatible walk}\\ \\Longrightarrow\\ (T^n)_{ji}=0.\n\\]\nIts converse is false. A zero entry may come from absent paths, zero coefficients, zero divisors in the coefficient ring, or cancellation among nonzero terms. The diamond exhibits the last possibility over $\\ZZ$, where no zero divisor is needed.","\\vpage{7}{Reachability is only an upper bound}{MODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONS}{Chapter III, mathematical text: Corollary 3.11}\nKeep the diamond and add an isolated packet $4$. Start with the single seed $e_0\\in\\QQ^5$. Direct multiplication gives\n\\[\nTe_0=e_1+e_2,\\qquad T^2e_0=0.\n\\]\nEvery later orbit vector is zero. The orbit hull generated by $T$ is therefore exactly the span of the first two displayed columns.","A vector belongs to this hull precisely when it has the form\n\\[\n(a,b,b,0,0)^\\mathsf T,\\qquad a,b\\in\\QQ.\n\\]\nPacket $4$ is excluded because it is unreachable. Packet $3$ is also absent, but for a different reason: the two contributions cancel. Even the reachable packets $1$ and $2$ remain linked. Powers of the single total operator do not separate them.","\\head{The admissible operator grammar matters}\nAllowing each individual edge operator is a different generation problem. For instance, applying the edge $0\\to1$ and then $1\\to3$ individually produces $e_3$. It does not follow that $e_3$ belongs to the orbit of the summed operator $T$. The chapter's theorem on orbit support concerns the explicitly stated operator action.","\\vpage{8}{Paths through changing packet lists}{MODEL B / RECTANGULAR PRODUCTS WITH TWO OR THREE STEPS}{Chapter III, mathematical text: Proposition 8.1}\nHere the packet lists have sizes $2,3,2,1$ in succession. Let the actual maps be\n\\[\n\\QQ^2\\xrightarrow{A}\\QQ^3\\xrightarrow{B}\\QQ^2\\xrightarrow{C}\\QQ.\n\\]\nUnlike a power of one matrix, each step has its own entries and its own source and target list.","\\vfig[.98]{rectangular_paths}\n\\platecaption{The lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are $C_{0j}B_{jk}A_{ki}$, with $k\\in\\{0,1,2\\}$ and $j\\in\\{0,1\\}$. Zero terms remain visible.}","The intermediate and total products are\n\\[\nBA=\\mat{-1&5\\\\2&4},\\qquad CBA=\\mat{-4&6}.\n\\]\nFor input packet $0$, the six contributions sum to $2-4+0+0-2+0=-4$. For input packet $1$, they sum to $0-2+12+0-1-3=6$. These are literal matrix entries, not probabilities or path counts.","2&4},\\qquad CBA=\\mat{-4&6}.\n\\]\nFor input packet $0$, the six contributions sum to $2-4+0+0-2+0=-4$. For input packet $1$, they sum to $0-2+12+0-1-3=6$. These are literal matrix entries, not probabilities or path counts.","\\vpage{9}{One vector, three exact frames}{MODEL C / CANONICAL TRANSPORT HAS NO HOLONOMY}{Chapter III, mathematical text: Theorem 4.1, Theorem 4.2}\nLet $F_a$ synthesize chart coordinates into one ambient $\\QQ^2$. Choose\n\\[\nF_0=I,\\qquad F_1=\\mat{1&1","\\vpage{9}{One vector, three exact frames}{MODEL C / CANONICAL TRANSPORT HAS NO HOLONOMY}{Chapter III, mathematical text: Theorem 4.1, Theorem 4.2}\nLet $F_a$ synthesize chart coordinates into one ambient $\\QQ^2$. Choose\n\\[\nF_0=I,\\qquad F_1=\\mat{1&1\\\\0&1},\\qquad\nF_2=\\mat{2&0\\\\1&1}.\n\\]\nThe same ambient vector $x=(3,2)^\\mathsf T$ has chart coordinates $u_a=F_a^{-1}x$.","1&1}.\n\\]\nThe same ambient vector $x=(3,2)^\\mathsf T$ has chart coordinates $u_a=F_a^{-1}x$.","\\vfig[.98]{frame_coordinates}\n\\platecaption{Each panel is a coordinate plane, not a different physical location. The numerical arrows differ because the frames differ. Multiplying each coordinate vector by its corresponding $F_a$ recovers the same $x$.}","Transport from chart $a$ to chart $b$ is forced:\n\\[\nJ_{b\\leftarrow a}=F_b^{-1}F_a.\n\\]\nThe route $0\\to1\\to2$ agrees with the direct route because the middle frame cancels:\n\\[\n(F_2^{-1}F_1)(F_1^{-1}F_0)=F_2^{-1}F_0.\n\\]\nReturning to chart $0$ cancels every remaining frame. The final coordinate vector is $(3,2)^\\mathsf T$, not a sheared or rotated variant.","\\vpage{10}{Nonlinear charts still cancel exactly}{MODEL C / NATURALITY DOES NOT REQUIRE DERIVATIVES}{Chapter III, mathematical text: Proposition 10.1}\nExact coordinate equivalences can be nonlinear. On $\\QQ^2$, use\n\\[\n\\mathcal A(x,y)=(x,y+x^2),\\qquad\n\\mathcal S(u,v)=(u,v-u^2).\n\\]\nBoth composites are the identity by direct substitution. The real drawing below is a geometric view of these same polynomial formulas.","Now represent the nonlinear operation $F(x,y)=(x+y,xy)$ in the new chart. Set $w=u+v-u^2$. Its correct coordinate formula is\n\\[\n\\widehat F_b(u,v)=\\mathcal A F\\mathcal S(u,v)\n=\\bigl(w,\\ u(v-u^2)+w^2\\bigr).\n\\]\nFor $(x,y)=(1,2)$, applying $F$ first gives $(3,2)$ and then $(3,11)$ in the new chart. Changing the input first gives $(u,v)=(1,3)$, and $\\widehat F_b(1,3)=(3,11)$ as well.","\\vpage{11}{Reversibility is a real hypothesis}{MODEL D / THE MINIMAL AND REVERSIBLE INTERFACES}{Chapter III, mathematical text: Definition 5.1, Definition 5.5}\nChoose the forward scalar transports $\\Gamma_{1\\leftarrow0}=2$, $\\Gamma_{2\\leftarrow1}=3$, and $\\Gamma_{2\\leftarrow0}=5$. Their multiplicative comparison between transport through two steps and direct transport is $D^\\times=6/5$. A loop also needs a declared return map $\\Gamma_{0\\leftarrow2}$.","eal hypothesis}{MODEL D / THE MINIMAL AND REVERSIBLE INTERFACES}{Chapter III, mathematical text: Definition 5.1, Definition 5.5}\nChoose the forward scalar transports $\\Gamma_{1\\leftarrow0}=2$, $\\Gamma_{2\\leftarrow1}=3$, and $\\Gamma_{2\\leftarrow0}=5$. Their multiplicative comparison between transport through two steps and direct transport is $D^\\times=6/5$. A loop also needs a declared return map $\\Gamma_{0\\leftarrow2}$.","In the first row, choose the return scalar $1/6$. Then\n\\[\nH=(1/6)\\cdot3\\cdot2=1,\n\\qquad \\Gamma_{2\\leftarrow1}\\Gamma_{1\\leftarrow0}-\\Gamma_{2\\leftarrow0}=1.\n\\]\nThis triangle loop is the identity even though its cocycle equation fails. The return is not the inverse of the direct transport: $(1/6)5\\ne1$. Thus one cannot transfer the reversible equivalence between defect and holonomy to the minimal interface.","In the reversible row, the return is $1/5$. Now the loop is $6/5$, and the additive defect of one is converted into the loop obstruction $1/5$ by that inverse return map.","\\takeaway{Self normalization means $\\Gamma_{a\\leftarrow a}=I$. Reversible additionally means $\\Gamma_{a\\leftarrow b}=\\Gamma_{b\\leftarrow a}^{-1}$. These are different assumptions.}","\\vpage{12}{Measure the mismatch at its base}{MODEL D / AN EXACT REVERSIBLE TRIANGLE}{Chapter III, mathematical text: Theorem 5.10, Section 9}\nRetain the reversible scalar model with forward values $2,3,5$ and all reverse values given by inverses. Two routes from chart $0$ to chart $2$ multiply the input by $6$ and $5$, respectively. The additive discrepancy equals the original input.","The defect lives at the target of the forward routes:\n\\[\n\\Delta_{2,1,0}=3\\cdot2-5=1.\n\\]\nThe holonomy lives back at chart $0$:\n\\[\nH_{2,1,0}=5^{-1}\\cdot3\\cdot2=\\frac65,\n\\qquad H-I=\\frac15=\\Gamma_{0\\leftarrow2}\\Delta.\n\\]\nFor input $5$, the direct output and the output after two steps are $25$ and $30$. Returning by $1/5$ gives $5$ and $6$. The target discrepancy of $5$ has become a base discrepancy of $1$. The types and locations of these differences matter even in an example in one dimension.","defect lives at the target of the forward routes:\n\\[\n\\Delta_{2,1,0}=3\\cdot2-5=1.\n\\]\nThe holonomy lives back at chart $0$:\n\\[\nH_{2,1,0}=5^{-1}\\cdot3\\cdot2=\\frac65,\n\\qquad H-I=\\frac15=\\Gamma_{0\\leftarrow2}\\Delta.\n\\]\nFor input $5$, the direct output and the output after two steps are $25$ and $30$. Returning by $1/5$ gives $5$ and $6$. The target discrepancy of $5$ has become a base discrepancy of $1$. The types and locations of these differences matter even in an example in one dimension.","Because the return map is invertible, it cannot turn a nonzero defect into zero. This is the mechanism behind $\\Delta=0\\Longleftrightarrow H=I$ in the reversible interface.","\\vpage{13}{The identity can hide in the spectrum}{MODEL E / A SHEAR WITH NONTRIVIAL HOLONOMY}{Chapter III, mathematical text: Definition 5.9, Definition 7.4}\nPut $\\Gamma_{1\\leftarrow0}=H$, $\\Gamma_{2\\leftarrow1}=I$, and $\\Gamma_{2\\leftarrow0}=I$, with reverse maps given by inverses, where\n\\[\nH=\\mat{1&1","\\vpage{13}{The identity can hide in the spectrum}{MODEL E / A SHEAR WITH NONTRIVIAL HOLONOMY}{Chapter III, mathematical text: Definition 5.9, Definition 7.4}\nPut $\\Gamma_{1\\leftarrow0}=H$, $\\Gamma_{2\\leftarrow1}=I$, and $\\Gamma_{2\\leftarrow0}=I$, with reverse maps given by inverses, where\n\\[\nH=\\mat{1&1\\\\0&1}.\n\\]\nThe loop $0\\to1\\to2\\to0$ then has exactly this shear as its holonomy.","0&1}.\n\\]\nThe loop $0\\to1\\to2\\to0$ then has exactly this shear as its holonomy.","\\vfig[.94]{shear_holonomy}\n\\platecaption{The dashed square is the original set. The solid parallelogram is its exact image. The second coordinate basis vector moves to $(1,1)^\\mathsf T$. Its nonzero displacement is the first basis vector.}","The usual scalar invariants can be misleading:\n\\[\n\\operatorname{tr}H=2,\\qquad \\det H=1,\\qquad\n\\chi_H(t)=(t-1)^2.\n\\]\nThese are also the invariants of the $2\\times2$ identity matrix. Nevertheless,\n\\[\n\\Ob=H-I=\\mat{0&1","The usual scalar invariants can be misleading:\n\\[\n\\operatorname{tr}H=2,\\qquad \\det H=1,\\qquad\n\\chi_H(t)=(t-1)^2.\n\\]\nThese are also the invariants of the $2\\times2$ identity matrix. Nevertheless,\n\\[\n\\Ob=H-I=\\mat{0&1\\\\0&0}\\ne0,\n\\qquad \\Ob e_1=e_0.\n\\]\nA complete obstruction test must therefore retain the operator or a certificate strong enough to detect its nonzero action. Trace, determinant, and eigenvalue lists alone do not distinguish this loop from the identity.","\\head{What the picture certifies, and what it does not}\nThe plotted corners are the images of the four corners of the unit square under the displayed matrix. The equation $\\Ob e_1=e_0$ is the exact witness. No smooth connection, curvature tensor, or continuum limit is used in this model. Its holonomy is the finite algebraic loop product. The mathematical text treats smooth transport separately under additional hypotheses.","\\takeaway{Flatness is the equality $H=I$, not agreement of a short list of spectral summaries. A single explicit moved vector can disprove that equality.}","\\vpage{14}{Gauge changes the picture, not flatness}{MODEL E / CONJUGATION AND ITS LIMITS}{Chapter III, mathematical text: Theorem 6.2}\nAt the base chart, change coordinates by $D=\\operatorname{diag}(2,1)$. Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving\n\\[\nH^D=DHD^{-1}=\\mat{1&2","\\vpage{14}{Gauge changes the picture, not flatness}{MODEL E / CONJUGATION AND ITS LIMITS}{Chapter III, mathematical text: Theorem 6.2}\nAt the base chart, change coordinates by $D=\\operatorname{diag}(2,1)$. Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving\n\\[\nH^D=DHD^{-1}=\\mat{1&2\\\\0&1}.\n\\]\nThe shear looks stronger in these fixed numerical axes, but the verdict on flatness is unchanged by gauge.","The rank of the obstruction is one on both sides. Trace, determinant, and characteristic polynomial are preserved under this conjugation. Conjugation also gives\n\\[\nD(H-I)D^{-1}=0\\quad\\Longleftrightarrow\\quad H-I=0.\n\\]\nAn invertible coordinate change cannot turn a nonidentity loop into the identity. The fact that a particular residual norm changes does not contradict the invariance of exact vanishing.","\\head{The base chart determines the outer gauge}\nIn a triangle product, adjacent intermediate gauges cancel:\n\\[\n(G_a\\Gamma_{a\\leftarrow c}G_c^{-1})\n(G_c\\Gamma_{c\\leftarrow b}G_b^{-1})\n(G_b\\Gamma_{b\\leftarrow a}G_a^{-1})\n=G_aHG_a^{-1}.\n\\]\nOnly the base gauge survives around the loop. For the additive defect, the surviving outer gauges instead belong to its different source and target fibers.","\\vpage{15}{Flat is not the same as calibrated}{MODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGES}{Chapter III, mathematical text: Theorem 6.3}\nMake all three canonical charts literally identical, so $J_{b\\leftarrow a}=1$ over $\\QQ$. Choose scalar gauges $g_0=1$, $g_1=2$, and $g_2=3$, and define\n\\[\n\\Gamma_{b\\leftarrow a}=g_b g_a^{-1}.\n\\]\nThe edge $0\\to1$ is multiplication by two, although its canonical counterpart is the identity.","\\vfig[.97]{flat_not_calibrated}\n\\platecaption{The rulers display the chosen gauge scales: the marks are $g_a n$ for integer $n$. They visualize the supplied scalar factors, not an intrinsic metric. Relative transport is the ratio of the target and source scales.}","The route through two steps $0\\to1\\to2$ has multiplier $(3/2)2=3$, equal to the direct route $0\\to2$. More generally,\n\\[\n(g_cg_b^{-1})(g_bg_a^{-1})=g_cg_a^{-1}\n\\]\nfor every triple. Thus this connection is flat and reversible, but it is not calibrated to the original $J$, whose values are identities.","\\head{Recovering the gauge from one root}\nChoose a base chart $o$. The theorem constructs\n\\[\nG_a=\\Gamma_{a\\leftarrow o}J_{o\\leftarrow a},\n\\qquad\n\\Gamma_{b\\leftarrow a}=G_bJ_{b\\leftarrow a}G_a^{-1}.\n\\]\nWith $o=0$ and $J=1$, the recovered $G_a$ are exactly $1,2,3$. The statement concerns the complete pairwise interface and its cocycle. It is not a topological claim about an arbitrary graph with missing direct comparisons.","\\vpage{16}{Resolve the entire obstruction}{MODEL F / PACKET BLOCKS FROM INPUT TO OUTPUT}{Chapter III, mathematical text: Theorem 7.5, Corollary 7.6}\nUse three coordinate packets with projectors $P_0,P_1,P_2$, and set\n\\[\nH=I+2E_{01}+3E_{12},\\qquad \\Ob=H-I.\n\\]\nHere $H$ is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside the diagonal encode two effects that mix packets.","\\vfig[.95]{all_obstruction_blocks}\n\\platecaption{The board on the right contains all nine actual embedded blocks $P_\\mu\\Ob P_\\nu$. Row labels are output packets and column labels are input packets. Only the addresses $(0,1)$ and $(1,2)$ carry nonzero maps.}","Completeness on both sides supplies the reconstruction:\n\\[\n\\Ob=(P_0+P_1+P_2)\\Ob(P_0+P_1+P_2)\n=\\sum_{\\mu,\\nu}P_\\mu\\Ob P_\\nu.\n\\]\nFor $x=(4,5,6)^\\mathsf T$, the input in packet $1$ contributes $10e_0$ and the input in packet $2$ contributes $18e_1$. Their sum is the full displacement $\\Ob x$.","\\head{A block becomes a witness when it acts}\nThe nonzero block $2E_{01}$ has the explicit source vector $e_1$, supported entirely in packet $1$, and $P_0\\Ob e_1=2e_0\\ne0$. Similarly, $P_1\\Ob e_2=3e_1\\ne0$. The general theorem supplies such a witness classically when a nonzero block is known. The example displays it concretely.","\\vpage{17}{Why diagonal checks are insufficient}{MODEL E / COMPLETENESS, TWO SIDES, AND OBLIQUE PACKETS}{Chapter III, mathematical text: Theorem 7.5}\nReturn to $\\Ob=E_{01}$. With the two standard coordinate projectors, both diagonal blocks vanish:\n\\[\nP_0\\Ob P_0=P_1\\Ob P_1=0.\n\\]\nThe obstruction nevertheless survives in the block outside the diagonal $P_0\\Ob P_1=E_{01}$.","There is a second failure mode. Supplying only $P_0$ gives a family of idempotents but not a complete one. The single tested block is zero, while the omitted packet carries the missing input. The implication ``all tested blocks vanish, therefore the operator vanishes'' requires the completeness hypothesis.","\\head{Exact reconstruction does not require right angles}\nFor the displayed $Q_0,Q_1$,\n\\[\nQ_0+Q_1=I,\\qquad Q_i^2=Q_i,\\qquad Q_0Q_1=Q_1Q_0=0,\n\\]\nbut $Q_0^\\mathsf T\\ne Q_0$. Expanding both copies of $I$ still proves\n\\[\n\\sum_{i,j}Q_i\\Ob Q_j=\\Ob.\n\\]\nThe block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a witness that remains supported in one packet after projection. The general formal reconstruction statement does not require pairwise annihilation.","\\vpage{18}{A gauge can move every packet label}{MODEL E / FIXED VERSUS TRANSPORTED PROJECTORS}{Chapter III, mathematical text: Proposition 7.7}\nThe diagonal gauge on Plate III.14 commuted with each standard projector. Now use the gauge that mixes packets\n\\[\nG=\\mat{1&0\\\\1&1},\\qquad \\Ob=E_{01}.\n\\]\nThe total obstruction transforms by conjugation, but its pattern in the \\emph{fixed} standard packets changes.","Direct multiplication gives\n\\[\nG\\Ob G^{-1}=\\mat{-1&1\\\\-1&1}.\n\\]\nFor example, the original $(0,0)$ block vanishes, while the new $(0,0)$ block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires $G P_i=P_iG$.","-1&1}.\n\\]\nFor example, the original $(0,0)$ block vanishes, while the new $(0,0)$ block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires $G P_i=P_iG$.","\\head{Transporting the packet frame as well}\nDefine $Q_i=GP_iG^{-1}$. Then the identities cancel exactly:\n\\[\nQ_i(G\\Ob G^{-1})Q_j=G(P_i\\Ob P_j)G^{-1}.\n\\]\nRelative to these transported packets, the corresponding labeled blocks again have matching vanishing status. These are the same oblique projectors displayed on Plate III.17. One must distinguish moving the operator alone in a fixed resolution from moving the entire resolved description.","\\vpage{19}{An obstruction rejects one candidate}{MODEL G / THE QUANTIFIERS ARE THE CONTENT}{Chapter III, mathematical text: Theorem 7.2}\nLet the candidates be rational numbers and the packet labels be $0,1$. Define\n\\[\nr(x,0)=x^2-1,\\qquad r(x,1)=x-1.\n\\]\nA candidate is a solution when both residuals vanish. The five candidate columns below are a display of this explicit residual function, not a search over an unspecified class.","\\vfig[.97]{candidate_residual_table}\n\\platecaption{A shaded cell is a nonzero exact residual. At $x=-1$, the first equation passes but the second supplies an obstruction. At $x=1$, both pass. The rejected candidate does not obstruct the successful one.}","The statement proved by the generic certificate has the form\n\\[\n\\neg\\bigl(\\forall p,\\ r(x,p)=0\\bigr)\n\\quad\\Longleftrightarrow\\quad\n\\exists p,\\ r(x,p)\\ne0.\n\\]\nThe same candidate $x$ appears on both sides. A certificate consists of the packet label and its nonvanishing proof. The general converse uses classical reasoning and does not itself provide an algorithm to find the label.","\\head{What would be required to rule out all solutions?}\nOne would need the stronger assertion $\\forall x\\,\\exists p,\\ r(x,p)\\ne0$. The value $x=1$ disproves that assertion in this example. A witness for $x=-1$ supplies no evidence for it. Likewise, finding an obstruction block for one triangle of one chosen connection does not classify every possible connection.","\\vpage{20}{A tree reads its named leaves}{MODEL H / ACTUAL SYNTAX, ARBITRARY SEMANTICS}{Chapter III, mathematical text: Definition 11.1, Theorem 11.2}\nUse the grammar's constant, input, unary, and binary constructors to build\n\\[\nt=(x^2+3)-y.\n\\]\nHere ``square,'' ``add,'' and ``subtract'' are explicitly supplied meanings of operation symbols. At $x=2,y=5$, every node value can be evaluated exactly.","The input set is $\\operatorname{inputs}(t)=\\{x,y\\}$. If two environments agree at $x$ and $y$, structural induction proves that the two evaluations agree: input leaves agree, constants agree, and every parent applies the same supplied operation to equal child values.","\\head{Syntactic locality is not minimal dependence}\nThe tree $x-x$ has input set $\\{x\\}$, although its value is always zero under ordinary subtraction. Locality provides a sufficient agreement condition. It does not compute the smallest set of semantically relevant variables. Nor does it prove that two different trees are equivalent.","The semantics may act on arbitrary values. For example, a binary concatenation symbol applied to strings ``ab'' and ``cd'' gives ``abcd.'' No linearity or differentiability is required. Conversely, swapping the ordered inputs of the subtraction root above changes the value from $2$ to $-2$.","\\vpage{21}{Where a nonlinear defect can enter}{MODEL H / EXACT COORDINATES VERSUS A CHOSEN RULE}{Chapter III, mathematical text: Proposition 10.1, Formalization target 11.7, Theorem 20.2}\nTake the unary operation $f(x)=x^2$ over $\\QQ$. Suppose both input and output transport are chosen as multiplication by two, while the same numerical formula $f$ is reused at the target. The two routes give\n\\[\n2f(x)=2x^2,\\qquad f(2x)=4x^2.\n\\]\nTheir difference is $-2x^2$, which is generally nonzero.","H / EXACT COORDINATES VERSUS A CHOSEN RULE}{Chapter III, mathematical text: Proposition 10.1, Formalization target 11.7, Theorem 20.2}\nTake the unary operation $f(x)=x^2$ over $\\QQ$. Suppose both input and output transport are chosen as multiplication by two, while the same numerical formula $f$ is reused at the target. The two routes give\n\\[\n2f(x)=2x^2,\\qquad f(2x)=4x^2.\n\\]\nTheir difference is $-2x^2$, which is generally nonzero.","\\vfig[.96]{node_defect_not_coordinate_defect}\n\\platecaption{Solid: $2x^2$. Dashed: $4x^2$. The curves are a real visualization of exact polynomial formulas. The table contains exact evaluations over the rationals. The mismatch is caused by the specified combination of transport and formula.}","For a canonical coordinate change with input and output chart $v=2x$, the target formula is not still $v^2$. It is\n\\[\n\\widehat f_b(v)=2f(v/2)=\\frac{v^2}{2}.\n\\]\nConsequently $\\widehat f_b(2x)=2x^2$, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on Plate III.10.","\\vpage{22}{Mixed terms are not optional}{FINITE EXAMPLES / WHY MULTILINEARITY MATTERS}{Chapter III, mathematical text: Formalization target 11.6, Theorem 18.3}\nA bilinear node distributes over two independently resolved inputs:\n\\[\n(a+b)(c+d)=ac+ad+bc+bd.\n\\]\nThe four terms correspond to all ordered choices of one input packet from each slot. Keeping only matching packet labels loses the mixed terms.","\\vfig[.95]{bilinear_vs_nonlinear}\n\\platecaption{The rectangle has width $2+3$ and height $5+7$, with differently scaled horizontal and vertical display units. Each cell is labeled by its actual product. The lighter diagonal pair contributes only $31$ of the total $60$.}","For a unary square, inserting $a+b$ does not give $a^2+b^2$. The extra $2ab$ arises because the operation is nonlinear in its single argument. It can be analyzed by representing the square using a binary multiplication node with a repeated input, but that is an additional choice of syntax and semantics.","\\vpage{23}{From loop data to a packet witness}{CAPSTONE / A COMPLETE FINITE DIAGNOSIS}{Chapter III, mathematical text: Theorem 5.10, Theorem 6.2, Corollary 7.6}\nSupply a reversible connection on three copies of $\\QQ^2$ by choosing\n\\[\n\\Gamma_{1\\leftarrow0}=A,\\qquad\\Gamma_{2\\leftarrow1}=B,\n\\qquad\\Gamma_{2\\leftarrow0}=C=I,\n\\]\nand declaring every reverse edge to be the inverse. This time both nontrivial forward matrices are retained.","\\vfig[.97]{worked_holonomy_diagnosis}\n\\platecaption{The vector row starts with the source witness $e_0$ and follows it through $A$, then $B$, then $C^{-1}$. Returning to the base produces $(1,1)^\\mathsf T$, whose displacement from the input is $e_1$.}","The calculation gives\n\\[\n\\Delta=BA-C=\\mat{0&1\\\\1&1},\\qquad H=C^{-1}BA=\\mat{1&1\\\\1&2}.\n\\]\nChoose the standard packet projectors. The $(1,0)$ block is nonzero, and the explicit candidate vector satisfies $P_1(H-I)e_0=e_1$. This proves nonflatness of the supplied connection at the supplied triangle.","1&2}.\n\\]\nChoose the standard packet projectors. The $(1,0)$ block is nonzero, and the explicit candidate vector satisfies $P_1(H-I)e_0=e_1$. This proves nonflatness of the supplied connection at the supplied triangle.","\\head{Repairing data is not the same as changing gauge}\nReplacing the direct map $C=I$ by $C=BA$, and updating its reverse, makes the new loop equal to $I$. That changes the connection. It cannot be a gauge of the original nonflat connection, because gauge preserves whether holonomy is the identity. For this complete example on three charts all cocycles then hold. Changing one edge in a larger network would require checking the other affected loops as well.","\\begin{abstract}\nFix a positive integer $N$. Root of unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and","\\begin{abstract}\nFix a positive integer $N$. Root of unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw y","\\begin{abstract}\nFix a positive integer $N$. Root of unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw yields zero-counting asymptotics with exp","\\begin{abstract}\nFix a positive integer $N$. Root of unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw yields zero-counting asymptotics with explicit leading constants and\nbounded er","\\begin{abstract}\nFix a positive integer $N$. Root of unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw yields zero-counting asymptotics with explicit leading constants and\nbounded error. These results give genus-ze","\\begin{abstract}\nFix a positive integer $N$. Root of unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw yields zero-counting asymptotics with explicit leading constants and\nbounded error. These results give genus-zero products, the standard diagon","$. Root of unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw yields zero-counting asymptotics with explicit leading constants and\nbounded error. These results give genus-zero products, the standard diagonal\nFredholm representation, zero-power-sum formulas for reciprocal and\nlogarithmic coefficients, nearest-zero coefficient asymptotics, and\nmeromorphic contin…","unity projection of the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw yields zero-counting asymptotics with explicit leading constants and\nbounded error. These results give genus-zero products, the standard diagonal\nFredholm representation, zero-power-sum formulas for reciprocal and\nlogarithmic coefficients, nearest-zero coefficient asymptotics, and\nmeromorphic continuation of t…","the exponential\nproduces residue-class sections that, for $\\lambda\\ne0$, form a basis for\n$y^{(N)}=\\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the\ndenominator $\\Dfun_N$ and its reduced form $\\Delta_N$. We derive reciprocal\ncoefficient polynomials, Bell and Newton recurrences, an additive parameter\nlaw, and residue section Appell and theta families. For $N\\geq2$,\n$\\Delta_N$ has order $1/N$. We specialize classical two term zero theory for\nexponential sums to the regular exponent polygon. In this equal-weight family,\nconjugation symmetry combined with unique near-model localization places all\nsufficiently large zeros exactly on the symmetry rays, and direct expansion\ngives their leading displacement and derivative sign. The classical perimeter\nlaw yields zero-counting asymptotics with explicit leading constants and\nbounded error. These results give genus-zero products, the standard diagonal\nFredholm representation, zero-power-sum formulas for reciprocal and\nlogarithmic coefficients, nearest-zero coefficient asymptotics, and\nmeromorphic continuation of the symmetry-orbit ze…","For a power series\n\\[\nF(z)=\\sum_{m\\geq 0}a_mz^m\n\\]\nand a positive integer $N$, the coefficients may be separated according to the residue of $m$ modulo $N$.  Root of unity averaging gives exact projection operators for this separation.  This elementary construction is usually called a multisection or a root filter.  It appears throughout the classical theory of generating functions and special functions.  Background on multisections, hypergeometric functions, and related coefficient systems may be found in \\cite{Rainville1960,AndrewsAskeyRoy1999,Comtet1974}.","This chapter follows the consequences of applying the cyclic projections to the exponential function.  The projected exponentials form a finite differential system.  After one sign is inserted around the derivative cycle, they form a basis for\n\\[\ny^{(N)}=-y.\n\\]\nThe section with initial value one is\n\\[\n\\Dfun_N(z)=\\sum_{n\\geq 0}(-1)^n\\frac{z^{Nn}}{(Nn)!}.\n\\]\nIts reciprocal powers produce polynomial coefficient families. Its logarithmic\nderivative produces a scalar higher Riccati identity. The same reciprocal\nconstruction applied to every exponential section gives normalized\nresidue section functions and ordinary Appell polynomial sequences. These\ncyclic exponential sections and their matrix addition laws belong to the\ngeneralized hyperbolic function literature\n\\cite{Schwaiger1992,Muldoon2005}. Appell and binomial-type sequences are\ndiscussed in \\cite{Roman1984,Comtet1974}.","The second half of this chapter concerns the zeros of $\\Dfun_N$.  The finite exponential sum\n\\[\n\\Dfun_N(z)=\\frac1N\\sum_{\\eta^N=-1}e^{\\eta z}\n\\]\nhas a regular polygon of exponents.  The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P\\'olya and Langer.  Later developments include the work of Moreno and the treatment in Levin's monograph \\cite{Polya1920,Langer1931,Moreno1973,Levin1980}. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficiently large zeros in two term strips and prove exponential localization and simplicity \\cite{BleherMallison2006}. The regular polygon and conjugation symmetry of the present family specialize that the","The second half of this chapter concerns the zeros of $\\Dfun_N$.  The finite exponential sum\n\\[\n\\Dfun_N(z)=\\frac1N\\sum_{\\eta^N=-1}e^{\\eta z}\n\\]\nhas a regular polygon of exponents.  The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P\\'olya and Langer.  Later developments include the work of Moreno and the treatment in Levin's monograph \\cite{Polya1920,Langer1931,Moreno1973,Levin1980}. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficiently large zeros in two term strips and prove exponential localization and simplicity \\cite{BleherMallison2006}. The regular polygon and conjugation symmetry of the present family specialize that theory to $N$ exact rays. We give a self-contained specialization because its nota","in Levin's monograph \\cite{Polya1920,Langer1931,Moreno1973,Levin1980}. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficiently large zeros in two term strips and prove exponential localization and simplicity \\cite{BleherMallison2006}. The regular polygon and conjugation symmetry of the present family specialize that theory to $N$ exact rays. We give a self-contained specialization because its notation is used throughout the product and zeta theory, and we compute the leading displacement term for this family. The resulting counting formulas have explicit leading constants and bounded error.","The rotational symmetry is more useful after the change of variable $w=z^N$.  There is an entire function\n\\[\n\\Delta_N(w)=\\sum_{n\\geq 0}(-1)^n\\frac{w^n}{(Nn)!}\n\\]\nwith\n\\[\n\\Dfun_N(z)=\\Delta_N(z^N).\n\\]\nFor $N\\geq2$, the order of $\\Delta_N$ is $1/N$.  Its Hadamard product therefore has genus zero.  The logarithm of this product identifies the transfer tangent coefficients with power sums of the reduced zeros.  The large zero formula also compares the reduced zero zeta with a Hurwitz zeta function.  Standard material on entire functions, canonical products, and exponential sums appears in \\cite{Boas1954,Levin1980,Titchmarsh1939}.  We use singularity analysis in the form described in \\cite{FlajoletSedgewick2009} and the classical continuation of the Hurwitz zeta functio","The rotational symmetry is more useful after the change of variable $w=z^N$.  There is an entire function\n\\[\n\\Delta_N(w)=\\sum_{n\\geq 0}(-1)^n\\frac{w^n}{(Nn)!}\n\\]\nwith\n\\[\n\\Dfun_N(z)=\\Delta_N(z^N).\n\\]\nFor $N\\geq2$, the order of $\\Delta_N$ is $1/N$.  Its Hadamard product therefore has genus zero.  The logarithm of this product identifies the transfer tangent coefficients with power sums of the reduced zeros.  The large zero formula also compares the reduced zero zeta with a Hurwitz zeta function.  Standard material on entire functions, canonical products, and exponential sums appears in \\cite{Boas1954,Levin1980,Titchmarsh1939}.  We use singularity analysis in the form described in \\cite{FlajoletSedgewick2009} and the classical continuation of the Hurwitz zeta function from \\cite{Apostol1976}.","\\begin{keyidea}\nThe same object is read in different coordinates. Selecting residues makes its Taylor coefficients visible. A signed shift makes the differential system finite. Passing to $w=z^N$ removes rotational repetition from its zeros. The formulas below identify these descriptions under their stated hypotheses.\n\\end{keyidea}","Two variables appear throughout this chapter.  The variable $z$ carries the differential equation and the rotational symmetry.  The reduced variable\n\\[\nw=z^N\n\\]\nrecords one representative from each rotational orbit.  Confusing these variables hides the main simplification.  The denominator $\\Dfun_N$ has order one as a function of $z$, while $\\Delta_N$ has order $1/N$ as a function of $w$.","The analytic objects are introduced in the following order.  First,\n\\[\n\\Efun_{N,j}(z)\n=\n\\sum_{n\\geq0}\\frac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the residue $j$ part of the exponential.  Second,\n\\[\n\\Dfun_{N,j}(z)\n=\n\\sum_{n\\geq0}(-1)^n\\frac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the signed basis for $y^{(N)}=-y$.  The section $\\Dfun_N=\\Dfun_{N,0}$ is the denominator used in the reciprocal family\n\\[\n\\Sfun_N^{(r)}=\\Dfun_N^{-r}.\n\\]\nSet $\\Tfun_N=-\\Dfun_N^{\\prime}/\\Dfun_N$. Equivalently, $\\Tfun_N$ is the logarithmic derivative of $\\Sfun_N^{(1)}$.","ic objects are introduced in the following order.  First,\n\\[\n\\Efun_{N,j}(z)\n=\n\\sum_{n\\geq0}\\frac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the residue $j$ part of the exponential.  Second,\n\\[\n\\Dfun_{N,j}(z)\n=\n\\sum_{n\\geq0}(-1)^n\\frac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the signed basis for $y^{(N)}=-y$.  The section $\\Dfun_N=\\Dfun_{N,0}$ is the denominator used in the reciprocal family\n\\[\n\\Sfun_N^{(r)}=\\Dfun_N^{-r}.\n\\]\nSet $\\Tfun_N=-\\Dfun_N^{\\prime}/\\Dfun_N$. Equivalently, $\\Tfun_N$ is the logarithmic derivative of $\\Sfun_N^{(1)}$.","First,\n\\[\n\\Efun_{N,j}(z)\n=\n\\sum_{n\\geq0}\\frac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the residue $j$ part of the exponential.  Second,\n\\[\n\\Dfun_{N,j}(z)\n=\n\\sum_{n\\geq0}(-1)^n\\frac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the signed basis for $y^{(N)}=-y$.  The section $\\Dfun_N=\\Dfun_{N,0}$ is the denominator used in the reciprocal family\n\\[\n\\Sfun_N^{(r)}=\\Dfun_N^{-r}.\n\\]\nSet $\\Tfun_N=-\\Dfun_N^{\\prime}/\\Dfun_N$. Equivalently, $\\Tfun_N$ is the logarithmic derivative of $\\Sfun_N^{(1)}$.","ac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the residue $j$ part of the exponential.  Second,\n\\[\n\\Dfun_{N,j}(z)\n=\n\\sum_{n\\geq0}(-1)^n\\frac{z^{Nn+j}}{(Nn+j)!}\n\\]\nis the signed basis for $y^{(N)}=-y$.  The section $\\Dfun_N=\\Dfun_{N,0}$ is the denominator used in the reciprocal family\n\\[\n\\Sfun_N^{(r)}=\\Dfun_N^{-r}.\n\\]\nSet $\\Tfun_N=-\\Dfun_N^{\\prime}/\\Dfun_N$. Equivalently, $\\Tfun_N$ is the logarithmic derivative of $\\Sfun_N^{(1)}$.","A different reciprocal family comes from normalizing each exponential section at the origin.  The function\n\\[\n\\Cfun_{N,j}(z)\n=\n\\frac{z^j/j!}{\\Efun_{N,j}(z)}\n\\]\nis a unit series in $z^N$.  Its powers, after multiplication by $e^{xz}$, give Appell polynomials.  The theta family couples a scaled argument with a reciprocal order.","The zero theory begins only after these algebraic relations have been established.  The factorization\n\\[\n\\Dfun_N(z)=\\Delta_N(z^N)\n\\]\nturns every orbit of $N$ zeros in the $z$ plane into one zero in the $w$ plane.  The large zeros in the $z$ plane are located by two dominant exponentials.  The zeros in the $w$ plane then give a genus zero product.  The logarithm of that product returns to the coefficient theory through\n\\[\n\\text{zero power sums}\n=\n\\text{logarithmic coefficients}.\n\\]\nThis equality is the bridge between the first and second halves of this chapter.","nly after these algebraic relations have been established.  The factorization\n\\[\n\\Dfun_N(z)=\\Delta_N(z^N)\n\\]\nturns every orbit of $N$ zeros in the $z$ plane into one zero in the $w$ plane.  The large zeros in the $z$ plane are located by two dominant exponentials.  The zeros in the $w$ plane then give a genus zero product.  The logarithm of that product returns to the coefficient theory through\n\\[\n\\text{zero power sums}\n=\n\\text{logarithmic coefficients}.\n\\]\nThis equality is the bridge between the first and second halves of this chapter.","\\begin{readerroadmap}\nThe root filter asks which coefficient addresses belong to a packet. It is an exact selector. Multiplication then asks how two selected packets interact. Their residue labels add modulo $N$, so mixed inputs remain part of the product.\n\\end{readerroadmap}\n\\modelfigure{root_filter}{An exact address model for $N=3$. Every displayed exponent belongs to one residue lane. The drawing records labels rather than coefficient magnitudes.}{fig:root-filter}","Fix $N\\geq1$ and write\n\\[\n\\omega=e^{2\\pi i/N}.\n\\]\nLet $\\Omega$ be a disk centered at the origin, with the whole plane allowed, and assume that $\\Omega$ is invariant under multiplication by $\\omega$.  For a holomorphic function $F$ on $\\Omega$, define\n\\[\n(\\Bcal_{N,j}F)(z)\n=\n\\frac1N\\sum_{\\ell=0}^{N-1}\\omega^{-j\\ell}F(\\omega^\\ell z),\n\\qquad 0\\leq j<N.\n\\]\nThe same formula is valid for formal power series.","\\begin{theorem}[Root filter]\\label{thm:root_filter}\nIf\n\\[\nF(z)=\\sum_{m\\geq0}a_mz^m,\n\\]\nthen\n\\[\n\\Bcal_{N,j}F(z)\n=\n\\sum_{m\\equiv j\\bmod N}a_mz^m.\n\\]\nThe operators satisfy\n\\[\n\\Bcal_{N,i}\\Bcal_{N,j}=\\delta_{ij}\\Bcal_{N,j}\n\\]\nand\n\\[\n\\sum_{j=0}^{N-1}\\Bcal_{N,j}=I.\n\\]\n\\end{theorem}","\\begin{proof}\nSubstitution of the power series gives\n\\[\n\\Bcal_{N,j}F(z)\n=\n\\sum_{m\\geq0}a_mz^m\n\\left(\n\\frac1N\\sum_{\\ell=0}^{N-1}\\omega^{(m-j)\\ell}\n\\right).\n\\]\nThe inner average is one when $m\\equiv j\\bmod N$ and zero otherwise.  The operator identities follow from the disjoint residue classes.\n\\end{proof}","\\begin{proposition}[Covariance characterization]\\label{prop:covariance}\nA holomorphic function $F$ belongs to the range of $\\Bcal_{N,j}$ if and only if\n\\[\nF(\\omega z)=\\omega^jF(z).\n\\]\n\\end{proposition}","\\begin{proof}\nEvery monomial selected by $\\Bcal_{N,j}$ has the stated covariance.  Conversely, write $F=\\sum_k\\Bcal_{N,k}F$.  Replacing $z$ by $\\omega z$ gives\n\\[\n\\sum_k\\omega^k\\Bcal_{N,k}F\n=\n\\omega^j\\sum_k\\Bcal_{N,k}F.\n\\]\nApplication of $\\Bcal_{N,k}$ gives\n\\[\n(\\omega^k-\\omega^j)\\Bcal_{N,k}F=0.\n\\]\nThe powers $1,\\omega,\\ldots,\\omega^{N-1}$ are distinct, so\n$\\Bcal_{N,k}F=0$ for $k\\ne j$.\n\\end{proof}","Multiplication is not diagonal in these coordinates. The residue labels add\nmodulo $N$. This is the scalar cyclic instance of the packet product from\nChapter I.","\\begin{theorem}[Cyclic product law]\\label{thm:cyclic_product}\nIf $F$ and $G$ are holomorphic on the same rotation-invariant domain, then\n\\[\n\\Bcal_{N,k}(FG)\n=\n\\sum_{\\substack{0\\leq i,j<N","\\begin{theorem}[Cyclic product law]\\label{thm:cyclic_product}\nIf $F$ and $G$ are holomorphic on the same rotation-invariant domain, then\n\\[\n\\Bcal_{N,k}(FG)\n=\n\\sum_{\\substack{0\\leq i,j<N\\\\ i+j\\equiv k\\bmod N}}\n(\\Bcal_{N,i}F)(\\Bcal_{N,j}G).\n\\]\n\\end{theorem}","\\begin{proof}\nWrite $F=\\sum_iF_i$ and $G=\\sum_jG_j$, where\n$F_i=\\Bcal_{N,i}F$ and $G_j=\\Bcal_{N,j}G$.  Their covariance gives\n\\[\n(F_iG_j)(\\omega z)=\\omega^{i+j}F_i(z)G_j(z),\n\\]\nso $F_iG_j$ belongs to the packet labeled by $i+j$ modulo $N$.\nExpanding the finite product and applying $\\Bcal_{N,k}$ gives the formula.\n\\end{proof}","\\begin{definition}\\label{def:transfer_exponential}\nFor $\\lambda\\in\\CC$ and $0\\leq j<N$, define\n\\[\n\\Efun_{N,j}(\\lambda z)\n=\n\\Bcal_{N,j}(e^{\\lambda z})\n=\n\\sum_{n\\geq0}\\frac{(\\lambda z)^{Nn+j}}{(Nn+j)!}.\n\\]\n\\end{definition}","\\begin{theorem}[Differential cycle and initial data]\\label{thm:exponential_cycle}\nFor $1\\leq j<N$,\n\\[\n\\frac{d}{dz}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda\\Efun_{N,j-1}(\\lambda z),\n\\]\nwhile\n\\[\n\\frac{d}{dz}\\Efun_{N,0}(\\lambda z)\n=\n\\lambda\\Efun_{N,N-1}(\\lambda z).\n\\]\nThus\n\\[\n\\frac{d^N}{dz^N}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda^N\\Efun_{N,j}(\\lambda z).\n\\]\nFor $0\\leq k<N$,\n\\[\n\\left.\\frac{d^k}{dz^k}\\Efun_{N,j}(\\lambda z)\\right|_{z=0}\n=\n\\delta_{kj}\\lambda^j.\n\\]\nIf $\\lambda\\ne0$, the $N$ functions $\\Efun_{N,j}(\\lambda z)$ form a basis of the solution space of $y^{(N)}=\\lambda^Ny$.\n\\end{theorem}","]\\label{thm:exponential_cycle}\nFor $1\\leq j<N$,\n\\[\n\\frac{d}{dz}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda\\Efun_{N,j-1}(\\lambda z),\n\\]\nwhile\n\\[\n\\frac{d}{dz}\\Efun_{N,0}(\\lambda z)\n=\n\\lambda\\Efun_{N,N-1}(\\lambda z).\n\\]\nThus\n\\[\n\\frac{d^N}{dz^N}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda^N\\Efun_{N,j}(\\lambda z).\n\\]\nFor $0\\leq k<N$,\n\\[\n\\left.\\frac{d^k}{dz^k}\\Efun_{N,j}(\\lambda z)\\right|_{z=0}\n=\n\\delta_{kj}\\lambda^j.\n\\]\nIf $\\lambda\\ne0$, the $N$ functions $\\Efun_{N,j}(\\lambda z)$ form a basis of the solution space of $y^{(N)}=\\lambda^Ny$.\n\\end{theorem}","tial_cycle}\nFor $1\\leq j<N$,\n\\[\n\\frac{d}{dz}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda\\Efun_{N,j-1}(\\lambda z),\n\\]\nwhile\n\\[\n\\frac{d}{dz}\\Efun_{N,0}(\\lambda z)\n=\n\\lambda\\Efun_{N,N-1}(\\lambda z).\n\\]\nThus\n\\[\n\\frac{d^N}{dz^N}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda^N\\Efun_{N,j}(\\lambda z).\n\\]\nFor $0\\leq k<N$,\n\\[\n\\left.\\frac{d^k}{dz^k}\\Efun_{N,j}(\\lambda z)\\right|_{z=0}\n=\n\\delta_{kj}\\lambda^j.\n\\]\nIf $\\lambda\\ne0$, the $N$ functions $\\Efun_{N,j}(\\lambda z)$ form a basis of the solution space of $y^{(N)}=\\lambda^Ny$.\n\\end{theorem}","r $1\\leq j<N$,\n\\[\n\\frac{d}{dz}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda\\Efun_{N,j-1}(\\lambda z),\n\\]\nwhile\n\\[\n\\frac{d}{dz}\\Efun_{N,0}(\\lambda z)\n=\n\\lambda\\Efun_{N,N-1}(\\lambda z).\n\\]\nThus\n\\[\n\\frac{d^N}{dz^N}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda^N\\Efun_{N,j}(\\lambda z).\n\\]\nFor $0\\leq k<N$,\n\\[\n\\left.\\frac{d^k}{dz^k}\\Efun_{N,j}(\\lambda z)\\right|_{z=0}\n=\n\\delta_{kj}\\lambda^j.\n\\]\nIf $\\lambda\\ne0$, the $N$ functions $\\Efun_{N,j}(\\lambda z)$ form a basis of the solution space of $y^{(N)}=\\lambda^Ny$.\n\\end{theorem}","Efun_{N,j-1}(\\lambda z),\n\\]\nwhile\n\\[\n\\frac{d}{dz}\\Efun_{N,0}(\\lambda z)\n=\n\\lambda\\Efun_{N,N-1}(\\lambda z).\n\\]\nThus\n\\[\n\\frac{d^N}{dz^N}\\Efun_{N,j}(\\lambda z)\n=\n\\lambda^N\\Efun_{N,j}(\\lambda z).\n\\]\nFor $0\\leq k<N$,\n\\[\n\\left.\\frac{d^k}{dz^k}\\Efun_{N,j}(\\lambda z)\\right|_{z=0}\n=\n\\delta_{kj}\\lambda^j.\n\\]\nIf $\\lambda\\ne0$, the $N$ functions $\\Efun_{N,j}(\\lambda z)$ form a basis of the solution space of $y^{(N)}=\\lambda^Ny$.\n\\end{theorem}","\\begin{proof}\nTermwise differentiation gives the cyclic derivative formulas.  One full cycle contributes $\\lambda^N$.  The first nonzero term of $\\Efun_{N,j}(\\lambda z)$ is $(\\lambda z)^j/j!$, which gives the initial data.  The initial value vectors are linearly independent when $\\lambda\\ne0$.  The solution space of an order $N$ constant coefficient equation has dimension $N$, so these functions form a basis.\n\\end{proof}","\\begin{corollary}[Wronskian of the transfer basis]\\label{cor:wronskians}\nAssume $N\\geq2$. For $\\lambda\\ne0$, the Wronskian of the ordered basis\n\\[\n\\Efun_{N,0}(\\lambda z),\\ldots,\\Efun_{N,N-1}(\\lambda z)\n\\]\nis the constant\n\\[\n\\lambda^{N(N-1)/2}.\n\\]\n\\end{corollary}","\\begin{proof}\nAt the origin, the derivative matrix is diagonal with entries\n\\[\n1,\\lambda,\\ldots,\\lambda^{N-1}.\n\\]\nFor an explicit determinant proof, write each basis function as a finite\nsum of exponentials with exponents $a_\\ell=\\lambda\\omega^\\ell$.\nIts derivative matrix factors as\n\\[\nW(z)=V\\,\\operatorname{diag}(e^{a_\\ell z})\\,C,\n\\qquad V_{k\\ell}=a_\\ell^k,\n\\]\nwhere $C$ is the constant matrix of Fourier weights. Thus\n\\[\n\\det W(z)=e^{z\\sum_\\ell a_\\ell}\\det W(0)=\\det W(0),\n\\]\nbecause $N\\geq2$ implies $\\sum_\\ell\\omega^\\ell=0$.\nThe product of the diagonal initial values is the stated power of $\\lambda$.\n\\end{proof}","igin, the derivative matrix is diagonal with entries\n\\[\n1,\\lambda,\\ldots,\\lambda^{N-1}.\n\\]\nFor an explicit determinant proof, write each basis function as a finite\nsum of exponentials with exponents $a_\\ell=\\lambda\\omega^\\ell$.\nIts derivative matrix factors as\n\\[\nW(z)=V\\,\\operatorname{diag}(e^{a_\\ell z})\\,C,\n\\qquad V_{k\\ell}=a_\\ell^k,\n\\]\nwhere $C$ is the constant matrix of Fourier weights. Thus\n\\[\n\\det W(z)=e^{z\\sum_\\ell a_\\ell}\\det W(0)=\\det W(0),\n\\]\nbecause $N\\geq2$ implies $\\sum_\\ell\\omega^\\ell=0$.\nThe product of the diagonal initial values is the stated power of $\\lambda$.\n\\end{proof}",",\\ldots,\\lambda^{N-1}.\n\\]\nFor an explicit determinant proof, write each basis function as a finite\nsum of exponentials with exponents $a_\\ell=\\lambda\\omega^\\ell$.\nIts derivative matrix factors as\n\\[\nW(z)=V\\,\\operatorname{diag}(e^{a_\\ell z})\\,C,\n\\qquad V_{k\\ell}=a_\\ell^k,\n\\]\nwhere $C$ is the constant matrix of Fourier weights. Thus\n\\[\n\\det W(z)=e^{z\\sum_\\ell a_\\ell}\\det W(0)=\\det W(0),\n\\]\nbecause $N\\geq2$ implies $\\sum_\\ell\\omega^\\ell=0$.\nThe product of the diagonal initial values is the stated power of $\\lambda$.\n\\end{proof}","N-1}.\n\\]\nFor an explicit determinant proof, write each basis function as a finite\nsum of exponentials with exponents $a_\\ell=\\lambda\\omega^\\ell$.\nIts derivative matrix factors as\n\\[\nW(z)=V\\,\\operatorname{diag}(e^{a_\\ell z})\\,C,\n\\qquad V_{k\\ell}=a_\\ell^k,\n\\]\nwhere $C$ is the constant matrix of Fourier weights. Thus\n\\[\n\\det W(z)=e^{z\\sum_\\ell a_\\ell}\\det W(0)=\\det W(0),\n\\]\nbecause $N\\geq2$ implies $\\sum_\\ell\\omega^\\ell=0$.\nThe product of the diagonal initial values is the stated power of $\\lambda$.\n\\end{proof}","onentials with exponents $a_\\ell=\\lambda\\omega^\\ell$.\nIts derivative matrix factors as\n\\[\nW(z)=V\\,\\operatorname{diag}(e^{a_\\ell z})\\,C,\n\\qquad V_{k\\ell}=a_\\ell^k,\n\\]\nwhere $C$ is the constant matrix of Fourier weights. Thus\n\\[\n\\det W(z)=e^{z\\sum_\\ell a_\\ell}\\det W(0)=\\det W(0),\n\\]\nbecause $N\\geq2$ implies $\\sum_\\ell\\omega^\\ell=0$.\nThe product of the diagonal initial values is the stated power of $\\lambda$.\n\\end{proof}","\\begin{corollary}[Addition law]\\label{cor:exponential_addition}\nFor $0\\leq k<N$,\n\\[\n\\Efun_{N,k}(\\lambda(z+w))\n=\n\\sum_{\\substack{0\\leq i,j<N","\\begin{corollary}[Addition law]\\label{cor:exponential_addition}\nFor $0\\leq k<N$,\n\\[\n\\Efun_{N,k}(\\lambda(z+w))\n=\n\\sum_{\\substack{0\\leq i,j<N\\\\ i+j\\equiv k\\bmod N}}\n\\Efun_{N,i}(\\lambda z)\\Efun_{N,j}(\\lambda w).\n\\]\n\\end{corollary}","\\begin{proof}\nExpand $e^{\\lambda(z+w)}=e^{\\lambda z}e^{\\lambda w}$ as a\nbivariate power series.  A term $z^a w^b$ contributes to the\n$k$th section exactly when $a+b\\equiv k\\bmod N$.  Grouping first by\n$i\\equiv a\\bmod N$ and $j\\equiv b\\bmod N$ gives the displayed sum.\n\\end{proof}","\\begin{theorem}[Hypergeometric form]\\label{thm:hypergeometric_form}\nLet\n\\[\nA_{N,j}\n=\n\\left\\{\n\\frac{j+m}{N}\n\\mathrel{:}\n1\\leq m\\leq N\n\\text{ and }\nj+m\\ne N\n\\right\\}.\n\\]\nThen\n\\[\n\\Efun_{N,j}(\\lambda z)\n=\n\\frac{(\\lambda z)^j}{j!}\n{}_0F_{N-1}\n\\left(\n\\begin{matrix}-","\\begin{theorem}[Hypergeometric form]\\label{thm:hypergeometric_form}\nLet\n\\[\nA_{N,j}\n=\n\\left\\{\n\\frac{j+m}{N}\n\\mathrel{:}\n1\\leq m\\leq N\n\\text{ and }\nj+m\\ne N\n\\right\\}.\n\\]\nThen\n\\[\n\\Efun_{N,j}(\\lambda z)\n=\n\\frac{(\\lambda z)^j}{j!}\n{}_0F_{N-1}\n\\left(\n\\begin{matrix}-\\\\ A_{N,j}\\end{matrix}\n\\mathrel{,}\n\\frac{(\\lambda z)^N}{N^N}\n\\right).\n\\]\n\\end{theorem}","\\begin{proof}\nThe multiplication formula for the gamma function gives\n\\[\n(Nn+j)!\n=\nj!N^{Nn}n!\n\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nIndeed, the $N$ Pochhammer factors supplied by the multiplication formula\nare $((j+m)/N)_n$ for $1\\leq m\\leq N$. The excluded factor is\n$((j+N-j)/N)_n=(1)_n=n!$.\nInsertion into the defining series gives the stated hypergeometric expansion.\n\\end{proof}","\\begin{example}\\label{ex:classical_exponential}\nFor $N=1$, the only section is $e^{\\lambda z}$.  For $N=2$,\n\\[\n\\Efun_{2,0}(z)=\\cosh z,\n\\qquad\n\\Efun_{2,1}(z)=\\sinh z.\n\\]\nThese are classical instances of the same cyclic basis.\n\\end{example}","\\begin{definition}\\label{def:signed_sections}\nFor $0\\leq j<N$, define\n\\[\n\\Dfun_{N,j}(z)\n=\n\\sum_{n\\geq0}(-1)^n\\frac{z^{Nn+j}}{(Nn+j)!}.\n\\]\nThe distinguished signed denominator is\n\\[\n\\Dfun_N(z)=\\Dfun_{N,0}(z).\n\\]\n\\end{definition}","Fix any $\\rho\\in\\CC$ with $\\rho^N=-1$.","\\begin{proposition}[Relation with the transfer exponential]\\label{prop:signed_exponential_relation}\nFor every $j$,\n\\[\n\\Dfun_{N,j}(z)\n=\n\\rho^{-j}\\Efun_{N,j}(\\rho z).\n\\]\n\\end{proposition}","\\begin{proof}\nThe coefficient of $z^{Nn+j}$ on the right is\n\\[\n\\rho^{-j}\\rho^{Nn+j}\\frac1{(Nn+j)!}\n=\n(-1)^n\\frac1{(Nn+j)!}.\n\\]\n\\end{proof}","\\begin{theorem}[Signed derivative cycle]\\label{thm:signed_cycle}\nFor $1\\leq j<N$,\n\\[\n\\Dfun_{N,j}'=\\Dfun_{N,j-1},\n\\]\nand\n\\[\n\\Dfun_{N,0}'=-\\Dfun_{N,N-1}.\n\\]\nEvery signed section satisfies\n\\[\n\\Dfun_{N,j}^{(N)}=-\\Dfun_{N,j}.\n\\]\nThe $N$ signed sections form the basis of $y^{(N)}=-y$ determined by\n\\[\n\\Dfun_{N,j}^{(k)}(0)=\\delta_{jk}\n\\]\nfor $0\\leq j,k<N$.\n\\end{theorem}","\\begin{proof}\nThe derivative formulas follow directly from the series.  The first nonzero term of $\\Dfun_{N,j}$ is $z^j/j!$, so the initial value matrix is the identity.  The basis statement follows from the dimension of the solution space.\n\\end{proof}","\\begin{corollary}[Wronskian of the signed basis]\\label{cor:signed_wronskian}\nAssume $N\\geq2$. The Wronskian of\n\\[\n\\Dfun_{N,0}(z),\\ldots,\\Dfun_{N,N-1}(z)\n\\]\nis one.\n\\end{corollary}","\\begin{proof}\nThe derivative matrix at the origin is the identity. The determinant\nfactorization in the proof of \\cref{cor:wronskians} applies with\n$a_\\ell=\\rho\\omega^\\ell$ and with the signed-section Fourier weights.\nSince $\\sum_\\ell a_\\ell=0$, the determinant is constant and equals one.\n\\end{proof}","\\begin{proposition}[Twisted addition law]\\label{prop:signed_addition}\nFor $0\\leq k<N$,\n\\[\n\\Dfun_{N,k}(z+w)\n=\n\\sum_{\\substack{0\\leq i,j<N","\\begin{proposition}[Twisted addition law]\\label{prop:signed_addition}\nFor $0\\leq k<N$,\n\\[\n\\Dfun_{N,k}(z+w)\n=\n\\sum_{\\substack{0\\leq i,j<N\\\\ i+j\\equiv k\\bmod N}}\n(-1)^{\\lfloor(i+j)/N\\rfloor}\n\\Dfun_{N,i}(z)\\Dfun_{N,j}(w).\n\\]\n\\end{proposition}","\\begin{proof}\nUse \\cref{prop:signed_exponential_relation,cor:exponential_addition}.  The factor introduced by a pair $(i,j)$ is\n\\[\n\\rho^{i+j-k}.\n\\]\nSince $0\\leq i+j\\leq2N-2$ and $i+j\\equiv k\\bmod N$, this factor is one when $i+j=k$ and $-1$ when $i+j=k+N$.\n\\end{proof}","\\begin{theorem}[Finite exponential representation]\\label{thm:finite_exponential_representation}\nLet\n\\[\n\\Rcal_N=\\{\\eta\\in\\CC\\mathrel{:}\\eta^N=-1\\}.\n\\]\nThen\n\\[\n\\Dfun_N(z)\n=\n\\frac1N\\sum_{\\eta\\in\\Rcal_N}e^{\\eta z}.\n\\]\nMore generally,\n\\[\n\\Dfun_{N,j}(z)\n=\n\\frac{\\rho^{-j}}N\n\\sum_{\\ell=0}^{N-1}\n\\omega^{-j\\ell}e^{\\rho\\omega^\\ell z}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe second formula is the root filter expression for $\\rho^{-j}\\Efun_{N,j}(\\rho z)$.  The first is the case $j=0$.\n\\end{proof}","\\begin{definition}\\label{def:reduced_denominator}\nDefine the reduced denominator\n\\[\n\\Delta_N(w)\n=\n\\sum_{n\\geq0}(-1)^n\\frac{w^n}{(Nn)!}.\n\\]\nThen\n\\[\n\\Dfun_N(z)=\\Delta_N(z^N).\n\\]\n\\end{definition}","\\begin{corollary}[Symmetries]\\label{cor:denominator_symmetries}\nThe denominator satisfies\n\\[\n\\Dfun_N(\\omega z)=\\Dfun_N(z)\n\\]\nand\n\\[\n\\Dfun_N(\\overline z)=\\overline{\\Dfun_N(z)}.\n\\]\nIts nonzero zeros occur in orbits under multiplication by the $N$th roots of unity and under complex conjugation.\n\\end{corollary}","\\begin{proof}\nThe first identity follows from the powers $z^{Nn}$.  The second follows from the real coefficients.  Both symmetries preserve the zero set and multiplicity.\n\\end{proof}","\\begin{example}\\label{ex:small_denominators}\nThe first denominators are\n\\[\n\\Dfun_1(z)=e^{-z},\n\\qquad\n\\Dfun_2(z)=\\cos z.\n\\]\nFor $N=3$,\n\\[\n\\Dfun_3(z)\n=\n\\frac13\\left(\ne^{-z}\n+2e^{z/2}\\cos\\left(\\frac{\\sqrt3z}{2}\\right)\n\\right).\n\\]\nFor $N=4$,\n\\[\n\\Dfun_4(z)\n=\n\\cos\\left(\\frac z{\\sqrt2}\\right)\n\\cosh\\left(\\frac z{\\sqrt2}\\right).\n\\]\n\\end{example}","\\begin{keyidea}\nThe signed shift is a finite carrier for the differential cycle. At order four, three basis steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression.\n\\end{keyidea}\n\\modelfigure{signed_cycle}{The exact signed shift $J_4$. Source columns and target rows describe the same action as the cycle on the left. Direct integer multiplication verifies $J_4^4=-I$.}{fig:signed-cycle}","steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression.\n\\end{keyidea}\n\\modelfigure{signed_cycle}{The exact signed shift $J_4$. Source columns and target rows describe the same action as the cycle on the left. Direct integer multiplication verifies $J_4^4=-I$.}{fig:signed-cycle}","\\begin{definition}\\label{def:cyclic_algebras}\nLet\n\\[\n\\mathfrak A_N^+=\\CC[\\xi]/(\\xi^N-1)\n\\]\nand\n\\[\n\\mathfrak A_N^-=\\CC[\\xi]/(\\xi^N+1).\n\\]\nEvery element has a unique representative of degree below $N$.\n\\end{definition}","\\begin{theorem}[Finite algebra exponential identities]\\label{thm:finite_algebra_exponentials}\nIn $\\mathfrak A_N^+$ one has\n\\[\ne^{\\lambda z\\xi}\n=\n\\sum_{j=0}^{N-1}\\Efun_{N,j}(\\lambda z)\\xi^j.\n\\]\nIn $\\mathfrak A_N^-$ one has\n\\[\ne^{z\\xi}\n=\n\\sum_{j=0}^{N-1}\\Dfun_{N,j}(z)\\xi^j.\n\\]\nDifferentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.\n\\end{theorem}","\\begin{proof}\nExpand the exponential and write every exponent uniquely as $Nn+j$.  In $\\mathfrak A_N^+$, the relation $\\xi^{Nn+j}=\\xi^j$ gives the transfer sections.  In $\\mathfrak A_N^-$, the relation $\\xi^{Nn+j}=(-1)^n\\xi^j$ gives the signed sections.","Differentiation multiplies by $\\lambda\\xi$ in the first algebra and by $\\xi$ in the second.  Reduction of the product by the defining relation gives the derivative cycles.  The identity\n\\[\ne^{(z+w)\\xi}=e^{z\\xi}e^{w\\xi}\n\\]\ngives the addition laws after comparison of the coefficients of $1,\\xi,\\ldots,\\xi^{N-1}$.\n\\end{proof}","\\begin{definition}\\label{def:signed_shift_matrix}\nLet $J_N$ be the signed cyclic shift on the standard basis $e_0,\\ldots,e_{N-1}$ of $\\CC^N$, defined by\n\\[\nJ_Ne_j=e_{j+1}\n\\quad (0\\leq j<N-1),\n\\qquad\nJ_Ne_{N-1}=-e_0.\n\\]\nThen $J_N^N=-I$.\n\\end{definition}","\\begin{theorem}[Companion matrix realization]\\label{thm:companion_matrix_realization}\nPut\n\\[\n\\mathbf D_N(z)\n=\n\\begin{pmatrix}\n\\Dfun_{N,0}(z)\\\\\n\\Dfun_{N,1}(z)\\\\\n\\vdots\\\\\n\\Dfun_{N,N-1}(z)\n\\end{pmatrix}.\n\\]\nThen\n\\[\n\\mathbf D_N(z)=e^{zJ_N}e_0\n\\]\nand\n\\[\n\\Dfun_N(z)=\\frac1N\\tr(e^{zJ_N}).\n\\]\nThe eigenvalues of $J_N$ are the $N$ roots of $-1$, each with multiplicity one.\n\\end{theorem}","\\Dfun_{N,N-1}(z)\n\\end{pmatrix}.\n\\]\nThen\n\\[\n\\mathbf D_N(z)=e^{zJ_N}e_0\n\\]\nand\n\\[\n\\Dfun_N(z)=\\frac1N\\tr(e^{zJ_N}).\n\\]\nThe eigenvalues of $J_N$ are the $N$ roots of $-1$, each with multiplicity one.\n\\end{theorem}","\\begin{proof}\nThe vector $e^{zJ_N}e_0$ has derivative $J_Ne^{zJ_N}e_0$ and initial value $e_0$.  Its components therefore satisfy the same signed derivative cycle and initial data as $\\mathbf D_N$.  Uniqueness for the linear system gives the first identity.","The vectors\n\\[\ne_0,J_Ne_0,\\ldots,J_N^{N-1}e_0\n\\]\nare the standard basis.  Thus $e_0$ is cyclic for $J_N$, so the\nminimal polynomial of $J_N$ has degree $N$.  Since $J_N^N=-I$, that\nminimal polynomial divides $x^N+1$ and therefore equals $x^N+1$.\nThe characteristic polynomial has the same degree and is divisible\nby the minimal polynomial, so it too equals $x^N+1$.  This polynomial\nhas distinct roots.  Consequently $J_N$ is diagonalizable and its\nspectrum consists of all $N$ roots of $-1$, each once.  Hence\n\\[\n\\tr(e^{zJ_N})\n=\n\\sum_{\\eta^N=-1}e^{\\eta z}\n=\nN\\Dfun_N(z)\n\\]\nby \\cref{thm:finite_exponential_representation}.\n\\end{proof}",",J_Ne_0,\\ldots,J_N^{N-1}e_0\n\\]\nare the standard basis.  Thus $e_0$ is cyclic for $J_N$, so the\nminimal polynomial of $J_N$ has degree $N$.  Since $J_N^N=-I$, that\nminimal polynomial divides $x^N+1$ and therefore equals $x^N+1$.\nThe characteristic polynomial has the same degree and is divisible\nby the minimal polynomial, so it too equals $x^N+1$.  This polynomial\nhas distinct roots.  Consequently $J_N$ is diagonalizable and its\nspectrum consists of all $N$ roots of $-1$, each once.  Hence\n\\[\n\\tr(e^{zJ_N})\n=\n\\sum_{\\eta^N=-1}e^{\\eta z}\n=\nN\\Dfun_N(z)\n\\]\nby \\cref{thm:finite_exponential_representation}.\n\\end{proof}","hus $e_0$ is cyclic for $J_N$, so the\nminimal polynomial of $J_N$ has degree $N$.  Since $J_N^N=-I$, that\nminimal polynomial divides $x^N+1$ and therefore equals $x^N+1$.\nThe characteristic polynomial has the same degree and is divisible\nby the minimal polynomial, so it too equals $x^N+1$.  This polynomial\nhas distinct roots.  Consequently $J_N$ is diagonalizable and its\nspectrum consists of all $N$ roots of $-1$, each once.  Hence\n\\[\n\\tr(e^{zJ_N})\n=\n\\sum_{\\eta^N=-1}e^{\\eta z}\n=\nN\\Dfun_N(z)\n\\]\nby \\cref{thm:finite_exponential_representation}.\n\\end{proof}","cyclic for $J_N$, so the\nminimal polynomial of $J_N$ has degree $N$.  Since $J_N^N=-I$, that\nminimal polynomial divides $x^N+1$ and therefore equals $x^N+1$.\nThe characteristic polynomial has the same degree and is divisible\nby the minimal polynomial, so it too equals $x^N+1$.  This polynomial\nhas distinct roots.  Consequently $J_N$ is diagonalizable and its\nspectrum consists of all $N$ roots of $-1$, each once.  Hence\n\\[\n\\tr(e^{zJ_N})\n=\n\\sum_{\\eta^N=-1}e^{\\eta z}\n=\nN\\Dfun_N(z)\n\\]\nby \\cref{thm:finite_exponential_representation}.\n\\end{proof}","minimal polynomial of $J_N$ has degree $N$.  Since $J_N^N=-I$, that\nminimal polynomial divides $x^N+1$ and therefore equals $x^N+1$.\nThe characteristic polynomial has the same degree and is divisible\nby the minimal polynomial, so it too equals $x^N+1$.  This polynomial\nhas distinct roots.  Consequently $J_N$ is diagonalizable and its\nspectrum consists of all $N$ roots of $-1$, each once.  Hence\n\\[\n\\tr(e^{zJ_N})\n=\n\\sum_{\\eta^N=-1}e^{\\eta z}\n=\nN\\Dfun_N(z)\n\\]\nby \\cref{thm:finite_exponential_representation}.\n\\end{proof}","\\begin{corollary}[Entries of the fundamental matrix]\\label{cor:fundamental_matrix_entries}\nFor $0\\leq p,q<N$, let $d_{pq}$ be the member of $\\{0,\\ldots,N-1\\}$ congruent to $p-q$ modulo $N$.  Put\n\\[\n\\varepsilon_{pq}\n=\n\\begin{cases}\n1,&p\\geq q,","\\begin{corollary}[Entries of the fundamental matrix]\\label{cor:fundamental_matrix_entries}\nFor $0\\leq p,q<N$, let $d_{pq}$ be the member of $\\{0,\\ldots,N-1\\}$ congruent to $p-q$ modulo $N$.  Put\n\\[\n\\varepsilon_{pq}\n=\n\\begin{cases}\n1,&p\\geq q,\\\\\n-1,&p<q.\n\\end{cases}\n\\]\nThen\n\\[\n\\bigl(e^{zJ_N}\\bigr)_{pq}\n=\n\\varepsilon_{pq}\\Dfun_{N,d_{pq}}(z).\n\\]\nThus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from $q$ to $p$ crosses the signed edge.\n\\end{corollary}","-1,&p<q.\n\\end{cases}\n\\]\nThen\n\\[\n\\bigl(e^{zJ_N}\\bigr)_{pq}\n=\n\\varepsilon_{pq}\\Dfun_{N,d_{pq}}(z).\n\\]\nThus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from $q$ to $p$ crosses the signed edge.\n\\end{corollary}","\\begin{proof}\nFor every nonnegative integer $n$,\n\\[\nJ_N^{Nn+d_{pq}}e_q\n=\n(-1)^n\\varepsilon_{pq}e_p.\n\\]\nOnly powers whose exponent is congruent to $d_{pq}$ modulo $N$ contribute to the $(p,q)$ entry of the exponential.  Therefore\n\\[\n\\bigl(e^{zJ_N}\\bigr)_{pq}\n=\n\\varepsilon_{pq}\n\\sum_{n\\geq0}(-1)^n\n\\frac{z^{Nn+d_{pq}}}{(Nn+d_{pq})!}.\n\\]\nThis is the stated formula.\n\\end{proof}","\\begin{corollary}[Addition and inversion laws for the companion matrix]\\label{cor:companion_addition_inversion}\nAssume $N\\geq2$.  The matrix\n\\[\nU_N(z)=e^{zJ_N}\n\\]\nsatisfies\n\\[\nU_N(z+w)=U_N(z)U_N(w),\n\\qquad\nU_N(z)^{-1}=U_N(-z),\n\\qquad\n\\det U_N(z)=1.\n\\]\nIts first column is $\\mathbf D_N(z)$, and all of its entries are given by \\cref{cor:fundamental_matrix_entries}.\n\\end{corollary}","\\begin{proof}\nThe first two identities are the exponential group law.  Since $\\tr J_N=0$,\n\\[\n\\det(e^{zJ_N})=e^{z\\tr J_N}=1.\n\\]\nThe first column statement is \\cref{thm:companion_matrix_realization}.\n\\end{proof}","\\begin{readerroadmap}\nA unit constant term makes the formal reciprocal unambiguous. To compute one coefficient, only finitely many earlier coefficients are needed. The later analytic theory explains how those exact coefficients relate to the full zero set.\n\\end{readerroadmap}\n\\modelfigure{reciprocal_normalization}{At $N=2$ and $r=1$, the reciprocal is $\\sec z$. The upper row gives block exponential coefficients and the lower row gives ordinary Taylor coefficients. Both rows encode the same five terms.}{fig:reciprocal-normalization}","Since $\\Dfun_N(0)=1$, its formal logarithm is defined at the origin.  On a\nsimply connected zero-free region containing the origin, there is a unique\nholomorphic branch normalized by $\\log\\Dfun_N(0)=0$, and its germ at the\norigin agrees with the formal logarithm.","\\begin{definition}\\label{def:reciprocal_family}\nFor $r\\in\\CC$, define\n\\[\n\\Sfun_N^{(r)}(z)\n=\n\\Dfun_N(z)^{-r}\n=\n\\exp\\bigl(-r\\log\\Dfun_N(z)\\bigr).\n\\]\nWrite\n\\[\n\\Sfun_N^{(r)}(z)\n=\n\\sum_{n\\geq0}P_{N,n}(r)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nDefine the logarithmic coefficients $\\kappa_{N,m}$ by\n\\[\n-\\log\\Delta_N(w)\n=\n\\sum_{m\\geq1}\\kappa_{N,m}\\frac{w^m}{m}.\n\\]\n\\end{definition}","The polynomial dependence on $r$ can be read directly from the formal logarithm.  We use the complete exponential Bell polynomials $B_n$ defined by\n\\[\n\\exp\\left(\\sum_{m\\geq1}x_m\\frac{t^m}{m!}\\right)\n=\n\\sum_{n\\geq0}B_n(x_1,\\ldots,x_n)\\frac{t^n}{n!}.\n\\]","\\begin{theorem}[Bell polynomial formula]\\label{thm:reciprocal_bell}\nFor every $n\\geq0$,\n\\[\nP_{N,n}(r)\n=\n\\frac{(Nn)!}{n!}\nB_n\\bigl(\n r\\kappa_{N,1},\n r\\,1!\\,\\kappa_{N,2},\n \\ldots,\n r\\,(n-1)!\\,\\kappa_{N,n}\n\\bigr).\n\\]\nFor $n\\geq1$, this is a polynomial in $r$ of exact degree $n$.  Its leading coefficient is\n\\[\n\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe identity\n\\[\n\\Delta_N(w)^{-r}\n=\n\\exp\\left(\n r\\sum_{m\\geq1}\\kappa_{N,m}\\frac{w^m}{m}\n\\right)\n\\]\nfits the defining generating function for $B_n$ with\n\\[\nx_m=r(m-1)!\\kappa_{N,m}.\n\\]\nThe coefficient of $w^n$ is therefore the displayed Bell polynomial divided by $n!$.  Multiplication by $(Nn)!$ gives $P_{N,n}(r)$.  Since\n\\[\n\\Delta_N(w)=1-\\frac{w}{N!}+O(w^2),\n\\]\nwe have $\\kappa_{N,1}=1/N!$.  The term $x_1^n$ gives the stated leading coefficient, which is nonzero.\n\\end{proof}","e identity\n\\[\n\\Delta_N(w)^{-r}\n=\n\\exp\\left(\n r\\sum_{m\\geq1}\\kappa_{N,m}\\frac{w^m}{m}\n\\right)\n\\]\nfits the defining generating function for $B_n$ with\n\\[\nx_m=r(m-1)!\\kappa_{N,m}.\n\\]\nThe coefficient of $w^n$ is therefore the displayed Bell polynomial divided by $n!$.  Multiplication by $(Nn)!$ gives $P_{N,n}(r)$.  Since\n\\[\n\\Delta_N(w)=1-\\frac{w}{N!}+O(w^2),\n\\]\nwe have $\\kappa_{N,1}=1/N!$.  The term $x_1^n$ gives the stated leading coefficient, which is nonzero.\n\\end{proof}","\\begin{theorem}[Parameter recurrence]\\label{thm:parameter_recurrence}\nWe have $P_{N,0}(r)=1$.  For $n\\geq1$,\n\\[\n\\begin{aligned}\nP_{N,n}(r)\n={}&-\n\\sum_{k=1}^{n-1}\n(-1)^k\n\\binom{Nn-1}{Nk}\nP_{N,n-k}(r)","\\begin{theorem}[Parameter recurrence]\\label{thm:parameter_recurrence}\nWe have $P_{N,0}(r)=1$.  For $n\\geq1$,\n\\[\n\\begin{aligned}\nP_{N,n}(r)\n={}&-\n\\sum_{k=1}^{n-1}\n(-1)^k\n\\binom{Nn-1}{Nk}\nP_{N,n-k}(r)\n\\\\\n&-r\n\\sum_{k=1}^{n}\n(-1)^k\n\\binom{Nn-1}{Nk-1}\nP_{N,n-k}(r).\n\\end{aligned}\n\\]\n\\end{theorem}","\\begin{proof}\nSet $S=\\Dfun_N^{-r}$.  Logarithmic differentiation gives\n\\[\n\\Dfun_NS'+r\\Dfun_N'S=0.\n\\]\nTaking the coefficient of $z^{Nn-1}/(Nn-1)!$ gives\n\\[\n\\sum_{k=0}^{n-1}(-1)^k\n\\binom{Nn-1}{Nk}P_{N,n-k}(r)\n+r\\sum_{k=1}^{n}(-1)^k\n\\binom{Nn-1}{Nk-1}P_{N,n-k}(r)=0.\n\\]\nThe term with $k=0$ in the first sum is $P_{N,n}(r)$, and every\nother term has a smaller block index.  Rearrangement gives the formula.\n\\end{proof}","\\begin{theorem}[Additive parameter law for reciprocal powers]\\label{thm:reciprocal_parameter_addition}\nFor all $r,s\\in\\CC$,\n\\[\n\\Sfun_N^{(r)}(z)\\Sfun_N^{(s)}(z)\n=\n\\Sfun_N^{(r+s)}(z)\n\\]\nas a formal identity and on every common zero-free region with a fixed logarithm.  Consequently,\n\\[\nP_{N,n}(r+s)\n=\n\\sum_{q=0}^{n}\n\\binom{Nn}{Nq}\nP_{N,q}(r)P_{N,n-q}(s).\n\\]\n\\end{theorem}","\\begin{corollary}[A sequence of binomial type]\\label{cor:binomial_type}\nDefine\n\\[\nA_{N,n}(r)\n=\n\\frac{n!}{(Nn)!}P_{N,n}(r).\n\\]\nThen\n\\[\nA_{N,n}(r+s)\n=\n\\sum_{q=0}^{n}\n\\binom nq\nA_{N,q}(r)A_{N,n-q}(s).\n\\]\nThus $(A_{N,n})_{n\\geq0}$ is a polynomial sequence of binomial type.\n\\end{corollary}","\\begin{proof}\nMultiply the coefficient identity in \\cref{thm:reciprocal_parameter_addition} by $n!/(Nn)!$ and simplify the factorials.\n\\end{proof}","\\begin{theorem}[Logarithmic Newton recurrence]\\label{thm:logarithmic_newton_recurrence}\nSet\n\\[\nb_{N,n}(r)=\\frac{P_{N,n}(r)}{(Nn)!}.\n\\]\nThen $b_{N,0}(r)=1$ and\nfor every $n\\geq1$,\n\\[\nnb_{N,n}(r)\n=\nr\\sum_{k=1}^{n}\\kappa_{N,k}b_{N,n-k}(r).\n\\]\nEquivalently,\n\\[\nP_{N,n}(r)\n=\n\\frac r n\n\\sum_{k=1}^{n}\n\\frac{(Nn)!}{(N(n-k))!}\n\\kappa_{N,k}P_{N,n-k}(r).\n\\]\nIf $p_k=r\\kappa_{N,k}$, then, for $n\\geq1$,\n\\[\nb_{N,n}(r)\n=\n\\frac1{n!}\n\\det\n\\begin{pmatrix}\np_1&-1&0&\\cdots&0","\\begin{theorem}[Logarithmic Newton recurrence]\\label{thm:logarithmic_newton_recurrence}\nSet\n\\[\nb_{N,n}(r)=\\frac{P_{N,n}(r)}{(Nn)!}.\n\\]\nThen $b_{N,0}(r)=1$ and\nfor every $n\\geq1$,\n\\[\nnb_{N,n}(r)\n=\nr\\sum_{k=1}^{n}\\kappa_{N,k}b_{N,n-k}(r).\n\\]\nEquivalently,\n\\[\nP_{N,n}(r)\n=\n\\frac r n\n\\sum_{k=1}^{n}\n\\frac{(Nn)!}{(N(n-k))!}\n\\kappa_{N,k}P_{N,n-k}(r).\n\\]\nIf $p_k=r\\kappa_{N,k}$, then, for $n\\geq1$,\n\\[\nb_{N,n}(r)\n=\n\\frac1{n!}\n\\det\n\\begin{pmatrix}\np_1&-1&0&\\cdots&0\\\\\np_2&p_1&-2&\\cdots&0\\\\\np_3&p_2&p_1&\\ddots&\\vdots\\\\\n\\vdots&\\vdots&\\ddots&\\ddots&-(n-1)\\\\\np_n&p_{n-1}&\\cdots&p_2&p_1\n\\end{pmatrix}.\n\\]\n\\end{theorem}","\\begin{proof}\nLet\n\\[\nB_r(w)=\\Delta_N(w)^{-r}\n=\n\\sum_{n\\geq0}b_{N,n}(r)w^n.\n\\]\nLogarithmic differentiation gives\n\\[\n\\frac{B_r'(w)}{B_r(w)}\n=\nr\\sum_{k\\geq1}\\kappa_{N,k}w^{k-1}.\n\\]\nComparison of the coefficient of $w^{n-1}$ proves the recurrence.","Let $D_n$ denote the displayed determinant, with $D_0=1$.  Expansion along the last row gives\n\\[\nD_n\n=\n\\sum_{k=1}^{n}\n\\frac{(n-1)!}{(n-k)!}p_kD_{n-k}.\n\\]\nHere the cofactor sign is canceled by the product of the negative\nsuperdiagonal entries $-(n-k+1),\\ldots,-(n-1)$.\nThus $D_n/n!$ satisfies the same recurrence and initial value as $b_{N,n}(r)$.\n\\end{proof}","\\begin{corollary}[The first three reciprocal polynomials]\\label{cor:first_reciprocal_polynomials}\nFor every $N\\geq1$,\n\\[\nP_{N,1}(r)=r,\n\\]\n\\[\nP_{N,2}(r)\n=\nr\\left[\n\\frac12\\binom{2N}{N}(r+1)-1\n\\right],\n\\]\nand\n\\[\nP_{N,3}(r)\n=\nr\\left[\n\\frac{(3N)!}{6(N!)^3}(r+1)(r+2)\n-\n\\binom{3N}{N}(r+1)\n+1\n\\right].\n\\]\n\\end{corollary}","\\begin{proof}\nThe first logarithmic coefficients are\n\\[\n\\kappa_{N,1}=\\frac1{N!},\n\\]\n\\[\n\\kappa_{N,2}=\\frac1{(N!)^2}-\\frac2{(2N)!},\n\\]\nand\n\\[\n\\kappa_{N,3}\n=\n\\frac1{(N!)^3}\n-\n\\frac3{N!(2N)!}\n+\n\\frac3{(3N)!}.\n\\]\nInsert these values into \\cref{thm:reciprocal_bell} and simplify.\n\\end{proof}","\\begin{corollary}\\label{cor:factor_r}\nFor $n\\geq1$, the polynomial $P_{N,n}(r)$ is divisible by $r$.\n\\end{corollary}","\\begin{proof}\nAt $r=0$, the reciprocal family is the constant function one.  Every positive-index block coefficient therefore vanishes.\n\\end{proof}","\\begin{example}\\label{ex:reciprocal_polynomials}\nFor $N=1$, one has $\\Dfun_1(z)^{-r}=e^{rz}$ and $P_{1,n}(r)=r^n$.  For $N=2$,\n\\[\nP_{2,1}(r)=r,\n\\]\n\\[\nP_{2,2}(r)=r(3r+2),\n\\]\nand\n\\[\nP_{2,3}(r)=r(15r^2+30r+16).\n\\]\nAt $r=1$, these are the positive secant coefficients.\n\\end{example}","\\begin{microexample}[title={A coefficient calculation at order two}]\nWrite $\\Delta_2(w)^{-r}=\\sum_{n\\geq0}b_n(r)w^n$. Since\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}+O(w^3),\n\\qquad -\\log\\Delta_2(w)=\\frac w2+\\frac{w^2}{12}+O(w^3),\n\\]\nthe first logarithmic coefficients are $\\kappa_{2,1}=1/2$ and $\\kappa_{2,2}=1/6$. Newton's recurrence gives\n\\[\nb_1(r)=\\frac r2,\\qquad\n2b_2(r)=r\\left(\\frac r4+\\frac16\\right).\n\\]\nRestoring the factorial normalization yields $P_{2,1}(r)=r$ and $P_{2,2}(r)=r(3r+2)$. At $r=1$, the second value is five. No zero-location assumption is needed for this finite calculation.\n\\end{microexample}\n\\modelfigure{parameter_polynomials}{The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at $r=1$ recover the first two pos","\\begin{microexample}[title={A coefficient calculation at order two}]\nWrite $\\Delta_2(w)^{-r}=\\sum_{n\\geq0}b_n(r)w^n$. Since\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}+O(w^3),\n\\qquad -\\log\\Delta_2(w)=\\frac w2+\\frac{w^2}{12}+O(w^3),\n\\]\nthe first logarithmic coefficients are $\\kappa_{2,1}=1/2$ and $\\kappa_{2,2}=1/6$. Newton's recurrence gives\n\\[\nb_1(r)=\\frac r2,\\qquad\n2b_2(r)=r\\left(\\frac r4+\\frac16\\right).\n\\]\nRestoring the factorial normalization yields $P_{2,1}(r)=r$ and $P_{2,2}(r)=r(3r+2)$. At $r=1$, the second value is five. No zero-location assumption is needed for this finite calculation.\n\\end{microexample}\n\\modelfigure{parameter_polynomials}{The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at $r=1$ recover the first two positive-index secant coefficients.}{fig:parameter-polynomials}","t order two}]\nWrite $\\Delta_2(w)^{-r}=\\sum_{n\\geq0}b_n(r)w^n$. Since\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}+O(w^3),\n\\qquad -\\log\\Delta_2(w)=\\frac w2+\\frac{w^2}{12}+O(w^3),\n\\]\nthe first logarithmic coefficients are $\\kappa_{2,1}=1/2$ and $\\kappa_{2,2}=1/6$. Newton's recurrence gives\n\\[\nb_1(r)=\\frac r2,\\qquad\n2b_2(r)=r\\left(\\frac r4+\\frac16\\right).\n\\]\nRestoring the factorial normalization yields $P_{2,1}(r)=r$ and $P_{2,2}(r)=r(3r+2)$. At $r=1$, the second value is five. No zero-location assumption is needed for this finite calculation.\n\\end{microexample}\n\\modelfigure{parameter_polynomials}{The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at $r=1$ recover the first two positive-index secant coefficients.}{fig:parameter-polynomials}","$\\Delta_2(w)^{-r}=\\sum_{n\\geq0}b_n(r)w^n$. Since\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}+O(w^3),\n\\qquad -\\log\\Delta_2(w)=\\frac w2+\\frac{w^2}{12}+O(w^3),\n\\]\nthe first logarithmic coefficients are $\\kappa_{2,1}=1/2$ and $\\kappa_{2,2}=1/6$. Newton's recurrence gives\n\\[\nb_1(r)=\\frac r2,\\qquad\n2b_2(r)=r\\left(\\frac r4+\\frac16\\right).\n\\]\nRestoring the factorial normalization yields $P_{2,1}(r)=r$ and $P_{2,2}(r)=r(3r+2)$. At $r=1$, the second value is five. No zero-location assumption is needed for this finite calculation.\n\\end{microexample}\n\\modelfigure{parameter_polynomials}{The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at $r=1$ recover the first two positive-index secant coefficients.}{fig:parameter-polynomials}","n\\geq0}b_n(r)w^n$. Since\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}+O(w^3),\n\\qquad -\\log\\Delta_2(w)=\\frac w2+\\frac{w^2}{12}+O(w^3),\n\\]\nthe first logarithmic coefficients are $\\kappa_{2,1}=1/2$ and $\\kappa_{2,2}=1/6$. Newton's recurrence gives\n\\[\nb_1(r)=\\frac r2,\\qquad\n2b_2(r)=r\\left(\\frac r4+\\frac16\\right).\n\\]\nRestoring the factorial normalization yields $P_{2,1}(r)=r$ and $P_{2,2}(r)=r(3r+2)$. At $r=1$, the second value is five. No zero-location assumption is needed for this finite calculation.\n\\end{microexample}\n\\modelfigure{parameter_polynomials}{The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at $r=1$ recover the first two positive-index secant coefficients.}{fig:parameter-polynomials}","c r4+\\frac16\\right).\n\\]\nRestoring the factorial normalization yields $P_{2,1}(r)=r$ and $P_{2,2}(r)=r(3r+2)$. At $r=1$, the second value is five. No zero-location assumption is needed for this finite calculation.\n\\end{microexample}\n\\modelfigure{parameter_polynomials}{The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at $r=1$ recover the first two positive-index secant coefficients.}{fig:parameter-polynomials}","\\begin{definition}\\label{def:transfer_tangent}\nDefine the transfer tangent\n\\[\n\\Tfun_N(z)\n=-\\frac{\\Dfun_N'(z)}{\\Dfun_N(z)}.\n\\]\nNear the origin, its convergent block expansion has the form\n\\[\n\\Tfun_N(z)\n=\n\\sum_{n\\geq0}T_N(n)\n\\frac{z^{Nn+N-1}}{(Nn+N-1)!}.\n\\]\n\\end{definition}","\\begin{proposition}[Logarithmic coefficient formula]\\label{prop:tangent_kappa}\nThe transfer tangent satisfies\n\\[\n\\Tfun_N(z)\n=\nN\\sum_{m\\geq1}\\kappa_{N,m}z^{Nm-1}.\n\\]\nEquivalently,\n\\[\nT_N(n)\n=\nN(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\n\\end{proposition}","\\begin{proof}\nSince $\\Dfun_N(z)=\\Delta_N(z^N)$,\n\\[\n-\\frac{\\Dfun_N'(z)}{\\Dfun_N(z)}\n=\n-Nz^{N-1}\\frac{\\Delta_N'(z^N)}{\\Delta_N(z^N)}.\n\\]\nDifferentiate the defining series for $-\\log\\Delta_N$ and compare coefficients.\n\\end{proof}","\\begin{theorem}[Tangent recurrence]\\label{thm:tangent_recurrence}\nThe coefficients satisfy $T_N(0)=1$ and\n\\[\nT_N(n)\n=\n(-1)^n\n-\n\\sum_{k=1}^{n}\n(-1)^k\n\\binom{Nn+N-1}{Nk}\nT_N(n-k).\n\\]\n\\end{theorem}","\\begin{proof}\nThe defining equation is\n\\[\n\\Dfun_N\\Tfun_N=-\\Dfun_N'.\n\\]\nExtracting the coefficient of $z^{Nn+N-1}/(Nn+N-1)!$ gives\n\\[\n\\sum_{k=0}^{n}(-1)^k\n\\binom{Nn+N-1}{Nk}T_N(n-k)=(-1)^n.\n\\]\nIsolating the $k=0$ term gives the recurrence.\n\\end{proof}","\\begin{corollary}[Derivative of the reciprocal family]\\label{cor:reciprocal_derivative}\nFor every $r\\in\\CC$, the formal identity\n\\[\n\\frac{d}{dz}\\Sfun_N^{(r)}(z)\n=\nr\\Tfun_N(z)\\Sfun_N^{(r)}(z)\n\\]\nholds.  The same identity holds analytically on every zero-free\ndomain on which the logarithm used to define $\\Sfun_N^{(r)}$ is fixed.\n\\end{corollary}","\\begin{proof}\nDifferentiate $\\Dfun_N^{-r}$ on a zero-free region.  The formal identity follows from the same logarithmic calculation.\n\\end{proof}","For a function $g$, define the complete differential Bell polynomial $Y_N$ by\n\\[\n\\frac{d^N}{dz^N}e^{g(z)}\n=\ne^{g(z)}Y_N\\bigl(g',g'',\\ldots,g^{(N)}\\bigr).\n\\]","\\begin{theorem}[Scalar higher Riccati identity]\\label{thm:higher_riccati}\nAt every point of the open set $\\{z:\\Dfun_N(z)\\ne0\\}$,\n\\[\nY_N\\bigl(\n-\\Tfun_N,\n-\\Tfun_N',\n\\ldots,\n-\\Tfun_N^{(N-1)}\n\\bigr)\n=-1.\n\\]\n\\end{theorem}","\\begin{proof}\nWrite $\\Dfun_N=e^g$ locally.  Then $g'=-\\Tfun_N$.  Divide the equation $\\Dfun_N^{(N)}=-\\Dfun_N$ by $\\Dfun_N$ and use the defining identity for $Y_N$.\n\\end{proof}","\\begin{example}\\label{ex:ordinary_tangent}\nFor $N=1$, $\\Tfun_1=1$.  For $N=2$, $\\Dfun_2=\\cos z$ and $\\Tfun_2=\\tan z$.  The Riccati equation becomes\n\\[\n\\Tfun_2'=1+\\Tfun_2^2.\n\\]\n\\end{example}","The section $\\Efun_{N,j}$ has a forced zero of order $j$ at the origin.  Removing it leaves a unit series in $z^N$.","\\begin{definition}\\label{def:normalized_section}\nFor $0\\leq j<N$, define\n\\[\n\\Phi_{N,j}(w)\n=\nj!\\sum_{n\\geq0}\\frac{w^n}{(Nn+j)!}.\n\\]\nThen\n\\[\n\\Efun_{N,j}(z)\n=\n\\frac{z^j}{j!}\\Phi_{N,j}(z^N).\n\\]\nDefine the residue-section reciprocal function\n\\[\n\\Cfun_{N,j}(z)\n=\n\\Phi_{N,j}(z^N)^{-1}\n=\n\\frac{z^j/j!}{\\Efun_{N,j}(z)}.\n\\]\nFor $j>0$, the last quotient at $z=0$ is interpreted by its removable\ncontinuation, namely the first expression in the display.\nWrite\n\\[\n\\Cfun_{N,j}(z)\n=\n\\sum_{n\\geq0}C_{N,j}(n)\\frac{z^{Nn}}{(Nn)!}.\n\\]\n\\end{definition}","or $0\\leq j<N$, define\n\\[\n\\Phi_{N,j}(w)\n=\nj!\\sum_{n\\geq0}\\frac{w^n}{(Nn+j)!}.\n\\]\nThen\n\\[\n\\Efun_{N,j}(z)\n=\n\\frac{z^j}{j!}\\Phi_{N,j}(z^N).\n\\]\nDefine the residue-section reciprocal function\n\\[\n\\Cfun_{N,j}(z)\n=\n\\Phi_{N,j}(z^N)^{-1}\n=\n\\frac{z^j/j!}{\\Efun_{N,j}(z)}.\n\\]\nFor $j>0$, the last quotient at $z=0$ is interpreted by its removable\ncontinuation, namely the first expression in the display.\nWrite\n\\[\n\\Cfun_{N,j}(z)\n=\n\\sum_{n\\geq0}C_{N,j}(n)\\frac{z^{Nn}}{(Nn)!}.\n\\]\n\\end{definition}","\\begin{theorem}[Triangular inverse system]\\label{thm:triangular_inverse}\nThe coefficients are determined by $C_{N,j}(0)=1$ and\n\\[\n\\sum_{q=0}^{n}\n\\binom{Nn+j}{Nq}\nC_{N,j}(q)\n=\n\\delta_{n0}.\n\\]\nFor $n\\geq1$,\n\\[\nC_{N,j}(n)\n=\n-\n\\frac{\n\\displaystyle\\sum_{q=0}^{n-1}\n\\binom{Nn+j}{Nq}C_{N,j}(q)\n}{\n\\displaystyle\\binom{Nn+j}{Nn}\n}.\n\\]\n\\end{theorem}","\\begin{proof}\nMultiply the defining series for $\\Cfun_{N,j}$ and $\\Efun_{N,j}$.  Their product is $z^j/j!$.  At degree $Nn+j$, multiplication by $(Nn+j)!$ gives the stated binomial coefficient.  The diagonal coefficient is nonzero, so the system is triangular.\n\\end{proof}","\\begin{definition}\\label{def:residue_section_power}\nFor $r\\in\\CC$, define formal coefficients $C_{N,j,n}(r)$ by\n\\[\n\\Cfun_{N,j}(z)^r\n=\n\\sum_{n\\geq0}C_{N,j,n}(r)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nOn a simply connected region containing $0$ and avoiding the zeros of $\\Phi_{N,j}(z^N)$, interpret this power as $e^{rL}$, where $L$ is the holomorphic logarithm of $\\Cfun_{N,j}$ normalized by $L(0)=0$.\n\\end{definition}","\\begin{theorem}[Parameter-addition law for residue-section reciprocals]\\label{thm:residue_section_parameter_addition}\nFor each $n\\geq1$, the coefficient $C_{N,j,n}(r)$ is a polynomial in $r$ of exact degree $n$.  Its leading coefficient is\n\\[\n\\frac{(Nn)!}{n!}\n\\left(-\\frac{j!}{(N+j)!}\\right)^n.\n\\]\nIt satisfies\n\\[\nC_{N,j,n}(r+s)\n=\n\\sum_{q=0}^{n}\n\\binom{Nn}{Nq}\nC_{N,j,q}(r)C_{N,j,n-q}(s).\n\\]\nIn particular,\n\\[\nC_{N,j,n}(1)=C_{N,j}(n).\n\\]\n\\end{theorem}","eorem}[Parameter-addition law for residue-section reciprocals]\\label{thm:residue_section_parameter_addition}\nFor each $n\\geq1$, the coefficient $C_{N,j,n}(r)$ is a polynomial in $r$ of exact degree $n$.  Its leading coefficient is\n\\[\n\\frac{(Nn)!}{n!}\n\\left(-\\frac{j!}{(N+j)!}\\right)^n.\n\\]\nIt satisfies\n\\[\nC_{N,j,n}(r+s)\n=\n\\sum_{q=0}^{n}\n\\binom{Nn}{Nq}\nC_{N,j,q}(r)C_{N,j,n-q}(s).\n\\]\nIn particular,\n\\[\nC_{N,j,n}(1)=C_{N,j}(n).\n\\]\n\\end{theorem}","\\begin{proof}\nThe unit series $\\Phi_{N,j}(w)^{-1}$ has a formal logarithm with zero constant term.  Its linear term is\n\\[\n-\\frac{j!}{(N+j)!}w.\n\\]\nExponentiation after multiplication by $r$ shows that the coefficient of $w^n$ is a polynomial of degree at most $n$.  The $n$th power of the linear logarithmic term gives the displayed nonzero leading coefficient, so the degree is exactly $n$.  Multiplication of the powers with parameters $r$ and $s$ gives the coefficient identity.\nMore explicitly, before the factorial normalization the coefficient of\n$r^nw^n$ is\n\\[\n\\frac1{n!}\\left(-\\frac{j!}{(N+j)!}\\right)^n;\n\\]\nthus multiplication by $(Nn)!$ gives the displayed leading coefficient.\n\\end{proof}",",j}(w)^{-1}$ has a formal logarithm with zero constant term.  Its linear term is\n\\[\n-\\frac{j!}{(N+j)!}w.\n\\]\nExponentiation after multiplication by $r$ shows that the coefficient of $w^n$ is a polynomial of degree at most $n$.  The $n$th power of the linear logarithmic term gives the displayed nonzero leading coefficient, so the degree is exactly $n$.  Multiplication of the powers with parameters $r$ and $s$ gives the coefficient identity.\nMore explicitly, before the factorial normalization the coefficient of\n$r^nw^n$ is\n\\[\n\\frac1{n!}\\left(-\\frac{j!}{(N+j)!}\\right)^n;\n\\]\nthus multiplication by $(Nn)!$ gives the displayed leading coefficient.\n\\end{proof}","-1}$ has a formal logarithm with zero constant term.  Its linear term is\n\\[\n-\\frac{j!}{(N+j)!}w.\n\\]\nExponentiation after multiplication by $r$ shows that the coefficient of $w^n$ is a polynomial of degree at most $n$.  The $n$th power of the linear logarithmic term gives the displayed nonzero leading coefficient, so the degree is exactly $n$.  Multiplication of the powers with parameters $r$ and $s$ gives the coefficient identity.\nMore explicitly, before the factorial normalization the coefficient of\n$r^nw^n$ is\n\\[\n\\frac1{n!}\\left(-\\frac{j!}{(N+j)!}\\right)^n;\n\\]\nthus multiplication by $(Nn)!$ gives the displayed leading coefficient.\n\\end{proof}","tiation after multiplication by $r$ shows that the coefficient of $w^n$ is a polynomial of degree at most $n$.  The $n$th power of the linear logarithmic term gives the displayed nonzero leading coefficient, so the degree is exactly $n$.  Multiplication of the powers with parameters $r$ and $s$ gives the coefficient identity.\nMore explicitly, before the factorial normalization the coefficient of\n$r^nw^n$ is\n\\[\n\\frac1{n!}\\left(-\\frac{j!}{(N+j)!}\\right)^n;\n\\]\nthus multiplication by $(Nn)!$ gives the displayed leading coefficient.\n\\end{proof}","multiplication by $r$ shows that the coefficient of $w^n$ is a polynomial of degree at most $n$.  The $n$th power of the linear logarithmic term gives the displayed nonzero leading coefficient, so the degree is exactly $n$.  Multiplication of the powers with parameters $r$ and $s$ gives the coefficient identity.\nMore explicitly, before the factorial normalization the coefficient of\n$r^nw^n$ is\n\\[\n\\frac1{n!}\\left(-\\frac{j!}{(N+j)!}\\right)^n;\n\\]\nthus multiplication by $(Nn)!$ gives the displayed leading coefficient.\n\\end{proof}","l of degree at most $n$.  The $n$th power of the linear logarithmic term gives the displayed nonzero leading coefficient, so the degree is exactly $n$.  Multiplication of the powers with parameters $r$ and $s$ gives the coefficient identity.\nMore explicitly, before the factorial normalization the coefficient of\n$r^nw^n$ is\n\\[\n\\frac1{n!}\\left(-\\frac{j!}{(N+j)!}\\right)^n;\n\\]\nthus multiplication by $(Nn)!$ gives the displayed leading coefficient.\n\\end{proof}","\\begin{definition}\\label{def:block_appell}\nDefine polynomials $E_m^{(N,j,r)}(x)$ by\n\\[\n\\Cfun_{N,j}(z)^re^{xz}\n=\n\\sum_{m\\geq0}E_m^{(N,j,r)}(x)\\frac{z^m}{m!}.\n\\]\n\\end{definition}","\\begin{theorem}[Appell laws]\\label{thm:appell_laws}\nFor $m\\geq1$, the polynomials satisfy\n\\[\n\\frac{d}{dx}E_m^{(N,j,r)}(x)\n=\nmE_{m-1}^{(N,j,r)}(x)\n\\]\nand, for $m\\geq0$,\n\\[\nE_m^{(N,j,r)}(x+y)\n=\n\\sum_{q=0}^{m}\n\\binom mq\nE_q^{(N,j,r)}(x)y^{m-q}.\n\\]\n\\end{theorem}","\\begin{proof}\nDifferentiate the generating function with respect to $x$ for the first identity.  Multiply it by $e^{yz}$ and compare coefficients for the second.\n\\end{proof}","\\begin{proposition}[Explicit Appell polynomial]\\label{prop:explicit_appell}\nFor every $m\\geq0$,\n\\[\nE_m^{(N,j,r)}(x)\n=\n\\sum_{0\\leq n\\leq m/N}\n\\binom{m}{Nn}\nC_{N,j,n}(r)x^{m-Nn}.\n\\]\n\\end{proposition}","\\begin{proof}\nMultiply the block series for $\\Cfun_{N,j}^r$ by the exponential series and collect the coefficient of total degree $m$.\n\\end{proof}","\\begin{theorem}[Joint Appell convolution in the space and order parameters]\\label{thm:appell_two_parameter_convolution}\nFor all $r,s,x,y\\in\\CC$ and $m\\geq0$,\n\\[\nE_m^{(N,j,r+s)}(x+y)\n=\n\\sum_{q=0}^{m}\n\\binom mq\nE_q^{(N,j,r)}(x)\nE_{m-q}^{(N,j,s)}(y).\n\\]\n\\end{theorem}","\\begin{proof}\nMultiply the generating functions\n\\[\n\\Cfun_{N,j}(z)^re^{xz}\n\\quad\\hbox{and}\\quad\n\\Cfun_{N,j}(z)^se^{yz}.\n\\]\nTheir product is the generating function with parameters $r+s$ and $x+y$.  Coefficient comparison gives the formula.\n\\end{proof}","\\begin{proposition}[Formal differential-operator representation on polynomials]\\label{prop:appell_operator_form}\nLet a formal series $A(z)=\\sum_{q\\geq0}a_qz^q/q!$ act on polynomials by\n\\[\nA(\\partial_x)f(x)\n=\n\\sum_{q\\geq0}\\frac{a_q}{q!}\\partial_x^qf(x).\n\\]\nThe power $r$ is applied to the formal symbol before the substitution $z=\\partial_x$.  Then\n\\[\nE_m^{(N,j,r)}(x)\n=\n\\Cfun_{N,j}(\\partial_x)^r x^m.\n\\]\nThe inverse relation is\n\\[\nx^m\n=\n\\Phi_{N,j}(\\partial_x^N)^r\nE_m^{(N,j,r)}(x).\n\\]\nBoth sums are finite on every polynomial.\n\\end{proposition}","ferential-operator representation on polynomials]\\label{prop:appell_operator_form}\nLet a formal series $A(z)=\\sum_{q\\geq0}a_qz^q/q!$ act on polynomials by\n\\[\nA(\\partial_x)f(x)\n=\n\\sum_{q\\geq0}\\frac{a_q}{q!}\\partial_x^qf(x).\n\\]\nThe power $r$ is applied to the formal symbol before the substitution $z=\\partial_x$.  Then\n\\[\nE_m^{(N,j,r)}(x)\n=\n\\Cfun_{N,j}(\\partial_x)^r x^m.\n\\]\nThe inverse relation is\n\\[\nx^m\n=\n\\Phi_{N,j}(\\partial_x^N)^r\nE_m^{(N,j,r)}(x).\n\\]\nBoth sums are finite on every polynomial.\n\\end{proposition}","\\begin{proof}\nApplying $A(\\partial_x)$ to $e^{xz}$ multiplies it by $A(z)$.  Taking\n$A=\\Cfun_{N,j}^r$ gives the first identity by the defining generating\nfunction.  Constant coefficient operators on polynomials satisfy\n\\[\n(AB)(\\partial_x)=A(\\partial_x)B(\\partial_x),\n\\]\nbecause all derivative sums are finite and powers of $\\partial_x$\ncommute.  The two formal symbols\n\\[\n\\Cfun_{N,j}(z)^r=\\Phi_{N,j}(z^N)^{-r}\n\\quad\\hbox{and}\\quad\n\\Phi_{N,j}(z^N)^r\n\\]\nmultiply to one.  Their substitutions at $z=\\partial_x$ are therefore\ninverse constant coefficient operators on polynomials, which proves\nthe second identity.\n\\end{proof}","N,j}^r$ gives the first identity by the defining generating\nfunction.  Constant coefficient operators on polynomials satisfy\n\\[\n(AB)(\\partial_x)=A(\\partial_x)B(\\partial_x),\n\\]\nbecause all derivative sums are finite and powers of $\\partial_x$\ncommute.  The two formal symbols\n\\[\n\\Cfun_{N,j}(z)^r=\\Phi_{N,j}(z^N)^{-r}\n\\quad\\hbox{and}\\quad\n\\Phi_{N,j}(z^N)^r\n\\]\nmultiply to one.  Their substitutions at $z=\\partial_x$ are therefore\ninverse constant coefficient operators on polynomials, which proves\nthe second identity.\n\\end{proof}","\\begin{corollary}[Appell symmetry]\\label{cor:appell_symmetry}\nEach polynomial $E_m^{(N,j,r)}$ is monic of degree $m$, and\n\\[\nE_m^{(N,j,r)}(\\omega x)\n=\n\\omega^mE_m^{(N,j,r)}(x).\n\\]\n\\end{corollary}","\\begin{proof}\nThe term $n=0$ in \\cref{prop:explicit_appell} is $x^m$.  Every other exponent is $m-Nn$, so replacement of $x$ by $\\omega x$ multiplies every term by $\\omega^m$.\n\\end{proof}","\\begin{example}\\label{ex:euler_boundaries}\nFor $(N,j)=(2,0)$,\n\\[\n\\Cfun_{2,0}(z)=\\sech z.\n\\]\nFor $(N,j)=(2,1)$,\n\\[\n\\Cfun_{2,1}(z)=\\frac z{\\sinh z}.\n\\]\nThe first case is a scaled and translated Euler family.  In fact, if $E_m(x)$ denotes the classical Euler polynomial, then\n\\[\nE_m^{(2,0,1)}(x)\n=\n2^mE_m\\left(\\frac{x+1}{2}\\right).\n\\]\nIf $B_m(x)$ denotes the classical Bernoulli polynomial, the second case has\nthe exact identification\n\\[\nE_m^{(2,1,1)}(x)\n=\n2^mB_m\\left(\\frac{x+1}{2}\\right),\n\\]\nobtained by substituting $t=2z$ in\n$t e^{yt}/(e^t-1)$.\n\\end{example}","daries}\nFor $(N,j)=(2,0)$,\n\\[\n\\Cfun_{2,0}(z)=\\sech z.\n\\]\nFor $(N,j)=(2,1)$,\n\\[\n\\Cfun_{2,1}(z)=\\frac z{\\sinh z}.\n\\]\nThe first case is a scaled and translated Euler family.  In fact, if $E_m(x)$ denotes the classical Euler polynomial, then\n\\[\nE_m^{(2,0,1)}(x)\n=\n2^mE_m\\left(\\frac{x+1}{2}\\right).\n\\]\nIf $B_m(x)$ denotes the classical Bernoulli polynomial, the second case has\nthe exact identification\n\\[\nE_m^{(2,1,1)}(x)\n=\n2^mB_m\\left(\\frac{x+1}{2}\\right),\n\\]\nobtained by substituting $t=2z$ in\n$t e^{yt}/(e^t-1)$.\n\\end{example}","n_{2,1}(z)=\\frac z{\\sinh z}.\n\\]\nThe first case is a scaled and translated Euler family.  In fact, if $E_m(x)$ denotes the classical Euler polynomial, then\n\\[\nE_m^{(2,0,1)}(x)\n=\n2^mE_m\\left(\\frac{x+1}{2}\\right).\n\\]\nIf $B_m(x)$ denotes the classical Bernoulli polynomial, the second case has\nthe exact identification\n\\[\nE_m^{(2,1,1)}(x)\n=\n2^mB_m\\left(\\frac{x+1}{2}\\right),\n\\]\nobtained by substituting $t=2z$ in\n$t e^{yt}/(e^t-1)$.\n\\end{example}","=\\frac z{\\sinh z}.\n\\]\nThe first case is a scaled and translated Euler family.  In fact, if $E_m(x)$ denotes the classical Euler polynomial, then\n\\[\nE_m^{(2,0,1)}(x)\n=\n2^mE_m\\left(\\frac{x+1}{2}\\right).\n\\]\nIf $B_m(x)$ denotes the classical Bernoulli polynomial, the second case has\nthe exact identification\n\\[\nE_m^{(2,1,1)}(x)\n=\n2^mB_m\\left(\\frac{x+1}{2}\\right),\n\\]\nobtained by substituting $t=2z$ in\n$t e^{yt}/(e^t-1)$.\n\\end{example}","\\begin{definition}\\label{def:unsigned_denominator}\nSet\n\\[\nH_N(z)=\\Efun_{N,0}(z)\n=\n\\sum_{n\\geq0}\\frac{z^{Nn}}{(Nn)!}.\n\\]\nFor $u\\in\\CC$, define $\\Thetafun_{N,n}(u)$ by\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\n\\sum_{n\\geq0}\\Thetafun_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nThis is first a formal identity at the origin.  Analytically, it is interpreted\non a common simply connected zero-free region with the branch of\n$\\log H_N$ normalized by $\\log H_N(0)=0$.\n\\end{definition}","el{def:unsigned_denominator}\nSet\n\\[\nH_N(z)=\\Efun_{N,0}(z)\n=\n\\sum_{n\\geq0}\\frac{z^{Nn}}{(Nn)!}.\n\\]\nFor $u\\in\\CC$, define $\\Thetafun_{N,n}(u)$ by\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\n\\sum_{n\\geq0}\\Thetafun_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nThis is first a formal identity at the origin.  Analytically, it is interpreted\non a common simply connected zero-free region with the branch of\n$\\log H_N$ normalized by $\\log H_N(0)=0$.\n\\end{definition}","\\begin{theorem}[Theta reconstruction]\\label{thm:theta_reconstruction}\nFor every $n\\geq0$,\n\\[\n\\Thetafun_{N,n}(u)\n=\n(-1)^n\n\\sum_{q=0}^{n}\n\\binom{Nn}{Nq}\nP_{N,q}(1)u^{Nq}P_{N,n-q}(u).\n\\]\nThus $\\Thetafun_{N,n}$ is a polynomial of degree at most $Nn$.\n\\end{theorem}","\\begin{proof}\nChoose $\\rho$ with $\\rho^N=-1$.  Then\n\\[\nH_N(z)=\\Dfun_N(\\rho z).\n\\]\nThe coefficient of $z^{Nn}$ in $H_N(z)^{-r}$ is therefore $(-1)^nP_{N,n}(r)/(Nn)!$.  Apply this to $H_N(uz)^{-1}$ and $H_N(z)^{-u}$, then multiply the two block series.\n\\end{proof}","\\begin{proposition}[Trivial parameter factors]\\label{prop:theta_factors}\nFor $n\\geq1$, the polynomial $\\Thetafun_{N,n}(u)$ is divisible by $u$.  If $N$ is even, it is also divisible by $u+1$.\n\\end{proposition}","\\begin{proof}\nAt $u=0$, the defining product is one.  Hence every positive-index coefficient vanishes.  If $N$ is even, then $H_N(-z)=H_N(z)$.  At $u=-1$, the product is again one, so every positive-index coefficient vanishes there as well.\n\\end{proof}","\\begin{proposition}[Theta Newton recurrence]\\label{prop:theta_newton_recurrence}\nPut\n\\[\n\\theta_{N,n}(u)=\\frac{\\Thetafun_{N,n}(u)}{(Nn)!}.\n\\]\nThen $\\theta_{N,0}(u)=1$ and\n\\[\nn\\theta_{N,n}(u)\n=\n\\sum_{m=1}^{n}\n(-1)^m\\kappa_{N,m}\n\\bigl(u^{Nm}+u\\bigr)\n\\theta_{N,n-m}(u).\n\\]\nIn particular,\n\\[\n\\Thetafun_{N,1}(u)=-(u^N+u).\n\\]\n\\end{proposition}","\\begin{proof}\nWrite $w=z^N$.  Since $H_N(z)=\\Delta_N(-w)$,\n\\[\n-\\log H_N(z)\n=\n\\sum_{m\\geq1}(-1)^m\\kappa_{N,m}\\frac{w^m}{m}.\n\\]\nThe logarithm of the theta generating function is therefore\n\\[\n\\sum_{m\\geq1}\n(-1)^m\\kappa_{N,m}\n\\bigl(u^{Nm}+u\\bigr)\n\\frac{w^m}{m}.\n\\]\nLogarithmic differentiation and coefficient comparison give the recurrence.  The case $n=1$ uses $\\kappa_{N,1}=1/N!$.\n\\end{proof}","\\begin{example}\\label{ex:theta_even}\nFor $N=2$,\n\\[\n\\Thetafun_{2,1}(u)=-u(u+1)\n\\]\nand\n\\[\n\\Thetafun_{2,2}(u)\n=\nu(u+1)(5u^2+u+2).\n\\]\nThe two factors predicted by \\cref{prop:theta_factors} are visible.\n\\end{example}","ion gives a polygon of exponents. Large zeros can occur where two dominant terms nearly cancel. The proof first localizes those zeros and then uses symmetry and uniqueness to place them on the rays. Finite exceptional zeros remain part of the later products and sums.\n\\end{readerroadmap}\n\\modelfigure{zero_orbits}{The exact order-four model. The first three zeros on each of four rays are mapped by $w=z^4$ to three positive reduced zeros. The reduced axis uses a logarithmic scale. This complete order-four example does not classify the finite exceptional set at other orders.}{fig:zero-orbits}","We now assume $N\\geq2$.  Write\n\\[\n\\eta_k=e^{(2k+1)\\pi i/N},\n\\qquad 0\\leq k<N.\n\\]\nThese are the vertices of a regular polygon and\n\\[\n\\Dfun_N(z)=\\frac1N\\sum_{k=0}^{N-1}e^{\\eta_kz}.\n\\]","\\begin{standardfact}[Order from Taylor coefficients]\\label{sf:coefficient_order}\nIf $f(w)=\\sum_{n\\geq0}a_nw^n$ is an entire function with infinitely many nonzero coefficients, then\n\\[\n\\ord f\n=\n\\limsup_{\\substack{n\\to\\infty","\\begin{standardfact}[Order from Taylor coefficients]\\label{sf:coefficient_order}\nIf $f(w)=\\sum_{n\\geq0}a_nw^n$ is an entire function with infinitely many nonzero coefficients, then\n\\[\n\\ord f\n=\n\\limsup_{\\substack{n\\to\\infty\\\\a_n\\ne0}}\n\\frac{n\\log n}{\\log(1/|a_n|)}.\n\\]\nCanonical products for entire functions of order below one have genus zero.  We use these standard facts in the form given in \\cite{Boas1954,Titchmarsh1939}.\n\\end{standardfact}","\\begin{proposition}[Orders and exponential type]\\label{prop:orders}\nFor $N\\geq2$,\n\\[\n\\ord\\Delta_N=\\frac1N.\n\\]\nThe function $\\Dfun_N$ has order one and exponential type one.\n\\end{proposition}","\\begin{proof}\nThe coefficient of $w^n$ in $\\Delta_N$ has modulus $1/(Nn)!$.\nStirling's formula and \\cref{sf:coefficient_order} give order $1/N$.\nSince $M_{f(\\,\\cdot\\,^N)}(r)=M_f(r^N)$ for the maximum modulus,\ncomposition with $z^N$ multiplies the order by $N$ and gives order one\nfor $\\Dfun_N$.","The finite exponential representation gives\n\\[\n|\\Dfun_N(z)|\\leq e^{|z|},\n\\]\nso the type is at most one.  Choose a root $\\eta_0$ and put $z=t\\overline{\\eta_0}$ with $t>0$.  The term $e^{\\eta_0z}=e^t$ has strictly larger real exponent than every other term.  Hence\n\\[\n\\Dfun_N(t\\overline{\\eta_0})\n=\n\\frac1Ne^t(1+o(1)).\n\\]\nThe type is therefore one.\n\\end{proof}","\\begin{definition}\\label{def:indicator_polygon}\nFor $\\theta\\in\\mathbb R$, define\n\\[\nh_N(\\theta)\n=\n\\max_{0\\leq k<N}\n\\cos\\left(\\theta+\\frac{(2k+1)\\pi}{N}\\right).\n\\]\nThe tie directions are\n\\[\n\\mathfrak T_N\n=\n\\left\\{\\frac{2\\pi k}{N}:0\\leq k<N\\right\\}\n\\quad\\bmod 2\\pi.\n\\]\n\\end{definition}","\\begin{theorem}[Indicator polygon and dominant sectors]\\label{thm:indicator_polygon}\nThe indicator of $\\Dfun_N$ equals $h_N$, with the limsup taken over radii where $\\Dfun_N(re^{i\\theta})\\ne0$:\n\\[\n\\limsup_{r\\to\\infty}\n\\frac1r\\log|\\Dfun_N(re^{i\\theta})|\n=\nh_N(\\theta).\n\\]\nIf $K$ is a compact set of directions disjoint from $\\mathfrak T_N$, then there are $\\delta>0$ and $R>0$ such that, for each $\\theta\\in K$, one root $\\eta(\\theta)$ of $-1$ satisfies\n\\[\n\\Dfun_N(re^{i\\theta})\n=\n\\frac1N\n\\exp\\bigl(\\eta(\\theta)re^{i\\theta}\\bigr)\n\\left(1+O(e^{-\\delta r})\\right)\n\\]\nuniformly for $r\\geq R$.  In particular, no sufficiently large zero can lie in those directions.\n\\end{theorem}","ator polygon and dominant sectors]\\label{thm:indicator_polygon}\nThe indicator of $\\Dfun_N$ equals $h_N$, with the limsup taken over radii where $\\Dfun_N(re^{i\\theta})\\ne0$:\n\\[\n\\limsup_{r\\to\\infty}\n\\frac1r\\log|\\Dfun_N(re^{i\\theta})|\n=\nh_N(\\theta).\n\\]\nIf $K$ is a compact set of directions disjoint from $\\mathfrak T_N$, then there are $\\delta>0$ and $R>0$ such that, for each $\\theta\\in K$, one root $\\eta(\\theta)$ of $-1$ satisfies\n\\[\n\\Dfun_N(re^{i\\theta})\n=\n\\frac1N\n\\exp\\bigl(\\eta(\\theta)re^{i\\theta}\\bigr)\n\\left(1+O(e^{-\\delta r})\\right)\n\\]\nuniformly for $r\\geq R$.  In particular, no sufficiently large zero can lie in those directions.\n\\end{theorem}","ominant sectors]\\label{thm:indicator_polygon}\nThe indicator of $\\Dfun_N$ equals $h_N$, with the limsup taken over radii where $\\Dfun_N(re^{i\\theta})\\ne0$:\n\\[\n\\limsup_{r\\to\\infty}\n\\frac1r\\log|\\Dfun_N(re^{i\\theta})|\n=\nh_N(\\theta).\n\\]\nIf $K$ is a compact set of directions disjoint from $\\mathfrak T_N$, then there are $\\delta>0$ and $R>0$ such that, for each $\\theta\\in K$, one root $\\eta(\\theta)$ of $-1$ satisfies\n\\[\n\\Dfun_N(re^{i\\theta})\n=\n\\frac1N\n\\exp\\bigl(\\eta(\\theta)re^{i\\theta}\\bigr)\n\\left(1+O(e^{-\\delta r})\\right)\n\\]\nuniformly for $r\\geq R$.  In particular, no sufficiently large zero can lie in those directions.\n\\end{theorem}","\\label{thm:indicator_polygon}\nThe indicator of $\\Dfun_N$ equals $h_N$, with the limsup taken over radii where $\\Dfun_N(re^{i\\theta})\\ne0$:\n\\[\n\\limsup_{r\\to\\infty}\n\\frac1r\\log|\\Dfun_N(re^{i\\theta})|\n=\nh_N(\\theta).\n\\]\nIf $K$ is a compact set of directions disjoint from $\\mathfrak T_N$, then there are $\\delta>0$ and $R>0$ such that, for each $\\theta\\in K$, one root $\\eta(\\theta)$ of $-1$ satisfies\n\\[\n\\Dfun_N(re^{i\\theta})\n=\n\\frac1N\n\\exp\\bigl(\\eta(\\theta)re^{i\\theta}\\bigr)\n\\left(1+O(e^{-\\delta r})\\right)\n\\]\nuniformly for $r\\geq R$.  In particular, no sufficiently large zero can lie in those directions.\n\\end{theorem}","\\theta})\\ne0$:\n\\[\n\\limsup_{r\\to\\infty}\n\\frac1r\\log|\\Dfun_N(re^{i\\theta})|\n=\nh_N(\\theta).\n\\]\nIf $K$ is a compact set of directions disjoint from $\\mathfrak T_N$, then there are $\\delta>0$ and $R>0$ such that, for each $\\theta\\in K$, one root $\\eta(\\theta)$ of $-1$ satisfies\n\\[\n\\Dfun_N(re^{i\\theta})\n=\n\\frac1N\n\\exp\\bigl(\\eta(\\theta)re^{i\\theta}\\bigr)\n\\left(1+O(e^{-\\delta r})\\right)\n\\]\nuniformly for $r\\geq R$.  In particular, no sufficiently large zero can lie in those directions.\n\\end{theorem}","\\begin{proof}\nFor $z=re^{i\\theta}$, the real part of the exponent $\\eta_kz$ is\n\\[\nr\\cos\\left(\\theta+\\frac{(2k+1)\\pi}{N}\\right).\n\\]\nThe triangle inequality gives the upper indicator bound $h_N(\\theta)$.","At a tie direction exactly two adjacent vertices attain the maximum.  After rotation it is enough to take $\\theta=0$.  With $c_N=\\cos(\\pi/N)$ and $s_N=\\sin(\\pi/N)$, the two dominant terms give\n\\[\n\\frac2N e^{c_Nr}\\cos(s_Nr)\n\\]\nand every other term is exponentially smaller.  Along a sequence on which $|\\cos(s_Nr)|\\geq1/2$, the lower indicator bound is $c_N=h_N(0)$.  Rotational symmetry treats every tie direction.\n\\end{proof}","The zero strings are governed by two adjacent exponents.  Put\n\\[\na_N=\\frac\\pi N,\n\\qquad\ns_N=\\sin a_N,\n\\qquad\nc_N=\\cos a_N,\n\\]\nand\n\\[\nA_N=\\frac\\pi{s_N},\n\\qquad\n t_m=A_N\\left(m+\\frac12\\right).\n\\]\nFor $N\\geq3$, also put\n\\[\n\\gamma_N=c_N-\\cos(3a_N)>0.\n\\]","\\begin{lemma}[Two term dominance]\\label{lem:two_term_dominance}\nAssume $N\\geq3$.  There are positive constants $Y$, $R$, and $C$ with the following properties.","If $z=x+iy$ lies in the sector $|\\arg z|\\leq a_N$, has $x\\geq R$, and is a zero of $\\Dfun_N$, then $|y|\\leq Y$.","On the strip $x\\geq R$ and $|y|\\leq Y$, define\n\\[\nR_N(z)\n=\n\\sum_{\\substack{\\eta^N=-1","On the strip $x\\geq R$ and $|y|\\leq Y$, define\n\\[\nR_N(z)\n=\n\\sum_{\\substack{\\eta^N=-1\\\\\n\\eta\\ne e^{ia_N},e^{-ia_N}}}\n\\exp\\bigl((\\eta-e^{-ia_N})z\\bigr).\n\\]\nThen\n\\[\nN e^{-e^{-ia_N}z}\\Dfun_N(z)\n=\n1+e^{2is_Nz}+R_N(z),\n\\]\nand, for $q=0,1,2$,\n\\[\n|R_N^{(q)}(z)|\n\\leq\nC e^{-\\gamma_Nx}.\n\\]\n\\end{lemma}","\\eta\\ne e^{ia_N},e^{-ia_N}}}\n\\exp\\bigl((\\eta-e^{-ia_N})z\\bigr).\n\\]\nThen\n\\[\nN e^{-e^{-ia_N}z}\\Dfun_N(z)\n=\n1+e^{2is_Nz}+R_N(z),\n\\]\nand, for $q=0,1,2$,\n\\[\n|R_N^{(q)}(z)|\n\\leq\nC e^{-\\gamma_Nx}.\n\\]\n\\end{lemma}","\\begin{proof}\nA direct projection estimate gives the stronger bound\n\\[\nY=\\frac{\\log(N-1)}{2s_N}\n\\]\nfor every zero in the sector, without a large-radius assumption.\nWrite $z=re^{i\\theta}$ with $0\\leq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound","\\begin{proof}\nA direct projection estimate gives the stronger bound\n\\[\nY=\\frac{\\log(N-1)}{2s_N}\n\\]\nfor every zero in the sector, without a large-radius assumption.\nWrite $z=re^{i\\theta}$ with $0\\leq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","begin{proof}\nA direct projection estimate gives the stronger bound\n\\[\nY=\\frac{\\log(N-1)}{2s_N}\n\\]\nfor every zero in the sector, without a large-radius assumption.\nWrite $z=re^{i\\theta}$ with $0\\leq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","ojection estimate gives the stronger bound\n\\[\nY=\\frac{\\log(N-1)}{2s_N}\n\\]\nfor every zero in the sector, without a large-radius assumption.\nWrite $z=re^{i\\theta}$ with $0\\leq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","s_N}\n\\]\nfor every zero in the sector, without a large-radius assumption.\nWrite $z=re^{i\\theta}$ with $0\\leq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","he sector, without a large-radius assumption.\nWrite $z=re^{i\\theta}$ with $0\\leq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","e $z=re^{i\\theta}$ with $0\\leq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","eq\\theta\\leq a_N$ and $r\\geq0$.\nFor $0\\leq\\ell\\leq N-2$, the angle\n$\\theta+(2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","2\\ell+1)a_N$ lies between $a_N+\\theta$ and\n$2\\pi-a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","a_N-\\theta$. Monotonicity of cosine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","ine on $[0,\\pi]$, followed by\nreflection about $\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","\\pi$, therefore gives\n\\[\n\\Re(\\eta_\\ell z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","l z)\\leq r\\cos(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","s(a_N+\\theta).\n\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","\\]\nThe remaining vertex $e^{-ia_N}$ has projection $r\\cos(a_N-\\theta)$.\nTheir difference is at least $2s_Nr\\sin\\theta=2s_Ny$.\nIf $\\Dfun_N(z)=0$, divide its finite exponential sum by the term\nwith exponent $e^{-ia_N}z$ and apply the triangle inequality:\n\\[\n1\\leq(N-1)e^{-2s_Ny}.\n\\]\nThus $y\\leq Y$. Conjugation gives the same bound for $-y$ on the lower\nhalf sector. Since $N\\geq3$ and $s_N>0$, this $Y$ is positive.","On the resulting strip, the roots $e^{ia_N}$ and $e^{-ia_N}$ have\nreal part $c_N$, while each remaining root $\\eta=u+iv$ satisfies\n$u\\leq\\cos(3a_N)=c_N-\\gamma_N$. Hence\n\\[\n\\Re\\bigl((\\eta-e^{-ia_N})z\\bigr)\n\\leq-\\gamma_Nx+2Y.\n\\]\nDivision by $e^{e^{-ia_N}z}$ gives the displayed normalization.\nEach differentiation multiplies a remainder term by\n$\\eta-e^{-ia_N}$, whose modulus is at most two. Thus, for $q=0,1,2$,\n\\[\n|R_N^{(q)}(z)|\\leq(N-2)2^qe^{2Y}e^{-\\gamma_Nx}.\n\\]\nOne may take $R=1$ and $C=4(N-2)e^{2Y}$.\n\\end{proof}","the resulting strip, the roots $e^{ia_N}$ and $e^{-ia_N}$ have\nreal part $c_N$, while each remaining root $\\eta=u+iv$ satisfies\n$u\\leq\\cos(3a_N)=c_N-\\gamma_N$. Hence\n\\[\n\\Re\\bigl((\\eta-e^{-ia_N})z\\bigr)\n\\leq-\\gamma_Nx+2Y.\n\\]\nDivision by $e^{e^{-ia_N}z}$ gives the displayed normalization.\nEach differentiation multiplies a remainder term by\n$\\eta-e^{-ia_N}$, whose modulus is at most two. Thus, for $q=0,1,2$,\n\\[\n|R_N^{(q)}(z)|\\leq(N-2)2^qe^{2Y}e^{-\\gamma_Nx}.\n\\]\nOne may take $R=1$ and $C=4(N-2)e^{2Y}$.\n\\end{proof}","ng strip, the roots $e^{ia_N}$ and $e^{-ia_N}$ have\nreal part $c_N$, while each remaining root $\\eta=u+iv$ satisfies\n$u\\leq\\cos(3a_N)=c_N-\\gamma_N$. Hence\n\\[\n\\Re\\bigl((\\eta-e^{-ia_N})z\\bigr)\n\\leq-\\gamma_Nx+2Y.\n\\]\nDivision by $e^{e^{-ia_N}z}$ gives the displayed normalization.\nEach differentiation multiplies a remainder term by\n$\\eta-e^{-ia_N}$, whose modulus is at most two. Thus, for $q=0,1,2$,\n\\[\n|R_N^{(q)}(z)|\\leq(N-2)2^qe^{2Y}e^{-\\gamma_Nx}.\n\\]\nOne may take $R=1$ and $C=4(N-2)e^{2Y}$.\n\\end{proof}","$, while each remaining root $\\eta=u+iv$ satisfies\n$u\\leq\\cos(3a_N)=c_N-\\gamma_N$. Hence\n\\[\n\\Re\\bigl((\\eta-e^{-ia_N})z\\bigr)\n\\leq-\\gamma_Nx+2Y.\n\\]\nDivision by $e^{e^{-ia_N}z}$ gives the displayed normalization.\nEach differentiation multiplies a remainder term by\n$\\eta-e^{-ia_N}$, whose modulus is at most two. Thus, for $q=0,1,2$,\n\\[\n|R_N^{(q)}(z)|\\leq(N-2)2^qe^{2Y}e^{-\\gamma_Nx}.\n\\]\nOne may take $R=1$ and $C=4(N-2)e^{2Y}$.\n\\end{proof}","ach remaining root $\\eta=u+iv$ satisfies\n$u\\leq\\cos(3a_N)=c_N-\\gamma_N$. Hence\n\\[\n\\Re\\bigl((\\eta-e^{-ia_N})z\\bigr)\n\\leq-\\gamma_Nx+2Y.\n\\]\nDivision by $e^{e^{-ia_N}z}$ gives the displayed normalization.\nEach differentiation multiplies a remainder term by\n$\\eta-e^{-ia_N}$, whose modulus is at most two. Thus, for $q=0,1,2$,\n\\[\n|R_N^{(q)}(z)|\\leq(N-2)2^qe^{2Y}e^{-\\gamma_Nx}.\n\\]\nOne may take $R=1$ and $C=4(N-2)e^{2Y}$.\n\\end{proof}","\\begin{lemma}[Near cancellation of two exponentials]\\label{lem:near_cancellation}\nFor every $M>0$, there are constants $\\varepsilon_M>0$ and $C_M>0$ such that the following holds.  If\n\\[\n|\\Re w|\\leq M\n\\]\nand\n\\[\n|1+e^w|\\leq\\varepsilon_M,\n\\]\nthen there is a unique integer $m$ with\n\\[\n|w-(2m+1)\\pi i|\n\\leq\nC_M|1+e^w|.\n\\]\n\\end{lemma}","\\begin{proof}\nIn fact, the constants $\\varepsilon_M=1/2$ and $C_M=3/2$ work\nfor every complex $w$, without the strip restriction. Put\n$u=-(1+e^w)$ and $L=\\log(1+u)=\\log(-e^w)$, using the principal\nlogarithm. The elementary logarithm estimate on $|u|\\leq1/2$ gives\n$|L|\\leq(3/2)|u|$. Since $e^{L+\\pi i}=e^w$, there is an integer\n$m$ such that\n\\[\nw=L+(2m+1)\\pi i.\n\\]\nThis gives the required distance bound. Distinct centers are separated\nby at least $2\\pi$, whereas two centers satisfying the bound would\nbe separated by at most $3|1+e^w|\\leq3/2$. Thus the integer is unique.\nThe formal proof uses the convergent logarithm estimate and the exact\nperiodicity theorem for the complex exponential.\n\\end{proof}","_M=1/2$ and $C_M=3/2$ work\nfor every complex $w$, without the strip restriction. Put\n$u=-(1+e^w)$ and $L=\\log(1+u)=\\log(-e^w)$, using the principal\nlogarithm. The elementary logarithm estimate on $|u|\\leq1/2$ gives\n$|L|\\leq(3/2)|u|$. Since $e^{L+\\pi i}=e^w$, there is an integer\n$m$ such that\n\\[\nw=L+(2m+1)\\pi i.\n\\]\nThis gives the required distance bound. Distinct centers are separated\nby at least $2\\pi$, whereas two centers satisfying the bound would\nbe separated by at most $3|1+e^w|\\leq3/2$. Thus the integer is unique.\nThe formal proof uses the convergent logarithm estimate and the exact\nperiodicity theorem for the complex exponential.\n\\end{proof}",". Put\n$u=-(1+e^w)$ and $L=\\log(1+u)=\\log(-e^w)$, using the principal\nlogarithm. The elementary logarithm estimate on $|u|\\leq1/2$ gives\n$|L|\\leq(3/2)|u|$. Since $e^{L+\\pi i}=e^w$, there is an integer\n$m$ such that\n\\[\nw=L+(2m+1)\\pi i.\n\\]\nThis gives the required distance bound. Distinct centers are separated\nby at least $2\\pi$, whereas two centers satisfying the bound would\nbe separated by at most $3|1+e^w|\\leq3/2$. Thus the integer is unique.\nThe formal proof uses the convergent logarithm estimate and the exact\nperiodicity theorem for the complex exponential.\n\\end{proof}","\\begin{theorem}[Large zero strings]\\label{thm:large_zero_strings}\nAssume $N\\geq3$.  There are constants $m_0$, $C$, and $\\gamma>0$ and a sequence of positive real simple zeros $(\\tau_m)_{m\\geq m_0}$ of $\\Dfun_N$ such that\n\\[\n\\tau_m\n=\n t_m+O(e^{-\\gamma t_m}).\n\\]\nEvery zero of sufficiently large modulus is of the form\n\\[\n\\omega^k\\tau_m\n\\]\nfor a unique $0\\leq k<N$ and a unique $m\\geq m_0$.","The derivatives satisfy\n\\[\n\\Dfun_N'(\\omega^k\\tau_m)\n=\n\\omega^{-k}\\Dfun_N'(\\tau_m)\n\\]\nand\n\\[\n|\\Dfun_N'(\\tau_m)|\n=\n\\frac{2s_N}{N}e^{c_N\\tau_m}\n\\left(1+O(e^{-\\gamma\\tau_m})\\right).\n\\]\n\\end{theorem}","\\begin{proof}\nWork first in the sector $|\\arg z|\\leq a_N$.  By \\cref{lem:two_term_dominance}, every sufficiently large zero lies in a fixed strip about the positive real axis.  Put\n\\[\nG(z)=N e^{-e^{-ia_N}z}\\Dfun_N(z).\n\\]\nThen\n\\[\nG(z)=1+e^{2is_Nz}+R_N(z)\n\\]\nwith an exponentially small remainder and derivative.","If $G(z)=0$, then\n\\[\n|1+e^{2is_Nz}|\n\\leq\nC e^{-\\gamma_N\\Re z}.\n\\]\nThe real part of $2is_Nz$ is bounded in the strip.  Apply\n\\cref{lem:near_cancellation} with $w=2is_Nz$.  There is a unique\ninteger $m$ such that\n\\[\n|z-t_m|\\leq C_1e^{-\\gamma_N\\Re z},\n\\]\nwhere\n\\[\n t_m=\\frac{(2m+1)\\pi}{2s_N}.\n\\]\nIn particular, $|\\Re z-t_m|=o(1)$.  For all sufficiently large zeros\nwe have $|\\Re z-t_m|\\leq1$, and hence\n\\[\n|z-t_m|\\leq C_2e^{-\\gamma_Nt_m}\n\\]\nwith a constant independent of $m$.\nBecause $\\Re z\\to+\\infty$ in this argument, the corresponding integers\n$m$ are eventually nonnegative and tend to infinity.","$G(z)=0$, then\n\\[\n|1+e^{2is_Nz}|\n\\leq\nC e^{-\\gamma_N\\Re z}.\n\\]\nThe real part of $2is_Nz$ is bounded in the strip.  Apply\n\\cref{lem:near_cancellation} with $w=2is_Nz$.  There is a unique\ninteger $m$ such that\n\\[\n|z-t_m|\\leq C_1e^{-\\gamma_N\\Re z},\n\\]\nwhere\n\\[\n t_m=\\frac{(2m+1)\\pi}{2s_N}.\n\\]\nIn particular, $|\\Re z-t_m|=o(1)$.  For all sufficiently large zeros\nwe have $|\\Re z-t_m|\\leq1$, and hence\n\\[\n|z-t_m|\\leq C_2e^{-\\gamma_Nt_m}\n\\]\nwith a constant independent of $m$.\nBecause $\\Re z\\to+\\infty$ in this argument, the corresponding integers\n$m$ are eventually nonnegative and tend to infinity.","real part of $2is_Nz$ is bounded in the strip.  Apply\n\\cref{lem:near_cancellation} with $w=2is_Nz$.  There is a unique\ninteger $m$ such that\n\\[\n|z-t_m|\\leq C_1e^{-\\gamma_N\\Re z},\n\\]\nwhere\n\\[\n t_m=\\frac{(2m+1)\\pi}{2s_N}.\n\\]\nIn particular, $|\\Re z-t_m|=o(1)$.  For all sufficiently large zeros\nwe have $|\\Re z-t_m|\\leq1$, and hence\n\\[\n|z-t_m|\\leq C_2e^{-\\gamma_Nt_m}\n\\]\nwith a constant independent of $m$.\nBecause $\\Re z\\to+\\infty$ in this argument, the corresponding integers\n$m$ are eventually nonnegative and tend to infinity.","$2is_Nz$ is bounded in the strip.  Apply\n\\cref{lem:near_cancellation} with $w=2is_Nz$.  There is a unique\ninteger $m$ such that\n\\[\n|z-t_m|\\leq C_1e^{-\\gamma_N\\Re z},\n\\]\nwhere\n\\[\n t_m=\\frac{(2m+1)\\pi}{2s_N}.\n\\]\nIn particular, $|\\Re z-t_m|=o(1)$.  For all sufficiently large zeros\nwe have $|\\Re z-t_m|\\leq1$, and hence\n\\[\n|z-t_m|\\leq C_2e^{-\\gamma_Nt_m}\n\\]\nwith a constant independent of $m$.\nBecause $\\Re z\\to+\\infty$ in this argument, the corresponding integers\n$m$ are eventually nonnegative and tend to infinity.","ation} with $w=2is_Nz$.  There is a unique\ninteger $m$ such that\n\\[\n|z-t_m|\\leq C_1e^{-\\gamma_N\\Re z},\n\\]\nwhere\n\\[\n t_m=\\frac{(2m+1)\\pi}{2s_N}.\n\\]\nIn particular, $|\\Re z-t_m|=o(1)$.  For all sufficiently large zeros\nwe have $|\\Re z-t_m|\\leq1$, and hence\n\\[\n|z-t_m|\\leq C_2e^{-\\gamma_Nt_m}\n\\]\nwith a constant independent of $m$.\nBecause $\\Re z\\to+\\infty$ in this argument, the corresponding integers\n$m$ are eventually nonnegative and tend to infinity.","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is small","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spac","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks li","Fix a large $m$, put $a=2is_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","_N$, and consider the closed disk\n$B_m=\\{z:|z-t_m|\\leq 1/(2s_N)\\}$. The remainder bounds hold\nuniformly on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","on these fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","se fixed-radius disks. For sufficiently large $m$,\n$|R_N|\\leq 1/2$ and $|R_N'|\\leq s_N/2$ there. Define\n\\[\n\\Phi_m(z)=t_m+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","+\\frac{\\log(1+R_N(z))}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","}{2is_N},\n\\]\nusing the principal logarithm. The estimate\n$|\\log(1+u)|\\leq(3/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","/2)|u|$ shows that $\\Phi_m$ maps $B_m$ into\nitself. The derivative bound $|\\Phi_m'|\\leq1/2$ makes it a contraction.\nBanach's fixed-point theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","nt theorem gives a unique fixed point.\nOn $B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","B_m$, $|a(z-t_m)|\\leq1<\\pi$, so the principal logarithm inverts\nthe exponential and the fixed-point equation is equivalent to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","t to $G(z)=0$.\nMoreover,\n\\[\n|z-t_m|\\leq\\frac{3}{4s_N}|R_N(z)|=O(e^{-\\gamma_Nt_m}).\n\\]\nAt this zero,\n\\[\n|G'(z)|\\geq |a|\\,|1+R_N(z)|-|R_N'(z)|\\geq s_N/2>0,\n\\]\nso the zero is simple. The disks are mutually disjoint because their\ndiameter $1/s_N$ is smaller than the center spacing $\\pi/s_N$.\nEnlarging the strip if necessary, these disks lie inside the fundamental\nsector and the fixed strip for large $m$.","The localization estimate puts every sufficiently large zero in the\nfundamental sector into one of these disks.  The zero sets of $G$ and\n$\\Dfun_N$ coincide because their prefactor is nonzero.  Each disk is\ninvariant under complex conjugation, and $\\Dfun_N$ has real\ncoefficients.  The conjugate of its unique zero is therefore the same\nzero.  It is real, positive for large $m$, and simple by the\nderivative bound above.  Denote it by $\\tau_m$.","sufficiently large zero in the\nfundamental sector into one of these disks.  The zero sets of $G$ and\n$\\Dfun_N$ coincide because their prefactor is nonzero.  Each disk is\ninvariant under complex conjugation, and $\\Dfun_N$ has real\ncoefficients.  The conjugate of its unique zero is therefore the same\nzero.  It is real, positive for large $m$, and simple by the\nderivative bound above.  Denote it by $\\tau_m$.","The parts of the fundamental sector away from the positive tie ray,\nincluding its boundary rays, were already excluded uniformly.\nRotational invariance now gives every remaining large zero as\n$\\omega^k\\tau_m$.  Distinct rays and the disjoint disks give uniqueness\nof $k$ and $m$.  Taking $\\gamma=\\gamma_N$ proves the zero location in\nthe statement.","Differentiate the normalized expression.  At $z=\\tau_m$,\n\\[\nG'(\\tau_m)\n=\n2is_Ne^{2is_N\\tau_m}+R_N'(\\tau_m)\n=\n-2is_N+O(e^{-\\gamma_N\\tau_m}).\n\\]\nSince $G(\\tau_m)=0$,\n\\[\n\\Dfun_N'(\\tau_m)\n=\n\\frac1N e^{e^{-ia_N}\\tau_m}G'(\\tau_m).\n\\]\nTaking absolute values gives the derivative asymptotic.  Differentiating $\\Dfun_N(\\omega z)=\\Dfun_N(z)$ gives the rotation formula.\n\\end{proof}","\\begin{theorem}[Leading displacement term and derivative sign]\\label{thm:first_zero_correction}\nWith the notation of \\cref{lem:two_term_dominance,thm:large_zero_strings},\n\\[\n\\tau_m\n=\nt_m+\n\\frac{R_N(t_m)}{2is_N}\n+\nO(e^{-2\\gamma_Nt_m}).\n\\]\nThe quotient $R_N(t_m)/(2is_N)$ is real.  One also has\n\\[\n\\Dfun_N'(\\tau_m)\n=\n(-1)^{m+1}\n\\frac{2s_N}{N}\ne^{c_N\\tau_m}\n\\left(1+O(e^{-\\gamma_N\\tau_m})\\right).\n\\]\n\\end{theorem}","\\begin{proof}\nPut $\\delta_m=\\tau_m-t_m$.  The large zero theorem gives $\\delta_m=O(e^{-\\gamma_Nt_m})$.  Since $e^{2is_Nt_m}=-1$,\n\\[\n1+e^{2is_N(t_m+\\delta_m)}\n=\n-2is_N\\delta_m+O(\\delta_m^2).\n\\]\nThe remainder estimate through the first derivative gives\n\\[\nR_N(t_m+\\delta_m)\n=\nR_N(t_m)+O(e^{-\\gamma_Nt_m}|\\delta_m|).\n\\]\nSubstitution in the zero equation gives the displacement formula.","At the real point $t_m$, the normalized value\n\\[\nN e^{-e^{-ia_N}t_m}\\Dfun_N(t_m)\n\\]\nis purely imaginary.  The two dominant terms cancel there, so $R_N(t_m)$ is purely imaginary and the displayed correction is real.","For the derivative, use the formula in the proof of \\cref{thm:large_zero_strings} and\n\\[\ne^{-is_N\\tau_m}\n=\n(-1)^m(-i)\n\\left(1+O(e^{-\\gamma_N\\tau_m})\\right).\n\\]\nThe stated real sign follows.\n\\end{proof}","\\begin{corollary}[Leading correction for $N=3$]\\label{cor:N3_zero_correction}\nFor $N=3$,\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}\\left(m+\\frac12\\right)\n\\]\nand\n\\[\n\\tau_m\n=\nt_m+\n\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}\n+\nO(e^{-3t_m}).\n\\]\n\\end{corollary}","\\begin{proof}\nThe only nondominant exponent is $-1$.  Hence\n\\[\nR_3(t)\n=\n\\exp\\left(\\left(-\\frac32+\\frac{\\sqrt3}{2}i\\right)t\\right).\n\\]\nAt $t=t_m$, its phase is $i(-1)^m$.  Insert this value into \\cref{thm:first_zero_correction}.\n\\end{proof}","\\begin{corollary}[The case $N=2$]\\label{cor:case_two_zeros}\nFor $N=2$,\n\\[\n\\Dfun_2(z)=\\cos z.\n\\]\nIts zeros are\n\\[\n\\pm\\left(m+\\frac12\\right)\\pi,\n\\qquad m\\geq0,\n\\]\nand all are simple.  They agree with the formula $t_m=A_2(m+1/2)$.\n\\end{corollary}","\\begin{corollary}[Eventual simplicity and exact rays]\\label{cor:eventual_simplicity}\nFor every $N\\geq2$, all but finitely many zeros of $\\Dfun_N$ are simple.  Every sufficiently large zero lies on one of the rays\n\\[\n\\omega^k(0,\\infty),\n\\qquad 0\\leq k<N.\n\\]\n\\end{corollary}","\\begin{proof}\nUse \\cref{thm:large_zero_strings} for $N\\geq3$ and \\cref{cor:case_two_zeros} for $N=2$.\n\\end{proof}","\\begin{example}[Exact fourth-order strings]\\label{ex:fourth_order_zeros}\nFor $N=4$,\n\\[\n\\Dfun_4(z)\n=\n\\cos\\left(\\frac z{\\sqrt2}\\right)\n\\cosh\\left(\\frac z{\\sqrt2}\\right).\n\\]\nThe positive zeros are exactly\n\\[\n\\tau_m\n=\n\\sqrt2\\pi\\left(m+\\frac12\\right)\n=\nA_4\\left(m+\\frac12\\right).\n\\]\nRotation by the fourth roots of unity gives the real and imaginary zero strings.  In this case the error term in \\cref{thm:large_zero_strings} is identically zero.\n\\end{example}","\\begin{corollary}[Reduced zeros]\\label{cor:reduced_zeros}\nLet $\\Lambda_N$ be the nonzero zero multiset of $\\Delta_N$.  All but finitely many members of $\\Lambda_N$ are positive and simple.  They can be indexed as\n\\[\n\\lambda_m=\\tau_m^N\n\\]\nfor all sufficiently large $m$.  If $N\\geq3$, the notation may be chosen\nas in \\cref{thm:large_zero_strings}, and\n\\[\n\\lambda_m\n=\n t_m^N\n+O\\left(t_m^{N-1}e^{-\\gamma t_m}\\right).\n\\]\nFor $N=2$, take $\\tau_m=t_m=\\pi(m+1/2)$ for $m\\geq0$. Then\n$\\lambda_m=t_m^2$ exactly.\n\\end{corollary}","ed zeros]\\label{cor:reduced_zeros}\nLet $\\Lambda_N$ be the nonzero zero multiset of $\\Delta_N$.  All but finitely many members of $\\Lambda_N$ are positive and simple.  They can be indexed as\n\\[\n\\lambda_m=\\tau_m^N\n\\]\nfor all sufficiently large $m$.  If $N\\geq3$, the notation may be chosen\nas in \\cref{thm:large_zero_strings}, and\n\\[\n\\lambda_m\n=\n t_m^N\n+O\\left(t_m^{N-1}e^{-\\gamma t_m}\\right).\n\\]\nFor $N=2$, take $\\tau_m=t_m=\\pi(m+1/2)$ for $m\\geq0$. Then\n$\\lambda_m=t_m^2$ exactly.\n\\end{corollary}","bel{cor:reduced_zeros}\nLet $\\Lambda_N$ be the nonzero zero multiset of $\\Delta_N$.  All but finitely many members of $\\Lambda_N$ are positive and simple.  They can be indexed as\n\\[\n\\lambda_m=\\tau_m^N\n\\]\nfor all sufficiently large $m$.  If $N\\geq3$, the notation may be chosen\nas in \\cref{thm:large_zero_strings}, and\n\\[\n\\lambda_m\n=\n t_m^N\n+O\\left(t_m^{N-1}e^{-\\gamma t_m}\\right).\n\\]\nFor $N=2$, take $\\tau_m=t_m=\\pi(m+1/2)$ for $m\\geq0$. Then\n$\\lambda_m=t_m^2$ exactly.\n\\end{corollary}","\\Lambda_N$ be the nonzero zero multiset of $\\Delta_N$.  All but finitely many members of $\\Lambda_N$ are positive and simple.  They can be indexed as\n\\[\n\\lambda_m=\\tau_m^N\n\\]\nfor all sufficiently large $m$.  If $N\\geq3$, the notation may be chosen\nas in \\cref{thm:large_zero_strings}, and\n\\[\n\\lambda_m\n=\n t_m^N\n+O\\left(t_m^{N-1}e^{-\\gamma t_m}\\right).\n\\]\nFor $N=2$, take $\\tau_m=t_m=\\pi(m+1/2)$ for $m\\geq0$. Then\n$\\lambda_m=t_m^2$ exactly.\n\\end{corollary}","he nonzero zero multiset of $\\Delta_N$.  All but finitely many members of $\\Lambda_N$ are positive and simple.  They can be indexed as\n\\[\n\\lambda_m=\\tau_m^N\n\\]\nfor all sufficiently large $m$.  If $N\\geq3$, the notation may be chosen\nas in \\cref{thm:large_zero_strings}, and\n\\[\n\\lambda_m\n=\n t_m^N\n+O\\left(t_m^{N-1}e^{-\\gamma t_m}\\right).\n\\]\nFor $N=2$, take $\\tau_m=t_m=\\pi(m+1/2)$ for $m\\geq0$. Then\n$\\lambda_m=t_m^2$ exactly.\n\\end{corollary}","\\begin{proof}\nThe identity $\\Dfun_N(z)=\\Delta_N(z^N)$ identifies each orbit of $N$ nonzero zeros of $\\Dfun_N$ with one zero of $\\Delta_N$.  For $N\\geq3$, the large orbit representatives are the positive zeros $\\tau_m$ from \\cref{thm:large_zero_strings}. For $N=2$, use \\cref{cor:case_two_zeros}.  Simplicity is preserved because\n\\[\n\\Dfun_N'(z)=Nz^{N-1}\\Delta_N'(z^N)\n\\]\nand $z\\ne0$.  The asymptotic follows by expanding $\\tau_m^N$ about $t_m$.\n\\end{proof}","ntity $\\Dfun_N(z)=\\Delta_N(z^N)$ identifies each orbit of $N$ nonzero zeros of $\\Dfun_N$ with one zero of $\\Delta_N$.  For $N\\geq3$, the large orbit representatives are the positive zeros $\\tau_m$ from \\cref{thm:large_zero_strings}. For $N=2$, use \\cref{cor:case_two_zeros}.  Simplicity is preserved because\n\\[\n\\Dfun_N'(z)=Nz^{N-1}\\Delta_N'(z^N)\n\\]\nand $z\\ne0$.  The asymptotic follows by expanding $\\tau_m^N$ about $t_m$.\n\\end{proof}","=\\Delta_N(z^N)$ identifies each orbit of $N$ nonzero zeros of $\\Dfun_N$ with one zero of $\\Delta_N$.  For $N\\geq3$, the large orbit representatives are the positive zeros $\\tau_m$ from \\cref{thm:large_zero_strings}. For $N=2$, use \\cref{cor:case_two_zeros}.  Simplicity is preserved because\n\\[\n\\Dfun_N'(z)=Nz^{N-1}\\Delta_N'(z^N)\n\\]\nand $z\\ne0$.  The asymptotic follows by expanding $\\tau_m^N$ about $t_m$.\n\\end{proof}","\\begin{theorem}[Zero counting asymptotics]\\label{thm:zero_counting}\nLet $n_{\\Dfun_N}(R)$ count the zeros of $\\Dfun_N$ in $|z|\\leq R$, with multiplicity.  Let $n_{\\Delta_N}(R)$ count the zeros of $\\Delta_N$ in $|w|\\leq R$.  Then, as $R\\to\\infty$,\n\\[\nn_{\\Dfun_N}(R)\n=\n\\frac{N\\sin(\\pi/N)}{\\pi}R+O(1)\n\\]\nand\n\\[\nn_{\\Delta_N}(R)\n=\n\\frac{\\sin(\\pi/N)}{\\pi}R^{1/N}+O(1).\n\\]\n\\end{theorem}","\\begin{proof}\nEach of the $N$ rays contains one large zero near\n\\[\n t_m=\\frac\\pi{\\sin(\\pi/N)}\\left(m+\\frac12\\right).\n\\]\nSince $\\tau_m-t_m=o(1)$ and the model spacing is the positive constant\n$A_N$, replacing $\\tau_m\\leq R$ by $t_m\\leq R$ changes the count by at\nmost a bounded amount.\nThe number of such indices with $t_m\\leq R$ is\n\\[\n\\frac{\\sin(\\pi/N)}\\pi R+O(1).\n\\]\nMultiplication by $N$ and addition of the finite exceptional set give the first formula.  The second follows from\n\\[\nn_{\\Dfun_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\n\\end{proof}","n(\\pi/N)}\\left(m+\\frac12\\right).\n\\]\nSince $\\tau_m-t_m=o(1)$ and the model spacing is the positive constant\n$A_N$, replacing $\\tau_m\\leq R$ by $t_m\\leq R$ changes the count by at\nmost a bounded amount.\nThe number of such indices with $t_m\\leq R$ is\n\\[\n\\frac{\\sin(\\pi/N)}\\pi R+O(1).\n\\]\nMultiplication by $N$ and addition of the finite exceptional set give the first formula.  The second follows from\n\\[\nn_{\\Dfun_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\n\\end{proof}","\\begin{corollary}[Zero exponents]\\label{cor:exponent_of_convergence}\nThe exponent of convergence of the nonzero zeros of $\\Dfun_N$ is one.  The exponent of convergence of the zeros of $\\Delta_N$ is $1/N$.\n\\end{corollary}","ct uses every reduced zero with its multiplicity. Its logarithm turns multiplication into addition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset.\n\\end{keyidea}\n\\modelfigure{zero_moments}{Order-two moment sums over the first forty reduced zeros $\\lambda_k=\\pi^2(k+1/2)^2$. The curves are finite numerical partial sums. The dashed limits $1/2$ and $1/6$ follow from the exact logarithmic coefficients and the product identity.}{fig:zero-moments}","dition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset.\n\\end{keyidea}\n\\modelfigure{zero_moments}{Order-two moment sums over the first forty reduced zeros $\\lambda_k=\\pi^2(k+1/2)^2$. The curves are finite numerical partial sums. The dashed limits $1/2$ and $1/6$ follow from the exact logarithmic coefficients and the product identity.}{fig:zero-moments}","reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset.\n\\end{keyidea}\n\\modelfigure{zero_moments}{Order-two moment sums over the first forty reduced zeros $\\lambda_k=\\pi^2(k+1/2)^2$. The curves are finite numerical partial sums. The dashed limits $1/2$ and $1/6$ follow from the exact logarithmic coefficients and the product identity.}{fig:zero-moments}","The reduction from $z$ to $w=z^N$ removes the rotational repetition of the zeros.","\\begin{theorem}[Genus zero product]\\label{thm:genus_zero_product}\nFor $N\\geq2$, let $\\Lambda_N$ be the nonzero zero multiset of $\\Delta_N$.  Then\n\\[\n\\Delta_N(w)\n=\n\\prod_{\\lambda\\in\\Lambda_N}\n\\left(1-\\frac w\\lambda\\right),\n\\]\nwhere the product converges locally uniformly.  Consequently,\n\\[\n\\Dfun_N(z)\n=\n\\prod_{\\lambda\\in\\Lambda_N}\n\\left(1-\\frac{z^N}{\\lambda}\\right).\n\\]\n\\end{theorem}","\\begin{proof}\nBy \\cref{prop:orders}, the order of $\\Delta_N$ is $1/N<1$.  By\n\\cref{cor:reduced_zeros}, it has infinitely many zeros, and\n\\cref{cor:exponent_of_convergence} gives\n\\[\n\\sum_{\\lambda\\in\\Lambda_N}|\\lambda|^{-1}<\\infty.\n\\]\nHadamard factorization therefore gives a canonical product of genus zero.\nThe only possible exponential factor is a nonzero constant because the order\nis below one.  Since $\\Delta_N(0)=1$, that constant is one.  Substitution of\n$w=z^N$ gives the second formula.\n\\end{proof}","{proof}\nBy \\cref{prop:orders}, the order of $\\Delta_N$ is $1/N<1$.  By\n\\cref{cor:reduced_zeros}, it has infinitely many zeros, and\n\\cref{cor:exponent_of_convergence} gives\n\\[\n\\sum_{\\lambda\\in\\Lambda_N}|\\lambda|^{-1}<\\infty.\n\\]\nHadamard factorization therefore gives a canonical product of genus zero.\nThe only possible exponential factor is a nonzero constant because the order\nis below one.  Since $\\Delta_N(0)=1$, that constant is one.  Substitution of\n$w=z^N$ gives the second formula.\n\\end{proof}","s $1/N<1$.  By\n\\cref{cor:reduced_zeros}, it has infinitely many zeros, and\n\\cref{cor:exponent_of_convergence} gives\n\\[\n\\sum_{\\lambda\\in\\Lambda_N}|\\lambda|^{-1}<\\infty.\n\\]\nHadamard factorization therefore gives a canonical product of genus zero.\nThe only possible exponential factor is a nonzero constant because the order\nis below one.  Since $\\Delta_N(0)=1$, that constant is one.  Substitution of\n$w=z^N$ gives the second formula.\n\\end{proof}","\\begin{theorem}[Diagonal Fredholm determinant representation]\\label{thm:trace_class_model}\nLet $I_N$ be a multiplicity-sensitive countable index set for $\\Lambda_N$\nand put $\\mathcal H_N=\\ell^2(I_N)$.  On its standard orthonormal basis,\ndefine the diagonal operator\n\\[\nK_Ne_\\lambda=\\lambda^{-1}e_\\lambda.\n\\]\nThen $K_N$ is trace class and\n\\[\n\\Delta_N(w)=\\Fdet(I-wK_N).\n\\]\nFor every $m\\geq1$,\n\\[\n\\kappa_{N,m}=\\tr(K_N^m).\n\\]\n\\end{theorem}","\\begin{proof}\nBy \\cref{cor:exponent_of_convergence},\n\\[\n\\sum_{\\lambda\\in\\Lambda_N}|\\lambda|^{-1}<\\infty.\n\\]\nThus the diagonal operator $K_N$ is trace class.  Its Fredholm determinant is the locally uniform product\n\\[\n\\Fdet(I-wK_N)\n=\n\\prod_{\\lambda\\in\\Lambda_N}\n\\left(1-\\frac w\\lambda\\right),\n\\]\nwhich equals $\\Delta_N(w)$ by \\cref{thm:genus_zero_product}.","Taking the logarithm gives\n\\[\n-\\log\\Delta_N(w)\n=\n\\sum_{m\\geq1}\\frac{w^m}{m}\\tr(K_N^m)\n\\]\nnear the origin.  Comparison with the definition of $\\kappa_{N,m}$ proves the trace identity.\n\\end{proof}","\\begin{corollary}[Complete symmetric function form]\\label{cor:complete_symmetric_form}\nFor a positive integer $r$, let each eigenvalue $\\lambda^{-1}$ of $K_N$ be repeated $r$ times.  If $h_n$ denotes the complete homogeneous symmetric function of degree $n$ in this absolutely summable multiset, then\n\\[\n\\frac{P_{N,n}(r)}{(Nn)!}=h_n.\n\\]\nFor arbitrary complex $r$, the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by $r$.\n\\end{corollary}","ete_symmetric_form}\nFor a positive integer $r$, let each eigenvalue $\\lambda^{-1}$ of $K_N$ be repeated $r$ times.  If $h_n$ denotes the complete homogeneous symmetric function of degree $n$ in this absolutely summable multiset, then\n\\[\n\\frac{P_{N,n}(r)}{(Nn)!}=h_n.\n\\]\nFor arbitrary complex $r$, the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by $r$.\n\\end{corollary}","\\begin{proof}\nThe generating function for the complete homogeneous symmetric functions is\n\\[\n\\prod_{\\lambda\\in\\Lambda_N}\n\\left(1-\\frac w\\lambda\\right)^{-r}\n=\n\\Delta_N(w)^{-r}.\n\\]\nCoefficient comparison gives the first identity.  Power sums determine the complete homogeneous functions through the Newton recurrence.  Replacing each power sum by $r$ times itself gives \\cref{thm:reciprocal_bell,thm:logarithmic_newton_recurrence}.\n\\end{proof}","\\begin{proposition}[Products for the normalized sections]\\label{prop:normalized_section_product}\nFor every $0\\leq j<N$, the entire function\n\\[\n\\Phi_{N,j}(w)\n=\nj!\\sum_{n\\geq0}\\frac{w^n}{(Nn+j)!}\n\\]\nhas order $1/N$.  If $N\\geq2$ and $\\Lambda_{N,j}$ is its nonzero zero multiset, then\n\\[\n\\Phi_{N,j}(w)\n=\n\\prod_{\\mu\\in\\Lambda_{N,j}}\n\\left(1-\\frac w\\mu\\right).\n\\]\nFor $m\\geq1$, put\n\\[\n\\sigma_{N,j,m}\n=\n\\sum_{\\mu\\in\\Lambda_{N,j}}\\mu^{-m}.\n\\]\nThen\n\\[\nC_{N,j,n}(r)\n=\n\\frac{(Nn)!}{n!}\nB_n\\bigl(\n r\\sigma_{N,j,1},\n r1!\\sigma_{N,j,2},\n \\ldots,\n r(n-1)!\\sigma_{N,j,n}\n\\bigr).\n\\]\n\\end{proposition}","ized sections]\\label{prop:normalized_section_product}\nFor every $0\\leq j<N$, the entire function\n\\[\n\\Phi_{N,j}(w)\n=\nj!\\sum_{n\\geq0}\\frac{w^n}{(Nn+j)!}\n\\]\nhas order $1/N$.  If $N\\geq2$ and $\\Lambda_{N,j}$ is its nonzero zero multiset, then\n\\[\n\\Phi_{N,j}(w)\n=\n\\prod_{\\mu\\in\\Lambda_{N,j}}\n\\left(1-\\frac w\\mu\\right).\n\\]\nFor $m\\geq1$, put\n\\[\n\\sigma_{N,j,m}\n=\n\\sum_{\\mu\\in\\Lambda_{N,j}}\\mu^{-m}.\n\\]\nThen\n\\[\nC_{N,j,n}(r)\n=\n\\frac{(Nn)!}{n!}\nB_n\\bigl(\n r\\sigma_{N,j,1},\n r1!\\sigma_{N,j,2},\n \\ldots,\n r(n-1)!\\sigma_{N,j,n}\n\\bigr).\n\\]\n\\end{proposition}","\\begin{proof}\nThe coefficient formula for the order gives $1/N$.  The function is not a polynomial.  An entire function of order below one with only finitely many zeros would be a polynomial times a constant by Hadamard factorization, which is impossible here.  The genus zero product follows as in \\cref{thm:genus_zero_product}.","Taking the logarithm gives\n\\[\n-\\log\\Phi_{N,j}(w)\n=\n\\sum_{m\\geq1}\\sigma_{N,j,m}\\frac{w^m}{m}.\n\\]\nSince $\\Cfun_{N,j}(z)^r=\\Phi_{N,j}(z^N)^{-r}$, the Bell polynomial calculation from \\cref{thm:reciprocal_bell} gives the coefficient formula.\n\\end{proof}","\\begin{definition}\\label{def:zero_power_sums}\nFor $m\\geq1$, define\n\\[\n\\sigma_{N,m}\n=\n\\sum_{\\lambda\\in\\Lambda_N}\\lambda^{-m},\n\\]\nwith multiplicities.\n\\end{definition}","The series converges absolutely because the exponent of convergence of $\\Lambda_N$ is $1/N<1$.","\\begin{theorem}[Newton sums for the reduced zeros]\\label{thm:newton_sums}\nFor $|w|$ below the least zero modulus, with the analytic logarithm\nnormalized to zero at the origin,\n\\[\n-\\log\\Delta_N(w)\n=\n\\sum_{m\\geq1}\\sigma_{N,m}\\frac{w^m}{m}.\n\\]\nThus\n\\[\n\\kappa_{N,m}=\\sigma_{N,m}.\n\\]\nIf\n\\[\na_m=\\frac{(-1)^m}{(Nm)!},\n\\qquad a_0=1,\n\\]\nthen\n\\[\n\\sigma_{N,m}\n=\n-ma_m-\n\\sum_{k=1}^{m-1}a_k\\sigma_{N,m-k}.\n\\]\n\\end{theorem}","\\begin{proof}\nTake the logarithm of the product in \\cref{thm:genus_zero_product}.  Absolute convergence in a disk below the first zero permits termwise expansion\n\\[\n-\\log\\left(1-\\frac w\\lambda\\right)\n=\n\\sum_{m\\geq1}\\frac{w^m}{m\\lambda^m}.\n\\]\nThis gives the logarithmic formula and identifies $\\kappa_{N,m}$ with $\\sigma_{N,m}$.","Differentiate the product identity in logarithmic form to obtain\n\\[\n\\Delta_N'(w)\n=\n-\\Delta_N(w)\n\\sum_{m\\geq1}\\sigma_{N,m}w^{m-1}.\n\\]\nComparison of the coefficient of $w^{m-1}$ gives\n\\[\nma_m\n=\n-\\sum_{k=0}^{m-1}a_k\\sigma_{N,m-k}.\n\\]\nSolving for $\\sigma_{N,m}$ gives the recurrence.\n\\end{proof}","\\begin{corollary}[Convergent Mittag--Leffler partial-fraction form]\\label{cor:mittag_leffler_tangent}\nAway from the zeros of $\\Dfun_N$,\n\\[\n\\Tfun_N(z)\n=\nNz^{N-1}\n\\sum_{\\lambda\\in\\Lambda_N}\n\\frac1{\\lambda-z^N}.\n\\]\nThe series converges absolutely and locally uniformly on compact sets\naway from the poles.  Thus no regularizing terms are required.\n\\end{corollary}","\\begin{proof}\nLogarithmic differentiation of the genus zero product gives\n\\[\n-\\frac{\\Delta_N'(w)}{\\Delta_N(w)}\n=\n\\sum_{\\lambda\\in\\Lambda_N}\\frac1{\\lambda-w}.\n\\]\nThe series is locally uniform because $\\sum_{\\lambda}|\\lambda|^{-1}<\\infty$.  Substitute $w=z^N$ and use the chain rule.\n\\end{proof}","\\begin{proposition}[Theta Bell polynomial formula]\\label{prop:theta_bell}\nPut\n\\[\n\\widetilde{\\sigma}_{N,m}=(-1)^m\\sigma_{N,m}.\n\\]\nThen\n\\[\n\\Thetafun_{N,n}(u)\n=\n\\frac{(Nn)!}{n!}\nB_n\\bigl(\n \\widetilde{\\sigma}_{N,1}(u^N+u),\n 1!\\widetilde{\\sigma}_{N,2}(u^{2N}+u),\n \\ldots,\n (n-1)!\\widetilde{\\sigma}_{N,n}(u^{nN}+u)\n\\bigr).\n\\]\n\\end{proposition}","\\begin{proof}\nThe reduced unsigned denominator is\n\\[\n\\Psi_N(w)=\\sum_{n\\geq0}\\frac{w^n}{(Nn)!}=\\Delta_N(-w).\n\\]\nBy \\cref{thm:newton_sums},\n\\[\n-\\log\\Psi_N(w)\n=\n\\sum_{m\\geq1}\\widetilde{\\sigma}_{N,m}\\frac{w^m}{m}.\n\\]\nWith $w=z^N$, the logarithm of the theta generating function is\n\\[\n\\sum_{m\\geq1}\n\\widetilde{\\sigma}_{N,m}(u^{Nm}+u)\\frac{w^m}{m}.\n\\]\nThe complete Bell polynomial formula follows.\n\\end{proof}","\\begin{corollary}[Tangent coefficients as zero sums]\\label{cor:tangent_zero_sums}\nFor every $n\\geq0$,\n\\[\nT_N(n)\n=\nN(Nn+N-1)!\\sigma_{N,n+1}.\n\\]\n\\end{corollary}","\\begin{corollary}[Reciprocal coefficients from zero data]\\label{cor:reciprocal_zero_data}\nFor every $n\\geq0$,\n\\[\nP_{N,n}(r)\n=\n\\frac{(Nn)!}{n!}\nB_n\\bigl(\n r\\sigma_{N,1},\n r1!\\sigma_{N,2},\n \\ldots,\n r(n-1)!\\sigma_{N,n}\n\\bigr).\n\\]\n\\end{corollary}","\\begin{example}[The cosine orbit sums]\\label{ex:cosine_orbit_sums}\nFor $N=2$,\n\\[\n\\Delta_2(w)=\\cos\\sqrt w.\n\\]\nIts zeros are\n\\[\n\\lambda_m=\\pi^2\\left(m+\\frac12\\right)^2,\n\\qquad m\\geq0.\n\\]\nHence\n\\[\n\\sigma_{2,q}\n=\n\\frac{2^{2q}-1}{\\pi^{2q}}\\zeta(2q).\n\\]\nThe identity in \\cref{cor:tangent_zero_sums} recovers the classical tangent coefficients from these zero power sums.\n\\end{example}","For an entire function $f$, let $\\Zcal(f)$ denote its multiset of nonzero zeros.  Let $r$ be a positive integer.  Then $\\Dfun_N^{-r}$ is meromorphic and a simple zero of $\\Dfun_N$ becomes a pole of order $r$.","\\begin{theorem}[Nearest-pole coefficient expansion]\\label{thm:nearest_circle_expansion}\nLet $r\\in\\ZZ_{>0}$.  Suppose the zeros of $\\Dfun_N$ of least modulus\n$R$ are simple and form the finite set $\\Zcal_R$.  Suppose the next\nzero modulus is $R_2>R$.  At each $\\rho\\in\\Zcal_R$, write the principal\npart\n\\[\n\\Dfun_N(z)^{-r}\n=\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n+\nO(1).\n\\]\nThen, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\\to\\infty$,\n\\[\nP_{N,n}(r)\n=\n m!\n\\sum_{\\rho\\in\\Zcal_R}\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\binom{m+\\ell-1}{\\ell-1}\n\\rho^{-m}\n+\nO_{R_*}(m!R_*^{-m}).\n\\]\nThe leading Laurent coefficient is\n\\[\na_{\\rho,r}\n=\n(-\\rho\\Dfun_N'(\\rho))^{-r}.\n\\]\n\\end{theorem}","rest-pole coefficient expansion]\\label{thm:nearest_circle_expansion}\nLet $r\\in\\ZZ_{>0}$.  Suppose the zeros of $\\Dfun_N$ of least modulus\n$R$ are simple and form the finite set $\\Zcal_R$.  Suppose the next\nzero modulus is $R_2>R$.  At each $\\rho\\in\\Zcal_R$, write the principal\npart\n\\[\n\\Dfun_N(z)^{-r}\n=\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n+\nO(1).\n\\]\nThen, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\\to\\infty$,\n\\[\nP_{N,n}(r)\n=\n m!\n\\sum_{\\rho\\in\\Zcal_R}\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\binom{m+\\ell-1}{\\ell-1}\n\\rho^{-m}\n+\nO_{R_*}(m!R_*^{-m}).\n\\]\nThe leading Laurent coefficient is\n\\[\na_{\\rho,r}\n=\n(-\\rho\\Dfun_N'(\\rho))^{-r}.\n\\]\n\\end{theorem}","oefficient expansion]\\label{thm:nearest_circle_expansion}\nLet $r\\in\\ZZ_{>0}$.  Suppose the zeros of $\\Dfun_N$ of least modulus\n$R$ are simple and form the finite set $\\Zcal_R$.  Suppose the next\nzero modulus is $R_2>R$.  At each $\\rho\\in\\Zcal_R$, write the principal\npart\n\\[\n\\Dfun_N(z)^{-r}\n=\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n+\nO(1).\n\\]\nThen, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\\to\\infty$,\n\\[\nP_{N,n}(r)\n=\n m!\n\\sum_{\\rho\\in\\Zcal_R}\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\binom{m+\\ell-1}{\\ell-1}\n\\rho^{-m}\n+\nO_{R_*}(m!R_*^{-m}).\n\\]\nThe leading Laurent coefficient is\n\\[\na_{\\rho,r}\n=\n(-\\rho\\Dfun_N'(\\rho))^{-r}.\n\\]\n\\end{theorem}","sion]\\label{thm:nearest_circle_expansion}\nLet $r\\in\\ZZ_{>0}$.  Suppose the zeros of $\\Dfun_N$ of least modulus\n$R$ are simple and form the finite set $\\Zcal_R$.  Suppose the next\nzero modulus is $R_2>R$.  At each $\\rho\\in\\Zcal_R$, write the principal\npart\n\\[\n\\Dfun_N(z)^{-r}\n=\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n+\nO(1).\n\\]\nThen, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\\to\\infty$,\n\\[\nP_{N,n}(r)\n=\n m!\n\\sum_{\\rho\\in\\Zcal_R}\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\binom{m+\\ell-1}{\\ell-1}\n\\rho^{-m}\n+\nO_{R_*}(m!R_*^{-m}).\n\\]\nThe leading Laurent coefficient is\n\\[\na_{\\rho,r}\n=\n(-\\rho\\Dfun_N'(\\rho))^{-r}.\n\\]\n\\end{theorem}","{thm:nearest_circle_expansion}\nLet $r\\in\\ZZ_{>0}$.  Suppose the zeros of $\\Dfun_N$ of least modulus\n$R$ are simple and form the finite set $\\Zcal_R$.  Suppose the next\nzero modulus is $R_2>R$.  At each $\\rho\\in\\Zcal_R$, write the principal\npart\n\\[\n\\Dfun_N(z)^{-r}\n=\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n+\nO(1).\n\\]\nThen, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\\to\\infty$,\n\\[\nP_{N,n}(r)\n=\n m!\n\\sum_{\\rho\\in\\Zcal_R}\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\binom{m+\\ell-1}{\\ell-1}\n\\rho^{-m}\n+\nO_{R_*}(m!R_*^{-m}).\n\\]\nThe leading Laurent coefficient is\n\\[\na_{\\rho,r}\n=\n(-\\rho\\Dfun_N'(\\rho))^{-r}.\n\\]\n\\end{theorem}","rcle_expansion}\nLet $r\\in\\ZZ_{>0}$.  Suppose the zeros of $\\Dfun_N$ of least modulus\n$R$ are simple and form the finite set $\\Zcal_R$.  Suppose the next\nzero modulus is $R_2>R$.  At each $\\rho\\in\\Zcal_R$, write the principal\npart\n\\[\n\\Dfun_N(z)^{-r}\n=\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n+\nO(1).\n\\]\nThen, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\\to\\infty$,\n\\[\nP_{N,n}(r)\n=\n m!\n\\sum_{\\rho\\in\\Zcal_R}\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\binom{m+\\ell-1}{\\ell-1}\n\\rho^{-m}\n+\nO_{R_*}(m!R_*^{-m}).\n\\]\nThe leading Laurent coefficient is\n\\[\na_{\\rho,r}\n=\n(-\\rho\\Dfun_N'(\\rho))^{-r}.\n\\]\n\\end{theorem}","_2>R$.  At each $\\rho\\in\\Zcal_R$, write the principal\npart\n\\[\n\\Dfun_N(z)^{-r}\n=\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n+\nO(1).\n\\]\nThen, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\\to\\infty$,\n\\[\nP_{N,n}(r)\n=\n m!\n\\sum_{\\rho\\in\\Zcal_R}\n\\sum_{\\ell=1}^{r}\n a_{\\rho,\\ell}\n\\binom{m+\\ell-1}{\\ell-1}\n\\rho^{-m}\n+\nO_{R_*}(m!R_*^{-m}).\n\\]\nThe leading Laurent coefficient is\n\\[\na_{\\rho,r}\n=\n(-\\rho\\Dfun_N'(\\rho))^{-r}.\n\\]\n\\end{theorem}","\\begin{proof}\nSubtract the complete principal parts at the zeros on $|z|=R$.  The remainder is holomorphic on $|z|<R_2$, hence on a neighborhood of every closed disk $|z|\\leq R_*<R_2$.  Cauchy's estimate gives an ordinary coefficient bound $O_{R_*}(R_*^{-m})$.  The coefficient of $z^m$ in\n\\[\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n\\]\nis\n\\[\n\\binom{m+\\ell-1}{\\ell-1}\\rho^{-m}.\n\\]\nMultiplication by $m!$ converts the ordinary coefficient to $P_{N,n}(r)$.","principal parts at the zeros on $|z|=R$.  The remainder is holomorphic on $|z|<R_2$, hence on a neighborhood of every closed disk $|z|\\leq R_*<R_2$.  Cauchy's estimate gives an ordinary coefficient bound $O_{R_*}(R_*^{-m})$.  The coefficient of $z^m$ in\n\\[\n\\left(1-\\frac z\\rho\\right)^{-\\ell}\n\\]\nis\n\\[\n\\binom{m+\\ell-1}{\\ell-1}\\rho^{-m}.\n\\]\nMultiplication by $m!$ converts the ordinary coefficient to $P_{N,n}(r)$.","Near a simple zero,\n\\[\n\\Dfun_N(z)\n=\n-\\rho\\Dfun_N'(\\rho)\n\\left(1-\\frac z\\rho\\right)\n+O\\left(\\left(1-\\frac z\\rho\\right)^2\\right).\n\\]\nRaising to the power $-r$ gives the formula for $a_{\\rho,r}$.\n\\end{proof}","\\begin{corollary}[Leading dominant zero term]\\label{cor:leading_dominant_zero}\nUnder the hypotheses of \\cref{thm:nearest_circle_expansion},\n\\[\n\\begin{aligned}\nP_{N,n}(r)\n={}&m!\\binom{m+r-1}{r-1}\n\\sum_{\\rho\\in\\Zcal_R}\n\\frac{\\rho^{-m}}{(-\\rho\\Dfun_N'(\\rho))^r}\n\\\\\n&+O\\left(m!m^{r-2}R^{-m}\\right)\n+O_{R_*}\\left(m!R_*^{-m}\\right).\n\\end{aligned}\n\\]\nFor $r=1$, the first error term is absent.\n\\end{corollary}","&+O\\left(m!m^{r-2}R^{-m}\\right)\n+O_{R_*}\\left(m!R_*^{-m}\\right).\n\\end{aligned}\n\\]\nFor $r=1$, the first error term is absent.\n\\end{corollary}","\\begin{proof}\nThe terms with $\\ell<r$ in \\cref{thm:nearest_circle_expansion} are bounded by a constant times $m^{r-2}R^{-m}$.  The term with $\\ell=r$ gives the displayed main term.\n\\end{proof}","\\begin{corollary}[Orbit reduced nearest circle term]\\label{cor:orbit_reduced_asymptotic}\nAssume the hypotheses of \\cref{thm:nearest_circle_expansion}.  Choose one representative from each orbit of the nearest zeros under multiplication by $\\omega$, and call the resulting set $\\Zcal_R^{\\mathrm{orb}}$.  With $m=Nn$,\n\\[\n\\begin{aligned}\nP_{N,n}(r)\n={}&N m!\\binom{m+r-1}{r-1}\n\\sum_{\\tau\\in\\Zcal_R^{\\mathrm{orb}}}\n\\frac{\\tau^{-m}}{(-\\tau\\Dfun_N'(\\tau))^r}","\\begin{corollary}[Orbit reduced nearest circle term]\\label{cor:orbit_reduced_asymptotic}\nAssume the hypotheses of \\cref{thm:nearest_circle_expansion}.  Choose one representative from each orbit of the nearest zeros under multiplication by $\\omega$, and call the resulting set $\\Zcal_R^{\\mathrm{orb}}$.  With $m=Nn$,\n\\[\n\\begin{aligned}\nP_{N,n}(r)\n={}&N m!\\binom{m+r-1}{r-1}\n\\sum_{\\tau\\in\\Zcal_R^{\\mathrm{orb}}}\n\\frac{\\tau^{-m}}{(-\\tau\\Dfun_N'(\\tau))^r}\n\\\\\n&+O\\left(m!m^{r-2}R^{-m}\\right)\n+O_{R_*}\\left(m!R_*^{-m}\\right).\n\\end{aligned}\n\\]\nFor $r=1$, the first error term is absent.\n\\end{corollary}","\\begin{proof}\nIf $\\rho=\\omega^k\\tau$, then\n\\[\n\\Dfun_N'(\\rho)=\\omega^{-k}\\Dfun_N'(\\tau).\n\\]\nTherefore\n\\[\n-\\rho\\Dfun_N'(\\rho)\n=-\\tau\\Dfun_N'(\\tau).\n\\]\nSince $m=Nn$, one also has $\\rho^{-m}=\\tau^{-m}$.  Every orbit therefore contributes $N$ equal terms in \\cref{cor:leading_dominant_zero}.\n\\end{proof}","\\begin{corollary}[Secant asymptotic]\\label{cor:secant_asymptotic}\nFor $N=2$ and $r=1$,\n\\[\nP_{2,n}(1)\n\\sim\n2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\n\\end{corollary}","\\begin{proof}\nThe nearest zeros of $\\cos z$ are $\\rho=\\pm\\pi/2$, both simple.  Their contributions in \\cref{cor:leading_dominant_zero} are equal.\n\\end{proof}","The assumption $N\\geq2$ remains in force throughout this section.  The zero counting theorem gives the exact abscissae of absolute convergence.  The exponentially accurate zero locations give more.  They compare the zero zeta with a Hurwitz zeta function.","\\begin{definition}\\label{def:absolute_zero_zetas}\nDefine\n\\[\n\\mathfrak Z_N^{\\mathrm{abs}}(s)\n=\n\\sum_{\\rho\\in\\Zcal(\\Dfun_N)}|\\rho|^{-s}\n\\]\nand\n\\[\n\\mathfrak Z_N^{\\mathrm{red}}(s)\n=\n\\sum_{\\lambda\\in\\Lambda_N}|\\lambda|^{-s},\n\\]\nwith multiplicities.  Each term means $e^{-s\\log|\\rho|}$ or $e^{-s\\log|\\lambda|}$.\n\\end{definition}","\\begin{theorem}[Abscissae of absolute convergence]\\label{thm:zeta_convergence}\nThe series $\\mathfrak Z_N^{\\mathrm{abs}}$ converges absolutely and locally uniformly when\n\\[\n\\Re s>1.\n\\]\nIt fails to converge absolutely when $\\Re s\\leq1$.  The series $\\mathfrak Z_N^{\\mathrm{red}}$ converges absolutely and locally uniformly when\n\\[\n\\Re s>\\frac1N.\n\\]\nIt fails to converge absolutely when $\\Re s\\leq1/N$.  Both functions are holomorphic on their open half-planes of absolute convergence.\n\\end{theorem}","\\begin{proof}\nBy \\cref{thm:zero_counting}, the zeros of $\\Dfun_N$ have linear counting density, while the reduced zeros have counting order $R^{1/N}$.  Partial summation gives absolute convergence in the stated open half-planes.  The positive terms obtained after taking absolute values diverge on the boundary and to its left.  Local uniform convergence follows by the Weierstrass test on compact subsets.\n\\end{proof}","\\begin{theorem}[Meromorphic continuation of the symmetry-orbit zero zeta]\\label{thm:zeta_continuation}\nPut\n\\[\nA_N=\\frac\\pi{\\sin(\\pi/N)}.\n\\]\nThere is an entire function $Q_N^{\\mathrm{red}}$ such that\n\\[\n\\mathfrak Z_N^{\\mathrm{red}}(s)\n=\nA_N^{-Ns}\\zeta\\left(Ns,\\frac12\\right)+Q_N^{\\mathrm{red}}(s)\n\\]\non $\\Re s>1/N$.  The right-hand side gives a meromorphic continuation to $\\CC$ with one simple pole at $s=1/N$.  Its residue is\n\\[\n\\frac{\\sin(\\pi/N)}{N\\pi}.\n\\]","gin{theorem}[Meromorphic continuation of the symmetry-orbit zero zeta]\\label{thm:zeta_continuation}\nPut\n\\[\nA_N=\\frac\\pi{\\sin(\\pi/N)}.\n\\]\nThere is an entire function $Q_N^{\\mathrm{red}}$ such that\n\\[\n\\mathfrak Z_N^{\\mathrm{red}}(s)\n=\nA_N^{-Ns}\\zeta\\left(Ns,\\frac12\\right)+Q_N^{\\mathrm{red}}(s)\n\\]\non $\\Re s>1/N$.  The right-hand side gives a meromorphic continuation to $\\CC$ with one simple pole at $s=1/N$.  Its residue is\n\\[\n\\frac{\\sin(\\pi/N)}{N\\pi}.\n\\]","ontinuation of the symmetry-orbit zero zeta]\\label{thm:zeta_continuation}\nPut\n\\[\nA_N=\\frac\\pi{\\sin(\\pi/N)}.\n\\]\nThere is an entire function $Q_N^{\\mathrm{red}}$ such that\n\\[\n\\mathfrak Z_N^{\\mathrm{red}}(s)\n=\nA_N^{-Ns}\\zeta\\left(Ns,\\frac12\\right)+Q_N^{\\mathrm{red}}(s)\n\\]\non $\\Re s>1/N$.  The right-hand side gives a meromorphic continuation to $\\CC$ with one simple pole at $s=1/N$.  Its residue is\n\\[\n\\frac{\\sin(\\pi/N)}{N\\pi}.\n\\]","There is an entire function $Q_N^{\\mathrm{abs}}$ such that, initially on\n$\\Re s>1$,\n\\[\n\\mathfrak Z_N^{\\mathrm{abs}}(s)\n=\nN A_N^{-s}\\zeta\\left(s,\\frac12\\right)+Q_N^{\\mathrm{abs}}(s).\n\\]\nThus $\\mathfrak Z_N^{\\mathrm{abs}}$ continues meromorphically to $\\CC$ with one simple pole at $s=1$, of residue\n\\[\n\\frac{N\\sin(\\pi/N)}\\pi.\n\\]\n\\end{theorem}","\\begin{proof}\nFor $N=2$, \\cref{cor:case_two_zeros} and\n$\\Dfun_2(z)=\\Delta_2(z^2)$ give\n\\[\n\\lambda_m=\\pi^2\\left(m+\\frac12\\right)^2,\n\\qquad m\\geq0.\n\\]\nThus the first formula holds with $Q_2^{\\mathrm{red}}=0$.","For $N\\geq3$, \\cref{cor:reduced_zeros} writes all but finitely many reduced zeros as\n\\[\n\\lambda_m\n=\n\\left(A_N\\left(m+\\frac12\\right)\\right)^N(1+\\varepsilon_m)\n\\]\nwhere $\\lambda_m>0$, $1+\\varepsilon_m>0$, and\n\\[\n\\varepsilon_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$","For $N\\geq3$, \\cref{cor:reduced_zeros} writes all but finitely many reduced zeros as\n\\[\n\\lambda_m\n=\n\\left(A_N\\left(m+\\frac12\\right)\\right)^N(1+\\varepsilon_m)\n\\]\nwhere $\\lambda_m>0$, $1+\\varepsilon_m>0$, and\n\\[\n\\varepsilon_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}","For $N\\geq3$, \\cref{cor:reduced_zeros} writes all but finitely many reduced zeros as\n\\[\n\\lambda_m\n=\n\\left(A_N\\left(m+\\frac12\\right)\\right)^N(1+\\varepsilon_m)\n\\]\nwhere $\\lambda_m>0$, $1+\\varepsilon_m>0$, and\n\\[\n\\varepsilon_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirs","For $N\\geq3$, \\cref{cor:reduced_zeros} writes all but finitely many reduced zeros as\n\\[\n\\lambda_m\n=\n\\left(A_N\\left(m+\\frac12\\right)\\right)^N(1+\\varepsilon_m)\n\\]\nwhere $\\lambda_m>0$, $1+\\varepsilon_m>0$, and\n\\[\n\\varepsilon_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that","For $N\\geq3$, \\cref{cor:reduced_zeros} writes all but finitely many reduced zeros as\n\\[\n\\lambda_m\n=\n\\left(A_N\\left(m+\\frac12\\right)\\right)^N(1+\\varepsilon_m)\n\\]\nwhere $\\lambda_m>0$, $1+\\varepsilon_m>0$, and\n\\[\n\\varepsilon_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.","ight)\\right)^N(1+\\varepsilon_m)\n\\]\nwhere $\\lambda_m>0$, $1+\\varepsilon_m>0$, and\n\\[\n\\varepsilon_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.","silon_m>0$, and\n\\[\n\\varepsilon_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.","on_m\n=\nO\\left(t_m^{-1}e^{-\\gamma t_m}\\right).\n\\]\nOn every compact set of $s$ values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.","values,\n\\[\n(1+\\varepsilon_m)^{-s}-1\n=\nO(|\\varepsilon_m|)\n\\]\nuniformly for large $m$.  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.",".  The difference\n\\[\n\\lambda_m^{-s}\n-\nA_N^{-Ns}\\left(m+\\frac12\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.","2\\right)^{-Ns}\n\\]\ntherefore has a normally convergent sum on every compact subset of\n$\\CC$.  Indeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.","ndeed, the model power has at most polynomial growth in $m$\non a compact set, while $|\\varepsilon_m|$ decays exponentially.  The\nfinitely many exceptional terms $|\\lambda|^{-s}$ and the finitely many\nomitted model terms are entire functions of $s$.  Their addition to the\nnormally convergent tail defines $Q_N^{\\mathrm{red}}$ and proves the\nfirst formula. Thus that formula is established for every $N\\geq2$.","In either case, the Hurwitz zeta function has a meromorphic continuation with one simple pole at its first argument equal to one and residue one.  Composition with $Ns$ gives the stated pole and residue.","For every $N\\geq2$, each nonzero zero $\\lambda$ of $\\Delta_N$ gives $N$ zeros of $\\Dfun_N$, each of modulus $|\\lambda|^{1/N}$.  Hence\n\\[\n\\mathfrak Z_N^{\\mathrm{abs}}(s)\n=\nN\\mathfrak Z_N^{\\mathrm{red}}(s/N).\n\\]\nSubstitution gives the second formula with\n\\[\nQ_N^{\\mathrm{abs}}(s)=NQ_N^{\\mathrm{red}}(s/N),\n\\]\nwhich is entire.\n\\end{proof}","\\begin{corollary}[Exact zeta models in orders two and four]\\label{cor:exact_zeta_models}\nFor $N=2$,\n\\[\n\\mathfrak Z_2^{\\mathrm{red}}(s)\n=\n\\pi^{-2s}\\zeta\\left(2s,\\frac12\\right)\n\\]\nand\n\\[\n\\mathfrak Z_2^{\\mathrm{abs}}(s)\n=\n2\\pi^{-s}\\zeta\\left(s,\\frac12\\right).\n\\]\nFor $N=4$, put $A_4=\\sqrt2\\pi$.  Then\n\\[\n\\mathfrak Z_4^{\\mathrm{red}}(s)\n=\nA_4^{-4s}\\zeta\\left(4s,\\frac12\\right)\n\\]\nand\n\\[\n\\mathfrak Z_4^{\\mathrm{abs}}(s)\n=\n4A_4^{-s}\\zeta\\left(s,\\frac12\\right).\n\\]\nIn these two cases the entire correction terms in \\cref{thm:zeta_continuation} vanish identically.\n\\end{corollary}","\\begin{proof}\nFor $N=2$, the reduced zeros are\n\\[\n\\pi^2\\left(m+\\frac12\\right)^2,\n\\qquad m\\geq0.\n\\]\nFor $N=4$, \\cref{ex:fourth_order_zeros} gives the reduced zeros\n\\[\nA_4^4\\left(m+\\frac12\\right)^4,\n\\qquad m\\geq0.\n\\]\nSumming their negative powers proves the reduced formulas.  The absolute formulas follow from\n\\[\n\\mathfrak Z_N^{\\mathrm{abs}}(s)\n=\nN\\mathfrak Z_N^{\\mathrm{red}}(s/N).\n\\]\n\\end{proof}","\\begin{definition}\\label{def:branch_orbit_zeta}\nChoose a logarithm for each member of the finite exceptional part of $\\Lambda_N$.  Use the real logarithm for the eventual positive sequence.  Define\n\\[\n\\mathfrak Z_N^{\\mathrm{orb}}(s)\n=\n\\sum_{\\lambda\\in\\Lambda_N}e^{-s\\Log\\lambda}.\n\\]\nThis series is initially defined on $\\Re s>1/N$.\n\\end{definition}","\\begin{corollary}[Branch-dependent symmetry-orbit continuation]\\label{cor:branch_grouped_continuation}\nThe branch-dependent orbit zeta $\\mathfrak Z_N^{\\mathrm{orb}}$ continues meromorphically to $\\CC$.  It has the same single pole and residue as $\\mathfrak Z_N^{\\mathrm{red}}$.  Changing the logarithms of the finite exceptional zeros changes the continuation by an entire function.\n\\end{corollary}","\\begin{definition}\\label{def:derivative_weighted_zeta}\nLet $q$ be a positive integer.  Choose logarithms of the simple zeros of $\\Dfun_N$ with arguments in a fixed bounded interval.  Define\n\\[\n\\mathfrak Z_{N,q}^{\\mathrm{der}}(s)\n=\n\\sum_{\\substack{\\Dfun_N(\\rho)=0","\\begin{definition}\\label{def:derivative_weighted_zeta}\nLet $q$ be a positive integer.  Choose logarithms of the simple zeros of $\\Dfun_N$ with arguments in a fixed bounded interval.  Define\n\\[\n\\mathfrak Z_{N,q}^{\\mathrm{der}}(s)\n=\n\\sum_{\\substack{\\Dfun_N(\\rho)=0\\\\ \\Dfun_N'(\\rho)\\ne0}}\n\\frac{e^{-s\\Log\\rho}}{\\Dfun_N'(\\rho)^q}.\n\\]\n\\end{definition}","\\begin{theorem}[Derivative-weighted convergence]\\label{thm:derivative_weighted_convergence}\nIf $N\\geq3$ and $q\\geq1$, the series $\\mathfrak Z_{N,q}^{\\mathrm{der}}(s)$ converges locally uniformly for every $s\\in\\CC$.  It therefore defines an entire function.","For $N=2$, the series converges absolutely and locally uniformly on\n$\\Re s>1$, where it defines a holomorphic function.\n\\end{theorem}","\\begin{proof}\nFor $N\\geq3$, \\cref{thm:large_zero_strings} gives\n\\[\n|\\Dfun_N'(\\rho)|^{-1}\n\\leq\nC e^{-c_N|\\rho|}\n\\]\nfor every sufficiently large simple zero.  Let $L$ be a compact subset\nof the $s$ plane.  Because the chosen arguments of the zeros lie in a\nfixed bounded interval, there are constants $C_L$ and $M_L$ such that\n\\[\n\\sup_{s\\in L}|e^{-s\\Log\\rho}|\n\\leq C_L|\\rho|^{M_L}\n\\]\nfor every sufficiently large zero.  Hence the absolute value of the\ncorresponding summand is at most\n\\[\nC_L'|\\rho|^{M_L}e^{-q c_N|\\rho|}.\n\\]\nThe linear zero counting estimate makes this majorant summable.  The\nWeierstrass test gives absolute local uniform convergence of the\nseries of entire summands, so its sum is entire.  The finite excepti","\\begin{proof}\nFor $N\\geq3$, \\cref{thm:large_zero_strings} gives\n\\[\n|\\Dfun_N'(\\rho)|^{-1}\n\\leq\nC e^{-c_N|\\rho|}\n\\]\nfor every sufficiently large simple zero.  Let $L$ be a compact subset\nof the $s$ plane.  Because the chosen arguments of the zeros lie in a\nfixed bounded interval, there are constants $C_L$ and $M_L$ such that\n\\[\n\\sup_{s\\in L}|e^{-s\\Log\\rho}|\n\\leq C_L|\\rho|^{M_L}\n\\]\nfor every sufficiently large zero.  Hence the absolute value of the\ncorresponding summand is at most\n\\[\nC_L'|\\rho|^{M_L}e^{-q c_N|\\rho|}.\n\\]\nThe linear zero counting estimate makes this majorant summable.  The\nWeierstrass test gives absolute local uniform convergence of the\nseries of entire summands, so its sum is entire.  The finite exceptional\nset causes no difficulty.","-1}\n\\leq\nC e^{-c_N|\\rho|}\n\\]\nfor every sufficiently large simple zero.  Let $L$ be a compact subset\nof the $s$ plane.  Because the chosen arguments of the zeros lie in a\nfixed bounded interval, there are constants $C_L$ and $M_L$ such that\n\\[\n\\sup_{s\\in L}|e^{-s\\Log\\rho}|\n\\leq C_L|\\rho|^{M_L}\n\\]\nfor every sufficiently large zero.  Hence the absolute value of the\ncorresponding summand is at most\n\\[\nC_L'|\\rho|^{M_L}e^{-q c_N|\\rho|}.\n\\]\nThe linear zero counting estimate makes this majorant summable.  The\nWeierstrass test gives absolute local uniform convergence of the\nseries of entire summands, so its sum is entire.  The finite exceptional\nset causes no difficulty.","For $N=2$, the derivative of the cosine has modulus one at every zero.  The assertion then follows from the linear zero spacing.\n\\end{proof}","\\begin{remark}\\label{rem:multiple_zeros}\nThe derivative weighted sum omits the finitely many zeros that might be multiple.  A treatment of those zeros requires the complete principal parts of $\\Dfun_N^{-q}$ rather than the single derivative weight.\n\\end{remark}","The large-zero theorem settles eventual simplicity but does not classify the\nfinite exceptional set. A remaining question is whether every zero of\n$\\Dfun_N$ is simple and every reduced zero of $\\Delta_N$ is positive. Both\nstatements hold for $N=2$ and $N=4$. The argument here proves them only outside\na compact set for general $N$.","The leading correction in \\cref{thm:first_zero_correction} may admit iteration\nto a full exponential expansion for $\\tau_m$. Such an expansion is an open\nproblem, not a result of this chapter. It could refine the entire correction in\nthe zero-zeta comparison and lead to explicit special values.","The symmetry-orbit zero zeta has a meromorphic continuation with one pole. No\nfunctional equation is obtained. A branch dependent sum over the $N$ roots of\neach reduced zero must specify logarithms on the rays.","The coefficient families in\n\\cref{sec:reciprocal_powers,sec:residue_section,sec:theta} have exact formal\nparameter-addition laws. Their arithmetic behavior, including odd-prime\ncongruences, dyadic weighting, and $p$-adic or Amice--Mahler interpolation,\nbelongs to Chapter IX. General bounded-operator semigroup, resolvent, and\nFeshbach theory belongs to Chapter VI. Chapter III covers general nonlinear\noperation trees, while abstract completion and occupancy questions belong to\nChapter X. None of those theories is inferred from the scalar complex-analytic\nresults proved here.","6.1--6.6 & Quotient-algebra power bases and actual exponential-series\nconvergence in the finite-dimensional coefficient topology. Both pointwise\nsection-coordinate identities. Signed-shift powers, all matrix-exponential\nentries, group laws and trace are checked. The characteristic polynomial\n$X^N+1$, its distinct roots, exact spectrum and determinant one are proved.","8.1--8.5 & The full tangent coefficient series has the exact kappa and\nfactorial normalization, including $N=1$. Reciprocal differentiation holds on\nevery stated fixed logarithm domain. The actual higher Riccati identity holds\nat every point where the denominator is nonzero.","9.1--9.10 & Arbitrary-parameter residue reciprocals and their analytic germs,\npolynomial degree and leading term, sparse Appell laws, joint convolution,\nmonicity and covariance. The Hasse and derivative-truncation encodings are\nproved equal. The named Appell polynomial satisfies both the forward action\nand the exact $\\Phi_{N,j}(\\partial_x^N)^r$ inverse formula.","10.1--10.4 & Exact formal theta reconstruction, polynomial lift and degree,\nliteral divisibility by $u$ and, for even $N$, by $u+1$, and the theta Newton\nrecurrence with its first coefficient.","11.2, 11.4--11.11 & Two-sided maximum-modulus bounds prove exact order\nand type. The indicator limsup formula holds in every direction, including ties.\nDominant-exponential errors and zero freeness are uniform on compact sets\navoiding ties. Near cancellation and the order-two zero set are checked.\nFor $N\\geq3$, gap and remainder bounds, sector confinement, global exhaustion\nand multiplicity identities give the full large-zero-string theorem.\nThe real leading correction, quadratic exponential error, signed derivative\nasymptotic, eventual alternating sign and cubic correction are checked.","11.13--11.15 & The reduced series is entire, and\n$\\Dfun_N(z)=\\Delta_N(z^N)$ preserves multiplicity away from zero. Finite\nexceptional sets, eventual positive simple reduced zeros, and their stated\ndisplacement are checked, including $N=2$. Both full disk-counting laws\nretain all exceptional multiplicities. The full-zero reciprocal sums have\nexact thresholds $s>1$ and $s>1/N$, including boundary divergence.","12.1 & For every $N\\geq2$, the complete multiplicity-indexed product\nconverges uniformly on compact sets and equals $\\Delta_N$. Jensen and Poisson\nestimates prove rigidity of the actual quotient from derived growth bounds.\nNo growth certificate remains. The original-variable product for $\\Dfun_N$\nis also checked.","12.2--12.4 & The actual diagonal operator is trace class, with a\nbasis-independent trace and determinant given by limits of actual finite\ncompressions. Infinite complete homogeneous coefficients are limits of finite\nmonomial sums. Every normalized section has exact order $1/N$. For $N\\geq2$\nits full zero product and zero-moment Bell formulas are proved. The actual\ncomplex-power germ at zero has the named residue Taylor coefficients.","12.6--12.10 & The normalized logarithm on each zero-free disk has the\nconvergent zero-moment series. Its coefficients equal $\\kappa_{N,m}$ and\nsatisfy Newton's recurrence. The Mittag--Leffler tangent expansion converges\nabsolutely and locally uniformly away from poles. Tangent coefficients and\nthe named reciprocal and theta families have their exact zero-moment Bell\nformulas. The full tangent series is identified with the analytic function.","13.1--13.4 & Analytic division constructs each simple-zero principal part.\nIts leading coefficient is uniquely determined. Removing the principal parts\ngives a holomorphic remainder and the exact factorial-scaled Cauchy estimate.\nBoth leading-term errors are bounded, with the first absent for $r=1$.\nRotation gives the orbit-reduced main term. The secant formula holds as a\nratio limit, with an exponentially vanishing relative error.","14.8 & The derivative-weighted series uses every actual simple zero and genuine\nlogarithms with bounded arguments. Absolute and local uniform convergence and\nan entire sum are proved for $N\\geq3$. For $N=2$, absolute/local uniform\nconvergence and holomorphy hold on $\\Re s>1$.","Cyclic projection turns the exponential into a finite differential basis, and\nthe signed cycle produces the distinguished denominator $\\Dfun_N$. Its\nreciprocal powers yield Bell and Newton recurrences, an additive parameter\nlaw, Appell systems, and theta polynomials. Reduction by $w=z^N$ then connects\nthe reciprocal coefficients to a genus zero product and reduced-zero power\nsums.","data}{READER'S GUIDE}{Chapter IV: Section 2, Section 7, Section 11, Section 12.}\nThis guide follows a concrete analytic family through three descriptions: selected Taylor coefficients, a finite differential system, and the zeros of an entire function. The formulas are linked, but the evidence needed for each description is different.\n\\vfig{three_views}\n\\head{Keep three small models in view}\nFor $N=2$, the signed denominator is $D_2(z)=\\cos z$. Its reciprocal is the familiar secant series. For $N=3$, a third exponential perturbs the zeros of a dominant cosine. For $N=4$, an exact factorization gives a complete zero picture:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2).\n\\]\nThe examples move between these orders deliberately. Order two makes the coefficient arithmetic transparent. Order three shows a genuine displacement. Order four makes orbit reduction completely explicit.\n\\head{Three routes through the guide}\n\\par\\noindent\\begin{tabularx}{\\linewidth}{@{}lY@{}}\\toprule\nRoute & Pages and question",": Section 2, Section 7, Section 11, Section 12.}\nThis guide follows a concrete analytic family through three descriptions: selected Taylor coefficients, a finite differential system, and the zeros of an entire function. The formulas are linked, but the evidence needed for each description is different.\n\\vfig{three_views}\n\\head{Keep three small models in view}\nFor $N=2$, the signed denominator is $D_2(z)=\\cos z$. Its reciprocal is the familiar secant series. For $N=3$, a third exponential perturbs the zeros of a dominant cosine. For $N=4$, an exact factorization gives a complete zero picture:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2).\n\\]\nThe examples move between these orders deliberately. Order two makes the coefficient arithmetic transparent. Order three shows a genuine displacement. Order four makes orbit reduction completely explicit.\n\\head{Three routes through the guide}\n\\par\\noindent\\begin{tabularx}{\\linewidth}{@{}lY@{}}\\toprule\nRoute & Pages and question","a concrete analytic family through three descriptions: selected Taylor coefficients, a finite differential system, and the zeros of an entire function. The formulas are linked, but the evidence needed for each description is different.\n\\vfig{three_views}\n\\head{Keep three small models in view}\nFor $N=2$, the signed denominator is $D_2(z)=\\cos z$. Its reciprocal is the familiar secant series. For $N=3$, a third exponential perturbs the zeros of a dominant cosine. For $N=4$, an exact factorization gives a complete zero picture:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2).\n\\]\nThe examples move between these orders deliberately. Order two makes the coefficient arithmetic transparent. Order three shows a genuine displacement. Order four makes orbit reduction completely explicit.\n\\head{Three routes through the guide}\n\\par\\noindent\\begin{tabularx}{\\linewidth}{@{}lY@{}}\\toprule\nRoute & Pages and question","coefficients, a finite differential system, and the zeros of an entire function. The formulas are linked, but the evidence needed for each description is different.\n\\vfig{three_views}\n\\head{Keep three small models in view}\nFor $N=2$, the signed denominator is $D_2(z)=\\cos z$. Its reciprocal is the familiar secant series. For $N=3$, a third exponential perturbs the zeros of a dominant cosine. For $N=4$, an exact factorization gives a complete zero picture:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2).\n\\]\nThe examples move between these orders deliberately. Order two makes the coefficient arithmetic transparent. Order three shows a genuine displacement. Order four makes orbit reduction completely explicit.\n\\head{Three routes through the guide}\n\\par\\noindent\\begin{tabularx}{\\linewidth}{@{}lY@{}}\\toprule\nRoute & Pages and question","n entire function. The formulas are linked, but the evidence needed for each description is different.\n\\vfig{three_views}\n\\head{Keep three small models in view}\nFor $N=2$, the signed denominator is $D_2(z)=\\cos z$. Its reciprocal is the familiar secant series. For $N=3$, a third exponential perturbs the zeros of a dominant cosine. For $N=4$, an exact factorization gives a complete zero picture:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2).\n\\]\nThe examples move between these orders deliberately. Order two makes the coefficient arithmetic transparent. Order three shows a genuine displacement. Order four makes orbit reduction completely explicit.\n\\head{Three routes through the guide}\n\\par\\noindent\\begin{tabularx}{\\linewidth}{@{}lY@{}}\\toprule\nRoute & Pages and question","One full calculation & \\pagerefV{25}: derive the same number, $69$, from denominator coefficients and from zero moments.","\\vpage{2}{Two variables, several normalizations}{OBJECT ATLAS}{Chapter IV: Section 2, Definition 5.7, Definition 7.1, Definition 9.1.}\nThe most important coordinate change is $w=z^N$. A term of degree $n$ in $w$ has degree $Nn$ in $z$. Its coefficient does not change, but an exponential-generating-function label does.\n\\[\n\\underbrace{b_{N,n}(r)\\,w^n}_{\\text{ordinary coefficient}}\n=\\underbrace{P_{N,n}(r)\\frac{z^{Nn}}{(Nn)!}}_{\\text{block exponential coefficient}},\n\\qquad P_{N,n}(r)=(Nn)!b_{N,n}(r).\n\\]\n\\par\\noindent\\begin{tabularx}{\\linewidth}{@{}lY@{}}\\toprule\nObject & Definition and role","midrule\n$E_{N,j}(z)$ & $\\displaystyle\\sum_{n\\ge0}\\frac{z^{Nn+j}}{(Nn+j)!}$ selects one residue class of the exponential.","$D_{N,j}(z)$ & The same series with $(-1)^n$. One full derivative cycle contributes a minus sign.","$D_N(z)$ & $D_{N,0}(z)$, the denominator whose zeros control the signed reciprocal family.","$\\Delta_N(w)$ & $\\displaystyle\\sum_{n\\ge0}\\frac{(-1)^nw^n}{(Nn)!}$. Thus $D_N(z)=\\Delta_N(z^N)$.","$P_{N,n}(r)$ & Block-EGF coefficients of $D_N^{-r}$. They are polynomials in $r$.","$\\kappa_{N,m}$ & $\\displaystyle-\\log\\Delta_N(w)=\\sum_{m\\ge1}\\kappa_{N,m}w^m/m$.","$T_N(z)$ & $-D_N'(z)/D_N(z)$, a function, not the coefficient $T_N(n)$.","$\\Phi_{N,j}(w)$ & $\\displaystyle j!\\sum_{n\\ge0}\\frac{w^n}{(Nn+j)!}$ removes the forced zero of $E_{N,j}$.","$C_{N,j}(z)$ & $1/\\Phi_{N,j}(z^N)$. This is generally not $1/D_N(z)$.","$\\Lambda_N$ & Zeros of $\\Delta_N$, counted with multiplicity, one per rotational orbit of zeros of $D_N$.","bottomrule\n\\end{tabularx}\n\\head{A normalization test}\nThe coefficient $5/24$ of $w^2$ in $\\Delta_2(w)^{-1}$ is the same term as $5z^4/4!$. It is not $5w^2/2!$. Keeping the two factorials separate prevents errors in Bell formulas, convolution weights, and zero-moment reconstruction.\n\\takeaway{Use $z$ for differentiation and rotational geometry. Use $w$ for reciprocal recurrences, canonical products, and orbit-zero sums.}","\\vpage{3}{A root filter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a sym","\\vpage{3}{A root filter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nF","\\vpage{3}{A root filter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomi","\\vpage{3}{A root filter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section s","\\vpage{3}{A root filter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes","\\vpage{3}{A root filter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient","\\vpage{3}{A root filter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This","ilter is an exact vote}{CYCLIC PROJECTION}{Chapter IV: Theorem 3.1, Proposition 3.2.}\nFor $N=4$, write $\\omega=i$. To extract residue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourie…","idue $j$, evaluate the function at four rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex ave…","r rotated arguments and attach the phase $i^{-j\\ell}$:\n\\[\n(\\mathcal B_{4,j}F)(z)=\\frac14\\sum_{\\ell=0}^{3}i^{-j\\ell}F(i^\\ell z).\n\\]\nA monomial $a_mz^m$ is multiplied by the finite average of $i^{(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient sele…","(m-j)\\ell}$. Matching phases all vote $1$. Nonmatching phases cancel exactly.\n\\vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient selection for this polynomial. The drawing shows the four complex phases, not a numerical convergence argument.}","vfig{root_filter}\n\\head{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient selection for this polynomial. The drawing shows the four complex phases, not a numerical convergence argument.}","d{An example that can be checked coefficient by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient selection for this polynomial. The drawing shows the four complex phases, not a numerical convergence argument.}","by coefficient}\nTake $F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient selection for this polynomial. The drawing shows the four complex phases, not a numerical convergence argument.}","F(z)=\\sum_{m=0}^{12}(m+1)z^m$. Then\n\\[\n\\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.\n\\]\nApplying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient selection for this polynomial. The drawing shows the four complex phases, not a numerical convergence argument.}","returns the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient selection for this polynomial. The drawing shows the four complex phases, not a numerical convergence argument.}","the original $F$.\n\\head{Selection is also a symmetry condition}\nFor the selected polynomial $F_2$, rotation gives $F_2(iz)=-F_2(z)$. More generally, a residue-$j$ section satisfies $F_j(\\omega z)=\\omega^jF_j(z)$. The phase label therefore describes both the coefficient addresses and the response to rotation.\n\\takeaway{This is not an approximate Fourier fit. The finite root sum is exactly $0$ or $1$ on every coefficient.}\n\\platecaption{The checker compares the actual complex average with independent coefficient selection for this polynomial. The drawing shows the four complex phases, not a numerical convergence argument.}","\\vpage{4}{Products mix the residue lanes}{CYCLIC CONVOLUTION}{Chapter IV: Theorem 3.3.}\nSelection separates a function additively. It does not make multiplication diagonal. A residue-$i$ term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cycl","\\vpage{4}{Products mix the residue lanes}{CYCLIC CONVOLUTION}{Chapter IV: Theorem 3.3.}\nSelection separates a function additively. It does not make multiplication diagonal. A residue-$i$ term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra beh","\\vpage{4}{Products mix the residue lanes}{CYCLIC CONVOLUTION}{Chapter IV: Theorem 3.3.}\nSelection separates a function additively. It does not make multiplication diagonal. A residue-$i$ term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra behind the later addition formulas and","\\vpage{4}{Products mix the residue lanes}{CYCLIC CONVOLUTION}{Chapter IV: Theorem 3.3.}\nSelection separates a function additively. It does not make multiplication diagonal. A residue-$i$ term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra behind the later addition formulas and block-coef","\\vpage{4}{Products mix the residue lanes}{CYCLIC CONVOLUTION}{Chapter IV: Theorem 3.3.}\nSelection separates a function additively. It does not make multiplication diagonal. A residue-$i$ term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra behind the later addition formulas and block-coefficient recurrences.\n\\takeaway{A packet labe","C CONVOLUTION}{Chapter IV: Theorem 3.3.}\nSelection separates a function additively. It does not make multiplication diagonal. A residue-$i$ term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra behind the later addition formulas and block-coefficient recurrences.\n\\takeaway{A packet label tells you where an interaction lands, not that interactions with other packets vanish.}\n\\platecaption{All nine table entries and all thre…","ltiplication diagonal. A residue-$i$ term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra behind the later addition formulas and block-coefficient recurrences.\n\\takeaway{A packet label tells you where an interaction lands, not that interactions with other packets vanish.}\n\\platecaption{All nine table entries and all three output sections are checked as exact polynomial identities. This is the scalar cyclic specialization …","term multiplied by a residue-$j$ term lands in residue $i+j\\pmod N$.\nFor $N=3$, use\n\\[\nF=1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra behind the later addition formulas and block-coefficient recurrences.\n\\takeaway{A packet label tells you where an interaction lands, not that interactions with other packets vanish.}\n\\platecaption{All nine table entries and all three output sections are checked as exact polynomial identities. This is the scalar cyclic specialization used in this chapter.}","1+2z+3z^2+z^3,\\qquad G=2+z+z^2.\n\\]\nThe table displays every product of their three selected sections.\n\\vfig{cyclic_product}\nThe shaded cells contribute to residue zero:\n\\[\n2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3.\n\\]\nKeeping only the product of the residue-zero inputs would give $2+2z^3$ and lose five units of the $z^3$ coefficient. The mixed cells are part of the exact result.\n\\head{The general rule}\n\\[\n\\mathcal B_{N,k}(FG)=\n\\sum_{i+j\\equiv k\\ (N)}(\\mathcal B_{N,i}F)(\\mathcal B_{N,j}G).\n\\]\nEvery pair belongs to exactly one output residue. No term is discarded and no coefficient is estimated. This finite cyclic convolution is the algebra behind the later addition formulas and block-coefficient recurrences.\n\\takeaway{A packet label tells you where an interaction lands, not that interactions with other packets vanish.}\n\\platecaption{All nine table entries and all three output sections are checked as exact polynomial identities. This is the scalar cyclic specialization used in this chapter.}","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a un","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear co","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exp","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spe","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takea","\\vpage{5}{One signed edge determines the cycle}{DIFFERENTIAL SYSTEM}{Chapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6.}\nLet $J_3e_0=e_1$, $J_3e_1=e_2$, and $J_3e_2=-e_0$. A complete turn changes sign, so $J_3^3=-I$. The signed sections are the coordinates of $e^{zJ_3}e_0$.\n\\vfig{signed_shift}\nHere $D_j$ inside the matrix means $D_{3,j}(z)$. For example,\n\\[\nD_{3,0}=1-\\frac{z^3}{3!}+\\frac{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takeaway{The functions, their initial data, the…","{z^6}{6!}-\\cdots,\n\\qquad D_{3,1}=z-\\frac{z^4}{4!}+\\cdots.\n\\]\nRows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takeaway{The functions, their initial data, the addition law, and the exponent polygon come from one signed shift.}\n\\platecaption{The exact tests check every matrix-entry Taylor coefficient through degree 14 for $N=1,\\ldots,5$. The displayed determinant-one law is for $N\\ge2$. $N=1$ is a separate boundary case.}","ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takeaway{The functions, their initial data, the addition law, and the exponent polygon come from one signed shift.}\n\\platecaption{The exact tests check every matrix-entry Taylor coefficient through degree 14 for $N=1,\\ldots,5$. The displayed determinant-one law is for $N\\ge2$. $N=1$ is a separate boundary case.}","igns record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices.\n\\head{Two checks connect the matrix and the scalar equation}\nThe initial derivative matrix is the identity. Each initial jet $(y(0),y'(0),y''(0))$ therefore determines a unique linear combination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takeaway{The functions, their initial data, the addition law, and the exponent polygon come from one signed shift.}\n\\platecaption{The exact tests check every matrix-entry Taylor coefficient through degree 14 for $N=1,\\ldots,5$. The displayed determinant-one law is for $N\\ge2$. $N=1$ is a separate boundary case.}","ombination. Also,\n\\[\n\\frac13\\Tr(e^{zJ_3})=D_3(z),\\qquad \\det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takeaway{The functions, their initial data, the addition law, and the exponent polygon come from one signed shift.}\n\\platecaption{The exact tests check every matrix-entry Taylor coefficient through degree 14 for $N=1,\\ldots,5$. The displayed determinant-one law is for $N\\ge2$. $N=1$ is a separate boundary case.}","det(XI-J_3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takeaway{The functions, their initial data, the addition law, and the exponent polygon come from one signed shift.}\n\\platecaption{The exact tests check every matrix-entry Taylor coefficient through degree 14 for $N=1,\\ldots,5$. The displayed determinant-one law is for $N\\ge2$. $N=1$ is a separate boundary case.}","3)=X^3+1.\n\\]\nThe roots of $-1$ appearing later in the exponential sum are already the spectrum of this concrete matrix.\n\\takeaway{The functions, their initial data, the addition law, and the exponent polygon come from one signed shift.}\n\\platecaption{The exact tests check every matrix-entry Taylor coefficient through degree 14 for $N=1,\\ldots,5$. The displayed determinant-one law is for $N\\ge2$. $N=1$ is a separate boundary case.}","\\vpage{6}{Addition is quotient multiplication}{FINITE ALGEBRA}{Chapter IV: Proposition 5.5, Theorem 6.2.}\nWork in $\\CC[\\xi]/(\\xi^3+1)$. Every element has coordinates in $1,\\xi,\\xi^2$. The entire exponential has the finite-coordinate form\n\\[\ne^{z\\xi}=D_{3,0}(z)+D_{3,1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differenti","\\vpage{6}{Addition is quotient multiplication}{FINITE ALGEBRA}{Chapter IV: Proposition 5.5, Theorem 6.2.}\nWork in $\\CC[\\xi]/(\\xi^3+1)$. Every element has coordinates in $1,\\xi,\\xi^2$. The entire exponential has the finite-coordinate form\n\\[\ne^{z\\xi}=D_{3,0}(z)+D_{3,1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a","\\vpage{6}{Addition is quotient multiplication}{FINITE ALGEBRA}{Chapter IV: Proposition 5.5, Theorem 6.2.}\nWork in $\\CC[\\xi]/(\\xi^3+1)$. Every element has coordinates in $1,\\xi,\\xi^2$. The entire exponential has the finite-coordinate form\n\\[\ne^{z\\xi}=D_{3,0}(z)+D_{3,1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\","ddition is quotient multiplication}{FINITE ALGEBRA}{Chapter IV: Proposition 5.5, Theorem 6.2.}\nWork in $\\CC[\\xi]/(\\xi^3+1)$. Every element has coordinates in $1,\\xi,\\xi^2$. The entire exponential has the finite-coordinate form\n\\[\ne^{z\\xi}=D_{3,0}(z)+D_{3,1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twi…","ient multiplication}{FINITE ALGEBRA}{Chapter IV: Proposition 5.5, Theorem 6.2.}\nWork in $\\CC[\\xi]/(\\xi^3+1)$. Every element has coordinates in $1,\\xi,\\xi^2$. The entire exponential has the finite-coordinate form\n\\[\ne^{z\\xi}=D_{3,0}(z)+D_{3,1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition l…","LGEBRA}{Chapter IV: Proposition 5.5, Theorem 6.2.}\nWork in $\\CC[\\xi]/(\\xi^3+1)$. Every element has coordinates in $1,\\xi,\\xi^2$. The entire exponential has the finite-coordinate form\n\\[\ne^{z\\xi}=D_{3,0}(z)+D_{3,1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition law is the ordinary exponentia…",", Theorem 6.2.}\nWork in $\\CC[\\xi]/(\\xi^3+1)$. Every element has coordinates in $1,\\xi,\\xi^2$. The entire exponential has the finite-coordinate form\n\\[\ne^{z\\xi}=D_{3,0}(z)+D_{3,1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition law is the ordinary exponential group law expressed in a signed c…","1}(z)\\xi+D_{3,2}(z)\\xi^2.\n\\]\nThe multiplication table makes the carry sign unavoidable.\n\\vfig{signed_addition}\nMultiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition law is the ordinary exponential group law expressed in a signed cyclic algebra.}\n\\platecaption{Bivariate Taylor identities are checked through total degree ten in every section for $N=2,3,4$. Omitting the carry sign is one of the deliberatel…","Multiply the coordinate forms of $e^{z\\xi}$ and $e^{w\\xi}$. Their product is $e^{(z+w)\\xi}$. The coefficient of $1$ receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition law is the ordinary exponential group law expressed in a signed cyclic algebra.}\n\\platecaption{Bivariate Taylor identities are checked through total degree ten in every section for $N=2,3,4$. Omitting the carry sign is one of the deliberately rejected controls.}","nd two contributions with a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition law is the ordinary exponential group law expressed in a signed cyclic algebra.}\n\\platecaption{Bivariate Taylor identities are checked through total degree ten in every section for $N=2,3,4$. Omitting the carry sign is one of the deliberately rejected controls.}","a wrap. Reading that coefficient gives the displayed addition law.\n\\head{Compare the unsigned model}\nIn $\\CC[\\xi]/(\\xi^3-1)$, the same table has no minus signs. Its exponential coordinates are the unsigned sections $E_{3,j}$. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition law is the ordinary exponential group law expressed in a signed cyclic algebra.}\n\\platecaption{Bivariate Taylor identities are checked through total degree ten in every section for $N=2,3,4$. Omitting the carry sign is one of the deliberately rejected controls.}","s are grouped.\nFor general $N$, the carry factor is\n\\[\n(-1)^{\\lfloor(i+j)/N\\rfloor},\\qquad 0\\le i,j<N.\n\\]\nOnly zero or one wrap is possible in a product of two coordinate basis elements.\n\\takeaway{The twisted addition law is the ordinary exponential group law expressed in a signed cyclic algebra.}\n\\platecaption{Bivariate Taylor identities are checked through total degree ten in every section for $N=2,3,4$. Omitting the carry sign is one of the deliberately rejected controls.}","\\vpage{7}{Why a hypergeometric name appears}{FACTORIAL NORMALIZATION}{Chapter IV: Theorem 4.5.}\nThe hypergeometric form is another way to organize the same coefficients. It does not introduce an extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takea","\\vpage{7}{Why a hypergeometric name appears}{FACTORIAL NORMALIZATION}{Chapter IV: Theorem 4.5.}\nThe hypergeometric form is another way to organize the same coefficients. It does not introduce an extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the pre","\\vpage{7}{Why a hypergeometric name appears}{FACTORIAL NORMALIZATION}{Chapter IV: Theorem 4.5.}\nThe hypergeometric form is another way to organize the same coefficients. It does not introduce an extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument sca","page{7}{Why a hypergeometric name appears}{FACTORIAL NORMALIZATION}{Chapter IV: Theorem 4.5.}\nThe hypergeometric form is another way to organize the same coefficients. It does not introduce an extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from …","N}{Chapter IV: Theorem 4.5.}\nThe hypergeometric form is another way to organize the same coefficients. It does not introduce an extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are…","organize the same coefficients. It does not introduce an extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These…","oduce an extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general …","extra function into the construction. Split a factorial into its residue classes and divide each factor by $N$:\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic …",":\n\\[\n(Nn+j)!=j!N^{Nn}n!\\prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","prod_{a\\in A_{N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","N,j}}(a)_n.\n\\]\nThe parameter equal to $1$ is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","is omitted from $A_{N,j}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","}$ because the hypergeometric convention already supplies $(1)_n=n!$.\n\\vfig{hypergeometric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","metric}\n\\head{Check the displayed case}\nFor $N=3,j=1,n=2$, the six factors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","ors after the initial $1!$ are $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","e $2,3,4,5,6,7$. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies $2!$. The others supply $(2/3)_2$ and $(4/3)_2$. The right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","right-hand side is exactly $5040$.\nConsequently,\n\\[\nE_{3,1}(z)=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}",")=z\\left(1+\\frac{z^3}{4!}+\\frac{z^6}{7!}+\\cdots\\right).\n\\]\nThe prefactor $z$ retains the forced zero at the origin. The hypergeometric argument $z^3/27$ retains the scaling $N^N$.\n\\takeaway{Parameters, the omitted unit parameter, the prefactor, and the argument scaling all come from factorial bookkeeping.}\n\\platecaption{Five choices of $(N,j)$ are checked against the exact factorial identity for $n=0,\\ldots,7$. These finite checks support the example. The general analytic identification belongs to the inherited manuscript.}","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nTh","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corr","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linea","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes","\\vpage{8}{Four zeros become one orbit zero}{THE REDUCED VARIABLE}{Chapter IV: Definition 5.7, Example 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetiti…","xample 11.12, Corollary 11.13.}\nFor $N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multip…","$N=4$ the factorization is exact:\n\\[\nD_4(z)=\\cos(z/\\sqrt2)\\cosh(z/\\sqrt2),\\qquad D_4(z)=\\Delta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multiplicity.}\n\\platecaption{The quartic f…","lta_4(z^4).\n\\]\nSet $A_4=\\sqrt2\\pi$. All the zeros are $A_4(m+1/2)i^k$, with $m\\ge0$ and $0\\le k<4$. Each zero is simple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multiplicity.}\n\\platecaption{The quartic factorization, its differential equation, and the displayed orbit powers are checked indepen…","ple.\n\\vfig{orbit_reduction}\nThe drawing uses $u=z/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multiplicity.}\n\\platecaption{The quartic factorization, its differential equation, and the displayed orbit powers are checked independently. Here the complete zero set is known exactly. A finite numerical root search is not being used to establish i…","/A_4$ and $v=w/A_4^4$, so its marked coordinates are rational. The first four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multiplicity.}\n\\platecaption{The quartic factorization, its differential equation, and the displayed orbit powers are checked independently. Here the complete zero set is known exactly. A finite numerical root search is not being used to establish it.}","st four-point orbit has fourth power $1/16$. The second has fourth power $81/16$.\n\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multiplicity.}\n\\platecaption{The quartic factorization, its differential equation, and the displayed orbit powers are checked independently. Here the complete zero set is known exactly. A finite numerical root search is not being used to establish it.}","\\head{Multiplicity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multiplicity.}\n\\platecaption{The quartic factorization, its differential equation, and the displayed orbit powers are checked independently. Here the complete zero set is known exactly. A finite numerical root search is not being used to establish it.}","icity is not multiplied in the reduced function}\nAt a nonzero point, $z\\mapsto z^4$ has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of $D_4$ therefore correspond to one simple zero of $\\Delta_4$, not one zero of multiplicity four.\nThe reduced zeros are\n\\[\n\\lambda_m=A_4^4(m+1/2)^4=4\\pi^4(m+1/2)^4.\n\\]\nThis is why an orbit factor in the $z$ plane is $1-z^4/\\lambda_m$. It is the product of the four corresponding linear factors, with their phases included.\n\\takeaway{Orbit reduction removes repetition. It changes growth and counting exponents, while preserving each nonzero zero's local multiplicity.}\n\\platecaption{The quartic factorization, its differential equation, and the displayed orbit powers are checked independently. Here the complete zero set is known exactly. A finite numerical root search is not being used to establish it.}","\\vpage{9}{A reciprocal is a triangular solve}{COEFFICIENT RECONSTRUCTION}{Chapter IV: Theorem 7.3, Theorem 7.6.}\nStart in the reduced variable with\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}-\\frac{w^3}{720}+\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncati","\\vpage{9}{A reciprocal is a triangular solve}{COEFFICIENT RECONSTRUCTION}{Chapter IV: Theorem 7.3, Theorem 7.6.}\nStart in the reduced variable with\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}-\\frac{w^3}{720}+\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating th","\\vpage{9}{A reciprocal is a triangular solve}{COEFFICIENT RECONSTRUCTION}{Chapter IV: Theorem 7.3, Theorem 7.6.}\nStart in the reduced variable with\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}-\\frac{w^3}{720}+\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an","is a triangular solve}{COEFFICIENT RECONSTRUCTION}{Chapter IV: Theorem 7.3, Theorem 7.6.}\nStart in the reduced variable with\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}-\\frac{w^3}{720}+\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does …","h\n\\[\n\\Delta_2(w)=1-\\frac w2+\\frac{w^2}{24}-\\frac{w^3}{720}+\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does not by itself specify the radius of analytic convergence.}\n\\platecaption{The checks compare an independent binomial-series …","_2(w)=1-\\frac w2+\\frac{w^2}{24}-\\frac{w^3}{720}+\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does not by itself specify the radius of analytic convergence.}\n\\platecaption{The checks compare an independent binomial-series expansion, …","+\\frac{w^2}{24}-\\frac{w^3}{720}+\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does not by itself specify the radius of analytic convergence.}\n\\platecaption{The checks compare an independent binomial-series expansion, the parameter re…","\\cdots.\n\\]\nWrite $1/\\Delta_2(w)=\\sum b_nw^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does not by itself specify the radius of analytic convergence.}\n\\platecaption{The checks compare an independent binomial-series expansion, the parameter recurrence, and the logarithmic Ne…","^n$. Matching the product with $1$ gives a unit lower-triangular system. At each row there is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does not by itself specify the radius of analytic convergence.}\n\\platecaption{The checks compare an independent binomial-series expansion, the parameter recurrence, and the logarithmic Newton recurrence through block index six. …","ere is exactly one new unknown.\n\\vfig{triangular_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does not by itself specify the radius of analytic convergence.}\n\\platecaption{The checks compare an independent binomial-series expansion, the parameter recurrence, and the logarithmic Newton recurrence through block index six. The table uses ordinary coefficients, with factorial conversion shown separately.}","lar_inverse}\nThe first three equations are\n\\[\nb_1-\\tfrac12=0,\\qquad b_2-\\tfrac12b_1+\\tfrac1{24}=0,\n\\qquad b_3-\\tfrac12b_2+\\tfrac1{24}b_1-\\tfrac1{720}=0.\n\\]\nThey yield $b_1=1/2$, $b_2=5/24$, and $b_3=61/720$. Multiplication by $(2n)!$ produces the secant coefficients $1,5,61$.\n\\head{Why this is an exact finite calculation}\nTo compute the coefficient of $w^n$, only denominator coefficients through $w^n$ are needed. Later terms of the entire series cannot affect it. A finite series inverse is therefore exact at its declared truncation order, even though evaluating that truncated series is only an approximation to the full analytic function away from the origin.\n\\takeaway{A unit constant term makes coefficient recovery triangular. It does not by itself specify the radius of analytic convergence.}\n\\platecaption{The checks compare an independent binomial-series expansion, the parameter recurrence, and the logarithmic Newton recurrence through block index six. The table uses ordinary coefficients, with factorial conversion shown separately.}","\\vpage{10}{The reciprocal order is a polynomial}{BELL PARTITIONS}{Chapter IV: Theorem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanis","\\vpage{10}{The reciprocal order is a polynomial}{BELL PARTITIONS}{Chapter IV: Theorem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear","\\vpage{10}{The reciprocal order is a polynomial}{BELL PARTITIONS}{Chapter IV: Theorem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contr","\\vpage{10}{The reciprocal order is a polynomial}{BELL PARTITIONS}{Chapter IV: Theorem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[","\\vpage{10}{The reciprocal order is a polynomial}{BELL PARTITIONS}{Chapter IV: Theorem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_","\\vpage{10}{The reciprocal order is a polynomial}{BELL PARTITIONS}{Chapter IV: Theorem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many lo","er is a polynomial}{BELL PARTITIONS}{Chapter IV: Theorem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. …","rem 7.2, Corollary 7.7.}\nLet $r$ vary instead of fixing the reciprocal power. Since\n\\[\n\\Delta_N(w)^{-r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal…","r}=\\exp\\!\\left(r\\sum_{m\\ge1}\\kappa_{N,m}\\frac{w^m}{m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verif…","m}\\right),\n\\]\nthe coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial …","coefficient of $w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block inde…","w^n$ is a polynomial of degree $n$ in $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five o…","n $r$. Bell polynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","ynomials collect the possible ways logarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","ogarithmic terms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","ms contribute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","bute total degree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","egree $n$.\n\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","\\vfig{bell_partitions}\nFor $N=2$, the first three logarithmic coefficients are\n\\[\n\\kappa_1=\\frac12,\\qquad\\kappa_2=\\frac16,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","6,\\qquad\\kappa_3=\\frac1{15}.\n\\]\nThe third Bell polynomial is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","is $B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","x_1^3+3x_1x_2+x_3$. Substituting $x_1=r\\kappa_1$, $x_2=r\\kappa_2$, $x_3=2r\\kappa_3$ and multiplying by $6!/3!$ gives\n\\[\nP_{2,3}(r)=15r^3+30r^2+16r.\n\\]\nAt $r=1$ this is $61$. At $r=2$ it is $272$. At $r=0$ it vanishes, as every positive-index reciprocal coefficient must.\n\\head{The leading term has a simple source}\nOnly $n$ copies of the linear logarithmic term can contribute $r^n w^n$. Therefore\n\\[\n[r^n]P_{N,n}(r)=\\frac{(Nn)!}{n!(N!)^n}.\n\\]\n\\takeaway{The polynomial parameter records how many logarithmic pieces are combined. It is not a numerical interpolation in the reciprocal order.}\n\\platecaption{The partition plate is a finite combinatorial expansion. The exact tests verify the leading coefficient and parameter polynomial through block index six for five orders.}","\\vpage{11}{Two recurrences reach the same answer}{NEWTON AND ADDITION}{Chapter IV: Theorem 7.3, Theorem 7.4, Theorem 7.6.}\nOne recurrence uses the denominator coefficients. Another uses its logarithm. With $b_n(r)=P_{N,n}(r)/(Nn)!$, the shorter logarithmic form is\n\\[\nnb_n(r)=r\\sum_{k=1}^{n}\\kappa_{N,k}b_{n-k}(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)","\\vpage{11}{Two recurrences reach the same answer}{NEWTON AND ADDITION}{Chapter IV: Theorem 7.3, Theorem 7.4, Theorem 7.6.}\nOne recurrence uses the denominator coefficients. Another uses its logarithm. With $b_n(r)=P_{N,n}(r)/(Nn)!$, the shorter logarithmic form is\n\\[\nnb_n(r)=r\\sum_{k=1}^{n}\\kappa_{N,k}b_{n-k}(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookk","rem 7.4, Theorem 7.6.}\nOne recurrence uses the denominator coefficients. Another uses its logarithm. With $b_n(r)=P_{N,n}(r)/(Nn)!$, the shorter logarithmic form is\n\\[\nnb_n(r)=r\\sum_{k=1}^{n}\\kappa_{N,k}b_{n-k}(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks…","ses the denominator coefficients. Another uses its logarithm. With $b_n(r)=P_{N,n}(r)/(Nn)!$, the shorter logarithmic form is\n\\[\nnb_n(r)=r\\sum_{k=1}^{n}\\kappa_{N,k}b_{n-k}(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbol…","=P_{N,n}(r)/(Nn)!$, the shorter logarithmic form is\n\\[\nnb_n(r)=r\\sum_{k=1}^{n}\\kappa_{N,k}b_{n-k}(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbolically. A deliberately incorrect ordinary-binomial substitution is rejecte…","thmic form is\n\\[\nnb_n(r)=r\\sum_{k=1}^{n}\\kappa_{N,k}b_{n-k}(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbolically. A deliberately incorrect ordinary-binomial substitution is rejected.}","n}\\kappa_{N,k}b_{n-k}(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbolically. A deliberately incorrect ordinary-binomial substitution is rejected.}","(r),\\qquad b_0=1.\n\\]\nThe negative superdiagonal in the Newton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbolically. A deliberately incorrect ordinary-binomial substitution is rejected.}","wton determinant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbolically. A deliberately incorrect ordinary-binomial substitution is rejected.}","ant is essential.\n\\vfig{newton_addition}\n\\head{The additive parameter law has block weights}\nUsing the same logarithm, $D_N^{-r}D_N^{-s}=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbolically. A deliberately incorrect ordinary-binomial substitution is rejected.}","=D_N^{-(r+s)}$. Thus\n\\[\nP_{N,n}(r+s)=\\sum_{q=0}^{n}\\binom{Nn}{Nq}P_{N,q}(r)P_{N,n-q}(s).\n\\]\nFor $N=2,n=3,r=s=1$, the four weights are $1,15,15,1$. The two middle terms are $15\\cdot1\\cdot5$ and $15\\cdot5\\cdot1$. Together with the endpoints, they give $272$.\nUsing $\\binom nq$ instead would be incorrect for these coefficients. That ordinary binomial law belongs to the separately normalized sequence $n!P_{N,n}/(Nn)!$.\n\\takeaway{Agreement of independent recurrences is a useful check precisely because their bookkeeping is different. Neither permits silently changing the factorial convention.}\n\\platecaption{The checker compares both recurrence routes and the determinant, then checks the parameter-addition identity symbolically. A deliberately incorrect ordinary-binomial substitution is rejected.}","\\vpage{12}{Compress the ODE into a tangent}{LOGARITHMIC DIFFERENTIATION}{Chapter IV: Proposition 8.2, Corollary 8.4, Theorem 8.5.}\nThe transfer tangent is $T_N=-D_N'/D_N$. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear","page{12}{Compress the ODE into a tangent}{LOGARITHMIC DIFFERENTIATION}{Chapter IV: Proposition 8.2, Corollary 8.4, Theorem 8.5.}\nThe transfer tangent is $T_N=-D_N'/D_N$. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{Th…","ollary 8.4, Theorem 8.5.}\nThe transfer tangent is $T_N=-D_N'/D_N$. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential …","}\nThe transfer tangent is $T_N=-D_N'/D_N$. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zero…","/D_N$. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify t…","esses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coef…","the zero-free set of the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. T…","the denominator.\n\\vfig{riccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-one tangent …","ccati}\nThe coefficients retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-one tangent is the separate constant …","retain the reduced logarithmic information:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-one tangent is the separate constant function $T_1=1$.}","nformation:\n\\[\nT_N(z)=N\\sum_{m\\ge1}\\kappa_{N,m}z^{Nm-1},\n\\qquad T_N(n)=N(Nn+N-1)!\\kappa_{N,n+1}.\n\\]\nFor order four, $\\kappa_{4,1}=1/24$ and $\\kappa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-one tangent is the separate constant function $T_1=1$.}","ppa_{4,2}=17/10080$. These give $T_4(0)=1$ and $T_4(1)=34$, explaining the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-one tangent is the separate constant function $T_1=1$.}","the last line of the plate.\n\\head{A nonlinear equation with a precise origin}\nIf $D=e^g$ locally, then $g'=-T$. Differentiating $e^g$ three times gives $D'''/D=-T''+3TT'-T^3$. Setting $D'''/D=-1$ yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed.\nThe reciprocal family also satisfies $(D_N^{-r})'=rT_ND_N^{-r}$ on a fixed-log domain. Behavior at zeros of $D_N$ depends on the reciprocal order and the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-one tangent is the separate constant function $T_1=1$.}","d the branch conventions stated on Plate~IV.13.\n\\takeaway{The nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros.}\n\\platecaption{The tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-one tangent is the separate constant function $T_1=1$.}","\\vpage{13}{A complex power needs a domain}{BRANCH BOOKKEEPING}{Chapter IV: Section 2, Definition 7.1, Corollary 8.4.}\nA formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing $0$ and normalize the logarithm to vanish there.\n\\[\nD_N(z)^{-r}=\\exp(-rL(z)),\\qquad e^{L(z)}=D_N(z),\\quad L(0)=0.\n\\]\n\\vfig{branch_domain}\nFor $D_2=\\cos z$, the open disk $|z|<\\pi/2$ is an admissible domain. A small loop around the simple zero $\\rho=\\pi/2$ is not contained in that disk. Analytic continuation around the zero changes the local logarithm by $2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot","\\vpage{13}{A complex power needs a domain}{BRANCH BOOKKEEPING}{Chapter IV: Section 2, Definition 7.1, Corollary 8.4.}\nA formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing $0$ and normalize the logarithm to vanish there.\n\\[\nD_N(z)^{-r}=\\exp(-rL(z)),\\qquad e^{L(z)}=D_N(z),\\quad L(0)=0.\n\\]\n\\vfig{branch_domain}\nFor $D_2=\\cos z$, the open disk $|z|<\\pi/2$ is an admissible domain. A small loop around the simple zero $\\rho=\\pi/2$ is not contained in that disk. Analytic continuation around the zero changes the local logarithm by $2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument.\nFor example, $D^{-r}D^{-s}=D^{-(r+s)}$ is immediate using one fixed log","A complex power needs a domain}{BRANCH BOOKKEEPING}{Chapter IV: Section 2, Definition 7.1, Corollary 8.4.}\nA formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing $0$ and normalize the logarithm to vanish there.\n\\[\nD_N(z)^{-r}=\\exp(-rL(z)),\\qquad e^{L(z)}=D_N(z),\\quad L(0)=0.\n\\]\n\\vfig{branch_domain}\nFor $D_2=\\cos z$, the open disk $|z|<\\pi/2$ is an admissible domain. A small loop around the simple zero $\\rho=\\pi/2$ is not contained in that disk. Analytic continuation around the zero changes the local logarithm by $2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument.\nFor example, $D^{-r}D^{-s}=D^{-(r+s)}$ is immediate using one fixed logarithm. Combining values buil…","in}{BRANCH BOOKKEEPING}{Chapter IV: Section 2, Definition 7.1, Corollary 8.4.}\nA formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing $0$ and normalize the logarithm to vanish there.\n\\[\nD_N(z)^{-r}=\\exp(-rL(z)),\\qquad e^{L(z)}=D_N(z),\\quad L(0)=0.\n\\]\n\\vfig{branch_domain}\nFor $D_2=\\cos z$, the open disk $|z|<\\pi/2$ is an admissible domain. A small loop around the simple zero $\\rho=\\pi/2$ is not contained in that disk. Analytic continuation around the zero changes the local logarithm by $2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument.\nFor example, $D^{-r}D^{-s}=D^{-(r+s)}$ is immediate using one fixed logarithm. Combining values built from unrelated branches ca…","8.4.}\nA formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing $0$ and normalize the logarithm to vanish there.\n\\[\nD_N(z)^{-r}=\\exp(-rL(z)),\\qquad e^{L(z)}=D_N(z),\\quad L(0)=0.\n\\]\n\\vfig{branch_domain}\nFor $D_2=\\cos z$, the open disk $|z|<\\pi/2$ is an admissible domain. A small loop around the simple zero $\\rho=\\pi/2$ is not contained in that disk. Analytic continuation around the zero changes the local logarithm by $2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument.\nFor example, $D^{-r}D^{-s}=D^{-(r+s)}$ is immediate using one fixed logarithm. Combining values built from unrelated branches can introduce additional phase factors.\n\\takeaway{The branch choice is math…","mply connected zero-free region containing $0$ and normalize the logarithm to vanish there.\n\\[\nD_N(z)^{-r}=\\exp(-rL(z)),\\qquad e^{L(z)}=D_N(z),\\quad L(0)=0.\n\\]\n\\vfig{branch_domain}\nFor $D_2=\\cos z$, the open disk $|z|<\\pi/2$ is an admissible domain. A small loop around the simple zero $\\rho=\\pi/2$ is not contained in that disk. Analytic continuation around the zero changes the local logarithm by $2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument.\nFor example, $D^{-r}D^{-s}=D^{-(r+s)}$ is immediate using one fixed logarithm. Combining values built from unrelated branches can introduce additional phase factors.\n\\takeaway{The branch choice is mathematical data. It is not a plotting convention that can be removed by drawing a cleaner curve.}\n\\platecaption{The…","gion containing $0$ and normalize the logarithm to vanish there.\n\\[\nD_N(z)^{-r}=\\exp(-rL(z)),\\qquad e^{L(z)}=D_N(z),\\quad L(0)=0.\n\\]\n\\vfig{branch_domain}\nFor $D_2=\\cos z$, the open disk $|z|<\\pi/2$ is an admissible domain. A small loop around the simple zero $\\rho=\\pi/2$ is not contained in that disk. Analytic continuation around the zero changes the local logarithm by $2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument.\nFor example, $D^{-r}D^{-s}=D^{-(r+s)}$ is immediate using one fixed logarithm. Combining values built from unrelated branches can introduce additional phase factors.\n\\takeaway{The branch choice is mathematical data. It is not a plotting convention that can be removed by drawing a cleaner curve.}\n\\platecaption{The domain picture is schemati…","2\\pi i$, and the power by $e^{-2\\pi ir}$.\n\\head{Three different kinds of statement}\nA \\emph{formal} identity compares coefficients at the origin. A \\emph{holomorphic} identity for complex powers uses a common specified logarithm. A positive \\emph{integer} reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument.\nFor example, $D^{-r}D^{-s}=D^{-(r+s)}$ is immediate using one fixed logarithm. Combining values built from unrelated branches can introduce additional phase factors.\n\\takeaway{The branch choice is mathematical data. It is not a plotting convention that can be removed by drawing a cleaner curve.}\n\\platecaption{The domain picture is schematic. The phase change comes from the local simple-zero model and winding number, not from a sampled loop. Later nearest-pole asymptotics retain positive integral reciprocal order.}","\\vpage{14}{Three similar reciprocals differ}{NORMALIZED SECTIONS}{Chapter IV: Definition 9.1, Theorem 9.2, Example 9.11.}\nThe signed denominator and the normalized unsigned sections produce different reciprocal families. In order two,\n\\[\nD_2^{-1}=\\sec z,\\qquad C_{2,0}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator se","pter IV: Definition 9.1, Theorem 9.2, Example 9.11.}\nThe signed denominator and the normalized unsigned sections produce different reciprocal families. In order two,\n\\[\nD_2^{-1}=\\sec z,\\qquad C_{2,0}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are ex…","1, Theorem 9.2, Example 9.11.}\nThe signed denominator and the normalized unsigned sections produce different reciprocal families. In order two,\n\\[\nD_2^{-1}=\\sec z,\\qquad C_{2,0}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed…","ample 9.11.}\nThe signed denominator and the normalized unsigned sections produce different reciprocal families. In order two,\n\\[\nD_2^{-1}=\\sec z,\\qquad C_{2,0}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent …","roduce different reciprocal families. In order two,\n\\[\nD_2^{-1}=\\sec z,\\qquad C_{2,0}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary…","order two,\n\\[\nD_2^{-1}=\\sec z,\\qquad C_{2,0}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by","\\sec z,\\qquad C_{2,0}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factoria…","}=\\sech z,\\qquad C_{2,1}=z/\\sinh z.\n\\]\nTheir first block-EGF coefficients make the difference visible.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","ble.\n\\vfig{different_reciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","eciprocals}\n\\head{Remove the forced zero before taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","taking a reciprocal}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","l}\nThe section $E_{N,j}$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","$ begins with $z^j/j!$. Define\n\\[\nE_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","E_{N,j}(z)=\\frac{z^j}{j!}\\Phi_{N,j}(z^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","^N),\\qquad\\Phi_{N,j}(0)=1.\n\\]\nThen $C_{N,j}(z)=1/\\Phi_{N,j}(z^N)$ has value $1$ at the origin. For $j>0$, the notation $(z^j/j!)/E_{N,j}(z)$ means this removable continuation. It does not mean evaluating an undefined $0/0$ expression.\nFor $N=2,j=1$, the first nontrivial coefficient is $-1/3$ in front of $z^2/2!$, equivalently $-z^2/6$. Dropping the numerator $z$ would instead produce the pole of $1/\\sinh z$.\n\\takeaway{A minus sign in the denominator series and removal of an initial zero are different operations. Their reciprocal coefficients should not be interchanged.}\n\\platecaption{All entries in the table are exact rationals computed from independent triangular inverses. The table explicitly uses block-EGF columns. Ordinary coefficients are obtained by dividing by the displayed factorials.}","\\vpage{15}{An infinite symbol acts finitely}{APPELL POLYNOMIALS}{Chapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9.}\nFor $C_{2,0}(z)=\\sech z$, define $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciproca","\\vpage{15}{An infinite symbol acts finitely}{APPELL POLYNOMIALS}{Chapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9.}\nFor $C_{2,0}(z)=\\sech z$, define $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol.","\\vpage{15}{An infinite symbol acts finitely}{APPELL POLYNOMIALS}{Chapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9.}\nFor $C_{2,0}(z)=\\sech z$, define $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities","\\vpage{15}{An infinite symbol acts finitely}{APPELL POLYNOMIALS}{Chapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9.}\nFor $C_{2,0}(z)=\\sech z$, define $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBo","\\vpage{15}{An infinite symbol acts finitely}{APPELL POLYNOMIALS}{Chapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9.}\nFor $C_{2,0}(z)=\\sech z$, define $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBoth follow from the generating function. No operator-series convergence as","\\vpage{15}{An infinite symbol acts finitely}{APPELL POLYNOMIALS}{Chapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9.}\nFor $C_{2,0}(z)=\\sech z$, define $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBoth follow from the generating function. No operator-series convergence assumption on an infinite-dimensional funct","ymbol acts finitely}{APPELL POLYNOMIALS}{Chapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9.}\nFor $C_{2,0}(z)=\\sech z$, define $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBoth follow from the generating function. No operator-series convergence assumption on an infinite-dimensional function space is needed for this po…","e $E_m(x)$ by $\\sech z\\,e^{xz}=\\sum E_m(x)z^m/m!$. The formal differential-operator formula is\n\\[\nE_m(x)=\\sech(\\partial_x)x^m.\n\\]\nAlthough the symbol is an infinite series, derivatives above degree $m$ vanish.\n\\vfig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBoth follow from the generating function. No operator-series convergence assumption on an infinite-dimensional function space is needed for this polynomial calculation.\n\\takeaway{Formal substitution into the derivative is safe here because every polynomial sees only finitely…","fig{appell_operator}\nFor $m=4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBoth follow from the generating function. No operator-series convergence assumption on an infinite-dimensional function space is needed for this polynomial calculation.\n\\takeaway{Formal substitution into the derivative is safe here because every polynomial sees only finitely many derivatives. That is a domain statement.}\n\\platecaption{The polynomial, its inverse image, and the Appell identities are checked symbolically through degree eight in four normalized-section families. The di…","4$, only three terms survive:\n\\[\n\\left(1-\\frac{\\partial_x^2}{2!}+5\\frac{\\partial_x^4}{4!}\\right)x^4\n=x^4-6x^2+5.\n\\]\nThe four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBoth follow from the generating function. No operator-series convergence assumption on an infinite-dimensional function space is needed for this polynomial calculation.\n\\takeaway{Formal substitution into the derivative is safe here because every polynomial sees only finitely many derivatives. That is a domain statement.}\n\\platecaption{The polynomial, its inverse image, and the Appell identities are checked symbolically through degree eight in four normalized-section families. The displayed curves are samples o…","The four real roots of this polynomial are $\\pm1,\\pm\\sqrt5$. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output.\n\\head{An inverse operation, not a truncation error}\nApplying $\\cosh(\\partial_x)$ returns $x^4$ exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are\n\\[\nE_m'=mE_{m-1},\\qquad E_m(x+y)=\\sum_{q=0}^m\\binom mqE_q(x)y^{m-q}.\n\\]\nBoth follow from the generating function. No operator-series convergence assumption on an infinite-dimensional function space is needed for this polynomial calculation.\n\\takeaway{Formal substitution into the derivative is safe here because every polynomial sees only finitely many derivatives. That is a domain statement.}\n\\platecaption{The polynomial, its inverse image, and the Appell identities are checked symbolically through degree eight in four normalized-section families. The displayed curves are samples of exact polynomials.}","\\vpage{16}{Scale and order enter differently}{THETA COUPLING}{Chapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4.}\nLet $H_N(z)=E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contras","\\vpage{16}{Scale and order enter differently}{THETA COUPLING}{Chapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4.}\nLet $H_N(z)=E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe","\\vpage{16}{Scale and order enter differently}{THETA COUPLING}{Chapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4.}\nLet $H_N(z)=E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial root","\\vpage{16}{Scale and order enter differently}{THETA COUPLING}{Chapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4.}\nLet $H_N(z)=E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does no","\\vpage{16}{Scale and order enter differently}{THETA COUPLING}{Chapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4.}\nLet $H_N(z)=E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling a","ge{16}{Scale and order enter differently}{THETA COUPLING}{Chapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4.}\nLet $H_N(z)=E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeab…","apter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4.}\nLet $H_N(z)=E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{…","E_{N,0}(z)$. The theta family uses the same parameter $u$ twice, but in two different roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton …","fferent roles:\n\\[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly …","[\nH_N(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an i…","(uz)^{-1}H_N(z)^{-u}\n=\\sum_{n\\ge0}\\Theta_{N,n}(u)\\frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an invali…","frac{z^{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an invalid control.}","{Nn}}{(Nn)!}.\n\\]\nArgument scaling contributes $u^{Nm}$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an invalid control.}","$ at logarithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an invalid control.}","garithmic block $m$. Reciprocal order contributes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an invalid control.}","ibutes $u$.\n\\vfig{theta}\nTheir combined logarithm has coefficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an invalid control.}","efficient\n\\[\n(-1)^m\\kappa_{N,m}\\frac{u^{Nm}+u}{m}.\n\\]\nAt the first block this gives $\\Theta_{N,1}(u)=-(u^N+u)$. Every positive-index theta polynomial vanishes at $u=0$. For even $N$, $H_N(-z)=H_N(z)$, so the defining product is also $1$ at $u=-1$.\n\\head{The odd-order boundary is visible}\nFor $N=3$, $\\Theta_{3,1}(-1)=2$, not zero. The factor $u+1$ is not a uniform law across orders. In contrast,\n\\[\n\\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).\n\\]\nThe two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots.\n\\takeaway{A common parameter does not make argument scaling and reciprocal order interchangeable. Their powers must be tracked separately.}\n\\platecaption{The exact checker compares theta convolution with its logarithmic Newton recurrence through block index four. The odd-order factor claim is explicitly rejected as an invalid control.}","\\vpage{17}{Growth comes from a polygon}{DOMINANT EXPONENTS}{Chapter IV: Definition 11.3, Theorem 11.4.}\nWrite the signed denominator as a finite exponential sum:\n\\[\nD_N(z)=\\frac1N\\sum_{\\eta^N=-1}e^{\\eta z}.\n\\]\nFor $z=re^{i\\theta}$, the size of one exponential is $e^{r\\Re(\\eta e^{i\\theta})}$. The maximal projection determines the directional exponential growth.\n\\vfig{indicator}\nThe indicator is\n\\[\nh_N(\\theta)=\\max_{0\\le k<N}\\cos\\left(\\theta+\\frac{(2k+1)\\pi}{N}\\right).\n\\]\nAway from a tie direction, one exponential dominates all the others by a uniform positive projection gap on compact angular sets. For sufficiently large radius, smaller terms cannot cancel it. Those sectors are eventually zero-free.\n\\head{Why zeros remain possible on a growing ray}\nAt a tie, two adjacent exponents have the same real projection. Their complex phases can be opposite. Their sum can therefore vanish even though each","\\vpage{17}{Growth comes from a polygon}{DOMINANT EXPONENTS}{Chapter IV: Definition 11.3, Theorem 11.4.}\nWrite the signed denominator as a finite exponential sum:\n\\[\nD_N(z)=\\frac1N\\sum_{\\eta^N=-1}e^{\\eta z}.\n\\]\nFor $z=re^{i\\theta}$, the size of one exponential is $e^{r\\Re(\\eta e^{i\\theta})}$. The maximal projection determines the directional exponential growth.\n\\vfig{indicator}\nThe indicator is\n\\[\nh_N(\\theta)=\\max_{0\\le k<N}\\cos\\left(\\theta+\\frac{(2k+1)\\pi}{N}\\right).\n\\]\nAway from a tie direction, one exponential dominates all the others by a uniform positive projection gap on compact angular sets. For sufficiently large radius, smaller terms cannot cancel it. Those sectors are eventually zero-free.\n\\head{Why zeros remain possible on a growing ray}\nAt a tie, two adjacent exponents have the same real projection. Their complex phases can be opposite. Their sum can therefore vanish even though each term is exponentially","\\vpage{17}{Growth comes from a polygon}{DOMINANT EXPONENTS}{Chapter IV: Definition 11.3, Theorem 11.4.}\nWrite the signed denominator as a finite exponential sum:\n\\[\nD_N(z)=\\frac1N\\sum_{\\eta^N=-1}e^{\\eta z}.\n\\]\nFor $z=re^{i\\theta}$, the size of one exponential is $e^{r\\Re(\\eta e^{i\\theta})}$. The maximal projection determines the directional exponential growth.\n\\vfig{indicator}\nThe indicator is\n\\[\nh_N(\\theta)=\\max_{0\\le k<N}\\cos\\left(\\theta+\\frac{(2k+1)\\pi}{N}\\right).\n\\]\nAway from a tie direction, one exponential dominates all the others by a uniform positive projection gap on compact angular sets. For sufficiently large radius, smaller terms cannot cancel it. Those sectors are eventually zero-free.\n\\head{Why zeros remain possible on a growing ray}\nAt a tie, two adjacent exponents have the same real projection. Their complex phases can be opposite. Their sum can therefore vanish even though each term is exponentially large. The minima in the order-four indicator plot occur pre","\\vpage{17}{Growth comes from a polygon}{DOMINANT EXPONENTS}{Chapter IV: Definition 11.3, Theorem 11.4.}\nWrite the signed denominator as a finite exponential sum:\n\\[\nD_N(z)=\\frac1N\\sum_{\\eta^N=-1}e^{\\eta z}.\n\\]\nFor $z=re^{i\\theta}$, the size of one exponential is $e^{r\\Re(\\eta e^{i\\theta})}$. The maximal projection determines the directional exponential growth.\n\\vfig{indicator}\nThe indicator is\n\\[\nh_N(\\theta)=\\max_{0\\le k<N}\\cos\\left(\\theta+\\frac{(2k+1)\\pi}{N}\\right).\n\\]\nAway from a tie direction, one exponential dominates all the others by a uniform positive projection gap on compact angular sets. For sufficiently large radius, smaller terms cannot cancel it. Those sectors are eventually zero-free.\n\\head{Why zeros remain possible on a growing ray}\nAt a tie, two adjacent exponents have the same real projection. Their complex phases can be opposite. Their sum can therefore vanish even though each term is exponentially large. The minima in the order-four indicator plot occur precisely at the tie directions $\\theta=k\\pi/2$.\nThe indicator uses a limsup in radius. It does not assert that $…","y large radius, smaller terms cannot cancel it. Those sectors are eventually zero-free.\n\\head{Why zeros remain possible on a growing ray}\nAt a tie, two adjacent exponents have the same real projection. Their complex phases can be opposite. Their sum can therefore vanish even though each term is exponentially large. The minima in the order-four indicator plot occur precisely at the tie directions $\\theta=k\\pi/2$.\nThe indicator uses a limsup in radius. It does not assert that $r^{-1}\\log|D_N(re^{i\\theta})|$ has an ordinary limit through the zeros on a tie ray.\n\\takeaway{The polygon predicts where large cancellation is possible. It does not, by itself, enumerate zeros or prove their simplicity.}\n\\platecaption{The order-four polygon and support-function formula are explicit. The curve is a sampled display of that formula. Eventual localization and uniqueness require the separate zero-analysis arguments.}","o-free.\n\\head{Why zeros remain possible on a growing ray}\nAt a tie, two adjacent exponents have the same real projection. Their complex phases can be opposite. Their sum can therefore vanish even though each term is exponentially large. The minima in the order-four indicator plot occur precisely at the tie directions $\\theta=k\\pi/2$.\nThe indicator uses a limsup in radius. It does not assert that $r^{-1}\\log|D_N(re^{i\\theta})|$ has an ordinary limit through the zeros on a tie ray.\n\\takeaway{The polygon predicts where large cancellation is possible. It does not, by itself, enumerate zeros or prove their simplicity.}\n\\platecaption{The order-four polygon and support-function formula are explicit. The curve is a sampled display of that formula. Eventual localization and uniqueness require the separate zero-analysis arguments.}","\\vpage{18}{Two large terms nearly cancel}{THE CUBIC MODEL}{Chapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9.}\nFor real $t$, order three has the exact formula\n\\[\nD_3(t)=\\frac{e^{-t}+2e^{t/2}\\cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs u","\\vpage{18}{Two large terms nearly cancel}{THE CUBIC MODEL}{Chapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9.}\nFor real $t$, order three has the exact formula\n\\[\nD_3(t)=\\frac{e^{-t}+2e^{t/2}\\cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs uniqueness}\nReal coefficients alone allow nonreal conjugate pairs. The manuscript first pr","\\vpage{18}{Two large terms nearly cancel}{THE CUBIC MODEL}{Chapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9.}\nFor real $t$, order three has the exact formula\n\\[\nD_3(t)=\\frac{e^{-t}+2e^{t/2}\\cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs uniqueness}\nReal coefficients alone allow nonreal conjugate pairs. The manuscript first proves that a near-model disk contains exactl","{18}{Two large terms nearly cancel}{THE CUBIC MODEL}{Chapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9.}\nFor real $t$, order three has the exact formula\n\\[\nD_3(t)=\\frac{e^{-t}+2e^{t/2}\\cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs uniqueness}\nReal coefficients alone allow nonreal conjugate pairs. The manuscript first proves that a near-model disk contains exactly one zero. Conjugation preserves that disk, so the unique zero must equal its conjugate. This additional uniqueness step, not the symmetry alone, pl…","CUBIC MODEL}{Chapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9.}\nFor real $t$, order three has the exact formula\n\\[\nD_3(t)=\\frac{e^{-t}+2e^{t/2}\\cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs uniqueness}\nReal coefficients alone allow nonreal conjugate pairs. The manuscript first proves that a near-model disk contains exactly one zero. Conjugation preserves that disk, so the unique zero must equal its conjugate. This additional uniqueness step, not the symmetry alone, places it exactly on the ray.\n\\takeaway{Th…","11.7, Corollary 11.9.}\nFor real $t$, order three has the exact formula\n\\[\nD_3(t)=\\frac{e^{-t}+2e^{t/2}\\cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs uniqueness}\nReal coefficients alone allow nonreal conjugate pairs. The manuscript first proves that a near-model disk contains exactly one zero. Conjugation preserves that disk, so the unique zero must equal its conjugate. This additional uniqueness step, not the symmetry alone, places it exactly on the ray.\n\\takeaway{The correction is caused by a real, nonzero rem…","eal $t$, order three has the exact formula\n\\[\nD_3(t)=\\frac{e^{-t}+2e^{t/2}\\cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs uniqueness}\nReal coefficients alone allow nonreal conjugate pairs. The manuscript first proves that a near-model disk contains exactly one zero. Conjugation preserves that disk, so the unique zero must equal its conjugate. This additional uniqueness step, not the symmetry alone, places it exactly on the ray.\n\\takeaway{The correction is caused by a real, nonzero remainder. A good cosine approx…","cos(\\sqrt3t/2)}{3}.\n\\]\nDivide by the nonzero factor $2e^{t/2}/3$. The resulting equation separates the dominant cosine from the small third exponential.\n\\vfig{two_term}\nIgnoring the last term gives model zeros\n\\[\nt_m=\\frac{2\\pi}{\\sqrt3}(m+1/2).\n\\]\nAt $t_m$, the full normalized expression equals $\\tfrac12e^{-3t_m/2}>0$, so $t_m$ is not an exact zero. Linearizing the cosine at that point predicts\n\\[\n\\delta_m^{(1)}=\\frac{(-1)^m}{\\sqrt3}e^{-3t_m/2}.\n\\]\nThe alternating slope explains why the true zero is to the right of the first model zero, then to the left of the second, and so on in the eventual regime.\n\\head{Symmetry needs uniqueness}\nReal coefficients alone allow nonreal conjugate pairs. The manuscript first proves that a near-model disk contains exactly one zero. Conjugation preserves that disk, so the unique zero must equal its conjugate. This additional uniqueness step, not the symmetry alone, places it exactly on the ray.\n\\takeaway{The correction is caused by a real, nonzero remainder. A good cosine approximation does not make its model zeros exact.}\n\\platecaption{The curves are …","\\vpage{19}{A root estimate becomes a certificate}{EXACT RATIONAL BRACKETS}{Chapter IV: Theorem 11.7, Theorem 11.8, Corollary 11.9.}\nThe root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for $D_3'$ excludes zero throughout the interval, proving uniqueness inside it.\n\\vfig{cubic_certificates}\n\\head{How the certificate works}\nEvaluate the Taylor polynomial\n\\[\nS_{90}(t)=\\sum_{n=0}^{90}\\frac{(-1)^nt^{3n}}{(3n)!}\n\\]\nat the two rational endpoints. From the next term onward, magnitudes decrease, because their successive ratio is\n\\[\n\\frac{t^3}{(3n+1)(3n+2)(3n+3)}<1\n\\]\nthere. The alternating-series remainder is bounded by the first omitted term. A separate outward interval evaluation of the differentiated polynomial, plus its tail bound, certifies the derivative sign.\nThe first bracket is\n\\[\n1.84981279919014347629<\\tau_0<1.84981279919014347630.\n\\]\nIts displacement differs from the leading asy","V: Theorem 11.7, Theorem 11.8, Corollary 11.9.}\nThe root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for $D_3'$ excludes zero throughout the interval, proving uniqueness inside it.\n\\vfig{cubic_certificates}\n\\head{How the certificate works}\nEvaluate the Taylor polynomial\n\\[\nS_{90}(t)=\\sum_{n=0}^{90}\\frac{(-1)^nt^{3n}}{(3n)!}\n\\]\nat the two rational endpoints. From the next term onward, magnitudes decrease, because their successive ratio is\n\\[\n\\frac{t^3}{(3n+1)(3n+2)(3n+3)}<1\n\\]\nthere. The alternating-series remainder is bounded by the first omitted term. A separate outward interval evaluation of the differentiated polynomial, plus its tail bound, certifies the derivative sign.\nThe first bracket is\n\\[\n1.84981279919014347629<\\tau_0<1.84981279919014347630.\n\\]\nIts displacement differs from the leading asymptotic prediction by about five percent. The later examples have much smaller relative discrepancies, as the plate records.\n\\takeaway{A small residual is not a ro…","te endpoint signs prove existence, and an interval bound for $D_3'$ excludes zero throughout the interval, proving uniqueness inside it.\n\\vfig{cubic_certificates}\n\\head{How the certificate works}\nEvaluate the Taylor polynomial\n\\[\nS_{90}(t)=\\sum_{n=0}^{90}\\frac{(-1)^nt^{3n}}{(3n)!}\n\\]\nat the two rational endpoints. From the next term onward, magnitudes decrease, because their successive ratio is\n\\[\n\\frac{t^3}{(3n+1)(3n+2)(3n+3)}<1\n\\]\nthere. The alternating-series remainder is bounded by the first omitted term. A separate outward interval evaluation of the differentiated polynomial, plus its tail bound, certifies the derivative sign.\nThe first bracket is\n\\[\n1.84981279919014347629<\\tau_0<1.84981279919014347630.\n\\]\nIts displacement differs from the leading asymptotic prediction by about five percent. The later examples have much smaller relative discrepancies, as the plate records.\n\\takeaway{A small residual is not a root certificate. Endpoint signs plus a derivative bound give a checkable local existence-and-uniqueness statement.}\n\\platecaption{Four certified intervals do not establish …","1)^nt^{3n}}{(3n)!}\n\\]\nat the two rational endpoints. From the next term onward, magnitudes decrease, because their successive ratio is\n\\[\n\\frac{t^3}{(3n+1)(3n+2)(3n+3)}<1\n\\]\nthere. The alternating-series remainder is bounded by the first omitted term. A separate outward interval evaluation of the differentiated polynomial, plus its tail bound, certifies the derivative sign.\nThe first bracket is\n\\[\n1.84981279919014347629<\\tau_0<1.84981279919014347630.\n\\]\nIts displacement differs from the leading asymptotic prediction by about five percent. The later examples have much smaller relative discrepancies, as the plate records.\n\\takeaway{A small residual is not a root certificate. Endpoint signs plus a derivative bound give a checkable local existence-and-uniqueness statement.}\n\\platecaption{Four certified intervals do not establish that no other real or complex zeros exist. The displacement columns are high-precision diagnostics, not a finite verification of a global big-$O$ theorem. All rational enclosures are included in the checks file.}","r relative discrepancies, as the plate records.\n\\takeaway{A small residual is not a root certificate. Endpoint signs plus a derivative bound give a checkable local existence-and-uniqueness statement.}\n\\platecaption{Four certified intervals do not establish that no other real or complex zeros exist. The displacement columns are high-precision diagnostics, not a finite verification of a global big-$O$ theorem. All rational enclosures are included in the checks file.}","\\vpage{20}{Counting changes under the power map}{EXACT FOURTH-ORDER STRINGS}{Chapter IV: Example 11.12, Theorem 11.14.}\nOrder four gives a case with no asymptotic displacement at all. In normalized coordinates the positive radial locations are $u=m+1/2$, while the reduced-zero locations are $v=(m+1/2)^4$.\n\\vfig{counting}\nThe two rows describe the same zero orbits. The upper row counts four zeros each time a new radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk.\n\\head{Separate radius conversion from orbit multiplicity}\nIn unscaled variables the general relation is\n\\[\nn_{D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+","\\vpage{20}{Counting changes under the power map}{EXACT FOURTH-ORDER STRINGS}{Chapter IV: Example 11.12, Theorem 11.14.}\nOrder four gives a case with no asymptotic displacement at all. In normalized coordinates the positive radial locations are $u=m+1/2$, while the reduced-zero locations are $v=(m+1/2)^4$.\n\\vfig{counting}\nThe two rows describe the same zero orbits. The upper row counts four zeros each time a new radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk.\n\\head{Separate radius conversion from orbit multiplicity}\nIn unscaled variables the general relation is\n\\[\nn_{D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+O(1),\\qquad\nn_{\\Delta_N}(R)=\\frac{\\sin(\\pi/N)}\\pi R^","ions are $v=(m+1/2)^4$.\n\\vfig{counting}\nThe two rows describe the same zero orbits. The upper row counts four zeros each time a new radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk.\n\\head{Separate radius conversion from orbit multiplicity}\nIn unscaled variables the general relation is\n\\[\nn_{D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+O(1),\\qquad\nn_{\\Delta_N}(R)=\\frac{\\sin(\\pi/N)}\\pi R^{1/N}+O(1).\n\\]\nThe different powers agree with the entire-function orders $1$ and $1/N$.\n\\takeaway{The reduced function has fewer repeated zeros and a different radius scale. Its counting law is not obtained by dividing a same-radius count alone.}\n\\platecaption{The exact order-four staircase formulas are tested at rational radii including jump points. The large-zero theorem fo…","counts four zeros each time a new radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk.\n\\head{Separate radius conversion from orbit multiplicity}\nIn unscaled variables the general relation is\n\\[\nn_{D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+O(1),\\qquad\nn_{\\Delta_N}(R)=\\frac{\\sin(\\pi/N)}\\pi R^{1/N}+O(1).\n\\]\nThe different powers agree with the entire-function orders $1$ and $1/N$.\n\\takeaway{The reduced function has fewer repeated zeros and a different radius scale. Its counting law is not obtained by dividing a same-radius count alone.}\n\\platecaption{The exact order-four staircase formulas are tested at rational radii including jump points. The large-zero theorem for general $N$ permits finitely many exceptions. These exact order-four identities do not justify d","radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk.\n\\head{Separate radius conversion from orbit multiplicity}\nIn unscaled variables the general relation is\n\\[\nn_{D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+O(1),\\qquad\nn_{\\Delta_N}(R)=\\frac{\\sin(\\pi/N)}\\pi R^{1/N}+O(1).\n\\]\nThe different powers agree with the entire-function orders $1$ and $1/N$.\n\\takeaway{The reduced function has fewer repeated zeros and a different radius scale. Its counting law is not obtained by dividing a same-radius count alone.}\n\\platecaption{The exact order-four staircase formulas are tested at rational radii including jump points. The large-zero theorem for general $N$ permits finitely many exceptions. These exact order-four identities do not justify discarding that boundary elsewhere.…","hed. Both formulas include zeros on the boundary of the disk.\n\\head{Separate radius conversion from orbit multiplicity}\nIn unscaled variables the general relation is\n\\[\nn_{D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+O(1),\\qquad\nn_{\\Delta_N}(R)=\\frac{\\sin(\\pi/N)}\\pi R^{1/N}+O(1).\n\\]\nThe different powers agree with the entire-function orders $1$ and $1/N$.\n\\takeaway{The reduced function has fewer repeated zeros and a different radius scale. Its counting law is not obtained by dividing a same-radius count alone.}\n\\platecaption{The exact order-four staircase formulas are tested at rational radii including jump points. The large-zero theorem for general $N$ permits finitely many exceptions. These exact order-four identities do not justify discarding that boundary elsewhere.}","s\n\\[\nn_{D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+O(1),\\qquad\nn_{\\Delta_N}(R)=\\frac{\\sin(\\pi/N)}\\pi R^{1/N}+O(1).\n\\]\nThe different powers agree with the entire-function orders $1$ and $1/N$.\n\\takeaway{The reduced function has fewer repeated zeros and a different radius scale. Its counting law is not obtained by dividing a same-radius count alone.}\n\\platecaption{The exact order-four staircase formulas are tested at rational radii including jump points. The large-zero theorem for general $N$ permits finitely many exceptions. These exact order-four identities do not justify discarding that boundary elsewhere.}","D_N}(R^{1/N})=Nn_{\\Delta_N}(R).\n\\]\nThere are two operations here: replace the radius by its $N$th root and divide the count by $N$. Forgetting either operation changes the leading constant.\nFor this family the asymptotics are\n\\[\nn_{D_N}(R)=\\frac{N\\sin(\\pi/N)}\\pi R+O(1),\\qquad\nn_{\\Delta_N}(R)=\\frac{\\sin(\\pi/N)}\\pi R^{1/N}+O(1).\n\\]\nThe different powers agree with the entire-function orders $1$ and $1/N$.\n\\takeaway{The reduced function has fewer repeated zeros and a different radius scale. Its counting law is not obtained by dividing a same-radius count alone.}\n\\platecaption{The exact order-four staircase formulas are tested at rational radii including jump points. The large-zero theorem for general $N$ permits finitely many exceptions. These exact order-four identities do not justify discarding that boundary elsewhere.}","/N)}\\pi R^{1/N}+O(1).\n\\]\nThe different powers agree with the entire-function orders $1$ and $1/N$.\n\\takeaway{The reduced function has fewer repeated zeros and a different radius scale. Its counting law is not obtained by dividing a same-radius count alone.}\n\\platecaption{The exact order-four staircase formulas are tested at rational radii including jump points. The large-zero theorem for general $N$ permits finitely many exceptions. These exact order-four identities do not justify discarding that boundary elsewhere.}","\\vpage{21}{A product and an operator share zeros}{CANONICAL PRODUCT}{Chapter IV: Theorem 12.1, Theorem 12.2.}\nFor $N\\ge2$, the reduced entire function has order $1/N<1$. Its multiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe las","\\vpage{21}{A product and an operator share zeros}{CANONICAL PRODUCT}{Chapter IV: Theorem 12.1, Theorem 12.2.}\nFor $N\\ge2$, the reduced entire function has order $1/N<1$. Its multiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. T","\\vpage{21}{A product and an operator share zeros}{CANONICAL PRODUCT}{Chapter IV: Theorem 12.1, Theorem 12.2.}\nFor $N\\ge2$, the reduced entire function has order $1/N<1$. Its multiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic t","\\vpage{21}{A product and an operator share zeros}{CANONICAL PRODUCT}{Chapter IV: Theorem 12.1, Theorem 12.2.}\nFor $N\\ge2$, the reduced entire function has order $1/N<1$. Its multiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled","\\vpage{21}{A product and an operator share zeros}{CANONICAL PRODUCT}{Chapter IV: Theorem 12.1, Theorem 12.2.}\nFor $N\\ge2$, the reduced entire function has order $1/N<1$. Its multiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not su","e2$, the reduced entire function has order $1/N<1$. Its multiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Ord…","e reduced entire function has order $1/N<1$. Its multiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one …","ltiplicity-sensitive genus-zero product is\n\\[\n\\Delta_N(w)=\\prod_{\\lambda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so …","bda\\in\\Lambda_N}(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so its empty product would be $1$, not the function.}","(1-w/\\lambda).\n\\]\nPutting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so its empty product would be $1$, not the function.}","Putting $K_Ne_\\lambda=\\lambda^{-1}e_\\lambda$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so its empty product would be $1$, not the function.}","$ on $\\ell^2(\\Lambda_N)$ gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so its empty product would be $1$, not the function.}","diagonal entries.\n\\vfig{product}\n\\head{Finite compression is an approximation}\nFor $N=2$, $\\lambda_m=\\pi^2(m+1/2)^2$. The finite determinant $P_L$ is exactly the displayed finite product, but it is not exactly $\\Delta_2$. For $0\\le w\\le1$ and $L\\ge1$, an elementary tail estimate gives\n\\[\n0\\le1-\\frac{\\Delta_2(w)}{P_L(w)}\n\\le w\\sum_{m\\ge L}\\lambda_m^{-1}\n\\le\\frac1{9(L-1/2)}.\n\\]\nThe last bound uses $\\pi>3$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so its empty product would be $1$, not the function.}","$ and integral comparison. Thus truncation has an explicit error mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so its empty product would be $1$, not the function.}","mechanism.\n\\takeaway{The infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem.}\n\\platecaption{The exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: $\\Delta_1(w)=e^{-w}$ has no zeros, so its empty product would be $1$, not the function.}","\\vpage{22}{Finite coefficients give infinite moments}{ZERO POWER SUMS}{Chapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10.}\nTaking the logarithm of the canonical product identifies\n\\[\n\\kappa_{N,q}=\\sigma_{N,q}:=\\sum_{\\lambda\\in\\Lambda_N}\\lambda^{-q}\n=\\Tr(K_N^q),\\qquad q\\ge1.\n\\]\nThe same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation.\n\\vfig{moments}\nFor order two,\n\\[\n\\sigma_{2,1}=\\frac12,\\quad\\sigma_{2,2}=\\frac16,\\quad\n\\sigma_{2,3}=\\frac1{15},\\quad\\sigma_{2,4}=\\frac{17}{630}.\n\\]\nThe plate shows how successive inverse powers emphasize the nearest zero. Its shaded pieces are contributions from the first five positive reduced zeros, followed by the remaining tail. The total in each row is the exact rational moment.\n\\head{The tangent is a sum of poles}\nLogarithmic differentiation also gives\n\\[\nT_N(z)=Nz^{N-1}\\sum_{\\lambda\\in\\Lambda_N}\\frac1{\\lambda-z^N}\n\\]\naway fr","Corollary 12.7, Corollary 12.9, Corollary 12.10.}\nTaking the logarithm of the canonical product identifies\n\\[\n\\kappa_{N,q}=\\sigma_{N,q}:=\\sum_{\\lambda\\in\\Lambda_N}\\lambda^{-q}\n=\\Tr(K_N^q),\\qquad q\\ge1.\n\\]\nThe same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation.\n\\vfig{moments}\nFor order two,\n\\[\n\\sigma_{2,1}=\\frac12,\\quad\\sigma_{2,2}=\\frac16,\\quad\n\\sigma_{2,3}=\\frac1{15},\\quad\\sigma_{2,4}=\\frac{17}{630}.\n\\]\nThe plate shows how successive inverse powers emphasize the nearest zero. Its shaded pieces are contributions from the first five positive reduced zeros, followed by the remaining tail. The total in each row is the exact rational moment.\n\\head{The tangent is a sum of poles}\nLogarithmic differentiation also gives\n\\[\nT_N(z)=Nz^{N-1}\\sum_{\\lambda\\in\\Lambda_N}\\frac1{\\lambda-z^N}\n\\]\naway from poles. Summability of $|\\lambda|^{-1}$ permits the locally uniform partial-fraction expansion without added regularizing terms.\n\\takeaway{A global analytic product theorem connects the infinite z…","ma_{2,2}=\\frac16,\\quad\n\\sigma_{2,3}=\\frac1{15},\\quad\\sigma_{2,4}=\\frac{17}{630}.\n\\]\nThe plate shows how successive inverse powers emphasize the nearest zero. Its shaded pieces are contributions from the first five positive reduced zeros, followed by the remaining tail. The total in each row is the exact rational moment.\n\\head{The tangent is a sum of poles}\nLogarithmic differentiation also gives\n\\[\nT_N(z)=Nz^{N-1}\\sum_{\\lambda\\in\\Lambda_N}\\frac1{\\lambda-z^N}\n\\]\naway from poles. Summability of $|\\lambda|^{-1}$ permits the locally uniform partial-fraction expansion without added regularizing terms.\n\\takeaway{A global analytic product theorem connects the infinite zero sum to a finite algebraic recurrence. The recurrence alone does not prove the product theorem.}\n\\platecaption{Displayed moments are checked against formal logarithm coefficients and exact order-two/order-four zeta values. Bar widths are numerical shares. General power sums use $\\lambda^{-q}$, not $|\\lambda|^{-q}$ when exceptional nonpositive zeros are present.}","s how successive inverse powers emphasize the nearest zero. Its shaded pieces are contributions from the first five positive reduced zeros, followed by the remaining tail. The total in each row is the exact rational moment.\n\\head{The tangent is a sum of poles}\nLogarithmic differentiation also gives\n\\[\nT_N(z)=Nz^{N-1}\\sum_{\\lambda\\in\\Lambda_N}\\frac1{\\lambda-z^N}\n\\]\naway from poles. Summability of $|\\lambda|^{-1}$ permits the locally uniform partial-fraction expansion without added regularizing terms.\n\\takeaway{A global analytic product theorem connects the infinite zero sum to a finite algebraic recurrence. The recurrence alone does not prove the product theorem.}\n\\platecaption{Displayed moments are checked against formal logarithm coefficients and exact order-two/order-four zeta values. Bar widths are numerical shares. General power sums use $\\lambda^{-q}$, not $|\\lambda|^{-q}$ when exceptional nonpositive zeros are present.}","ction expansion without added regularizing terms.\n\\takeaway{A global analytic product theorem connects the infinite zero sum to a finite algebraic recurrence. The recurrence alone does not prove the product theorem.}\n\\platecaption{Displayed moments are checked against formal logarithm coefficients and exact order-two/order-four zeta values. Bar widths are numerical shares. General power sums use $\\lambda^{-q}$, not $|\\lambda|^{-q}$ when exceptional nonpositive zeros are present.}","out added regularizing terms.\n\\takeaway{A global analytic product theorem connects the infinite zero sum to a finite algebraic recurrence. The recurrence alone does not prove the product theorem.}\n\\platecaption{Displayed moments are checked against formal logarithm coefficients and exact order-two/order-four zeta values. Bar widths are numerical shares. General power sums use $\\lambda^{-q}$, not $|\\lambda|^{-q}$ when exceptional nonpositive zeros are present.}","\\vpage{23}{Nearest poles set coefficient growth}{SINGULARITY ANALYSIS}{Chapter IV: Theorem 13.1, Corollary 13.2, Corollary 13.3, Corollary 13.4.}\nFor $1/\\cos z$, the nearest poles are $\\rho=\\pm\\pi/2$. They contribute equally to even-degree coefficients. Including both poles gives\n\\[\nP_{2,n}(1)\\sim2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\nThe factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator","\\vpage{23}{Nearest poles set coefficient growth}{SINGULARITY ANALYSIS}{Chapter IV: Theorem 13.1, Corollary 13.2, Corollary 13.3, Corollary 13.4.}\nFor $1/\\cos z$, the nearest poles are $\\rho=\\pm\\pi/2$. They contribute equally to even-degree coefficients. Including both poles gives\n\\[\nP_{2,n}(1)\\sim2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\nThe factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The","\\vpage{23}{Nearest poles set coefficient growth}{SINGULARITY ANALYSIS}{Chapter IV: Theorem 13.1, Corollary 13.2, Corollary 13.3, Corollary 13.4.}\nFor $1/\\cos z$, the nearest poles are $\\rho=\\pm\\pi/2$. They contribute equally to even-degree coefficients. Including both poles gives\n\\[\nP_{2,n}(1)\\sim2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\nThe factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. S","pter IV: Theorem 13.1, Corollary 13.2, Corollary 13.3, Corollary 13.4.}\nFor $1/\\cos z$, the nearest poles are $\\rho=\\pm\\pi/2$. They contribute equally to even-degree coefficients. Including both poles gives\n\\[\nP_{2,n}(1)\\sim2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\nThe factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geo…","$1/\\cos z$, the nearest poles are $\\rho=\\pm\\pi/2$. They contribute equally to even-degree coefficients. Including both poles gives\n\\[\nP_{2,n}(1)\\sim2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\nThe factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal …","t poles are $\\rho=\\pm\\pi/2$. They contribute equally to even-degree coefficients. Including both poles gives\n\\[\nP_{2,n}(1)\\sim2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\nThe factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal order, and lower princ…","n-degree coefficients. Including both poles gives\n\\[\nP_{2,n}(1)\\sim2(2n)!\\left(\\frac2\\pi\\right)^{2n+1}.\n\\]\nThe factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal order, and lower principal parts determine its precise normalization.}\n\\platecapt…","factor two is orbit multiplicity, not an adjustable fitted constant.\n\\vfig{pole_asymptotic}\n\\head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal order, and lower principal parts determine its precise normalization.}\n\\platecaption{The ratio certificates use a rational Machin-formula enclosure for $\\pi$. The inherited general theorem ret","head{A finite quantitative check}\nFor $n=1,\\ldots,8$, the exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal order, and lower principal parts determine its precise normalization.}\n\\platecaption{The ratio certificates use a rational Machin-formula enclosure for $\\pi$. The inherited general theorem retains simple nearest zeros, a gap to the next modulus, and positive integral reciprocal order.","he exact integer coefficients and a rational enclosure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal order, and lower principal parts determine its precise normalization.}\n\\platecaption{The ratio certificates use a rational Machin-formula enclosure for $\\pi$. The inherited general theorem retains simple nearest zeros, a gap to the next modulus, and positive integral reciprocal order. The plot does not remove those hypotheses.}","ure for $\\pi$ bound the ratio $R_n$ strictly between zero and one. In this tested range, $1-R_n<3^{-(2n+1)}$. The plot displays the logarithm of the enclosed error, not an independently asserted all-$n$ bound.\n\\head{Higher reciprocal powers need more Laurent data}\nFor positive integral order $r$, a simple denominator zero becomes a pole of order $r$. The highest Laurent term contributes a factor $\\binom{m+r-1}{r-1}$, but lower Laurent terms still contribute on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal order, and lower principal parts determine its precise normalization.}\n\\platecaption{The ratio certificates use a rational Machin-formula enclosure for $\\pi$. The inherited general theorem retains simple nearest zeros, a gap to the next modulus, and positive integral reciprocal order. The plot does not remove those hypotheses.}","on the same exponential scale. Subtracting \\emph{all} principal parts is what leaves a holomorphic remainder controlled by the next zero radius.\n\\takeaway{Nearest-zero geometry explains factorial-times-exponential growth. Multiplicity, reciprocal order, and lower principal parts determine its precise normalization.}\n\\platecaption{The ratio certificates use a rational Machin-formula enclosure for $\\pi$. The inherited general theorem retains simple nearest zeros, a gap to the next modulus, and positive integral reciprocal order. The plot does not remove those hypotheses.}","\\vpage{24}{Convergence is not continuation}{ZERO ZETA FUNCTIONS}{Chapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4.}\nReduced zeros grow like $m^N$. Their absolute zero-zeta series therefore has half-plane of absolute convergence $\\Re s>1/N$. The absolute zero zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(","\\vpage{24}{Convergence is not continuation}{ZERO ZETA FUNCTIONS}{Chapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4.}\nReduced zeros grow like $m^N$. Their absolute zero-zeta series therefore has half-plane of absolute convergence $\\Re s>1/N$. The absolute zero zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to","\\vpage{24}{Convergence is not continuation}{ZERO ZETA FUNCTIONS}{Chapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4.}\nReduced zeros grow like $m^N$. Their absolute zero-zeta series therefore has half-plane of absolute convergence $\\Re s>1/N$. The absolute zero zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire ter","\\vpage{24}{Convergence is not continuation}{ZERO ZETA FUNCTIONS}{Chapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4.}\nReduced zeros grow like $m^N$. Their absolute zero-zeta series therefore has half-plane of absolute convergence $\\Re s>1/N$. The absolute zero zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishe","\\vpage{24}{Convergence is not continuation}{ZERO ZETA FUNCTIONS}{Chapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4.}\nReduced zeros grow like $m^N$. Their absolute zero-zeta series therefore has half-plane of absolute convergence $\\Re s>1/N$. The absolute zero zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continua","\\vpage{24}{Convergence is not continuation}{ZERO ZETA FUNCTIONS}{Chapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4.}\nReduced zeros grow like $m^N$. Their absolute zero-zeta series therefore has half-plane of absolute convergence $\\Re s>1/N$. The absolute zero zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\ta…","solute zero zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\takeaway{An analytic continuation can exist at points where the original infinite sum diverges. It must not be evaluated there by naively summing that series.}\n\\platecaption{The shaded regions show convergence, and the open circles mark poles of the conti…","ro zeta in the original variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\takeaway{An analytic continuation can exist at points where the original infinite sum diverges. It must not be evaluated there by naively summing that series.}\n\\platecaption{The shaded regions show convergence, and the open circles mark poles of the continued func…","nal variable instead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\takeaway{An analytic continuation can exist at points where the original infinite sum diverges. It must not be evaluated there by naively summing that series.}\n\\platecaption{The shaded regions show convergence, and the open circles mark poles of the continued functions. They are not …","nstead requires $\\Re s>1$.\n\\vfig{zeta_plane}\nFor $N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\takeaway{An analytic continuation can exist at points where the original infinite sum diverges. It must not be evaluated there by naively summing that series.}\n\\platecaption{The shaded regions show convergence, and the open circles mark poles of the continued functions. They are not zeros of the z…","$N=2$ and $N=4$, the lattices are exact:\n\\[\n\\mathfrak Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\takeaway{An analytic continuation can exist at points where the original infinite sum diverges. It must not be evaluated there by naively summing that series.}\n\\platecaption{The shaded regions show convergence, and the open circles mark poles of the continued functions. They are not zeros of the zeta functions. Branch-dependent orbit zetas and d","k Z_2^{\\rm red}(s)=\\pi^{-2s}\\zeta(2s,1/2),\\qquad\n\\mathfrak Z_4^{\\rm red}(s)=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\takeaway{An analytic continuation can exist at points where the original infinite sum diverges. It must not be evaluated there by naively summing that series.}\n\\platecaption{The shaded regions show convergence, and the open circles mark poles of the continued functions. They are not zeros of the zeta functions. Branch-dependent orbit zetas and derivative-weighted zetas retain their separately sta…","=(\\sqrt2\\pi)^{-4s}\\zeta(4s,1/2).\n\\]\nEach identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at $1/2$ or $1/4$, with residue $1/(2\\pi)$ or $1/(4\\sqrt2\\pi)$ respectively.\n\\head{The factor $N$ matters twice}\nFor every order in this chapter,\n\\[\n\\mathfrak Z_N^{\\rm abs}(s)=N\\mathfrak Z_N^{\\rm red}(s/N).\n\\]\nFor general $N$, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero set also contributes entire terms. This reasoning establishes continuation, not a zeta functional equation.\n\\takeaway{An analytic continuation can exist at points where the original infinite sum diverges. It must not be evaluated there by naively summing that series.}\n\\platecaption{The shaded regions show convergence, and the open circles mark poles of the continued functions. They are not zeros of the zeta functions. Branch-dependent orbit zetas and derivative-weighted zetas retain their separately stated conventions.}","\\vpage{25}{One closed calculation: the number 69}{COEFFICIENT AND ZERO WORKED MODEL}{Chapter IV: Corollary 7.7, Example 11.12, Theorem 12.6.}\nUse $N=4$ throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off t","5}{One closed calculation: the number 69}{COEFFICIENT AND ZERO WORKED MODEL}{Chapter IV: Corollary 7.7, Example 11.12, Theorem 12.6.}\nUse $N=4$ throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrat…","tion: the number 69}{COEFFICIENT AND ZERO WORKED MODEL}{Chapter IV: Corollary 7.7, Example 11.12, Theorem 12.6.}\nUse $N=4$ throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridge…","{COEFFICIENT AND ZERO WORKED MODEL}{Chapter IV: Corollary 7.7, Example 11.12, Theorem 12.6.}\nUse $N=4$ throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof…","orollary 7.7, Example 11.12, Theorem 12.6.}\nUse $N=4$ throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption…","12, Theorem 12.6.}\nUse $N=4$ throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption{Every rational value and","Use $N=4$ throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption{Every rational value and polynomial identit…","recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption{Every rational value and polynomial identity on this page is checked exactly. The infinite zero-sum interpretation uses the inherited c…","cient. All symbols in the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption{Every rational value and polynomial identity on this page is checked exactly. The infinite zero-sum interpretation uses the inherited canonical-product theorem and the exact order-four…","n the plate refer to this one order.\n\\vfig{worked_circuit}\n\\head{Route one: invert the denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption{Every rational value and polynomial identity on this page is checked exactly. The infinite zero-sum interpretation uses the inherited canonical-product theorem and the exact order-four zero formula.}","he denominator}\nThe $w^2$ coefficient of $\\Delta_4^{-1}$ is\n\\[\n\\frac1{24^2}-\\frac1{8!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption{Every rational value and polynomial identity on this page is checked exactly. The infinite zero-sum interpretation uses the inherited canonical-product theorem and the exact order-four zero formula.}","!}=\\frac{69}{8!}.\n\\]\nThus $P_{4,2}(1)=69$. Keeping $r$ variable gives $P_{4,2}(r)=35r^2+34r$.\n\\head{Route two: use the zero moments}\nThe exact reduced zeros are $4\\pi^4(m+1/2)^4$. Their first two power sums are $\\sigma_1=1/24$ and $\\sigma_2=17/10080$. The Bell/Newton identity gives the same reciprocal coefficient $(\\sigma_1^2+\\sigma_2)/2$. The tangent independently reads off the second moment as $4\\cdot7!\\sigma_2=34$.\nThis links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.\n\\takeaway{The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.}\n\\platecaption{Every rational value and polynomial identity on this page is checked exactly. The infinite zero-sum interpretation uses the inherited canonical-product theorem and the exact order-four zero formula.}","Certified zero brackets & Four rational intervals of width $10^{-20}$ establish local existence and uniqueness only. The full suite reproduced all four rational brackets, opposite endpoint signs, and derivative exclusion throughout each interval. These are local certificates, not a global zero census.","\\section*{Principal results}\n\\addcontentsline{toc}{section}{Principal results}\nThe original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If $p$ is prime, $m$ is positive with $m\\mid p-1$ or $m=2$, $0\\le j<mp$, and $q=\\operatorname{lcm}(2,p-1)$, then the original Euler coefficients are $p$-adically integral and\n\\[\n E_{mpn+qp^r}^{(mp,j)}\\equiv E_{mpn}^{(mp,j)}\\pmod{p^r}\n \\qquad (r\\ge1,\\ n\\ge r).\n\\]\nThe chapter states the generating function and the conversion between ordinary degree and packet row before using this formula. The normalized coefficient is $C_{N,j}(n)$ and the original coefficient is $E_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The constant coefficient of the original series is $j!$.","s positive with $m\\mid p-1$ or $m=2$, $0\\le j<mp$, and $q=\\operatorname{lcm}(2,p-1)$, then the original Euler coefficients are $p$-adically integral and\n\\[\n E_{mpn+qp^r}^{(mp,j)}\\equiv E_{mpn}^{(mp,j)}\\pmod{p^r}\n \\qquad (r\\ge1,\\ n\\ge r).\n\\]\nThe chapter states the generating function and the conversion between ordinary degree and packet row before using this formula. The normalized coefficient is $C_{N,j}(n)$ and the original coefficient is $E_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The constant coefficient of the original series is $j!$.","<mp$, and $q=\\operatorname{lcm}(2,p-1)$, then the original Euler coefficients are $p$-adically integral and\n\\[\n E_{mpn+qp^r}^{(mp,j)}\\equiv E_{mpn}^{(mp,j)}\\pmod{p^r}\n \\qquad (r\\ge1,\\ n\\ge r).\n\\]\nThe chapter states the generating function and the conversion between ordinary degree and packet row before using this formula. The normalized coefficient is $C_{N,j}(n)$ and the original coefficient is $E_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The constant coefficient of the original series is $j!$.","$p$-adically integral and\n\\[\n E_{mpn+qp^r}^{(mp,j)}\\equiv E_{mpn}^{(mp,j)}\\pmod{p^r}\n \\qquad (r\\ge1,\\ n\\ge r).\n\\]\nThe chapter states the generating function and the conversion between ordinary degree and packet row before using this formula. The normalized coefficient is $C_{N,j}(n)$ and the original coefficient is $E_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The constant coefficient of the original series is $j!$.","The strongest common-step conclusion in width four is compact. For $n\\ge0$ and $k,H\\ge1$,\n\\begin{equation*}\n \\sum_{i=0}^k(-1)^{k-i}\\binom ki E_{4(n+iH)}^{(4,j)}\n \\equiv0\\pmod{2^{\\,v_2(k!)+kv_2(H)+A_j(k)}},\n\\end{equation*}\nwhere\n\\[\n A_0(k)=A_1(k)=k,\\qquad\n A_2(k)=\\max\\{k+4,\\,2k+\\lfloor k/2\\rfloor+1\\},\\qquad\n A_3(k)=k+2.\n\\]\nThere is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addresses zero and one, and exact first-displacement laws.","\\midrule\nNishibuchi Conjecture 3.9 & Every parameter in the original statement, with explicit $n\\ge r$ & Inherited Lean proof","Taylor radii and broader dyadic first differences & Exact radii in addresses $0,1,2$ and stated all-rank cases & Written proof and audit","\\[\n(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).\n\\]","nge and develop sharper arithmetic consequences. For every prime-power packet width, a charged Hurwitz differential operator transfers integral difference estimates through the reciprocal generating series. In the dyadic case this gives mixed finite-difference bounds at every row, without a starting threshold. For the four addresses of width four, a divided-power argument adds the factorial gain $v_2(k!)$, determines every repeated-step valuation in addresses zero and one, and gives a stronger address-two estimate with exact values at all power-of-two orders. The corresponding first differences yield least periods and bijective isometric continuations onto explicit dyadic residue classes. We also determine exact Taylor radii for three addresses. The manuscript retains the complete earlier arithmetic, lifting, deformation, and analytic-regularity development, with detailed mathematical appendices. A separate appendix states the precise remaining simplicity problem for ordinary Euler tails. Formal coverage is explicit: the external conjecture and several four-address results have inher…","\\clearpage\n\\appendix\n\\input{../root_scope.tex}\n\\input{parts/70_continuation}\n\\input{parts/71_native}\n\\input{parts/72_recovered}\n\\input{parts/73_supplements}\n\\clearpage\\def\\IXMainAtlas{1}\\def\\AtlasFigDir{../../figures/worked}\\input{../worked_atlas.tex}\\clearpage\n\\begin{thebibliography}{99}\n\\bibitem{Nishibuchi}\nYuta Nishibuchi,\n\\emph{$p$-adic Congruences of Generalized Euler Numbers and Relations to Even\nZeta Values}, arXiv:2605.10677v2, 1 June 2026.\n\\url{https://arxiv.org/abs/2605.10677v2}.\nThe comparison to Conjecture 3.9 uses the source review and fixed-version\nrecord preserved with this edition.","% Generated from references_historical.bib using amsplain, with full repeated authors and DOI links.\n\\bibitem{Amice1964}\nYvette Amice, \\emph{{Interpolation {$p$}-adique}}, Bulletin de la Soci\\'et\\'e\n  Math\\'ematique de France \\textbf{92} (1964), 117--180.\n\\url{https://doi.org/10.24033/bsmf.1606}.","\\bibitem{Gouvea1997}\nFernando~Q. Gouv\\^ea, \\emph{{{$p$}-adic Numbers: An Introduction}}, second ed.,\n  Universitext, Springer, Berlin, 1997.\n\\url{https://doi.org/10.1007/978-3-642-59058-0}.","\\bibitem{Koblitz1984}\nNeal Koblitz, \\emph{{{$p$}-adic Numbers, {$p$}-adic Analysis, and\n  Zeta-Functions}}, second ed., Graduate Texts in Mathematics, vol.~58,\n  Springer, New York, 1984.\n\\url{https://doi.org/10.1007/978-1-4612-1112-9}.","\\bibitem{Mahler1958}\nKurt Mahler, \\emph{{An Interpolation Series for Continuous Functions of a\n  {$p$}-adic Variable}}, Journal f\\\"ur die Reine und Angewandte Mathematik\n  \\textbf{199} (1958), 23--34.\n\\url{https://doi.org/10.1515/crll.1958.199.23}.","\\bibitem{Robert2000}\nAlain~M. Robert, \\emph{{A Course in {$p$}-adic Analysis}}, Graduate Texts in\n  Mathematics, vol. 198, Springer, New York, 2000.\n\\url{https://doi.org/10.1007/978-1-4757-3254-2}.","\\section{Scope and conventions}\n\\label{sec:scope}\nThe central objects are sequences and polynomial equations equipped with\narithmetic coordinates. Residue projectors select addresses in a sequence,\nfinite differences measure compatibility between nearby addresses, and\nvaluation bounds control the passage from finite precision to a $p$-adic\nlimit. These constructions provide a common language for the Euler packet\nresults and the later lifting results. Each theorem nevertheless retains its\nown coefficient ring, topology and nonsingularity hypotheses.","Throughout, $\\N=\\{0,1,2,\\ldots\\}$. When $p$ is prime, $\\Zp$ denotes the\n$p$-adic integers and $\\Qp$ their fraction field. We normalize $v_p(p)=1$.\nFor a rational or $p$-adic quantity, a displayed lower valuation bound is\nunderstood with the mathematical convention $v_p(0)=+\\infty$.\nThe corresponding formal predicates are zero-safe: the quantity is zero or\nits valuation meets the stated bound. An exact finite valuation assertion\nrequires a nonzero quantity, and this is stated where used. Congruence in\n$\\Zp$ means ideal divisibility. Thus\n\\[\n x\\equiv y\\pmod{p^r}\\quad\\Longleftrightarrow\\quad\n x-y\\in p^r\\Zp.\n\\]\nFor vectors, ideal membership and congruence are coordinatewise.","raction field. We normalize $v_p(p)=1$.\nFor a rational or $p$-adic quantity, a displayed lower valuation bound is\nunderstood with the mathematical convention $v_p(0)=+\\infty$.\nThe corresponding formal predicates are zero-safe: the quantity is zero or\nits valuation meets the stated bound. An exact finite valuation assertion\nrequires a nonzero quantity, and this is stated where used. Congruence in\n$\\Zp$ means ideal divisibility. Thus\n\\[\n x\\equiv y\\pmod{p^r}\\quad\\Longleftrightarrow\\quad\n x-y\\in p^r\\Zp.\n\\]\nFor vectors, ideal membership and congruence are coordinatewise.",".\nFor a rational or $p$-adic quantity, a displayed lower valuation bound is\nunderstood with the mathematical convention $v_p(0)=+\\infty$.\nThe corresponding formal predicates are zero-safe: the quantity is zero or\nits valuation meets the stated bound. An exact finite valuation assertion\nrequires a nonzero quantity, and this is stated where used. Congruence in\n$\\Zp$ means ideal divisibility. Thus\n\\[\n x\\equiv y\\pmod{p^r}\\quad\\Longleftrightarrow\\quad\n x-y\\in p^r\\Zp.\n\\]\nFor vectors, ideal membership and congruence are coordinatewise.","Write $\\Delta_u f(n)=f(n+u)-f(n)$ and use $\\Delta_u^k$ for its $k$-fold\niterate. Euler coefficients indexed by ordinary degree and by packet row are\ndistinguished explicitly. A packet displacement by $u$ rows is a degree\ndisplacement by $Nu$ in a packet of width $N$.","The original arithmetic seed notation\n\\[\n\\mathcal S=(A,a_0,\\mathcal O,\\mathcal R,\\{P_{N,j}\\},w,\\tau_p)\n\\]\nrecords a coefficient ring or module, a seed, allowed operators, relations,\nresidue selectors, a depth weight, and a topology. This notation describes the\ningredients of the construction. A category of seed presentations and a\nnotion of equivalence still require separate definitions. We therefore state\neach theorem for a concrete object. Section~\\ref{sec:open} discusses the\nadditional structure needed for the original native conjectures.","Let $R$ be a commutative ring, $a:\\N\\to R$, $N\\geq1$, and $0\\leq j<N$.\nDefine\n\\[\n(P_{N,j}a)(n)=\n\\begin{cases}\na(n),&n\\equiv j\\pmod N,","Let $R$ be a commutative ring, $a:\\N\\to R$, $N\\geq1$, and $0\\leq j<N$.\nDefine\n\\[\n(P_{N,j}a)(n)=\n\\begin{cases}\na(n),&n\\equiv j\\pmod N,\\\\\n0,&\\text{otherwise}.\n\\end{cases}\n\\]\nThis definition keeps the original index set. The compressed subsequence\n$a(Nq+j)$, used below for recurrence packets, is a different representation of\nthe selected coefficients.","0,&\\text{otherwise}.\n\\end{cases}\n\\]\nThis definition keeps the original index set. The compressed subsequence\n$a(Nq+j)$, used below for recurrence packets, is a different representation of\nthe selected coefficients.","\\begin{theorem}[Residue reconstruction]\n\\label{thm:residue-reconstruction}\nThe selectors satisfy\n\\[\nP_{N,j}^2=P_{N,j},\\qquad P_{N,i}P_{N,j}=0\\quad(i\\ne j),\n\\qquad \\sum_{j=0}^{N-1}P_{N,j}=I.\n\\]\nIf $N\\mid M$ and $M>0$, then\n\\[\nP_{N,j}a=\\sum_{\\substack{0\\leq r<M","\\begin{theorem}[Residue reconstruction]\n\\label{thm:residue-reconstruction}\nThe selectors satisfy\n\\[\nP_{N,j}^2=P_{N,j},\\qquad P_{N,i}P_{N,j}=0\\quad(i\\ne j),\n\\qquad \\sum_{j=0}^{N-1}P_{N,j}=I.\n\\]\nIf $N\\mid M$ and $M>0$, then\n\\[\nP_{N,j}a=\\sum_{\\substack{0\\leq r<M\\\\r\\equiv j\\pmod N}}P_{M,r}a.\n\\]\nThe same identities hold coefficientwise for formal power series.\n\\end{theorem}","\\begin{proof}\nAt each index $n$, exactly one standard residue label equals $n\\bmod N$.\nConsequently a repeated selector changes nothing, two different selectors have\nzero composite, and the sum retains $a(n)$ exactly once. If $N\\mid M$, reducing\n$n\\bmod M$ modulo $N$ gives $n\\bmod N$. This proves the refinement identity\nat every coefficient. Equality of all coefficients proves the formal-series\nversions.\n\\end{proof}","Over a characteristic-zero field $K$, put\n\\[\n\\mathcal E(a)=\\sum_{n\\geq0}\\frac{a_n}{n!}t^n,\n\\qquad \\mathcal D(A)_n=n![t^n]A.\n\\]\nThese inverse maps between sequences and $K[[t]]$ provide \\emph{formal}\nexponential coordinates. No analytic convergence is needed for the coefficient\nidentities below.","\\begin{theorem}[Constructed unit-triangular coordinates]\n\\label{thm:egf-coordinates}\nLet $h:\\N\\to K$ satisfy $h_0=1$, write $H=\\mathcal E(h)$ and\n$H^{-1}=\\mathcal E(k)$, and define\n\\[\nU_Ha=\\mathcal D(H\\mathcal E(a)).\n\\]\nThen $U_H$ is an equivalence, with inverse\n$U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a))$. Its coefficient formulas are\n\\[\n(U_Ha)_n=\\sum_{r=0}^{n}\\binom nr h_r a_{n-r},\\qquad\n(U_H^{-1}a)_n=\\sum_{r=0}^{n}\\binom nr k_r a_{n-r}.\n\\]\nThe resulting triangular matrix has diagonal entries $1$. The transported\nselectors\n\\[\n\\widehat P_{N,j}=U_HP_{N,j}U_H^{-1}\n\\]\nare idempotent and pairwise orthogonal, and their sum is the identity.\n\\end{theorem}","satisfy $h_0=1$, write $H=\\mathcal E(h)$ and\n$H^{-1}=\\mathcal E(k)$, and define\n\\[\nU_Ha=\\mathcal D(H\\mathcal E(a)).\n\\]\nThen $U_H$ is an equivalence, with inverse\n$U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a))$. Its coefficient formulas are\n\\[\n(U_Ha)_n=\\sum_{r=0}^{n}\\binom nr h_r a_{n-r},\\qquad\n(U_H^{-1}a)_n=\\sum_{r=0}^{n}\\binom nr k_r a_{n-r}.\n\\]\nThe resulting triangular matrix has diagonal entries $1$. The transported\nselectors\n\\[\n\\widehat P_{N,j}=U_HP_{N,j}U_H^{-1}\n\\]\nare idempotent and pairwise orthogonal, and their sum is the identity.\n\\end{theorem}","cal E(h)$ and\n$H^{-1}=\\mathcal E(k)$, and define\n\\[\nU_Ha=\\mathcal D(H\\mathcal E(a)).\n\\]\nThen $U_H$ is an equivalence, with inverse\n$U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a))$. Its coefficient formulas are\n\\[\n(U_Ha)_n=\\sum_{r=0}^{n}\\binom nr h_r a_{n-r},\\qquad\n(U_H^{-1}a)_n=\\sum_{r=0}^{n}\\binom nr k_r a_{n-r}.\n\\]\nThe resulting triangular matrix has diagonal entries $1$. The transported\nselectors\n\\[\n\\widehat P_{N,j}=U_HP_{N,j}U_H^{-1}\n\\]\nare idempotent and pairwise orthogonal, and their sum is the identity.\n\\end{theorem}","\\begin{proof}\nThe factorials are nonzero in $K$, so encoding and decoding cancel\ncoefficientwise. The constant coefficient of $H$ is $1$. Hence its formal\npower-series inverse exists. Multiplication by $H$ and $H^{-1}$ gives the two\ninverse laws. The ordinary product coefficient formula, multiplied by $n!$,\ngives the displayed binomial sums. In row $n$, the coefficient of $a_n$ is\n$h_0=1$ and coefficients of $a_q$ with $q>n$ vanish. Conjugating the identities\nof Theorem~\\ref{thm:residue-reconstruction} by the constructed equivalence proves\nthe transported projector identities.\n\\end{proof}","The factorials are nonzero in $K$, so encoding and decoding cancel\ncoefficientwise. The constant coefficient of $H$ is $1$. Hence its formal\npower-series inverse exists. Multiplication by $H$ and $H^{-1}$ gives the two\ninverse laws. The ordinary product coefficient formula, multiplied by $n!$,\ngives the displayed binomial sums. In row $n$, the coefficient of $a_n$ is\n$h_0=1$ and coefficients of $a_q$ with $q>n$ vanish. Conjugating the identities\nof Theorem~\\ref{thm:residue-reconstruction} by the constructed equivalence proves\nthe transported projector identities.\n\\end{proof}","ials are nonzero in $K$, so encoding and decoding cancel\ncoefficientwise. The constant coefficient of $H$ is $1$. Hence its formal\npower-series inverse exists. Multiplication by $H$ and $H^{-1}$ gives the two\ninverse laws. The ordinary product coefficient formula, multiplied by $n!$,\ngives the displayed binomial sums. In row $n$, the coefficient of $a_n$ is\n$h_0=1$ and coefficients of $a_q$ with $q>n$ vanish. Conjugating the identities\nof Theorem~\\ref{thm:residue-reconstruction} by the constructed equivalence proves\nthe transported projector identities.\n\\end{proof}","In characteristic $p$, factorial division is unavailable in general, but the\ncoefficient operation remains meaningful over every commutative ring:\n\\[\n(f\\star g)_n=\\sum_{r=0}^{n}\\binom nr f_r g_{n-r},\\qquad\n(D^d f)_n=f_{n+d},\\qquad \\mathbf1_n=\\begin{cases}1,&n=0,","In characteristic $p$, factorial division is unavailable in general, but the\ncoefficient operation remains meaningful over every commutative ring:\n\\[\n(f\\star g)_n=\\sum_{r=0}^{n}\\binom nr f_r g_{n-r},\\qquad\n(D^d f)_n=f_{n+d},\\qquad \\mathbf1_n=\\begin{cases}1,&n=0,\\\\0,&n>0.\\end{cases}\n\\]\nHere $\\star$ is Hurwitz multiplication. The symbol $D^d$ denotes a coefficient\nshift, which agrees with formal differentiation after characteristic-zero\nencoding.","0,&n>0.\\end{cases}\n\\]\nHere $\\star$ is Hurwitz multiplication. The symbol $D^d$ denotes a coefficient\nshift, which agrees with formal differentiation after characteristic-zero\nencoding.","\\begin{theorem}[Hurwitz product rules]\n\\label{thm:hurwitz-product}\nFor $d\\geq0$,\n\\[\nD^d(f\\star g)=\\sum_{r=0}^{d}\\binom dr(D^r f)\\star(D^{d-r}g).\n\\]\nIf $R$ has prime characteristic $p$, then\n\\[\nD^p(f\\star g)=f\\star D^p g+D^p f\\star g.\n\\]\nOver a field, if $g_0=1$, a sequence $b$ is constructed recursively with\n$b\\star g=\\mathbf1$. If also $D^p g=Dg$ in characteristic $p$, then\n$D^p b=Db$.\n\\end{theorem}","\\begin{proof}\nPascal's identity proves the first product rule, and induction proves the\nhigher rule. The intermediate binomial coefficients $\\binom pr$ vanish in\ncharacteristic $p$. For the inverse, solve row $n$ using the coefficient of\n$b_n$, which is $g_0=1$:\n\\[\nb_n=\\mathbf1_n-\\sum_{r<n}\\binom nr b_r g_{n-r}.\n\\]\nSubtract the first and $p$th product rules for $b\\star g=\\mathbf1$.\nThe assumed identity for $g$ leaves\n$(D^p b-Db)\\star g=0$. A second triangular induction shows that convolution\nwith a sequence having constant coefficient $1$ has trivial kernel.\n\\end{proof}","f}\nPascal's identity proves the first product rule, and induction proves the\nhigher rule. The intermediate binomial coefficients $\\binom pr$ vanish in\ncharacteristic $p$. For the inverse, solve row $n$ using the coefficient of\n$b_n$, which is $g_0=1$:\n\\[\nb_n=\\mathbf1_n-\\sum_{r<n}\\binom nr b_r g_{n-r}.\n\\]\nSubtract the first and $p$th product rules for $b\\star g=\\mathbf1$.\nThe assumed identity for $g$ leaves\n$(D^p b-Db)\\star g=0$. A second triangular induction shows that convolution\nwith a sequence having constant coefficient $1$ has trivial kernel.\n\\end{proof}","identity proves the first product rule, and induction proves the\nhigher rule. The intermediate binomial coefficients $\\binom pr$ vanish in\ncharacteristic $p$. For the inverse, solve row $n$ using the coefficient of\n$b_n$, which is $g_0=1$:\n\\[\nb_n=\\mathbf1_n-\\sum_{r<n}\\binom nr b_r g_{n-r}.\n\\]\nSubtract the first and $p$th product rules for $b\\star g=\\mathbf1$.\nThe assumed identity for $g$ leaves\n$(D^p b-Db)\\star g=0$. A second triangular induction shows that convolution\nwith a sequence having constant coefficient $1$ has trivial kernel.\n\\end{proof}","binom pr$ vanish in\ncharacteristic $p$. For the inverse, solve row $n$ using the coefficient of\n$b_n$, which is $g_0=1$:\n\\[\nb_n=\\mathbf1_n-\\sum_{r<n}\\binom nr b_r g_{n-r}.\n\\]\nSubtract the first and $p$th product rules for $b\\star g=\\mathbf1$.\nThe assumed identity for $g$ leaves\n$(D^p b-Db)\\star g=0$. A second triangular induction shows that convolution\nwith a sequence having constant coefficient $1$ has trivial kernel.\n\\end{proof}","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{Hurwitz.shift_convolution}, \\lean{Hurwitz.prime_shift_convolution},\n\\lean{Hurwitz.convolution_left_eq_zero},\n\\lean{HurwitzInverse.inverse_convolution}, and\n\\lean{HurwitzInverse.inverse_prime_difference}.\nWorking directly with coefficient sequences avoids the undefined expression\n$1/n!$ in $\\F_p$ when $p\\mid n!$.\n\\par\\endgroup","\\begin{theorem}[Forward substitution and denominator clearing]\n\\label{thm:triangular-foundation}\nLet $T_{nq}$ be coefficients over a field with $T_{nn}\\ne0$ for every $n$.\nFor any sequence $b$, define\n\\[\na_n=\\frac{b_n-\\sum_{q<n}T_{nq}a_q}{T_{nn}}.\n\\]\nThis constructs the unique solution of\n$\\sum_{q=0}^{n}T_{nq}a_q=b_n$ for all $n$. If $T_{nq}$ and $b_n$ are integers\nand the solution is taken over $\\Q$, then\n\\[\n\\left(\\prod_{r=0}^{n}T_{rr}\\right)a_n\\in\\Z.\n\\]\n\\end{theorem}","em}[Forward substitution and denominator clearing]\n\\label{thm:triangular-foundation}\nLet $T_{nq}$ be coefficients over a field with $T_{nn}\\ne0$ for every $n$.\nFor any sequence $b$, define\n\\[\na_n=\\frac{b_n-\\sum_{q<n}T_{nq}a_q}{T_{nn}}.\n\\]\nThis constructs the unique solution of\n$\\sum_{q=0}^{n}T_{nq}a_q=b_n$ for all $n$. If $T_{nq}$ and $b_n$ are integers\nand the solution is taken over $\\Q$, then\n\\[\n\\left(\\prod_{r=0}^{n}T_{rr}\\right)a_n\\in\\Z.\n\\]\n\\end{theorem}","\\begin{proof}\nThe recursion uses only smaller indices, and substitution verifies each row.\nIf a second solution agrees through row $n-1$, subtracting row $n$ and\ncancelling $T_{nn}$ proves equality at $n$. For integrality, use induction and\npromote every earlier clearing product to $\\prod_{r<n}T_{rr}$. Multiplication\nby the additional factor $T_{nn}$ cancels the only new division.\n\\end{proof}","To isolate the filtration argument, let $p$ be any element of a commutative\nring and let $w:\\N\\to\\N\\cup\\{\\infty\\}$. Set $p^{[r]}=p^r$ for finite $r$\nand $p^{[\\infty]}=0$. The latter notation means the coefficient vanishes.","\\begin{theorem}[Filtered finite reconstruction]\n\\label{thm:filtered-foundation}\nFor every $n,L\\in\\N$ and sequences $c,S$ in that ring,\n\\[\n\\begin{split}\n&\\sum_{r=0}^{n}\\binom nr c_rp^{[w(r)]}S_{n-r}","\\begin{theorem}[Filtered finite reconstruction]\n\\label{thm:filtered-foundation}\nFor every $n,L\\in\\N$ and sequences $c,S$ in that ring,\n\\[\n\\begin{split}\n&\\sum_{r=0}^{n}\\binom nr c_rp^{[w(r)]}S_{n-r}\\\\\n&\\hspace{8mm}\\equiv\n\\sum_{\\substack{0\\leq r\\leq n\\\\w(r)<L}}\n\\binom nr c_rp^{[w(r)]}S_{n-r}\\pmod{p^L}.\n\\end{split}\n\\]\nHere congruence means divisibility of the difference by $p^L$ in the\ncoefficient ring. In $\\Zp$, it means membership in $p^L\\Zp$.\n\\end{theorem}","w(r)<L}}\n\\binom nr c_rp^{[w(r)]}S_{n-r}\\pmod{p^L}.\n\\end{split}\n\\]\nHere congruence means divisibility of the difference by $p^L$ in the\ncoefficient ring. In $\\Zp$, it means membership in $p^L\\Zp$.\n\\end{theorem}","\\begin{proof}\nThe difference of the two sums consists exactly of terms with $w(r)\\geq L$.\nEvery such term is divisible by $p^L$, including the zero terms with infinite\nweight. Their finite sum has the same divisibility.\n\\end{proof}","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{Filtered.filtered_triangular_reconstruction} and\n\\lean{Filtered.padic_filtered_reconstruction}.\nFor weights $2r$, retaining $r\\leq M$ gives the bound $p^{2M+2}$ through\n\\lean{Filtered.even_weight_tail}. A sharper kernel weight can improve this\nbound, as in the classical dyadic reconstruction later in the chapter. Each\ncoefficient row is finite, so no hypothesis $w(r)\\to\\infty$ is needed here.\n\\par\\endgroup","emph{Formal correspondence.}\n\\lean{Filtered.filtered_triangular_reconstruction} and\n\\lean{Filtered.padic_filtered_reconstruction}.\nFor weights $2r$, retaining $r\\leq M$ gives the bound $p^{2M+2}$ through\n\\lean{Filtered.even_weight_tail}. A sharper kernel weight can improve this\nbound, as in the classical dyadic reconstruction later in the chapter. Each\ncoefficient row is finite, so no hypothesis $w(r)\\to\\infty$ is needed here.\n\\par\\endgroup","For $f:\\N\\to R$, define the step-$h$ forward difference and its iterates by\n\\[\n\\Delta_h f(n)=f(n+h)-f(n),\\qquad \\Delta_h^0 f=f,\n\\]\nand put $c_r(f)=\\Delta_1^r f(0)$. Thus the coefficients are determined by the\ndifferences of $f$.","\\begin{theorem}[Finite Newton--Mahler identities]\n\\label{thm:mahler-foundation}\nFor every $n,s,h\\in\\N$,\n\\[\n\\begin{aligned}\nf(n)&=\\sum_{r=0}^{n}c_r(f)\\binom nr,","\\begin{theorem}[Finite Newton--Mahler identities]\n\\label{thm:mahler-foundation}\nFor every $n,s,h\\in\\N$,\n\\[\n\\begin{aligned}\nf(n)&=\\sum_{r=0}^{n}c_r(f)\\binom nr,\\\\\nc_r(\\Delta_1^s f)&=c_{s+r}(f),\\\\\n\\Delta_1^s f(n)&=\\sum_{j=0}^{n}c_{s+j}(f)\\binom nj,\\\\\n\\Delta_h^s f(n)&=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\end{aligned}\n\\]\nThe finite interpolant\n$I_Nf(n)=\\sum_{r=0}^{N}c_r(f)\\binom nr$ agrees with $f$ for $n\\leq N$.\nIt has the same order-$s$ unit-step difference at $n$ whenever $n+s\\leq N$.\nFor arbitrary coefficients $c_0,\\ldots,c_N$, the corresponding binomial\ninterpolant has exactly those coefficients and zero coefficients above $N$.\n\\end{theorem}","\\Delta_h^s f(n)&=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\end{aligned}\n\\]\nThe finite interpolant\n$I_Nf(n)=\\sum_{r=0}^{N}c_r(f)\\binom nr$ agrees with $f$ for $n\\leq N$.\nIt has the same order-$s$ unit-step difference at $n$ whenever $n+s\\leq N$.\nFor arbitrary coefficients $c_0,\\ldots,c_N$, the corresponding binomial\ninterpolant has exactly those coefficients and zero coefficients above $N$.\n\\end{theorem}","\\begin{proof}\nWrite $E_h f(n)=f(n+h)$ for the shift operator. The unit shift is\n$I+\\Delta_1$. Expanding its $n$th power and evaluating\nat zero gives reconstruction. Composing difference operators gives the\ncoefficient shift and applying reconstruction to $\\Delta_1^s f$ gives the\nthird identity. The binomial expansion of $(E_h-I)^s$ gives the explicit\nstencil. Binomial coefficients vanish when their lower index exceeds $n$,\nwhich proves the interpolation claim. Equality of every point of the stencil\nproves equality of differences. Repeated finite differences of $\\binom nr$\nat zero give the exact coefficient extraction.\n\\end{proof}","h f(n)=f(n+h)$ for the shift operator. The unit shift is\n$I+\\Delta_1$. Expanding its $n$th power and evaluating\nat zero gives reconstruction. Composing difference operators gives the\ncoefficient shift and applying reconstruction to $\\Delta_1^s f$ gives the\nthird identity. The binomial expansion of $(E_h-I)^s$ gives the explicit\nstencil. Binomial coefficients vanish when their lower index exceeds $n$,\nwhich proves the interpolation claim. Equality of every point of the stencil\nproves equality of differences. Repeated finite differences of $\\binom nr$\nat zero give the exact coefficient extraction.\n\\end{proof}","posing difference operators gives the\ncoefficient shift and applying reconstruction to $\\Delta_1^s f$ gives the\nthird identity. The binomial expansion of $(E_h-I)^s$ gives the explicit\nstencil. Binomial coefficients vanish when their lower index exceeds $n$,\nwhich proves the interpolation claim. Equality of every point of the stencil\nproves equality of differences. Repeated finite differences of $\\binom nr$\nat zero give the exact coefficient extraction.\n\\end{proof}","For $x\\in\\Zp$, extend the valuation to zero by setting\n\\[\n\\operatorname{ord}_p(x)=\n\\begin{cases}v_p(x),&x\\ne0,","For $x\\in\\Zp$, extend the valuation to zero by setting\n\\[\n\\operatorname{ord}_p(x)=\n\\begin{cases}v_p(x),&x\\ne0,\\\\\\infty,&x=0.\\end{cases}\n\\]\nThus $L\\leq\\operatorname{ord}_p(x)$ is equivalent to $x\\in p^L\\Zp$.","infty,&x=0.\\end{cases}\n\\]\nThus $L\\leq\\operatorname{ord}_p(x)$ is equivalent to $x\\in p^L\\Zp$.","\\begin{theorem}[Weighted difference bound]\n\\label{thm:weighted-mahler-foundation}\nLet $f:\\N\\to\\Zp$ and let $w:\\N\\to\\N\\cup\\{\\infty\\}$ satisfy\n$w(s+j)\\leq\\operatorname{ord}_p(c_{s+j}(f))$ for $0\\leq j\\leq n$.\nThen\n\\[\n\\operatorname{ord}_p(\\Delta_1^s f(n))\\geq\n\\min_{0\\leq j\\leq n}\n\\left\\{w(s+j)+v_p\\binom nj\\right\\}.\n\\]\n\\end{theorem}","\\begin{proof}\nApply the nonarchimedean inequality to the finite coefficient formula in\nTheorem~\\ref{thm:mahler-foundation}. For $0\\leq j\\leq n$, the integer\n$\\binom nj$ is positive. The order of its product with $c_{s+j}(f)$ is the\nsum of the two orders, including the case where that coefficient is zero.\n\\end{proof}","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{PadicBounds.nat_le_order_iff},\n\\lean{PadicBounds.weighted_difference_bound}, and\n\\lean{PadicBounds.weighted_difference_divisibility}.\nAn expansion on all of $\\Zp$ requires a convergence theorem in addition to\nthese finite algebraic identities and bounds.\n\\par\\endgroup","\\begin{theorem}[Differences of unit modes]\n\\label{thm:unit-modes-foundation}\nOver every commutative ring,\n\\[\n\\Delta_h^s(u^n)=u^n(u^h-1)^s.\n\\]\nFor any prime $p$, units $u_i\\in\\Zp^\\times$, coefficients $a_i\\in\\Zp$, and\na finite index set $I$, the sequence $f(n)=\\sum_{i\\in I}a_i u_i^n$ satisfies\n\\[\n\\Delta_{p-1}^s f(n)\\in p^s\\Zp\n\\qquad(n,s\\in\\N).\n\\]\n\\end{theorem}","\\begin{proof}\nInduction gives the geometric identity. Reduction of a unit to $\\F_p^\\times$\ngives $u_i^{p-1}=1$ modulo $p$. Taking $s$ powers supplies $p^s$ in each\nsummand. The difference operator is linear over finite sums.\n\\end{proof}","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{UnitModes.geometric_difference},\n\\lean{UnitModes.unit_fermat_divisibility}, and\n\\lean{UnitModes.finite_unit_modes_divisibility}.\nThe result includes $p=2$, where the displayed step is $1$. Its scope here\nis finite sums of modes.\n\\par\\endgroup","\\begin{theorem}[Finite unit coordinates]\n\\label{thm:finite-unit-foundation}\nLet $p$ be prime, $L\\geq0$, and\n$a\\in(\\Z/p^{L+1}\\Z)^\\times$. Define\n\\[\n\\omega_L(a)=a^{p^L},\\qquad \\langle a\\rangle_L=a\\omega_L(a)^{-1}.\n\\]\nThen\n\\[\na=\\omega_L(a)\\langle a\\rangle_L,\\qquad\n\\omega_L(a)^{p-1}=1,\\qquad \\langle a\\rangle_L\\equiv1\\pmod p.\n\\]\nThe two factors are uniquely determined by these properties and are compatible\nwith reduction to lower precision.\nFor $a\\in(\\Z/2^{L+2}\\Z)^\\times$, there is instead a unique decomposition\n\\[\na=\\epsilon(a)\\langle a\\rangle_4,\n\\qquad \\epsilon(a)\\in\\{1,-1\\},\\qquad\n\\langle a\\rangle_4\\equiv1\\pmod4.\n\\]\n\\end{theorem}","el{thm:finite-unit-foundation}\nLet $p$ be prime, $L\\geq0$, and\n$a\\in(\\Z/p^{L+1}\\Z)^\\times$. Define\n\\[\n\\omega_L(a)=a^{p^L},\\qquad \\langle a\\rangle_L=a\\omega_L(a)^{-1}.\n\\]\nThen\n\\[\na=\\omega_L(a)\\langle a\\rangle_L,\\qquad\n\\omega_L(a)^{p-1}=1,\\qquad \\langle a\\rangle_L\\equiv1\\pmod p.\n\\]\nThe two factors are uniquely determined by these properties and are compatible\nwith reduction to lower precision.\nFor $a\\in(\\Z/2^{L+2}\\Z)^\\times$, there is instead a unique decomposition\n\\[\na=\\epsilon(a)\\langle a\\rangle_4,\n\\qquad \\epsilon(a)\\in\\{1,-1\\},\\qquad\n\\langle a\\rangle_4\\equiv1\\pmod4.\n\\]\n\\end{theorem}","Z/p^{L+1}\\Z)^\\times$. Define\n\\[\n\\omega_L(a)=a^{p^L},\\qquad \\langle a\\rangle_L=a\\omega_L(a)^{-1}.\n\\]\nThen\n\\[\na=\\omega_L(a)\\langle a\\rangle_L,\\qquad\n\\omega_L(a)^{p-1}=1,\\qquad \\langle a\\rangle_L\\equiv1\\pmod p.\n\\]\nThe two factors are uniquely determined by these properties and are compatible\nwith reduction to lower precision.\nFor $a\\in(\\Z/2^{L+2}\\Z)^\\times$, there is instead a unique decomposition\n\\[\na=\\epsilon(a)\\langle a\\rangle_4,\n\\qquad \\epsilon(a)\\in\\{1,-1\\},\\qquad\n\\langle a\\rangle_4\\equiv1\\pmod4.\n\\]\n\\end{theorem}","\\begin{proof}\nThe unit-group exponent $p^L(p-1)$ gives the root identity for $\\omega_L$.\nIts residue is that of $a$, so the remaining factor has residue $1$.\nIf two $(p-1)$st roots have equal residue, raising them to $p^L$ makes them\nequal modulo $p^{L+1}$, while each root is fixed by this exponent. This proves\nuniqueness and then compatibility under reduction. In the dyadic construction,\nan odd residue is exactly $1$ or $-1$ modulo $4$. Choose that sign and multiply\n$a$ by it to obtain the principal factor. The two signs remain distinct\nmodulo $4$, which proves uniqueness.\n\\end{proof}","nit-group exponent $p^L(p-1)$ gives the root identity for $\\omega_L$.\nIts residue is that of $a$, so the remaining factor has residue $1$.\nIf two $(p-1)$st roots have equal residue, raising them to $p^L$ makes them\nequal modulo $p^{L+1}$, while each root is fixed by this exponent. This proves\nuniqueness and then compatibility under reduction. In the dyadic construction,\nan odd residue is exactly $1$ or $-1$ modulo $4$. Choose that sign and multiply\n$a$ by it to obtain the principal factor. The two signs remain distinct\nmodulo $4$, which proves uniqueness.\n\\end{proof}","up exponent $p^L(p-1)$ gives the root identity for $\\omega_L$.\nIts residue is that of $a$, so the remaining factor has residue $1$.\nIf two $(p-1)$st roots have equal residue, raising them to $p^L$ makes them\nequal modulo $p^{L+1}$, while each root is fixed by this exponent. This proves\nuniqueness and then compatibility under reduction. In the dyadic construction,\nan odd residue is exactly $1$ or $-1$ modulo $4$. Choose that sign and multiply\n$a$ by it to obtain the principal factor. The two signs remain distinct\nmodulo $4$, which proves uniqueness.\n\\end{proof}","$p^L(p-1)$ gives the root identity for $\\omega_L$.\nIts residue is that of $a$, so the remaining factor has residue $1$.\nIf two $(p-1)$st roots have equal residue, raising them to $p^L$ makes them\nequal modulo $p^{L+1}$, while each root is fixed by this exponent. This proves\nuniqueness and then compatibility under reduction. In the dyadic construction,\nan odd residue is exactly $1$ or $-1$ modulo $4$. Choose that sign and multiply\n$a$ by it to obtain the principal factor. The two signs remain distinct\nmodulo $4$, which proves uniqueness.\n\\end{proof}","for $\\omega_L$.\nIts residue is that of $a$, so the remaining factor has residue $1$.\nIf two $(p-1)$st roots have equal residue, raising them to $p^L$ makes them\nequal modulo $p^{L+1}$, while each root is fixed by this exponent. This proves\nuniqueness and then compatibility under reduction. In the dyadic construction,\nan odd residue is exactly $1$ or $-1$ modulo $4$. Choose that sign and multiply\n$a$ by it to obtain the principal factor. The two signs remain distinct\nmodulo $4$, which proves uniqueness.\n\\end{proof}","$1$.\nIf two $(p-1)$st roots have equal residue, raising them to $p^L$ makes them\nequal modulo $p^{L+1}$, while each root is fixed by this exponent. This proves\nuniqueness and then compatibility under reduction. In the dyadic construction,\nan odd residue is exactly $1$ or $-1$ modulo $4$. Choose that sign and multiply\n$a$ by it to obtain the principal factor. The two signs remain distinct\nmodulo $4$, which proves uniqueness.\n\\end{proof}","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{FiniteUnits.finite_unit_decomposition},\n\\lean{FiniteUnits.decomposition_unique},\n\\lean{UnitRefinement.omega_compatible},\n\\lean{UnitRefinement.principal_compatible}, and\n\\lean{DyadicUnits.dyadic_unit_decomposition} with\n\\lean{DyadicUnits.decomposition_unique}.\nThe finite $p=2$ version of the first construction has trivial root-of-unity\nfactor. The separate modulo-$4$ construction retains the sign that reduction\nmodulo $2$ cannot detect.\n\\par\\endgroup","mph{Formal correspondence.}\n\\lean{FiniteUnits.finite_unit_decomposition},\n\\lean{FiniteUnits.decomposition_unique},\n\\lean{UnitRefinement.omega_compatible},\n\\lean{UnitRefinement.principal_compatible}, and\n\\lean{DyadicUnits.dyadic_unit_decomposition} with\n\\lean{DyadicUnits.decomposition_unique}.\nThe finite $p=2$ version of the first construction has trivial root-of-unity\nfactor. The separate modulo-$4$ construction retains the sign that reduction\nmodulo $2$ cannot detect.\n\\par\\endgroup","t_decomposition},\n\\lean{FiniteUnits.decomposition_unique},\n\\lean{UnitRefinement.omega_compatible},\n\\lean{UnitRefinement.principal_compatible}, and\n\\lean{DyadicUnits.dyadic_unit_decomposition} with\n\\lean{DyadicUnits.decomposition_unique}.\nThe finite $p=2$ version of the first construction has trivial root-of-unity\nfactor. The separate modulo-$4$ construction retains the sign that reduction\nmodulo $2$ cannot detect.\n\\par\\endgroup","\\begin{theorem}[Companion jumps and packet recurrences]\n\\label{thm:companion-foundation}\nLet $C$ be a $d\\times d$ matrix over a commutative ring and\n$x_n=C^n x_0$. For $N,j,k\\in\\N$, put $b_k=x_{Nk+j}$. Then\n\\[\nb_k=C^j(C^N)^kx_0,\\qquad b_{k+1}=C^Nb_k.\n\\]\nIf $\\chi_{C^N}(X)=X^d+\\sum_{r<d}q_rX^r$, then\n\\[\nb_{k+d}=-\\sum_{r=0}^{d-1}q_r b_{k+r}.\n\\]\nEvery coordinate and every linear scalar observable satisfies the same\nrecurrence. No invertibility or diagonalization hypothesis is required.\n\\end{theorem}","\\begin{proof}\nThe jump identity follows from the power law for one matrix. Apply the\nCayley--Hamilton identity $\\chi_{C^N}(C^N)=0$ to the state $b_k$.\nThe leading coefficient is $1$, so the final state can be isolated. A linear\nobservable preserves the resulting finite sum.\n\\end{proof}","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{Companion.companion_jump},\n\\lean{Companion.packet_charpoly_recurrence}, and\n\\lean{Companion.packet_observable_recurrence}.\nThe source-sequence versions derive the packet recurrence from the relation\n$x_{n+1}=Cx_n$.\nFor the Fibonacci matrix $C=\\left(\\begin{smallmatrix}1&1","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{Companion.companion_jump},\n\\lean{Companion.packet_charpoly_recurrence}, and\n\\lean{Companion.packet_observable_recurrence}.\nThe source-sequence versions derive the packet recurrence from the relation\n$x_{n+1}=Cx_n$.\nFor the Fibonacci matrix $C=\\left(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\right)$,\nits jump trace is the Lucas number $L_N$ and its determinant is $(-1)^N$.\nThus\n\\[\nF_{N(k+2)+j}=L_NF_{N(k+1)+j}-(-1)^NF_{Nk+j}.\n\\]\nThis specialization is\n\\lean{FibonacciPackets.fibonacci_packet_recurrence}. It is derived from the\nFibonacci state.\n\\par\\endgroup","1&0\\end{smallmatrix}\\right)$,\nits jump trace is the Lucas number $L_N$ and its determinant is $(-1)^N$.\nThus\n\\[\nF_{N(k+2)+j}=L_NF_{N(k+1)+j}-(-1)^NF_{Nk+j}.\n\\]\nThis specialization is\n\\lean{FibonacciPackets.fibonacci_packet_recurrence}. It is derived from the\nFibonacci state.\n\\par\\endgroup","\\begin{theorem}[Dyadic--odd packet factorization]\n\\label{thm:crt-foundation}\nLet $a\\geq0$ and let $M$ be a positive odd integer. The Chinese remainder map\nis a ring isomorphism\n\\[\n\\Z/(2^aM)\\Z\\simeq\\Z/2^a\\Z\\times\\Z/M\\Z.\n\\]\nFor $0\\leq r<2^aM$ and any coefficient sequence $f$,\n\\[\nP_{2^aM,r}f\n=P_{2^a,r\\bmod 2^a}\\,P_{M,r\\bmod M}f.\n\\]\nThe same identity holds for formal-series coefficient selectors.\n\\end{theorem}","\\begingroup\\small\\raggedright\n\\noindent\\emph{Formal correspondence.}\n\\lean{CRTPackets.addressEquiv}, \\lean{CRTPackets.residue_equivalence},\n\\lean{CRTPackets.packet_factorization}, and\n\\lean{CRTPackets.series_packet_factorization}.\nThis theorem synchronizes finite arithmetic addresses. The passage from a\nfamily of $p$-adic objects to a global integer object is a separate question,\ndiscussed in Section~\\ref{subsec:native133}.\n\\par\\endgroup","The recovered Annals text develops a broader polyphase vocabulary. We explain\nit through elementary identities with explicit algebraic derivations. These\nidentities have not been separately formalized in Lean for this consolidation.\nFor a sequence $a$ and $M\\geq1$, let $a_r^{[M]}(q)=a_{Mq+r}$ for $0\\leq r<M$.\nIf $A(z)=\\sum_n a_nz^n$ and $A_r(w)=\\sum_q a_{Mq+r}w^q$ are ordinary generating\nfunctions, then unique quotient and remainder give\n\\[\nA(z)=\\sum_{r=0}^{M-1}z^rA_r(z^M),\\qquad\n\\bigl(a_r^{[M]}\\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.\n\\]\nFor ordinary Cauchy multiplication, the phase-$r$ product has a term without a\ncarry and a term with one carry:\n\\[\nC_r(w)=\\sum_{u+v=r}A_u(w)B_v(w)\n+w\\sum_{u+v=r+M}A_u(w)B_v(w),\\qquad 0\\leq u,v<M.\n\\]\nAdding the two phase addresses gives the formula. It describes ordinary\nCauchy multiplication. Hurwitz convolution has its own binomial weights.","polyphase vocabulary. We explain\nit through elementary identities with explicit algebraic derivations. These\nidentities have not been separately formalized in Lean for this consolidation.\nFor a sequence $a$ and $M\\geq1$, let $a_r^{[M]}(q)=a_{Mq+r}$ for $0\\leq r<M$.\nIf $A(z)=\\sum_n a_nz^n$ and $A_r(w)=\\sum_q a_{Mq+r}w^q$ are ordinary generating\nfunctions, then unique quotient and remainder give\n\\[\nA(z)=\\sum_{r=0}^{M-1}z^rA_r(z^M),\\qquad\n\\bigl(a_r^{[M]}\\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.\n\\]\nFor ordinary Cauchy multiplication, the phase-$r$ product has a term without a\ncarry and a term with one carry:\n\\[\nC_r(w)=\\sum_{u+v=r}A_u(w)B_v(w)\n+w\\sum_{u+v=r+M}A_u(w)B_v(w),\\qquad 0\\leq u,v<M.\n\\]\nAdding the two phase addresses gives the formula. It describes ordinary\nCauchy multiplication. Hurwitz convolution has its own binomial weights.","arately formalized in Lean for this consolidation.\nFor a sequence $a$ and $M\\geq1$, let $a_r^{[M]}(q)=a_{Mq+r}$ for $0\\leq r<M$.\nIf $A(z)=\\sum_n a_nz^n$ and $A_r(w)=\\sum_q a_{Mq+r}w^q$ are ordinary generating\nfunctions, then unique quotient and remainder give\n\\[\nA(z)=\\sum_{r=0}^{M-1}z^rA_r(z^M),\\qquad\n\\bigl(a_r^{[M]}\\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.\n\\]\nFor ordinary Cauchy multiplication, the phase-$r$ product has a term without a\ncarry and a term with one carry:\n\\[\nC_r(w)=\\sum_{u+v=r}A_u(w)B_v(w)\n+w\\sum_{u+v=r+M}A_u(w)B_v(w),\\qquad 0\\leq u,v<M.\n\\]\nAdding the two phase addresses gives the formula. It describes ordinary\nCauchy multiplication. Hurwitz convolution has its own binomial weights.","a$ and $M\\geq1$, let $a_r^{[M]}(q)=a_{Mq+r}$ for $0\\leq r<M$.\nIf $A(z)=\\sum_n a_nz^n$ and $A_r(w)=\\sum_q a_{Mq+r}w^q$ are ordinary generating\nfunctions, then unique quotient and remainder give\n\\[\nA(z)=\\sum_{r=0}^{M-1}z^rA_r(z^M),\\qquad\n\\bigl(a_r^{[M]}\\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.\n\\]\nFor ordinary Cauchy multiplication, the phase-$r$ product has a term without a\ncarry and a term with one carry:\n\\[\nC_r(w)=\\sum_{u+v=r}A_u(w)B_v(w)\n+w\\sum_{u+v=r+M}A_u(w)B_v(w),\\qquad 0\\leq u,v<M.\n\\]\nAdding the two phase addresses gives the formula. It describes ordinary\nCauchy multiplication. Hurwitz convolution has its own binomial weights.","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient ext","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the bi","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nEx","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The\nrecovered","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The\nrecovered infinite analytic extensions and stron","For $M\\geq1$ and $s,r,k\\in\\N$, define the affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The\nrecovered infinite analytic extensions and stronger historical identities\nlie beyond these finite derivations. Their sta","affine Mahler kernel by\n\\[\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s\\bigl((1+z)^M-1\\bigr)^k\n=\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The\nrecovered infinite analytic extensions and stronger historical identities\nlie beyond these finite derivations. Their status is discussed in\nSection~\\ref{subsec:historical-scope}","binom kj\\binom{Mj+s}{r}.\n\\]\nIt vanishes unless $k\\leq r\\leq Mk+s$. For a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The\nrecovered infinite analytic extensions and stronger historical identities\nlie beyond these finite derivations. Their status is discussed in\nSection~\\ref{subsec:historical-scope}.","or a finite binomial expansion\n$f(n)=\\sum_{r=0}^{D}c_r\\binom nr$, the $k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The\nrecovered infinite analytic extensions and stronger historical identities\nlie beyond these finite derivations. Their status is discussed in\nSection~\\ref{subsec:historical-scope}.","$k$th Newton coefficient of\n$g(q)=f(Mq+s)$ is therefore\n\\[\nc_k(g)=\\sum_{r=k}^{\\min(D,Mk+s)}c_r\\Theta_{M,s}(r,k).\n\\]\nIndeed, insert the finite stencil for $\\Delta_1^k g(0)$ and interchange its\ntwo finite sums. Coefficient extraction of the binomial expansion of\n$((1+z)^M-1)^k$ gives the kernel. In the binary zero-address case,\n\\[\n\\Theta_{2,0}(k+j,k)=2^{k-j}\\binom kj\\qquad(0\\leq j\\leq k),\n\\]\nsince $(2z+z^2)^k=z^k(2+z)^k$.\nExact valuations for the general kernel remain to be determined. The\nrecovered infinite analytic extensions and stronger historical identities\nlie beyond these finite derivations. Their status is discussed in\nSection~\\ref{subsec:historical-scope}.","Two related constructions run through this section.  The congruential\nEuler family is indexed by a modulus $N$ and an address $j$.  Its\nexceptional dyadic member has $N=4$ and four addresses.  The classical\nsigned Euler numbers arise from $N=2$, $j=0$.  Keeping these indices\nvisible distinguishes the four-packet period theorems from the classical\nedge truncations.","\\begin{definition}[Congruential Euler coefficients]\\label{def:arith-euler}\nFor $N\\ge1$ and $0\\le j<N$, define rational coefficients by the formal identity\n\\begin{equation}\\label{eq:arith-euler-egf}\n \\sum_{d\\ge0}E_d^{(N,j)}\\frac{t^d}{d!}\n =\\left(\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!}\\right)^{-1}.\n\\end{equation}\nThe reciprocal exists because its denominator has constant coefficient\n$1/j!\\ne0$.  Set\n\\[\n U_{N,j}(n)=E_{Nn}^{(N,j)},\\qquad C_{N,j}(n)=\\frac{U_{N,j}(n)}{j!}.\n\\]\nThus $C_{N,j}(0)=1$ and $U_{N,j}(0)=j!$.  All coefficients outside the\nmultiples of $N$ vanish.\n\\end{definition}","or $N\\ge1$ and $0\\le j<N$, define rational coefficients by the formal identity\n\\begin{equation}\\label{eq:arith-euler-egf}\n \\sum_{d\\ge0}E_d^{(N,j)}\\frac{t^d}{d!}\n =\\left(\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!}\\right)^{-1}.\n\\end{equation}\nThe reciprocal exists because its denominator has constant coefficient\n$1/j!\\ne0$.  Set\n\\[\n U_{N,j}(n)=E_{Nn}^{(N,j)},\\qquad C_{N,j}(n)=\\frac{U_{N,j}(n)}{j!}.\n\\]\nThus $C_{N,j}(0)=1$ and $U_{N,j}(0)=j!$.  All coefficients outside the\nmultiples of $N$ vanish.\n\\end{definition}","e j<N$, define rational coefficients by the formal identity\n\\begin{equation}\\label{eq:arith-euler-egf}\n \\sum_{d\\ge0}E_d^{(N,j)}\\frac{t^d}{d!}\n =\\left(\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!}\\right)^{-1}.\n\\end{equation}\nThe reciprocal exists because its denominator has constant coefficient\n$1/j!\\ne0$.  Set\n\\[\n U_{N,j}(n)=E_{Nn}^{(N,j)},\\qquad C_{N,j}(n)=\\frac{U_{N,j}(n)}{j!}.\n\\]\nThus $C_{N,j}(0)=1$ and $U_{N,j}(0)=j!$.  All coefficients outside the\nmultiples of $N$ vanish.\n\\end{definition}","\\begin{equation}\\label{eq:arith-euler-egf}\n \\sum_{d\\ge0}E_d^{(N,j)}\\frac{t^d}{d!}\n =\\left(\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!}\\right)^{-1}.\n\\end{equation}\nThe reciprocal exists because its denominator has constant coefficient\n$1/j!\\ne0$.  Set\n\\[\n U_{N,j}(n)=E_{Nn}^{(N,j)},\\qquad C_{N,j}(n)=\\frac{U_{N,j}(n)}{j!}.\n\\]\nThus $C_{N,j}(0)=1$ and $U_{N,j}(0)=j!$.  All coefficients outside the\nmultiples of $N$ vanish.\n\\end{definition}","The normalized source and its inverse are\n\\begin{equation}\\label{eq:arith-kernel}\n K_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n \\mathcal P_{N,j}(t)=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\end{equation}\nTheir product is one.  Clearing factorials in its coefficient of $t^{Nn}$\ngives the triangular system\n\\begin{equation}\\label{eq:arith-triangular}\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0}.\n\\end{equation}\nIn particular, for $n>0$,\n\\[\n C_{N,j}(n)=-\\binom{Nn+j}{Nn}^{-1}\n       \\sum_{q=0}^{n-1}\\binom{Nn+j}{Nq}C_{N,j}(q).\n\\]\nThe diagonal binomial coefficient is positive, so this recurrence determines\nthe sequence uniquely.  The Lean construction begins with this recurrence\nand derives the reciprocal identity from it.  The resulting sequence is\nthen identified with the degree-indexed coefficients of\n\\eqref{eq:arith-euler-egf}, so both descriptions refer to the same famil","The normalized source and its inverse are\n\\begin{equation}\\label{eq:arith-kernel}\n K_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n \\mathcal P_{N,j}(t)=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\end{equation}\nTheir product is one.  Clearing factorials in its coefficient of $t^{Nn}$\ngives the triangular system\n\\begin{equation}\\label{eq:arith-triangular}\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0}.\n\\end{equation}\nIn particular, for $n>0$,\n\\[\n C_{N,j}(n)=-\\binom{Nn+j}{Nn}^{-1}\n       \\sum_{q=0}^{n-1}\\binom{Nn+j}{Nq}C_{N,j}(q).\n\\]\nThe diagonal binomial coefficient is positive, so this recurrence determines\nthe sequence uniquely.  The Lean construction begins with this recurrence\nand derives the reciprocal identity from it.  The resulting sequence is\nthen identified with the degree-indexed coefficients of\n\\eqref{eq:arith-euler-egf}, so both descriptions refer to the same family.","ion}\\label{eq:arith-kernel}\n K_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n \\mathcal P_{N,j}(t)=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\end{equation}\nTheir product is one.  Clearing factorials in its coefficient of $t^{Nn}$\ngives the triangular system\n\\begin{equation}\\label{eq:arith-triangular}\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0}.\n\\end{equation}\nIn particular, for $n>0$,\n\\[\n C_{N,j}(n)=-\\binom{Nn+j}{Nn}^{-1}\n       \\sum_{q=0}^{n-1}\\binom{Nn+j}{Nq}C_{N,j}(q).\n\\]\nThe diagonal binomial coefficient is positive, so this recurrence determines\nthe sequence uniquely.  The Lean construction begins with this recurrence\nand derives the reciprocal identity from it.  The resulting sequence is\nthen identified with the degree-indexed coefficients of\n\\eqref{eq:arith-euler-egf}, so both descriptions refer to the same family.","el{eq:arith-kernel}\n K_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n \\mathcal P_{N,j}(t)=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\end{equation}\nTheir product is one.  Clearing factorials in its coefficient of $t^{Nn}$\ngives the triangular system\n\\begin{equation}\\label{eq:arith-triangular}\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0}.\n\\end{equation}\nIn particular, for $n>0$,\n\\[\n C_{N,j}(n)=-\\binom{Nn+j}{Nn}^{-1}\n       \\sum_{q=0}^{n-1}\\binom{Nn+j}{Nq}C_{N,j}(q).\n\\]\nThe diagonal binomial coefficient is positive, so this recurrence determines\nthe sequence uniquely.  The Lean construction begins with this recurrence\nand derives the reciprocal identity from it.  The resulting sequence is\nthen identified with the degree-indexed coefficients of\n\\eqref{eq:arith-euler-egf}, so both descriptions refer to the same family.","\\begin{proposition}[A denominator bound for every address]\n\\label{prop:arith-denominator}\nFor $N\\ge1$, $0\\le j<N$, and $n\\in\\N$, set\n\\[\n D_{N,j}(n)=\\prod_{r=1}^{n}\\binom{Nr+j}{Nr},\\qquad D_{N,j}(0)=1.\n\\]\nThen $D_{N,j}(n)C_{N,j}(n)\\in\\Z$.\n\\end{proposition}","\\begin{proof}\nProceed by induction in the recurrence \\eqref{eq:arith-triangular}.\nFor every $q<n$, the product $D_{N,j}(q)$ divides $D_{N,j}(n-1)$.\nMultiplying the recurrence by $D_{N,j}(n-1)$ therefore makes every\nprevious term integral.  The diagonal factor completes the product\n$D_{N,j}(n)$, proving the claim.  At $n=0$, both factors are one.\nThis gives a uniform denominator bound without requiring integrality of\nthe normalized coefficients themselves.  The Lean statements are\n\\lean{KomorebiPaperIX.Congruential.clearing_product} and\n\\lean{KomorebiPaperIX.Congruential.universal_denominator_clearing}.\n\\end{proof}","\\begin{example}[The Bernoulli boundary address]\\label{ex:arith-bernoulli}\nAt $N=2$, the address $j=1$ gives a second classical family.  Define the\nBernoulli numbers by\n\\[\n B(z)=\\frac{z}{e^z-1}=\\sum_{r\\ge0}B_r\\frac{z^r}{r!},\n \\qquad B_0=1,\\quad B_1=-\\frac12.\n\\]\nAll quotients here are regular formal series.  The identity\n\\[\n 2B(z)-B(2z)=\\frac{z}{\\sinh z}\n            =\\sum_{n\\ge0}C_{2,1}(n)\\frac{z^{2n}}{(2n)!}\n\\]\nfollows by putting the two fractions on a common denominator and using\n\\eqref{eq:arith-euler-egf}.  Comparing even coefficients yields\n\\[\n C_{2,1}(n)=(2-2^{2n})B_{2n}\\qquad(n\\in\\N).\n\\]\nThis classical derivation restores the boundary example from the original\nmanuscript.  No dedicated Lean bridge for this Bernoulli specialization\nwas identified in the preserved sour","\\begin{example}[The Bernoulli boundary address]\\label{ex:arith-bernoulli}\nAt $N=2$, the address $j=1$ gives a second classical family.  Define the\nBernoulli numbers by\n\\[\n B(z)=\\frac{z}{e^z-1}=\\sum_{r\\ge0}B_r\\frac{z^r}{r!},\n \\qquad B_0=1,\\quad B_1=-\\frac12.\n\\]\nAll quotients here are regular formal series.  The identity\n\\[\n 2B(z)-B(2z)=\\frac{z}{\\sinh z}\n            =\\sum_{n\\ge0}C_{2,1}(n)\\frac{z^{2n}}{(2n)!}\n\\]\nfollows by putting the two fractions on a common denominator and using\n\\eqref{eq:arith-euler-egf}.  Comparing even coefficients yields\n\\[\n C_{2,1}(n)=(2-2^{2n})B_{2n}\\qquad(n\\in\\N).\n\\]\nThis classical derivation restores the boundary example from the original\nmanuscript.  No dedicated Lean bridge for this Bernoulli specialization\nwas identified in the preserved sources.\n\\end{example}","oulli numbers by\n\\[\n B(z)=\\frac{z}{e^z-1}=\\sum_{r\\ge0}B_r\\frac{z^r}{r!},\n \\qquad B_0=1,\\quad B_1=-\\frac12.\n\\]\nAll quotients here are regular formal series.  The identity\n\\[\n 2B(z)-B(2z)=\\frac{z}{\\sinh z}\n            =\\sum_{n\\ge0}C_{2,1}(n)\\frac{z^{2n}}{(2n)!}\n\\]\nfollows by putting the two fractions on a common denominator and using\n\\eqref{eq:arith-euler-egf}.  Comparing even coefficients yields\n\\[\n C_{2,1}(n)=(2-2^{2n})B_{2n}\\qquad(n\\in\\N).\n\\]\nThis classical derivation restores the boundary example from the original\nmanuscript.  No dedicated Lean bridge for this Bernoulli specialization\nwas identified in the preserved sources.\n\\end{example}","\\begin{definition}[Differences and valuation conventions]\\label{def:arith-precision}\nFor a sequence $f$ and $h,k,n\\in\\N$, put\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad\n \\Delta_h^0 f(n)=f(n),\\qquad\n \\Delta_h^{k+1}f(n)=\\Delta_h^k f(n+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","ion conventions]\\label{def:arith-precision}\nFor a sequence $f$ and $h,k,n\\in\\N$, put\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad\n \\Delta_h^0 f(n)=f(n),\\qquad\n \\Delta_h^{k+1}f(n)=\\Delta_h^k f(n+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","\\label{def:arith-precision}\nFor a sequence $f$ and $h,k,n\\in\\N$, put\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad\n \\Delta_h^0 f(n)=f(n),\\qquad\n \\Delta_h^{k+1}f(n)=\\Delta_h^k f(n+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","equence $f$ and $h,k,n\\in\\N$, put\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad\n \\Delta_h^0 f(n)=f(n),\\qquad\n \\Delta_h^{k+1}f(n)=\\Delta_h^k f(n+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","put\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad\n \\Delta_h^0 f(n)=f(n),\\qquad\n \\Delta_h^{k+1}f(n)=\\Delta_h^k f(n+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","d\n \\Delta_h^{k+1}f(n)=\\Delta_h^k f(n+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","\\Delta_h^k f(n+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","+h)-\\Delta_h^k f(n).\n\\]\nWe use the extended valuation $v_p(0)=+\\infty$ in mathematical statements.\nThus $v_p(x)\\ge L$, including negative integers $L$, means\n$x=0$ or the ordinary valuation of the nonzero rational $x$ is at least $L$.\nFor integral coefficients and $L\\ge0$, it is equivalent to\n$x\\in p^L\\Z_p$.  In expressions involving a step multiplier we write\n$\\nu_p(u)=v_p(u)$ for $u>0$ and set $\\nu_p(0)=0$.  This finite convention\nmakes the displayed exponents meaningful at step zero, where all\npositive-order differences vanish.\n\\end{definition}","The valuation convention at zero is encoded by\n\\lean{KomorebiPaperIX.GlobalEulerBoundary.precision}.  For $p$-adic\ncongruences, divisibility is interpreted in $\\Z_p$. Statements over other\ncoefficient rings specify the ring explicitly.  Field divisibility in\n$\\Q_p$ alone does not encode precision.","\\begin{proposition}[Lucas reduction and address independence]\n\\label{prop:arith-lucas-window}\nLet $p$ be prime, $m>0$, and $N=mp$.  For $0\\le j<p$ and $n,q\\in\\N$,\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p.\n\\]\nEvery $C_{mp,j}(n)$ belongs to $\\Z_p$.  Its reduction is the unique\nsequence $\\overline C_m:\\N\\to\\F_p$ satisfying\n\\[\n \\sum_{q=0}^{n}\\binom{mn}{mq}\\overline C_m(q)=\\delta_{n0}\n \\quad\\text{in }\\F_p.\n\\]\nConsequently, for any two addresses $0\\le j,k<p$,\n\\[\n C_{mp,j}(n)\\equiv C_{mp,k}(n)\\pmod p\\qquad(n\\in\\N).\n\\]\n\\end{proposition}","ndence]\n\\label{prop:arith-lucas-window}\nLet $p$ be prime, $m>0$, and $N=mp$.  For $0\\le j<p$ and $n,q\\in\\N$,\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p.\n\\]\nEvery $C_{mp,j}(n)$ belongs to $\\Z_p$.  Its reduction is the unique\nsequence $\\overline C_m:\\N\\to\\F_p$ satisfying\n\\[\n \\sum_{q=0}^{n}\\binom{mn}{mq}\\overline C_m(q)=\\delta_{n0}\n \\quad\\text{in }\\F_p.\n\\]\nConsequently, for any two addresses $0\\le j,k<p$,\n\\[\n C_{mp,j}(n)\\equiv C_{mp,k}(n)\\pmod p\\qquad(n\\in\\N).\n\\]\n\\end{proposition}","prop:arith-lucas-window}\nLet $p$ be prime, $m>0$, and $N=mp$.  For $0\\le j<p$ and $n,q\\in\\N$,\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p.\n\\]\nEvery $C_{mp,j}(n)$ belongs to $\\Z_p$.  Its reduction is the unique\nsequence $\\overline C_m:\\N\\to\\F_p$ satisfying\n\\[\n \\sum_{q=0}^{n}\\binom{mn}{mq}\\overline C_m(q)=\\delta_{n0}\n \\quad\\text{in }\\F_p.\n\\]\nConsequently, for any two addresses $0\\le j,k<p$,\n\\[\n C_{mp,j}(n)\\equiv C_{mp,k}(n)\\pmod p\\qquad(n\\in\\N).\n\\]\n\\end{proposition}","\\begin{proof}\nLucas reduction separates the lowest base-$p$ digits, which are $j$ and\nzero in the upper and lower arguments.  Their binomial contribution is\none, leaving the displayed reduced row.  On the diagonal the reduced\ncoefficient is one.  The recurrence \\eqref{eq:arith-triangular} therefore\nconstructs an integral packet by successive division by units in $\\Z_p$.\nAfter reduction, every address satisfies the same triangular system over\n$\\F_p$.  Its diagonal entries are one, so its solution is unique.  This\nproves both the common reduction and the congruence between addresses.\nThe row reduction is\n\\lean{KomorebiPaperIX.StandardWindow.lucas_row_reduction}.\nThe integral identification and the two reduction statements are\n\\lean{KomorebiPaperIX.Wi","\\begin{proof}\nLucas reduction separates the lowest base-$p$ digits, which are $j$ and\nzero in the upper and lower arguments.  Their binomial contribution is\none, leaving the displayed reduced row.  On the diagonal the reduced\ncoefficient is one.  The recurrence \\eqref{eq:arith-triangular} therefore\nconstructs an integral packet by successive division by units in $\\Z_p$.\nAfter reduction, every address satisfies the same triangular system over\n$\\F_p$.  Its diagonal entries are one, so its solution is unique.  This\nproves both the common reduction and the congruence between addresses.\nThe row reduction is\n\\lean{KomorebiPaperIX.StandardWindow.lucas_row_reduction}.\nThe integral identification and the two reduction statements are\n\\lean{KomorebiPaperIX.WindowIndependence.stand","\\begin{proof}\nLucas reduction separates the lowest base-$p$ digits, which are $j$ and\nzero in the upper and lower arguments.  Their binomial contribution is\none, leaving the displayed reduced row.  On the diagonal the reduced\ncoefficient is one.  The recurrence \\eqref{eq:arith-triangular} therefore\nconstructs an integral packet by successive division by units in $\\Z_p$.\nAfter reduction, every address satisfies the same triangular system over\n$\\F_p$.  Its diagonal entries are one, so its solution is unique.  This\nproves both the common reduction and the congruence between addresses.\nThe row reduction is\n\\lean{KomorebiPaperIX.StandardWindow.lucas_row_reduction}.\nThe integral identification and the two reduction statements are\n\\lean{KomorebiPaperIX.WindowIndependence.standard_window_integral},\n\\lean{KomorebiPaperIX.WindowIndependence.coeffPadic_coe},\n\\lean{KomorebiPaperIX.WindowIndependence.residue_independence}, and\n\\lean{KomorebiPaperIX.WindowIndependence.two_window_addresses_congruent}.\n\\end{proof}","est base-$p$ digits, which are $j$ and\nzero in the upper and lower arguments.  Their binomial contribution is\none, leaving the displayed reduced row.  On the diagonal the reduced\ncoefficient is one.  The recurrence \\eqref{eq:arith-triangular} therefore\nconstructs an integral packet by successive division by units in $\\Z_p$.\nAfter reduction, every address satisfies the same triangular system over\n$\\F_p$.  Its diagonal entries are one, so its solution is unique.  This\nproves both the common reduction and the congruence between addresses.\nThe row reduction is\n\\lean{KomorebiPaperIX.StandardWindow.lucas_row_reduction}.\nThe integral identification and the two reduction statements are\n\\lean{KomorebiPaperIX.WindowIndependence.standard_window_integral},\n\\lean{KomorebiPaperIX.WindowIndependence.coeffPadic_coe},\n\\lean{KomorebiPaperIX.WindowIndependence.residue_independence}, and\n\\lean{KomorebiPaperIX.WindowIndependence.two_window_addresses_congruent}.\n\\end{proof}","\\begin{theorem}[Nishibuchi congruence]\\label{thm:arith-nishibuchi}\nLet $p$ be prime, let $m>0$, and suppose that $m\\mid p-1$ or $m=2$.\nPut $N=mp$ and $q=\\operatorname{lcm}(2,p-1)$.  For every $0\\le j<N$,\nall $U_{N,j}(n)$ belong to $\\Z_p$.  For $r\\ge1$ and $n\\ge r$,\n\\begin{equation}\\label{eq:arith-nishibuchi}\n E_{Nn+qp^r}^{(N,j)}\\equiv E_{Nn}^{(N,j)}\\pmod{p^r}.\n\\end{equation}\nThus the positive sufficient starting index $n_0=r$ works uniformly.\nIf $m\\mid p-1$, the stronger sufficient condition $r\\le mn$ is valid.\n\\end{theorem}","chi congruence]\\label{thm:arith-nishibuchi}\nLet $p$ be prime, let $m>0$, and suppose that $m\\mid p-1$ or $m=2$.\nPut $N=mp$ and $q=\\operatorname{lcm}(2,p-1)$.  For every $0\\le j<N$,\nall $U_{N,j}(n)$ belong to $\\Z_p$.  For $r\\ge1$ and $n\\ge r$,\n\\begin{equation}\\label{eq:arith-nishibuchi}\n E_{Nn+qp^r}^{(N,j)}\\equiv E_{Nn}^{(N,j)}\\pmod{p^r}.\n\\end{equation}\nThus the positive sufficient starting index $n_0=r$ works uniformly.\nIf $m\\mid p-1$, the stronger sufficient condition $r\\le mn$ is valid.\n\\end{theorem}","ith-nishibuchi}\nLet $p$ be prime, let $m>0$, and suppose that $m\\mid p-1$ or $m=2$.\nPut $N=mp$ and $q=\\operatorname{lcm}(2,p-1)$.  For every $0\\le j<N$,\nall $U_{N,j}(n)$ belong to $\\Z_p$.  For $r\\ge1$ and $n\\ge r$,\n\\begin{equation}\\label{eq:arith-nishibuchi}\n E_{Nn+qp^r}^{(N,j)}\\equiv E_{Nn}^{(N,j)}\\pmod{p^r}.\n\\end{equation}\nThus the positive sufficient starting index $n_0=r$ works uniformly.\nIf $m\\mid p-1$, the stronger sufficient condition $r\\le mn$ is valid.\n\\end{theorem}","d suppose that $m\\mid p-1$ or $m=2$.\nPut $N=mp$ and $q=\\operatorname{lcm}(2,p-1)$.  For every $0\\le j<N$,\nall $U_{N,j}(n)$ belong to $\\Z_p$.  For $r\\ge1$ and $n\\ge r$,\n\\begin{equation}\\label{eq:arith-nishibuchi}\n E_{Nn+qp^r}^{(N,j)}\\equiv E_{Nn}^{(N,j)}\\pmod{p^r}.\n\\end{equation}\nThus the positive sufficient starting index $n_0=r$ works uniformly.\nIf $m\\mid p-1$, the stronger sufficient condition $r\\le mn$ is valid.\n\\end{theorem}","\\begin{theorem}[All-address differences on the divisor branch]\n\\label{thm:arith-divisor-base}\nLet $p$ be prime, $m>0$, $m\\mid p-1$, $N=mp$, and $0\\le j<N$.\nWrite $h=(p-1)/m$.  For every $u,k,n\\in\\N$,\n\\begin{align}\n U_{N,j}(n)&\\in\\Z_p,\\label{eq:arith-divisor-integral}","\\begin{theorem}[All-address differences on the divisor branch]\n\\label{thm:arith-divisor-base}\nLet $p$ be prime, $m>0$, $m\\mid p-1$, $N=mp$, and $0\\le j<N$.\nWrite $h=(p-1)/m$.  For every $u,k,n\\in\\N$,\n\\begin{align}\n U_{N,j}(n)&\\in\\Z_p,\\label{eq:arith-divisor-integral}\\\\\n \\Delta_{uh}^{k}U_{N,j}(n)&\\in p^{\\min(k,mn)}\\Z_p,\\label{eq:arith-divisor-bound}\\\\\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\n   &\\ge\\min(k,mn)-v_p(j!).\\label{eq:arith-divisor-normalized}\n\\end{align}\nThe factorial correction is part of the normalized statement, including\naddresses $j\\ge p$.\n\\end{theorem}","v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\n   &\\ge\\min(k,mn)-v_p(j!).\\label{eq:arith-divisor-normalized}\n\\end{align}\nThe factorial correction is part of the normalized statement, including\naddresses $j\\ge p$.\n\\end{theorem}","\\begin{proof}\nWork in the Hurwitz integral ring of formal series\n\\[\n \\mathcal H_p=\\left\\{F\\in\\Q_p[[t]]:\n            d![t^d]F\\in\\Z_p\\text{ for every }d\\right\\}.\n\\]\nFor $D=d/dt$, let $B_p=(D^{p^2}-D^p)/p$.  Call $F$ flat if\n$B_p^sF\\in\\mathcal H_p$ for every $s\\ge0$.\nWe will prove that $j!\\mathcal P_{N,j}$ is flat at every address.  The main\ndifficulty is to include the addresses at which the usual unit-pivot\nargument fails.","Set\n\\[\n H_c(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\qquad\n A_c(t)=\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+c)!},\\qquad\n R_c(t)=\\frac{A_{c-1}(t)}{A_c(t)}\\quad(1\\le c<N).\n\\]\nThe denominator of this last quotient has nonzero constant coefficient.\nThe section identities give\n\\[\n R_cH_c=tH_{c-1},\\qquad R_c(0)=c,\\qquad\n j!\\mathcal P_{N,j}=\\mathcal P_{N,0}\\prod_{c=1}^{j}R_c.\n\\]\nCancellation of consecutive source sections proves the product identity\nand ties the ratios directly to the family under study.","At the singular addresses $c=ap$, $1\\le a<m$, use the weighted ring\n\\[\n \\mathcal W_p=\\left\\{\\sum_{d\\ge0}a_dT^d:\n          p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\right\\}.\n\\]\nThe key estimate is $R_{ap}(\\log(1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the","At the singular addresses $c=ap$, $1\\le a<m$, use the weighted ring\n\\[\n \\mathcal W_p=\\left\\{\\sum_{d\\ge0}a_dT^d:\n          p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\right\\}.\n\\]\nThe key estimate is $R_{ap}(\\log(1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\","At the singular addresses $c=ap$, $1\\le a<m$, use the weighted ring\n\\[\n \\mathcal W_p=\\left\\{\\sum_{d\\ge0}a_dT^d:\n          p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\right\\}.\n\\]\nThe key estimate is $R_{ap}(\\log(1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates wi","At the singular addresses $c=ap$, $1\\le a<m$, use the weighted ring\n\\[\n \\mathcal W_p=\\left\\{\\sum_{d\\ge0}a_dT^d:\n          p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\right\\}.\n\\]\nThe key estimate is $R_{ap}(\\log(1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","sses $c=ap$, $1\\le a<m$, use the weighted ring\n\\[\n \\mathcal W_p=\\left\\{\\sum_{d\\ge0}a_dT^d:\n          p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\right\\}.\n\\]\nThe key estimate is $R_{ap}(\\log(1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","_p=\\left\\{\\sum_{d\\ge0}a_dT^d:\n          p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\right\\}.\n\\]\nThe key estimate is $R_{ap}(\\log(1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","\\in\\Z_p\\right\\}.\n\\]\nThe key estimate is $R_{ap}(\\log(1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","1+T))\\in\\mathcal W_p$.  Its arithmetic\ninput comes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","mes from the cyclic group ring $\\Z[\\Z/N\\Z]$.\nFor its canonical element $x$ and $I=(p,x^{p-1}-1)$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","$, the relation\n$(x^{p-1})^p=1$ implies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","mplies $I^p\\subseteq(p)I$.  Induction yields\n$I^q\\subseteq(p^{v_p(q!)})$.  Grouping the falling factorial\n$(x)_d=\\prod_{i=0}^{d-1}(x-i)$ into consecutive blocks of length $p$\nthen gives an integral residue-coordinate factor\n$p^{v_p(\\lfloor d/p\\rfloor!)}$.  The identity\n\\[\n v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)\n\\]\naccounts for precisely the weight in $\\mathcal W_p$ after dividing by $d!$.\nComparison of the section and group-ring recurrences identifies these\ncoordinates with the logarithmically substituted section.","To pass from the section to the singular ratio, factor out its initial zero:\nfor $c=ap$, the section has the form\n\\[\n H_c(\\log(1+T))=\\frac{T^c}{c!}\\,V(T),\\qquad V(0)=1,\n \\qquad V\\in\\mathcal W_p.\n\\]\nThe weight of $c!$ compensates for the coefficient shift by $c$.\nThe weighted ring is closed under inversion of series with constant\ncoefficient one and under $L_T=\\log(1+T)(1+T)d/dT$.  With\n$\\ell(T)=\\log(1+T)/T$, interpreted by coefficient shifting, differentiation\nof the displayed factorization gives\n\\[\n R_c(\\log(1+T))=c(1+T)\\ell(T)+(L_TV)V^{-1}.\n\\]\nSince $p\\ell(T)\\in\\mathcal W_p$ and $p\\mid c$, both terms are weighted.\nThis is the singular estimate\n\\lean{KomorebiPaperIX.SingularRatioAmice.paper_singular_amice_estimate}.","the form\n\\[\n H_c(\\log(1+T))=\\frac{T^c}{c!}\\,V(T),\\qquad V(0)=1,\n \\qquad V\\in\\mathcal W_p.\n\\]\nThe weight of $c!$ compensates for the coefficient shift by $c$.\nThe weighted ring is closed under inversion of series with constant\ncoefficient one and under $L_T=\\log(1+T)(1+T)d/dT$.  With\n$\\ell(T)=\\log(1+T)/T$, interpreted by coefficient shifting, differentiation\nof the displayed factorization gives\n\\[\n R_c(\\log(1+T))=c(1+T)\\ell(T)+(L_TV)V^{-1}.\n\\]\nSince $p\\ell(T)\\in\\mathcal W_p$ and $p\\mid c$, both terms are weighted.\nThis is the singular estimate\n\\lean{KomorebiPaperIX.SingularRatioAmice.paper_singular_amice_estimate}.","d V(0)=1,\n \\qquad V\\in\\mathcal W_p.\n\\]\nThe weight of $c!$ compensates for the coefficient shift by $c$.\nThe weighted ring is closed under inversion of series with constant\ncoefficient one and under $L_T=\\log(1+T)(1+T)d/dT$.  With\n$\\ell(T)=\\log(1+T)/T$, interpreted by coefficient shifting, differentiation\nof the displayed factorization gives\n\\[\n R_c(\\log(1+T))=c(1+T)\\ell(T)+(L_TV)V^{-1}.\n\\]\nSince $p\\ell(T)\\in\\mathcal W_p$ and $p\\mid c$, both terms are weighted.\nThis is the singular estimate\n\\lean{KomorebiPaperIX.SingularRatioAmice.paper_singular_amice_estimate}.","W_p.\n\\]\nThe weight of $c!$ compensates for the coefficient shift by $c$.\nThe weighted ring is closed under inversion of series with constant\ncoefficient one and under $L_T=\\log(1+T)(1+T)d/dT$.  With\n$\\ell(T)=\\log(1+T)/T$, interpreted by coefficient shifting, differentiation\nof the displayed factorization gives\n\\[\n R_c(\\log(1+T))=c(1+T)\\ell(T)+(L_TV)V^{-1}.\n\\]\nSince $p\\ell(T)\\in\\mathcal W_p$ and $p\\mid c$, both terms are weighted.\nThis is the singular estimate\n\\lean{KomorebiPaperIX.SingularRatioAmice.paper_singular_amice_estimate}.","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intert","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, bec","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.Weighted","To deduce flatness from the weighted estimate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","mate, use\n$\\Theta=(1+T)d/dT$.  Both $\\Theta$ and\n$(\\Theta^{p^2}-\\Theta^p)/p$ preserve $\\mathcal W_p$.\nFor the first operator, its coefficient formula is\n$(d+1)a_{d+1}+da_d$. Whenever the floor weight rises, the factor $d+1$\nprovides the missing $p$.  For the second operator, the proof uses its\nfull Leibniz correction and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","ection and the scalar congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","congruence\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","e\n$p^2\\mid b^{p^2}-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","-b^p$.  Weighted monomials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","omials factor as\n\\[\n a_dT^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","^d=(p^{\\lfloor d/p\\rfloor}a_d)\n          (T^p/p)^{\\lfloor d/p\\rfloor}T^{d\\bmod p}.\n\\]\nThe operator preserves these monomials and hence finite weighted\npolynomials.  Each fixed output coefficient depends on only finitely many\ninput coefficients, which extends the result to the formal series.\nSubstitution $T=e^t-1$ intertwines $D$ with $\\Theta$ and maps\n$\\mathcal W_p$ into $\\mathcal H_p$, because\n\\[\n d![t^d]F(e^t-1)=[T^0]\\Theta^dF.\n\\]\nIt follows that the singular ratios are flat.  This transfer is proved in\n\\lean{KomorebiPaperIX.WeightedDworkTransfer.singular_ratio_flat}.","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On th","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the d","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrali","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the in","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserve","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n S","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two ri","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its se","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the","At a nonsingular address $2\\le c<N$ with $p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteratio","$p\\nmid c$, differentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery…","fferentiation of the section\nidentities gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient o…","s gives\n\\[\n tR_c'=R_c(1+R_{c-1}-R_c).\n\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from…","\\]\nWriting $q=R_{c-1}-(c-1)$ and $y=R_c-c$, this becomes\n\\[\n (tD+c)y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  …","y=cq+yq-y^2.\n\\]\nOn degrees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{Komor…","ees divisible by $p$, the inverse of $tD+c$ divides the degree-$d$\ncoefficient by the unit $d+c$.  Starting with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatP…","ing with $y_0=0$, form\n\\[\n y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.","-1}(cq+y_sq-y_s^2).\n\\]\nFlatness and support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.","support are preserved. Indeed, put $E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.","$E_c=tD+c$ and $C_p=pD^{p^2}-D^p$.\nThe commutator is $[B_p,E_c]=C_p$, and iteration gives\n\\[\n B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.","r-1}y.\n\\]\nFor the inverse equation $E_cy=f$, induction on $r$ makes the\ncorrection term Hurwitz-integral. Both operators preserve support in\ndegrees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.","grees divisible by $p$. On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.",". On that support the degree multiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.","ltiplier $d+c$\nis a unit, so integrality of $E_cB_p^ry$ implies integrality of\n$B_p^ry$. Thus the inverse preserves full flatness, as well as support.\n Since the difference of two right\nsides factors as $(z-y)(q-z-y)$ and its second factor has zero constant\nterm, agreement with the solution gains one coefficient at each iteration.\nConsequently $[t^d]y_s=[t^d]y$ for $d\\le s$.\nEvery coefficient of $B_p^r y$ can therefore be read from a sufficiently\nlong finite iterate.  The argument is entirely formal.\nThis is the content of\n\\lean{KomorebiPaperIX.RiccatiFlatPropagation.riccati_solution_flat}.","The standard window supplies the initial ratio $R_1$.  The weighted\nestimate handles $p\\mid c$, and Riccati propagation handles $p\\nmid c$.\nFlatness is closed under products, so the exact factorization establishes\nflatness of $j!\\mathcal P_{N,j}$.  Extracting its supported Hurwitz coefficients\nand applying the proved saturation and step-multiple lemmas yields\n\\eqref{eq:arith-divisor-integral}--\\eqref{eq:arith-divisor-bound}.","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\D","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalize","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retai","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resul","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements a","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllRe","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiP","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalize","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, an","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.Al","Here is the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_preci","s the coefficient extraction explicitly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","ly. If $a_d=d![t^d]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","]F$,\nthen\n\\[\n d![t^d]B_p^kF=p^{-k}\\Delta_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","_{p(p-1)}^ka_{d+pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","pk}.\n\\]\nFor $F=j!\\mathcal P_{N,j}$ and $k\\le mn$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","n$, take $d=mpn-pk$.\nSupport in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","in multiples of $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","f $mp$ turns this into\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","o\n$p^{-k}\\Delta_h^kU_{N,j}(n)$, so flatness gives divisibility by $p^k$.\nFor $k>mn$, expand $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","d $\\Delta_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","a_h^{k-mn}$ applied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","pplied to\n$\\Delta_h^{mn}U_{N,j}(n)$. Every shifted row is at least $n$, so\neach summand is divisible by $p^{mn}$. Finally\n$\\Delta_{uh}=((I+\\Delta_h)^u-I)$, whose $k$th power for $u>0$\nis an integral combination of powers of $\\Delta_h$ of order at least $k$.\nThe case $u=0$ gives zero for $k>0$, and order zero is the original\nintegral coefficient. This proves the bound for all $u$ and $k$.\nFinally $\\Delta_{uh}^kU_{N,j}=j!\\Delta_{uh}^kC_{N,j}$ gives\n\\eqref{eq:arith-divisor-normalized}, with the zero case retained.\nThe resulting statements are\n\\lean{KomorebiPaperIX.AllResidueEuler.all_residue_flat},\n\\lean{KomorebiPaperIX.AllResidueEuler.global_unnormalized_bound}, and\n\\lean{KomorebiPaperIX.AllResidueEuler.global_normalized_precision}.\n\\end{proof}","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exa","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact valu","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutat","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutation of a universal bound without loss\nare formalized in\n\\lean{KomorebiPaperIX.Normalization.normalized_first},\n\\lean{KomorebiPaperIX.N","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutation of a universal bound without loss\nare formalized in\n\\lean{KomorebiPaperIX.Normalization.normalized_first},\n\\lean{KomorebiPaperIX.Normalization.normalized_secon","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutation of a universal bound without loss\nare formalized in\n\\lean{KomorebiPaperIX.Normalization.normalized_first},\n\\lean{KomorebiPaperIX.Normalization.normalized_second},\n\\lean{KomorebiPaperIX.Normali","\\begin{example}[Why the factorial loss is necessary]\n\\label{ex:arith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutation of a universal bound without loss\nare formalized in\n\\lean{KomorebiPaperIX.Normalization.normalized_first},\n\\lean{KomorebiPaperIX.Normalization.normalized_second},\n\\lean{KomorebiPaperIX.Normalization.normal","ith-normalization-loss}\nTake $p=3$, $m=2$, $N=6$, and $j=3$.  The recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutation of a universal bound without loss\nare formalized in\n\\lean{KomorebiPaperIX.Normalization.normalized_first},\n\\lean{KomorebiPaperIX.Normalization.normalized_second},\n\\lean{KomorebiPaperIX.Normalization.normalized_difference},\n\\lean{KomorebiPaperIX.Normalization.difference_valuation},\n\\lean{KomorebiPaperIX.Normalization.factorial_loss_is_sh…","ce gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutation of a universal bound without loss\nare formalized in\n\\lean{KomorebiPaperIX.Normalization.normalized_first},\n\\lean{KomorebiPaperIX.Normalization.normalized_second},\n\\lean{KomorebiPaperIX.Normalization.normalized_difference},\n\\lean{KomorebiPaperIX.Normalization.difference_valuation},\n\\lean{KomorebiPaperIX.Normalization.factorial_loss_is_sharp},\n\\lean{KomorebiPaperIX.Normalization.unnormalized_difference},\n\\lean{…","C_{6,3}(1)=-\\frac1{84},\\qquad\n C_{6,3}(2)=\\frac{703}{5460},\\qquad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\nThe last difference has $3$-adic valuation zero.  Here $h=1$, so at\n$n=1$ the order-one bound without factorial loss would require valuation\nat least one.  The corrected normalized bound is sharp, since\n\\[\n v_3\\bigl(C_{6,3}(2)-C_{6,3}(1)\\bigr)=0=1-v_3(3!).\n\\]\nMultiplication by $3!$ gives\n\\[\n U_{6,3}(2)-U_{6,3}(1)=\\frac{384}{455},\\qquad\n v_3\\bigl(U_{6,3}(2)-U_{6,3}(1)\\bigr)=1,\n\\]\nconsistent with the unnormalized theorem.  Thus the factorial correction\nis already necessary at a single step in the all-address parameter range.\nThese exact values and the refutation of a universal bound without loss\nare formalized in\n\\lean{KomorebiPaperIX.Normalization.normalized_first},\n\\lean{KomorebiPaperIX.Normalization.normalized_second},\n\\lean{KomorebiPaperIX.Normalization.normalized_difference},\n\\lean{KomorebiPaperIX.Normalization.difference_valuation},\n\\lean{KomorebiPaperIX.Normalization.factorial_loss_is_sharp},\n\\lean{KomorebiPaperIX.Normalization.unnormalized_difference},\n\\lean{KomorebiPaper…","\\begin{corollary}[A common period in the standard window]\n\\label{cor:arith-window-period}\nLet $p$ be prime, $m>0$, $m\\mid p-1$, and $0\\le j<p$.  Put\n$h=(p-1)/m$.  For every $n\\ge1$ and $u\\in\\N$,\n\\[\n \\overline C_m(n+uh)=\\overline C_m(n),\\qquad\n C_{mp,j}(n+uh)\\equiv C_{mp,j}(n)\\pmod p.\n\\]\nThus $h$ is a sufficient period from row one, common to all addresses\nin the standard window.  It need not be the least period.\n\\end{corollary}","\\begin{proof}\nSince $j<p$, the factorial $j!$ is a unit in $\\Z_p$.  Also $mn\\ge1$.\nThe order-one case of Theorem~\\ref{thm:arith-divisor-base} therefore gives\nvaluation at least one for the normalized difference at every step $uh$.\nThis proves the congruence, and Proposition~\\ref{prop:arith-lucas-window}\nidentifies its reduction with the equality for $\\overline C_m$.\nThe preserved Lean development also proves this period directly from the\nHurwitz inverse in characteristic $p$:\n\\lean{KomorebiPaperIX.WindowPeriod.uniform_standard_window_period},\n\\lean{KomorebiPaperIX.WindowPeriod.multiple_period}, and\n\\lean{KomorebiPaperIX.WindowPeriod.standard_window_period_divisibility}.\n\\end{proof}","ge1$.\nThe order-one case of Theorem~\\ref{thm:arith-divisor-base} therefore gives\nvaluation at least one for the normalized difference at every step $uh$.\nThis proves the congruence, and Proposition~\\ref{prop:arith-lucas-window}\nidentifies its reduction with the equality for $\\overline C_m$.\nThe preserved Lean development also proves this period directly from the\nHurwitz inverse in characteristic $p$:\n\\lean{KomorebiPaperIX.WindowPeriod.uniform_standard_window_period},\n\\lean{KomorebiPaperIX.WindowPeriod.multiple_period}, and\n\\lean{KomorebiPaperIX.WindowPeriod.standard_window_period_divisibility}.\n\\end{proof}","\\begin{lemma}[Prime-power period from finite differences]\n\\label{lem:arith-newton-period}\nLet $R$ be a commutative ring, $p$ a prime, $f:\\N\\to R$, $h,n\\in\\N$,\nand $r\\ge1$.  If\n\\[\n p^{\\min(k,r)}\\mid\\Delta_h^k f(n)\\quad\\text{for every }k\\in\\N,\n\\]\nthen $p^r\\mid f(n+p^{r-1}h)-f(n)$.\n\\end{lemma}","\\begin{proof}\nThe finite Newton expansion gives\n\\[\n f(n+p^{r-1}h)-f(n)\n =\\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nTerms with $k\\ge r$ have the required factor from the difference.\nFor $1\\le k<r$, the identity\n\\[\n k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}\n\\]\nand $v_p(k)<k$ imply\n$p^r\\mid\\binom{p^{r-1}}k p^k$ in $\\Z$.\nMapping that integer divisibility to $R$ proves the claim term by term.\nThe argument includes $r=1$.\nThe declaration is\n\\lean{KomorebiPaperIX.PeriodFromDifferences.prime_power_period}.\n\\end{proof}","pansion gives\n\\[\n f(n+p^{r-1}h)-f(n)\n =\\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nTerms with $k\\ge r$ have the required factor from the difference.\nFor $1\\le k<r$, the identity\n\\[\n k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}\n\\]\nand $v_p(k)<k$ imply\n$p^r\\mid\\binom{p^{r-1}}k p^k$ in $\\Z$.\nMapping that integer divisibility to $R$ proves the claim term by term.\nThe argument includes $r=1$.\nThe declaration is\n\\lean{KomorebiPaperIX.PeriodFromDifferences.prime_power_period}.\n\\end{proof}","\\begin{proof}[Proof of Theorem~\\ref{thm:arith-nishibuchi}]\nOn the divisor branch, $m\\mid q$ and the packet step $q/m$ is an integer\nmultiple of $h=(p-1)/m$.  If $r\\le mn$, Theorem~\\ref{thm:arith-divisor-base}\nsupplies the hypotheses of Lemma~\\ref{lem:arith-newton-period}.\nIts packet displacement is $p^{r-1}q/m$, which converts to the original\ndegree as follows:\n\\[\n mp\\left(n+p^{r-1}\\frac qm\\right)=mpn+qp^r.\n\\]\nThe choice $n\\ge r$ implies $mn\\ge r$ because $m\\ge1$.","orem~\\ref{thm:arith-nishibuchi}]\nOn the divisor branch, $m\\mid q$ and the packet step $q/m$ is an integer\nmultiple of $h=(p-1)/m$.  If $r\\le mn$, Theorem~\\ref{thm:arith-divisor-base}\nsupplies the hypotheses of Lemma~\\ref{lem:arith-newton-period}.\nIts packet displacement is $p^{r-1}q/m$, which converts to the original\ndegree as follows:\n\\[\n mp\\left(n+p^{r-1}\\frac qm\\right)=mpn+qp^r.\n\\]\nThe choice $n\\ge r$ implies $mn\\ge r$ because $m\\ge1$.","th-nishibuchi}]\nOn the divisor branch, $m\\mid q$ and the packet step $q/m$ is an integer\nmultiple of $h=(p-1)/m$.  If $r\\le mn$, Theorem~\\ref{thm:arith-divisor-base}\nsupplies the hypotheses of Lemma~\\ref{lem:arith-newton-period}.\nIts packet displacement is $p^{r-1}q/m$, which converts to the original\ndegree as follows:\n\\[\n mp\\left(n+p^{r-1}\\frac qm\\right)=mpn+qp^r.\n\\]\nThe choice $n\\ge r$ implies $mn\\ge r$ because $m\\ge1$.","the divisor branch, $m\\mid q$ and the packet step $q/m$ is an integer\nmultiple of $h=(p-1)/m$.  If $r\\le mn$, Theorem~\\ref{thm:arith-divisor-base}\nsupplies the hypotheses of Lemma~\\ref{lem:arith-newton-period}.\nIts packet displacement is $p^{r-1}q/m$, which converts to the original\ndegree as follows:\n\\[\n mp\\left(n+p^{r-1}\\frac qm\\right)=mpn+qp^r.\n\\]\nThe choice $n\\ge r$ implies $mn\\ge r$ because $m\\ge1$.","For an odd prime, $2\\mid p-1$, so the option $m=2$ introduces no further\ncase.  The sole remaining branch is $p=2,m=2,N=4$.  The all-row theorem\nproved next gives $2^k\\mid\\Delta_1^kC_{4,j}(n)$ at every row and address.\nAfter multiplying by $j!$, Lemma~\\ref{lem:arith-newton-period} gives packet\nperiod $2^{r-1}$ and degree displacement $4\\cdot2^{r-1}=2^{r+1}=qp^r$.\nIt also gives integrality.  Hence the complete statement follows.\nThe formal degree conversion is\n\\lean{KomorebiPaperIX.NishibuchiCompletion.original_degree_shift}.\nThe all-parameter degree theorem is\n\\lean{KomorebiPaperIX.NishibuchiCompletion.full_period_degree}.\n\\end{proof}","For the remainder of the four-packet discussion abbreviate\n$C_j=C_{4,j}$ and $U_j=U_{4,j}$.  Removing the original row cap requires\nan unshifted coefficient operator.","\\begin{theorem}[All-row dyadic higher differences]\\label{thm:arith-dyadic-differences}\nFor $0\\le j<4$ and all $u,k,n\\in\\N$,\n\\begin{align}\n C_j(n)&\\in\\Z_2,","\\begin{theorem}[All-row dyadic higher differences]\\label{thm:arith-dyadic-differences}\nFor $0\\le j<4$ and all $u,k,n\\in\\N$,\n\\begin{align}\n C_j(n)&\\in\\Z_2,\\\\\n \\Delta_u^kC_j(n)&\\in 2^{k(1+\\nu_2(u))}\\Z_2,\\label{eq:arith-allrow-c}\\\\\n \\Delta_u^kU_j(n)&\\in\n 2^{k(1+\\nu_2(u))+v_2(j!)}\\Z_2.\\label{eq:arith-allrow-u}\n\\end{align}\nThere is no restriction $k\\le n$ and no minimum with the row $n$.\n\\end{theorem}","\\Delta_u^kU_j(n)&\\in\n 2^{k(1+\\nu_2(u))+v_2(j!)}\\Z_2.\\label{eq:arith-allrow-u}\n\\end{align}\nThere is no restriction $k\\le n$ and no minimum with the row $n$.\n\\end{theorem}","\\begin{proof}\nThe Hurwitz coefficients of the normalized source $K_{4,j}$ at degrees\n$4n$ are\n\\[\n g_j(n)=\\binom{4n+j}{4n}^{-1}.\n\\]\nTheir explicit expressions are\n\\[\n \\begin{aligned}\n g_0(n)&=1,&\n g_1(n)&=\\frac1{4n+1},","\\begin{proof}\nThe Hurwitz coefficients of the normalized source $K_{4,j}$ at degrees\n$4n$ are\n\\[\n g_j(n)=\\binom{4n+j}{4n}^{-1}.\n\\]\nTheir explicit expressions are\n\\[\n \\begin{aligned}\n g_0(n)&=1,&\n g_1(n)&=\\frac1{4n+1},\\\\\n g_2(n)&=\\frac1{(4n+1)(2n+1)},&\n g_3(n)&=\\frac3{(4n+1)(2n+1)(4n+3)}.\n \\end{aligned}\n\\]\nEvery denominator is odd.  An affine sequence with even slope satisfies\n$2^k\\mid\\Delta_1^k f(n)$ for every $k,n$.  The exact product identity\n\\[\n \\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n          \\Delta_1^if(n)\\Delta_1^{s-i}g(n+i)\n\\]\nproves closure of this property under products.  Applying the formula\nwith the factors interchanged to $fg=1$ isolates $f(n+s)\\Delta_1^sg(n)$.\nMultiplication by $f(n+s)^{-1}$ and induction then prove closure under\npointwise unit inverses.\nThus all four $g_j$ have the property.  The explicit formulas and difference\nbounds are formalized in\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_coe} and\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_kummer}.","g_2(n)&=\\frac1{(4n+1)(2n+1)},&\n g_3(n)&=\\frac3{(4n+1)(2n+1)(4n+3)}.\n \\end{aligned}\n\\]\nEvery denominator is odd.  An affine sequence with even slope satisfies\n$2^k\\mid\\Delta_1^k f(n)$ for every $k,n$.  The exact product identity\n\\[\n \\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n          \\Delta_1^if(n)\\Delta_1^{s-i}g(n+i)\n\\]\nproves closure of this property under products.  Applying the formula\nwith the factors interchanged to $fg=1$ isolates $f(n+s)\\Delta_1^sg(n)$.\nMultiplication by $f(n+s)^{-1}$ and induction then prove closure under\npointwise unit inverses.\nThus all four $g_j$ have the property.  The explicit formulas and difference\nbounds are formalized in\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_coe} and\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_kummer}.","+1)},&\n g_3(n)&=\\frac3{(4n+1)(2n+1)(4n+3)}.\n \\end{aligned}\n\\]\nEvery denominator is odd.  An affine sequence with even slope satisfies\n$2^k\\mid\\Delta_1^k f(n)$ for every $k,n$.  The exact product identity\n\\[\n \\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n          \\Delta_1^if(n)\\Delta_1^{s-i}g(n+i)\n\\]\nproves closure of this property under products.  Applying the formula\nwith the factors interchanged to $fg=1$ isolates $f(n+s)\\Delta_1^sg(n)$.\nMultiplication by $f(n+s)^{-1}$ and induction then prove closure under\npointwise unit inverses.\nThus all four $g_j$ have the property.  The explicit formulas and difference\nbounds are formalized in\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_coe} and\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_kummer}.","\\frac3{(4n+1)(2n+1)(4n+3)}.\n \\end{aligned}\n\\]\nEvery denominator is odd.  An affine sequence with even slope satisfies\n$2^k\\mid\\Delta_1^k f(n)$ for every $k,n$.  The exact product identity\n\\[\n \\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n          \\Delta_1^if(n)\\Delta_1^{s-i}g(n+i)\n\\]\nproves closure of this property under products.  Applying the formula\nwith the factors interchanged to $fg=1$ isolates $f(n+s)\\Delta_1^sg(n)$.\nMultiplication by $f(n+s)^{-1}$ and induction then prove closure under\npointwise unit inverses.\nThus all four $g_j$ have the property.  The explicit formulas and difference\nbounds are formalized in\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_coe} and\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_kummer}.","\\]\nEvery denominator is odd.  An affine sequence with even slope satisfies\n$2^k\\mid\\Delta_1^k f(n)$ for every $k,n$.  The exact product identity\n\\[\n \\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n          \\Delta_1^if(n)\\Delta_1^{s-i}g(n+i)\n\\]\nproves closure of this property under products.  Applying the formula\nwith the factors interchanged to $fg=1$ isolates $f(n+s)\\Delta_1^sg(n)$.\nMultiplication by $f(n+s)^{-1}$ and induction then prove closure under\npointwise unit inverses.\nThus all four $g_j$ have the property.  The explicit formulas and difference\nbounds are formalized in\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_coe} and\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_kummer}.","^k f(n)$ for every $k,n$.  The exact product identity\n\\[\n \\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n          \\Delta_1^if(n)\\Delta_1^{s-i}g(n+i)\n\\]\nproves closure of this property under products.  Applying the formula\nwith the factors interchanged to $fg=1$ isolates $f(n+s)\\Delta_1^sg(n)$.\nMultiplication by $f(n+s)^{-1}$ and induction then prove closure under\npointwise unit inverses.\nThus all four $g_j$ have the property.  The explicit formulas and difference\nbounds are formalized in\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_coe} and\n\\lean{KomorebiPaperIX.DyadicFourKernel.source_kummer}.","For an integer charge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves t","For an integer charge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge","For an integer charge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so in","For an integer charge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserv","For an integer charge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPape","For an integer charge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPaperIX.ChargedDworkClosure.B_mul},\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_inverse}, and\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_charge_","For an integer charge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPaperIX.ChargedDworkClosure.B_mul},\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_inverse}, and\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_charge_shift}.","harge $c$, put\n\\[\n A_c=\\frac{D^4-c}{2}.\n\\]\nA series is flat of charge $c$ when applying any power $A_c^s$ to it gives\nan element of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPaperIX.ChargedDworkClosure.B_mul},\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_inverse}, and\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_charge_shift}.","t of $\\mathcal H_2$.  The required product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPaperIX.ChargedDworkClosure.B_mul},\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_inverse}, and\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_charge_shift}.","d product formula is\n\\begin{equation}\\label{eq:arith-charged-product}\n A_{c+d}(FG)=(A_cF)G+F(A_dG)\n       +2F'G'''+3F''G''+2F'''G'.\n\\end{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPaperIX.ChargedDworkClosure.B_mul},\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_inverse}, and\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_charge_shift}.","{equation}\nAll middle coefficients are integral.  Derivatives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPaperIX.ChargedDworkClosure.B_mul},\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_inverse}, and\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_charge_shift}.","atives preserve\n$\\mathcal H_2$ and commute with $A_c$.  Induction on the number of\niterations in \\eqref{eq:arith-charged-product} proves that multiplication\nadds charges.  If $FG=1$, the same formula yields\n\\[\n A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').\n\\]\nInduction, starting from the independently established integrality of\nthe reciprocal $G$, proves that inversion negates the charge.\nFinally $A_{c+2}=A_c-1$, so increasing a charge by two preserves flatness.\nThese are\n\\lean{KomorebiPaperIX.ChargedDworkClosure.B_mul},\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_inverse}, and\n\\lean{KomorebiPaperIX.ChargedDworkClosure.flat_charge_shift}.","Packing $g_j$ into degrees divisible by four gives $K_{4,j}$.\nIts source property is precisely charge-one flatness.  Inversion gives\ncharge $-1$, and the charge shift restores charge one for the\nreciprocal $\\mathcal P_{4,j}$.  If $\\operatorname{dec}(F)(d)=d![t^d]F$, the exact\ncoefficient identity is\n\\[\n \\operatorname{dec}(A_1^sF)(d)\n       =2^{-s}\\Delta_4^s\\operatorname{dec}(F)(d).\n\\]\nAt $d=4n$, packing turns the right side into\n$2^{-s}\\Delta_1^sC_j(n)$.  This extraction requires no negative degree\nand therefore holds at every $n$, including zero.  It proves the unit-step\nbound \\lean{KomorebiPaperIX.DyadicFourAllRows.base_step_bound}.","le by four gives $K_{4,j}$.\nIts source property is precisely charge-one flatness.  Inversion gives\ncharge $-1$, and the charge shift restores charge one for the\nreciprocal $\\mathcal P_{4,j}$.  If $\\operatorname{dec}(F)(d)=d![t^d]F$, the exact\ncoefficient identity is\n\\[\n \\operatorname{dec}(A_1^sF)(d)\n       =2^{-s}\\Delta_4^s\\operatorname{dec}(F)(d).\n\\]\nAt $d=4n$, packing turns the right side into\n$2^{-s}\\Delta_1^sC_j(n)$.  This extraction requires no negative degree\nand therefore holds at every $n$, including zero.  It proves the unit-step\nbound \\lean{KomorebiPaperIX.DyadicFourAllRows.base_step_bound}.","harge $-1$, and the charge shift restores charge one for the\nreciprocal $\\mathcal P_{4,j}$.  If $\\operatorname{dec}(F)(d)=d![t^d]F$, the exact\ncoefficient identity is\n\\[\n \\operatorname{dec}(A_1^sF)(d)\n       =2^{-s}\\Delta_4^s\\operatorname{dec}(F)(d).\n\\]\nAt $d=4n$, packing turns the right side into\n$2^{-s}\\Delta_1^sC_j(n)$.  This extraction requires no negative degree\nand therefore holds at every $n$, including zero.  It proves the unit-step\nbound \\lean{KomorebiPaperIX.DyadicFourAllRows.base_step_bound}.","To incorporate $v_2(u)$, expand a displacement using finite Newton\ncoefficients.  For any prime $p$, $u>0$, and $1\\le a\\le u$,\n\\[\n p^{v_p(u)+1}\\mid\\binom ua p^a,\n\\]\nas follows from $a\\binom ua=u\\binom{u-1}{a-1}$ and $v_p(a)<a$.\nRepeated displacement, retaining the remaining base-difference order\nat each induction step, gives a factor $p^{k(1+v_p(u))}$ from $k$\ndisplacements.  This induction is formalized in\n\\lean{KomorebiPaperIX.StepAmplification.repeated_precision}.\nApplying it with $p=2$ proves \\eqref{eq:arith-allrow-c}. Multiplication\nby $j!$ proves \\eqref{eq:arith-allrow-u}.  Zero step and zero order follow\nfrom their definitions.  The final statements are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.no","To incorporate $v_2(u)$, expand a displacement using finite Newton\ncoefficients.  For any prime $p$, $u>0$, and $1\\le a\\le u$,\n\\[\n p^{v_p(u)+1}\\mid\\binom ua p^a,\n\\]\nas follows from $a\\binom ua=u\\binom{u-1}{a-1}$ and $v_p(a)<a$.\nRepeated displacement, retaining the remaining base-difference order\nat each induction step, gives a factor $p^{k(1+v_p(u))}$ from $k$\ndisplacements.  This induction is formalized in\n\\lean{KomorebiPaperIX.StepAmplification.repeated_precision}.\nApplying it with $p=2$ proves \\eqref{eq:arith-allrow-c}. Multiplication\nby $j!$ proves \\eqref{eq:arith-allrow-u}.  Zero step and zero order follow\nfrom their definitions.  The final statements are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.normalized_step_bound}\nand\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.","To incorporate $v_2(u)$, expand a displacement using finite Newton\ncoefficients.  For any prime $p$, $u>0$, and $1\\le a\\le u$,\n\\[\n p^{v_p(u)+1}\\mid\\binom ua p^a,\n\\]\nas follows from $a\\binom ua=u\\binom{u-1}{a-1}$ and $v_p(a)<a$.\nRepeated displacement, retaining the remaining base-difference order\nat each induction step, gives a factor $p^{k(1+v_p(u))}$ from $k$\ndisplacements.  This induction is formalized in\n\\lean{KomorebiPaperIX.StepAmplification.repeated_precision}.\nApplying it with $p=2$ proves \\eqref{eq:arith-allrow-c}. Multiplication\nby $j!$ proves \\eqref{eq:arith-allrow-u}.  Zero step and zero order follow\nfrom their definitions.  The final statements are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.normalized_step_bound}\nand\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.unnorma","To incorporate $v_2(u)$, expand a displacement using finite Newton\ncoefficients.  For any prime $p$, $u>0$, and $1\\le a\\le u$,\n\\[\n p^{v_p(u)+1}\\mid\\binom ua p^a,\n\\]\nas follows from $a\\binom ua=u\\binom{u-1}{a-1}$ and $v_p(a)<a$.\nRepeated displacement, retaining the remaining base-difference order\nat each induction step, gives a factor $p^{k(1+v_p(u))}$ from $k$\ndisplacements.  This induction is formalized in\n\\lean{KomorebiPaperIX.StepAmplification.repeated_precision}.\nApplying it with $p=2$ proves \\eqref{eq:arith-allrow-c}. Multiplication\nby $j!$ proves \\eqref{eq:arith-allrow-u}.  Zero step and zero order follow\nfrom their definitions.  The final statements are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.normalized_step_bound}\nand\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.unnormalized_step_bound}.","To incorporate $v_2(u)$, expand a displacement using finite Newton\ncoefficients.  For any prime $p$, $u>0$, and $1\\le a\\le u$,\n\\[\n p^{v_p(u)+1}\\mid\\binom ua p^a,\n\\]\nas follows from $a\\binom ua=u\\binom{u-1}{a-1}$ and $v_p(a)<a$.\nRepeated displacement, retaining the remaining base-difference order\nat each induction step, gives a factor $p^{k(1+v_p(u))}$ from $k$\ndisplacements.  This induction is formalized in\n\\lean{KomorebiPaperIX.StepAmplification.repeated_precision}.\nApplying it with $p=2$ proves \\eqref{eq:arith-allrow-c}. Multiplication\nby $j!$ proves \\eqref{eq:arith-allrow-u}.  Zero step and zero order follow\nfrom their definitions.  The final statements are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.normalized_step_bound}\nand\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.unnormalized_step_bound}.\n\\end{proof}","prime $p$, $u>0$, and $1\\le a\\le u$,\n\\[\n p^{v_p(u)+1}\\mid\\binom ua p^a,\n\\]\nas follows from $a\\binom ua=u\\binom{u-1}{a-1}$ and $v_p(a)<a$.\nRepeated displacement, retaining the remaining base-difference order\nat each induction step, gives a factor $p^{k(1+v_p(u))}$ from $k$\ndisplacements.  This induction is formalized in\n\\lean{KomorebiPaperIX.StepAmplification.repeated_precision}.\nApplying it with $p=2$ proves \\eqref{eq:arith-allrow-c}. Multiplication\nby $j!$ proves \\eqref{eq:arith-allrow-u}.  Zero step and zero order follow\nfrom their definitions.  The final statements are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.normalized_step_bound}\nand\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.unnormalized_step_bound}.\n\\end{proof}","binom ua p^a,\n\\]\nas follows from $a\\binom ua=u\\binom{u-1}{a-1}$ and $v_p(a)<a$.\nRepeated displacement, retaining the remaining base-difference order\nat each induction step, gives a factor $p^{k(1+v_p(u))}$ from $k$\ndisplacements.  This induction is formalized in\n\\lean{KomorebiPaperIX.StepAmplification.repeated_precision}.\nApplying it with $p=2$ proves \\eqref{eq:arith-allrow-c}. Multiplication\nby $j!$ proves \\eqref{eq:arith-allrow-u}.  Zero step and zero order follow\nfrom their definitions.  The final statements are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.normalized_step_bound}\nand\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.unnormalized_step_bound}.\n\\end{proof}","\\begin{corollary}[Step-sensitive divisor-branch bound]\\label{cor:arith-amplified}\nUnder the hypotheses of Theorem~\\ref{thm:arith-divisor-base}, for all\n$u,k,n\\in\\N$,\n\\[\n \\Delta_{u(p-1)/m}^{k}U_{mp,j}(n)\n       \\in p^{\\min(k(1+\\nu_p(u)),mn)}\\Z_p.\n\\]\nFor $C_{mp,j}$ the corresponding valuation lower bound is\n$\\min(k(1+\\nu_p(u)),mn)-v_p(j!)$.\n\\end{corollary}","\\begin{proof}\nApply the same repeated-displacement theorem with the fixed cap $mn$ to\n\\eqref{eq:arith-divisor-bound}.  The argument preserves this cap and\nimproves the contribution from the order of the difference.\nDivision by the fixed factorial gives the rational valuation statement,\nwith the zero case included.\nThese are\n\\lean{KomorebiPaperIX.AllResidueStepAmplification.unnormalized_precision}\nand\n\\lean{KomorebiPaperIX.AllResidueStepAmplification.normalized_precision}.\n\\end{proof}","\\begin{corollary}[Dyadic periods from row zero]\\label{cor:arith-allrow-period}\nFor $0\\le j<4$, $r\\ge1$, and $n\\ge0$,\n\\[\n U_j(n+2^{r-1})\\equiv U_j(n)\\pmod{2^r}.\n\\]\nFor $j=2,3$ and $r\\ge2$, $2^{r-2}$ is also a sufficient packet period.\nThe respective original degree displacements are $2^{r+1}$ and $2^r$.\n\\end{corollary}","\\begin{proof}\nUse Lemma~\\ref{lem:arith-newton-period} and the unit-step case of\nTheorem~\\ref{thm:arith-dyadic-differences}.  At $j=2,3$ the factorial\ncontributes an additional power of two.  The original degree versions are\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.all_rows_period_degree}\nand\n\\lean{KomorebiPaperIX.DyadicFourFinalStrengthening.all_rows_half_period_degree}.\nThe next theorem sharpens these sufficient periods to a complete\nclassification.\n\\end{proof}","\\begin{lemma}[Finite Newton dominance]\\label{lem:arith-dominance}\nLet $f:\\N\\to\\Q$ and write $a_k=\\Delta_1^kf(0)$.  Suppose that $c\\in\\N$,\n$a_1\\ne0$, $v_2(a_1)=c$, and\n\\[\n v_2(a_k)\\ge c+1+\\lfloor\\log_2k\\rfloor\\qquad(k\\ge2).\n\\]\nThen for every $n\\ge0$ and $T>0$,\n\\[\n f(n+T)-f(n)\\ne0,\\qquad\n v_2\\bigl(f(n+T)-f(n)\\bigr)=v_2(T)+c.\n\\]\n\\end{lemma}","\\begin{proof}\nFinite Newton reconstruction at an arbitrary row gives\n\\[\n \\Delta_1^kf(n)=\\sum_{i=0}^{n}\\binom ni a_{k+i}.\n\\]\nThe monotonicity of $\\lfloor\\log_2k\\rfloor$ therefore preserves the\nhigher-coefficient bound after translation.  In addition,\n$\\Delta_1f(n)-a_1\\in2^{c+1}\\Z_2$, so the first difference is nonzero\nof exact valuation $c$.","Now expand the displacement:\n\\[\n f(n+T)-f(n)=T\\Delta_1f(n)\n          +\\sum_{k=2}^{T}\\binom Tk\\Delta_1^kf(n).\n\\]\nFor $1\\le k\\le T$, the binomial denominator identity implies\n\\[\n v_2\\binom Tk\\ge v_2(T)-v_2(k)\n            \\ge v_2(T)-\\lfloor\\log_2k\\rfloor.\n\\]\nEvery term of the remaining sum consequently has valuation at least\n$v_2(T)+c+1$.  Its sum cannot cancel the leading term, which has exact\nvaluation $v_2(T)+c$.  This proves nonvanishing and the claimed valuation\nusing only finite sums.\nThe general theorem is\n\\lean{KomorebiPaperIX.NewtonDominance.exact_displacement}.\n\\end{proof}","pand the displacement:\n\\[\n f(n+T)-f(n)=T\\Delta_1f(n)\n          +\\sum_{k=2}^{T}\\binom Tk\\Delta_1^kf(n).\n\\]\nFor $1\\le k\\le T$, the binomial denominator identity implies\n\\[\n v_2\\binom Tk\\ge v_2(T)-v_2(k)\n            \\ge v_2(T)-\\lfloor\\log_2k\\rfloor.\n\\]\nEvery term of the remaining sum consequently has valuation at least\n$v_2(T)+c+1$.  Its sum cannot cancel the leading term, which has exact\nvaluation $v_2(T)+c$.  This proves nonvanishing and the claimed valuation\nusing only finite sums.\nThe general theorem is\n\\lean{KomorebiPaperIX.NewtonDominance.exact_displacement}.\n\\end{proof}","\\begin{theorem}[Exact four-packet displacement law]\\label{thm:arith-exact}\nSet\n\\[\n (c_0,c_1,c_2,c_3)=(1,1,4,2),\\qquad\n (d_0,d_1,d_2,d_3)=(1,1,5,3).\n\\]\nFor $0\\le j<4$, $n\\ge0$, and every positive integer $T$,\n\\begin{align*}\n C_j(n+T)-C_j(n)&\\ne0,&\n v_2\\bigl(C_j(n+T)-C_j(n)\\bigr)&=v_2(T)+c_j,","\\begin{theorem}[Exact four-packet displacement law]\\label{thm:arith-exact}\nSet\n\\[\n (c_0,c_1,c_2,c_3)=(1,1,4,2),\\qquad\n (d_0,d_1,d_2,d_3)=(1,1,5,3).\n\\]\nFor $0\\le j<4$, $n\\ge0$, and every positive integer $T$,\n\\begin{align*}\n C_j(n+T)-C_j(n)&\\ne0,&\n v_2\\bigl(C_j(n+T)-C_j(n)\\bigr)&=v_2(T)+c_j,\\\\\n U_j(n+T)-U_j(n)&\\ne0,&\n v_2\\bigl(U_j(n+T)-U_j(n)\\bigr)&=v_2(T)+d_j.\n\\end{align*}\n\\end{theorem}","\\begin{proof}\nThe first Newton coefficients, derived from\n\\eqref{eq:arith-triangular}, are\n\\[\n a_{0,1}=-2,\\quad a_{1,1}=-\\frac65,\\quad\n a_{2,1}=-\\frac{16}{15},\\quad a_{3,1}=-\\frac{36}{35}.\n\\]\nTheir exact valuations are $c_j$.  Theorem~\\ref{thm:arith-dyadic-differences}\ngives $v_2(a_{j,k})\\ge k$ for all orders.  To obtain the stronger\nlogarithmic envelope required by Lemma~\\ref{lem:arith-dominance}, only\na finite initial segment needs improvement.  The recurrence proves the\nfollowing exact valuations:\n\\begin{center}\n\\begin{tabular}{cll}\\toprule\nAddress $j$ & Further orders $k$ & $v_2(a_{j,k})$ in that order","\\quad a_{3,1}=-\\frac{36}{35}.\n\\]\nTheir exact valuations are $c_j$.  Theorem~\\ref{thm:arith-dyadic-differences}\ngives $v_2(a_{j,k})\\ge k$ for all orders.  To obtain the stronger\nlogarithmic envelope required by Lemma~\\ref{lem:arith-dominance}, only\na finite initial segment needs improvement.  The recurrence proves the\nfollowing exact valuations:\n\\begin{center}\n\\begin{tabular}{cll}\\toprule\nAddress $j$ & Further orders $k$ & $v_2(a_{j,k})$ in that order","36}{35}.\n\\]\nTheir exact valuations are $c_j$.  Theorem~\\ref{thm:arith-dyadic-differences}\ngives $v_2(a_{j,k})\\ge k$ for all orders.  To obtain the stronger\nlogarithmic envelope required by Lemma~\\ref{lem:arith-dominance}, only\na finite initial segment needs improvement.  The recurrence proves the\nfollowing exact valuations:\n\\begin{center}\n\\begin{tabular}{cll}\\toprule\nAddress $j$ & Further orders $k$ & $v_2(a_{j,k})$ in that order","}.\n\\]\nTheir exact valuations are $c_j$.  Theorem~\\ref{thm:arith-dyadic-differences}\ngives $v_2(a_{j,k})\\ge k$ for all orders.  To obtain the stronger\nlogarithmic envelope required by Lemma~\\ref{lem:arith-dominance}, only\na finite initial segment needs improvement.  The recurrence proves the\nfollowing exact valuations:\n\\begin{center}\n\\begin{tabular}{cll}\\toprule\nAddress $j$ & Further orders $k$ & $v_2(a_{j,k})$ in that order","midrule\n$0$ & $2$ & $3$","$1$ & $2$ & $3$","$2$ & $2,3,4,5,6$ & $6,9,13,16,22$","$3$ & $2,3,4$ & $8,8,15$","bottomrule\n\\end{tabular}\n\\end{center}\nFor all subsequent $k$, the elementary inequality\n$c_j+1+\\lfloor\\log_2k\\rfloor\\le k$ completes the envelope.\nThe entries follow by exact rational computation from the defining\nrecurrence, and their valuations are proved in Lean.  Together with the\ninequality for all subsequent orders, they establish the bound at every\norder.\nThe uniform envelope is\n\\lean{KomorebiPaperIX.DyadicFourPrefix.tail_precision}.","Apply Lemma~\\ref{lem:arith-dominance} to $C_j$.  Multiplication by the\nnonzero factorial $j!$ adds $v_2(j!)$ and preserves nonzeroness.\nHence $d_j=c_j+v_2(j!)$, giving the four valuation gains for the original\ncoefficients.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicFourLeastPeriods.normalized_exact_displacement}\nand \\lean{KomorebiPaperIX.DyadicFourLeastPeriods.exact_displacement}.\n\\end{proof}","\\begin{corollary}[All positive periods and their least element]\n\\label{cor:arith-least-period}\nFor $r\\in\\N$ and $0\\le j<4$, a positive integer $T$ is a packet period\nof $U_j$ modulo $2^r$ if and only if\n\\[\n 2^{\\max(0,r-d_j)}\\mid T.\n\\]\nHere a packet period means $U_j(n+T)\\equiv U_j(n)\\pmod{2^r}$ for every\n$n\\ge0$.  Consequently the least positive packet periods are\n\\[\n \\begin{array}{c|cccc}\n j&0&1&2&3","\\begin{corollary}[All positive periods and their least element]\n\\label{cor:arith-least-period}\nFor $r\\in\\N$ and $0\\le j<4$, a positive integer $T$ is a packet period\nof $U_j$ modulo $2^r$ if and only if\n\\[\n 2^{\\max(0,r-d_j)}\\mid T.\n\\]\nHere a packet period means $U_j(n+T)\\equiv U_j(n)\\pmod{2^r}$ for every\n$n\\ge0$.  Consequently the least positive packet periods are\n\\[\n \\begin{array}{c|cccc}\n j&0&1&2&3\\\\\\hline\n \\text{least period}&2^{\\max(0,r-1)}&2^{\\max(0,r-1)}&\n                 2^{\\max(0,r-5)}&2^{\\max(0,r-3)}.\n \\end{array}\n\\]\n\\end{corollary}","\\begin{proof}\nBy the exact displacement law, the congruence at any row is equivalent to\n$v_2(T)+d_j\\ge r$.  This is exactly the displayed integer\ndivisibility criterion.  Its least positive solution is the indicated\npower of two, which also covers $r=0$ and $r\\le d_j$.\nThe all-period and least-element statements are\n\\lean{KomorebiPaperIX.DyadicFourLeastPeriods.period_iff_dvd} and\n\\lean{KomorebiPaperIX.DyadicFourLeastPeriods.least_packet_period}.\n\\end{proof}","For example, $U_2(n)\\equiv2\\pmod{32}$ and $U_3(n)\\equiv6\\pmod8$\nfor every row.  At precision $2^{10}$ the least packet periods are\n$512,512,32,128$.  A packet displacement $T$ has original degree\ndisplacement $4T$, as proved by\n\\lean{KomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement}.\nMinimality here concerns packet shifts.  Shifts of the full degree-indexed\nsequence, including its zero coefficients off the packet support, require\na separate analysis.  The new strengthening chapters determine all repeated-step higher differences in addresses zero and one and a power-of-two-order family in address two. The full exact classification of arbitrary higher-order differences\nis also a separate question from th","For example, $U_2(n)\\equiv2\\pmod{32}$ and $U_3(n)\\equiv6\\pmod8$\nfor every row.  At precision $2^{10}$ the least packet periods are\n$512,512,32,128$.  A packet displacement $T$ has original degree\ndisplacement $4T$, as proved by\n\\lean{KomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement}.\nMinimality here concerns packet shifts.  Shifts of the full degree-indexed\nsequence, including its zero coefficients off the packet support, require\na separate analysis.  The new strengthening chapters determine all repeated-step higher differences in addresses zero and one and a power-of-two-order family in address two. The full exact classification of arbitrary higher-order differences\nis also a separate question from the first-displacement law.","The classical signed Euler numbers have generating function\n$\\operatorname{sech}t=\\sum_{n\\ge0}E_{2n}t^{2n}/(2n)!$.\nThey are the $N=2$, $j=0$ member of\n\\eqref{eq:arith-euler-egf}.  The coordinate theorem reconstructs\nthem from finite alternating sums of odd powers.","\\begin{definition}[Odd-length source and inverse coefficients]\n\\label{def:arith-odd-source}\nFor a positive odd integer $p$, define\n\\[\n S_d(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.\n\\]\nLet $\\gamma_m$ be the integral Hurwitz coefficients of $\\operatorname{sech}^2$:\n\\[\n \\operatorname{sech}^2t=\\sum_{m\\ge0}\\gamma_m\\frac{t^{2m}}{(2m)!},\n \\qquad\n \\gamma_m=\\sum_{b=0}^{2m}\\binom{2m}{b}E_bE_{2m-b},\n\\]\nwhere $E_b=0$ for odd $b$.  Then $\\gamma_0=1$.  Define\n\\[\n T^{\\mathrm{odd}}_{p,M}(n)=\\sum_{m=0}^{\\min(M,n)}\n       \\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nHere $p$ need only be odd. Primality is needed only to interpret the\nresult as a prime-adic congruence ladder.\n\\end{definition}","nition}[Odd-length source and inverse coefficients]\n\\label{def:arith-odd-source}\nFor a positive odd integer $p$, define\n\\[\n S_d(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.\n\\]\nLet $\\gamma_m$ be the integral Hurwitz coefficients of $\\operatorname{sech}^2$:\n\\[\n \\operatorname{sech}^2t=\\sum_{m\\ge0}\\gamma_m\\frac{t^{2m}}{(2m)!},\n \\qquad\n \\gamma_m=\\sum_{b=0}^{2m}\\binom{2m}{b}E_bE_{2m-b},\n\\]\nwhere $E_b=0$ for odd $b$.  Then $\\gamma_0=1$.  Define\n\\[\n T^{\\mathrm{odd}}_{p,M}(n)=\\sum_{m=0}^{\\min(M,n)}\n       \\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nHere $p$ need only be odd. Primality is needed only to interpret the\nresult as a prime-adic congruence ladder.\n\\end{definition}","-length source and inverse coefficients]\n\\label{def:arith-odd-source}\nFor a positive odd integer $p$, define\n\\[\n S_d(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.\n\\]\nLet $\\gamma_m$ be the integral Hurwitz coefficients of $\\operatorname{sech}^2$:\n\\[\n \\operatorname{sech}^2t=\\sum_{m\\ge0}\\gamma_m\\frac{t^{2m}}{(2m)!},\n \\qquad\n \\gamma_m=\\sum_{b=0}^{2m}\\binom{2m}{b}E_bE_{2m-b},\n\\]\nwhere $E_b=0$ for odd $b$.  Then $\\gamma_0=1$.  Define\n\\[\n T^{\\mathrm{odd}}_{p,M}(n)=\\sum_{m=0}^{\\min(M,n)}\n       \\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nHere $p$ need only be odd. Primality is needed only to interpret the\nresult as a prime-adic congruence ladder.\n\\end{definition}","rse coefficients]\n\\label{def:arith-odd-source}\nFor a positive odd integer $p$, define\n\\[\n S_d(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.\n\\]\nLet $\\gamma_m$ be the integral Hurwitz coefficients of $\\operatorname{sech}^2$:\n\\[\n \\operatorname{sech}^2t=\\sum_{m\\ge0}\\gamma_m\\frac{t^{2m}}{(2m)!},\n \\qquad\n \\gamma_m=\\sum_{b=0}^{2m}\\binom{2m}{b}E_bE_{2m-b},\n\\]\nwhere $E_b=0$ for odd $b$.  Then $\\gamma_0=1$.  Define\n\\[\n T^{\\mathrm{odd}}_{p,M}(n)=\\sum_{m=0}^{\\min(M,n)}\n       \\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nHere $p$ need only be odd. Primality is needed only to interpret the\nresult as a prime-adic congruence ladder.\n\\end{definition}","1}(-1)^a(2a+1)^d.\n\\]\nLet $\\gamma_m$ be the integral Hurwitz coefficients of $\\operatorname{sech}^2$:\n\\[\n \\operatorname{sech}^2t=\\sum_{m\\ge0}\\gamma_m\\frac{t^{2m}}{(2m)!},\n \\qquad\n \\gamma_m=\\sum_{b=0}^{2m}\\binom{2m}{b}E_bE_{2m-b},\n\\]\nwhere $E_b=0$ for odd $b$.  Then $\\gamma_0=1$.  Define\n\\[\n T^{\\mathrm{odd}}_{p,M}(n)=\\sum_{m=0}^{\\min(M,n)}\n       \\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nHere $p$ need only be odd. Primality is needed only to interpret the\nresult as a prime-adic congruence ladder.\n\\end{definition}","The same inverse coefficients can be written in the signed zigzag notation\nof the original manuscript.  Define $Z_r$ by\n\\[\n \\sec t+\\tan t=\\sum_{r\\ge0}Z_r\\frac{t^r}{r!}.\n\\]\nThen\n\\begin{equation}\\label{eq:arith-signed-zigzag}\n \\gamma_m=(-1)^m Z_{2m+1}\\qquad(m\\in\\N).\n\\end{equation}\nIndeed, the odd coefficients come from $\\tan t$, whose derivative is\n$\\sec^2t$.  Changing the sign of the squared variable in the even series\nfor $\\operatorname{sech}^2t$ gives $\\sec^2t$, so its degree-$2m$\nHurwitz coefficient is $(-1)^m\\gamma_m$.  Differentiation identifies\nthis coefficient with $Z_{2m+1}$, proving the formula.  The Lean bridge is\n\\lean{KomorebiPaperIX.Zigzag.inverse_is_signed_zigzag}. It uses the\nformal generating-function definition of the zigzag numbers.","ts can be written in the signed zigzag notation\nof the original manuscript.  Define $Z_r$ by\n\\[\n \\sec t+\\tan t=\\sum_{r\\ge0}Z_r\\frac{t^r}{r!}.\n\\]\nThen\n\\begin{equation}\\label{eq:arith-signed-zigzag}\n \\gamma_m=(-1)^m Z_{2m+1}\\qquad(m\\in\\N).\n\\end{equation}\nIndeed, the odd coefficients come from $\\tan t$, whose derivative is\n$\\sec^2t$.  Changing the sign of the squared variable in the even series\nfor $\\operatorname{sech}^2t$ gives $\\sec^2t$, so its degree-$2m$\nHurwitz coefficient is $(-1)^m\\gamma_m$.  Differentiation identifies\nthis coefficient with $Z_{2m+1}$, proving the formula.  The Lean bridge is\n\\lean{KomorebiPaperIX.Zigzag.inverse_is_signed_zigzag}. It uses the\nformal generating-function definition of the zigzag numbers.","zigzag notation\nof the original manuscript.  Define $Z_r$ by\n\\[\n \\sec t+\\tan t=\\sum_{r\\ge0}Z_r\\frac{t^r}{r!}.\n\\]\nThen\n\\begin{equation}\\label{eq:arith-signed-zigzag}\n \\gamma_m=(-1)^m Z_{2m+1}\\qquad(m\\in\\N).\n\\end{equation}\nIndeed, the odd coefficients come from $\\tan t$, whose derivative is\n$\\sec^2t$.  Changing the sign of the squared variable in the even series\nfor $\\operatorname{sech}^2t$ gives $\\sec^2t$, so its degree-$2m$\nHurwitz coefficient is $(-1)^m\\gamma_m$.  Differentiation identifies\nthis coefficient with $Z_{2m+1}$, proving the formula.  The Lean bridge is\n\\lean{KomorebiPaperIX.Zigzag.inverse_is_signed_zigzag}. It uses the\nformal generating-function definition of the zigzag numbers.","riginal manuscript.  Define $Z_r$ by\n\\[\n \\sec t+\\tan t=\\sum_{r\\ge0}Z_r\\frac{t^r}{r!}.\n\\]\nThen\n\\begin{equation}\\label{eq:arith-signed-zigzag}\n \\gamma_m=(-1)^m Z_{2m+1}\\qquad(m\\in\\N).\n\\end{equation}\nIndeed, the odd coefficients come from $\\tan t$, whose derivative is\n$\\sec^2t$.  Changing the sign of the squared variable in the even series\nfor $\\operatorname{sech}^2t$ gives $\\sec^2t$, so its degree-$2m$\nHurwitz coefficient is $(-1)^m\\gamma_m$.  Differentiation identifies\nthis coefficient with $Z_{2m+1}$, proving the formula.  The Lean bridge is\n\\lean{KomorebiPaperIX.Zigzag.inverse_is_signed_zigzag}. It uses the\nformal generating-function definition of the zigzag numbers.","$Z_r$ by\n\\[\n \\sec t+\\tan t=\\sum_{r\\ge0}Z_r\\frac{t^r}{r!}.\n\\]\nThen\n\\begin{equation}\\label{eq:arith-signed-zigzag}\n \\gamma_m=(-1)^m Z_{2m+1}\\qquad(m\\in\\N).\n\\end{equation}\nIndeed, the odd coefficients come from $\\tan t$, whose derivative is\n$\\sec^2t$.  Changing the sign of the squared variable in the even series\nfor $\\operatorname{sech}^2t$ gives $\\sec^2t$, so its degree-$2m$\nHurwitz coefficient is $(-1)^m\\gamma_m$.  Differentiation identifies\nthis coefficient with $Z_{2m+1}$, proving the formula.  The Lean bridge is\n\\lean{KomorebiPaperIX.Zigzag.inverse_is_signed_zigzag}. It uses the\nformal generating-function definition of the zigzag numbers.","egin{equation}\\label{eq:arith-signed-zigzag}\n \\gamma_m=(-1)^m Z_{2m+1}\\qquad(m\\in\\N).\n\\end{equation}\nIndeed, the odd coefficients come from $\\tan t$, whose derivative is\n$\\sec^2t$.  Changing the sign of the squared variable in the even series\nfor $\\operatorname{sech}^2t$ gives $\\sec^2t$, so its degree-$2m$\nHurwitz coefficient is $(-1)^m\\gamma_m$.  Differentiation identifies\nthis coefficient with $Z_{2m+1}$, proving the formula.  The Lean bridge is\n\\lean{KomorebiPaperIX.Zigzag.inverse_is_signed_zigzag}. It uses the\nformal generating-function definition of the zigzag numbers.","\\begin{theorem}[Classical odd-length reconstruction]\n\\label{thm:arith-odd-reconstruction}\nFor every positive odd integer $p$ and every $n\\in\\N$,\n\\[\n E_{2n}=\\sum_{m=0}^{n}\\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nEquivalently, $E_{2n}=\\sum_{q=0}^n V_{nq}S_{2q}(p)$ for the lower\ntriangular matrix\n\\[\n V_{nq}=\\binom{2n}{2q}\\gamma_{n-q}p^{2(n-q)}\\quad(q\\le n),\n \\qquad V_{nq}=0\\quad(q>n).\n\\]\nIt has $V_{nn}=1$ and $p^{2(n-q)}\\mid V_{nq}$ when $q\\le n$.\n\\end{theorem}","egin{theorem}[Classical odd-length reconstruction]\n\\label{thm:arith-odd-reconstruction}\nFor every positive odd integer $p$ and every $n\\in\\N$,\n\\[\n E_{2n}=\\sum_{m=0}^{n}\\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nEquivalently, $E_{2n}=\\sum_{q=0}^n V_{nq}S_{2q}(p)$ for the lower\ntriangular matrix\n\\[\n V_{nq}=\\binom{2n}{2q}\\gamma_{n-q}p^{2(n-q)}\\quad(q\\le n),\n \\qquad V_{nq}=0\\quad(q>n).\n\\]\nIt has $V_{nn}=1$ and $p^{2(n-q)}\\mid V_{nq}$ when $q\\le n$.\n\\end{theorem}","lassical odd-length reconstruction]\n\\label{thm:arith-odd-reconstruction}\nFor every positive odd integer $p$ and every $n\\in\\N$,\n\\[\n E_{2n}=\\sum_{m=0}^{n}\\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nEquivalently, $E_{2n}=\\sum_{q=0}^n V_{nq}S_{2q}(p)$ for the lower\ntriangular matrix\n\\[\n V_{nq}=\\binom{2n}{2q}\\gamma_{n-q}p^{2(n-q)}\\quad(q\\le n),\n \\qquad V_{nq}=0\\quad(q>n).\n\\]\nIt has $V_{nn}=1$ and $p^{2(n-q)}\\mid V_{nq}$ when $q\\le n$.\n\\end{theorem}","ction]\n\\label{thm:arith-odd-reconstruction}\nFor every positive odd integer $p$ and every $n\\in\\N$,\n\\[\n E_{2n}=\\sum_{m=0}^{n}\\binom{2n}{2m}\\gamma_m p^{2m}S_{2(n-m)}(p).\n\\]\nEquivalently, $E_{2n}=\\sum_{q=0}^n V_{nq}S_{2q}(p)$ for the lower\ntriangular matrix\n\\[\n V_{nq}=\\binom{2n}{2q}\\gamma_{n-q}p^{2(n-q)}\\quad(q\\le n),\n \\qquad V_{nq}=0\\quad(q>n).\n\\]\nIt has $V_{nn}=1$ and $p^{2(n-q)}\\mid V_{nq}$ when $q\\le n$.\n\\end{theorem}","\\begin{proof}\nThe exponential generating series of the source is the finite sum\n\\[\n \\mathcal S_p(t)=\\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.\n\\]\nMultiplication by $\\cosh t$ telescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lea","\\begin{proof}\nThe exponential generating series of the source is the finite sum\n\\[\n \\mathcal S_p(t)=\\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.\n\\]\nMultiplication by $\\cosh t$ telescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.od","\\begin{proof}\nThe exponential generating series of the source is the finite sum\n\\[\n \\mathcal S_p(t)=\\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.\n\\]\nMultiplication by $\\cosh t$ telescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction}","\\begin{proof}\nThe exponential generating series of the source is the finite sum\n\\[\n \\mathcal S_p(t)=\\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.\n\\]\nMultiplication by $\\cosh t$ telescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction} and\n\\lean{KomorebiPaperIX.OddEuler.row_action}.\n\\end{proof}","The exponential generating series of the source is the finite sum\n\\[\n \\mathcal S_p(t)=\\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.\n\\]\nMultiplication by $\\cosh t$ telescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction} and\n\\lean{KomorebiPaperIX.OddEuler.row_action}.\n\\end{proof}","of the source is the finite sum\n\\[\n \\mathcal S_p(t)=\\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.\n\\]\nMultiplication by $\\cosh t$ telescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction} and\n\\lean{KomorebiPaperIX.OddEuler.row_action}.\n\\end{proof}","h t$ telescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction} and\n\\lean{KomorebiPaperIX.OddEuler.row_action}.\n\\end{proof}","lescopes.  Since $p$ is odd,\n\\[\n \\mathcal S_p(t)=\\frac{1+e^{2pt}}{2}\\operatorname{sech}t.\n\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction} and\n\\lean{KomorebiPaperIX.OddEuler.row_action}.\n\\end{proof}","\\]\nTaking its even part gives\n\\[\n \\sum_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction} and\n\\lean{KomorebiPaperIX.OddEuler.row_action}.\n\\end{proof}","um_{n\\ge0}S_{2n}(p)\\frac{t^{2n}}{(2n)!}\n       =\\cosh^2(pt)\\operatorname{sech}t.\n\\]\nMultiplication by the reciprocal $\\operatorname{sech}^2(pt)$\nyields $\\operatorname{sech}t$.  The EGF product rule now gives the\nreconstruction coefficient by coefficient.  Replacing $m$ by $n-q$\nproduces the displayed lower triangular row.  Its diagonal and weights\nfollow from $\\gamma_0=1$ and the explicit factor $p^{2(n-q)}$.\nThe source-series identity is\n\\lean{KomorebiPaperIX.OddPower.odd_source_even_egf}. The reconstruction\nand row statement are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_reconstruction} and\n\\lean{KomorebiPaperIX.OddEuler.row_action}.\n\\end{proof}","\\begin{corollary}[Odd-prime truncation ladder]\\label{cor:arith-odd-ladder}\nFor every positive odd integer $p$ and all $M,n\\in\\N$,\n\\[\n p^{2M+2}\\mid E_{2n}-T^{\\mathrm{odd}}_{p,M}(n)\\quad\\text{in }\\Z.\n\\]\nIf $M\\ge n$, the truncation equals $E_{2n}$ exactly.  In particular,\nfor an odd prime $p$ this is a congruence modulo $p^{2M+2}$ in $\\Z_p$.\n\\end{corollary}","\\begin{proof}\nEvery omitted summand has $m\\ge M+1$, hence contains the integer factor\n$p^{2m}$ divisible by $p^{2M+2}$.  All its other factors are integral.\nThe finite sum of these omitted terms proves the assertion, with the\nempty-tail case giving exact equality.  These are\n\\lean{KomorebiPaperIX.OddEuler.odd_euler_supercongruence} and\n\\lean{KomorebiPaperIX.OddEuler.full_depth_exact}.\n\\end{proof}","The reconstruction and its truncation bound already hold over the integers.\nFor the dyadic construction we use a different source.  Its original bound\nof $2^{2M+2}$ follows from integral reciprocal coefficients, and the next\nsubsection sharpens that bound to the optimal uniform exponent $4M+3$.","Return now to the classical signed Euler numbers\n\\[\n \\operatorname{sech}t=\\frac1{\\cosh t}\n       =\\sum_{n\\ge0}E_{2n}\\frac{t^{2n}}{(2n)!}.\n\\]\nThus $E_{2n}=E_{2n}^{(2,0)}$ in the convention\n\\eqref{eq:arith-euler-egf}.  In particular $E_0=1$, $E_2=-1$, and $E_4=5$.\nThe following construction concerns these $N=2$ coefficients, not $C_{4,0}$.","\\begin{definition}[Dyadic truncation]\\label{def:arith-truncation}\nDefine $\\beta_m$ by the formal unit-series identity\n\\[\n \\sum_{m\\ge0}\\beta_m z^{2m}=\\left(\\frac{z}{\\sinh z}\\right)^2,\n \\qquad \\beta_0=1,\n\\]\nwhere the quotient means the inverse of the unit series\n$(\\sinh z/z)^2$.  Define\n\\[\n A(n)=\\frac{3^{2n+2}-1}{4(2n+1)(2n+2)},\\qquad\n S_m(n)=\\beta_m2^{2m}\\frac{(2n)!}{(2n-2m)!}A(n-m)\n       \\quad(0\\le m\\le n),\n\\]\nand\n\\begin{equation}\\label{eq:arith-tm}\n T_M(n)=\\sum_{m=0}^{\\min(M,n)}S_m(n)\\qquad(M,n\\in\\N).\n\\end{equation}\n\\end{definition}","n]\\label{def:arith-truncation}\nDefine $\\beta_m$ by the formal unit-series identity\n\\[\n \\sum_{m\\ge0}\\beta_m z^{2m}=\\left(\\frac{z}{\\sinh z}\\right)^2,\n \\qquad \\beta_0=1,\n\\]\nwhere the quotient means the inverse of the unit series\n$(\\sinh z/z)^2$.  Define\n\\[\n A(n)=\\frac{3^{2n+2}-1}{4(2n+1)(2n+2)},\\qquad\n S_m(n)=\\beta_m2^{2m}\\frac{(2n)!}{(2n-2m)!}A(n-m)\n       \\quad(0\\le m\\le n),\n\\]\nand\n\\begin{equation}\\label{eq:arith-tm}\n T_M(n)=\\sum_{m=0}^{\\min(M,n)}S_m(n)\\qquad(M,n\\in\\N).\n\\end{equation}\n\\end{definition}","\\begin{lemma}[Exact reconstruction and source integrality]\n\\label{lem:arith-reconstruction}\nFor every $n\\in\\N$,\n\\[\n E_{2n}=\\sum_{m=0}^{n}S_m(n).\n\\]\nEvery $\\beta_m$ is $2$-adically integral and every $A(n)$ is a nonzero\n$2$-adic unit.  If $M\\ge n$, then $T_M(n)=E_{2n}$.\n\\end{lemma}","\\begin{proof}\nThe ordinary kernel coefficients of $(\\sinh z/z)^2$, in $z^2$, are\n\\[\n \\frac{2^{2m+1}}{(2m+2)!}.\n\\]\nLegendre's strict factorial bound\n$v_2((2m+2)!)<2m+2$ makes them integral.  Their constant coefficient is\none.  The coefficient recurrence for its reciprocal therefore\nproves integrality of every $\\beta_m$.\nFor the source factor, lifting the exponent at the even power $2n+2$\ngives\n\\[\n v_2(3^{2n+2}-1)=2+v_2(2n+2).\n\\]\nIts denominator has the same valuation, since $2n+1$ is odd.  Thus\n$A(n)$ is a nonzero unit.","e\n\\[\n \\frac{2^{2m+1}}{(2m+2)!}.\n\\]\nLegendre's strict factorial bound\n$v_2((2m+2)!)<2m+2$ makes them integral.  Their constant coefficient is\none.  The coefficient recurrence for its reciprocal therefore\nproves integrality of every $\\beta_m$.\nFor the source factor, lifting the exponent at the even power $2n+2$\ngives\n\\[\n v_2(3^{2n+2}-1)=2+v_2(2n+2).\n\\]\nIts denominator has the same valuation, since $2n+1$ is odd.  Thus\n$A(n)$ is a nonzero unit.","{(2m+2)!}.\n\\]\nLegendre's strict factorial bound\n$v_2((2m+2)!)<2m+2$ makes them integral.  Their constant coefficient is\none.  The coefficient recurrence for its reciprocal therefore\nproves integrality of every $\\beta_m$.\nFor the source factor, lifting the exponent at the even power $2n+2$\ngives\n\\[\n v_2(3^{2n+2}-1)=2+v_2(2n+2).\n\\]\nIts denominator has the same valuation, since $2n+1$ is odd.  Thus\n$A(n)$ is a nonzero unit.","Its exponential generating function is\n\\[\n \\mathcal A(t)=\\sum_{n\\ge0}A(n)\\frac{t^{2n}}{(2n)!}\n             =\\frac{\\cosh(3t)-\\cosh t}{4t^2}.\n\\]\nThe quotients by powers of $t$ are well defined because the required\ninitial coefficients vanish.  The hyperbolic identity\n\\[\n \\mathcal A(t)\\cosh t=\\left(\\frac{\\sinh(2t)}{2t}\\right)^2\n\\]\nthen gives the exact formal-series product\n\\[\n \\left(\\sum_{m\\ge0}\\beta_m(2t)^{2m}\\right)\\mathcal A(t)\n       =\\operatorname{sech}t.\n\\]\nTaking the factorial-scaled coefficient of $t^{2n}$ produces the stated\nfinite sum.  Full depth leaves no omitted terms.\nThe relevant declarations are\n\\lean{KomorebiPaperIX.Dyadic.beta_integral},\n\\lean{KomorebiPaperIX.Dyadic.source_valuation},\n\\lean{KomorebiPaperIX.DyadicSeries.dyadic_reconstruction_series}, and\n\\lean{KomorebiPaperIX.DyadicReconstruction.dyadic_reconstruct","Its exponential generating function is\n\\[\n \\mathcal A(t)=\\sum_{n\\ge0}A(n)\\frac{t^{2n}}{(2n)!}\n             =\\frac{\\cosh(3t)-\\cosh t}{4t^2}.\n\\]\nThe quotients by powers of $t$ are well defined because the required\ninitial coefficients vanish.  The hyperbolic identity\n\\[\n \\mathcal A(t)\\cosh t=\\left(\\frac{\\sinh(2t)}{2t}\\right)^2\n\\]\nthen gives the exact formal-series product\n\\[\n \\left(\\sum_{m\\ge0}\\beta_m(2t)^{2m}\\right)\\mathcal A(t)\n       =\\operatorname{sech}t.\n\\]\nTaking the factorial-scaled coefficient of $t^{2n}$ produces the stated\nfinite sum.  Full depth leaves no omitted terms.\nThe relevant declarations are\n\\lean{KomorebiPaperIX.Dyadic.beta_integral},\n\\lean{KomorebiPaperIX.Dyadic.source_valuation},\n\\lean{KomorebiPaperIX.DyadicSeries.dyadic_reconstruction_series}, and\n\\lean{KomorebiPaperIX.DyadicReconstruction.dyadic_reconstruction}.\n\\end{proof}","frac{\\cosh(3t)-\\cosh t}{4t^2}.\n\\]\nThe quotients by powers of $t$ are well defined because the required\ninitial coefficients vanish.  The hyperbolic identity\n\\[\n \\mathcal A(t)\\cosh t=\\left(\\frac{\\sinh(2t)}{2t}\\right)^2\n\\]\nthen gives the exact formal-series product\n\\[\n \\left(\\sum_{m\\ge0}\\beta_m(2t)^{2m}\\right)\\mathcal A(t)\n       =\\operatorname{sech}t.\n\\]\nTaking the factorial-scaled coefficient of $t^{2n}$ produces the stated\nfinite sum.  Full depth leaves no omitted terms.\nThe relevant declarations are\n\\lean{KomorebiPaperIX.Dyadic.beta_integral},\n\\lean{KomorebiPaperIX.Dyadic.source_valuation},\n\\lean{KomorebiPaperIX.DyadicSeries.dyadic_reconstruction_series}, and\n\\lean{KomorebiPaperIX.DyadicReconstruction.dyadic_reconstruction}.\n\\end{proof}","\\begin{lemma}[Exact weighted inverse coefficient]\\label{lem:arith-kappa}\nLet $\\kappa_m=\\beta_m2^{2m}(2m)!$.  For every $m\\ge1$,\n\\[\n \\kappa_m=-2^{4m}(2m-1)B_{2m}\\ne0,\\qquad\n v_2(\\kappa_m)=4m-1,\n\\]\nwhere $B_{2m}$ is the usual even Bernoulli number.\n\\end{lemma}","\\begin{proof}\nLet $Q(z)=z\\coth z$, the regular formal series with constant coefficient\none.  The elementary differential identity\n\\[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli","\\begin{proof}\nLet $Q(z)=z\\coth z$, the regular formal series with constant coefficient\none.  The elementary differential identity\n\\[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{Komore","\\begin{proof}\nLet $Q(z)=z\\coth z$, the regular formal series with constant coefficient\none.  The elementary differential identity\n\\[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","Q(z)=z\\coth z$, the regular formal series with constant coefficient\none.  The elementary differential identity\n\\[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","egular formal series with constant coefficient\none.  The elementary differential identity\n\\[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","ent\none.  The elementary differential identity\n\\[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","The elementary differential identity\n\\[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","[\n \\left(\\frac z{\\sinh z}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","}\\right)^2=Q(z)-zQ'(z)\n\\]\nand the Bernoulli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","lli expansion\n\\[\n Q(z)=1+\\sum_{m\\ge1}\\frac{2^{2m}B_{2m}}{(2m)!}z^{2m}\n\\]\ngive\n$\\beta_m=-2^{2m}(2m-1)B_{2m}/(2m)!$.\nMultiplication by $2^{2m}(2m)!$ proves the identity for $\\kappa_m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","m$.\nThe Bernoulli denominator theorem gives $v_2(B_{2m})=-1$ for $m>0$: its denominator contains each prime $q$ with $(q-1)\\mid 2m$ exactly once, including $2$\\cite{DLMF24}.\nSince $2m-1$ is odd, the valuation is exactly $4m-1$.\nThis also proves nonvanishing.  In Lean, the hyperbolic identity is proved\nafter clearing removable zeros.  The library Bernoulli theorem then\ndetermines the valuation of $\\kappa_m$.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_bernoulli} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.kappa_valuation}.\n\\end{proof}","\\begin{theorem}[Optimal uniform dyadic tower]\\label{thm:arith-sharp-tower}\nFor all $M,n\\in\\N$,\n\\begin{equation}\\label{eq:arith-sharp-tail}\n E_{2n}-T_M(n)\\in2^{4M+3}\\Z_2.\n\\end{equation}\nAt the first omitted row $n=M+1$, the error is nonzero and\n\\[\n v_2\\bigl(E_{2M+2}-T_M(M+1)\\bigr)=4M+3.\n\\]\nThus the exponent $4M+3$ cannot be increased uniformly in the Euler row.\n\\end{theorem}","\\begin{proof}\nRewrite the summand exactly as\n\\[\n S_m(n)=\\kappa_m\\binom{2n}{2m}A(n-m).\n\\]\nFor $m>M$, its first factor is divisible by $2^{4m-1}$, the binomial\ncoefficient is integral, and $A(n-m)$ is a unit.  Hence every omitted\nsummand is divisible by $2^{4M+3}$.  Summing the finite tail proves\n\\eqref{eq:arith-sharp-tail}.  If $M\\ge n$, the tail is empty and the\nstatement includes the resulting zero error.","At $n=M+1$ there is exactly one omitted summand.  Here\n$\\binom{2n}{2n}=1$ and $A(0)=1$, so the error is $\\kappa_{M+1}$.\nLemma~\\ref{lem:arith-kappa} gives its exact valuation $4M+3$ and\nnonvanishing.  The statements are\n\\lean{KomorebiPaperIX.DyadicSharpTower.dyadic_supercongruence_sharp},\n\\lean{KomorebiPaperIX.DyadicSharpTower.first_omitted_error}, and\n\\lean{KomorebiPaperIX.DyadicSharpTower.first_omitted_error_valuation}.\n\\end{proof}","\\begin{proposition}[Row-dependent summand weights]\\label{prop:arith-row-weight}\nFor $1\\le m\\le n$, the summand is nonzero and\n\\[\n v_2(S_m(n))=4m-1+v_2\\binom{2n}{2m}.\n\\]\nMore generally, if $L\\in\\N$ satisfies\n\\[\n L\\le4m-1+v_2\\binom{2n}{2m}\\qquad\\text{for every }M<m\\le n,\n\\]\nthen $E_{2n}-T_M(n)\\in2^L\\Z_2$.\n\\end{proposition}","\\begin{proof}\nUse the same factorization of $S_m(n)$, the nonzero binomial coefficient\nfor $m\\le n$, the exact valuation of $\\kappa_m$, and the unit property\nof $A(n-m)$.  The tail conclusion follows by addition in $2^L\\Z_2$.\nThese are \\lean{KomorebiPaperIX.DyadicSharpTower.summand_valuation} and\n\\lean{KomorebiPaperIX.DyadicSharpTower.dyadic_supercongruence_row_weight}.\n\\end{proof}","The minimum summand weight bounds the tail valuation from below, but\ncancellation may raise that valuation. Appendix~\\ref{app:root:scope} gives a later exact factorization and finite-polynomial reduction for each fixed depth. A universal explicit root and simplicity classification for that factor remains open.  Uniform optimality follows\nfrom the equality already established at $n=M+1$.","\\begin{theorem}[Isometric packet bijections]\\label{thm:arith-isometry}\nFor $0\\le j<4$, define on natural numbers\n\\[\n f_j(n)=\\frac{U_j(n)-j!}{2^{d_j}},\\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).\n\\]\nThere is a unique continuous map $\\Phi_j:\\Z_2\\to\\Z_2$ whose value at\nevery $n\\in\\N$ equals the embedded rational $f_j(n)$.\nIt is an isometric bijection and $\\Phi_j(0)=0$:\n\\[\n |\\Phi_j(x)-\\Phi_j(y)|_2=|x-y|_2\\qquad(x,y\\in\\Z_2).\n\\]\nIn particular, every element of $\\Z_2$ has exactly one preimage.\n\\end{theorem}","thm:arith-isometry}\nFor $0\\le j<4$, define on natural numbers\n\\[\n f_j(n)=\\frac{U_j(n)-j!}{2^{d_j}},\\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).\n\\]\nThere is a unique continuous map $\\Phi_j:\\Z_2\\to\\Z_2$ whose value at\nevery $n\\in\\N$ equals the embedded rational $f_j(n)$.\nIt is an isometric bijection and $\\Phi_j(0)=0$:\n\\[\n |\\Phi_j(x)-\\Phi_j(y)|_2=|x-y|_2\\qquad(x,y\\in\\Z_2).\n\\]\nIn particular, every element of $\\Z_2$ has exactly one preimage.\n\\end{theorem}","\\begin{proof}\nFor $n>0$, the displacement law at row zero gives\n$v_2(U_j(n)-j!)=v_2(n)+d_j$, so $f_j(n)\\in\\Z_2$.\nThe value at zero is zero.  Applying the same law to the smaller of two\nunequal natural numbers shows\n\\[\n |f_j(n)-f_j(m)|_2=|n-m|_2.\n\\]\nThe natural numbers are dense in $\\Z_2$, and the target is complete.\nThe map therefore extends continuously.  The distance identity extends\non the dense product of natural-number points, making the extension\nan isometry.  Density also proves uniqueness among all continuous maps\nwith these prescribed values.","To prove surjectivity, use compactness.  For an isometry $g$ of a compact\nmetric space into itself, the sequence of iterates $g^n(x)$ has a\nconvergent subsequence.  Two sufficiently close iterates, with exponents\n$a<b$, yield $d(x,g^{b-a}(x))$ equally small after removing $a$ isometric\niterations.  Since $b-a>0$, the latter point is in the image of $g$.\nThus the image is dense.  It is also compact and hence closed, so it is\nthe whole space.  Apply this to $\\Phi_j$ on $\\Z_2$.","of a compact\nmetric space into itself, the sequence of iterates $g^n(x)$ has a\nconvergent subsequence.  Two sufficiently close iterates, with exponents\n$a<b$, yield $d(x,g^{b-a}(x))$ equally small after removing $a$ isometric\niterations.  Since $b-a>0$, the latter point is in the image of $g$.\nThus the image is dense.  It is also compact and hence closed, so it is\nthe whole space.  Apply this to $\\Phi_j$ on $\\Z_2$.","metric space into itself, the sequence of iterates $g^n(x)$ has a\nconvergent subsequence.  Two sufficiently close iterates, with exponents\n$a<b$, yield $d(x,g^{b-a}(x))$ equally small after removing $a$ isometric\niterations.  Since $b-a>0$, the latter point is in the image of $g$.\nThus the image is dense.  It is also compact and hence closed, so it is\nthe whole space.  Apply this to $\\Phi_j$ on $\\Z_2$.","\\begin{corollary}[Exact image of the original coefficients]\n\\label{cor:arith-affine-image}\nThe continuous extension of the original sequence is\n\\[\n \\widetilde U_j(x)=j!+2^{d_j}\\Phi_j(x).\n\\]\nIt is injective, its image is exactly $j!+2^{d_j}\\Z_2$, and\n\\[\n |\\widetilde U_j(x)-\\widetilde U_j(y)|_2\n     =2^{-d_j}|x-y|_2.\n\\]\nEvery point of that affine image has exactly one preimage in $\\Z_2$.\n\\end{corollary}","\\begin{proof}\nThe formula agrees with $U_j$ on every natural number.  Continuity and\nthe distance equality follow from the corresponding properties of\n$\\Phi_j$.  Its surjectivity gives precisely the displayed affine image,\nand multiplication by the nonzero scalar $2^{d_j}$ preserves injectivity.\nThe declarations are\n\\lean{KomorebiPaperIX.DyadicFourIsometry.originalExtension_nat},\n\\lean{KomorebiPaperIX.DyadicFourIsometry.originalExtension_dist}, and\n\\lean{KomorebiPaperIX.DyadicFourIsometry.originalExtension_range}.\n\\end{proof}","Unless a different coefficient ring is specified, a polynomial system is a\nfinite list of integer polynomials. All divisibility conditions on vectors are\ncoordinatewise. For a positive integer $q$, write $S_q=\\Z/q\\Z$ and use an\noverline for reduction modulo $q$. For a prime $p$, write\n$\\F_p=S_p$, normalize $|p|_p=p^{-1}$, and put\n\\[\n B_L(x)=x+p^L\\Zp^d.\n\\]\nAn integer center is included in this notation by its natural image in\n$\\Zp^d$. All dimensions below are finite and may be zero. A statement over\n$S_q$ allows composite $q$ and $q=1$ when explicitly stated. A cardinality\nformula in terms of vector-space rank uses prime $p$.","ring is specified, a polynomial system is a\nfinite list of integer polynomials. All divisibility conditions on vectors are\ncoordinatewise. For a positive integer $q$, write $S_q=\\Z/q\\Z$ and use an\noverline for reduction modulo $q$. For a prime $p$, write\n$\\F_p=S_p$, normalize $|p|_p=p^{-1}$, and put\n\\[\n B_L(x)=x+p^L\\Zp^d.\n\\]\nAn integer center is included in this notation by its natural image in\n$\\Zp^d$. All dimensions below are finite and may be zero. A statement over\n$S_q$ allows composite $q$ and $q=1$ when explicitly stated. A cardinality\nformula in terms of vector-space rank uses prime $p$.","ials. All divisibility conditions on vectors are\ncoordinatewise. For a positive integer $q$, write $S_q=\\Z/q\\Z$ and use an\noverline for reduction modulo $q$. For a prime $p$, write\n$\\F_p=S_p$, normalize $|p|_p=p^{-1}$, and put\n\\[\n B_L(x)=x+p^L\\Zp^d.\n\\]\nAn integer center is included in this notation by its natural image in\n$\\Zp^d$. All dimensions below are finite and may be zero. A statement over\n$S_q$ allows composite $q$ and $q=1$ when explicitly stated. A cardinality\nformula in terms of vector-space rank uses prime $p$.","conditions on vectors are\ncoordinatewise. For a positive integer $q$, write $S_q=\\Z/q\\Z$ and use an\noverline for reduction modulo $q$. For a prime $p$, write\n$\\F_p=S_p$, normalize $|p|_p=p^{-1}$, and put\n\\[\n B_L(x)=x+p^L\\Zp^d.\n\\]\nAn integer center is included in this notation by its natural image in\n$\\Zp^d$. All dimensions below are finite and may be zero. A statement over\n$S_q$ allows composite $q$ and $q=1$ when explicitly stated. A cardinality\nformula in terms of vector-space rank uses prime $p$.","ns on vectors are\ncoordinatewise. For a positive integer $q$, write $S_q=\\Z/q\\Z$ and use an\noverline for reduction modulo $q$. For a prime $p$, write\n$\\F_p=S_p$, normalize $|p|_p=p^{-1}$, and put\n\\[\n B_L(x)=x+p^L\\Zp^d.\n\\]\nAn integer center is included in this notation by its natural image in\n$\\Zp^d$. All dimensions below are finite and may be zero. A statement over\n$S_q$ allows composite $q$ and $q=1$ when explicitly stated. A cardinality\nformula in terms of vector-space rank uses prime $p$.","verline for reduction modulo $q$. For a prime $p$, write\n$\\F_p=S_p$, normalize $|p|_p=p^{-1}$, and put\n\\[\n B_L(x)=x+p^L\\Zp^d.\n\\]\nAn integer center is included in this notation by its natural image in\n$\\Zp^d$. All dimensions below are finite and may be zero. A statement over\n$S_q$ allows composite $q$ and $q=1$ when explicitly stated. A cardinality\nformula in terms of vector-space rank uses prime $p$.","For $F=(F_1,\\ldots,F_e)\\in\\Z[X_1,\\ldots,X_d]^e$, define its evaluated\nJacobian by the formal partial derivatives:\n\\[\n DF(x)_{ij}=(\\partial_jF_i)(x),\\qquad\n J_x=DF(x)\\bmod q:S_q^d\\longrightarrow S_q^e.\n\\]\nThe obstruction therefore uses the Jacobian of the given polynomials. The\nfollowing algebraic Taylor identity relates this linear map to polynomial\nevaluation.","\\begin{lemma}[Polynomial Taylor remainder]\\label{lem:polynomial-taylor}\nLet $R$ be a commutative ring, $P\\in R[X_1,\\ldots,X_d]$, $a\\in R^d$, and\n$c\\in R$. There is a polynomial $Q\\in R[U_1,\\ldots,U_d]$, with $Q(0)=0$,\nsuch that for every $u\\in R^d$,\n\\begin{equation}\\label{eq:polynomial-taylor}\n P(a+cu)=P(a)+c\\sum_{j=1}^d(\\partial_jP)(a)u_j+c^2Q(u).\n\\end{equation}\nIn particular, when $R=\\Z$ and $a,u,c$ are integral, subtracting the first\ntwo terms on the right from $P(a+cu)$ gives a multiple of $c^2$.\n\\end{lemma}","}\nLet $R$ be a commutative ring, $P\\in R[X_1,\\ldots,X_d]$, $a\\in R^d$, and\n$c\\in R$. There is a polynomial $Q\\in R[U_1,\\ldots,U_d]$, with $Q(0)=0$,\nsuch that for every $u\\in R^d$,\n\\begin{equation}\\label{eq:polynomial-taylor}\n P(a+cu)=P(a)+c\\sum_{j=1}^d(\\partial_jP)(a)u_j+c^2Q(u).\n\\end{equation}\nIn particular, when $R=\\Z$ and $a,u,c$ are integral, subtracting the first\ntwo terms on the right from $P(a+cu)$ gives a multiple of $c^2$.\n\\end{lemma}","g, $P\\in R[X_1,\\ldots,X_d]$, $a\\in R^d$, and\n$c\\in R$. There is a polynomial $Q\\in R[U_1,\\ldots,U_d]$, with $Q(0)=0$,\nsuch that for every $u\\in R^d$,\n\\begin{equation}\\label{eq:polynomial-taylor}\n P(a+cu)=P(a)+c\\sum_{j=1}^d(\\partial_jP)(a)u_j+c^2Q(u).\n\\end{equation}\nIn particular, when $R=\\Z$ and $a,u,c$ are integral, subtracting the first\ntwo terms on the right from $P(a+cu)$ gives a multiple of $c^2$.\n\\end{lemma}","\\begin{proof}\nApply polynomial induction using constants, addition, and multiplication by a\nvariable. Constants have zero remainder, and remainders add. For the\nmultiplication step, suppose the assertion is known for $P$ and write\n$D_P(U)=\\sum_j(\\partial_jP)(a)U_j$. Multiplication of its expansion by\n$a_i+cU_i$ gives constant term $P(a)a_i$, linear term\n$c(a_iD_P(U)+P(a)U_i)$, and remainder\n\\[\n c^2\\bigl(Q(U)(a_i+cU_i)+D_P(U)U_i\\bigr).\n\\]\nThe product rule identifies the linear term with the derivative of\n$PX_i$. The displayed remainder vanishes at $U=0$. This constructs the\npolynomial remainder before evaluation at $u$, even over rings where\ndivision by $c$ would be invalid.\n\\end{proof}","s have zero remainder, and remainders add. For the\nmultiplication step, suppose the assertion is known for $P$ and write\n$D_P(U)=\\sum_j(\\partial_jP)(a)U_j$. Multiplication of its expansion by\n$a_i+cU_i$ gives constant term $P(a)a_i$, linear term\n$c(a_iD_P(U)+P(a)U_i)$, and remainder\n\\[\n c^2\\bigl(Q(U)(a_i+cU_i)+D_P(U)U_i\\bigr).\n\\]\nThe product rule identifies the linear term with the derivative of\n$PX_i$. The displayed remainder vanishes at $U=0$. This constructs the\npolynomial remainder before evaluation at $u$, even over rings where\ndivision by $c$ would be invalid.\n\\end{proof}","or the\nmultiplication step, suppose the assertion is known for $P$ and write\n$D_P(U)=\\sum_j(\\partial_jP)(a)U_j$. Multiplication of its expansion by\n$a_i+cU_i$ gives constant term $P(a)a_i$, linear term\n$c(a_iD_P(U)+P(a)U_i)$, and remainder\n\\[\n c^2\\bigl(Q(U)(a_i+cU_i)+D_P(U)U_i\\bigr).\n\\]\nThe product rule identifies the linear term with the derivative of\n$PX_i$. The displayed remainder vanishes at $U=0$. This constructs the\npolynomial remainder before evaluation at $u$, even over rings where\ndivision by $c$ would be invalid.\n\\end{proof}","d write\n$D_P(U)=\\sum_j(\\partial_jP)(a)U_j$. Multiplication of its expansion by\n$a_i+cU_i$ gives constant term $P(a)a_i$, linear term\n$c(a_iD_P(U)+P(a)U_i)$, and remainder\n\\[\n c^2\\bigl(Q(U)(a_i+cU_i)+D_P(U)U_i\\bigr).\n\\]\nThe product rule identifies the linear term with the derivative of\n$PX_i$. The displayed remainder vanishes at $U=0$. This constructs the\npolynomial remainder before evaluation at $u$, even over rings where\ndivision by $c$ would be invalid.\n\\end{proof}","_j$. Multiplication of its expansion by\n$a_i+cU_i$ gives constant term $P(a)a_i$, linear term\n$c(a_iD_P(U)+P(a)U_i)$, and remainder\n\\[\n c^2\\bigl(Q(U)(a_i+cU_i)+D_P(U)U_i\\bigr).\n\\]\nThe product rule identifies the linear term with the derivative of\n$PX_i$. The displayed remainder vanishes at $U=0$. This constructs the\npolynomial remainder before evaluation at $u$, even over rings where\ndivision by $c$ would be invalid.\n\\end{proof}","\\begin{theorem}[Finite polynomial obstruction]\\label{thm:finite-polynomial-obstruction}\nLet $q\\ge1$, $L\\ge1$, $x\\in\\Z^d$, and suppose\n$F(x)\\in q^L\\Z^e$. Define the exact integer residual $r_x$ by\n$F(x)=q^Lr_x$, and put\n\\[\n \\mathcal O_x=S_q^e/\\im J_x,\\qquad o_x=[\\overline{r_x}].\n\\]\nThen $\\mathcal O_x$ is finite and\n\\begin{equation}\\label{eq:finite-obstruction}\n o_x=0\\quad\\Longleftrightarrow\\quad\n \\exists y\\in\\Z^d:\\quad\n y\\equiv x\\pmod{q^L},\\qquad F(y)\\equiv0\\pmod{q^{L+1}}.\n\\end{equation}\nNo surjectivity or nonsingularity assumption is required.\n\\end{theorem}","\\begin{proof}\nEvery point in the lower residue class has the form $y=x+q^Lt$ with\n$t\\in\\Z^d$. By Lemma~\\ref{lem:polynomial-taylor} and $2L\\ge L+1$,\n\\[\n F(x+q^Lt)\\equiv q^L\\bigl(r_x+DF(x)t\\bigr)\\pmod{q^{L+1}}.\n\\]\nThe integer $q^L$ is nonzero. Cancelling it in the corresponding divisibility\ncondition proves\n\\begin{equation}\\label{eq:digit-correction}\n F(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n \\quad\\Longleftrightarrow\\quad\n \\overline{r_x}+J_x\\overline t=0.\n\\end{equation}\nInteger vectors reduce onto $S_q^d$. The equation on the right is solvable\nexactly when $\\overline{r_x}$ belongs to $\\im J_x$, since that image is\nclosed under negation. This is precisely $o_x=0$. Finiteness follows because\n$\\mathcal O_x$ is a quotient of the finite module $S_q^e$.\n\\end{proof}",". By Lemma~\\ref{lem:polynomial-taylor} and $2L\\ge L+1$,\n\\[\n F(x+q^Lt)\\equiv q^L\\bigl(r_x+DF(x)t\\bigr)\\pmod{q^{L+1}}.\n\\]\nThe integer $q^L$ is nonzero. Cancelling it in the corresponding divisibility\ncondition proves\n\\begin{equation}\\label{eq:digit-correction}\n F(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n \\quad\\Longleftrightarrow\\quad\n \\overline{r_x}+J_x\\overline t=0.\n\\end{equation}\nInteger vectors reduce onto $S_q^d$. The equation on the right is solvable\nexactly when $\\overline{r_x}$ belongs to $\\im J_x$, since that image is\nclosed under negation. This is precisely $o_x=0$. Finiteness follows because\n$\\mathcal O_x$ is a quotient of the finite module $S_q^e$.\n\\end{proof}","F(x+q^Lt)\\equiv q^L\\bigl(r_x+DF(x)t\\bigr)\\pmod{q^{L+1}}.\n\\]\nThe integer $q^L$ is nonzero. Cancelling it in the corresponding divisibility\ncondition proves\n\\begin{equation}\\label{eq:digit-correction}\n F(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n \\quad\\Longleftrightarrow\\quad\n \\overline{r_x}+J_x\\overline t=0.\n\\end{equation}\nInteger vectors reduce onto $S_q^d$. The equation on the right is solvable\nexactly when $\\overline{r_x}$ belongs to $\\im J_x$, since that image is\nclosed under negation. This is precisely $o_x=0$. Finiteness follows because\n$\\mathcal O_x$ is a quotient of the finite module $S_q^e$.\n\\end{proof}","+DF(x)t\\bigr)\\pmod{q^{L+1}}.\n\\]\nThe integer $q^L$ is nonzero. Cancelling it in the corresponding divisibility\ncondition proves\n\\begin{equation}\\label{eq:digit-correction}\n F(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n \\quad\\Longleftrightarrow\\quad\n \\overline{r_x}+J_x\\overline t=0.\n\\end{equation}\nInteger vectors reduce onto $S_q^d$. The equation on the right is solvable\nexactly when $\\overline{r_x}$ belongs to $\\im J_x$, since that image is\nclosed under negation. This is precisely $o_x=0$. Finiteness follows because\n$\\mathcal O_x$ is a quotient of the finite module $S_q^e$.\n\\end{proof}","it in the corresponding divisibility\ncondition proves\n\\begin{equation}\\label{eq:digit-correction}\n F(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n \\quad\\Longleftrightarrow\\quad\n \\overline{r_x}+J_x\\overline t=0.\n\\end{equation}\nInteger vectors reduce onto $S_q^d$. The equation on the right is solvable\nexactly when $\\overline{r_x}$ belongs to $\\im J_x$, since that image is\nclosed under negation. This is precisely $o_x=0$. Finiteness follows because\n$\\mathcal O_x$ is a quotient of the finite module $S_q^e$.\n\\end{proof}","y\ncondition proves\n\\begin{equation}\\label{eq:digit-correction}\n F(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n \\quad\\Longleftrightarrow\\quad\n \\overline{r_x}+J_x\\overline t=0.\n\\end{equation}\nInteger vectors reduce onto $S_q^d$. The equation on the right is solvable\nexactly when $\\overline{r_x}$ belongs to $\\im J_x$, since that image is\nclosed under negation. This is precisely $o_x=0$. Finiteness follows because\n$\\mathcal O_x$ is a quotient of the finite module $S_q^e$.\n\\end{proof}","-correction}\n F(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n \\quad\\Longleftrightarrow\\quad\n \\overline{r_x}+J_x\\overline t=0.\n\\end{equation}\nInteger vectors reduce onto $S_q^d$. The equation on the right is solvable\nexactly when $\\overline{r_x}$ belongs to $\\im J_x$, since that image is\nclosed under negation. This is precisely $o_x=0$. Finiteness follows because\n$\\mathcal O_x$ is a quotient of the finite module $S_q^e$.\n\\end{proof}","The formal theorem\n\\lean{Komorebi.PolynomialObstruction.finite_polynomial_obstruction}\npresents the image using the integer-domain map\n$\\Z^d\\xrightarrow{DF(x)}\\Z^e\\to S_q^e$. Reduction onto $S_q^d$ identifies\nthat image with $\\im J_x$. The class-preserving comparison is explicit in\n\\lean{Komorebi.PolynomialLiftCounts.obstructionEquiv}. The change of\npresentation does not change the distinguished residual class.","\\begin{proposition}[Affine depth zero]\\label{prop:affine-obstruction}\nLet $F(x)=Ax-b$ for an integer linear map $A:\\Z^d\\to\\Z^e$. The criterion\nof Theorem~\\ref{thm:finite-polynomial-obstruction} holds for every $L\\ge0$,\nwith $J=A\\bmod q$.\n\\end{proposition}","\\begin{proof}\nThe affine identity\n$F(x+q^Lt)=F(x)+q^LAt$ has no quadratic remainder. The proof of\n\\eqref{eq:digit-correction} therefore needs no inequality involving $2L$.\nThe residual and the finite quotient are obtained as above.\n\\end{proof}","The affine statement is formalized by\n\\lean{Komorebi.NativeAffine.finite_affine_obstruction}.\nPositive depth is essential in the nonlinear theorem: for $F(X)=X^2-1$,\n$q=2$, $L=0$, and $x=0$, the Jacobian is zero and the residual class is\nnonzero, yet $y=1$ is a root modulo two in the same residue modulo one.","\\begin{proposition}[Independence of the integer representative]\n\\label{prop:representative-independence}\nUnder the hypotheses of Theorem~\\ref{thm:finite-polynomial-obstruction},\nlet $x'=x+q^Ls$ and write $F(x')=q^Lr_{x'}$. Then\n\\[\n J_{x'}=J_x,\\qquad\n \\overline{r_{x'}}=\\overline{r_x}+J_x\\overline s.\n\\]\nConsequently the two residual classes coincide in the common quotient\n$S_q^e/\\im J_x$.\n\\end{proposition}","\\begin{proof}\nPolynomial evaluation respects coordinatewise congruence. Apply this first\nto every formal partial derivative: $x'\\equiv x\\pmod q$ gives\n$J_{x'}=J_x$. Next apply \\eqref{eq:polynomial-taylor} to $F$ at scale $q^L$.\nSubtract the two exact residual identities and cancel $q^L$ to obtain the\ndisplayed relation modulo $q$. Its additional term belongs to the common\nJacobian image, so it vanishes in the quotient.\n\\end{proof}","This comparison is formalized by\n\\lean{Komorebi.PolynomialRepresentative.representative_independence}.\nThis proves independence of the integer representative within a fixed\npolynomial presentation. It makes no identification of the raw cokernels\nof different equation lists. For example, adjoining a zero equation leaves\nthe roots unchanged while adding an $S_q$ summand to the raw cokernel.","The next obstruction concerns compatibility between individually attainable\ncorrections. Its algebraic form works over an arbitrary commutative ring.\nLet $U,V,W$ be $R$-modules and let $J:U\\to V$, $K:U\\to W$ be linear.\nDefine\n\\begin{equation}\\label{eq:common-quotient}\n C(J,K)=U/(\\ker J+\\ker K).\n\\end{equation}\nAn attainable value $Ju$ has label $[u]\\in C(J,K)$. This is independent of\nthe choice of its preimage because $\\ker J$ is killed in the quotient.\nSimilarly, $Kv$ has label $[v]$. Denote these maps by\n$\\lambda_J:\\im J\\to C(J,K)$ and $\\lambda_K:\\im K\\to C(J,K)$.","oncerns compatibility between individually attainable\ncorrections. Its algebraic form works over an arbitrary commutative ring.\nLet $U,V,W$ be $R$-modules and let $J:U\\to V$, $K:U\\to W$ be linear.\nDefine\n\\begin{equation}\\label{eq:common-quotient}\n C(J,K)=U/(\\ker J+\\ker K).\n\\end{equation}\nAn attainable value $Ju$ has label $[u]\\in C(J,K)$. This is independent of\nthe choice of its preimage because $\\ker J$ is killed in the quotient.\nSimilarly, $Kv$ has label $[v]$. Denote these maps by\n$\\lambda_J:\\im J\\to C(J,K)$ and $\\lambda_K:\\im K\\to C(J,K)$.","inable\ncorrections. Its algebraic form works over an arbitrary commutative ring.\nLet $U,V,W$ be $R$-modules and let $J:U\\to V$, $K:U\\to W$ be linear.\nDefine\n\\begin{equation}\\label{eq:common-quotient}\n C(J,K)=U/(\\ker J+\\ker K).\n\\end{equation}\nAn attainable value $Ju$ has label $[u]\\in C(J,K)$. This is independent of\nthe choice of its preimage because $\\ker J$ is killed in the quotient.\nSimilarly, $Kv$ has label $[v]$. Denote these maps by\n$\\lambda_J:\\im J\\to C(J,K)$ and $\\lambda_K:\\im K\\to C(J,K)$.","ons. Its algebraic form works over an arbitrary commutative ring.\nLet $U,V,W$ be $R$-modules and let $J:U\\to V$, $K:U\\to W$ be linear.\nDefine\n\\begin{equation}\\label{eq:common-quotient}\n C(J,K)=U/(\\ker J+\\ker K).\n\\end{equation}\nAn attainable value $Ju$ has label $[u]\\in C(J,K)$. This is independent of\nthe choice of its preimage because $\\ker J$ is killed in the quotient.\nSimilarly, $Kv$ has label $[v]$. Denote these maps by\n$\\lambda_J:\\im J\\to C(J,K)$ and $\\lambda_K:\\im K\\to C(J,K)$.","s over an arbitrary commutative ring.\nLet $U,V,W$ be $R$-modules and let $J:U\\to V$, $K:U\\to W$ be linear.\nDefine\n\\begin{equation}\\label{eq:common-quotient}\n C(J,K)=U/(\\ker J+\\ker K).\n\\end{equation}\nAn attainable value $Ju$ has label $[u]\\in C(J,K)$. This is independent of\nthe choice of its preimage because $\\ker J$ is killed in the quotient.\nSimilarly, $Kv$ has label $[v]$. Denote these maps by\n$\\lambda_J:\\im J\\to C(J,K)$ and $\\lambda_K:\\im K\\to C(J,K)$.","ring.\nLet $U,V,W$ be $R$-modules and let $J:U\\to V$, $K:U\\to W$ be linear.\nDefine\n\\begin{equation}\\label{eq:common-quotient}\n C(J,K)=U/(\\ker J+\\ker K).\n\\end{equation}\nAn attainable value $Ju$ has label $[u]\\in C(J,K)$. This is independent of\nthe choice of its preimage because $\\ker J$ is killed in the quotient.\nSimilarly, $Kv$ has label $[v]$. Denote these maps by\n$\\lambda_J:\\im J\\to C(J,K)$ and $\\lambda_K:\\im K\\to C(J,K)$.","\\begin{theorem}[Common correction and fiber product]\\label{thm:common-correction}\nFor every $u,v\\in U$,\n\\begin{equation}\\label{eq:common-correction}\n [u]=[v]\\text{ in }C(J,K)\n \\quad\\Longleftrightarrow\\quad\n \\exists w\\in U:\\ Jw=Ju,\\quad Kw=Kv.\n\\end{equation}\nBoth label maps are surjective, and the image of the joint map is exactly\nthe fiber product:\n\\begin{equation}\\label{eq:joint-fiber-product}\n \\im\\bigl(U\\to\\im J\\times\\im K,\\ u\\mapsto(Ju,Ku)\\bigr)\n =\\im J\\mathbin{\\times}_{C(J,K)}\\im K.\n\\end{equation}\nIf either $V$ or $W$ is finite, then $C(J,K)$ is finite.\n\\end{theorem}","n}\\label{eq:common-correction}\n [u]=[v]\\text{ in }C(J,K)\n \\quad\\Longleftrightarrow\\quad\n \\exists w\\in U:\\ Jw=Ju,\\quad Kw=Kv.\n\\end{equation}\nBoth label maps are surjective, and the image of the joint map is exactly\nthe fiber product:\n\\begin{equation}\\label{eq:joint-fiber-product}\n \\im\\bigl(U\\to\\im J\\times\\im K,\\ u\\mapsto(Ju,Ku)\\bigr)\n =\\im J\\mathbin{\\times}_{C(J,K)}\\im K.\n\\end{equation}\nIf either $V$ or $W$ is finite, then $C(J,K)$ is finite.\n\\end{theorem}","l{eq:common-correction}\n [u]=[v]\\text{ in }C(J,K)\n \\quad\\Longleftrightarrow\\quad\n \\exists w\\in U:\\ Jw=Ju,\\quad Kw=Kv.\n\\end{equation}\nBoth label maps are surjective, and the image of the joint map is exactly\nthe fiber product:\n\\begin{equation}\\label{eq:joint-fiber-product}\n \\im\\bigl(U\\to\\im J\\times\\im K,\\ u\\mapsto(Ju,Ku)\\bigr)\n =\\im J\\mathbin{\\times}_{C(J,K)}\\im K.\n\\end{equation}\nIf either $V$ or $W$ is finite, then $C(J,K)$ is finite.\n\\end{theorem}","on}\n [u]=[v]\\text{ in }C(J,K)\n \\quad\\Longleftrightarrow\\quad\n \\exists w\\in U:\\ Jw=Ju,\\quad Kw=Kv.\n\\end{equation}\nBoth label maps are surjective, and the image of the joint map is exactly\nthe fiber product:\n\\begin{equation}\\label{eq:joint-fiber-product}\n \\im\\bigl(U\\to\\im J\\times\\im K,\\ u\\mapsto(Ju,Ku)\\bigr)\n =\\im J\\mathbin{\\times}_{C(J,K)}\\im K.\n\\end{equation}\nIf either $V$ or $W$ is finite, then $C(J,K)$ is finite.\n\\end{theorem}","\\begin{proof}\nIf $[u]=[v]$, write $u-v=s+t$ with $s\\in\\ker J$ and $t\\in\\ker K$.\nThen $w=u-s=v+t$ satisfies both required equalities. Conversely, a common\n$w$ gives\n\\[\n u-v=(u-w)+(w-v)\\in\\ker J+\\ker K.\n\\]\nEvery class $[u]$ is the label of $Ju$ and of $Ku$, proving surjectivity.\nFor attainable $a=Ju$ and $b=Kv$, the first equivalence says exactly that\nequal labels are equivalent to a simultaneous preimage. This proves\n\\eqref{eq:joint-fiber-product}. Finally, a finite target has finite image,\nand the associated surjective label map makes $C(J,K)$ finite.\n\\end{proof}","y, a common\n$w$ gives\n\\[\n u-v=(u-w)+(w-v)\\in\\ker J+\\ker K.\n\\]\nEvery class $[u]$ is the label of $Ju$ and of $Ku$, proving surjectivity.\nFor attainable $a=Ju$ and $b=Kv$, the first equivalence says exactly that\nequal labels are equivalent to a simultaneous preimage. This proves\n\\eqref{eq:joint-fiber-product}. Finally, a finite target has finite image,\nand the associated surjective label map makes $C(J,K)$ finite.\n\\end{proof}","These statements are\n\\lean{Komorebi.SharedCorrection.quotient_mk_eq_iff_common} and\n\\lean{Komorebi.SharedCorrection.joint_range_eq_fiber}, together with the\nfinite-target corollaries in the same module. The quotient also has the\nexpected universal property: every linear map from $U$ annihilating both\nkernels factors uniquely through $C(J,K)$. Equivalently, compatible linear\nobservations on $\\im J$ and $\\im K$ factor uniquely through their label\nmaps. This is formalized by\n\\lean{Komorebi.SharedCorrection.common_paired_universal}.","statements are\n\\lean{Komorebi.SharedCorrection.quotient_mk_eq_iff_common} and\n\\lean{Komorebi.SharedCorrection.joint_range_eq_fiber}, together with the\nfinite-target corollaries in the same module. The quotient also has the\nexpected universal property: every linear map from $U$ annihilating both\nkernels factors uniquely through $C(J,K)$. Equivalently, compatible linear\nobservations on $\\im J$ and $\\im K$ factor uniquely through their label\nmaps. This is formalized by\n\\lean{Komorebi.SharedCorrection.common_paired_universal}.","\\begin{corollary}[Simultaneous polynomial lifting]\\label{cor:shared-polynomial}\nLet $F\\in\\Z[X_1,\\ldots,X_d]^e$ and\n$G\\in\\Z[X_1,\\ldots,X_d]^f$ have a common lower root $x$ modulo $q^L$,\nwhere $q\\ge1$ and $L\\ge1$. Use the integer-domain reduced Jacobians\n$J:\\Z^d\\to S_q^e$ and $K:\\Z^d\\to S_q^f$. Suppose $u,v\\in\\Z^d$ are\nseparately valid corrections for $F$ and $G$, respectively. Then\n\\[\n [u]=[v]\\text{ in }\\Z^d/(\\ker J+\\ker K)\n\\]\nif and only if a single $y\\equiv x\\pmod{q^L}$ solves both systems modulo\n$q^{L+1}$. Each separate label is independent of the chosen valid\ncorrection. The common quotient is finite.\n\\end{corollary}","l{cor:shared-polynomial}\nLet $F\\in\\Z[X_1,\\ldots,X_d]^e$ and\n$G\\in\\Z[X_1,\\ldots,X_d]^f$ have a common lower root $x$ modulo $q^L$,\nwhere $q\\ge1$ and $L\\ge1$. Use the integer-domain reduced Jacobians\n$J:\\Z^d\\to S_q^e$ and $K:\\Z^d\\to S_q^f$. Suppose $u,v\\in\\Z^d$ are\nseparately valid corrections for $F$ and $G$, respectively. Then\n\\[\n [u]=[v]\\text{ in }\\Z^d/(\\ker J+\\ker K)\n\\]\nif and only if a single $y\\equiv x\\pmod{q^L}$ solves both systems modulo\n$q^{L+1}$. Each separate label is independent of the chosen valid\ncorrection. The common quotient is finite.\n\\end{corollary}","_d]^e$ and\n$G\\in\\Z[X_1,\\ldots,X_d]^f$ have a common lower root $x$ modulo $q^L$,\nwhere $q\\ge1$ and $L\\ge1$. Use the integer-domain reduced Jacobians\n$J:\\Z^d\\to S_q^e$ and $K:\\Z^d\\to S_q^f$. Suppose $u,v\\in\\Z^d$ are\nseparately valid corrections for $F$ and $G$, respectively. Then\n\\[\n [u]=[v]\\text{ in }\\Z^d/(\\ker J+\\ker K)\n\\]\nif and only if a single $y\\equiv x\\pmod{q^L}$ solves both systems modulo\n$q^{L+1}$. Each separate label is independent of the chosen valid\ncorrection. The common quotient is finite.\n\\end{corollary}","\\begin{proof}\nFor each system, \\eqref{eq:digit-correction} translates a valid next root\ninto a linear correction equation. Once $u$ corrects $F$, another digit\n$w$ corrects $F$ precisely when $Jw=Ju$. The analogous condition for $G$\nis $Kw=Kv$. Theorem~\\ref{thm:common-correction} now gives the criterion.\nTwo valid corrections of one block differ by a vector in its Jacobian\nkernel, so they have equal labels. Finiteness follows from either finite\nresidue target. An unconditional formulation quantifies over separately\nvalid $u,v$ with equal labels, thereby including their existence in the\ncriterion rather than assuming it.\n\\end{proof}","$u$ corrects $F$, another digit\n$w$ corrects $F$ precisely when $Jw=Ju$. The analogous condition for $G$\nis $Kw=Kv$. Theorem~\\ref{thm:common-correction} now gives the criterion.\nTwo valid corrections of one block differ by a vector in its Jacobian\nkernel, so they have equal labels. Finiteness follows from either finite\nresidue target. An unconditional formulation quantifies over separately\nvalid $u,v$ with equal labels, thereby including their existence in the\ncriterion rather than assuming it.\n\\end{proof}","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged.","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstate","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lif","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komoreb","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffineP","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.fini","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibili","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSep","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability a","For affine integer systems $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability alone is insufficient. At $x=0$ modulo two,","ms $F(X)=AX-b$ and $G(X)=BX-c$, where\n$A:\\Z^d\\to\\Z^e$ and $B:\\Z^d\\to\\Z^f$ are linear,\nCorollary~\\ref{cor:shared-polynomial} holds for $q\\ge1$ and every\n$L\\ge0$, including depth zero. Here $x$ is a common root modulo $q^L$,\nand $J$ and $K$ are the reductions of $A$ and $B$ with domain $\\Z^d$.\nGiven separately valid corrections $u,v$, a simultaneous next lift exists\nif and only if their classes in $\\Z^d/(\\ker J+\\ker K)$ agree. The quotient\nis finite, and each label is independent of the chosen valid correction.\nIndeed, Proposition~\\ref{prop:affine-obstruction} supplies the correction\nequation without a positive-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability alone is insufficient. At $x=0$ modulo two,\n$F(X)=X$ and $G(X)=X-2$ both lift to modulus four, but their correc…","-depth restriction, and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability alone is insufficient. At $x=0$ modulo two,\n$F(X)=X$ and $G(X)=X-2$ both lift to modulus four, but their correcting\ndigits are respectively $0$ and $1$ modulo two. No single residue solves\nboth equations modulo four.",", and the proof of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability alone is insufficient. At $x=0$ modulo two,\n$F(X)=X$ and $G(X)=X-2$ both lift to modulus four, but their correcting\ndigits are respectively $0$ and $1$ modulo two. No single residue solves\nboth equations modulo four.","of of\nTheorem~\\ref{thm:common-correction} then applies unchanged. The formal\nstatement is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability alone is insufficient. At $x=0$ modulo two,\n$F(X)=X$ and $G(X)=X-2$ both lift to modulus four, but their correcting\ndigits are respectively $0$ and $1$ modulo two. No single residue solves\nboth equations modulo four.","ent is \\lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability alone is insufficient. At $x=0$ modulo two,\n$F(X)=X$ and $G(X)=X-2$ both lift to modulus four, but their correcting\ndigits are respectively $0$ and $1$ modulo two. No single residue solves\nboth equations modulo four.","lean{Komorebi.AffinePair.joint_lift_iff_class_eq}.\nQuantifying over separately valid corrections with equal labels also\nincludes their existence in the criterion, as in\n\\lean{Komorebi.AffinePair.finite_affine_pair_compatibility}.\nSeparate liftability alone is insufficient. At $x=0$ modulo two,\n$F(X)=X$ and $G(X)=X-2$ both lift to modulus four, but their correcting\ndigits are respectively $0$ and $1$ modulo two. No single residue solves\nboth equations modulo four.","Integer representatives and residue classes at the next level must also be\ndistinguished. For $q\\ge1$ and $L\\ge0$, the map\n\\[\n S_q^d\\longrightarrow\n \\{y\\in(\\Z/q^{L+1}\\Z)^d:y\\equiv x\\pmod{q^L}\\},\n \\qquad t\\longmapsto x+q^L\\widetilde t\n\\]\nis a bijection, where $\\widetilde t$ is any integer representative.\nIndependence of the representative and injectivity both follow by cancelling\n$q^L$ in integer divisibility. Surjectivity follows by expressing the lower\ncongruence as a multiple of $q^L$. This is the equivalence\n\\lean{Komorebi.ResidueLiftDigits.digitEquivResidue}.","d. For $q\\ge1$ and $L\\ge0$, the map\n\\[\n S_q^d\\longrightarrow\n \\{y\\in(\\Z/q^{L+1}\\Z)^d:y\\equiv x\\pmod{q^L}\\},\n \\qquad t\\longmapsto x+q^L\\widetilde t\n\\]\nis a bijection, where $\\widetilde t$ is any integer representative.\nIndependence of the representative and injectivity both follow by cancelling\n$q^L$ in integer divisibility. Surjectivity follows by expressing the lower\ncongruence as a multiple of $q^L$. This is the equivalence\n\\lean{Komorebi.ResidueLiftDigits.digitEquivResidue}.","At positive polynomial depth, restrict it to the correction fiber\n\\[\n \\mathcal T_x=\\{t\\in S_q^d:\\overline{r_x}+J_xt=0\\}\n\\]\nand the lift set\n\\[\n \\mathcal L_{x,L}=\n \\{y\\in(\\Z/q^{L+1}\\Z)^d:y\\equiv x\\pmod{q^L},\\ F(y)=0\\}.\n\\]\nThe polynomial predicate in the second display is evaluation in the residue\nring. Evaluation on canonical integer representatives gives the same\npredicate, because integer polynomial evaluation respects congruence.","\\begin{theorem}[Torsor and lift cardinalities]\\label{thm:lift-torsor-count}\nUnder the hypotheses of Theorem~\\ref{thm:finite-polynomial-obstruction},\nthere is an equivalence $\\mathcal T_x\\simeq\\mathcal L_{x,L}$.\nEvery nonempty fiber is a torsor for $\\ker J_x$ under addition of\ncorrection digits. If $q=p$ is prime and\n$\\rho_x=\\rank_{\\F_p}J_x$, then\n\\begin{equation}\\label{eq:lift-count}\n \\#\\mathcal L_{x,L}=\n \\begin{cases}p^{d-\\rho_x},&o_x=0,","\\begin{theorem}[Torsor and lift cardinalities]\\label{thm:lift-torsor-count}\nUnder the hypotheses of Theorem~\\ref{thm:finite-polynomial-obstruction},\nthere is an equivalence $\\mathcal T_x\\simeq\\mathcal L_{x,L}$.\nEvery nonempty fiber is a torsor for $\\ker J_x$ under addition of\ncorrection digits. If $q=p$ is prime and\n$\\rho_x=\\rank_{\\F_p}J_x$, then\n\\begin{equation}\\label{eq:lift-count}\n \\#\\mathcal L_{x,L}=\n \\begin{cases}p^{d-\\rho_x},&o_x=0,\\\\0,&o_x\\ne0,\\end{cases}\n \\qquad\n \\#\\mathcal O_x=p^{e-\\rho_x}.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nEquation~\\eqref{eq:digit-correction} identifies exactly which points of the\nresidue fiber satisfy the equations, proving the restricted equivalence.\nIf $u,t\\in\\mathcal T_x$, subtraction gives $J_x(t-u)=0$. Conversely,\nadding any element of $\\ker J_x$ preserves the correction equation. The\nunique kernel vector carrying $u$ to $t$ is $t-u$, so the action is free\nand transitive. Choosing one valid digit identifies the fiber with the\nkernel, without making that choice part of its translation action.","For prime $p$, rank-nullity gives\n$\\dim\\ker J_x=d-\\rho_x$ and\n$\\dim(\\F_p^e/\\im J_x)=e-\\rho_x$. The cardinality of an\n$n$-dimensional $\\F_p$-space is $p^n$. Combine these identities with the\nobstruction criterion and the residue equivalence.\n\\end{proof}","The cardinality formula is formalized by\n\\lean{Komorebi.PolynomialLiftCounts.finite_polynomial_lift_count}.\nThe explicit torsor and chosen-point equivalence are developed in\n\\lean{Komorebi.FiniteFiberCounts.translate_exists_unique} and\n\\lean{Komorebi.PolynomialLiftCounts.liftEquivKernel}.\nFor composite $q$, the torsor and kernel cardinality remain valid, while the\nvector-space rank formula is unavailable. Multiplication by two on\n$\\Z/4\\Z$ already has a kernel of size two.","ed by\n\\lean{Komorebi.PolynomialLiftCounts.finite_polynomial_lift_count}.\nThe explicit torsor and chosen-point equivalence are developed in\n\\lean{Komorebi.FiniteFiberCounts.translate_exists_unique} and\n\\lean{Komorebi.PolynomialLiftCounts.liftEquivKernel}.\nFor composite $q$, the torsor and kernel cardinality remain valid, while the\nvector-space rank formula is unavailable. Multiplication by two on\n$\\Z/4\\Z$ already has a kernel of size two.","\\begin{theorem}[Compatible polynomial tower]\\label{thm:smooth-tower}\nLet $q\\ge1$, $L\\ge1$, $F\\in\\Z[X_1,\\ldots,X_d]^e$, and\n$x\\in\\Z^d$ satisfy $F(x)\\in q^L\\Z^e$. Suppose $J_x:S_q^d\\to S_q^e$\nis surjective. There is a single sequence $z_n\\in\\Z^d$ such that\n\\begin{align*}\n z_0&=x,& F(z_n)&\\in q^{L+n}\\Z^e,","\\begin{theorem}[Compatible polynomial tower]\\label{thm:smooth-tower}\nLet $q\\ge1$, $L\\ge1$, $F\\in\\Z[X_1,\\ldots,X_d]^e$, and\n$x\\in\\Z^d$ satisfy $F(x)\\in q^L\\Z^e$. Suppose $J_x:S_q^d\\to S_q^e$\nis surjective. There is a single sequence $z_n\\in\\Z^d$ such that\n\\begin{align*}\n z_0&=x,& F(z_n)&\\in q^{L+n}\\Z^e,\\\\\n z_{n+1}-z_n&\\in q^{L+n}\\Z^d,& J_{z_n}&=J_x.\n\\end{align*}\nFor $a\\le b$, it also satisfies $z_b-z_a\\in q^{L+a}\\Z^d$.\n\\end{theorem}","z_{n+1}-z_n&\\in q^{L+n}\\Z^d,& J_{z_n}&=J_x.\n\\end{align*}\nFor $a\\le b$, it also satisfies $z_b-z_a\\in q^{L+a}\\Z^d$.\n\\end{theorem}","\\begin{proof}\nAt level $N=L+n$, divide the residual by $q^N$. Surjectivity supplies\n$t\\in S_q^d$ with $J_{z_n}t=-\\overline{F(z_n)/q^N}$. Choose an integer\nrepresentative and apply \\eqref{eq:digit-correction} to obtain $z_{n+1}$.\nThe new point is congruent to $z_n$ modulo $q$. Evaluation of each formal\npartial derivative therefore gives $J_{z_{n+1}}=J_{z_n}$.","Construct the sequence recursively, retaining the root condition and the\nfixed Jacobian at each step. Surjectivity at every later stage then follows\nfrom the single initial assumption. The cross-level congruence follows by\ninduction, adding consecutive differences whose divisibilities all imply divisibility\nby $q^{L+a}$.\n\\end{proof}","\\begin{theorem}[Completion and smooth branching]\\label{thm:smooth-padic-root}\nIn Theorem~\\ref{thm:smooth-tower}, let $q=p$ be prime. The tower can be\nchosen with a limit $a\\in\\Zp^d$ such that\n\\[\n F(a)=0,\\qquad z_n\\longrightarrow a,\\qquad\n a-z_n\\in p^{L+n}\\Zp^d\\quad(n\\ge0).\n\\]\nAt each stage, the number of distinct next-level residue roots over $z_n$\nis exactly $p^{d-e}$, and the finite obstruction module has one element.\nIf $d=e$, there is exactly one root of $F$ in $B_L(x)$.\n\\end{theorem}","nd smooth branching]\\label{thm:smooth-padic-root}\nIn Theorem~\\ref{thm:smooth-tower}, let $q=p$ be prime. The tower can be\nchosen with a limit $a\\in\\Zp^d$ such that\n\\[\n F(a)=0,\\qquad z_n\\longrightarrow a,\\qquad\n a-z_n\\in p^{L+n}\\Zp^d\\quad(n\\ge0).\n\\]\nAt each stage, the number of distinct next-level residue roots over $z_n$\nis exactly $p^{d-e}$, and the finite obstruction module has one element.\nIf $d=e$, there is exactly one root of $F$ in $B_L(x)$.\n\\end{theorem}","h-padic-root}\nIn Theorem~\\ref{thm:smooth-tower}, let $q=p$ be prime. The tower can be\nchosen with a limit $a\\in\\Zp^d$ such that\n\\[\n F(a)=0,\\qquad z_n\\longrightarrow a,\\qquad\n a-z_n\\in p^{L+n}\\Zp^d\\quad(n\\ge0).\n\\]\nAt each stage, the number of distinct next-level residue roots over $z_n$\nis exactly $p^{d-e}$, and the finite obstruction module has one element.\nIf $d=e$, there is exactly one root of $F$ in $B_L(x)$.\n\\end{theorem}","-root}\nIn Theorem~\\ref{thm:smooth-tower}, let $q=p$ be prime. The tower can be\nchosen with a limit $a\\in\\Zp^d$ such that\n\\[\n F(a)=0,\\qquad z_n\\longrightarrow a,\\qquad\n a-z_n\\in p^{L+n}\\Zp^d\\quad(n\\ge0).\n\\]\nAt each stage, the number of distinct next-level residue roots over $z_n$\nis exactly $p^{d-e}$, and the finite obstruction module has one element.\nIf $d=e$, there is exactly one root of $F$ in $B_L(x)$.\n\\end{theorem}","\\begin{proof}\nCross-level compatibility bounds the supremum distance between later terms\nby $p^{-(L+n)}$. Completeness gives a limit, and closed norm bounds retain\nthe congruence to every $z_n$. View the coefficients of the original integer\npolynomials in $\\Zp$. Their evaluations satisfy\nboth $F_i(z_n)\\to F_i(a)$, by continuity, and $F_i(z_n)\\to0$, by their\ngrowing divisibility. Uniqueness of limits gives exact vanishing.","Surjectivity gives $\\rank J_x=e\\le d$. Theorem~\\ref{thm:lift-torsor-count}\nthen gives $p^{d-e}$ next residue roots and obstruction cardinality one at\nevery stage.","For square uniqueness, let $a,b\\in B_L(x)$ be arbitrary exact roots.\nTheir canonical integer truncations are genuine roots in the respective\nfinite residue rings, since evaluation commutes with reduction. They agree\nat depth $L$. If they agree at depth $N\\ge L$, both next residues belong\nto the same lift set, which has cardinality $p^{d-d}=1$. Induction gives\nagreement at every depth. The residue maps separate $p$-adic integers, so\n$a=b$. Existence was already supplied by the tower.\n\\end{proof}","ueness, let $a,b\\in B_L(x)$ be arbitrary exact roots.\nTheir canonical integer truncations are genuine roots in the respective\nfinite residue rings, since evaluation commutes with reduction. They agree\nat depth $L$. If they agree at depth $N\\ge L$, both next residues belong\nto the same lift set, which has cardinality $p^{d-d}=1$. Induction gives\nagreement at every depth. The residue maps separate $p$-adic integers, so\n$a=b$. Existence was already supplied by the tower.\n\\end{proof}","L(x)$ be arbitrary exact roots.\nTheir canonical integer truncations are genuine roots in the respective\nfinite residue rings, since evaluation commutes with reduction. They agree\nat depth $L$. If they agree at depth $N\\ge L$, both next residues belong\nto the same lift set, which has cardinality $p^{d-d}=1$. Induction gives\nagreement at every depth. The residue maps separate $p$-adic integers, so\n$a=b$. Existence was already supplied by the tower.\n\\end{proof}","\\begin{remark}\\label{rem:singular-tower-boundary}\nThe one-step singular criterion and the smooth infinite theorem have\ndifferent hypotheses. The polynomial $X^2-5$ has odd roots modulo two and\nfour but has no root modulo eight. Thus a previous zero obstruction does\nnot force the next one to vanish. Also, rectangular surjectivity alone\ndoes not imply uniqueness: $F(X,Y)=X$ has the distinct dyadic roots\n$(0,0)$ and $(0,2)$ in the same residue ball modulo two. These examples show\nwhy the hypotheses of the tower theorem and its uniqueness clause are needed.\n\\end{remark}","{remark}\\label{rem:singular-tower-boundary}\nThe one-step singular criterion and the smooth infinite theorem have\ndifferent hypotheses. The polynomial $X^2-5$ has odd roots modulo two and\nfour but has no root modulo eight. Thus a previous zero obstruction does\nnot force the next one to vanish. Also, rectangular surjectivity alone\ndoes not imply uniqueness: $F(X,Y)=X$ has the distinct dyadic roots\n$(0,0)$ and $(0,2)$ in the same residue ball modulo two. These examples show\nwhy the hypotheses of the tower theorem and its uniqueness clause are needed.\n\\end{remark}","el{rem:singular-tower-boundary}\nThe one-step singular criterion and the smooth infinite theorem have\ndifferent hypotheses. The polynomial $X^2-5$ has odd roots modulo two and\nfour but has no root modulo eight. Thus a previous zero obstruction does\nnot force the next one to vanish. Also, rectangular surjectivity alone\ndoes not imply uniqueness: $F(X,Y)=X$ has the distinct dyadic roots\n$(0,0)$ and $(0,2)$ in the same residue ball modulo two. These examples show\nwhy the hypotheses of the tower theorem and its uniqueness clause are needed.\n\\end{remark}","\\begin{proposition}[Smooth coefficient stability]\\label{prop:smooth-coefficient-stability}\nLet $p$ be prime, $M\\ge1$, $F,H\\in\\Z[X_1,\\ldots,X_d]^e$, and\n$c\\in p^M\\Z$. Suppose $DF(x)\\bmod p$ is surjective for an integer\n$x\\in\\Z^d$. If $a\\in\\Zp^d$ satisfies $F(a)=0$ and\n$a\\equiv x\\pmod p$, then there is a root $b$ of $G=F+cH$ with\n$b-a\\in p^M\\Zp^d$.","If $d=e$ and $M\\ge L\\ge1$, every pair of roots $a,b\\in B_L(x)$ of\n$F$ and $G$, respectively, satisfies $b-a\\in p^M\\Zp^d$. The perturbed\nroot is unique in that ball whenever it exists.\n\\end{proposition}","\\begin{proof}\nSince $p\\mid c$, differentiation gives\n$DG(x)\\equiv DF(x)\\pmod p$. Truncate $a$ to an integer vector $y$ at\ndepth $M$. Then $F(y)\\equiv0\\pmod{p^M}$, hence also\n$G(y)\\equiv0\\pmod{p^M}$. Congruence of $y$ and $x$ modulo $p$ preserves\nthe reduced Jacobian. The smooth root theorem gives\n$b\\equiv y\\equiv a\\pmod{p^M}$.","In the square case this constructed root lies in $B_L(x)$ because $M\\ge L$.\nSquare uniqueness for $G$ identifies it with every competing root in that\nball. The construction therefore gives the precision assertion for an\narbitrary root there.\n\\end{proof}","See \\lean{Komorebi.PolynomialRootStability.exists_nearby_perturbed_root}\nand \\lean{Komorebi.PolynomialRootStability.perturbed_roots_congruent}.\nThe scalar $c$ may be negative or zero. The weighted theorems below allow\narbitrary integral $p$-adic coefficients and give an explicit first-order\ndisplacement.","\\begin{theorem}[Continuous charts with a selected Jacobian minor]\n\\label{thm:continuous-parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv","\\begin{theorem}[Continuous charts with a selected Jacobian minor]\n\\label{thm:continuous-parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end","\\begin{theorem}[Continuous charts with a selected Jacobian minor]\n\\label{thm:continuous-parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma","\\begin{theorem}[Continuous charts with a selected Jacobian minor]\n\\label{thm:continuous-parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ f","\\begin{theorem}[Continuous charts with a selected Jacobian minor]\n\\label{thm:continuous-parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to","\\begin{theorem}[Continuous charts with a selected Jacobian minor]\n\\label{thm:continuous-parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","arts with a selected Jacobian minor]\n\\label{thm:continuous-parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","parameter-chart}\nLet $p$ be prime, $L\\ge1$,\n$F\\in\\Z[X_1,\\ldots,X_{e+k}]^e$, and $x\\in\\Z^{e+k}$ satisfy\n$F(x)\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","\\in p^L\\Z^e$. Suppose the full reduced Jacobian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","ian is surjective.\nThere is a permutation $\\sigma$ of the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","the coordinates such that, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","t, writing\n$\\sigma x=(y_0,t_0)$ with $e$ solved coordinates and $k$ parameter\ncoordinates, the parameter ball\n$T=t_0+p^L\\Zp^k$ is homeomorphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","rphic to the entire root locus\n\\[\n \\mathcal R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","R=\\{z\\in B_L(x):F(z)=0\\}.\n\\]\nThe inverse homeomorphism is the actual free-coordinate projection\n$z\\mapsto\\operatorname{free}(\\sigma z)$. If $h:T\\to\\mathcal R$ is the\nforward map, then for every $N\\ge0$,\n\\begin{equation}\\label{eq:chart-congruence}\n h(t)\\equiv h(u)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad t\\equiv u\\pmod{p^N}.\n\\end{equation}\nEquivalently, $\\sigma h(t)=(\\phi(t),t)$ for a continuous\n$\\phi:T\\to y_0+p^L\\Zp^e$ whose graph is the entire reordered root locus.\n\\end{theorem}","\\begin{proof}\nFirst consider a square integer polynomial map $P$ whose reduced Jacobian\nat $x$ is surjective. Its Jacobian is invertible over $\\F_p$. A digit\ncomparison using Taylor expansion shows that equal outputs modulo\n$p^{N+1}$ force equal inputs modulo $p^{N+1}$ whenever the inputs already\nagree modulo $p^N$ and belong to $B_1(x)$. Induction and integer truncation\ngive, for arbitrary $a,b\\in B_1(x)$,\n\\[\n P(a)\\equiv P(b)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad a\\equiv b\\pmod{p^N}.\n\\]","First consider a square integer polynomial map $P$ whose reduced Jacobian\nat $x$ is surjective. Its Jacobian is invertible over $\\F_p$. A digit\ncomparison using Taylor expansion shows that equal outputs modulo\n$p^{N+1}$ force equal inputs modulo $p^{N+1}$ whenever the inputs already\nagree modulo $p^N$ and belong to $B_1(x)$. Induction and integer truncation\ngive, for arbitrary $a,b\\in B_1(x)$,\n\\[\n P(a)\\equiv P(b)\\pmod{p^N}\n \\quad\\Longleftrightarrow\\quad a\\equiv b\\pmod{p^N}.\n\\]","More generally, for a rectangular map with surjective reduced Jacobian,\n\\begin{equation}\\label{eq:polynomial-ball-image}\n P(B_L(x))=P(x)+p^L\\Zp^e.\n\\end{equation}\nOne inclusion follows from polynomial congruence. For integer targets in\nthe right-hand ball, subtract the target from the constant terms and use\nthe smooth root theorem. For arbitrary $p$-adic targets, approximate by\ninteger truncations. The input ball is compact, so its continuous image is\nclosed. It contains all these approximants and hence their limit. This\nargument does not assume that independently chosen preimages form a\ncompatible sequence.","For square $P$, the resulting bijection between the two compact residue\nballs is continuous, and its target is Hausdorff. It is therefore a\nhomeomorphism, with the same congruence preservation for its inverse.","Now suppose a chosen set of $e$ columns of $DF(x)\\bmod p$ is independent,\nand put those coordinates first. Apply the square result to the augmented\npolynomial map\n\\[\n \\Psi(Y,T)=(F(Y,T),T).\n\\]\nIts reduced Jacobian has upper blocks $(D_YF,D_TF)$ and lower blocks\n$(0,I_k)$, so it is invertible. For $t\\in T$, the target $(0,t)$ belongs\nto $\\Psi(x)+p^L\\Zp^{e+k}$ because $F(x)\\equiv0\\pmod{p^L}$.\nThe inverse of $\\Psi$ at that target gives the unique root with parameters\n$t$. The inverse congruence property gives one direction of\n\\eqref{eq:chart-congruence}. Projection onto the parameter coordinates\ngives the other.","se a chosen set of $e$ columns of $DF(x)\\bmod p$ is independent,\nand put those coordinates first. Apply the square result to the augmented\npolynomial map\n\\[\n \\Psi(Y,T)=(F(Y,T),T).\n\\]\nIts reduced Jacobian has upper blocks $(D_YF,D_TF)$ and lower blocks\n$(0,I_k)$, so it is invertible. For $t\\in T$, the target $(0,t)$ belongs\nto $\\Psi(x)+p^L\\Zp^{e+k}$ because $F(x)\\equiv0\\pmod{p^L}$.\nThe inverse of $\\Psi$ at that target gives the unique root with parameters\n$t$. The inverse congruence property gives one direction of\n\\eqref{eq:chart-congruence}. Projection onto the parameter coordinates\ngives the other.","ent,\nand put those coordinates first. Apply the square result to the augmented\npolynomial map\n\\[\n \\Psi(Y,T)=(F(Y,T),T).\n\\]\nIts reduced Jacobian has upper blocks $(D_YF,D_TF)$ and lower blocks\n$(0,I_k)$, so it is invertible. For $t\\in T$, the target $(0,t)$ belongs\nto $\\Psi(x)+p^L\\Zp^{e+k}$ because $F(x)\\equiv0\\pmod{p^L}$.\nThe inverse of $\\Psi$ at that target gives the unique root with parameters\n$t$. The inverse congruence property gives one direction of\n\\eqref{eq:chart-congruence}. Projection onto the parameter coordinates\ngives the other.","Finally, surjectivity of the original rectangular Jacobian guarantees a\nbasis among its columns. Extend the corresponding injection of coordinate\nindices to a permutation $\\sigma$. Renaming the variables by\n$\\sigma^{-1}$ transports polynomial evaluation and every formal partial\nderivative. The fixed-coordinate construction therefore applies to the\noriginal system after undoing the permutation.\n\\end{proof}","The main theorem is\n\\lean{Komorebi.PolynomialGeneralChart.exists_parameter_homeomorph}.\nThe square local inverse and surjectivity for arbitrary targets are\n\\lean{Komorebi.PolynomialLocalInverse.localHomeomorph} and\n\\lean{Komorebi.PolynomialLocalSurjective.exists_padic_preimage}.\nThe center $x$ is only an approximate root: the chart need not satisfy\n$\\phi(t_0)=y_0$ unless $F(x)=0$ exactly. The assertion selects suitable\ncoordinates. It does not permit an arbitrary prescribed solved block.","Suppose now that the split into solved variables and parameter variables\nis fixed in advance. Let\n\\[\n F\\in\\Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\\qquad x=(y_0,t_0),\n\\]\nand assume $F(x)=q^Lr_x$ with $q\\ge1$ and $L\\ge1$. Write the reduced\nJacobian as\n\\[\n DF(x)\\bmod q=(J_x\\ K_x),\\qquad\n J_x:S_q^m\\to S_q^e,\\quad K_x:S_q^k\\to S_q^e.\n\\]\nFor a prescribed parameter digit $v\\in S_q^k$, let $\\mathcal L_{x,L,v}$\nbe the set of next residue roots over $x$ whose parameter coordinates are\n$t_0+q^L\\widetilde v$ modulo $q^{L+1}$.","riables and parameter variables\nis fixed in advance. Let\n\\[\n F\\in\\Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\\qquad x=(y_0,t_0),\n\\]\nand assume $F(x)=q^Lr_x$ with $q\\ge1$ and $L\\ge1$. Write the reduced\nJacobian as\n\\[\n DF(x)\\bmod q=(J_x\\ K_x),\\qquad\n J_x:S_q^m\\to S_q^e,\\quad K_x:S_q^k\\to S_q^e.\n\\]\nFor a prescribed parameter digit $v\\in S_q^k$, let $\\mathcal L_{x,L,v}$\nbe the set of next residue roots over $x$ whose parameter coordinates are\n$t_0+q^L\\widetilde v$ modulo $q^{L+1}$.","ixed in advance. Let\n\\[\n F\\in\\Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\\qquad x=(y_0,t_0),\n\\]\nand assume $F(x)=q^Lr_x$ with $q\\ge1$ and $L\\ge1$. Write the reduced\nJacobian as\n\\[\n DF(x)\\bmod q=(J_x\\ K_x),\\qquad\n J_x:S_q^m\\to S_q^e,\\quad K_x:S_q^k\\to S_q^e.\n\\]\nFor a prescribed parameter digit $v\\in S_q^k$, let $\\mathcal L_{x,L,v}$\nbe the set of next residue roots over $x$ whose parameter coordinates are\n$t_0+q^L\\widetilde v$ modulo $q^{L+1}$.","\\Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\\qquad x=(y_0,t_0),\n\\]\nand assume $F(x)=q^Lr_x$ with $q\\ge1$ and $L\\ge1$. Write the reduced\nJacobian as\n\\[\n DF(x)\\bmod q=(J_x\\ K_x),\\qquad\n J_x:S_q^m\\to S_q^e,\\quad K_x:S_q^k\\to S_q^e.\n\\]\nFor a prescribed parameter digit $v\\in S_q^k$, let $\\mathcal L_{x,L,v}$\nbe the set of next residue roots over $x$ whose parameter coordinates are\n$t_0+q^L\\widetilde v$ modulo $q^{L+1}$.","\\begin{theorem}[Prescribed-parameter obstruction and count]\n\\label{thm:prescribed-obstruction}\nDefine\n\\[\n o_x(v)=[\\overline{r_x}+K_xv]\\in S_q^e/\\im J_x.\n\\]\nThen $\\mathcal L_{x,L,v}$ is nonempty if and only if $o_x(v)=0$.\nWhen nonempty, it is a torsor for $\\ker J_x$. If $q=p$ is prime and\n$\\rho_x=\\rank_{\\F_p}J_x$, then\n\\begin{equation}\\label{eq:prescribed-count}\n \\#\\mathcal L_{x,L,v}\n =\\begin{cases}p^{m-\\rho_x},&o_x(v)=0,","\\begin{theorem}[Prescribed-parameter obstruction and count]\n\\label{thm:prescribed-obstruction}\nDefine\n\\[\n o_x(v)=[\\overline{r_x}+K_xv]\\in S_q^e/\\im J_x.\n\\]\nThen $\\mathcal L_{x,L,v}$ is nonempty if and only if $o_x(v)=0$.\nWhen nonempty, it is a torsor for $\\ker J_x$. If $q=p$ is prime and\n$\\rho_x=\\rank_{\\F_p}J_x$, then\n\\begin{equation}\\label{eq:prescribed-count}\n \\#\\mathcal L_{x,L,v}\n =\\begin{cases}p^{m-\\rho_x},&o_x(v)=0,\\\\0,&o_x(v)\\ne0,\\end{cases}\n \\qquad \\#(\\F_p^e/\\im J_x)=p^{e-\\rho_x}.\n\\end{equation}\nThe criterion holds uniformly for every prescribed $v$ without a\nsurjectivity hypothesis.\n\\end{theorem}","0,&o_x(v)\\ne0,\\end{cases}\n \\qquad \\#(\\F_p^e/\\im J_x)=p^{e-\\rho_x}.\n\\end{equation}\nThe criterion holds uniformly for every prescribed $v$ without a\nsurjectivity hypothesis.\n\\end{theorem}","\\begin{proof}\nA candidate for the next lift has the form\n$x+q^L(\\widetilde u,\\widetilde v)$, with solved digit $u\\in S_q^m$.\nIts Taylor correction equation is\n\\[\n \\overline{r_x}+J_xu+K_xv=0.\n\\]\nSolvability is exactly vanishing of the displayed cokernel class. Restrict\nthe residue-digit equivalence to the prescribed parameter coordinates.\nThis identifies the lift fiber with the solved-digit fiber.\nTranslation by $\\ker J_x$ is free and transitive on any nonempty such\nfiber. The prime-field count follows from rank-nullity as before.\n\\end{proof}","candidate for the next lift has the form\n$x+q^L(\\widetilde u,\\widetilde v)$, with solved digit $u\\in S_q^m$.\nIts Taylor correction equation is\n\\[\n \\overline{r_x}+J_xu+K_xv=0.\n\\]\nSolvability is exactly vanishing of the displayed cokernel class. Restrict\nthe residue-digit equivalence to the prescribed parameter coordinates.\nThis identifies the lift fiber with the solved-digit fiber.\nTranslation by $\\ker J_x$ is free and transitive on any nonempty such\nfiber. The prime-field count follows from rank-nullity as before.\n\\end{proof}","The formal statements start from the lower root and construct its residual:\n\\lean{Komorebi.PolynomialPrescribedObstruction.finite_prescribed_obstruction}\nand \\lean{Komorebi.PolynomialPrescribedObstruction.finite_prescribed_lift_count}.\nFor $F(Y,T)=T$ at the dyadic origin, the full Jacobian $(0\\ 1)$ is\nsurjective but $J_x=0$. Prescribing $v=1$ forces $T=2\\pmod4$, so no root\nexists. Prescribing $v=0$ gives two choices of $Y$ modulo four over the\ninitial zero residue. This example explains why the criterion depends on\nthe solved block, rather than merely on the full Jacobian.","art from the lower root and construct its residual:\n\\lean{Komorebi.PolynomialPrescribedObstruction.finite_prescribed_obstruction}\nand \\lean{Komorebi.PolynomialPrescribedObstruction.finite_prescribed_lift_count}.\nFor $F(Y,T)=T$ at the dyadic origin, the full Jacobian $(0\\ 1)$ is\nsurjective but $J_x=0$. Prescribing $v=1$ forces $T=2\\pmod4$, so no root\nexists. Prescribing $v=0$ gives two choices of $Y$ modulo four over the\ninitial zero residue. This example explains why the criterion depends on\nthe solved block, rather than merely on the full Jacobian.","\\begin{theorem}[Every compatible integer parameter path]\n\\label{thm:prescribed-integer-tower}\nIn the preceding notation, assume $J_x$ is surjective. For every supplied\ninteger path $t_n\\in\\Z^k$ satisfying\n\\[\n t_0=\\operatorname{free}(x),\\qquad\n t_{n+1}-t_n\\in q^{L+n}\\Z^k,\n\\]\nthere is a compatible polynomial tower $z_n\\in\\Z^{m+k}$ with\n\\[\n z_0=x,\\quad \\operatorname{free}(z_n)=t_n,\\quad\n F(z_n)\\in q^{L+n}\\Z^e,\\quad\n z_{n+1}-z_n\\in q^{L+n}\\Z^{m+k}.\n\\]\nThe free-coordinate equality is exact in integers. Both reduced Jacobian\nblocks remain fixed, and $z_b-z_a\\in q^{L+a}\\Z^{m+k}$ whenever $a\\le b$.\n\\end{theorem}","r every supplied\ninteger path $t_n\\in\\Z^k$ satisfying\n\\[\n t_0=\\operatorname{free}(x),\\qquad\n t_{n+1}-t_n\\in q^{L+n}\\Z^k,\n\\]\nthere is a compatible polynomial tower $z_n\\in\\Z^{m+k}$ with\n\\[\n z_0=x,\\quad \\operatorname{free}(z_n)=t_n,\\quad\n F(z_n)\\in q^{L+n}\\Z^e,\\quad\n z_{n+1}-z_n\\in q^{L+n}\\Z^{m+k}.\n\\]\nThe free-coordinate equality is exact in integers. Both reduced Jacobian\nblocks remain fixed, and $z_b-z_a\\in q^{L+a}\\Z^{m+k}$ whenever $a\\le b$.\n\\end{theorem}","Z^k$ satisfying\n\\[\n t_0=\\operatorname{free}(x),\\qquad\n t_{n+1}-t_n\\in q^{L+n}\\Z^k,\n\\]\nthere is a compatible polynomial tower $z_n\\in\\Z^{m+k}$ with\n\\[\n z_0=x,\\quad \\operatorname{free}(z_n)=t_n,\\quad\n F(z_n)\\in q^{L+n}\\Z^e,\\quad\n z_{n+1}-z_n\\in q^{L+n}\\Z^{m+k}.\n\\]\nThe free-coordinate equality is exact in integers. Both reduced Jacobian\nblocks remain fixed, and $z_b-z_a\\in q^{L+a}\\Z^{m+k}$ whenever $a\\le b$.\n\\end{theorem}","\\begin{proof}\nAt level $N=L+n$, write the supplied parameter increment exactly as\n$t_{n+1}-t_n=q^Nw$ with $w\\in\\Z^k$. Surjectivity of the solved block\ngives a digit $u$ satisfying\n$J_{z_n}u=-\\overline{F(z_n)/q^N}-K_{z_n}\\overline w$.\nChoose an integer representative $\\widetilde u$ and set\n$z_{n+1}=z_n+q^N(\\widetilde u,w)$. Its parameter coordinates equal the\nsupplied target exactly. Taylor expansion proves the next root condition,\nwhile congruence modulo $q$ preserves the full reduced Jacobian.\nConstruct the sequence recursively, preserving the exact parameter\ncoordinates and the fixed solved Jacobian. The earlier cross-level\nargument proves all the stated integer congruences.\n\\end{proof}","increment exactly as\n$t_{n+1}-t_n=q^Nw$ with $w\\in\\Z^k$. Surjectivity of the solved block\ngives a digit $u$ satisfying\n$J_{z_n}u=-\\overline{F(z_n)/q^N}-K_{z_n}\\overline w$.\nChoose an integer representative $\\widetilde u$ and set\n$z_{n+1}=z_n+q^N(\\widetilde u,w)$. Its parameter coordinates equal the\nsupplied target exactly. Taylor expansion proves the next root condition,\nwhile congruence modulo $q$ preserves the full reduced Jacobian.\nConstruct the sequence recursively, preserving the exact parameter\ncoordinates and the fixed solved Jacobian. The earlier cross-level\nargument proves all the stated integer congruences.\n\\end{proof}","\\begin{theorem}[Arbitrary $p$-adic parameter endpoints]\n\\label{thm:prescribed-padic-tower}\nLet $q=p$ be prime in Theorem~\\ref{thm:prescribed-integer-tower}.\nFor every $\\tau\\in t_0+p^L\\Zp^k$, there are an integer tower $z_n$ and\n$b\\in\\Zp^{m+k}$ such that\n\\[\n F(b)=0,\\qquad\\operatorname{free}(b)=\\tau,\\qquad\n z_n\\longrightarrow b,\\qquad b-z_n\\in p^{L+n}\\Zp^{m+k}.\n\\]\nThe initial value is $z_0=x$. At every stage and for every proposed\nparameter digit $v\\in\\F_p^k$, the corresponding next residue fiber has\nexactly $p^{m-e}$ elements. If $m=e$, then for each $\\tau$ there is exactly\none such root in $B_L(x)$, independently of all choices of integer\napproximants and correcting digits.\n\\end{theorem}","ints]\n\\label{thm:prescribed-padic-tower}\nLet $q=p$ be prime in Theorem~\\ref{thm:prescribed-integer-tower}.\nFor every $\\tau\\in t_0+p^L\\Zp^k$, there are an integer tower $z_n$ and\n$b\\in\\Zp^{m+k}$ such that\n\\[\n F(b)=0,\\qquad\\operatorname{free}(b)=\\tau,\\qquad\n z_n\\longrightarrow b,\\qquad b-z_n\\in p^{L+n}\\Zp^{m+k}.\n\\]\nThe initial value is $z_0=x$. At every stage and for every proposed\nparameter digit $v\\in\\F_p^k$, the corresponding next residue fiber has\nexactly $p^{m-e}$ elements. If $m=e$, then for each $\\tau$ there is exactly\none such root in $B_L(x)$, independently of all choices of integer\napproximants and correcting digits.\n\\end{theorem}","Theorem~\\ref{thm:prescribed-integer-tower}.\nFor every $\\tau\\in t_0+p^L\\Zp^k$, there are an integer tower $z_n$ and\n$b\\in\\Zp^{m+k}$ such that\n\\[\n F(b)=0,\\qquad\\operatorname{free}(b)=\\tau,\\qquad\n z_n\\longrightarrow b,\\qquad b-z_n\\in p^{L+n}\\Zp^{m+k}.\n\\]\nThe initial value is $z_0=x$. At every stage and for every proposed\nparameter digit $v\\in\\F_p^k$, the corresponding next residue fiber has\nexactly $p^{m-e}$ elements. If $m=e$, then for each $\\tau$ there is exactly\none such root in $B_L(x)$, independently of all choices of integer\napproximants and correcting digits.\n\\end{theorem}","ibed-integer-tower}.\nFor every $\\tau\\in t_0+p^L\\Zp^k$, there are an integer tower $z_n$ and\n$b\\in\\Zp^{m+k}$ such that\n\\[\n F(b)=0,\\qquad\\operatorname{free}(b)=\\tau,\\qquad\n z_n\\longrightarrow b,\\qquad b-z_n\\in p^{L+n}\\Zp^{m+k}.\n\\]\nThe initial value is $z_0=x$. At every stage and for every proposed\nparameter digit $v\\in\\F_p^k$, the corresponding next residue fiber has\nexactly $p^{m-e}$ elements. If $m=e$, then for each $\\tau$ there is exactly\none such root in $B_L(x)$, independently of all choices of integer\napproximants and correcting digits.\n\\end{theorem}","For every $\\tau\\in t_0+p^L\\Zp^k$, there are an integer tower $z_n$ and\n$b\\in\\Zp^{m+k}$ such that\n\\[\n F(b)=0,\\qquad\\operatorname{free}(b)=\\tau,\\qquad\n z_n\\longrightarrow b,\\qquad b-z_n\\in p^{L+n}\\Zp^{m+k}.\n\\]\nThe initial value is $z_0=x$. At every stage and for every proposed\nparameter digit $v\\in\\F_p^k$, the corresponding next residue fiber has\nexactly $p^{m-e}$ elements. If $m=e$, then for each $\\tau$ there is exactly\none such root in $B_L(x)$, independently of all choices of integer\napproximants and correcting digits.\n\\end{theorem}","re an integer tower $z_n$ and\n$b\\in\\Zp^{m+k}$ such that\n\\[\n F(b)=0,\\qquad\\operatorname{free}(b)=\\tau,\\qquad\n z_n\\longrightarrow b,\\qquad b-z_n\\in p^{L+n}\\Zp^{m+k}.\n\\]\nThe initial value is $z_0=x$. At every stage and for every proposed\nparameter digit $v\\in\\F_p^k$, the corresponding next residue fiber has\nexactly $p^{m-e}$ elements. If $m=e$, then for each $\\tau$ there is exactly\none such root in $B_L(x)$, independently of all choices of integer\napproximants and correcting digits.\n\\end{theorem}","\\begin{proof}\nSet the first integer parameter approximation equal to the prescribed\n$t_0$. For $n\\ge1$, use canonical representatives of\n$\\tau\\bmod p^{L+n}$. The initial congruence and truncation precision give\n\\[\n \\tau-t_n\\in p^{L+n}\\Zp^k.\n\\]\nSubtracting consecutive relations gives compatibility in $\\Zp$.\nFaithfulness of integer divisibility under $\\Z\\hookrightarrow\\Zp$\ngives compatibility in $\\Z$, and the same norm bounds give $t_n\\to\\tau$.\nTheorem~\\ref{thm:prescribed-integer-tower} and completion now produce $b$\nwith exact polynomial vanishing and all retained precisions. Each free\ncoordinate has both the corresponding coordinate of $b$ and that of\n$\\tau$ as its limit, proving exact parameter equality.","r parameter approximation equal to the prescribed\n$t_0$. For $n\\ge1$, use canonical representatives of\n$\\tau\\bmod p^{L+n}$. The initial congruence and truncation precision give\n\\[\n \\tau-t_n\\in p^{L+n}\\Zp^k.\n\\]\nSubtracting consecutive relations gives compatibility in $\\Zp$.\nFaithfulness of integer divisibility under $\\Z\\hookrightarrow\\Zp$\ngives compatibility in $\\Z$, and the same norm bounds give $t_n\\to\\tau$.\nTheorem~\\ref{thm:prescribed-integer-tower} and completion now produce $b$\nwith exact polynomial vanishing and all retained precisions. Each free\ncoordinate has both the corresponding coordinate of $b$ and that of\n$\\tau$ as its limit, proving exact parameter equality.","representatives of\n$\\tau\\bmod p^{L+n}$. The initial congruence and truncation precision give\n\\[\n \\tau-t_n\\in p^{L+n}\\Zp^k.\n\\]\nSubtracting consecutive relations gives compatibility in $\\Zp$.\nFaithfulness of integer divisibility under $\\Z\\hookrightarrow\\Zp$\ngives compatibility in $\\Z$, and the same norm bounds give $t_n\\to\\tau$.\nTheorem~\\ref{thm:prescribed-integer-tower} and completion now produce $b$\nwith exact polynomial vanishing and all retained precisions. Each free\ncoordinate has both the corresponding coordinate of $b$ and that of\n$\\tau$ as its limit, proving exact parameter equality.","ubtracting consecutive relations gives compatibility in $\\Zp$.\nFaithfulness of integer divisibility under $\\Z\\hookrightarrow\\Zp$\ngives compatibility in $\\Z$, and the same norm bounds give $t_n\\to\\tau$.\nTheorem~\\ref{thm:prescribed-integer-tower} and completion now produce $b$\nwith exact polynomial vanishing and all retained precisions. Each free\ncoordinate has both the corresponding coordinate of $b$ and that of\n$\\tau$ as its limit, proving exact parameter equality.","cutive relations gives compatibility in $\\Zp$.\nFaithfulness of integer divisibility under $\\Z\\hookrightarrow\\Zp$\ngives compatibility in $\\Z$, and the same norm bounds give $t_n\\to\\tau$.\nTheorem~\\ref{thm:prescribed-integer-tower} and completion now produce $b$\nwith exact polynomial vanishing and all retained precisions. Each free\ncoordinate has both the corresponding coordinate of $b$ and that of\n$\\tau$ as its limit, proving exact parameter equality.","The solved Jacobian remains surjective, so every shifted obstruction\n$[\\overline r+K_{z_n}v]$ vanishes. Formula~\\eqref{eq:prescribed-count}\ntherefore gives $p^{m-e}$ choices for every parameter digit at every stage.\nWhen $m=e$, the fixed-coordinate chart in the proof of\nTheorem~\\ref{thm:continuous-parameter-chart} contains every root in the\ninitial ball with the specified parameters. Its uniqueness proves the\nendpoint assertion. This is uniqueness of the actual endpoint, not of the\ninteger representatives used to approach it.\n\\end{proof}","The endpoint theorems are\n\\lean{Komorebi.PolynomialPrescribedPadicTower.exists_prescribed_padic_root_with_tower}\nand \\lean{Komorebi.PolynomialPrescribedPadicTower.exists_unique_prescribed_padic_root}.\nThe separate supplied-path theorem\n\\lean{Komorebi.PolynomialPrescribedPadicTower.exists_root_with_prescribed_path}\naccepts any compatible integer parameter path with limit $\\tau$.\nFor $m>e$, endpoint uniqueness is false in general: the system\n$F(Y_1,Y_2,T)=Y_1$ has two distinct roots $(0,0,0)$ and $(0,2,0)$ in the\nsame dyadic ball with identical parameter value.","nomialPrescribedPadicTower.exists_prescribed_padic_root_with_tower}\nand \\lean{Komorebi.PolynomialPrescribedPadicTower.exists_unique_prescribed_padic_root}.\nThe separate supplied-path theorem\n\\lean{Komorebi.PolynomialPrescribedPadicTower.exists_root_with_prescribed_path}\naccepts any compatible integer parameter path with limit $\\tau$.\nFor $m>e$, endpoint uniqueness is false in general: the system\n$F(Y_1,Y_2,T)=Y_1$ has two distinct roots $(0,0,0)$ and $(0,2,0)$ in the\nsame dyadic ball with identical parameter value.","ed_padic_root_with_tower}\nand \\lean{Komorebi.PolynomialPrescribedPadicTower.exists_unique_prescribed_padic_root}.\nThe separate supplied-path theorem\n\\lean{Komorebi.PolynomialPrescribedPadicTower.exists_root_with_prescribed_path}\naccepts any compatible integer parameter path with limit $\\tau$.\nFor $m>e$, endpoint uniqueness is false in general: the system\n$F(Y_1,Y_2,T)=Y_1$ has two distinct roots $(0,0,0)$ and $(0,2,0)$ in the\nsame dyadic ball with identical parameter value.","with_tower}\nand \\lean{Komorebi.PolynomialPrescribedPadicTower.exists_unique_prescribed_padic_root}.\nThe separate supplied-path theorem\n\\lean{Komorebi.PolynomialPrescribedPadicTower.exists_root_with_prescribed_path}\naccepts any compatible integer parameter path with limit $\\tau$.\nFor $m>e$, endpoint uniqueness is false in general: the system\n$F(Y_1,Y_2,T)=Y_1$ has two distinct roots $(0,0,0)$ and $(0,2,0)$ in the\nsame dyadic ball with identical parameter value.","The remaining theorems in this section allow arbitrary coefficients and\ncenters in $R=\\Zp$. They concern square solved Jacobians invertible over\n$\\Qp$, even when their reductions modulo $p$ are singular. The exponent\ncontrolling an integral inverse is retained explicitly.","\\begin{theorem}[Weighted nonlinear lifting]\\label{thm:weighted-lifting}\nLet $p$ be prime, $d,s,r\\in\\N$, $F\\in R[X_1,\\ldots,X_d]^d$, and\n$a\\in R^d$. Set $J=DF(a)$. Suppose\n\\begin{equation}\\label{eq:weighted-hypotheses}\n J\\in\\operatorname{GL}_d(\\Qp),\\qquad\n p^sJ^{-1}\\in M_d(R),\\qquad F(a)\\in p^rR^d,\\qquad r>2s.\n\\end{equation}\nThen there is exactly one root $b$ of $F$ in $a+p^{s+1}R^d$, and it\nsatisfies\n\\begin{equation}\\label{eq:weighted-displacement}\n b-a\\in p^{r-s}R^d.\n\\end{equation}\nUniqueness holds throughout the full ball of depth $s+1$.\n\\end{theorem}","hted nonlinear lifting]\\label{thm:weighted-lifting}\nLet $p$ be prime, $d,s,r\\in\\N$, $F\\in R[X_1,\\ldots,X_d]^d$, and\n$a\\in R^d$. Set $J=DF(a)$. Suppose\n\\begin{equation}\\label{eq:weighted-hypotheses}\n J\\in\\operatorname{GL}_d(\\Qp),\\qquad\n p^sJ^{-1}\\in M_d(R),\\qquad F(a)\\in p^rR^d,\\qquad r>2s.\n\\end{equation}\nThen there is exactly one root $b$ of $F$ in $a+p^{s+1}R^d$, and it\nsatisfies\n\\begin{equation}\\label{eq:weighted-displacement}\n b-a\\in p^{r-s}R^d.\n\\end{equation}\nUniqueness holds throughout the full ball of depth $s+1$.\n\\end{theorem}","n R^d$. Set $J=DF(a)$. Suppose\n\\begin{equation}\\label{eq:weighted-hypotheses}\n J\\in\\operatorname{GL}_d(\\Qp),\\qquad\n p^sJ^{-1}\\in M_d(R),\\qquad F(a)\\in p^rR^d,\\qquad r>2s.\n\\end{equation}\nThen there is exactly one root $b$ of $F$ in $a+p^{s+1}R^d$, and it\nsatisfies\n\\begin{equation}\\label{eq:weighted-displacement}\n b-a\\in p^{r-s}R^d.\n\\end{equation}\nUniqueness holds throughout the full ball of depth $s+1$.\n\\end{theorem}","\\begin{proof}\nLet $A=p^sJ^{-1}\\in M_d(R)$. Inclusion into $\\Qp$ and its injectivity\ngive the two integral identities\n\\[\n AJ=JA=p^sI_d.\n\\]\nThese identities are also an equivalent integral certificate for the\nfirst two hypotheses of \\eqref{eq:weighted-hypotheses}. In particular,\n$A$ is injective: $Aw=0$ implies $p^sw=JAw=0$, hence $w=0$.","Precondition the polynomial system by $A$ and set $c=p^{s+1}$.\nIts derivative at $a$ is $AJ=p^sI_d$. Lemma~\\ref{lem:polynomial-taylor}\ngives a polynomial vector $Q$ with $Q(0)=0$ and\n\\[\n (AF)(a+cu)=(AF)(a)+p^{2s+1}u+p^{2s+2}Q(u).\n\\]\nSince $A$ is integral, $(AF)(a)\\in p^rR^d$. The strict integer inequality\n$r>2s$ permits $k=r-(2s+1)\\ge0$ and a vector $C\\in p^kR^d$ such that\n$(AF)(a)=p^{2s+1}C$. The normalized root equation is therefore\n\\begin{equation}\\label{eq:weighted-fixed-point}\n u=-C-pQ(u).\n\\end{equation}","{s+1}$.\nIts derivative at $a$ is $AJ=p^sI_d$. Lemma~\\ref{lem:polynomial-taylor}\ngives a polynomial vector $Q$ with $Q(0)=0$ and\n\\[\n (AF)(a+cu)=(AF)(a)+p^{2s+1}u+p^{2s+2}Q(u).\n\\]\nSince $A$ is integral, $(AF)(a)\\in p^rR^d$. The strict integer inequality\n$r>2s$ permits $k=r-(2s+1)\\ge0$ and a vector $C\\in p^kR^d$ such that\n$(AF)(a)=p^{2s+1}C$. The normalized root equation is therefore\n\\begin{equation}\\label{eq:weighted-fixed-point}\n u=-C-pQ(u).\n\\end{equation}","Integral polynomial evaluation is $1$-Lipschitz for the supremum\n$p$-adic metric: a common divisibility of input differences is preserved\nunder addition and multiplication by integral elements. Consequently the\nright side of \\eqref{eq:weighted-fixed-point} has contraction constant\n$|p|_p<1$ on the complete space $R^d$. It has a unique fixed point $u$.","To obtain the sharper displacement, begin with $u\\in R^d$. If\n$u\\in p^nR^d$, then $Q(0)=0$ and polynomial congruence give\n$Q(u)\\in p^nR^d$. For $n<k$, both $C$ and $pQ(u)$ are divisible by\n$p^{n+1}$. Equation~\\eqref{eq:weighted-fixed-point} therefore improves\nthe divisibility of $u$ by one digit. Induction gives $u\\in p^kR^d$.\nThus $b=a+p^{s+1}u$ satisfies \\eqref{eq:weighted-displacement}, because\n$s+1+k=r-s$. Its normalized preconditioned value vanishes, and injectivity\nof $A$ gives $F(b)=0$.","obtain the sharper displacement, begin with $u\\in R^d$. If\n$u\\in p^nR^d$, then $Q(0)=0$ and polynomial congruence give\n$Q(u)\\in p^nR^d$. For $n<k$, both $C$ and $pQ(u)$ are divisible by\n$p^{n+1}$. Equation~\\eqref{eq:weighted-fixed-point} therefore improves\nthe divisibility of $u$ by one digit. Induction gives $u\\in p^kR^d$.\nThus $b=a+p^{s+1}u$ satisfies \\eqref{eq:weighted-displacement}, because\n$s+1+k=r-s$. Its normalized preconditioned value vanishes, and injectivity\nof $A$ gives $F(b)=0$.","n $Q(0)=0$ and polynomial congruence give\n$Q(u)\\in p^nR^d$. For $n<k$, both $C$ and $pQ(u)$ are divisible by\n$p^{n+1}$. Equation~\\eqref{eq:weighted-fixed-point} therefore improves\nthe divisibility of $u$ by one digit. Induction gives $u\\in p^kR^d$.\nThus $b=a+p^{s+1}u$ satisfies \\eqref{eq:weighted-displacement}, because\n$s+1+k=r-s$. Its normalized preconditioned value vanishes, and injectivity\nof $A$ gives $F(b)=0$.","and polynomial congruence give\n$Q(u)\\in p^nR^d$. For $n<k$, both $C$ and $pQ(u)$ are divisible by\n$p^{n+1}$. Equation~\\eqref{eq:weighted-fixed-point} therefore improves\nthe divisibility of $u$ by one digit. Induction gives $u\\in p^kR^d$.\nThus $b=a+p^{s+1}u$ satisfies \\eqref{eq:weighted-displacement}, because\n$s+1+k=r-s$. Its normalized preconditioned value vanishes, and injectivity\nof $A$ gives $F(b)=0$.","Every competing root $z\\in a+p^{s+1}R^d$ can be written\n$z=a+p^{s+1}w$. Its root equation implies the same fixed-point\nequation after cancellation of the nonzero scalar $p^{2s+1}$.\nContraction uniqueness gives $w=u$, proving uniqueness on the entire\nstated ball.\n\\end{proof}","The existence and uniqueness statements are\n\\lean{Komorebi.PolynomialWeightedLift.exists_weighted_root} and\n\\lean{Komorebi.PolynomialWeightedLift.exists_unique_weighted_root}.\nThe inverse hypothesis in Lean specifies an integral matrix whose image\nis exactly $p^sJ^{-1}$. It does not substitute a potentially larger\ndeterminant valuation for $s$.","\\begin{example}[Strict threshold and displacement boundary]\n\\label{ex:weighted-boundary}\nFor $p=2$, $F(X)=X^2+3$, and $a=1$, one has $F(a)=4$ and $DF(a)=2$.\nThus $s=1$ and $r=2$ satisfy every displayed hypothesis except the strict\ninequality. No root exists in $\\Z_2$, since none exists modulo eight.\nHence $r\\ge2s$ cannot replace $r>2s$ uniformly. Also,\n$F(X)=p^sX+p^r$ has root $-p^{r-s}$, whose displacement is not divisible\nby $p^{r-s+1}$. The guaranteed displacement cannot uniformly gain a digit.\n\\end{example}","and displacement boundary]\n\\label{ex:weighted-boundary}\nFor $p=2$, $F(X)=X^2+3$, and $a=1$, one has $F(a)=4$ and $DF(a)=2$.\nThus $s=1$ and $r=2$ satisfy every displayed hypothesis except the strict\ninequality. No root exists in $\\Z_2$, since none exists modulo eight.\nHence $r\\ge2s$ cannot replace $r>2s$ uniformly. Also,\n$F(X)=p^sX+p^r$ has root $-p^{r-s}$, whose displacement is not divisible\nby $p^{r-s+1}$. The guaranteed displacement cannot uniformly gain a digit.\n\\end{example}","\\begin{theorem}[Weighted coefficient deformation]\\label{thm:coefficient-motion}\nLet $F,H\\in R[X_1,\\ldots,X_d]^d$, $F(a)=0$, and $J=DF(a)$. Suppose\n$J\\in\\operatorname{GL}_d(\\Qp)$, $p^sJ^{-1}\\in M_d(R)$, and $M>2s$.\nFor $G=F+p^MH$, there is exactly one root $b\\in a+p^{s+1}R^d$.\nIt satisfies $b-a\\in p^{M-s}R^d$ and, in $\\Qp^d$,\n\\begin{equation}\\label{eq:coefficient-first-order}\n b-a=-p^MJ^{-1}H(a)+\\varepsilon,\n \\qquad\\varepsilon\\in p^{2M-3s}R^d.\n\\end{equation}\nOnly the original Jacobian $DF(a)$ is subject to an inverse hypothesis.\n\\end{theorem}","n R[X_1,\\ldots,X_d]^d$, $F(a)=0$, and $J=DF(a)$. Suppose\n$J\\in\\operatorname{GL}_d(\\Qp)$, $p^sJ^{-1}\\in M_d(R)$, and $M>2s$.\nFor $G=F+p^MH$, there is exactly one root $b\\in a+p^{s+1}R^d$.\nIt satisfies $b-a\\in p^{M-s}R^d$ and, in $\\Qp^d$,\n\\begin{equation}\\label{eq:coefficient-first-order}\n b-a=-p^MJ^{-1}H(a)+\\varepsilon,\n \\qquad\\varepsilon\\in p^{2M-3s}R^d.\n\\end{equation}\nOnly the original Jacobian $DF(a)$ is subject to an inverse hypothesis.\n\\end{theorem}","\\begin{proof}\nRetain the original certificate $A=p^sJ^{-1}$ and put $c=p^{s+1}$.\nTaylor expansion and a polynomial difference identity yield polynomial\nvectors $Q_F,Q_H$, both zero at the origin, such that\n\\[\n (AF)(a+cu)=p^{2s+1}u+p^{2s+2}Q_F(u),\\qquad\n (AH)(a+cu)=AH(a)+cQ_H(u).\n\\]\nConsequently\n\\[\n (AG)(a+cu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\]\nwhere\n\\[\n C=p^{M-(2s+1)}AH(a),\\qquad\n Q=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\]\nAll exponents are nonnegative. The same contraction and divisibility\nargument as in Theorem~\\ref{thm:weighted-lifting} gives the root,\nthe displacement $p^{M-s}$, and full-ball uniqueness. This constructs the\nroot without introducing an inverse hypothesis for $DG(a)$.","th zero at the origin, such that\n\\[\n (AF)(a+cu)=p^{2s+1}u+p^{2s+2}Q_F(u),\\qquad\n (AH)(a+cu)=AH(a)+cQ_H(u).\n\\]\nConsequently\n\\[\n (AG)(a+cu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\]\nwhere\n\\[\n C=p^{M-(2s+1)}AH(a),\\qquad\n Q=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\]\nAll exponents are nonnegative. The same contraction and divisibility\nargument as in Theorem~\\ref{thm:weighted-lifting} gives the root,\nthe displacement $p^{M-s}$, and full-ball uniqueness. This constructs the\nroot without introducing an inverse hypothesis for $DG(a)$.","]\nConsequently\n\\[\n (AG)(a+cu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\]\nwhere\n\\[\n C=p^{M-(2s+1)}AH(a),\\qquad\n Q=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\]\nAll exponents are nonnegative. The same contraction and divisibility\nargument as in Theorem~\\ref{thm:weighted-lifting} gives the root,\nthe displacement $p^{M-s}$, and full-ball uniqueness. This constructs the\nroot without introducing an inverse hypothesis for $DG(a)$.","For the error estimate, let $\\delta=b-a$ and $D=M-s$. Taylor's theorem\nfor the preconditioned polynomial gives\n\\[\n E_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2D}R^d.\n\\]\nPolynomial congruence gives\n$E_H=(AH)(b)-(AH)(a)\\in p^DR^d$.\nThe equations $F(a)=0$ and $G(b)=0$ imply\n\\[\n p^s\\delta+p^MAH(a)=-E_F-p^ME_H\\in p^{2D}R^d,\n\\]\nbecause $M+D\\ge2D$. Cancel $p^s$ in the integral domain $R$ to obtain\n\\[\n \\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\]\nThis integral vector is $\\varepsilon$. After inclusion into $\\Qp$,\n$p^{M-s}A=p^MJ^{-1}$ gives \\eqref{eq:coefficient-first-order}.\n\\end{proof}","ylor's theorem\nfor the preconditioned polynomial gives\n\\[\n E_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2D}R^d.\n\\]\nPolynomial congruence gives\n$E_H=(AH)(b)-(AH)(a)\\in p^DR^d$.\nThe equations $F(a)=0$ and $G(b)=0$ imply\n\\[\n p^s\\delta+p^MAH(a)=-E_F-p^ME_H\\in p^{2D}R^d,\n\\]\nbecause $M+D\\ge2D$. Cancel $p^s$ in the integral domain $R$ to obtain\n\\[\n \\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\]\nThis integral vector is $\\varepsilon$. After inclusion into $\\Qp$,\n$p^{M-s}A=p^MJ^{-1}$ gives \\eqref{eq:coefficient-first-order}.\n\\end{proof}","d polynomial gives\n\\[\n E_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2D}R^d.\n\\]\nPolynomial congruence gives\n$E_H=(AH)(b)-(AH)(a)\\in p^DR^d$.\nThe equations $F(a)=0$ and $G(b)=0$ imply\n\\[\n p^s\\delta+p^MAH(a)=-E_F-p^ME_H\\in p^{2D}R^d,\n\\]\nbecause $M+D\\ge2D$. Cancel $p^s$ in the integral domain $R$ to obtain\n\\[\n \\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\]\nThis integral vector is $\\varepsilon$. After inclusion into $\\Qp$,\n$p^{M-s}A=p^MJ^{-1}$ gives \\eqref{eq:coefficient-first-order}.\n\\end{proof}","omial gives\n\\[\n E_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2D}R^d.\n\\]\nPolynomial congruence gives\n$E_H=(AH)(b)-(AH)(a)\\in p^DR^d$.\nThe equations $F(a)=0$ and $G(b)=0$ imply\n\\[\n p^s\\delta+p^MAH(a)=-E_F-p^ME_H\\in p^{2D}R^d,\n\\]\nbecause $M+D\\ge2D$. Cancel $p^s$ in the integral domain $R$ to obtain\n\\[\n \\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\]\nThis integral vector is $\\varepsilon$. After inclusion into $\\Qp$,\n$p^{M-s}A=p^MJ^{-1}$ gives \\eqref{eq:coefficient-first-order}.\n\\end{proof}","The complete theorem is\n\\lean{Komorebi.PolynomialRootMotion.exists_unique_root_motion}.\nThe error depth exceeds the displacement depth by $M-2s\\ge1$. It is a\nguaranteed divisibility, not an assertion of exact valuation for every\nsystem.","\\begin{theorem}[Joint coefficient and parameter deformation]\n\\label{thm:joint-deformation}\\leavevmode\\par\nLet $F,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e$ and\n$a=(y_0,t_0)\\in R^{e+k}$ satisfy $F(a)=0$. Define the actual blocks\n\\[\n J=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\]\nSuppose $J\\in\\operatorname{GL}_e(\\Qp)$,\n$p^sJ^{-1}\\in M_e(R)$, and $M>2s$. For every $v\\in R^k$, set\n\\[\n \\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\]\nThere is exactly one $y\\in y_0+p^{s+1}R^e$ with $G(y,\\tau)=0$.\nIt satisfies\n\\begin{equation}\\label{eq:joint-first-order}\n \\begin{aligned}\n  y-y_0&\\in p^{M-s}R^e,","rmation]\n\\label{thm:joint-deformation}\\leavevmode\\par\nLet $F,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e$ and\n$a=(y_0,t_0)\\in R^{e+k}$ satisfy $F(a)=0$. Define the actual blocks\n\\[\n J=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\]\nSuppose $J\\in\\operatorname{GL}_e(\\Qp)$,\n$p^sJ^{-1}\\in M_e(R)$, and $M>2s$. For every $v\\in R^k$, set\n\\[\n \\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\]\nThere is exactly one $y\\in y_0+p^{s+1}R^e$ with $G(y,\\tau)=0$.\nIt satisfies\n\\begin{equation}\\label{eq:joint-first-order}\n \\begin{aligned}\n  y-y_0&\\in p^{M-s}R^e,","eformation}\\leavevmode\\par\nLet $F,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e$ and\n$a=(y_0,t_0)\\in R^{e+k}$ satisfy $F(a)=0$. Define the actual blocks\n\\[\n J=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\]\nSuppose $J\\in\\operatorname{GL}_e(\\Qp)$,\n$p^sJ^{-1}\\in M_e(R)$, and $M>2s$. For every $v\\in R^k$, set\n\\[\n \\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\]\nThere is exactly one $y\\in y_0+p^{s+1}R^e$ with $G(y,\\tau)=0$.\nIt satisfies\n\\begin{equation}\\label{eq:joint-first-order}\n \\begin{aligned}\n  y-y_0&\\in p^{M-s}R^e,","1,\\ldots,Y_e,T_1,\\ldots,T_k]^e$ and\n$a=(y_0,t_0)\\in R^{e+k}$ satisfy $F(a)=0$. Define the actual blocks\n\\[\n J=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\]\nSuppose $J\\in\\operatorname{GL}_e(\\Qp)$,\n$p^sJ^{-1}\\in M_e(R)$, and $M>2s$. For every $v\\in R^k$, set\n\\[\n \\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\]\nThere is exactly one $y\\in y_0+p^{s+1}R^e$ with $G(y,\\tau)=0$.\nIt satisfies\n\\begin{equation}\\label{eq:joint-first-order}\n \\begin{aligned}\n  y-y_0&\\in p^{M-s}R^e,\\\\\n  y-y_0&=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon,\n  \\qquad\\varepsilon\\in p^{2M-3s}R^e.\n \\end{aligned}\n\\end{equation}\nThe equality involving $J^{-1}$ is in $\\Qp^e$, with an integral error\nvector. The parameter equality is exact.\n\\end{theorem}","y-y_0&=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon,\n  \\qquad\\varepsilon\\in p^{2M-3s}R^e.\n \\end{aligned}\n\\end{equation}\nThe equality involving $J^{-1}$ is in $\\Qp^e$, with an integral error\nvector. The parameter equality is exact.\n\\end{theorem}","\\begin{proof}\nUse the augmented polynomial systems\n\\[\n \\mathcal F(Y,T)=(F(Y,T),T-t_0),\\qquad\n \\mathcal H(Y,T)=(H(Y,T),-v).\n\\]\nThen $\\mathcal F(a)=0$, and its derivative is\n\\[\n B=D\\mathcal F(a)=\\begin{pmatrix}J&K","\\begin{proof}\nUse the augmented polynomial systems\n\\[\n \\mathcal F(Y,T)=(F(Y,T),T-t_0),\\qquad\n \\mathcal H(Y,T)=(H(Y,T),-v).\n\\]\nThen $\\mathcal F(a)=0$, and its derivative is\n\\[\n B=D\\mathcal F(a)=\\begin{pmatrix}J&K\\\\0&I_k\\end{pmatrix}.\n\\]\nFor $A=p^sJ^{-1}\\in M_e(R)$ and $c=p^s$, define\n\\[\n C=\\begin{pmatrix}A&-AK","0&I_k\\end{pmatrix}.\n\\]\nFor $A=p^sJ^{-1}\\in M_e(R)$ and $c=p^s$, define\n\\[\n C=\\begin{pmatrix}A&-AK","0&I_k\\end{pmatrix}.\n\\]\nFor $A=p^sJ^{-1}\\in M_e(R)$ and $c=p^s$, define\n\\[\n C=\\begin{pmatrix}A&-AK\\\\0&cI_k\\end{pmatrix}.\n\\]\nThe identities $AJ=JA=cI_e$ give $CB=BC=cI_{e+k}$. In the second\nproduct, the upper right block is $-JAK+cK=0$. In the first, it is\n$AK-AK=0$. Thus augmentation preserves the original inverse bound $s$.","0&cI_k\\end{pmatrix}.\n\\]\nThe identities $AJ=JA=cI_e$ give $CB=BC=cI_{e+k}$. In the second\nproduct, the upper right block is $-JAK+cK=0$. In the first, it is\n$AK-AK=0$. Thus augmentation preserves the original inverse bound $s$.","Theorem~\\ref{thm:coefficient-motion} gives a unique root $b$ of\n$\\mathcal F+p^M\\mathcal H$ in $a+p^{s+1}R^{e+k}$, with displacement\ndepth $M-s$. Its last equations say exactly\n$b_T=t_0+p^Mv=\\tau$. Its first equations are $G(b)=0$.\nThe integral first-order expression uses\n\\[\n C\\binom{H(a)}{-v}\n =\\binom{A(H(a)+Kv)}{-p^sv}.\n\\]\nProjection onto the solved coordinates gives\n\\[\n b_Y-y_0+p^{M-s}A(H(a)+Kv)\\in p^{2M-3s}R^e.\n\\]\nThe same inclusion identity as before converts this to\n\\eqref{eq:joint-first-order}.","Finally, if $z\\in y_0+p^{s+1}R^e$ is any competing solved root with\nparameters $\\tau$, then $(z,\\tau)$ belongs to the full augmented ball.\nIndeed, $M>2s$ implies $M\\ge s+1$. Full-point uniqueness therefore gives\n$(z,\\tau)=b$. This comparison proves uniqueness on exactly the solved\nball stated in the theorem.\n\\end{proof}","The exact solved-coordinate statement is\n\\lean{Komorebi.PolynomialJointDeformation.exists_joint_deformation},\nwith its unique-existence interface\n\\lean{Komorebi.PolynomialJointDeformation.exists_unique_joint_deformation}.\nThe triangular inverse identities hold over arbitrary commutative rings,\nso they apply in particular over the $p$-adic integers.","\\begin{example}[First-order sign and error precision]\\label{ex:motion-boundary}\nFor $p=2$, $s=0$, $M=2$, $F(Y,T)=Y+T$, $H=1$, $a=0$, and $v=1$,\nthe prescribed parameter is $\\tau=4$ and the solved root is $y=-8$.\nHere $J=K=1$, and the forcing term is $H(a)+Kv=2$. Omitting the parameter\nterm would predict $-4$, and reversing its sign would predict $0$.\nThe corresponding errors $-4$ and $-8$ are both outside\n$2^{2M}\\Z_2=16\\Z_2$. Thus either change would violate the stated error\nbound.","mple}[First-order sign and error precision]\\label{ex:motion-boundary}\nFor $p=2$, $s=0$, $M=2$, $F(Y,T)=Y+T$, $H=1$, $a=0$, and $v=1$,\nthe prescribed parameter is $\\tau=4$ and the solved root is $y=-8$.\nHere $J=K=1$, and the forcing term is $H(a)+Kv=2$. Omitting the parameter\nterm would predict $-4$, and reversing its sign would predict $0$.\nThe corresponding errors $-4$ and $-8$ are both outside\n$2^{2M}\\Z_2=16\\Z_2$. Thus either change would violate the stated error\nbound.","The error exponent cannot uniformly be increased. For\n$F(Y,T)=Y+Y^2+T$, $H=0$, $a=0$, $p=2$, $s=0$, $M=1$, and $v=1$,\nthere is an even root $y$ of $y+y^2+2=0$. Its equation forces\n$y\\equiv2\\pmod4$ and $y+2=-y^2$, so\n\\[\n 4\\mid y+2,\\qquad 8\\nmid y+2.\n\\]\nThus the error depth $2M-3s=2$ is attained even for parameter-only motion.\nThe same quadratic with a constant coefficient perturbation gives the\ncoefficient-only boundary. These examples refute a uniform extra error\ndigit. They do not assert optimality for every individual choice of\n$p,s,M$.\n\\end{example}","ot $y$ of $y+y^2+2=0$. Its equation forces\n$y\\equiv2\\pmod4$ and $y+2=-y^2$, so\n\\[\n 4\\mid y+2,\\qquad 8\\nmid y+2.\n\\]\nThus the error depth $2M-3s=2$ is attained even for parameter-only motion.\nThe same quadratic with a constant coefficient perturbation gives the\ncoefficient-only boundary. These examples refute a uniform extra error\ndigit. They do not assert optimality for every individual choice of\n$p,s,M$.\n\\end{example}","\\section{Analytic regularity of the local solution map}\n\\label{sec:regularity}\nThe deformation estimates control how far an exact root moves under a small\nperturbation. We now show that its local dependence on the parameters is\nanalytic, and compute the derivative from the polynomial Jacobian. The\nanalytic argument takes place over a complete normed field. For integral\n$p$-adic data, we first include the coefficients and base point in $\\Qp$.\n$\\Zp$ itself is not a vector space over $\\Qp$.","l solution map}\n\\label{sec:regularity}\nThe deformation estimates control how far an exact root moves under a small\nperturbation. We now show that its local dependence on the parameters is\nanalytic, and compute the derivative from the polynomial Jacobian. The\nanalytic argument takes place over a complete normed field. For integral\n$p$-adic data, we first include the coefficients and base point in $\\Qp$.\n$\\Zp$ itself is not a vector space over $\\Qp$.","ion map}\n\\label{sec:regularity}\nThe deformation estimates control how far an exact root moves under a small\nperturbation. We now show that its local dependence on the parameters is\nanalytic, and compute the derivative from the polynomial Jacobian. The\nanalytic argument takes place over a complete normed field. For integral\n$p$-adic data, we first include the coefficients and base point in $\\Qp$.\n$\\Zp$ itself is not a vector space over $\\Qp$.","rmation estimates control how far an exact root moves under a small\nperturbation. We now show that its local dependence on the parameters is\nanalytic, and compute the derivative from the polynomial Jacobian. The\nanalytic argument takes place over a complete normed field. For integral\n$p$-adic data, we first include the coefficients and base point in $\\Qp$.\n$\\Zp$ itself is not a vector space over $\\Qp$.","Let $\\mathbb K$ be a complete nontrivially normed field. Let $e,k\\in\\N$, and\nwrite the variables as $(Y,T)\\in\\mathbb K^e\\times\\mathbb K^k$. For\n\\[\n F\\in\\mathbb K[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n \\qquad a=(y_0,t_0),\n\\]\ndefine the matrices of evaluated formal partial derivatives\n\\begin{equation}\n J=D_YF(a)\\in M_e(\\mathbb K),\\qquad\n K=D_TF(a)\\in M_{e\\times k}(\\mathbb K).\n \\label{eq:regularity-blocks}\n\\end{equation}\nRows index equations and columns index variables. We first establish that\nthese formal derivatives are the derivatives of the evaluation map.","\\begin{lemma}[Derivative of polynomial evaluation]\n\\label{lem:actual-polynomial-derivative}\nFor $P\\in\\mathbb K[X_1,\\ldots,X_n]$, polynomial evaluation is analytic at\nevery point. At $a\\in\\mathbb K^n$ its strict Fr\\'echet derivative is\n\\[\n v\\longmapsto\\sum_{j=1}^n (\\partial_jP)(a)v_j.\n\\]\nFor a finite system, its strict derivative is the continuous linear map\nrepresented by the matrix of evaluated formal partial derivatives.\n\\end{lemma}\n\\begin{proof}\nFor the derivative statement, use polynomial induction on constants,\naddition, and multiplication by a variable. Constants have zero derivative.\nThe sum rule agrees with the additivity of formal partial differentiation.\nFor $PX_j$, the product rule gives\n\\[\n d(PX_j)_a(v)=a_j\\,dP_a(v)+P(a)v_j,\n\\]\nwhich agrees with the evaluated formal product rule for every coordinate.\nTh","\\begin{lemma}[Derivative of polynomial evaluation]\n\\label{lem:actual-polynomial-derivative}\nFor $P\\in\\mathbb K[X_1,\\ldots,X_n]$, polynomial evaluation is analytic at\nevery point. At $a\\in\\mathbb K^n$ its strict Fr\\'echet derivative is\n\\[\n v\\longmapsto\\sum_{j=1}^n (\\partial_jP)(a)v_j.\n\\]\nFor a finite system, its strict derivative is the continuous linear map\nrepresented by the matrix of evaluated formal partial derivatives.\n\\end{lemma}\n\\begin{proof}\nFor the derivative statement, use polynomial induction on constants,\naddition, and multiplication by a variable. Constants have zero derivative.\nThe sum rule agrees with the additivity of formal partial differentiation.\nFor $PX_j$, the product rule gives\n\\[\n d(PX_j)_a(v)=a_j\\,dP_a(v)+P(a)v_j,\n\\]\nwhich agrees with the evaluated formal product rule for every coordinate.\nThe finite product rule for maps to coordinate spaces then assembles the\ns","\\begin{lemma}[Derivative of polynomial evaluation]\n\\label{lem:actual-polynomial-derivative}\nFor $P\\in\\mathbb K[X_1,\\ldots,X_n]$, polynomial evaluation is analytic at\nevery point. At $a\\in\\mathbb K^n$ its strict Fr\\'echet derivative is\n\\[\n v\\longmapsto\\sum_{j=1}^n (\\partial_jP)(a)v_j.\n\\]\nFor a finite system, its strict derivative is the continuous linear map\nrepresented by the matrix of evaluated formal partial derivatives.\n\\end{lemma}\n\\begin{proof}\nFor the derivative statement, use polynomial induction on constants,\naddition, and multiplication by a variable. Constants have zero derivative.\nThe sum rule agrees with the additivity of formal partial differentiation.\nFor $PX_j$, the product rule gives\n\\[\n d(PX_j)_a(v)=a_j\\,dP_a(v)+P(a)v_j,\n\\]\nwhich agrees with the evaluated formal product rule for every coordinate.\nThe finite product rule for maps to coordinate spaces then assembles the\nsystem derivative. Each matrix defines a continuous linear map because the\ncoordinate sp","a)v_j.\n\\]\nFor a finite system, its strict derivative is the continuous linear map\nrepresented by the matrix of evaluated formal partial derivatives.\n\\end{lemma}\n\\begin{proof}\nFor the derivative statement, use polynomial induction on constants,\naddition, and multiplication by a variable. Constants have zero derivative.\nThe sum rule agrees with the additivity of formal partial differentiation.\nFor $PX_j$, the product rule gives\n\\[\n d(PX_j)_a(v)=a_j\\,dP_a(v)+P(a)v_j,\n\\]\nwhich agrees with the evaluated formal product rule for every coordinate.\nThe finite product rule for maps to coordinate spaces then assembles the\nsystem derivative. Each matrix defines a continuous linear map because the\ncoordinate spaces are finite-dimensional over the complete field.\nPolynomial analyticity follows from the corresponding polynomial induction\nin Mathlib and its finite-coordinate assembly.\n\\end{proof}\nThe derivative bridge and analytic statements are\n\\lean{Komorebi.PolynomialAnalyticEvaluation.hasStrictFDerivAt_evalSystem},\n\\lean{Komorebi.PolynomialAnalyticEvaluation.analyticAt_evalSystem}, and\n\\lean{Kom…","ts strict derivative is the continuous linear map\nrepresented by the matrix of evaluated formal partial derivatives.\n\\end{lemma}\n\\begin{proof}\nFor the derivative statement, use polynomial induction on constants,\naddition, and multiplication by a variable. Constants have zero derivative.\nThe sum rule agrees with the additivity of formal partial differentiation.\nFor $PX_j$, the product rule gives\n\\[\n d(PX_j)_a(v)=a_j\\,dP_a(v)+P(a)v_j,\n\\]\nwhich agrees with the evaluated formal product rule for every coordinate.\nThe finite product rule for maps to coordinate spaces then assembles the\nsystem derivative. Each matrix defines a continuous linear map because the\ncoordinate spaces are finite-dimensional over the complete field.\nPolynomial analyticity follows from the corresponding polynomial induction\nin Mathlib and its finite-coordinate assembly.\n\\end{proof}\nThe derivative bridge and analytic statements are\n\\lean{Komorebi.PolynomialAnalyticEvaluation.hasStrictFDerivAt_evalSystem},\n\\lean{Komorebi.PolynomialAnalyticEvaluation.analyticAt_evalSystem}, and\n\\lean{Komorebi.PolynomialAnalyticEvaluati…","\\begin{lemma}[The augmented derivative]\n\\label{lem:augmented-analytic-inverse}\nIf $J\\in\\operatorname{GL}_e(\\mathbb K)$, the polynomial map\n\\[\n \\Phi(Y,T)=(F(Y,T),T)\n\\]\nhas invertible strict derivative at $a$, with\n\\begin{equation}\n D\\Phi(a)=\\begin{pmatrix}J&K","\\begin{lemma}[The augmented derivative]\n\\label{lem:augmented-analytic-inverse}\nIf $J\\in\\operatorname{GL}_e(\\mathbb K)$, the polynomial map\n\\[\n \\Phi(Y,T)=(F(Y,T),T)\n\\]\nhas invertible strict derivative at $a$, with\n\\begin{equation}\n D\\Phi(a)=\\begin{pmatrix}J&K\\\\0&I_k\\end{pmatrix},\\qquad\n D\\Phi(a)^{-1}=\\begin{pmatrix}J^{-1}&-J^{-1}K\\\\0&I_k\\end{pmatrix}.\n \\label{eq:analytic-block-inverse}\n\\end{equation}\nThe local inverse supplied by this derivative is analytic near $\\Phi(a)$.\n\\end{lemma}\n\\begin{proof}\nThe lower coordinates of $\\Phi$ are the parameter variables. Differentiating\nthem gives the blocks $0$ and $I_k$. Lemma~\\ref{lem:actual-polynomial-derivative}\nidentifies the remaining blocks with~\\eqref{eq:regularity-blocks}.\nMultiplication of the two matrices in~\\eqref{eq:analytic-block-inverse},\nin both orders, gives the identity. The resulting linear equivalence is\ncontinuous, as is its inverse. Apply the strict inverse-function theorem to\n$\\Phi$. The analytic inverse theorem applies to the same open partial\nhomeomorphism, because $\\Phi$ is analytic and its derivative is that","0&I_k\\end{pmatrix}.\n \\label{eq:analytic-block-inverse}\n\\end{equation}\nThe local inverse supplied by this derivative is analytic near $\\Phi(a)$.\n\\end{lemma}\n\\begin{proof}\nThe lower coordinates of $\\Phi$ are the parameter variables. Differentiating\nthem gives the blocks $0$ and $I_k$. Lemma~\\ref{lem:actual-polynomial-derivative}\nidentifies the remaining blocks with~\\eqref{eq:regularity-blocks}.\nMultiplication of the two matrices in~\\eqref{eq:analytic-block-inverse},\nin both orders, gives the identity. The resulting linear equivalence is\ncontinuous, as is its inverse. Apply the strict inverse-function theorem to\n$\\Phi$. The analytic inverse theorem applies to the same open partial\nhomeomorphism, because $\\Phi$ is analytic and its derivative is that\ncontinuous linear equivalence. It supplies a convergent local power-series\nrepresenta","0&I_k\\end{pmatrix}.\n \\label{eq:analytic-block-inverse}\n\\end{equation}\nThe local inverse supplied by this derivative is analytic near $\\Phi(a)$.\n\\end{lemma}\n\\begin{proof}\nThe lower coordinates of $\\Phi$ are the parameter variables. Differentiating\nthem gives the blocks $0$ and $I_k$. Lemma~\\ref{lem:actual-polynomial-derivative}\nidentifies the remaining blocks with~\\eqref{eq:regularity-blocks}.\nMultiplication of the two matrices in~\\eqref{eq:analytic-block-inverse},\nin both orders, gives the identity. The resulting linear equivalence is\ncontinuous, as is its inverse. Apply the strict inverse-function theorem to\n$\\Phi$. The analytic inverse theorem applies to the same open partial\nhomeomorphism, because $\\Phi$ is analytic and its derivative is that\ncontinuous linear equivalence. It supplies a convergent local power-series\nrepresentation of this inverse.\n\\end{proof}\nThe augmented Jacobian is","0&I_k\\end{pmatrix}.\n \\label{eq:analytic-block-inverse}\n\\end{equation}\nThe local inverse supplied by this derivative is analytic near $\\Phi(a)$.\n\\end{lemma}\n\\begin{proof}\nThe lower coordinates of $\\Phi$ are the parameter variables. Differentiating\nthem gives the blocks $0$ and $I_k$. Lemma~\\ref{lem:actual-polynomial-derivative}\nidentifies the remaining blocks with~\\eqref{eq:regularity-blocks}.\nMultiplication of the two matrices in~\\eqref{eq:analytic-block-inverse},\nin both orders, gives the identity. The resulting linear equivalence is\ncontinuous, as is its inverse. Apply the strict inverse-function theorem to\n$\\Phi$. The analytic inverse theorem applies to the same open partial\nhomeomorphism, because $\\Phi$ is analytic and its derivative is that\ncontinuous linear equivalence. It supplies a convergent local power-series\nrepresentation of this inverse.\n\\end{proof}\nThe augmented Jacobian is identified in\n\\lean{Komorebi.PolynomialJointSystem.jacobian_augmented}","0&I_k\\end{pmatrix}.\n \\label{eq:analytic-block-inverse}\n\\end{equation}\nThe local inverse supplied by this derivative is analytic near $\\Phi(a)$.\n\\end{lemma}\n\\begin{proof}\nThe lower coordinates of $\\Phi$ are the parameter variables. Differentiating\nthem gives the blocks $0$ and $I_k$. Lemma~\\ref{lem:actual-polynomial-derivative}\nidentifies the remaining blocks with~\\eqref{eq:regularity-blocks}.\nMultiplication of the two matrices in~\\eqref{eq:analytic-block-inverse},\nin both orders, gives the identity. The resulting linear equivalence is\ncontinuous, as is its inverse. Apply the strict inverse-function theorem to\n$\\Phi$. The analytic inverse theorem applies to the same open partial\nhomeomorphism, because $\\Phi$ is analytic and its derivative is that\ncontinuous linear equivalence. It supplies a convergent local power-series\nrepresentation of this inverse.\n\\end{proof}\nThe augmented Jacobian is identified in\n\\lean{Komorebi.PolynomialJointSystem.jacobian_augmented}.\nThe cont","r variables. Differentiating\nthem gives the blocks $0$ and $I_k$. Lemma~\\ref{lem:actual-polynomial-derivative}\nidentifies the remaining blocks with~\\eqref{eq:regularity-blocks}.\nMultiplication of the two matrices in~\\eqref{eq:analytic-block-inverse},\nin both orders, gives the identity. The resulting linear equivalence is\ncontinuous, as is its inverse. Apply the strict inverse-function theorem to\n$\\Phi$. The analytic inverse theorem applies to the same open partial\nhomeomorphism, because $\\Phi$ is analytic and its derivative is that\ncontinuous linear equivalence. It supplies a convergent local power-series\nrepresentation of this inverse.\n\\end{proof}\nThe augmented Jacobian is identified in\n\\lean{Komorebi.PolynomialJointSystem.jacobian_augmented}.\nThe continuous inverse block and its restriction are\n\\lean{Komorebi.PolynomialAnalyticBlocks.upperEquiv} and\n\\lean{Komorebi.PolynomialAnalyticBlocks.inverse_solved_zero}.\nAnalyticity of the same constructed inverse is\n\\lean{Komorebi.PolynomialAnalyticInverse.analyticAt_inverse}.","omial-derivative}\nidentifies the remaining blocks with~\\eqref{eq:regularity-blocks}.\nMultiplication of the two matrices in~\\eqref{eq:analytic-block-inverse},\nin both orders, gives the identity. The resulting linear equivalence is\ncontinuous, as is its inverse. Apply the strict inverse-function theorem to\n$\\Phi$. The analytic inverse theorem applies to the same open partial\nhomeomorphism, because $\\Phi$ is analytic and its derivative is that\ncontinuous linear equivalence. It supplies a convergent local power-series\nrepresentation of this inverse.\n\\end{proof}\nThe augmented Jacobian is identified in\n\\lean{Komorebi.PolynomialJointSystem.jacobian_augmented}.\nThe continuous inverse block and its restriction are\n\\lean{Komorebi.PolynomialAnalyticBlocks.upperEquiv} and\n\\lean{Komorebi.PolynomialAnalyticBlocks.inverse_solved_zero}.\nAnalyticity of the same constructed inverse is\n\\lean{Komorebi.PolynomialAnalyticInverse.analyticAt_inverse}.","\\begin{theorem}[Analytic local root graph]\n\\label{thm:analytic-local-graph}\nSuppose $F(a)=0$ and $J\\in\\operatorname{GL}_e(\\mathbb K)$. There are open\nneighborhoods $t_0\\in U\\subseteq\\mathbb K^k$ and\n$y_0\\in V\\subseteq\\mathbb K^e$, and a map\n$\\psi:\\mathbb K^k\\to\\mathbb K^e$, such that\n\\begin{gather}\n \\psi(t_0)=y_0,\\qquad \\psi(U)\\subseteq V,\\qquad\n \\psi\\text{ is analytic at every point of }U,\\label{eq:analytic-psi}","\\begin{theorem}[Analytic local root graph]\n\\label{thm:analytic-local-graph}\nSuppose $F(a)=0$ and $J\\in\\operatorname{GL}_e(\\mathbb K)$. There are open\nneighborhoods $t_0\\in U\\subseteq\\mathbb K^k$ and\n$y_0\\in V\\subseteq\\mathbb K^e$, and a map\n$\\psi:\\mathbb K^k\\to\\mathbb K^e$, such that\n\\begin{gather}\n \\psi(t_0)=y_0,\\qquad \\psi(U)\\subseteq V,\\qquad\n \\psi\\text{ is analytic at every point of }U,\\label{eq:analytic-psi}\\\\\n F(y,t)=0\\quad\\Longleftrightarrow\\quad y=\\psi(t)\n \\qquad(t\\in U,\\ y\\in V).\\label{eq:analytic-root-graph}\n\\end{gather}\nAt the base parameter, the strict Fr\\'echet derivative is\n\\begin{equation}\n \\boxed{D\\psi(t_0)=-J^{-1}K.}\n \\label{eq:implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-bloc…","F(y,t)=0\\quad\\Longleftrightarrow\\quad y=\\psi(t)\n \\qquad(t\\in U,\\ y\\in V).\\label{eq:analytic-root-graph}\n\\end{gather}\nAt the base parameter, the strict Fr\\'echet derivative is\n\\begin{equation}\n \\boxed{D\\psi(t_0)=-J^{-1}K.}\n \\label{eq:implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K","t\\in U,\\ y\\in V).\\label{eq:analytic-root-graph}\n\\end{gather}\nAt the base parameter, the strict Fr\\'echet derivative is\n\\begin{equation}\n \\boxed{D\\psi(t_0)=-J^{-1}K.}\n \\label{eq:implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K","\\in V).\\label{eq:analytic-root-graph}\n\\end{gather}\nAt the base parameter, the strict Fr\\'echet derivative is\n\\begin{equation}\n \\boxed{D\\psi(t_0)=-J^{-1}K.}\n \\label{eq:implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K","tic-root-graph}\n\\end{gather}\nAt the base parameter, the strict Fr\\'echet derivative is\n\\begin{equation}\n \\boxed{D\\psi(t_0)=-J^{-1}K.}\n \\label{eq:implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K","ather}\nAt the base parameter, the strict Fr\\'echet derivative is\n\\begin{equation}\n \\boxed{D\\psi(t_0)=-J^{-1}K.}\n \\label{eq:implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K","strict Fr\\'echet derivative is\n\\begin{equation}\n \\boxed{D\\psi(t_0)=-J^{-1}K.}\n \\label{eq:implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K",":implicit-derivative}\n\\end{equation}\nIn particular, $\\psi$ is differentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K","erentiable on $U$.\n\\end{theorem}\n\\begin{proof}\nLet $g$ be the analytic local inverse of $\\Phi$ supplied by\nLemma~\\ref{lem:augmented-analytic-inverse}. Since $F(a)=0$,\n$\\Phi(a)=(0,t_0)$ and $g(0,t_0)=a$. Define\n\\[\n \\psi(t)=\\pi_Yg(0,t),\n\\]\nwhere $\\pi_Y$ projects to the solved coordinates. This gives the base value\nand analyticity at $t_0$. The chain rule and\n\\eqref{eq:analytic-block-inverse} give\n\\[\n D\\psi(t_0)v\n =\\pi_Y\\begin{pmatrix}J^{-1}&-J^{-1}K\\\\0&I_k\\end{pmatrix}\n       \\binom{0}{v}\n =-J^{-1}Kv.\n\\]","To obtain the root equivalence on a product neighborhood, we choose the\nneighborhoods as follows. The source of the open partial homeomorphism defined by\n$\\Phi$ is open around $a$. Its inverse is $g$. Choose open sets $V$ and $U_0$\ncontaining $y_0$ and $t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","duct neighborhood, we choose the\nneighborhoods as follows. The source of the open partial homeomorphism defined by\n$\\Phi$ is open around $a$. Its inverse is $g$. Choose open sets $V$ and $U_0$\ncontaining $y_0$ and $t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","rhoods as follows. The source of the open partial homeomorphism defined by\n$\\Phi$ is open around $a$. Its inverse is $g$. Choose open sets $V$ and $U_0$\ncontaining $y_0$ and $t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","partial homeomorphism defined by\n$\\Phi$ is open around $a$. Its inverse is $g$. Choose open sets $V$ and $U_0$\ncontaining $y_0$ and $t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","around $a$. Its inverse is $g$. Choose open sets $V$ and $U_0$\ncontaining $y_0$ and $t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","$. Its inverse is $g$. Choose open sets $V$ and $U_0$\ncontaining $y_0$ and $t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","erse is $g$. Choose open sets $V$ and $U_0$\ncontaining $y_0$ and $t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","$t_0$ such that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","h that\n$V\\times U_0$ lies in that source. Restrict $U_0$ further by the preimage of\nan open analytic inverse neighborhood under $t\\mapsto(0,t)$, the open locus\nwhere this same $\\psi$ is analytic, and the interior of $\\psi^{-1}(V)$.\nAll of these sets contain $t_0$. For the last set, use the continuity of\n$\\psi$ at $t_0$ and $\\psi(t_0)=y_0\\in V$. Their intersection is an open\nneighborhood $U$ with $\\psi(U)\\subseteq V$.","If $t\\in U$, $y\\in V$ and $F(y,t)=0$, then\n$\\Phi(y,t)=(0,t)$. The left inverse identity forces\n$g(0,t)=(y,t)$, hence $y=\\psi(t)$. Conversely, the right inverse identity\n$\\Phi(g(0,t))=(0,t)$ says both that its solved equations vanish and that its\nparameter coordinates are exactly $t$. Thus\n$g(0,t)=(\\psi(t),t)$ and $F(\\psi(t),t)=0$.\nAnalyticity at every point of $U$ implies differentiability there.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialAnalyticImplicit.exists_analytic_implicit}.\nDifferentiability is also included explicitly in the statement contract\n\\lean{Komorebi.PolynomialRegularityContracts.actual_padic_analytic_root_graph}.","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.Pol","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelectio","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_select","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts fo","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation deri","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equ","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\np","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base v","\\begin{theorem}[Regularity of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","ty of a supplied root selection]\n\\label{thm:selection-regularity}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","ty}\nUnder the hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","hypotheses of Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","Theorem~\\ref{thm:analytic-local-graph}, let\n$h:\\mathbb K^k\\to\\mathbb K^e$ be continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","e continuous at $t_0$, with\n$h(t_0)=y_0$. Suppose $F(h(t),t)=0$ for every $t$ in some neighborhood of\n$t_0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","0$. Then $h$ and $\\psi$ agree in a neighborhood of $t_0$. Consequently\n$h$ is analytic there as a germ and has strict derivative\n$Dh(t_0)=-J^{-1}K$.\n\\end{theorem}\n\\begin{proof}\nContinuity at $t_0$ places $h(t)$ inside $V$ for all sufficiently close\nparameters. Intersect this neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","neighborhood with $U$ and the neighborhood on\nwhich the root equation holds. Equation~\\eqref{eq:analytic-root-graph} gives\n$h(t)=\\psi(t)$ throughout the intersection. Analyticity and strict\ndifferentiability at a point are unchanged by agreement on a neighborhood.\n\\end{proof}\nThe formal result is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_regularity}.\nIts formulation derives $F(a)=0$ from the root equation near the base\nparameter and the specified base value.","\\begin{corollary}[Regularity throughout an open parameter set]\n\\label{cor:selection-open-regularity}\nLet $U\\subseteq\\mathbb K^k$ be open, and let $h$ be continuous on $U$.\nSuppose that for every $t\\in U$,\n\\[\n F(h(t),t)=0,\n \\qquad J_t=D_YF(h(t),t)\\in\\operatorname{GL}_e(\\mathbb K).\n\\]\nThen $h$ is analytic at every point of $U$ and its strict derivative is\n\\[\n Dh(t)=-J_t^{-1}D_TF(h(t),t)\\qquad(t\\in U).\n\\]\n\\end{corollary}\n\\begin{proof}\nAt each $t\\in U$, openness turns continuity on $U$ into continuity at $t$\nand makes the root equation hold on a neighborhood of $t$. Apply\nTheorem~\\ref{thm:selection-regularity} at $(h(t),t)$.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_analyticOnNhd}.\nThe pointwise invertibility assumption is part of this corollary.","en parameter set]\n\\label{cor:selection-open-regularity}\nLet $U\\subseteq\\mathbb K^k$ be open, and let $h$ be continuous on $U$.\nSuppose that for every $t\\in U$,\n\\[\n F(h(t),t)=0,\n \\qquad J_t=D_YF(h(t),t)\\in\\operatorname{GL}_e(\\mathbb K).\n\\]\nThen $h$ is analytic at every point of $U$ and its strict derivative is\n\\[\n Dh(t)=-J_t^{-1}D_TF(h(t),t)\\qquad(t\\in U).\n\\]\n\\end{corollary}\n\\begin{proof}\nAt each $t\\in U$, openness turns continuity on $U$ into continuity at $t$\nand makes the root equation hold on a neighborhood of $t$. Apply\nTheorem~\\ref{thm:selection-regularity} at $(h(t),t)$.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_analyticOnNhd}.\nThe pointwise invertibility assumption is part of this corollary.","open-regularity}\nLet $U\\subseteq\\mathbb K^k$ be open, and let $h$ be continuous on $U$.\nSuppose that for every $t\\in U$,\n\\[\n F(h(t),t)=0,\n \\qquad J_t=D_YF(h(t),t)\\in\\operatorname{GL}_e(\\mathbb K).\n\\]\nThen $h$ is analytic at every point of $U$ and its strict derivative is\n\\[\n Dh(t)=-J_t^{-1}D_TF(h(t),t)\\qquad(t\\in U).\n\\]\n\\end{corollary}\n\\begin{proof}\nAt each $t\\in U$, openness turns continuity on $U$ into continuity at $t$\nand makes the root equation hold on a neighborhood of $t$. Apply\nTheorem~\\ref{thm:selection-regularity} at $(h(t),t)$.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_analyticOnNhd}.\nThe pointwise invertibility assumption is part of this corollary.","\\subseteq\\mathbb K^k$ be open, and let $h$ be continuous on $U$.\nSuppose that for every $t\\in U$,\n\\[\n F(h(t),t)=0,\n \\qquad J_t=D_YF(h(t),t)\\in\\operatorname{GL}_e(\\mathbb K).\n\\]\nThen $h$ is analytic at every point of $U$ and its strict derivative is\n\\[\n Dh(t)=-J_t^{-1}D_TF(h(t),t)\\qquad(t\\in U).\n\\]\n\\end{corollary}\n\\begin{proof}\nAt each $t\\in U$, openness turns continuity on $U$ into continuity at $t$\nand makes the root equation hold on a neighborhood of $t$. Apply\nTheorem~\\ref{thm:selection-regularity} at $(h(t),t)$.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_analyticOnNhd}.\nThe pointwise invertibility assumption is part of this corollary.","n $U$.\nSuppose that for every $t\\in U$,\n\\[\n F(h(t),t)=0,\n \\qquad J_t=D_YF(h(t),t)\\in\\operatorname{GL}_e(\\mathbb K).\n\\]\nThen $h$ is analytic at every point of $U$ and its strict derivative is\n\\[\n Dh(t)=-J_t^{-1}D_TF(h(t),t)\\qquad(t\\in U).\n\\]\n\\end{corollary}\n\\begin{proof}\nAt each $t\\in U$, openness turns continuity on $U$ into continuity at $t$\nand makes the root equation hold on a neighborhood of $t$. Apply\nTheorem~\\ref{thm:selection-regularity} at $(h(t),t)$.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_analyticOnNhd}.\nThe pointwise invertibility assumption is part of this corollary.","=0,\n \\qquad J_t=D_YF(h(t),t)\\in\\operatorname{GL}_e(\\mathbb K).\n\\]\nThen $h$ is analytic at every point of $U$ and its strict derivative is\n\\[\n Dh(t)=-J_t^{-1}D_TF(h(t),t)\\qquad(t\\in U).\n\\]\n\\end{corollary}\n\\begin{proof}\nAt each $t\\in U$, openness turns continuity on $U$ into continuity at $t$\nand makes the root equation hold on a neighborhood of $t$. Apply\nTheorem~\\ref{thm:selection-regularity} at $(h(t),t)$.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialAnalyticSelection.continuous_selection_analyticOnNhd}.\nThe pointwise invertibility assumption is part of this corollary.","\\begin{corollary}[Integral coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of th","\\begin{corollary}[Integral coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exa","\\begin{corollary}[Integral coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains a","\\begin{corollary}[Integral coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root.","\\begin{corollary}[Integral coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-loc","\\begin{corollary}[Integral coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-local-graph} to the system with included\ncoefficients.\n\\end{proof}\nT","\\begin{corollary}[Integral coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-local-graph} to the system with included\ncoefficients.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialRegularityIntegral.exists_analytic_integral_impl","coefficients and centers]\n\\label{cor:integral-analytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-local-graph} to the system with included\ncoefficients.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialRegularityIntegral.exists_analytic_integral_implicit}.\nIt requires no invertibility modulo $p$. In particular, it includes solved\n…","ytic-graph}\nLet $p$ be any prime. Suppose $F\\in\\Zp[Y,T]^e$, $a\\in\\Zp^{e+k}$,\nand $F(a)=0$. Let $\\iota:\\Zp\\hookrightarrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-local-graph} to the system with included\ncoefficients.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialRegularityIntegral.exists_analytic_integral_implicit}.\nIt requires no invertibility modulo $p$. In particular, it includes solved\nJacobians singular modulo $p$ but invertible over …","arrow\\Qp$ be the inclusion.\nIf $\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\Qp)$, the polynomial system\nobtained by including the original coefficients in $\\Qp$ has the analytic\nlocal root graph of Theorem~\\ref{thm:analytic-local-graph} at $\\iota(a)$.\nThe strict derivative at the base is\n\\[\n -\\iota(D_YF(a))^{-1}\\iota(D_TF(a)).\n\\]\nThe open neighborhoods and the output map are in the ambient spaces over\n$\\Qp$.\n\\end{corollary}\n\\begin{proof}\nCoefficient inclusion commutes with evaluation and formal partial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-local-graph} to the system with included\ncoefficients.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialRegularityIntegral.exists_analytic_integral_implicit}.\nIt requires no invertibility modulo $p$. In particular, it includes solved\nJacobians singular modulo $p$ but invertible over the fraction field.","tial\ndifferentiation. Consequently both Jacobian blocks after inclusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-local-graph} to the system with included\ncoefficients.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialRegularityIntegral.exists_analytic_integral_implicit}.\nIt requires no invertibility modulo $p$. In particular, it includes solved\nJacobians singular modulo $p$ but invertible over the fraction field.","lusion are the\nentrywise inclusions of the original blocks, and the original exact root\nremains an exact root. Apply Theorem~\\ref{thm:analytic-local-graph} to the system with included\ncoefficients.\n\\end{proof}\nThis is\n\\lean{Komorebi.PolynomialRegularityIntegral.exists_analytic_integral_implicit}.\nIt requires no invertibility modulo $p$. In particular, it includes solved\nJacobians singular modulo $p$ but invertible over the fraction field.","\\begin{example}[The derivative sign and a nonlinear control]\nFor $F(Y,T)=Y+T$, the exact branch is $\\psi(T)=-T$ and $J=K=1$.\nThe derivative is $-1$. Uniqueness of the derivative rules out $+1$ over\n$\\Qp$, including $p=2$. The nonlinear polynomial $F(Y,T)=Y+Y^2+T$\nhas $F(0,0)=0$ and $J=K=1$ there. Applying the formal theorem to this polynomial constructs an analytic\nlocal root graph with derivative $-1$. Thus the example checks the existence\nconclusion as well as the derivative formula.\nThe declarations and the deliberately rejected positive-sign statement are\nin the regularity control and statement-contract files.\n\\end{example}","egin{example}[The derivative sign and a nonlinear control]\nFor $F(Y,T)=Y+T$, the exact branch is $\\psi(T)=-T$ and $J=K=1$.\nThe derivative is $-1$. Uniqueness of the derivative rules out $+1$ over\n$\\Qp$, including $p=2$. The nonlinear polynomial $F(Y,T)=Y+Y^2+T$\nhas $F(0,0)=0$ and $J=K=1$ there. Applying the formal theorem to this polynomial constructs an analytic\nlocal root graph with derivative $-1$. Thus the example checks the existence\nconclusion as well as the derivative formula.\nThe declarations and the deliberately rejected positive-sign statement are\nin the regularity control and statement-contract files.\n\\end{example}","The results cover arbitrary finite $e$ and $k$, including zero. The local\ninverse machinery explicitly handles a zero-dimensional domain. The analytic theorem does not assign a uniform numerical radius to its\nneighborhoods. The earlier\nweighted lifting and joint deformation theorems retain their quantitative\nradii and error bounds, while the present statements establish regularity\nof the local solution germ. A direct Lean identification with each\nhistorical subtype-valued chart is an additional interface task. General\npersistence a","The results cover arbitrary finite $e$ and $k$, including zero. The local\ninverse machinery explicitly handles a zero-dimensional domain. The analytic theorem does not assign a uniform numerical radius to its\nneighborhoods. The earlier\nweighted lifting and joint deformation theorems retain their quantitative\nradii and error bounds, while the present statements establish regularity\nof the local solution germ. A direct Lean identification with each\nhistorical subtype-valued chart is an additional interface task. General\npersistence at a frac","The weighted lifting theorem allows the solved Jacobian to be noninvertible\nmodulo $p$ provided that it is invertible over $\\Qp$ with a controlled inverse.\nThis includes a substantial class of lifts often called singular modulo $p$.\nThe next problem is to understand Jacobians that remain rank-deficient over\n$\\Qp$, where the inverse construction is no longer available.","For a fixed polynomial system $F\\in\\Z[X_1,\\ldots,X_d]^e$, a precision $L\\geq1$,\nand a root $\\bar x$ modulo $p^L$, let\n\\[\n\\mathcal L_{F,L}(\\bar x)\n=\\{x'\\in(\\Z/p^{L+1}\\Z)^d:\nF(x')=0,\\ x'\\equiv\\bar x\\pmod{p^L}\\}.\n\\]\nThe finite obstruction theorem decides whether this set is empty from the\nresidual digit and reduced Jacobian. When it is nonempty, the theorem also\ndetermines its cardinality and affine kernel structure. The conclusion\nconcerns one level: vanishing of the obstruction permits the next step,\nbut does not guarantee that subsequent steps are possible.","For example, $X^2-5$ has the root $1$ modulo $2$, and both $1$ and $3$ lift it\nto roots modulo $4$. Every odd square is $1$ modulo $8$, so neither branch\ncontinues to a root modulo $8$. Thus even a successful first correction can\nlead to branches that stop at the following level.","The original conjecture asks for a canonical finite obstruction packet whose\nvanishing is necessary and sufficient for a one-level lift in a finitely\npresented arithmetic seed system. For a fixed finite polynomial presentation,\nthe chapter supplies the precise obstruction\n\\[\no_{F,\\bar x}=[r]\\in\n\\coker\\!\\left(J_{F,\\bar x}:\\F_p^d\\longrightarrow\\F_p^e\\right),\n\\]\nwhere $r$ is the residual digit. We have proved independence from the chosen\ninteger representative. Invariance under a change of equations or presentation\nrequires more.","The raw cokernel already reveals the difficulty.\nThe presentations $F=(X)$ and $G=(X,0)$ describe the same zero locus over every\ncoefficient ring. At zero their Jacobians are, respectively,\n\\[\n\\F_p\\xrightarrow{\\,1\\,}\\F_p,\n\\qquad\n\\F_p\\xrightarrow{\\,(1,0)\\,}\\F_p^2.\n\\]\nTheir raw cokernels are $0$ and $\\F_p$. Both distinguished obstruction classes\nvanish, but the ambient obstruction groups are not isomorphic. The example\nrules out the raw cokernel as a presentation-independent object while leaving\nopen the possibility of an intrinsic obstruction construction.","The original proposed form is\n\\[\nX\\simeq X_2^{\\mathrm{wt}}\\times_C\\prod_{p>2}'X_p.\n\\]\nIt leaves $X$, the distinguished subsets defining the restricted product,\nthe maps to $C$, the ambient category, and the universal meaning of\n``smallest'' unspecified. In particular, one must decide whether $X$ is a\nglobal object or a completed object.","The linear compatibility theorem provides one concrete part of this program.\nIf two local observations\n$J:U\\to V$ and $K:U\\to W$ are linear maps from the \\emph{same} correction\nmodule, define\n\\[\nC_{J,K}=U/(\\ker J+\\ker K).\n\\]\nThe attainable observations $\\im J$ and $\\im K$ have canonical labels in this\nquotient. Equality of labels is equivalent to simultaneous realization by a\nsingle correction. The joint image is exactly the fiber product over\n$C_{J,K}$, with the proved linear universal property.\nThe declarations are\n\\begingroup\\small\\raggedright\n\\lean{Komorebi.SharedCorrection.labels_eq_iff_common},\n\\lean{Komorebi.SharedCorrection.joint_range_eq_fiber}, and\n\\lean{Komorebi.SharedCorrection.common_universal}.\n\\par\\endgroup\nTo apply this result arithmetically, one must construct the common correction\nmo","The linear compatibility theorem provides one concrete part of this program.\nIf two local observations\n$J:U\\to V$ and $K:U\\to W$ are linear maps from the \\emph{same} correction\nmodule, define\n\\[\nC_{J,K}=U/(\\ker J+\\ker K).\n\\]\nThe attainable observations $\\im J$ and $\\im K$ have canonical labels in this\nquotient. Equality of labels is equivalent to simultaneous realization by a\nsingle correction. The joint image is exactly the fiber product over\n$C_{J,K}$, with the proved linear universal property.\nThe declarations are\n\\begingroup\\small\\raggedright\n\\lean{Komorebi.SharedCorrection.labels_eq_iff_common},\n\\lean{Komorebi.SharedCorrection.joint_range_eq_fiber}, and\n\\lean{Komorebi.SharedCorrection.common_universal}.\n\\par\\endgroup\nTo apply this result arithmetically, one must construct the common correction\nmodule and the ob","The linear compatibility theorem provides one concrete part of this program.\nIf two local observations\n$J:U\\to V$ and $K:U\\to W$ are linear maps from the \\emph{same} correction\nmodule, define\n\\[\nC_{J,K}=U/(\\ker J+\\ker K).\n\\]\nThe attainable observations $\\im J$ and $\\im K$ have canonical labels in this\nquotient. Equality of labels is equivalent to simultaneous realization by a\nsingle correction. The joint image is exactly the fiber product over\n$C_{J,K}$, with the proved linear universal property.\nThe declarations are\n\\begingroup\\small\\raggedright\n\\lean{Komorebi.SharedCorrection.labels_eq_iff_common},\n\\lean{Komorebi.SharedCorrection.joint_range_eq_fiber}, and\n\\lean{Komorebi.SharedCorrection.common_universal}.\n\\par\\endgroup\nTo apply this result arithmetically, one must construct the common correction\nmodule and the observation maps, then show that the resulting compatibility\ncondition agrees with the intended a","The linear compatibility theorem provides one concrete part of this program.\nIf two local observations\n$J:U\\to V$ and $K:U\\to W$ are linear maps from the \\emph{same} correction\nmodule, define\n\\[\nC_{J,K}=U/(\\ker J+\\ker K).\n\\]\nThe attainable observations $\\im J$ and $\\im K$ have canonical labels in this\nquotient. Equality of labels is equivalent to simultaneous realization by a\nsingle correction. The joint image is exactly the fiber product over\n$C_{J,K}$, with the proved linear universal property.\nThe declarations are\n\\begingroup\\small\\raggedright\n\\lean{Komorebi.SharedCorrection.labels_eq_iff_common},\n\\lean{Komorebi.SharedCorrection.joint_range_eq_fiber}, and\n\\lean{Komorebi.SharedCorrection.common_universal}.\n\\par\\endgroup\nTo apply this result arithmetically, one must construct the common correction\nmodule and the observation maps, then show that the resulting compatibility\ncondition agrees with the intended arithmetic one.","crete part of this program.\nIf two local observations\n$J:U\\to V$ and $K:U\\to W$ are linear maps from the \\emph{same} correction\nmodule, define\n\\[\nC_{J,K}=U/(\\ker J+\\ker K).\n\\]\nThe attainable observations $\\im J$ and $\\im K$ have canonical labels in this\nquotient. Equality of labels is equivalent to simultaneous realization by a\nsingle correction. The joint image is exactly the fiber product over\n$C_{J,K}$, with the proved linear universal property.\nThe declarations are\n\\begingroup\\small\\raggedright\n\\lean{Komorebi.SharedCorrection.labels_eq_iff_common},\n\\lean{Komorebi.SharedCorrection.joint_range_eq_fiber}, and\n\\lean{Komorebi.SharedCorrection.common_universal}.\n\\par\\endgroup\nTo apply this result arithmetically, one must construct the common correction\nmodule and the observation maps, then show that the resulting compatibility\ncondition agrees with the intended arithmetic one.","\\paragraph{Why the global/completed distinction is necessary.}\nConsider constant sequences with integer initial value. Their global parameter\nspace is $\\Z$, while their local parameter spaces are $\\Z_p$. Under the\ncanonical local maps, the image of the global objects is the diagonal\n\\[\nD=\\{(n)_p:n\\in\\Z\\}\\subseteq\\prod_p\\Z_p.\n\\]\nThis diagonal is dense: a basic open set specifies finitely many congruences,\nand the Chinese remainder theorem provides an integer satisfying them. It is\nproper: the tuple with $2$-adic coordinate zero and all odd-adic coordinates\none is not the image of an integer. Indeed an integer zero in $\\Z_2$ must be\nzero, contradicting its $3$-adic coordinate.","al value. Their global parameter\nspace is $\\Z$, while their local parameter spaces are $\\Z_p$. Under the\ncanonical local maps, the image of the global objects is the diagonal\n\\[\nD=\\{(n)_p:n\\in\\Z\\}\\subseteq\\prod_p\\Z_p.\n\\]\nThis diagonal is dense: a basic open set specifies finitely many congruences,\nand the Chinese remainder theorem provides an integer satisfying them. It is\nproper: the tuple with $2$-adic coordinate zero and all odd-adic coordinates\none is not the image of an integer. Indeed an integer zero in $\\Z_2$ must be\nzero, contradicting its $3$-adic coordinate.","ocal maps, the image of the global objects is the diagonal\n\\[\nD=\\{(n)_p:n\\in\\Z\\}\\subseteq\\prod_p\\Z_p.\n\\]\nThis diagonal is dense: a basic open set specifies finitely many congruences,\nand the Chinese remainder theorem provides an integer satisfying them. It is\nproper: the tuple with $2$-adic coordinate zero and all odd-adic coordinates\none is not the image of an integer. Indeed an integer zero in $\\Z_2$ must be\nzero, contradicting its $3$-adic coordinate.","cts is the diagonal\n\\[\nD=\\{(n)_p:n\\in\\Z\\}\\subseteq\\prod_p\\Z_p.\n\\]\nThis diagonal is dense: a basic open set specifies finitely many congruences,\nand the Chinese remainder theorem provides an integer satisfying them. It is\nproper: the tuple with $2$-adic coordinate zero and all odd-adic coordinates\none is not the image of an integer. Indeed an integer zero in $\\Z_2$ must be\nzero, contradicting its $3$-adic coordinate.","Consequently this canonical global diagonal cannot be an equality locus\n\\[\n\\{(a,b):f(a)=g(b)\\}\n\\subseteq \\Z_2\\times\\prod_{p>2}\\Z_p\n\\]\nfor continuous maps to a Hausdorff space $C$. Such an equality locus is closed,\nas the inverse image of the closed diagonal in $C\\times C$, whereas $D$ is\nproper and dense. This topological argument, retained from the native progress\nnote, concerns the stated topological interpretation. It has not been\nformalized in Lean for this manuscript.","Finite CRT reconstruction leads to the familiar completed object\n$\\widehat\\Z=\\prod_p\\Z_p$, but does not identify which local tuples come from a\nsingle global integer. A complete form of Native 13.3 must therefore specify\nfinite-level compatibility, transition maps and completion, and any additional\nglobal descent condition. If recurrence, spectral, or Frobenius information\nis to supply that condition, it must be expressed through defined maps or\nconstraints on the local data, with a precise account of how they enter $C$.","l tuples come from a\nsingle global integer. A complete form of Native 13.3 must therefore specify\nfinite-level compatibility, transition maps and completion, and any additional\nglobal descent condition. If recurrence, spectral, or Frobenius information\nis to supply that condition, it must be expressed through defined maps or\nconstraints on the local data, with a precise account of how they enter $C$.","The accompanying \\nolinkurl{native/Native_Conjectures_Progress.md} records\nfurther results in restricted settings: an intrinsic finite\n$\\operatorname{Ext}^1$ obstruction for finitely presented commutative integer\nalgebras, finite CRT reconstruction for polynomial solution sets, and a\ncharacterization of rectangular finite relations by equality of common\nlabels. The obstruction result is a stated specialization of a classical\nexternal theorem. The CRT and rectangularity results have prose proofs in\nthe note. These are preserved mathematical progress, with the restrictions\nand source records given there, rather than Lean completions of the general\nNative 13.2 and 13.3 statements.","Write $U_{N,j}(n)=j!C_{N,j}(n)$ and $U_j=U_{4,j}$, and use\n$v_p(0)=\\infty$ when discussing valuation bounds. The following classification\nquestions remain open in this program.","\\begin{enumerate}\n\\item \\textbf{The remaining higher-difference classification.}\nThe new strengthening chapters determine every repeated-step valuation in addresses $j=0,1$, give the exact criterion for equality in the common mixed-step lower bound there, and determine address-two valuations at all power-of-two orders. What remains is a complete exact formula in address two at the other orders, a full classification in address three, and the amount of additional cancellation in mixed-step cases where the leading term vanishes. Numerical patterns through a finite range do not settle these questions.","\\item \\textbf{Upper-half addresses at larger dyadic rank.}\nThe exact first-difference laws now include $N=2,4,8$ and every lower-half address for $N=2^a\\ge4$. Larger upper-half addresses need a finer classification. Explicit counterexamples show that extrapolating one low-order Newton coefficient to a universal metric law is false. A least period is measured in packet rows. Its Euler-degree displacement is $NP$.","bf{Upper-half addresses at larger dyadic rank.}\nThe exact first-difference laws now include $N=2,4,8$ and every lower-half address for $N=2^a\\ge4$. Larger upper-half addresses need a finer classification. Explicit counterexamples show that extrapolating one low-order Newton coefficient to a universal metric law is false. A least period is measured in packet rows. Its Euler-degree displacement is $NP$.","\\item \\textbf{Exact affine Mahler-kernel valuations.}\nFor the kernel of Section~\\ref{subsec:polyphase-exposition}, determine exact\nvalues of\n\\[\nv_p\\bigl(\\Theta_{M,s}(r,k)\\bigr),\\qquad\n\\Theta_{M,s}(r,k)=[z^r](1+z)^s((1+z)^M-1)^k,\n\\]\nbeyond the explicit special families. The minimum valuation of the summands\ngives a lower bound. Equality requires a separate argument excluding\ncancellation.","One further task is to connect the older chart constructions to the new\nregularity theorem. That theorem applies to polynomials with arbitrary\ncoefficients in the ambient field $\\Qp$ and identifies the analytic\ngerm of every continuous ambient-field root selection under its hypotheses.\nSeveral older rectangular charts were defined using integer coefficients,\ninteger centers, or subtype-valued maps. Comparing each such historical\ninterface explicitly with the new ambient formulation remains to be done.\nThese comparisons would connect the existing constructions to the proved\nanalytic implicit-function theorem. General singular persistence remains the\nseparate problem describ","\\midrule\nOriginal completion baseline & 79 & $79+37+3$ & 1000","retained logs. In particular, the following interfaces distinguish\nthe principal completed results from generic auxiliary lemmas:\n\\begin{itemize}\n\\item The external conjecture is assembled by\n\\lean{NishibuchiCompletion.conjecture_3_9} in the original completion\nbaseline. Its coefficient identification, integrality, parameter split and\nexplicit threshold are part of that proof chain.\n\\item Exact $N=4$ packet periods are supplied by\n\\lean{DyadicFourLeastPeriods.least_packet_period}. The isometric extension\nis supplied by \\lean{DyadicFourIsometry.existsUnique_isometry}. The\nnormalization of the original coefficients is an explicit part of these\ninterfaces.\n\\item The classical sharp edge tower is supplied by\n\\lean{DyadicSharpTower.dyadic_supercongruence_sharp}, together with its\nfirst omitted error valuation. Its optimality concerns the stated uniform\ntruncation bound.\n\\item Prescribed parameter lifting uses the actual residual and actual\npartial Jacobians in\n\\lean{PolynomialPrescribedObstruction.prescribed_lift_iff_obstruction_zero}.\nThe quotient criterion and exact lift counts are not …","ion.prescribed_lift_iff_obstruction_zero}.\nThe quotient criterion and exact lift counts are not assumptions of the\nlater tower results.\n\\item Joint deformation is supplied by\n\\lean{PolynomialJointDeformation.exists_unique_joint_deformation}.\nLocal regularity is supplied by\n\\lean{PolynomialAnalyticImplicit.exists_analytic_implicit}, with the\nactual polynomial derivative and the derivative formula $-J^{-1}K$.\n\\item \\lean{PolynomialAnalyticSelection.continuous_selection_regularity}\ntransfers the analytic germ and strict derivative to any continuous local\nroot selection satisfying the stated root and base-point conditions.\n\\end{itemize}\nHere arithmetic declaration names have namespace prefix\n\\texttt{KomorebiPaperIX.}. Polynomial and native declarations have prefix\n\\texttt{Komorebi.}. The section maps and CSV retain the full names.","The original conjecture correspondence was reviewed against Nishibuchi's\narXiv version \\texttt{2605.10677v2}, dated 1 June 2026. The retained source\nrecord identifies the official version, the full title, the PDF checksum and\nthe extracted-text checksum. The detailed review compares Definition~2.1,\nNotation~3.2 and Conjecture~3.9 with the actual reciprocal EGF coefficients,\nthe factor $j!$, the residue range, the degree displacement, and the threshold\nin the formal capstone. Nishibuchi's conjecture is the conclusion to be\nproved. It is not an axiom imported into Lean.","texttt{Classical.choice} and \\texttt{Quot.sound}.\nThe reports do not describe a rebuild of Mathlib from source. The local\nregularity audit separately records the applicable Mathlib analytic inverse\nand strict derivative interfaces and their hypotheses. In particular, the\nfield-valued analysis is performed over a complete nontrivially normed field.\nIts integral specialization embeds the data into $\\Qp$. It does not use\n$\\Zp$ as a vector space over $\\Qp$.","} and \\texttt{Quot.sound}.\nThe reports do not describe a rebuild of Mathlib from source. The local\nregularity audit separately records the applicable Mathlib analytic inverse\nand strict derivative interfaces and their hypotheses. In particular, the\nfield-valued analysis is performed over a complete nontrivially normed field.\nIts integral specialization embeds the data into $\\Qp$. It does not use\n$\\Zp$ as a vector space over $\\Qp$.","e reports do not describe a rebuild of Mathlib from source. The local\nregularity audit separately records the applicable Mathlib analytic inverse\nand strict derivative interfaces and their hypotheses. In particular, the\nfield-valued analysis is performed over a complete nontrivially normed field.\nIts integral specialization embeds the data into $\\Qp$. It does not use\n$\\Zp$ as a vector space over $\\Qp$.","ly-row script separately reproduced 312 finite period\ncases and its odd-prime row-zero boundary control. Other research-script\nreplays, unchanged baseline regression gates and recovered-text controls\nare recorded separately in the same audit directory. Their totals are\nnot added to the independent gate. Lean and Lake were unavailable in this\nexecution environment. No fresh compiler run, infinite $p$-adic computation,\nanalytic-continuation computation or hardware benchmark is asserted.","The dedicated arithmetic experiment computed the four reciprocal rows through index 512 modulo $2^{4096}$, with independent exact-rational inversion through index 16. It checked 525,312 first differences, 1,141,424 repeated-step factorial bounds, 139,000 mixed-step bounds, and 2,048 Newton coefficients. The supplemental experiments cover additional mixed-step and sharpness cases and retain their separate tested ranges. Their outputs and source hashes are preserved under \\nolinkurl{preserved/threshold_free/computation}. All these are bounded implementation checks. The written arguments establish the universal claims.","\\textbf{Scope.} Fix a prime \\(p\\), an integer \\(L\\ge1\\), finite natural\nnumbers \\(d,r\\), and polynomials\n\\(F_1,\\ldots,F_r\\in\\mathbb Z[X_1,\\ldots,X_d]\\). Let\n\\(\\bar x\\in(\\mathbb Z/p^L\\mathbb Z)^d\\) satisfy every equation\n\\(F_i(\\bar x)=0\\). Choose any integer representative\n\\(x\\in\\mathbb Z^d\\). Set","\\[e_i=F_i(x)/p^L\\pmod p,\\qquad J_{ij}=\\frac{\\partial F_i}{\\partial X_j}(x)\\pmod p.\\]","The divisions are exact integer divisions. Regard \\(J\\) as a linear map\n\\(\\mathbb F_p^d\\to\\mathbb F_p^r\\). Define","\\[O_{F,\\bar x}=\\mathbb F_p^r/\\operatorname{im}J,\\qquad o_{F,\\bar x}=[e].\\]","\\textbf{Theorem.} For this fixed polynomial presentation, the group and\ndistinguished class are independent of the chosen integer\nrepresentative. The class vanishes if and only if \\(\\bar x\\) lifts to a\nsolution modulo \\(p^{L+1}\\). The obstruction group has\n\\(p^{r-\\operatorname{rank}J}\\) elements. When lifts exist, their set is\na torsor for \\(\\ker J\\) and contains exactly\n\\(p^{d-\\operatorname{rank}J}\\) elements.","\\textbf{Proof.} For every integer correction vector \\(t\\), expansion of\neach monomial gives","\\[F(x+p^Lt)\\equiv F(x)+p^LJ(x)t\\pmod {p^{L+1}}.\\tag{1}\\]","Indeed, each term involving at least two correction factors is divisible\nby \\(p^{2L}\\), and \\(2L\\ge L+1\\). Constant and linear correction terms\ngive the two displayed summands. Adding the monomial expansions proves\n(1) for every polynomial, with no denominator or analytic convergence\nissue.","Every element reducing to \\(\\bar x\\) is uniquely expressible modulo\n\\(p^{L+1}\\) as \\(x+p^Lt\\) with \\(t\\in\\mathbb F_p^d\\). By (1), it solves\nthe equations exactly when","\\[e+Jt=0\\quad\\text{in }\\mathbb F_p^r.\\tag{2}\\]","Equation (2) is soluble exactly when \\(e\\in\\operatorname{im}J\\): the\nsign makes no difference because the image is a subspace. This is\nequivalent to \\([e]=0\\).","Another representative has the form \\(x'=x+p^Ls\\). Since \\(L\\ge1\\), its\nJacobian is the same modulo \\(p\\). Dividing (1) by \\(p^L\\) shows that\nits residual digit is \\(e'=e+Js\\), so it gives exactly the same quotient\nclass. If \\(t_0\\) solves (2), all its solutions are \\(t_0+\\ker J\\), and\nno others. The finite cardinalities follow from rank-nullity. This also\ncovers no variables, no equations, zero Jacobian and singular solutions.\n\\(\\square\\)","m \\(x'=x+p^Ls\\). Since \\(L\\ge1\\), its\nJacobian is the same modulo \\(p\\). Dividing (1) by \\(p^L\\) shows that\nits residual digit is \\(e'=e+Js\\), so it gives exactly the same quotient\nclass. If \\(t_0\\) solves (2), all its solutions are \\(t_0+\\ker J\\), and\nno others. The finite cardinalities follow from rank-nullity. This also\ncovers no variables, no equations, zero Jacobian and singular solutions.\n\\(\\square\\)","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruc","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at t","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step.","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infin","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\i","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solutio","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system.","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their ra","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\ma","\\begin{itemize}\n\\tightlist\n\\item\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\)","em\n  \\textbf{A failed singular lift:} \\(p=3\\), \\(L=1\\), \\(F(X)=X^2-3\\),\n  \\(x=0\\). Here \\(e=-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  o…","-1\\ne0\\) in \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw group…","n \\(\\mathbb F_3\\) and \\(J=0\\). There are\n  no lifts of zero modulo 3 to a root modulo 9.\n\\item\n  \\textbf{Representative independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely b…","tive independence is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely because the equations define the same\n  seed.\n\\end{itemize}","e is a quotient assertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely because the equations define the same\n  seed.\n\\end{itemize}","sertion:} for\n  \\(F(X)=X^2-1\\), \\(p=3\\), \\(L=1\\), representatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely because the equations define the same\n  seed.\n\\end{itemize}","epresentatives \\(1\\) and \\(4\\) have\n  residual digits \\(0\\) and \\(2\\). Their common Jacobian is \\(2\\), with\n  full image, and both give zero obstruction. Their corrected\n  representatives \\(1\\) and \\(10\\) are the same root modulo 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely because the equations define the same\n  seed.\n\\end{itemize}","o 9.\n\\item\n  \\textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely because the equations define the same\n  seed.\n\\end{itemize}","textbf{A successful step need not give indefinite continuation:} for\n  \\(F(X)=X^2-5\\), \\(p=2\\), the root \\(1\\) modulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely because the equations define the same\n  seed.\n\\end{itemize}","odulo 2 has zero obstruction\n  and lifts to \\(1,3\\) modulo 4. Both have nonzero obstruction at the\n  next step. There is no root modulo 8 and no compatible infinite lift.\n\\item\n  \\textbf{The raw Jacobian cokernel need not be presentation\n  independent:} the presentations \\(F=(X)\\) and \\(G=(X,0)\\) define the\n  same solution system. Their raw obstruction groups at zero are\n  respectively zero and \\(\\mathbb F_p\\). Both actual distinguished\n  obstructions are zero. Thus one cannot claim a canonical isomorphism\n  of these two raw groups merely because the equations define the same\n  seed.\n\\end{itemize}","Let \\(B\\) be a finitely presented commutative \\(\\mathbb Z\\)-algebra,\n\\(p\\) prime and \\(L\\ge1\\). For a ring map\n\\(f:B\\to\\mathbb Z/p^L\\mathbb Z\\), use its reduction to make\n\\(\\mathbb F_p\\) a \\(B\\)-module. There is a canonical class","\\[o(f)\\in\\operatorname{Ext}^1_B(\\mathrm{NL}_{B/\\mathbb Z},\\mathbb F_p)\\]","whose vanishing is equivalent to a lift\n\\(B\\to\\mathbb Z/p^{L+1}\\mathbb Z\\). This follows from the ring-map\nlifting theorem in\n\\href{https://stacks.math.columbia.edu/tag/08S5}{Stacks Project, Lemma\n91.2.1, tag 08S5}. Specialize the source extension to\n\\(0\\to0\\to B\\to B\\) and the target extension to the kernel\n\\(p^L/p^{L+1}\\cong\\mathbb F_p\\). The base maps are identities of\n\\(\\mathbb Z\\). The kernel is square zero because \\(L\\ge1\\).","ent to a lift\n\\(B\\to\\mathbb Z/p^{L+1}\\mathbb Z\\). This follows from the ring-map\nlifting theorem in\n\\href{https://stacks.math.columbia.edu/tag/08S5}{Stacks Project, Lemma\n91.2.1, tag 08S5}. Specialize the source extension to\n\\(0\\to0\\to B\\to B\\) and the target extension to the kernel\n\\(p^L/p^{L+1}\\cong\\mathbb F_p\\). The base maps are identities of\n\\(\\mathbb Z\\). The kernel is square zero because \\(L\\ge1\\).","The group is finite. Choose a finite presentation\n\\(P=\\mathbb Z[X_1,\\ldots,X_d]\\twoheadrightarrow B\\) with finitely\ngenerated kernel \\(I\\). The presentation description in\n\\href{https://stacks.math.columbia.edu/tag/0GPU}{Stacks Project, Remark\n91.2.4, tag 0GPU} identifies the group as","\\[\\operatorname{coker}\\left(\\operatorname{Hom}_B(\\Omega_{P/\\mathbb Z}\\otimes_P B,\\mathbb F_p)\\longrightarrow\\operatorname{Hom}_B(I/I^2,\\mathbb F_p)\\right).\\]","If \\(I\\) has \\(r\\) chosen generators, evaluation on their classes\ninjects the latter Hom module into \\(\\mathbb F_p^r\\). Hence its quotient\nis finite. The relation module \\(I/I^2\\) retains dependencies between\nequations, including redundant zero equations, which the raw target\n\\(\\mathbb F_p^r\\) does not.","Consider the recurrence \\(a_{n+1}=a_n\\) with integer initial value. Its\nglobal parameter space is \\(X=\\mathbb Z\\). Its local parameter spaces\nare \\(X_p=\\mathbb Z_p\\), and every local realization has the same\nprime-independent recurrence. Give the local factors their usual\n\\(p\\)-adic topologies and the product its product topology. For these\nintegral local factors, the restricted product relative to all\n\\(\\mathbb Z_p\\) is the ordinary product. Write","the recurrence \\(a_{n+1}=a_n\\) with integer initial value. Its\nglobal parameter space is \\(X=\\mathbb Z\\). Its local parameter spaces\nare \\(X_p=\\mathbb Z_p\\), and every local realization has the same\nprime-independent recurrence. Give the local factors their usual\n\\(p\\)-adic topologies and the product its product topology. For these\nintegral local factors, the restricted product relative to all\n\\(\\mathbb Z_p\\) is the ordinary product. Write","\\[A=\\mathbb Z_2,\\qquad B=\\prod_{p>2}\\mathbb Z_p,\\qquad D=\\{(n,(n)_{p>2}):n\\in\\mathbb Z\\}\\subset A\\times B.\\]","\\textbf{Theorem.} There are no Hausdorff space \\(C\\) and continuous maps\n\\(f:A\\to C\\), \\(g:B\\to C\\) for which the equality locus","\\[\\{(a,b):f(a)=g(b)\\}\\]","is exactly \\(D\\). Thus the canonical local realizations of global\ninteger constant sequences cannot satisfy the manuscript's fiber-product\nformula in this Hausdorff topological interpretation.","\\textbf{Proof.} The equality locus is closed: it is the inverse image of\nthe diagonal of \\(C\\times C\\) under the continuous map \\(f\\times g\\),\nand a Hausdorff space has closed diagonal.","The set \\(D\\) is dense in \\(A\\times B\\). A nonempty basic open set\nprescribes congruences at only finitely many primes and finite precision\nat each. The moduli are pairwise coprime. The finite Chinese remainder\ntheorem supplies an integer meeting all those congruences, so every such\nopen set meets \\(D\\).","However, \\(D\\) is not the whole product. The element with 2-adic\ncoordinate zero and every odd-adic coordinate one is not diagonal. If an\ninteger had zero 2-adic coordinate, it would be divisible by every power\nof two, forcing it to be zero. Its 3-adic coordinate would then be zero,\nnot one. Consequently \\(D\\) is a proper dense subset and is not closed.\nIt cannot be the equality locus above. \\(\\square\\)","For a finite set \\(S\\) of primes, positive precisions \\(L_p\\), and a\nfixed integer polynomial system \\(F\\), put","\\[N=\\prod_{p\\in S}p^{L_p},\\qquad\\operatorname{Sol}_F(R)=\\{x\\in R^d:F(x)=0\\}.\\]","\\[\\operatorname{Sol}_F(\\mathbb Z/N\\mathbb Z)\\cong\\prod_{p\\in S}\\operatorname{Sol}_F(\\mathbb Z/p^{L_p}\\mathbb Z).\\]","The reason is exact: the CRT ring isomorphism preserves polynomial\nevaluation, and zero in a finite product is equivalent to zero in every\ncomponent. For \\(F\\) empty, passing to all compatible finite congruences\nidentifies the profinite completion \\(\\widehat{\\mathbb Z}\\) with\n\\(\\prod_p\\mathbb Z_p\\). The same finite reconstruction is compatible\nwith coefficient projections and integer recurrence matrices because\nthese operations commute with reduction and CRT.","Here the polynomial equations or recurrence matrices are fixed common\ndata. The theorem reconstructs finite or profinite objects. It does not\nidentify a ``smallest'' spine, impose additional spectral descent, or\nrecover global \\(\\mathbb Z\\) from arbitrary tuples of local points. A\ncompleted version of 13.3 must say exactly which object is completed and\nspecify its universal property.","Fix finite sets \\(A,B\\) of local packets and an explicitly given\nrelation \\(R\\subseteq A\\times B\\). Say that \\(R\\) is rectangular when","\\[R(a,b),\\ R(a',b),\\ R(a',b')\\quad\\Longrightarrow\\quad R(a,b')\\]","for all \\(a,a'\\in A\\) and \\(b,b'\\in B\\).","\\textbf{Theorem.} The relation \\(R\\) can be represented exactly as","\\[R(a,b)\\quad\\Longleftrightarrow\\quad f(a)=g(b)\\]","for maps from \\(A\\) and \\(B\\) to a common set if and only if \\(R\\) is\nrectangular. In that case a canonical finite choice of common set is the\nconnected-component set \\(C_R\\) of the bipartite graph with vertex set\n\\(A\\sqcup B\\) and edges given by \\(R\\).","The component maps \\(f_R:A\\to C_R\\) and \\(g_R:B\\to C_R\\) have the\nfollowing universal property. For every set \\(Y\\) and maps \\(h:A\\to Y\\),\n\\(k:B\\to Y\\) satisfying \\(h(a)=k(b)\\) whenever \\(R(a,b)\\), there is a\nunique map \\(u:C_R\\to Y\\) with \\(u\\circ f_R=h\\) and \\(u\\circ g_R=k\\).","\\textbf{Proof.} Any equality representation is rectangular, since the\nthree assumed equalities imply \\(f(a)=g(b)=f(a')=g(b')\\).","Conversely, suppose \\(R\\) is rectangular. An edge always gives equal\ncomponent labels. If \\(a\\in A\\) and \\(b\\in B\\) have equal component\nlabels, there is a finite path between them. Its length is odd. A path\nof length one is already an \\(R\\)-edge. A longer path begins with three\nedges \\(a-b_1-a_1-b_2\\). Rectangularity gives the edge \\(a-b_2\\).\nReplacing these three edges by that edge shortens the path by two.\nInduction therefore yields \\(R(a,b)\\). This proves the exact equality\nrepresentation. Finiteness follows because the component set is a\nquotient of the finite set \\(A\\sqcup B\\).","An edge always gives equal\ncomponent labels. If \\(a\\in A\\) and \\(b\\in B\\) have equal component\nlabels, there is a finite path between them. Its length is odd. A path\nof length one is already an \\(R\\)-edge. A longer path begins with three\nedges \\(a-b_1-a_1-b_2\\). Rectangularity gives the edge \\(a-b_2\\).\nReplacing these three edges by that edge shortens the path by two.\nInduction therefore yields \\(R(a,b)\\). This proves the exact equality\nrepresentation. Finiteness follows because the component set is a\nquotient of the finite set \\(A\\sqcup B\\).","em. Its length is odd. A path\nof length one is already an \\(R\\)-edge. A longer path begins with three\nedges \\(a-b_1-a_1-b_2\\). Rectangularity gives the edge \\(a-b_2\\).\nReplacing these three edges by that edge shortens the path by two.\nInduction therefore yields \\(R(a,b)\\). This proves the exact equality\nrepresentation. Finiteness follows because the component set is a\nquotient of the finite set \\(A\\sqcup B\\).","For the universal property, combine \\(h\\) and \\(k\\) into a function on\n\\(A\\sqcup B\\). It is constant across every edge by hypothesis, hence\nalong every finite path. It therefore descends to a well-defined\nfunction on components. The two required factorization identities hold\nby construction. They determine the value on every component, proving\nuniqueness. \\(\\square\\)","For example, the three pairs \\((a_0,b_0),(a_1,b_0),(a_1,b_1)\\) without\n\\((a_0,b_1)\\) fail rectangularity. Every equality-label representation\nagreeing on those three pairs also admits the fourth pair, so it cannot\nrepresent that relation exactly.","This is a precise finite gluing obligation: future arithmetic\napplications must define \\(A\\), \\(B\\) and \\(R\\) and prove\nrectangularity. The universal property holds for the component quotient\neven when rectangularity fails, but then the equality locus strictly\nenlarges \\(R\\). The theorem establishes neither a minimum number of\nlabels nor a smallest spine in the manuscript's unspecified category. It\ncarries no automatic topology, spectral information or Frobenius\ncompatibility. \\textbf{Evidence: prose proof in this note, not a Lean\nformalization.}","reconstruction lacked all 14 original figure files. This\nreading edition supplies new reconstructions at their retained labels and\ncaptions. These are explanatory schematics and exact finite examples, not\nrecovered original artwork. The reproducible program\n\\nolinkurl{tools/reconstructed_figures.py} checks the displayed finite data.\nThe modulo-seven inverse plot uses the corrected positive tail\n$6,5,2,6,5,2,\\ldots$. A reconstructed illustration does not upgrade the\nhistorical proof or formalization status. The original historical source\npackage, including all 14 vector figures and the bibliography, has now been\nrecovered and is preserved unchanged in the accompanying release. The current\nreading edition retains its explicitly identified reconstructions and corrected\nfinite data. The 22 historical references now have recovered bibliographic\nentries and checked identities. Superseded claims about the divisor-branch\nfinite-difference problem have local notices directing the reader to the\ncurrent theorem. The higher kernel-valuation and unrestricted\nlocal-to-global questions remain open …","This article begins with the Hurwitz product\n\\[\n(a\\starh b)_n\n=\n\\sum_{r=0}^{n}\n\\binom nr a_rb_{n-r}.\n\\]\nOver a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes \\(\\epsilon_u=(u^n)_{n\\geq0}\\). This finite algebra gives an exact proof of the standard window period. Hurwitz series rings are treated in \\cite{Keigher1997,BarberoCerrutiMurru2018}. The finite difference and umbral background may be found in \\cite{Roman1984,Comtet1974,GrahamKnuthPatashnik1994}.","Fix an odd prime \\(p\\), a divisor \\(m\\mid p-1\\), and put\n\\[\nh=\\frac{p-1}{m}.\n\\]\nThe reduced standard window inverse is defined by\n\\[\n\\sum_{q=0}^{n}\n\\binom{mn}{mq}e_q^{(m,p)}\n=\n\\delta_{n0}\n\\quad\\text{in }\\FF_p.\n\\]\nWe prove that\n\\[\ne_{n+h}^{(m,p)}=e_n^{(m,p)}\n\\qquad(n\\geq1).\n\\]\nThe result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are \\(h\\) periodic after their constant term. The two arguments expose different parts of the same finite character structure.","\\frac{p-1}{m}.\n\\]\nThe reduced standard window inverse is defined by\n\\[\n\\sum_{q=0}^{n}\n\\binom{mn}{mq}e_q^{(m,p)}\n=\n\\delta_{n0}\n\\quad\\text{in }\\FF_p.\n\\]\nWe prove that\n\\[\ne_{n+h}^{(m,p)}=e_n^{(m,p)}\n\\qquad(n\\geq1).\n\\]\nThe result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are \\(h\\) periodic after their constant term. The two arguments expose different parts of the same finite character structure.","The odd and dyadic Euler formulas share a common reconstruction framework. For each prime \\(p\\) we define an even source series \\(H_p\\) and an inverse kernel \\(K_p\\) satisfying\n\\[\nK_p(t)H_p(t)=\\operatorname{sech}t.\n\\]\nAt an odd prime,\n\\[\nH_p(t)=\\cosh^2(pt)\\operatorname{sech}t,\n\\qquad\nK_p(t)=\\operatorname{sech}^2(pt).\n\\]\nAt the dyadic prime,\n\\[\nH_2(t)=\\frac{\\sinh(2t)\\sinh t}{2t^2},\n\\qquad\nK_2(t)=\\frac{4t^2}{\\sinh^2(2t)}.\n\\]\nThe coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.","The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If\n\\[\n\\frac{z^2}{\\sinh^2z}\n=\n\\sum_{r\\geq0}\\beta_rz^{2r}\n\\]\nand\n\\[\n\\kappa_{2,r}=\\beta_r2^{2r}(2r)!,\n\\]\nthen\n\\[\n\\vTwo(\\kappa_{2,r})=4r-1\n\\qquad(r\\geq1).\n\\]\nConsequently, truncation after depth \\(M\\) is uniformly valid modulo\n\\[\n2^{4M+3},\n\\]\nand this exponent is optimal. The rowwise estimate records the additional binary carries in the binomial coefficient. Classical congruence theory for Euler families is extensive \\cite{CarlitzLevine1960,Young1999,Young2001,Sun2008,Chen2004}. The formulas here are stated as exact coordinate identities. Historical priority requires a separate comparison of the individual theorems.","The second expansion concerns local interpolation. The exponent of the finite unit group moves every unit mode into a principal unit neighborhood. Binomial expansion then gives a Mahler series whose depth \\(r\\) coefficient gains at least \\(Lr\\) digits when the chosen jump is the identity modulo \\(p^L\\). The same argument applies to polynomial exponential modes and to invertible matrix state systems. Mahler interpolation and the Amice transform are classical \\cite{Mahler1958,Amice1964,Robert2000}. The local unit structure and valuation tools used here are standard \\cite{Koblitz1984,Serre1979,Washington1997,Gouvea1997}.","The odd Euler source contains one nonunit base. Removing that base gives an analytic unit source and an interpolation of a depleted Euler sequence on every even character branch. The dyadic source is already analytic on \\(\\Zp\\) after a removable singularity is filled. It yields a direct analytic interpolation of \\(E_{2n}\\) on \\(\\mathbb Z_2\\). These branches are then refined by the substitution\n\\[\nx=Mq+s.\n\\]\nThe induced Mahler coefficient transform is finite and explicit. Its kernels compose exactly. At a prime power radix, they become sparse modulo the residue prime. Every monotone valuation ladder survives the refinement.","The remaining sections treat arithmetic lift obstructions, recurrence packets, and Chinese remainder reconstruction. The lifting obstruction is the residual defect modulo the image of the declared correction map. A recurrence packet is governed by a jump matrix \\(C^N\\). If one integral recurrence is fixed, all of its local realizations have one common source. These statements do not prove that arbitrary unrelated local objects have a global origin.","Four boundaries remain explicit. A period modulo \\(p\\) does not by itself lift to all powers of \\(p\\). The full finite difference estimate for every congruential reciprocal packet still requires control of the triangular inverse. Exact higher valuations of the affine polyphase kernel are not yet known in general. Chinese remainder reconstruction and finite adelic synchronization do not prove a global source for arbitrary local systems.\n\\subsection{Hurwitz series and finite field subgroup inversion}\\label{hist:sec:hurwitz}\n\\par\\smallskip\\noindent\\textbf{Current status.} This dated statement is superseded for the stated divisor-branch family by Theorem~\\ref{thm:arith-divisor-base}, with the $j!$ normalization and factorial valuation loss explic","Four boundaries remain explicit. A period modulo \\(p\\) does not by itself lift to all powers of \\(p\\). The full finite difference estimate for every congruential reciprocal packet still requires control of the triangular inverse. Exact higher valuations of the affine polyphase kernel are not yet known in general. Chinese remainder reconstruction and finite adelic synchronization do not prove a global source for arbitrary local systems.\n\\subsection{Hurwitz series and finite field subgroup inversion}\\label{hist:sec:hurwitz}\n\\par\\smallskip\\noindent\\textbf{Current status.} This dated statement is superseded for the stated divisor-branch family by Theorem~\\ref{thm:arith-divisor-base}, with the $j!$ normalization and factorial valuation loss explicitly retained. It makes no claim for unrestrict","own in general. Chinese remainder reconstruction and finite adelic synchronization do not prove a global source for arbitrary local systems.\n\\subsection{Hurwitz series and finite field subgroup inversion}\\label{hist:sec:hurwitz}\n\\par\\smallskip\\noindent\\textbf{Current status.} This dated statement is superseded for the stated divisor-branch family by Theorem~\\ref{thm:arith-divisor-base}, with the $j!$ normalization and factorial valuation loss explicitly retained. It makes no claim for unrestricted families.\\par","\\begin{histdefinition}[Hurwitz series ring]\\label{hist:def:hurwitz_ring}\nLet \\(R\\) be a commutative ring with identity. The Hurwitz series ring \\(\\Hcal(R)\\) is the set \\(R^{\\NN}\\) with componentwise addition and product\n\\[\n(a\\starh b)_n\n=\n\\sum_{r=0}^{n}\n\\binom nr a_r b_{n-r}.\n\\]\nIts identity is\n\\[\n\\delta=(1,0,0,\\ldots).\n\\]\n\\end{histdefinition}","When the factorials are invertible, the sequence \\(a\\) may be written as the exponential generating series\n\\[\nA(t)=\\sum_{n\\geq0}a_n\\frac{t^n}{n!},\n\\]\nand the Hurwitz product becomes ordinary multiplication. The sequence definition remains valid in every characteristic.","\\begin{histdefinition}[Exponential mode]\\label{hist:def:exponential_mode}\nFor \\(u\\in R\\), define\n\\[\n\\epsilon_u=(u^n)_{n\\geq0},\n\\]\nwhere \\(0^0=1\\).\n\\end{histdefinition}","\\begin{histproposition}[Addition law for modes]\\label{hist:prop:mode_addition}\nFor every \\(u,v\\in R\\),\n\\[\n\\boxed{\n\\epsilon_u\\starh\\epsilon_v\n=\n\\epsilon_{u+v}.\n}\n\\]\n\\end{histproposition}","\\begin{proof}\nThe \\(n\\)th coefficient is\n\\[\n\\sum_{r=0}^{n}\n\\binom nr u^rv^{n-r}\n=\n(u+v)^n\n\\]\nby the binomial theorem.\n\\end{proof}","Let \\(p\\) be prime. Write \\(\\FF_p^+\\) for the additive group of the finite field.","\\begin{histtheorem}[Finite field mode algebra]\\label{hist:thm:mode_group_algebra}\nThe map\n\\[\n\\Theta:\n\\FF_p[\\FF_p^+]\n\\longrightarrow\n\\Hcal(\\FF_p),\n\\qquad\n[u]\\longmapsto\\epsilon_u,\n\\]\nis an injective algebra homomorphism. Its image is the \\(p\\) dimensional subalgebra spanned by the modes \\(\\epsilon_u\\).\n\\end{histtheorem}","\\begin{proof}\nThe product law follows from \\cref{hist:prop:mode_addition}. To prove injectivity, suppose\n\\[\n\\sum_{u\\in\\FF_p}c_u\\epsilon_u=0.\n\\]\nLooking at the coordinates \\(n=0,1,\\ldots,p-1\\) gives a Vandermonde system with matrix\n\\[\n(u^n)_{0\\leq n\\leq p-1,\\ u\\in\\FF_p}.\n\\]\nIts determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every \\(c_u\\) vanishes.\n\\end{proof}","\\begin{histproposition}[Augmentation criterion]\\label{hist:prop:augmentation_criterion}\nLet\n\\[\nx=\\sum_{u\\in\\FF_p}c_u[u]\n\\in\n\\FF_p[\\FF_p^+].\n\\]\nThen \\(x\\) is a unit if and only if\n\\[\n\\operatorname{aug}(x)\n=\n\\sum_uc_u\n\\]\nis nonzero. If \\(J\\) is the augmentation ideal, then\n\\[\nJ^p=0.\n\\]\n\\end{histproposition}","\\begin{proof}\nChoose a generator \\(g\\) of the cyclic additive group. Then\n\\[\n\\FF_p[\\FF_p^+]\n\\cong\n\\FF_p[y]/(y^p-1)\n=\n\\FF_p[y]/(y-1)^p.\n\\]\nThe augmentation ideal is generated by \\(y-1\\), so its \\(p\\)th power is zero. Every element has the form \\(a+n\\) with \\(a\\in\\FF_p\\) and \\(n\\in J\\). It is invertible exactly when \\(a\\neq0\\), which is the augmentation condition.\n\\end{proof}","\\begin{histcorollary}[Finite inverse formula]\\label{hist:cor:finite_inverse_formula}\nIf \\(x\\) has augmentation one and \\(n=x-1\\), then\n\\[\n\\boxed{\nx^{-1}\n=\n\\sum_{r=0}^{p-1}(-n)^r.\n}\n\\]\n\\end{histcorollary}","\\begin{proof}\nBy \\cref{hist:prop:augmentation_criterion}, \\(n^p=0\\). The finite geometric sum gives the inverse.\n\\end{proof}","Fix a divisor \\(m\\) of \\(p-1\\). Let\n\\[\nH_m\n=\n\\{\\zeta\\in\\FF_p^\\times:\\zeta^m=1\\}.\n\\]\nThis group has order \\(m\\).","\\begin{histdefinition}[Multiple selector]\\label{hist:def:multiple_selector}\nDefine \\(a^{(m)}\\in\\Hcal(\\FF_p)\\) by\n\\[\na_n^{(m)}\n=\n\\begin{cases}\n1,&m\\mid n,","\\begin{histdefinition}[Multiple selector]\\label{hist:def:multiple_selector}\nDefine \\(a^{(m)}\\in\\Hcal(\\FF_p)\\) by\n\\[\na_n^{(m)}\n=\n\\begin{cases}\n1,&m\\mid n,\\\\\n0,&m\\nmid n.\n\\end{cases}\n\\]\n\\end{histdefinition}","\\begin{histlemma}[Subgroup average]\\label{hist:lem:subgroup_average}\nThe multiple selector has the mode expansion\n\\[\n\\boxed{\na^{(m)}\n=\n\\frac1m\n\\sum_{\\zeta\\in H_m}\n\\epsilon_\\zeta.\n}\n\\]\n\\end{histlemma}","\\begin{proof}\nThe \\(n\\)th coefficient on the right is\n\\[\n\\frac1m\n\\sum_{\\zeta\\in H_m}\\zeta^n.\n\\]\nThe character sum is one when \\(m\\mid n\\) and zero otherwise.\n\\end{proof}","Since the subgroup average has augmentation one, it has a unique Hurwitz inverse. Write\n\\[\nb^{(m,p)}\n=\n(a^{(m)})^{-1}.\n\\]","\\begin{histtheorem}[Support and positive degree period]\\label{hist:thm:support_period}\nThe inverse \\(b^{(m,p)}\\) has the following properties.\n\\begin{enumerate}\n\\item \\(b_n^{(m,p)}=0\\) unless \\(m\\mid n\\).\n\\item For every positive \\(n\\),\n\\[\n\\boxed{\nb_{n+p-1}^{(m,p)}\n=\nb_n^{(m,p)}.\n}\n\\]\n\\end{enumerate}\n\\end{histtheorem}","\\begin{proof}\nMultiplication by an element \\(\\eta\\in H_m\\) gives an automorphism of the additive group and therefore an automorphism\n\\[\n\\epsilon_u\\longmapsto\\epsilon_{\\eta u}\n\\]\nof the mode algebra. The selector \\(a^{(m)}\\) is fixed by every such automorphism, so its unique inverse is fixed as well.","Write\n\\[\nb^{(m,p)}\n=\n\\sum_{u\\in\\FF_p}c_u\\epsilon_u.\n\\]\nInvariance gives \\(c_{\\eta u}=c_u\\). After the change of variable \\(u\\mapsto\\eta u\\),\n\\[\nb_n^{(m,p)}\n=\n\\eta^n b_n^{(m,p)}\n\\]\nfor every \\(\\eta\\in H_m\\). If \\(m\\nmid n\\), some \\(\\eta\\) satisfies \\(\\eta^n\\neq1\\), so the coefficient vanishes.","For positive \\(n\\), the zero mode contributes nothing. Every nonzero mode satisfies \\(u^{p-1}=1\\). Hence\n\\[\nb_{n+p-1}^{(m,p)}\n=\n\\sum_{u\\neq0}c_uu^{n+p-1}\n=\n\\sum_{u\\neq0}c_uu^n\n=\nb_n^{(m,p)}.\n\\]\n\\end{proof}","\\begin{histdefinition}[Compressed inverse]\\label{hist:def:compressed_inverse}\nSet\n\\[\ne_n^{(m,p)}\n=\nb_{mn}^{(m,p)}.\n\\]\n\\end{histdefinition}","\\begin{histproposition}[Compressed inverse recurrence]\\label{hist:prop:compressed_inverse_recurrence}\nThe compressed coefficients satisfy\n\\[\n\\boxed{\n\\sum_{q=0}^{n}\n\\binom{mn}{mq}\ne_q^{(m,p)}\n=\n\\delta_{n0}.\n}\n\\]\nIn particular, \\(e_0^{(m,p)}=1\\), and the positive coefficients are determined recursively because the diagonal is one.\n\\end{histproposition}","\\begin{proof}\nEvaluate the Hurwitz identity\n\\[\na^{(m)}\\starh b^{(m,p)}=\\delta\n\\]\nat degree \\(mn\\). Both factors vanish outside degrees divisible by \\(m\\). Reindexing the surviving terms gives the displayed recurrence.\n\\end{proof}","\\begin{histcorollary}[Uniform compressed period]\\label{hist:cor:compressed_period}\nLet\n\\[\nh=\\frac{p-1}{m}.\n\\]\nThen\n\\[\n\\boxed{\ne_{n+h}^{(m,p)}\n=\ne_n^{(m,p)}\n\\qquad(n\\geq1).\n}\n\\]\n\\end{histcorollary}","\\begin{proof}\nSince \\(m(n+h)=mn+p-1\\), the conclusion follows from \\cref{hist:thm:support_period}.\n\\end{proof}","\\begin{histcorollary}[Rational ordinary generating function]\\label{hist:cor:periodic_ogf}\nIn \\(\\FF_p[[z]]\\),\n\\[\n\\boxed{\n\\sum_{n\\geq0}e_n^{(m,p)}z^n\n=\n1+\n\\frac{\n\\sum_{r=1}^{h}e_r^{(m,p)}z^r\n}{1-z^h}.\n}\n\\]\n\\end{histcorollary}","\\begin{proof}\nSeparate the constant coefficient and group the positive indices into residue classes modulo \\(h\\).\n\\end{proof}","\\begin{histtheorem}[Exponential form and minimal period]\\label{hist:thm:minimal_period}\nLet\n\\[\nG_m=(\\FF_p^\\times)^m.\n\\]\nThis cyclic group has order \\(h\\). There are uniquely determined coefficients \\(d_v\\in\\FF_p\\), indexed by \\(v\\in G_m\\), such that\n\\[\n\\boxed{\ne_n^{(m,p)}\n=\n\\sum_{v\\in G_m}d_vv^n\n\\qquad(n\\geq1).\n}\n\\]\nThe minimal positive period is the least common multiple of the orders of those \\(v\\) for which \\(d_v\\neq0\\). In particular, it divides \\(h\\).\n\\end{histtheorem}","em}[Exponential form and minimal period]\\label{hist:thm:minimal_period}\nLet\n\\[\nG_m=(\\FF_p^\\times)^m.\n\\]\nThis cyclic group has order \\(h\\). There are uniquely determined coefficients \\(d_v\\in\\FF_p\\), indexed by \\(v\\in G_m\\), such that\n\\[\n\\boxed{\ne_n^{(m,p)}\n=\n\\sum_{v\\in G_m}d_vv^n\n\\qquad(n\\geq1).\n}\n\\]\nThe minimal positive period is the least common multiple of the orders of those \\(v\\) for which \\(d_v\\neq0\\). In particular, it divides \\(h\\).\n\\end{histtheorem}","\\label{hist:thm:minimal_period}\nLet\n\\[\nG_m=(\\FF_p^\\times)^m.\n\\]\nThis cyclic group has order \\(h\\). There are uniquely determined coefficients \\(d_v\\in\\FF_p\\), indexed by \\(v\\in G_m\\), such that\n\\[\n\\boxed{\ne_n^{(m,p)}\n=\n\\sum_{v\\in G_m}d_vv^n\n\\qquad(n\\geq1).\n}\n\\]\nThe minimal positive period is the least common multiple of the orders of those \\(v\\) for which \\(d_v\\neq0\\). In particular, it divides \\(h\\).\n\\end{histtheorem}","\\begin{proof}\nThe coefficients \\(c_u\\) in the proof of \\cref{hist:thm:support_period} are constant on the cosets of \\(H_m\\). For \\(u\\neq0\\), the value \\(u^m\\) depends only on the coset, and every element of \\(G_m\\) arises. Grouping the modes by cosets gives the displayed expansion.","Distinct elements of \\(G_m\\) define linearly independent functions on one full period, by the Vandermonde determinant. Therefore a positive integer \\(r\\) is a period exactly when \\(v^r=1\\) for every active mode. The least such \\(r\\) is the stated least common multiple.\n\\end{proof}","\\begin{histexample}[A compressed inverse modulo seven]\\label{hist:ex:compressed_mod_seven}\n\\par\\noindent\\textbf{Editorial numerical correction.} The positive-index standard-window tail printed below is erroneous. The defining recurrence gives $e_1=-1\\equiv6$, $e_2=5$ and $e_3=-61\\equiv2\\pmod7$, so the correct positive tail is $6,5,2,6,5,2,\\ldots$. The period-three conclusion remains correct. The original line is retained only as a corrected historical record.\\par\\medskip\nTake \\(p=7\\) and \\(m=2\\), so \\(h=3\\). The recurrence gives\n\\[\ne_n^{(2,7)}\n=\n1,6,4,1,6,4,\\ldots\n\\qquad(n\\geq1).\n\\]\nThe positive tail has exact period three.\n\\end{histexample}","_seven}\n\\par\\noindent\\textbf{Editorial numerical correction.} The positive-index standard-window tail printed below is erroneous. The defining recurrence gives $e_1=-1\\equiv6$, $e_2=5$ and $e_3=-61\\equiv2\\pmod7$, so the correct positive tail is $6,5,2,6,5,2,\\ldots$. The period-three conclusion remains correct. The original line is retained only as a corrected historical record.\\par\\medskip\nTake \\(p=7\\) and \\(m=2\\), so \\(h=3\\). The recurrence gives\n\\[\ne_n^{(2,7)}\n=\n1,6,4,1,6,4,\\ldots\n\\qquad(n\\geq1).\n\\]\nThe positive tail has exact period three.\n\\end{histexample}","\\noindent\\textbf{Editorial numerical correction.} The positive-index standard-window tail printed below is erroneous. The defining recurrence gives $e_1=-1\\equiv6$, $e_2=5$ and $e_3=-61\\equiv2\\pmod7$, so the correct positive tail is $6,5,2,6,5,2,\\ldots$. The period-three conclusion remains correct. The original line is retained only as a corrected historical record.\\par\\medskip\nTake \\(p=7\\) and \\(m=2\\), so \\(h=3\\). The recurrence gives\n\\[\ne_n^{(2,7)}\n=\n1,6,4,1,6,4,\\ldots\n\\qquad(n\\geq1).\n\\]\nThe positive tail has exact period three.\n\\end{histexample}","xtbf{Editorial numerical correction.} The positive-index standard-window tail printed below is erroneous. The defining recurrence gives $e_1=-1\\equiv6$, $e_2=5$ and $e_3=-61\\equiv2\\pmod7$, so the correct positive tail is $6,5,2,6,5,2,\\ldots$. The period-three conclusion remains correct. The original line is retained only as a corrected historical record.\\par\\medskip\nTake \\(p=7\\) and \\(m=2\\), so \\(h=3\\). The recurrence gives\n\\[\ne_n^{(2,7)}\n=\n1,6,4,1,6,4,\\ldots\n\\qquad(n\\geq1).\n\\]\nThe positive tail has exact period three.\n\\end{histexample}","ction.} The positive-index standard-window tail printed below is erroneous. The defining recurrence gives $e_1=-1\\equiv6$, $e_2=5$ and $e_3=-61\\equiv2\\pmod7$, so the correct positive tail is $6,5,2,6,5,2,\\ldots$. The period-three conclusion remains correct. The original line is retained only as a corrected historical record.\\par\\medskip\nTake \\(p=7\\) and \\(m=2\\), so \\(h=3\\). The recurrence gives\n\\[\ne_n^{(2,7)}\n=\n1,6,4,1,6,4,\\ldots\n\\qquad(n\\geq1).\n\\]\nThe positive tail has exact period three.\n\\end{histexample}","The examples later in the chapter use different kernels, but their coefficient algebra has one common form. This section records that form once. Let \\(R\\) be a commutative algebra over \\(\\QQ\\), and take all coefficients in \\(R\\).","Let\n\\[\nH(t)=\\sum_{r\\geq0}h_r\\frac{t^r}{r!},\n\\qquad\nh_0=1.\n\\]\nThere is a unique inverse\n\\[\nK(t)=H(t)^{-1}\n=\\sum_{r\\geq0}k_r\\frac{t^r}{r!}.\n\\]","\\begin{histtheorem}[Exact coefficient transport]\\label{hist:thm:exact_coefficient_transport}\nLet\n\\[\nA(t)=\\sum_{n\\geq0}A_n\\frac{t^n}{n!},\n\\qquad\nS(t)=H(t)A(t).\n\\]\nThen\n\\[\n\\boxed{\nS_n\n=\n\\sum_{r=0}^{n}\n\\binom nr h_rA_{n-r}.\n}\n\\]\nThe inverse coordinate formula is\n\\[\n\\boxed{\nA_n\n=\n\\sum_{r=0}^{n}\n\\binom nr k_rS_{n-r}.\n}\n\\]\nBoth transforms are lower triangular with diagonal one.\n\\end{histtheorem}","\\begin{proof}\nThe first identity is the coefficient rule for the product of exponential generating functions. Multiply by \\(K(t)\\) and use \\(K(t)H(t)=1\\) for the inverse identity. The diagonal coefficient in each transform is the constant coefficient of its kernel, which is one.\n\\end{proof}","\\begin{histcorollary}[Composition of coefficient transports]\\label{hist:cor:transport_composition}\nIf \\(U_H\\) and \\(U_G\\) are the triangular transforms associated with unit kernels \\(H\\) and \\(G\\), then\n\\[\n\\boxed{\nU_HU_G=U_{HG}.\n}\n\\]\nIn particular, the inverse of \\(U_H\\) is \\(U_{H^{-1}}\\).\n\\end{histcorollary}","\\begin{proof}\nMultiplication by \\(G\\), followed by multiplication by \\(H\\), is multiplication by \\(HG\\).\n\\end{proof}","\\begin{histtheorem}[Filtered triangular reconstruction]\\label{hist:thm:filtered_triangular_reconstruction}\nLet \\(p\\) be prime. Suppose\n\\[\nA_n\n=\n\\sum_{r=0}^{n}\n\\binom nr\nc_rp^{w(r)}S_{n-r},\n\\]\nwhere \\(c_r\\in\\Zp\\), \\(S_n\\in\\Zp\\), and\n\\[\nw(r)\\in\\NN.\n\\]\nFor every precision \\(L\\geq1\\),\n\\[\n\\boxed{\nA_n\n\\equiv\n\\sum_{\\substack{0\\leq r\\leq n","\\begin{histtheorem}[Filtered triangular reconstruction]\\label{hist:thm:filtered_triangular_reconstruction}\nLet \\(p\\) be prime. Suppose\n\\[\nA_n\n=\n\\sum_{r=0}^{n}\n\\binom nr\nc_rp^{w(r)}S_{n-r},\n\\]\nwhere \\(c_r\\in\\Zp\\), \\(S_n\\in\\Zp\\), and\n\\[\nw(r)\\in\\NN.\n\\]\nFor every precision \\(L\\geq1\\),\n\\[\n\\boxed{\nA_n\n\\equiv\n\\sum_{\\substack{0\\leq r\\leq n\\\\w(r)<L}}\n\\binom nr\nc_rp^{w(r)}S_{n-r}\n\\pmod {p^L}.\n}\n\\]\nWhen the omitted set is nonempty, the tail has valuation at least\n\\[\n\\boxed{\n\\min_{\\substack{0\\leq r\\leq n","w(r)<L}}\n\\binom nr\nc_rp^{w(r)}S_{n-r}\n\\pmod {p^L}.\n}\n\\]\nWhen the omitted set is nonempty, the tail has valuation at least\n\\[\n\\boxed{\n\\min_{\\substack{0\\leq r\\leq n\\\\w(r)\\geq L}}\n\\left(\nw(r)+\\vp\\binom nr+\\vp(S_{n-r})\n\\right).\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nEvery omitted summand belongs to \\(p^L\\Zp\\). The sharper statement follows by taking the minimum of the exact lower bounds for the summands and applying the nonarchimedean triangle inequality.\n\\end{proof}","For integers \\(N\\geq1\\) and \\(0\\leq j<N\\), define\n\\[\n\\Ecal_{N,j}(t)\n=\n\\sum_{n\\geq0}\n\\frac{t^{Nn+j}}{(Nn+j)!}.\n\\]\nThe forced factor \\(t^j\\) may be removed without changing the residue address.","\\begin{histdefinition}[Congruential reciprocal packet]\\label{hist:def:congruential_packet}\nDefine\n\\[\n\\boxed{\n\\Ccal_{N,j}(t)\n=\n\\frac{t^j/j!}{\\Ecal_{N,j}(t)}\n=\n\\sum_{n\\geq0}\nC_{N,j}(n)\n\\frac{t^{Nn}}{(Nn)!}.\n}\n\\]\n\\end{histdefinition}","\\begin{histtheorem}[Triangular inverse system]\\label{hist:thm:triangular_inverse}\nThe coefficients satisfy \\(C_{N,j}(0)=1\\) and\n\\[\n\\boxed{\n\\sum_{q=0}^{n}\n\\binom{Nn+j}{Nq}\nC_{N,j}(q)\n=\n\\delta_{n0}.\n}\n\\]\nFor \\(n\\geq1\\),\n\\[\n\\boxed{\nC_{N,j}(n)\n=\n-\n\\frac{\n\\sum_{q=0}^{n-1}\n\\binom{Nn+j}{Nq}C_{N,j}(q)\n}{\n\\binom{Nn+j}{Nn}\n}.\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nMultiply \\(\\Ccal_{N,j}(t)\\) by \\(\\Ecal_{N,j}(t)\\) and compare the coefficient of \\(t^{Nn+j}/(Nn+j)!\\). The last summand is the diagonal term, which is nonzero over \\(\\QQ\\).\n\\end{proof}","\\begin{histproposition}[A denominator bound]\\label{hist:prop:denominator_bound}\nLet\n\\[\nD_{N,j}(n)\n=\n\\prod_{r=1}^{n}\n\\binom{Nr+j}{Nr},\n\\qquad\nD_{N,j}(0)=1.\n\\]\nThen\n\\[\n\\boxed{\nD_{N,j}(n)C_{N,j}(n)\n\\in\n\\ZZ.\n}\n\\]\nFor every prime \\(p\\),\n\\[\n\\vp(C_{N,j}(n))\n\\geq\n-\n\\sum_{r=1}^{n}\n\\vp\\binom{Nr+j}{Nr}.\n\\]\n\\end{histproposition}","\\begin{proof}\nUse induction on \\(n\\) in the recurrence of \\cref{hist:thm:triangular_inverse}. Every earlier denominator divides \\(D_{N,j}(n-1)\\), while the new diagonal denominator is the final factor of \\(D_{N,j}(n)\\). The valuation statement follows.\n\\end{proof}","\\begin{histexample}[Euler and Bernoulli boundaries]\\label{hist:ex:euler_bernoulli_boundaries}\nFor \\((N,j)=(2,0)\\),\n\\[\n\\Ccal_{2,0}(t)=\\operatorname{sech}t,\n\\]\nso\n\\[\nC_{2,0}(n)=E_{2n}.\n\\]\nFor \\((N,j)=(2,1)\\),\n\\[\n\\Ccal_{2,1}(t)=\\frac{t}{\\sinh t},\n\\]\nand\n\\[\nC_{2,1}(n)\n=\n2\\left(1-2^{2n-1}\\right)B_{2n}.\n\\]\nThe usual convention \\(B_1=-1/2\\) is used. Euler and Bernoulli number conventions are reviewed in \\cite{Comtet1974,Stanley2012}.\n\\end{histexample}","Fix a prime \\(p\\), an integer \\(m\\geq1\\), and an address \\(0\\leq j<p\\). Put\n\\[\nN=mp.\n\\]\nFor arbitrary \\(m\\), let \\(\\widetilde e_n^{(m,p)}\\in\\FF_p\\) be the unique sequence satisfying\n\\[\n\\boxed{\n\\sum_{q=0}^{n}\n\\binom{mn}{mq}\n\\widetilde e_q^{(m,p)}\n=\n\\delta_{n0}.\n}\n\\]\nWhen \\(m\\mid p-1\\), this sequence agrees with the compressed Hurwitz inverse in \\cref{hist:prop:compressed_inverse_recurrence}.","\\begin{histstandardfact}[Lucas congruence]\\label{hist:fact:lucas}\nIf \\(a=a_0+a_1p+\\cdots\\) and \\(b=b_0+b_1p+\\cdots\\) are base \\(p\\) expansions, then\n\\[\n\\binom ab\n\\equiv\n\\prod_r\\binom{a_r}{b_r}\n\\pmod p.\n\\]\nThis is the classical theorem of Lucas \\cite{Lucas1878a}.\n\\end{histstandardfact}","\\begin{histtheorem}[Standard window reduction]\\label{hist:thm:standard_window_reduction}\nFor \\(0\\leq q\\leq n\\),\n\\[\n\\boxed{\n\\binom{mpn+j}{mpq}\n\\equiv\n\\binom{mn}{mq}\n\\pmod p.\n}\n\\]\nThe diagonal coefficient is a unit modulo \\(p\\):\n\\[\n\\binom{mpn+j}{mpn}\n\\equiv1\n\\pmod p.\n\\]\n\\end{histtheorem}","\\begin{proof}\nWrite\n\\[\nmpn+j=p(mn)+j,\n\\qquad\nmpq=p(mq).\n\\]\nLucas reduction separates the lowest base \\(p\\) digit and gives\n\\[\n\\binom{mpn+j}{mpq}\n\\equiv\n\\binom j0\n\\binom{mn}{mq}\n\\pmod p.\n\\]\nThe diagonal case is obtained by setting \\(q=n\\).\n\\end{proof}","\\begin{histcorollary}[Standard window integrality]\\label{hist:cor:standard_integrality}\nEvery coefficient \\(C_{mp,j}(n)\\) belongs to \\(\\Zp\\).\n\\end{histcorollary}","\\begin{proof}\nThe recurrence in \\cref{hist:thm:triangular_inverse} starts with a \\(p\\) integral constant term. The binomial coefficients are integers, and every diagonal term is a \\(p\\) adic unit by \\cref{hist:thm:standard_window_reduction}. Induction gives the result.\n\\end{proof}","\\begin{histcorollary}[Address independence modulo the prime]\\label{hist:cor:address_independence}\nLet \\(\\widetilde e_n^{(m,p)}\\) be the reduced standard inverse above. Then\n\\[\n\\boxed{\nC_{mp,j}(n)\n\\equiv\n\\widetilde e_n^{(m,p)}\n\\pmod p\n}\n\\]\nfor every \\(0\\leq j<p\\).\n\\end{histcorollary}","\\begin{proof}\nModulo \\(p\\), the recurrence in \\cref{hist:thm:triangular_inverse} becomes\n\\[\n\\sum_{q=0}^{n}\n\\binom{mn}{mq}\nC_{mp,j}(q)\n=\n\\delta_{n0}.\n\\]\nThis is the defining recurrence for \\(\\widetilde e_n^{(m,p)}\\). The diagonal is one, so the solution is unique.\n\\end{proof}","\\begin{histtheorem}[Uniform standard window period]\\label{hist:thm:standard_window_period}\nAssume that \\(p\\) is odd and \\(m\\mid p-1\\). Put\n\\[\nh=\\frac{p-1}{m}.\n\\]\nThen for every standard window address \\(0\\leq j<p\\),\n\\[\n\\boxed{\nC_{mp,j}(n+h)\n\\equiv\nC_{mp,j}(n)\n\\pmod p\n\\qquad(n\\geq1).\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nBy \\cref{hist:cor:address_independence}, every address reduces to the same reduced standard inverse. Under the divisibility assumption, that inverse agrees with the compressed Hurwitz inverse by \\cref{hist:prop:compressed_inverse_recurrence}. Its positive tail has period \\(h\\) by \\cref{hist:cor:compressed_period}.\n\\end{proof}","\\begin{histcorollary}[Rational residual generating function]\\label{hist:cor:standard_rational_ogf}\nUnder the hypotheses of \\cref{hist:thm:standard_window_period}, for every standard address \\(j\\),\n\\[\n\\sum_{n\\geq0}\n\\left(C_{mp,j}(n)\\bmod p\\right)z^n\n=\n1+\n\\frac{\n\\sum_{r=1}^{h}\\widetilde e_r^{(m,p)}z^r\n}{1-z^h}.\n\\]\n\\end{histcorollary}","\\begin{histtheorem}[Fourier invariance of the standard window operator]\\label{hist:thm:standard_window_fourier}\nLet \\(p\\) be odd, let \\(m\\mid p-1\\), and let \\(h=(p-1)/m\\). Define\n\\[\n(T_{m,p}f)_n\n=\n\\sum_{q=0}^{n}\\binom{mn}{mq}f_q\n\\quad\\text{in }\\FF_p.\n\\]\nLet \\(V_h\\) be the vector space of sequences satisfying\n\\[\nf_{n+h}=f_n\n\\qquad(n\\geq1).\n\\]\nThen\n\\[\n\\boxed{T_{m,p}(V_h)=V_h.}\n\\]\nThe restriction of \\(T_{m,p}\\) to \\(V_h\\) is an automorphism.\n\\end{histtheorem}","egin{histtheorem}[Fourier invariance of the standard window operator]\\label{hist:thm:standard_window_fourier}\nLet \\(p\\) be odd, let \\(m\\mid p-1\\), and let \\(h=(p-1)/m\\). Define\n\\[\n(T_{m,p}f)_n\n=\n\\sum_{q=0}^{n}\\binom{mn}{mq}f_q\n\\quad\\text{in }\\FF_p.\n\\]\nLet \\(V_h\\) be the vector space of sequences satisfying\n\\[\nf_{n+h}=f_n\n\\qquad(n\\geq1).\n\\]\nThen\n\\[\n\\boxed{T_{m,p}(V_h)=V_h.}\n\\]\nThe restriction of \\(T_{m,p}\\) to \\(V_h\\) is an automorphism.\n\\end{histtheorem}","m}[Fourier invariance of the standard window operator]\\label{hist:thm:standard_window_fourier}\nLet \\(p\\) be odd, let \\(m\\mid p-1\\), and let \\(h=(p-1)/m\\). Define\n\\[\n(T_{m,p}f)_n\n=\n\\sum_{q=0}^{n}\\binom{mn}{mq}f_q\n\\quad\\text{in }\\FF_p.\n\\]\nLet \\(V_h\\) be the vector space of sequences satisfying\n\\[\nf_{n+h}=f_n\n\\qquad(n\\geq1).\n\\]\nThen\n\\[\n\\boxed{T_{m,p}(V_h)=V_h.}\n\\]\nThe restriction of \\(T_{m,p}\\) to \\(V_h\\) is an automorphism.\n\\end{histtheorem}","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower tr","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on ever","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initia","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. I","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is inje","\\begin{proof}\nChoose \\(d_0=f_h\\) and \\(d_n=f_n\\) for \\(n\\geq1\\). Then \\(d\\) is fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","s fully \\(h\\) periodic and \\(f=d+c\\delta_0\\) for \\(c=f_0-f_h\\). Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}",". Since \\(T_{m,p}\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","\\delta_0\\) is the constant sequence one, it is enough to treat \\(d\\). Expand\n\\[\nd_q=\\sum_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","um_{\\chi\\in\\mu_h}c_\\chi\\chi^q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","q.\n\\]\nThe \\(m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","m\\)th power map on \\(\\FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","FF_p^\\times\\) has image \\(\\mu_h\\). Choose \\(y\\) with \\(y^m=\\chi\\). The root filter gives\n\\[\nT_{m,p}(\\chi^\\bullet)_n\n=\n\\frac1m\n\\sum_{\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","\\zeta\\in\\mu_m}\n(1+\\zeta y)^{mn}.\n\\]\nIf \\(1+\\zeta y=0\\), the corresponding term vanishes for positive \\(n\\). Otherwise\n\\[\n\\left((1+\\zeta y)^m\\right)^h\n=\n(1+\\zeta y)^{p-1}\n=1.\n\\]\nEvery summand is therefore \\(h\\) periodic for positive \\(n\\). Thus \\(T_{m,p}\\) preserves \\(V_h\\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \\(f_0,f_1,\\ldots,f_h\\) is injective and hence bijective.\n\\end{proof}","\\begin{histcorollary}[A second proof of the standard window period]\\label{hist:cor:standard_window_fourier_period}\nThe reduced inverse \\(e^{(m,p)}\\) and every standard address satisfy the period in \\cref{hist:thm:standard_window_period}.\n\\end{histcorollary}","\\begin{proof}\nThe sequence \\(\\delta=(1,0,0,\\ldots)\\) belongs to \\(V_h\\). Its inverse image under the automorphism in \\cref{hist:thm:standard_window_fourier} also belongs to \\(V_h\\). Address independence modulo \\(p\\) gives the statement for every standard address.\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_fourier.pdf}\n\\caption{New reconstruction. Finite field Fourier transfer for a standard window. Each character becomes a finite sum of positive index modes with period dividing \\(h\\).}\n\\label{hist:fig:standard_fourier}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_matrix.pdf}\n\\caption{New reconstruction. The standard window matrix for $p=7$, $m=2$, and $0\\le j<7$. Reduction removes the address and retains diagonal one.}\n\\label{hist:fig:standard_matrix}\n\\end{figure}","on. Finite field Fourier transfer for a standard window. Each character becomes a finite sum of positive index modes with period dividing \\(h\\).}\n\\label{hist:fig:standard_fourier}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_matrix.pdf}\n\\caption{New reconstruction. The standard window matrix for $p=7$, $m=2$, and $0\\le j<7$. Reduction removes the address and retains diagonal one.}\n\\label{hist:fig:standard_matrix}\n\\end{figure}","ite field Fourier transfer for a standard window. Each character becomes a finite sum of positive index modes with period dividing \\(h\\).}\n\\label{hist:fig:standard_fourier}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_matrix.pdf}\n\\caption{New reconstruction. The standard window matrix for $p=7$, $m=2$, and $0\\le j<7$. Reduction removes the address and retains diagonal one.}\n\\label{hist:fig:standard_matrix}\n\\end{figure}","ourier transfer for a standard window. Each character becomes a finite sum of positive index modes with period dividing \\(h\\).}\n\\label{hist:fig:standard_fourier}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_matrix.pdf}\n\\caption{New reconstruction. The standard window matrix for $p=7$, $m=2$, and $0\\le j<7$. Reduction removes the address and retains diagonal one.}\n\\label{hist:fig:standard_matrix}\n\\end{figure}","\\begin{histremark}\\label{hist:rem:period_boundary}\nThe theorem gives a period modulo \\(p\\). A period modulo higher powers requires additional control of the triangular inverse and cannot be inferred from the finite field argument alone.","Let \\(q\\) be a positive odd integer. Define\n\\[\nS_n(q)\n=\n\\sum_{a=0}^{q-1}\n(-1)^a(2a+1)^n.\n\\]\nLet the signed Euler numbers and Euler zigzag numbers be defined by\n\\[\n\\operatorname{sech}t\n=\n\\sum_{n\\geq0}E_n\\frac{t^n}{n!}\n\\]\nand\n\\[\n\\sec t+\\tan t\n=\n\\sum_{n\\geq0}Z_n\\frac{t^n}{n!}.\n\\]","\\begin{histlemma}[Odd source generating function]\\label{hist:lem:odd_source_egf}\nFor every positive odd \\(q\\),\n\\[\n\\boxed{\n\\sum_{n\\geq0}S_n(q)\\frac{t^n}{n!}\n=\n\\e^{qt}\\cosh(qt)\\operatorname{sech}t.\n}\n\\]\nIts even part is\n\\[\n\\boxed{\n\\sum_{n\\geq0}S_{2n}(q)\\frac{t^{2n}}{(2n)!}\n=\n\\cosh^2(qt)\\operatorname{sech}t.\n}\n\\]\n\\end{histlemma}","\\begin{proof}\nThe finite geometric sum gives\n\\[\n\\sum_{a=0}^{q-1}(-1)^a\\e^{(2a+1)t}\n=\n\\e^t\\frac{1+\\e^{2qt}}{1+\\e^{2t}}\n=\n\\e^{qt}\\cosh(qt)\\operatorname{sech}t.\n\\]\nAveraging at \\(t\\) and \\(-t\\) gives the even part.\n\\end{proof}","\\begin{histlemma}[Zigzag inverse kernel]\\label{hist:lem:local_zigzag_kernel}\n\\[\n\\boxed{\n\\operatorname{sech}^2t\n=\n\\sum_{r\\geq0}\n(-1)^rZ_{2r+1}\n\\frac{t^{2r}}{(2r)!}.\n}\n\\]\n\\end{histlemma}","\\begin{proof}\nThe identity \\(\\tanh t=-i\\tan(it)\\) gives the odd zigzag expansion of \\(\\tanh t\\). Differentiate.\n\\end{proof}","\\begin{histtheorem}[Odd modulus Euler coordinate pair]\\label{hist:thm:odd_coordinate_pair}\nFor every positive odd \\(q\\) and every even \\(n\\geq0\\),\n\\[\n\\boxed{\nE_n\n=\n\\sum_{r=0}^{n/2}\n(-1)^rZ_{2r+1}\n\\binom n{2r}\nq^{2r}S_{n-2r}(q).\n}\n\\]\nThe inverse transform is\n\\[\n\\boxed{\nS_n(q)\n=\nE_n\n+\n\\sum_{r=1}^{n/2}\n2^{2r-1}\n\\binom n{2r}\nq^{2r}E_{n-2r}.\n}\n\\]\nBoth transforms are lower triangular with diagonal one.\n\\end{histtheorem}","\\begin{proof}\nThe even source identity gives\n\\[\n\\operatorname{sech}t\n=\n\\operatorname{sech}^2(qt)\n\\sum_{n\\geq0}S_{2n}(q)\\frac{t^{2n}}{(2n)!}.\n\\]\nInsert \\cref{hist:lem:local_zigzag_kernel} and compare coefficients. For the inverse, multiply \\(\\operatorname{sech}t\\) by\n\\[\n\\cosh^2(qt)\n=\n1+\n\\sum_{r\\geq1}2^{2r-1}q^{2r}\\frac{t^{2r}}{(2r)!}.\n\\]\n\\end{proof}","\\begin{histcorollary}[Odd modulus truncation]\\label{hist:cor:odd_modulus_truncation}\nFor every \\(M\\geq0\\), every positive odd \\(q\\), and every even \\(n\\),\n\\[\n\\boxed{\nE_n\n\\equiv\n\\sum_{r=0}^{\\min(M,n/2)}\n(-1)^rZ_{2r+1}\n\\binom n{2r}q^{2r}S_{n-2r}(q)\n\\pmod {q^{2M+2}}.\n}\n\\]\nIf \\(p\\mid q\\) and \\(a=\\vp(q)\\), the same truncation is valid modulo \\(p^{2a(M+1)}\\).\n\\end{histcorollary}","\\begin{proof}\nEvery omitted term contains \\(q^{2r}\\) with \\(r\\geq M+1\\). The local statement follows by taking the \\(p\\) adic valuation.\n\\end{proof}","\\begin{histproposition}[Odd row weight]\\label{hist:prop:odd_row_weight}\nFor an odd prime \\(p\\), the \\(r\\)th nonzero term in the reconstruction has valuation\n\\[\n\\boxed{\nW_p(n,r)\n=\n2r+\\vp(Z_{2r+1})\n+\\vp\\binom n{2r}\n+\\vp\\bigl(S_{n-2r}(p)\\bigr).\n}\n\\]\nThe binomial contribution is\n\\[\n\\boxed{\n\\vp\\binom n{2r}\n=\n\\frac{s_p(2r)+s_p(n-2r)-s_p(n)}{p-1},\n}\n\\]\nwhere \\(s_p\\) is the base \\(p\\) digit sum.\n\\end{histproposition}","For even \\(n\\geq0\\), define\n\\[\n\\boxed{\nR_n\n=\n\\frac{3^{n+2}-1}{4(n+1)(n+2)}.\n}\n\\]\nIts even exponential generating function is\n\\[\n\\boxed{\n\\sum_{n\\text{ even}}R_n\\frac{t^n}{n!}\n=\n\\frac{\\sinh(2t)\\sinh t}{2t^2}.\n}\n\\]\nDefine \\(\\beta_r\\) by\n\\[\n\\boxed{\n\\frac{z^2}{\\sinh^2z}\n=\n\\sum_{r\\geq0}\\beta_rz^{2r}.\n}\n\\]","\\begin{histproposition}[Dyadic inverse coefficients]\\label{hist:prop:dyadic_beta_unified}\nFor \\(r\\geq1\\),\n\\[\n\\boxed{\n\\beta_r\n=\n-\n\\frac{2^{2r}(2r-1)B_{2r}}{(2r)!}.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\vTwo(\\beta_r)=\\sTwo(r)-1.\n}\n\\]\nEvery \\(R_n\\) is a dyadic unit.\n\\end{histproposition}","\\begin{proof}\nDifferentiate the Bernoulli expansion of \\(\\coth z\\) and use \\(-(\\coth z)'=\\sinh(z)^{-2}\\). The von Staudt and Clausen theorem gives \\(\\vTwo(B_{2r})=-1\\). Legendre's formula gives\n\\[\n\\vTwo((2r)!)=2r-\\sTwo(r).\n\\]\nFor \\(R_n\\), put \\(k=n+2\\). Since \\(k\\) is even,\n\\[\n\\vTwo(3^k-1)=2+\\vTwo(k).\n\\]\nThe denominator has the same valuation because \\(n+1\\) is odd.\n\\end{proof}","\\begin{histtheorem}[Dyadic Euler coordinate pair]\\label{hist:thm:dyadic_coordinate_pair}\nFor every even \\(n\\geq0\\),\n\\[\n\\boxed{\nE_n\n=\n\\sum_{r=0}^{n/2}\n\\beta_r2^{2r}\n\\fall n{2r}\nR_{n-2r}.\n}\n\\]\nThe inverse transform is\n\\[\n\\boxed{\nR_n\n=\n\\sum_{r=0}^{n/2}\n\\frac{2^{4r+1}}{(2r+2)!}\n\\fall n{2r}\nE_{n-2r}.\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nSubstitute \\(z=2t\\) in the inverse series and multiply\n\\[\n\\frac{4t^2}{\\sinh^2(2t)}\n\\frac{\\sinh(2t)\\sinh t}{2t^2}\n=\n\\operatorname{sech}t.\n\\]\nCoefficient extraction gives the forward transform. The inverse follows from\n\\[\n\\left(\\frac{\\sinh(2t)}{2t}\\right)^2\n=\n\\sum_{r\\geq0}\n\\frac{2^{4r+1}}{(2r+2)!}t^{2r}.\n\\]\n\\end{proof}","For a prime \\(p\\), define\n\\[\nA_{p,n}\n=\n\\begin{cases}\nS_n(p),&p\\text{ odd},","For a prime \\(p\\), define\n\\[\nA_{p,n}\n=\n\\begin{cases}\nS_n(p),&p\\text{ odd},\\\\\nR_n,&p=2\\text{ and }n\\text{ even},\\\\\n0,&p=2\\text{ and }n\\text{ odd},\n\\end{cases}\n\\]\nand\n\\[\n\\kappa_{p,r}\n=\n\\begin{cases}\n(-1)^rZ_{2r+1}p^{2r},&p\\text{ odd},","0,&p=2\\text{ and }n\\text{ odd},\n\\end{cases}\n\\]\nand\n\\[\n\\kappa_{p,r}\n=\n\\begin{cases}\n(-1)^rZ_{2r+1}p^{2r},&p\\text{ odd},\\\\\n\\beta_r2^{2r}(2r)!,&p=2.\n\\end{cases}\n\\]","\\begin{histtheorem}[Unified local Euler transfer]\\label{hist:thm:unified_local_euler}\nFor every prime \\(p\\) and every even \\(n\\geq0\\),\n\\[\n\\boxed{\nE_n\n=\n\\sum_{r=0}^{n/2}\n\\binom n{2r}\n\\kappa_{p,r}\nA_{p,n-2r}.\n}\n\\]\nIf\n\\[\nH_p(t)\n=\n\\sum_{m\\geq0}A_{p,2m}\\frac{t^{2m}}{(2m)!}\n\\]\nand\n\\[\nK_p(t)\n=\n\\sum_{r\\geq0}\\kappa_{p,r}\\frac{t^{2r}}{(2r)!},\n\\]\nthen\n\\[\n\\boxed{K_p(t)H_p(t)=\\operatorname{sech}t.}\n\\]\nFor odd \\(p\\),\n\\[\nH_p(t)=\\cosh^2(pt)\\operatorname{sech}t,\n\\qquad\nK_p(t)=\\operatorname{sech}^2(pt).\n\\]\nFor \\(p=2\\),\n\\[\nH_2(t)=\\frac{\\sinh(2t)\\sinh t}{2t^2},\n\\qquad\nK_2(t)=\\frac{4t^2}{\\sinh^2(2t)}.\n\\]\n\\end{histtheorem}","egin{histtheorem}[Unified local Euler transfer]\\label{hist:thm:unified_local_euler}\nFor every prime \\(p\\) and every even \\(n\\geq0\\),\n\\[\n\\boxed{\nE_n\n=\n\\sum_{r=0}^{n/2}\n\\binom n{2r}\n\\kappa_{p,r}\nA_{p,n-2r}.\n}\n\\]\nIf\n\\[\nH_p(t)\n=\n\\sum_{m\\geq0}A_{p,2m}\\frac{t^{2m}}{(2m)!}\n\\]\nand\n\\[\nK_p(t)\n=\n\\sum_{r\\geq0}\\kappa_{p,r}\\frac{t^{2r}}{(2r)!},\n\\]\nthen\n\\[\n\\boxed{K_p(t)H_p(t)=\\operatorname{sech}t.}\n\\]\nFor odd \\(p\\),\n\\[\nH_p(t)=\\cosh^2(pt)\\operatorname{sech}t,\n\\qquad\nK_p(t)=\\operatorname{sech}^2(pt).\n\\]\nFor \\(p=2\\),\n\\[\nH_2(t)=\\frac{\\sinh(2t)\\sinh t}{2t^2},\n\\qquad\nK_2(t)=\\frac{4t^2}{\\sinh^2(2t)}.\n\\]\n\\end{histtheorem}","sttheorem}[Unified local Euler transfer]\\label{hist:thm:unified_local_euler}\nFor every prime \\(p\\) and every even \\(n\\geq0\\),\n\\[\n\\boxed{\nE_n\n=\n\\sum_{r=0}^{n/2}\n\\binom n{2r}\n\\kappa_{p,r}\nA_{p,n-2r}.\n}\n\\]\nIf\n\\[\nH_p(t)\n=\n\\sum_{m\\geq0}A_{p,2m}\\frac{t^{2m}}{(2m)!}\n\\]\nand\n\\[\nK_p(t)\n=\n\\sum_{r\\geq0}\\kappa_{p,r}\\frac{t^{2r}}{(2r)!},\n\\]\nthen\n\\[\n\\boxed{K_p(t)H_p(t)=\\operatorname{sech}t.}\n\\]\nFor odd \\(p\\),\n\\[\nH_p(t)=\\cosh^2(pt)\\operatorname{sech}t,\n\\qquad\nK_p(t)=\\operatorname{sech}^2(pt).\n\\]\nFor \\(p=2\\),\n\\[\nH_2(t)=\\frac{\\sinh(2t)\\sinh t}{2t^2},\n\\qquad\nK_2(t)=\\frac{4t^2}{\\sinh^2(2t)}.\n\\]\n\\end{histtheorem}","e \\(p\\) and every even \\(n\\geq0\\),\n\\[\n\\boxed{\nE_n\n=\n\\sum_{r=0}^{n/2}\n\\binom n{2r}\n\\kappa_{p,r}\nA_{p,n-2r}.\n}\n\\]\nIf\n\\[\nH_p(t)\n=\n\\sum_{m\\geq0}A_{p,2m}\\frac{t^{2m}}{(2m)!}\n\\]\nand\n\\[\nK_p(t)\n=\n\\sum_{r\\geq0}\\kappa_{p,r}\\frac{t^{2r}}{(2r)!},\n\\]\nthen\n\\[\n\\boxed{K_p(t)H_p(t)=\\operatorname{sech}t.}\n\\]\nFor odd \\(p\\),\n\\[\nH_p(t)=\\cosh^2(pt)\\operatorname{sech}t,\n\\qquad\nK_p(t)=\\operatorname{sech}^2(pt).\n\\]\nFor \\(p=2\\),\n\\[\nH_2(t)=\\frac{\\sinh(2t)\\sinh t}{2t^2},\n\\qquad\nK_2(t)=\\frac{4t^2}{\\sinh^2(2t)}.\n\\]\n\\end{histtheorem}","nd every even \\(n\\geq0\\),\n\\[\n\\boxed{\nE_n\n=\n\\sum_{r=0}^{n/2}\n\\binom n{2r}\n\\kappa_{p,r}\nA_{p,n-2r}.\n}\n\\]\nIf\n\\[\nH_p(t)\n=\n\\sum_{m\\geq0}A_{p,2m}\\frac{t^{2m}}{(2m)!}\n\\]\nand\n\\[\nK_p(t)\n=\n\\sum_{r\\geq0}\\kappa_{p,r}\\frac{t^{2r}}{(2r)!},\n\\]\nthen\n\\[\n\\boxed{K_p(t)H_p(t)=\\operatorname{sech}t.}\n\\]\nFor odd \\(p\\),\n\\[\nH_p(t)=\\cosh^2(pt)\\operatorname{sech}t,\n\\qquad\nK_p(t)=\\operatorname{sech}^2(pt).\n\\]\nFor \\(p=2\\),\n\\[\nH_2(t)=\\frac{\\sinh(2t)\\sinh t}{2t^2},\n\\qquad\nK_2(t)=\\frac{4t^2}{\\sinh^2(2t)}.\n\\]\n\\end{histtheorem}","\\begin{histtheorem}[Unified inverse local transfer]\\label{hist:thm:unified_inverse_local_euler}\nDefine\n\\[\n\\lambda_{p,r}\n=\n\\begin{cases}\n1,&p\\text{ odd and }r=0,\\\\\n2^{2r-1}p^{2r},&p\\text{ odd and }r\\geq1,\\\\\n\\dfrac{2^{4r+1}}{(2r+1)(2r+2)},&p=2.\n\\end{cases}\n\\]\nThen for every prime \\(p\\) and every even \\(n\\geq0\\),\n\\[\n\\boxed{\nA_{p,n}\n=\n\\sum_{r=0}^{n/2}\n\\binom n{2r}\n\\lambda_{p,r}\nE_{n-2r}.\n}\n\\]\nEquivalently, if\n\\[\nL_p(t)=K_p(t)^{-1},\n\\]\nthen\n\\[\n\\boxed{H_p(t)=L_p(t)\\operatorname{sech}t,}\n\\]\nwhere\n\\[\nL_p(t)=\\cosh^2(pt)\n\\]\nfor odd \\(p\\), and\n\\[\nL_2(t)=\\left(\\frac{\\sinh(2t)}{2t}\\right)^2\n\\]\nfor \\(p=2\\).\n\\end{histtheorem}","\\dfrac{2^{4r+1}}{(2r+1)(2r+2)},&p=2.\n\\end{cases}\n\\]\nThen for every prime \\(p\\) and every even \\(n\\geq0\\),\n\\[\n\\boxed{\nA_{p,n}\n=\n\\sum_{r=0}^{n/2}\n\\binom n{2r}\n\\lambda_{p,r}\nE_{n-2r}.\n}\n\\]\nEquivalently, if\n\\[\nL_p(t)=K_p(t)^{-1},\n\\]\nthen\n\\[\n\\boxed{H_p(t)=L_p(t)\\operatorname{sech}t,}\n\\]\nwhere\n\\[\nL_p(t)=\\cosh^2(pt)\n\\]\nfor odd \\(p\\), and\n\\[\nL_2(t)=\\left(\\frac{\\sinh(2t)}{2t}\\right)^2\n\\]\nfor \\(p=2\\).\n\\end{histtheorem}","\\begin{proof}\nFor odd \\(p\\), this is the inverse identity in \\cref{hist:thm:odd_coordinate_pair}. For \\(p=2\\), rewrite the inverse identity in \\cref{hist:thm:dyadic_coordinate_pair} by using\n\\[\n\\fall n{2r}=(2r)!\\binom n{2r}.\n\\]\nThe coefficient becomes\n\\[\n\\frac{2^{4r+1}(2r)!}{(2r+2)!}\n=\n\\frac{2^{4r+1}}{(2r+1)(2r+2)}.\n\\]\nThe generating functions are the inverses of the kernels in \\cref{hist:thm:unified_local_euler}.\n\\end{proof}","\\begin{histcorollary}[Exact change between local source charts]\\label{hist:cor:local_source_change}\n\\par\\noindent\\textbf{Editorial range correction.} This comparison holds for the even source subsequences, equivalently the evenized generating functions $H_p$. It does not identify their full odd-degree source values. For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero. The even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","l source charts]\\label{hist:cor:local_source_change}\n\\par\\noindent\\textbf{Editorial range correction.} This comparison holds for the even source subsequences, equivalently the evenized generating functions $H_p$. It does not identify their full odd-degree source values. For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero. The even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","the even source subsequences, equivalently the evenized generating functions $H_p$. It does not identify their full odd-degree source values. For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero. The even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","It does not identify their full odd-degree source values. For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero. The even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","ot identify their full odd-degree source values. For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero. The even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","ee source values. For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero. The even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}",". For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero. The even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","t odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\\ell H_\\ell$ and its proof remain valid.\\par\\medskip\nFor any two primes \\(p\\) and \\(\\ell\\), the source sequences \\(A_{p,n}\\) and \\(A_{\\ell,n}\\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \\(L_p\\) with the reconstruction kernel \\(K_\\ell\\).\n\\end{histcorollary}","\\begin{proof}\nBoth source series reconstruct \\(\\operatorname{sech}t\\). Hence\n\\[\nH_p(t)=L_p(t)K_\\ell(t)H_\\ell(t).\n\\]\nThe multiplier has constant coefficient one, so \\cref{hist:thm:exact_coefficient_transport} gives an exact unit lower triangular coordinate change.\n\\end{proof}","\\midrule\nFinite torsion address & Modulo \\(p-1\\) & Sign address modulo two","Principal units & \\(1+p\\mathbb Z_p\\) & \\(1+4\\mathbb Z_2\\)","Source & Alternating odd power sum \\(S_n(p)\\) & Weighted source \\(R_n\\)","Kernel & \\(\\operatorname{sech}^2(pt)\\) & \\(4t^2/\\sinh^2(2t)\\)","Positive depth weight & At least \\(2r\\), with zigzag and row factors & Exactly \\(4r-1\\), before the binomial carry","Natural refined step & \\((p-1)p^a\\) & \\(2^{a+1}\\)","\\begin{histexample}[The first odd reconstruction]\\label{hist:ex:first_odd_reconstruction}\nFor \\(p=3\\) and \\(n=2\\),\n\\[\nS_2(3)=1-9+25=17.\n\\]\nSince \\(Z_1=1\\), \\(Z_3=2\\), and \\(S_0(3)=1\\), the odd coordinate formula gives\n\\[\nE_2\n=17-2\\cdot9\n=-1.\n\\]\nThe large source value and the depth one correction are both necessary.\n\\end{histexample}","\\begin{histexample}[The first nontrivial dyadic row]\\label{hist:ex:first_dyadic_reconstruction}\nFor \\(n=4\\),\n\\[\nR_4=\\frac{91}{15},\n\\qquad\nR_2=\\frac53,\n\\qquad\nR_0=1.\n\\]\nThe exact kernel coefficients are\n\\[\n\\kappa_{2,1}=-\\frac83,\n\\qquad\n\\kappa_{2,2}=\\frac{128}{5}.\n\\]\nTherefore\n\\[\nE_4\n=\n\\frac{91}{15}\n-\\frac83\\binom42\\frac53\n+\\frac{128}{5}\n=5.\n\\]\nThe rational source values are dyadic units, while the transfer depth is carried by the kernel.\n\\end{histexample}","\\begin{histremark}\\label{hist:rem:no_dyadic_substitution}\nThe common convolution does not identify the integral charts. At an odd prime the finite character coordinate has order \\(p-1\\). At the dyadic prime it begins with the sign coordinate, and the principal unit subgroup begins at \\(1+4\\mathbb Z_2\\).\n\\end{histremark}","For \\(r\\geq1\\), write\n\\[\n\\kappa_{2,r}\n=\n\\beta_r2^{2r}(2r)!\n=\n-2^{4r}(2r-1)B_{2r}.\n\\]","\\begin{histtheorem}[Exact dyadic kernel weight]\\label{hist:thm:exact_dyadic_kernel_weight}\n\\par\\noindent\\textbf{Editorial index conventions.} The standalone kernel valuation holds for all $r\\ge1$. Its reconstruction-term formula requires even $n\\ge0$ and $1\\le r\\le n/2$. In the following truncation formulas only terms with $r\\le n/2$ occur. Terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","noindent\\textbf{Editorial index conventions.} The standalone kernel valuation holds for all $r\\ge1$. Its reconstruction-term formula requires even $n\\ge0$ and $1\\le r\\le n/2$. In the following truncation formulas only terms with $r\\le n/2$ occur. Terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","ntions.} The standalone kernel valuation holds for all $r\\ge1$. Its reconstruction-term formula requires even $n\\ge0$ and $1\\le r\\le n/2$. In the following truncation formulas only terms with $r\\le n/2$ occur. Terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","ation holds for all $r\\ge1$. Its reconstruction-term formula requires even $n\\ge0$ and $1\\le r\\le n/2$. In the following truncation formulas only terms with $r\\le n/2$ occur. Terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","for all $r\\ge1$. Its reconstruction-term formula requires even $n\\ge0$ and $1\\le r\\le n/2$. In the following truncation formulas only terms with $r\\le n/2$ occur. Terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","mula requires even $n\\ge0$ and $1\\le r\\le n/2$. In the following truncation formulas only terms with $r\\le n/2$ occur. Terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","In the following truncation formulas only terms with $r\\le n/2$ occur. Terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","egative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\\infty$. These conventions include $M\\ge n/2$.\\par\\medskip\nFor every \\(r\\geq1\\),\n\\[\n\\boxed{\\vTwo(\\kappa_{2,r})=4r-1.}\n\\]\nThe \\(r\\)th reconstruction term has exact valuation\n\\[\n\\boxed{\n\\vTwo\\left(\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}\n\\right)\n=\n4r-1+\\vTwo\\binom n{2r}.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\nW_2(n,r)\n=\n4r-1+\\sTwo(r)+\\sTwo(n-2r)-\\sTwo(n).\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nThe Bernoulli formula gives \\(\\vTwo(B_{2r})=-1\\). The source value is a unit. Kummer's binary carry formula gives the last expression.\n\\end{proof}","\\begin{histtheorem}[Optimal dyadic truncation]\\label{hist:thm:optimal_dyadic_truncation}\nLet\n\\[\nT_{2,M}(n)\n=\n\\sum_{r=0}^{\\min(M,n/2)}\n\\kappa_{2,r}\\binom n{2r}R_{n-2r}.\n\\]\nThen\n\\[\n\\boxed{\nE_n\\equiv T_{2,M}(n)\\pmod {2^{4M+3}}.\n}\n\\]\nMore precisely,\n\\[\n\\boxed{\n\\vTwo\\bigl(E_n-T_{2,M}(n)\\bigr)\n\\geq\n\\min_{M+1\\leq r\\leq n/2}\n\\left(\n4r-1+\\vTwo\\binom n{2r}\n\\right).\n}\n\\]\nThe uniform exponent \\(4M+3\\) is best possible.\n\\end{histtheorem}","\\begin{proof}\nEvery omitted term has valuation at least \\(4(M+1)-1\\). For optimality, take \\(n=2M+2\\). There is one omitted term. Its binomial coefficient and source value are one, so its valuation is exactly \\(4M+3\\).\n\\end{proof}","The first three sharp truncations are\n\\[\n\\begin{aligned}\nE_n&\\equiv R_n&&\\pmod 8,\\\\\nE_n&\\equiv R_n-\\frac83\\binom n2R_{n-2}&&\\pmod {128},\\\\\nE_n&\\equiv R_n-\\frac83\\binom n2R_{n-2}\n+\\frac{128}{5}\\binom n4R_{n-4}&&\\pmod {2048}.\n\\end{aligned}\n\\]","For a function \\(f\\) on the nonnegative integers, define\n\\[\n\\Delta f(n)=f(n+1)-f(n).\n\\]","\\begin{histtheorem}[Finite Mahler reconstruction]\\label{hist:thm:finite_mahler_unified}\nGiven \\(f(0),\\ldots,f(N)\\), put \\(c_r=\\Delta^rf(0)\\). Then\n\\[\n\\boxed{\nf(n)=\\sum_{r=0}^{n}c_r\\binom nr\n\\qquad(0\\leq n\\leq N).\n}\n\\]\nFor every \\(s\\) with \\(0\\leq s\\leq N\\),\n\\[\n\\boxed{\n\\Delta^sf(n)\n=\n\\sum_{r=s}^{N}c_r\\binom n{r-s}\n\\qquad(0\\leq n\\leq N-s).\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nLet\n\\[\nP(x)=\\sum_{r=0}^{N}c_r\\binom xr.\n\\]\nNewton interpolation gives \\(P(n)=f(n)\\) for \\(0\\leq n\\leq N\\). Iterating\n\\[\n\\Delta\\binom xr=\\binom x{r-1}\n\\]\ngives\n\\[\n\\Delta^sP(n)=\\sum_{r=s}^{N}c_r\\binom n{r-s}.\n\\]\nWhen \\(0\\leq n\\leq N-s\\), every value used in \\(\\Delta^sf(n)\\) lies among the prescribed data, so \\(\\Delta^sf(n)=\\Delta^sP(n)\\).\n\\end{proof}","\\begin{histcorollary}[Mahler valuation ledger]\\label{hist:cor:mahler_valuation_ledger}\nIf \\(c_r\\in\\Zp\\) and \\(\\vp(c_r)\\geq w(r)\\), then\n\\[\n\\boxed{\n\\vp(\\Delta^sf(n))\n\\geq\n\\min_{r\\geq s}\n\\left(\nw(r)+\\vp\\binom n{r-s}\n\\right).\n}\n\\]\n\\end{histcorollary}","The infinite interpolation theorem states that every continuous function on \\(\\Zp\\) has a unique convergent Mahler expansion \\cite{Mahler1958,Robert2000}. We use that theorem as a standard input when the local variable is allowed to range over all of \\(\\Zp\\).","\\begin{histstandardfact}[Local unit decomposition]\\label{hist:fact:local_unit_decomposition}\nFor odd \\(p\\), every unit has a unique factorization\n\\[\nu=\\omega(u)\\langle u\\rangle,\n\\qquad\n\\omega(u)^{p-1}=1,\n\\qquad\n\\langle u\\rangle\\in1+p\\mathbb Z_p.\n\\]\nEvery dyadic unit has a unique factorization\n\\[\nu=(-1)^{\\epsilon(u)}\\langle u\\rangle,\n\\qquad\n\\epsilon(u)\\in\\{0,1\\},\n\\qquad\n\\langle u\\rangle\\in1+4\\mathbb Z_2.\n\\]\nThe first factor gives the finite character address. The second factor admits logarithmic and Mahler coordinates.\n\\end{histstandardfact}","For odd \\(p\\) and \\(L\\geq1\\), put\n\\[\n\\Lambda_{p,L}=(p-1)p^{L-1}.\n\\]\nFor \\(p=2\\) and \\(L\\geq3\\), put\n\\[\n\\Lambda_{2,L}=2^{L-2}.\n\\]\nThese are exponents of the unit groups modulo the declared prime powers.","\\begin{histtheorem}[One unit mode]\\label{hist:thm:carmichael_mahler_scalar}\nLet \\(u\\in\\Zp^\\times\\), let \\(H=\\Lambda_{p,L}\\), and put\n\\[\nq_{u,L}=u^H-1\\in p^L\\Zp.\n\\]\nFor every integer address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\nu^{Hx+j}\n=\nu^j\\sum_{r\\geq0}q_{u,L}^{\\,r}\\binom xr.\n}\n\\]\nThe \\(r\\)th Mahler coefficient has valuation at least \\(Lr\\).\n\\end{histtheorem}","\\begin{proof}\nWrite \\(u^{Hx+j}=u^j(1+q_{u,L})^x\\) and use the convergent binomial series on \\(\\Zp\\).\n\\end{proof}","\\begin{histexample}[Paired principal unit expansions]\\label{hist:ex:paired_unit_expansions}\nAt the odd prime three,\n\\[\n2^{2x+j}\n=\n2^j4^x\n=\n2^j\\sum_{r\\geq0}3^r\\binom xr.\n\\]\nAt the dyadic prime,\n\\[\n3^{2x+j}\n=\n3^j9^x\n=\n3^j\\sum_{r\\geq0}8^r\\binom xr.\n\\]\nThe formal expansions are identical. Their first local slopes are one and three because \\(4-1=3\\) while \\(9-1=8\\).\n\\end{histexample}","\\begin{histcorollary}[Basic odd and dyadic difference gains]\\label{hist:cor:basic_local_difference_gains}\nFor odd \\(p\\),\n\\[\n\\boxed{\n\\Delta_{(p-1)p^a}^{\\,r}u^n\n=\nu^n\\left(u^{(p-1)p^a}-1\\right)^r,\n\\qquad\n\\vp\\left(\\Delta_{(p-1)p^a}^{\\,r}u^n\\right)\\geq(a+1)r.\n}\n\\]\nFor odd \\(u\\in\\mathbb Z_2^\\times\\),\n\\[\n\\boxed{\n\\Delta_{2^{a+1}}^{\\,r}u^n\n=\nu^n\\left(u^{2^{a+1}}-1\\right)^r,\n\\qquad\n\\vTwo\\left(\\Delta_{2^{a+1}}^{\\,r}u^n\\right)\\geq(a+3)r.\n}\n\\]\n\\end{histcorollary}","\\begin{proof}\nOne step difference multiplies a unit mode by \\(u^H-1\\). Fermat reduction and lifting the exponent give the odd estimate. For odd \\(u\\),\n\\[\n\\vTwo\\left(u^{2^{a+1}}-1\\right)\n=\n\\vTwo(u^2-1)+a\n\\geq a+3.\n\\]\n\\end{proof}","\\begin{histtheorem}[Polynomial exponential Mahler transfer]\\label{hist:thm:polynomial_exponential_mahler}\nLet\n\\[\nf(n)=P(n)u^n\n\\]\nwith \\(P\\in\\Zp[n]\\) of degree \\(d\\). Let \\(H=\\Lambda_{p,L}\\), fix an address \\(j\\), and write\n\\[\nQ_j(x)=P(Hx+j),\n\\qquad\ny=u^H,\n\\qquad q=y-1.\n\\]\nThen\n\\[\n\\boxed{\nf(Hx+j)=\\sum_{r\\geq0}c_{j,r}\\binom xr,\n}\n\\]\nwhere\n\\[\n\\boxed{\nc_{j,r}\n=\nu^j\n\\sum_{k=0}^{\\min(d,r)}\n\\binom rk\n\\Delta^kQ_j(0)y^kq^{r-k}.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\vp(c_{j,r})\\geq L\\max(r-d,0).\n}\n\\]\nFor \\(R\\geq d\\), truncation at depth \\(R\\) is valid modulo \\(p^{L(R+1-d)}\\) uniformly on \\(\\Zp\\).\n\\end{histtheorem}","ahler transfer]\\label{hist:thm:polynomial_exponential_mahler}\nLet\n\\[\nf(n)=P(n)u^n\n\\]\nwith \\(P\\in\\Zp[n]\\) of degree \\(d\\). Let \\(H=\\Lambda_{p,L}\\), fix an address \\(j\\), and write\n\\[\nQ_j(x)=P(Hx+j),\n\\qquad\ny=u^H,\n\\qquad q=y-1.\n\\]\nThen\n\\[\n\\boxed{\nf(Hx+j)=\\sum_{r\\geq0}c_{j,r}\\binom xr,\n}\n\\]\nwhere\n\\[\n\\boxed{\nc_{j,r}\n=\nu^j\n\\sum_{k=0}^{\\min(d,r)}\n\\binom rk\n\\Delta^kQ_j(0)y^kq^{r-k}.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\vp(c_{j,r})\\geq L\\max(r-d,0).\n}\n\\]\nFor \\(R\\geq d\\), truncation at depth \\(R\\) is valid modulo \\(p^{L(R+1-d)}\\) uniformly on \\(\\Zp\\).\n\\end{histtheorem}","al_mahler}\nLet\n\\[\nf(n)=P(n)u^n\n\\]\nwith \\(P\\in\\Zp[n]\\) of degree \\(d\\). Let \\(H=\\Lambda_{p,L}\\), fix an address \\(j\\), and write\n\\[\nQ_j(x)=P(Hx+j),\n\\qquad\ny=u^H,\n\\qquad q=y-1.\n\\]\nThen\n\\[\n\\boxed{\nf(Hx+j)=\\sum_{r\\geq0}c_{j,r}\\binom xr,\n}\n\\]\nwhere\n\\[\n\\boxed{\nc_{j,r}\n=\nu^j\n\\sum_{k=0}^{\\min(d,r)}\n\\binom rk\n\\Delta^kQ_j(0)y^kq^{r-k}.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\vp(c_{j,r})\\geq L\\max(r-d,0).\n}\n\\]\nFor \\(R\\geq d\\), truncation at depth \\(R\\) is valid modulo \\(p^{L(R+1-d)}\\) uniformly on \\(\\Zp\\).\n\\end{histtheorem}","with \\(P\\in\\Zp[n]\\) of degree \\(d\\). Let \\(H=\\Lambda_{p,L}\\), fix an address \\(j\\), and write\n\\[\nQ_j(x)=P(Hx+j),\n\\qquad\ny=u^H,\n\\qquad q=y-1.\n\\]\nThen\n\\[\n\\boxed{\nf(Hx+j)=\\sum_{r\\geq0}c_{j,r}\\binom xr,\n}\n\\]\nwhere\n\\[\n\\boxed{\nc_{j,r}\n=\nu^j\n\\sum_{k=0}^{\\min(d,r)}\n\\binom rk\n\\Delta^kQ_j(0)y^kq^{r-k}.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\vp(c_{j,r})\\geq L\\max(r-d,0).\n}\n\\]\nFor \\(R\\geq d\\), truncation at depth \\(R\\) is valid modulo \\(p^{L(R+1-d)}\\) uniformly on \\(\\Zp\\).\n\\end{histtheorem}","f degree \\(d\\). Let \\(H=\\Lambda_{p,L}\\), fix an address \\(j\\), and write\n\\[\nQ_j(x)=P(Hx+j),\n\\qquad\ny=u^H,\n\\qquad q=y-1.\n\\]\nThen\n\\[\n\\boxed{\nf(Hx+j)=\\sum_{r\\geq0}c_{j,r}\\binom xr,\n}\n\\]\nwhere\n\\[\n\\boxed{\nc_{j,r}\n=\nu^j\n\\sum_{k=0}^{\\min(d,r)}\n\\binom rk\n\\Delta^kQ_j(0)y^kq^{r-k}.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\vp(c_{j,r})\\geq L\\max(r-d,0).\n}\n\\]\nFor \\(R\\geq d\\), truncation at depth \\(R\\) is valid modulo \\(p^{L(R+1-d)}\\) uniformly on \\(\\Zp\\).\n\\end{histtheorem}","a_{p,L}\\), fix an address \\(j\\), and write\n\\[\nQ_j(x)=P(Hx+j),\n\\qquad\ny=u^H,\n\\qquad q=y-1.\n\\]\nThen\n\\[\n\\boxed{\nf(Hx+j)=\\sum_{r\\geq0}c_{j,r}\\binom xr,\n}\n\\]\nwhere\n\\[\n\\boxed{\nc_{j,r}\n=\nu^j\n\\sum_{k=0}^{\\min(d,r)}\n\\binom rk\n\\Delta^kQ_j(0)y^kq^{r-k}.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\vp(c_{j,r})\\geq L\\max(r-d,0).\n}\n\\]\nFor \\(R\\geq d\\), truncation at depth \\(R\\) is valid modulo \\(p^{L(R+1-d)}\\) uniformly on \\(\\Zp\\).\n\\end{histtheorem}","\\begin{proof}\nOn the polynomial factor the difference operator acts as \\(q+y\\Delta\\). The two terms commute. Expand \\((q+y\\Delta)^r\\) and evaluate at zero. Since \\(\\Delta^kQ_j=0\\) for \\(k>d\\), every surviving term contains at least \\(q^{r-d}\\).\n\\end{proof}","Let\n\\[\nv_{n+1}=Cv_n,\n\\qquad\na_n=w^TC^nv_0,\n\\]\nwith \\(C\\in M_d(\\Zp)\\) invertible modulo \\(p\\).","\\begin{histtheorem}[Matrix Carmichael and Mahler transfer]\\label{hist:thm:matrix_carmichael_mahler}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The original displayed statement leaves the vectors untyped. Its integrality conclusion requires $v_0,w\\in\\Z_p^d$, $L\\ge1$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","in{histtheorem}[Matrix Carmichael and Mahler transfer]\\label{hist:thm:matrix_carmichael_mahler}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The original displayed statement leaves the vectors untyped. Its integrality conclusion requires $v_0,w\\in\\Z_p^d$, $L\\ge1$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","theorem}[Matrix Carmichael and Mahler transfer]\\label{hist:thm:matrix_carmichael_mahler}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The original displayed statement leaves the vectors untyped. Its integrality conclusion requires $v_0,w\\in\\Z_p^d$, $L\\ge1$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","atrix Carmichael and Mahler transfer]\\label{hist:thm:matrix_carmichael_mahler}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The original displayed statement leaves the vectors untyped. Its integrality conclusion requires $v_0,w\\in\\Z_p^d$, $L\\ge1$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","d Mahler transfer]\\label{hist:thm:matrix_carmichael_mahler}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The original displayed statement leaves the vectors untyped. Its integrality conclusion requires $v_0,w\\in\\Z_p^d$, $L\\ge1$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}",":matrix_carmichael_mahler}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The original displayed statement leaves the vectors untyped. Its integrality conclusion requires $v_0,w\\in\\Z_p^d$, $L\\ge1$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","_0,w\\in\\Z_p^d$, $L\\ge1$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","$, and $H,j,n\\in\\N$ with $H>0$. The value at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","alue at $Hx+j$ for $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","or $x\\in\\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","s $\\Delta_1a_0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","0=1$, which is not in $3\\Z_3$. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\). For every address \\(j\\) and every \\(x\\in\\Zp\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r\\geq0}\np^{Lr}\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nThe series converges on \\(\\Zp\\), and\n\\[\n\\boxed{\n\\Delta_H^ra_n\n\\equiv0\n\\pmod {p^{Lr}}.\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nUse\n\\[\nC^{Hx+j}=C^j(I+p^LB)^x\n\\]\nand the matrix binomial series. One difference in the coarse variable multiplies the series by \\(p^LB\\).\n\\end{proof}","\\begin{histtheorem}[Finite adelic transfer]\\label{hist:thm:finite_adelic_transfer}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Read this statement with $v_0,w\\in\\Z^d$, $H>0$ and $j,n,r\\in\\N$, and positive integer $L_p$. These restrictions make the integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","f{Editorial hypothesis correction.} Read this statement with $v_0,w\\in\\Z^d$, $H>0$ and $j,n,r\\in\\N$, and positive integer $L_p$. These restrictions make the integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","v_0,w\\in\\Z^d$, $H>0$ and $j,n,r\\in\\N$, and positive integer $L_p$. These restrictions make the integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","and positive integer $L_p$. These restrictions make the integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","er $L_p$. These restrictions make the integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","restrictions make the integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","e integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","$v_0=1/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","/3$ in the one-prime example above loses the advertised digit. Invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","f $C$ alone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","lone does not restore it. The original proof is retained under these hypotheses.\\par\\medskip\nLet\n\\[\nQ=\\prod_{p\\in S}p^{L_p}\n\\]\nfor a finite set of primes. Let \\(C\\in M_d(\\ZZ)\\) be invertible modulo \\(Q\\), and choose \\(H\\) such that\n\\[\nC^H=I+QB\n\\]\nwith \\(B\\in M_d(\\ZZ)\\). Then for every nonnegative integer \\(x\\),\n\\[\n\\boxed{\na_{Hx+j}\n=\n\\sum_{r=0}^{x}Q^r\\binom xr\nw^TC^jB^rv_0.\n}\n\\]\nFor every \\(r\\),\n\\[\n\\boxed{\n\\Delta_H^ra_n\\equiv0\\pmod {Q^r}.\n}\n\\]\n\\end{histtheorem}","\\begin{histexample}[A simultaneous Fibonacci jump]\\label{hist:ex:fibonacci_finite_adelic}\nLet\n\\[\nC=\n\\begin{pmatrix}\n1&1\\\\\n1&0\n\\end{pmatrix}.\n\\]\nThe matrix \\(C\\) is invertible modulo twenty four, and\n\\[\nC^{24}-I\n=\n24\n\\begin{pmatrix}\n3126&1932","1&0\n\\end{pmatrix}.\n\\]\nThe matrix \\(C\\) is invertible modulo twenty four, and\n\\[\nC^{24}-I\n=\n24\n\\begin{pmatrix}\n3126&1932","1&0\n\\end{pmatrix}.\n\\]\nThe matrix \\(C\\) is invertible modulo twenty four, and\n\\[\nC^{24}-I\n=\n24\n\\begin{pmatrix}\n3126&1932\\\\\n1932&1194\n\\end{pmatrix}.\n\\]\nFor every scalar coordinate of the state sequence \\(C^nv_0\\),\n\\[\n\\boxed{\n\\Delta_{24}^{\\,r}a_n\n\\equiv0\n\\pmod {24^r}.\n}\n\\]\nOne integral jump therefore gives simultaneous congruence ladders at the primes two and three.\n\\end{histexample}","1932&1194\n\\end{pmatrix}.\n\\]\nFor every scalar coordinate of the state sequence \\(C^nv_0\\),\n\\[\n\\boxed{\n\\Delta_{24}^{\\,r}a_n\n\\equiv0\n\\pmod {24^r}.\n}\n\\]\nOne integral jump therefore gives simultaneous congruence ladders at the primes two and three.\n\\end{histexample}","\\begin{histdefinition}[Amice pullback]\\label{hist:def:amice_pullback_unified}\nFor a formal exponential generating series\n\\[\nF(z)=\\sum_{n\\geq0}a_n\\frac{z^n}{n!},\n\\]\ndefine\n\\[\n\\widehat F(T)=F(\\log(1+T)).\n\\]\n\\end{histdefinition}","\\begin{histtheorem}[Derivative conjugacy]\\label{hist:thm:amice_derivative_unified}\n\\[\n\\boxed{\n\\widehat{F'}(T)\n=\n(1+T)\\frac{d}{dT}\\widehat F(T).\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nDifferentiate the composition \\(F(\\log(1+T))\\).\n\\end{proof}","\\begin{histproposition}[Weighted convergence]\\label{hist:prop:amice_weighted_convergence_unified}\nIf\n\\[\n\\widehat F(T)=\\sum_{r\\geq0}c_rT^r\n\\]\nand\n\\[\n\\vp(c_r)+r\\alpha\\longrightarrow+\\infty,\n\\]\nthen the series converges whenever \\(\\vp(T)\\geq\\alpha\\).\n\\end{histproposition}","\\begin{proof}\nThe valuation of the \\(r\\)th term is \\(\\vp(c_r)+r\\vp(T)\\), which tends to positive infinity.\n\\end{proof}","For an odd prime \\(p\\), the source \\(S_n(p)\\) contains one nonunit base. Put\n\\[\n\\sigma_p=(-1)^{(p-1)/2}\n\\]\nand\n\\[\nS_n^\\times(p)=S_n(p)-\\sigma_pp^n.\n\\]","\\begin{histtheorem}[Depleted odd prime reconstruction]\\label{hist:thm:depleted_odd_reconstruction}\nFor every even \\(n\\),\n\\[\n\\boxed{\n\\left(1-\\sigma_pp^n\\right)E_n\n=\n\\sum_{r=0}^{n/2}\n(-1)^rZ_{2r+1}\n\\binom n{2r}p^{2r}S_{n-2r}^\\times(p).\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nThe removed source mode contributes \\(\\sigma_p\\cosh(pt)\\). Multiplication by \\(\\operatorname{sech}^2(pt)\\) gives \\(\\sigma_p\\operatorname{sech}(pt)\\). Subtract it from the full source and kernel product.\n\\end{proof}","Fix an even address \\(j\\in\\{0,2,\\ldots,p-3\\}\\). For \\(x\\in\\Zp\\), define\n\\[\n\\begin{aligned}\n\\mathscr E_{p,j}^\\times(x)\n=\n\\sum_{r\\geq0}\n&(-1)^rZ_{2r+1}p^{2r}\n\\binom{(p-1)x+j}{2r}","Fix an even address \\(j\\in\\{0,2,\\ldots,p-3\\}\\). For \\(x\\in\\Zp\\), define\n\\[\n\\begin{aligned}\n\\mathscr E_{p,j}^\\times(x)\n=\n\\sum_{r\\geq0}\n&(-1)^rZ_{2r+1}p^{2r}\n\\binom{(p-1)x+j}{2r}\\\\\n&\\times\n\\sum_{\\substack{0\\leq a<p\\\\2a+1\\neq p}}\n(-1)^a\n(2a+1)^{j-2r}\n\\left((2a+1)^{p-1}\\right)^x.\n\\end{aligned}\n\\]","\\begin{histtheorem}[Odd prime analytic Euler branch]\\label{hist:thm:odd_analytic_branch}\nThe series defining \\(\\mathscr E_{p,j}^\\times\\) converges in the Tate algebra on \\(\\Zp\\). For every nonnegative integer \\(x\\),\n\\[\n\\boxed{\n\\mathscr E_{p,j}^\\times(x)\n=\n\\left(1-\\sigma_pp^{(p-1)x+j}\\right)\nE_{(p-1)x+j}.\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nEvery retained base is a unit, and its \\((p-1)\\)st power lies in \\(1+p\\Zp\\). The polynomial\n\\[\np^{2r}\\binom{(p-1)x+j}{2r}\n\\]\nhas Gauss norm at most\n\\[\np^{-2r+\\vp((2r)!)}.\n\\]\nThis tends to zero because \\(p\\) is odd. The remaining factors have bounded Gauss norm. Thus the series converges in the Tate algebra. At a nonnegative integer \\(x\\), the binomial factor vanishes for large \\(r\\), and \\cref{hist:thm:depleted_odd_reconstruction} gives the value.\n\\end{proof}","\\begin{histremark}\\label{hist:rem:depleted_factor_boundary}\nThe analytic function is defined by its unit source series. The expression \\(p^{(p-1)x+j}\\) in the interpolation identity is evaluated only at nonnegative integers. No analytic function \\(x\\mapsto p^x\\) on all of \\(\\Zp\\) is used.\n\\end{histremark}","For \\(x\\in\\mathbb Z_2\\), define\n\\[\n\\boxed{\n\\mathscr R(x)\n=\n\\frac{9^{x+1}-1}{8(x+1)(2x+1)}.\n}\n\\]\nThe apparent singularity at \\(x=-1\\) is removable, with value \\(-\\log(9)/8\\).","\\begin{histlemma}[Analytic dyadic source]\\label{hist:lem:analytic_dyadic_source}\nThe function \\(\\mathscr R\\) belongs to the dyadic Tate algebra and takes values in \\(\\mathbb Z_2\\). For every nonnegative integer \\(n\\),\n\\[\n\\boxed{\\mathscr R(n)=R_{2n}.}\n\\]\n\\end{histlemma}","\\begin{proof}\nPut \\(y=x+1\\) and let \\(\\ell=\\log 9\\). Since \\(\\vTwo(\\ell)=3\\),\n\\[\n\\frac{9^y-1}{8y}\n=\n\\frac{\\ell}{8}\n\\sum_{m\\geq0}\n\\frac{\\ell^m}{(m+1)!}y^m.\n\\]\nThe coefficient of \\(y^m\\) has valuation\n\\[\n3m-\\vTwo((m+1)!),\n\\]\nwhich is nonnegative and tends to positive infinity. The quotient therefore belongs to \\(\\mathbb Z_2\\langle y\\rangle\\). The factor \\(2x+1=1+2x\\) is a unit in \\(\\mathbb Z_2\\langle x\\rangle\\), with inverse \\(\\sum_{m\\geq0}(-2x)^m\\). Hence \\(\\mathscr R\\) is integral and analytic. The integer values follow from the definition of \\(R_{2n}\\).\n\\end{proof}","nd let \\(\\ell=\\log 9\\). Since \\(\\vTwo(\\ell)=3\\),\n\\[\n\\frac{9^y-1}{8y}\n=\n\\frac{\\ell}{8}\n\\sum_{m\\geq0}\n\\frac{\\ell^m}{(m+1)!}y^m.\n\\]\nThe coefficient of \\(y^m\\) has valuation\n\\[\n3m-\\vTwo((m+1)!),\n\\]\nwhich is nonnegative and tends to positive infinity. The quotient therefore belongs to \\(\\mathbb Z_2\\langle y\\rangle\\). The factor \\(2x+1=1+2x\\) is a unit in \\(\\mathbb Z_2\\langle x\\rangle\\), with inverse \\(\\sum_{m\\geq0}(-2x)^m\\). Hence \\(\\mathscr R\\) is integral and analytic. The integer values follow from the definition of \\(R_{2n}\\).\n\\end{proof}","\\vTwo(\\ell)=3\\),\n\\[\n\\frac{9^y-1}{8y}\n=\n\\frac{\\ell}{8}\n\\sum_{m\\geq0}\n\\frac{\\ell^m}{(m+1)!}y^m.\n\\]\nThe coefficient of \\(y^m\\) has valuation\n\\[\n3m-\\vTwo((m+1)!),\n\\]\nwhich is nonnegative and tends to positive infinity. The quotient therefore belongs to \\(\\mathbb Z_2\\langle y\\rangle\\). The factor \\(2x+1=1+2x\\) is a unit in \\(\\mathbb Z_2\\langle x\\rangle\\), with inverse \\(\\sum_{m\\geq0}(-2x)^m\\). Hence \\(\\mathscr R\\) is integral and analytic. The integer values follow from the definition of \\(R_{2n}\\).\n\\end{proof}","^m.\n\\]\nThe coefficient of \\(y^m\\) has valuation\n\\[\n3m-\\vTwo((m+1)!),\n\\]\nwhich is nonnegative and tends to positive infinity. The quotient therefore belongs to \\(\\mathbb Z_2\\langle y\\rangle\\). The factor \\(2x+1=1+2x\\) is a unit in \\(\\mathbb Z_2\\langle x\\rangle\\), with inverse \\(\\sum_{m\\geq0}(-2x)^m\\). Hence \\(\\mathscr R\\) is integral and analytic. The integer values follow from the definition of \\(R_{2n}\\).\n\\end{proof}","Define\n\\[\n\\boxed{\n\\mathscr E_2(x)\n=\n\\sum_{r\\geq0}\n\\kappa_{2,r}\n\\binom{2x}{2r}\n\\mathscr R(x-r).\n}\n\\]","\\begin{histtheorem}[Direct dyadic Euler interpolation]\\label{hist:thm:direct_dyadic_interpolation}\nThe series defining \\(\\mathscr E_2\\) converges in the dyadic Tate algebra. For every nonnegative integer \\(n\\),\n\\[\n\\boxed{\\mathscr E_2(n)=E_{2n}.}\n\\]\nTruncation after depth \\(M\\) is uniformly valid modulo \\(2^{4M+3}\\) on \\(\\mathbb Z_2\\).\n\\end{histtheorem}","\\begin{proof}\nThe monomial Gauss norm of \\(\\binom{2x}{2r}\\) is at most \\(2^{\\vTwo((2r)!)}\\). Since\n\\[\n\\vTwo(\\kappa_{2,r})=4r-1,\n\\]\nthe product tends to zero in Gauss norm. Translation by the integer \\(r\\) preserves the unit disc, so \\(\\mathscr R(x-r)\\) remains an integral analytic function with uniformly bounded supremum norm. The series therefore converges in the Tate algebra.","At a nonnegative integer \\(n\\), every term with \\(r>n\\) vanishes and \\cref{hist:thm:dyadic_coordinate_pair} remains. For arbitrary \\(x\\in\\mathbb Z_2\\), the integer valued polynomial \\(\\binom{2x}{2r}\\) and the source value \\(\\mathscr R(x-r)\\) lie in \\(\\mathbb Z_2\\). Every omitted term after depth \\(M\\) is therefore divisible by \\(2^{4M+3}\\). This proves the uniform truncation statement.\n\\end{proof}","\\begin{histtheorem}[Unified local interpolation statement]\\label{hist:thm:unified_local_interpolation}\nThe odd branch \\(\\mathscr E_{p,j}^\\times\\) and the dyadic branch \\(\\mathscr E_2\\) are analytic local realizations of the source and kernel law in \\cref{hist:thm:unified_local_euler}. At integer arguments they recover, respectively,\n\\[\n\\left(1-\\sigma_pp^n\\right)E_n\n\\]\non each even odd prime character branch and\n\\[\nE_{2n}\n\\]\non the dyadic even branch. The two realizations have distinct source corrections and distinct kernel valuation slopes.\n\\end{histtheorem}","m}[Unified local interpolation statement]\\label{hist:thm:unified_local_interpolation}\nThe odd branch \\(\\mathscr E_{p,j}^\\times\\) and the dyadic branch \\(\\mathscr E_2\\) are analytic local realizations of the source and kernel law in \\cref{hist:thm:unified_local_euler}. At integer arguments they recover, respectively,\n\\[\n\\left(1-\\sigma_pp^n\\right)E_n\n\\]\non each even odd prime character branch and\n\\[\nE_{2n}\n\\]\non the dyadic even branch. The two realizations have distinct source corrections and distinct kernel valuation slopes.\n\\end{histtheorem}","Polyphase refinement replaces one local branch by finitely many digit branches while preserving exact synthesis. For sequences it is residue analysis. For local functions it is the digit tree of \\(\\mathbb Z_p\\). In Mahler coordinates it becomes a finite triangular kernel.","For a sequence \\(a=(a_n)_{n\\geq0}\\) and \\(M\\geq2\\), define\n\\[\na_r^{[M]}(q)=a_{Mq+r}\n\\qquad(0\\leq r<M).\n\\]\nThe analysis map is\n\\[\n\\mathcal P_Ma\n=\n\\left(a_0^{[M]},\\ldots,a_{M-1}^{[M]}\\right).\n\\]\nSynthesis interleaves the branches by the rule\n\\[\na_{Mq+r}=a_r^{[M]}(q).\n\\]","\\begin{histtheorem}[Exact discrete polyphase reconstruction]\\label{hist:thm:discrete_polyphase_reconstruction}\nThe analysis map \\(\\mathcal P_M\\) is a bijection between sequences and ordered families of \\(M\\) sequences. If\n\\[\nA(z)=\\sum_{n\\geq0}a_nz^n\n\\]\nand\n\\[\nA_r^{[M]}(w)=\\sum_{q\\geq0}a_{Mq+r}w^q,\n\\]\nthen\n\\[\n\\boxed{\nA(z)=\\sum_{r=0}^{M-1}z^rA_r^{[M]}(z^M).\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nEvery nonnegative integer has a unique expression \\(Mq+r\\) with \\(0\\leq r<M\\). Group the terms of the ordinary generating function by this residue.\n\\end{proof}","\\begin{histtheorem}[Nested discrete refinement]\\label{hist:thm:nested_discrete_polyphase}\nFor positive integers \\(M,N\\),\n\\[\n\\boxed{\n\\left(a_r^{[M]}\\right)_s^{[N]}\n=\na_{r+Ms}^{[MN]}.\n}\n\\]\nFor a mixed radix path \\((M_1,\\ldots,M_\\ell)\\) with digits \\(0\\leq s_i<M_i\\), the final address is\n\\[\n\\boxed{\nr(\\boldsymbol s)\n=\ns_1+M_1s_2+\\cdots+\n\\left(\\prod_{i=1}^{\\ell-1}M_i\\right)s_\\ell.\n}\n\\]\nOnly the product radix and accumulated address determine the final branch.\n\\end{histtheorem}","\\begin{proof}\nDirect substitution gives\n\\[\n\\left(a_r^{[M]}\\right)_s^{[N]}(q)\n=\na_{M(Nq+s)+r}.\n\\]\nIterate this identity along the mixed radix path.\n\\end{proof}","Let \\(c=a*b\\) be the ordinary Cauchy convolution. Write \\(A_r\\), \\(B_r\\), and \\(C_r\\) for the phase ordinary generating functions.","\\begin{histproposition}[One carry convolution law]\\label{hist:prop:polyphase_carry_convolution}\nFor every residue \\(r\\),\n\\[\n\\boxed{\nC_r(w)\n=\n\\sum_{u+v=r}A_u(w)B_v(w)\n+\nw\\sum_{u+v=r+M}A_u(w)B_v(w).\n}\n\\]\nThe first sum has no phase carry. The second has one coarse index carry.\n\\end{histproposition}","\\begin{proof}\nWrite the input indices as \\(Mq+u\\) and \\(Mq'+v\\). Since \\(0\\leq u,v<M\\), their sum has residue \\(u+v\\) when \\(u+v<M\\), and residue \\(u+v-M\\) with one additional coarse index when \\(u+v\\geq M\\).\n\\end{proof}","\\begin{histtheorem}[Polyphase shift matrix]\\label{hist:thm:polyphase_shift_matrix}\nLet \\(E\\) be the scalar shift \\((Ea)_n=a_{n+1}\\), and let \\(E_q\\) shift the coarse index. Then\n\\[\n\\boxed{\n\\mathcal P_M(Ea)\n=\n\\mathsf S_M(E_q)\\mathcal P_Ma,\n}\n\\]\nwhere\n\\[\n\\mathsf S_M(E_q)\n=\n\\begin{pmatrix}\n0&1&0&\\cdots&0","\\begin{histtheorem}[Polyphase shift matrix]\\label{hist:thm:polyphase_shift_matrix}\nLet \\(E\\) be the scalar shift \\((Ea)_n=a_{n+1}\\), and let \\(E_q\\) shift the coarse index. Then\n\\[\n\\boxed{\n\\mathcal P_M(Ea)\n=\n\\mathsf S_M(E_q)\\mathcal P_Ma,\n}\n\\]\nwhere\n\\[\n\\mathsf S_M(E_q)\n=\n\\begin{pmatrix}\n0&1&0&\\cdots&0\\\\\n0&0&1&\\cdots&0\\\\\n\\vdots&&&\\ddots&\\vdots\\\\\n0&0&0&\\cdots&1\\\\\nE_q&0&0&\\cdots&0\n\\end{pmatrix}.\n\\]\nIt also satisfies\n\\[\n\\boxed{\\mathsf S_M(E_q)^M=E_qI_M.}\n\\]\n\\end{histtheorem}","E_q&0&0&\\cdots&0\n\\end{pmatrix}.\n\\]\nIt also satisfies\n\\[\n\\boxed{\\mathsf S_M(E_q)^M=E_qI_M.}\n\\]\n\\end{histtheorem}","\\begin{proof}\nFor \\(r<M-1\\),\n\\[\n(Ea)_{Mq+r}=a_{Mq+r+1}=a_{r+1}^{[M]}(q).\n\\]\nFor \\(r=M-1\\),\n\\[\n(Ea)_{Mq+M-1}=a_{M(q+1)}=E_qa_0^{[M]}(q).\n\\]\nAfter \\(M\\) shifts every phase has advanced by one coarse index.\n\\end{proof}","For \\(a\\geq1\\) and \\(0\\leq s<p^a\\), define\n\\[\nf_{a,s}(x)=f(p^ax+s).\n\\]","\\begin{histtheorem}[Exact local branch reconstruction]\\label{hist:thm:local_branch_reconstruction}\nThe family \\(\\{f_{a,s}\\}_{0\\leq s<p^a}\\) determines \\(f\\) uniquely on \\(\\mathbb Z_p\\). More precisely, if \\(y\\in s+p^a\\mathbb Z_p\\), then\n\\[\n\\boxed{\nf(y)=f_{a,s}\\left(\\frac{y-s}{p^a}\\right).\n}\n\\]\nFor \\(a,b\\geq1\\),\n\\[\n\\boxed{\n(f_{a,s})_{b,t}\n=\nf_{a+b,s+p^at}.\n}\n\\]\nThus local refinement is exactly the \\(p\\) adic digit tree.\n\\end{histtheorem}","egin{histtheorem}[Exact local branch reconstruction]\\label{hist:thm:local_branch_reconstruction}\nThe family \\(\\{f_{a,s}\\}_{0\\leq s<p^a}\\) determines \\(f\\) uniquely on \\(\\mathbb Z_p\\). More precisely, if \\(y\\in s+p^a\\mathbb Z_p\\), then\n\\[\n\\boxed{\nf(y)=f_{a,s}\\left(\\frac{y-s}{p^a}\\right).\n}\n\\]\nFor \\(a,b\\geq1\\),\n\\[\n\\boxed{\n(f_{a,s})_{b,t}\n=\nf_{a+b,s+p^at}.\n}\n\\]\nThus local refinement is exactly the \\(p\\) adic digit tree.\n\\end{histtheorem}","\\begin{proof}\nThe cosets \\(s+p^a\\mathbb Z_p\\) form a disjoint partition of \\(\\mathbb Z_p\\). The first formula is the inverse affine change of variables on one coset. The second follows from\n\\[\np^a(p^bx+t)+s\n=\np^{a+b}x+p^at+s.\n\\]\n\\end{proof}","Let\n\\[\n\\Phi_M(z)=(1+z)^M-1.\n\\]\nFor \\(M\\geq1\\), \\(s\\geq0\\), and \\(r,k\\geq0\\), define\n\\[\n\\boxed{\n\\Theta_{M,s}(r,k)\n=\n[z^r](1+z)^s\\Phi_M(z)^k.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Theta_{M,s}(r,k)\n=\n\\sum_{j=0}^{k}\n(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.\n}\n\\]\nIt vanishes unless\n\\[\nk\\leq r\\leq Mk+s.\n\\]","\\begin{histproposition}[Edge coefficients of the affine kernel]\\label{hist:prop:affine_kernel_edges}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(k\\geq0\\),\n\\[\n\\boxed{\n\\Theta_{M,s}(k,k)=M^k,\n\\qquad\n\\Theta_{M,s}(Mk+s,k)=1.\n}\n\\]\nFor binary radix and zero address,\n\\[\n\\boxed{\n\\Theta_{2,0}(k+j,k)\n=\n2^{k-j}\\binom kj\n\\qquad(0\\leq j\\leq k).\n}\n\\]\nHence\n\\[\n\\boxed{\n\\vTwo\\left(\\Theta_{2,0}(k+j,k)\\right)\n=\nk-j+\\vTwo\\binom kj.\n}\n\\]\n\\end{histproposition}","\\begin{proof}\nThe lowest degree term of \\(\\Phi_M(z)\\) is \\(Mz\\), and its highest degree term is \\(z^M\\). Multiplication by \\((1+z)^s\\) gives the two edge coefficients. When \\(M=2\\) and \\(s=0\\),\n\\[\n\\Phi_2(z)^k\n=\n(2z+z^2)^k\n=\nz^k(2+z)^k.\n\\]\nTaking the coefficient of \\(z^{k+j}\\) gives the binary formula.\n\\end{proof}","\\begin{histtheorem}[Mahler polyphase transfer]\\label{hist:thm:mahler_polyphase_transfer}\nLet\n\\[\nf(x)=\\sum_{r\\geq0}c_r\\binom xr\n\\]\nbe a convergent Mahler expansion. For\n\\[\ng_s(q)=f(Mq+s),\n\\]\none has\n\\[\n\\boxed{\ng_s(q)=\\sum_{k\\geq0}d_{s,k}\\binom qk,}\n\\]\nwhere\n\\[\n\\boxed{\nd_{s,k}\n=\n\\sum_{r=k}^{Mk+s}\nc_r\\Theta_{M,s}(r,k).\n}\n\\]\nEvery refined coefficient uses a finite window of the original Mahler coefficients.\n\\end{histtheorem}","\\begin{proof}\nUse\n\\[\n(1+z)^{Mq+s}\n=\n(1+z)^s\\left(1+\\Phi_M(z)\\right)^q\n=\n\\sum_{k\\geq0}\\binom qk(1+z)^s\\Phi_M(z)^k.\n\\]\nTaking the coefficient of \\(z^r\\) gives the affine expansion of \\(\\binom{Mq+s}{r}\\). Since \\(c_r\\to0\\) and every \\(d_{s,k}\\) uses indices \\(r\\geq k\\), the transformed coefficients also tend to zero.\n\\end{proof}","\\begin{histtheorem}[Kernel composition]\\label{hist:thm:polyphase_kernel_composition}\nFor positive integers \\(M,N\\),\n\\[\n\\boxed{\n\\sum_{k\\geq0}\n\\Theta_{M,s}(r,k)\n\\Theta_{N,t}(k,\\ell)\n=\n\\Theta_{MN,Mt+s}(r,\\ell).\n}\n\\]\nThe sum is finite.\n\\end{histtheorem}","\\begin{proof}\nApply \\cref{hist:thm:mahler_polyphase_transfer} first to \\(x=Mq+s\\) and then to \\(q=Ny+t\\). The composite substitution is \\(x=MNy+Mt+s\\). Uniqueness of Mahler coefficients gives the identity.\n\\end{proof}","\\begin{histtheorem}[Polyphase valuation ledger]\\label{hist:thm:polyphase_valuation_ledger}\nFor every refined coefficient,\n\\[\n\\boxed{\n\\vp(d_{s,k})\n\\geq\n\\min_{k\\leq r\\leq Mk+s}\n\\left(\n\\vp(c_r)+\\vp(\\Theta_{M,s}(r,k))\n\\right).\n}\n\\]\nIf \\(\\vp(c_r)\\geq w(r)\\) for a nondecreasing function \\(w\\), then\n\\[\n\\boxed{\\vp(d_{s,k})\\geq w(k).}\n\\]\nThus every monotone Mahler precision ladder survives arbitrary polyphase refinement.\n\\end{histtheorem}","\\begin{proof}\nApply the nonarchimedean triangle inequality to the finite coefficient formula. Since \\(r\\geq k\\), monotonicity gives \\(w(r)\\geq w(k)\\), and the affine kernel is integral.\n\\end{proof}","\\begin{histtheorem}[Prime power sparsification]\\label{hist:thm:prime_power_sparsification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","theorem}[Prime power sparsification]\\label{hist:thm:prime_power_sparsification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","ime power sparsification]\\label{hist:thm:prime_power_sparsification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","ification]\\label{hist:thm:prime_power_sparsification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","bel{hist:thm:prime_power_sparsification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","rime_power_sparsification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","needs no change, but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","but its consequence for transformed Mahler coefficients requires $c_r\\in\\Z_p$ for every $r$. Convergence of a $\\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","(q)=q$ has $d_1=1$. Thus $d_1\\not\\equiv c_2\\pmod2$. The coefficient consequence is read with integral original coefficients.\\par\\medskip\nLet \\(M=p^a\\). Then\n\\[\n\\boxed{\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p,\n}\n\\]\nwhere the binomial coefficient is zero outside its natural range. Consequently,\n\\[\n\\boxed{\nd_{s,k}\n\\equiv\n\\sum_{u=0}^{s}\n\\binom su\nc_{p^ak+u}\n\\pmod p.\n}\n\\]\nIn particular,\n\\[\n\\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{histtheorem}","\\begin{proof}\nIn characteristic \\(p\\),\n\\[\n(1+z)^{p^a}-1\\equiv z^{p^a}.\n\\]\nSubstitute this congruence in the definition of \\(\\Theta_{p^a,s}\\).\n\\end{proof}","\\begin{histtheorem}[Scalar polyphase precision amplification]\\label{hist:thm:scalar_polyphase_precision}\nLet \\(u\\in\\mathbb Z_p^\\times\\), let \\(a,r\\geq0\\), and put\n\\[\nH_a=\n\\begin{cases}\n(p-1)p^a,&p\\text{ odd},","\\begin{histtheorem}[Scalar polyphase precision amplification]\\label{hist:thm:scalar_polyphase_precision}\nLet \\(u\\in\\mathbb Z_p^\\times\\), let \\(a,r\\geq0\\), and put\n\\[\nH_a=\n\\begin{cases}\n(p-1)p^a,&p\\text{ odd},\\\\\n2^{a+1},&p=2.\n\\end{cases}\n\\]\nThen\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}u^n\n=\nu^n\\left(u^{H_a}-1\\right)^r.\n}\n\\]\nFor odd \\(p\\),\n\\[\n\\boxed{\n\\vp\\left(\\Delta_{H_a}^{\\,r}u^n\\right)\n\\geq(a+1)r.\n}\n\\]\nFor \\(p=2\\) and odd \\(u\\),\n\\[\n\\boxed{\n\\vTwo\\left(\\Delta_{H_a}^{\\,r}u^n\\right)\n\\geq(a+3)r.\n}\n\\]\nEquivalently, every additional prime power digit increases the certified Mahler slope by one.\n\\end{histtheorem}","2^{a+1},&p=2.\n\\end{cases}\n\\]\nThen\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}u^n\n=\nu^n\\left(u^{H_a}-1\\right)^r.\n}\n\\]\nFor odd \\(p\\),\n\\[\n\\boxed{\n\\vp\\left(\\Delta_{H_a}^{\\,r}u^n\\right)\n\\geq(a+1)r.\n}\n\\]\nFor \\(p=2\\) and odd \\(u\\),\n\\[\n\\boxed{\n\\vTwo\\left(\\Delta_{H_a}^{\\,r}u^n\\right)\n\\geq(a+3)r.\n}\n\\]\nEquivalently, every additional prime power digit increases the certified Mahler slope by one.\n\\end{histtheorem}","\\begin{proof}\nOne difference with step \\(H_a\\) multiplies the unit mode by \\(u^{H_a}-1\\). Iterate. The valuation estimates are those of \\cref{hist:cor:basic_local_difference_gains}.\n\\end{proof}","\\begin{histlemma}[Matrix power amplification]\\label{hist:lem:matrix_power_amplification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$. At $L=0$ the standalone claim is false: for $p=3$, $d=1$ and $X=2=1+B$, one has $X^3-1=7\\notin3\\Z_3$. With $L\\ge1$, every higher binomial term in the retained proof has the required valuation, including $p=2$.\\par\\medskip\nIf\n\\[\nX=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\), then for every \\(a\\geq0\\) there is \\(B_a\\in M_d(\\Zp)\\) such that\n\\[\n\\boxed{X^{p^a}=I+p^{L+a}B_a.}\n\\]\n\\end{histlemma}","istlemma}[Matrix power amplification]\\label{hist:lem:matrix_power_amplification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$. At $L=0$ the standalone claim is false: for $p=3$, $d=1$ and $X=2=1+B$, one has $X^3-1=7\\notin3\\Z_3$. With $L\\ge1$, every higher binomial term in the retained proof has the required valuation, including $p=2$.\\par\\medskip\nIf\n\\[\nX=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\), then for every \\(a\\geq0\\) there is \\(B_a\\in M_d(\\Zp)\\) such that\n\\[\n\\boxed{X^{p^a}=I+p^{L+a}B_a.}\n\\]\n\\end{histlemma}","on]\\label{hist:lem:matrix_power_amplification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$. At $L=0$ the standalone claim is false: for $p=3$, $d=1$ and $X=2=1+B$, one has $X^3-1=7\\notin3\\Z_3$. With $L\\ge1$, every higher binomial term in the retained proof has the required valuation, including $p=2$.\\par\\medskip\nIf\n\\[\nX=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\), then for every \\(a\\geq0\\) there is \\(B_a\\in M_d(\\Zp)\\) such that\n\\[\n\\boxed{X^{p^a}=I+p^{L+a}B_a.}\n\\]\n\\end{histlemma}","atrix_power_amplification}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$. At $L=0$ the standalone claim is false: for $p=3$, $d=1$ and $X=2=1+B$, one has $X^3-1=7\\notin3\\Z_3$. With $L\\ge1$, every higher binomial term in the retained proof has the required valuation, including $p=2$.\\par\\medskip\nIf\n\\[\nX=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\), then for every \\(a\\geq0\\) there is \\(B_a\\in M_d(\\Zp)\\) such that\n\\[\n\\boxed{X^{p^a}=I+p^{L+a}B_a.}\n\\]\n\\end{histlemma}","r\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$. At $L=0$ the standalone claim is false: for $p=3$, $d=1$ and $X=2=1+B$, one has $X^3-1=7\\notin3\\Z_3$. With $L\\ge1$, every higher binomial term in the retained proof has the required valuation, including $p=2$.\\par\\medskip\nIf\n\\[\nX=I+p^LB\n\\]\nwith \\(B\\in M_d(\\Zp)\\), then for every \\(a\\geq0\\) there is \\(B_a\\in M_d(\\Zp)\\) such that\n\\[\n\\boxed{X^{p^a}=I+p^{L+a}B_a.}\n\\]\n\\end{histlemma}","\\begin{proof}\nInduct on \\(a\\). Expand the \\(p\\)th power of \\(I+p^{L+a}B_a\\). The linear term has valuation at least \\(L+a+1\\), and every higher term has at least that valuation.\n\\end{proof}","\\begin{histtheorem}[Matrix polyphase precision amplification]\\label{hist:thm:matrix_polyphase_precision}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$, $C,B\\in M_d(\\Z_p)$, $v_0,w\\in\\Z_p^d$, $H>0$, and $a,j,n,r\\in\\N$. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","thm:matrix_polyphase_precision}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$, $C,B\\in M_d(\\Z_p)$, $v_0,w\\in\\Z_p^d$, $H>0$, and $a,j,n,r\\in\\N$. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","recision}\n\\par\\noindent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$, $C,B\\in M_d(\\Z_p)$, $v_0,w\\in\\Z_p^d$, $H>0$, and $a,j,n,r\\in\\N$. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","dent\\textbf{Editorial hypothesis correction.} Require $L\\ge1$, $C,B\\in M_d(\\Z_p)$, $v_0,w\\in\\Z_p^d$, $H>0$, and $a,j,n,r\\in\\N$. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","ire $L\\ge1$, $C,B\\in M_d(\\Z_p)$, $v_0,w\\in\\Z_p^d$, $H>0$, and $a,j,n,r\\in\\N$. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","$, $v_0,w\\in\\Z_p^d$, $H>0$, and $a,j,n,r\\in\\N$. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","ypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\\par\\medskip\nAssume\n\\[\nC^H=I+p^LB.\n\\]\nPut \\(H_a=Hp^a\\). Then\n\\[\nC^{H_a}=I+p^{L+a}B_a\n\\]\nfor an integral matrix \\(B_a\\). Every refined packet\n\\[\nb_{a,j}(x)=a_{H_ax+j}\n\\]\nhas Mahler coefficients satisfying\n\\[\n\\boxed{\\vp(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Delta_{H_a}^{\\,r}a_n\n\\equiv0\n\\pmod {p^{(L+a)r}}.\n}\n\\]\n\\end{histtheorem}","\\begin{histexample}[Refined dyadic Fibonacci ladder]\\label{hist:ex:refined_fibonacci_ladder}\nFor the Fibonacci companion matrix\n\\[\nC=\n\\begin{pmatrix}\n1&1\\\\\n1&0\n\\end{pmatrix},\n\\]\none has\n\\[\nC^{12}\n=I+8\n\\begin{pmatrix}\n29&18","1&0\n\\end{pmatrix},\n\\]\none has\n\\[\nC^{12}\n=I+8\n\\begin{pmatrix}\n29&18\\\\\n18&11\n\\end{pmatrix}.\n\\]\nTherefore\n\\[\n\\boxed{\n\\Delta_{12\\cdot2^a}^{\\,r}F_n\n\\equiv0\n\\pmod {2^{(a+3)r}}.\n}\n\\]\nEvery dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.\n\\end{histexample}","18&11\n\\end{pmatrix}.\n\\]\nTherefore\n\\[\n\\boxed{\n\\Delta_{12\\cdot2^a}^{\\,r}F_n\n\\equiv0\n\\pmod {2^{(a+3)r}}.\n}\n\\]\nEvery dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.\n\\end{histexample}","For odd \\(p\\), write\n\\[\n\\boxed{\nn=j+(p-1)(s+p^aq),\n}\n\\]\nwith \\(0\\leq j<p-1\\) and \\(0\\leq s<p^a\\). The address \\(j\\) records the finite character. The address \\(s\\) records the principal unit digit. The variable \\(q\\) carries the remaining Mahler expansion.","At the dyadic prime, write\n\\[\n\\boxed{\nn=j+2(s+2^aq),\n}\n\\]\nwith \\(j\\in\\{0,1\\}\\) and \\(0\\leq s<2^a\\). The sign coordinate replaces the odd prime character coordinate.","\\begin{histproposition}[Local phase lattice]\\label{hist:prop:local_phase_lattice}\nFor a unit mode \\(u^n\\), the odd prime coordinate gives\n\\[\n\\boxed{\nu^n\n=\nu^{j+(p-1)s}\n\\left(u^{(p-1)p^a}\\right)^q.\n}\n\\]\nThe dyadic coordinate gives\n\\[\n\\boxed{\nu^n\n=\nu^{j+2s}\n\\left(u^{2^{a+1}}\\right)^q.\n}\n\\]\nThus finite character phase, local digit address, and Mahler depth are separate coordinates.\n\\end{histproposition}","\\begin{proof}\nSubstitute the two displayed decompositions of \\(n\\).\n\\end{proof}","\\begin{histcorollary}[Polyphase refinement of the Euler branches]\\label{hist:cor:euler_polyphase_refinement}\nThe analytic branches in \\cref{hist:thm:odd_analytic_branch,hist:thm:direct_dyadic_interpolation} admit exact refinements under \\(x=p^aq+s\\). Their refined Mahler coefficients are obtained from \\(\\Theta_{p^a,s}\\). The original monotone valuation ladders are preserved, and unit mode or invertible matrix representations gain one certified digit per refinement level.\n\\end{histcorollary}","Let \\(p\\) be a prime. Let\n\\[\nM\\in M_{r,d}(\\Zp),\n\\qquad\nb\\in\\Zp^r.\n\\]\nSuppose that \\(x_L\\in\\Zp^d\\) satisfies\n\\[\nMx_L\\equiv b\\pmod {p^L}\n\\]\nfor some \\(L\\geq1\\). Define the reduced residual\n\\[\n\\rho_L(x_L)\n=\n\\frac{Mx_L-b}{p^L}\n\\pmod p\n\\in\\FF_p^r.\n\\]\nWrite \\(\\overline M\\) for the reduction of \\(M\\) modulo \\(p\\).","\\begin{histdefinition}[Linear lift obstruction]\\label{hist:def:linear_lift_obstruction}\nThe linear obstruction at depth \\(L\\) is\n\\[\n\\boxed{\n\\operatorname{Ob}^{\\mathrm{lin}}_L(x_L)\n=\n[\\rho_L(x_L)]\n\\in\n\\coker(\\overline M).\n}\n\\]\n\\end{histdefinition}","\\begin{histtheorem}[One level linear lifting]\\label{hist:thm:linear_lifting}\nA class represented by \\(x_L\\) lifts to a solution modulo \\(p^{L+1}\\) if and only if\n\\[\n\\boxed{\n\\operatorname{Ob}^{\\mathrm{lin}}_L(x_L)=0.\n}\n\\]\nMore precisely, a correction \\(y\\in\\FF_p^d\\) gives a lift\n\\[\nx_{L+1}=x_L+p^Ly\n\\pmod {p^{L+1}}\n\\]\nif and only if\n\\[\n\\boxed{\n\\overline My=-\\rho_L(x_L).\n}\n\\]\nWhen one correction exists, the set of all correction classes is a coset of \\(\\ker\\overline M\\).\n\\end{histtheorem}","st:thm:linear_lifting}\nA class represented by \\(x_L\\) lifts to a solution modulo \\(p^{L+1}\\) if and only if\n\\[\n\\boxed{\n\\operatorname{Ob}^{\\mathrm{lin}}_L(x_L)=0.\n}\n\\]\nMore precisely, a correction \\(y\\in\\FF_p^d\\) gives a lift\n\\[\nx_{L+1}=x_L+p^Ly\n\\pmod {p^{L+1}}\n\\]\nif and only if\n\\[\n\\boxed{\n\\overline My=-\\rho_L(x_L).\n}\n\\]\nWhen one correction exists, the set of all correction classes is a coset of \\(\\ker\\overline M\\).\n\\end{histtheorem}","\\begin{proof}\nModulo \\(p^{L+1}\\),\n\\[\nM(x_L+p^Ly)-b\n=\nMx_L-b+p^LMy.\n\\]\nAfter division by \\(p^L\\) and reduction modulo \\(p\\), the lift condition becomes\n\\[\n\\rho_L(x_L)+\\overline My=0.\n\\]\nThis equation is solvable exactly when the residual lies in the image of \\(\\overline M\\). If \\(y_0\\) is one solution, every solution is \\(y_0+z\\) with \\(z\\in\\ker\\overline M\\).\n\\end{proof}","\\begin{histproposition}[Representative independence]\\label{hist:prop:linear_obstruction_independence}\nThe class \\(\\operatorname{Ob}^{\\mathrm{lin}}_L(x_L)\\) depends only on the solution class of \\(x_L\\) modulo \\(p^L\\).\n\\end{histproposition}","\\begin{proof}\nReplace \\(x_L\\) by \\(x_L+p^Lz\\). The reduced residual changes by \\(\\overline Mz\\). Its class in the cokernel is unchanged.\n\\end{proof}","\\begin{histcorollary}[Exact number of linear children]\\label{hist:cor:linear_child_count}\nIf \\(\\rho=\\rank_{\\FF_p}\\overline M\\), then a solvable depth \\(L\\) class has exactly\n\\[\n\\boxed{\np^{d-\\rho}\n}\n\\]\nchildren modulo \\(p^{L+1}\\).\n\\end{histcorollary}","\\begin{proof}\nThe correction set is a coset of a vector space of dimension \\(d-\\rho\\).\n\\end{proof}","\\begin{histexample}[A visible linear obstruction]\\label{hist:ex:linear_lift_obstruction}\nTake \\(p=3\\), \\(L=1\\), and\n\\[\nM=\n\\begin{pmatrix}\n1&1","\\begin{histexample}[A visible linear obstruction]\\label{hist:ex:linear_lift_obstruction}\nTake \\(p=3\\), \\(L=1\\), and\n\\[\nM=\n\\begin{pmatrix}\n1&1\\\\\n0&0\n\\end{pmatrix}.\n\\]\nThe class \\(x_1=(0,0)\\) solves\n\\[\nMx_1\\equiv b\\pmod3\n\\]\nfor both\n\\[\nb_1=(-3,0)\n\\qquad\\text{and}\\qquad\nb_2=(0,-3).\n\\]\nFor \\(b_1\\), the reduced residual is \\((1,0)\\), which belongs to the image of \\(\\overline M\\). There are three correction classes, given by\n\\[\ny_1+y_2=2\n\\pmod3.\n\\]\nFor \\(b_2\\), the residual is \\((0,1)\\), whose cokernel class is nonzero. No lift modulo nine exists because the second row of \\(M\\) remains zero.\n\\end{histexample}","0&0\n\\end{pmatrix}.\n\\]\nThe class \\(x_1=(0,0)\\) solves\n\\[\nMx_1\\equiv b\\pmod3\n\\]\nfor both\n\\[\nb_1=(-3,0)\n\\qquad\\text{and}\\qquad\nb_2=(0,-3).\n\\]\nFor \\(b_1\\), the reduced residual is \\((1,0)\\), which belongs to the image of \\(\\overline M\\). There are three correction classes, given by\n\\[\ny_1+y_2=2\n\\pmod3.\n\\]\nFor \\(b_2\\), the residual is \\((0,1)\\), whose cokernel class is nonzero. No lift modulo nine exists because the second row of \\(M\\) remains zero.\n\\end{histexample}","\\(x_1=(0,0)\\) solves\n\\[\nMx_1\\equiv b\\pmod3\n\\]\nfor both\n\\[\nb_1=(-3,0)\n\\qquad\\text{and}\\qquad\nb_2=(0,-3).\n\\]\nFor \\(b_1\\), the reduced residual is \\((1,0)\\), which belongs to the image of \\(\\overline M\\). There are three correction classes, given by\n\\[\ny_1+y_2=2\n\\pmod3.\n\\]\nFor \\(b_2\\), the residual is \\((0,1)\\), whose cokernel class is nonzero. No lift modulo nine exists because the second row of \\(M\\) remains zero.\n\\end{histexample}","Let\n\\[\nF=(F_1,\\ldots,F_r):\\Zp^d\\longrightarrow\\Zp^r\n\\]\nbe a polynomial map.\\par\\noindent\\textbf{Editorial coefficient clarification.} Here polynomial means $F_i\\in\\Z_p[X_1,\\ldots,X_d]$, not merely an integer-valued polynomial with coefficients in $\\Q_p$. This makes the Jacobian and Taylor remainder integral and their reduction well typed.\\par Suppose\n\\[\nF(x_L)\\in p^L\\Zp^r.\n\\]\nDefine\n\\[\n\\rho_L^F(x_L)\n=\n\\frac{F(x_L)}{p^L}\n\\pmod p.\n\\]\nLet \\(J_F(x_L)\\) denote the Jacobian matrix and write \\(\\overline J_F(x_L)\\) for its reduction modulo \\(p\\).","rrow\\Zp^r\n\\]\nbe a polynomial map.\\par\\noindent\\textbf{Editorial coefficient clarification.} Here polynomial means $F_i\\in\\Z_p[X_1,\\ldots,X_d]$, not merely an integer-valued polynomial with coefficients in $\\Q_p$. This makes the Jacobian and Taylor remainder integral and their reduction well typed.\\par Suppose\n\\[\nF(x_L)\\in p^L\\Zp^r.\n\\]\nDefine\n\\[\n\\rho_L^F(x_L)\n=\n\\frac{F(x_L)}{p^L}\n\\pmod p.\n\\]\nLet \\(J_F(x_L)\\) denote the Jacobian matrix and write \\(\\overline J_F(x_L)\\) for its reduction modulo \\(p\\).","tbf{Editorial coefficient clarification.} Here polynomial means $F_i\\in\\Z_p[X_1,\\ldots,X_d]$, not merely an integer-valued polynomial with coefficients in $\\Q_p$. This makes the Jacobian and Taylor remainder integral and their reduction well typed.\\par Suppose\n\\[\nF(x_L)\\in p^L\\Zp^r.\n\\]\nDefine\n\\[\n\\rho_L^F(x_L)\n=\n\\frac{F(x_L)}{p^L}\n\\pmod p.\n\\]\nLet \\(J_F(x_L)\\) denote the Jacobian matrix and write \\(\\overline J_F(x_L)\\) for its reduction modulo \\(p\\).","olynomial means $F_i\\in\\Z_p[X_1,\\ldots,X_d]$, not merely an integer-valued polynomial with coefficients in $\\Q_p$. This makes the Jacobian and Taylor remainder integral and their reduction well typed.\\par Suppose\n\\[\nF(x_L)\\in p^L\\Zp^r.\n\\]\nDefine\n\\[\n\\rho_L^F(x_L)\n=\n\\frac{F(x_L)}{p^L}\n\\pmod p.\n\\]\nLet \\(J_F(x_L)\\) denote the Jacobian matrix and write \\(\\overline J_F(x_L)\\) for its reduction modulo \\(p\\).","\\begin{histdefinition}[Polynomial lift obstruction]\\label{hist:def:polynomial_lift_obstruction}\nThe polynomial obstruction at depth \\(L\\) is\n\\[\n\\boxed{\n\\operatorname{Ob}^{F}_L(x_L)\n=\n[\\rho_L^F(x_L)]\n\\in\n\\coker\\bigl(\\overline J_F(x_L)\\bigr).\n}\n\\]\n\\end{histdefinition}","\\begin{histtheorem}[One level polynomial lifting]\\label{hist:thm:polynomial_lifting}\nLet \\(L\\geq1\\). A class represented by \\(x_L\\) lifts to a zero of \\(F\\) modulo \\(p^{L+1}\\) if and only if\n\\[\n\\boxed{\n\\operatorname{Ob}^{F}_L(x_L)=0.\n}\n\\]\nA correction \\(y\\in\\FF_p^d\\) gives a lift if and only if\n\\[\n\\boxed{\n\\overline J_F(x_L)y\n=\n-\\rho_L^F(x_L).\n}\n\\]\nWhen a correction exists, the correction classes form a coset of\n\\[\n\\ker\\overline J_F(x_L).\n\\]\n\\end{histtheorem}","m}[One level polynomial lifting]\\label{hist:thm:polynomial_lifting}\nLet \\(L\\geq1\\). A class represented by \\(x_L\\) lifts to a zero of \\(F\\) modulo \\(p^{L+1}\\) if and only if\n\\[\n\\boxed{\n\\operatorname{Ob}^{F}_L(x_L)=0.\n}\n\\]\nA correction \\(y\\in\\FF_p^d\\) gives a lift if and only if\n\\[\n\\boxed{\n\\overline J_F(x_L)y\n=\n-\\rho_L^F(x_L).\n}\n\\]\nWhen a correction exists, the correction classes form a coset of\n\\[\n\\ker\\overline J_F(x_L).\n\\]\n\\end{histtheorem}","\\begin{proof}\nThe polynomial Taylor expansion gives\n\\[\nF(x_L+p^Ly)\n\\equiv\nF(x_L)+p^LJ_F(x_L)y\n\\pmod {p^{L+1}}.\n\\]\nEvery term of degree at least two in the increment contains \\(p^{2L}\\), which is divisible by \\(p^{L+1}\\). Divide by \\(p^L\\), reduce modulo \\(p\\), and apply the same cokernel criterion as in \\cref{hist:thm:linear_lifting}.\n\\end{proof}","\\begin{histproposition}[Polynomial representative independence]\\label{hist:prop:polynomial_obstruction_independence}\nThe class \\(\\operatorname{Ob}^{F}_L(x_L)\\) depends only on the class of \\(x_L\\) modulo \\(p^L\\).\n\\end{histproposition}","\\begin{proof}\nIf \\(x_L\\) is replaced by \\(x_L+p^Lz\\), Taylor expansion shows that the reduced residual changes by an element of the image of \\(\\overline J_F(x_L)\\). The reduced Jacobian itself is unchanged because \\(L\\geq1\\).\n\\end{proof}","\\begin{histcorollary}[Simple root lifting]\\label{hist:cor:simple_root_lifting}\nAssume \\(r=d\\) and\n\\[\n\\det J_F(x_L)\\not\\equiv0\\pmod p.\n\\]\nThen the depth \\(L\\) solution has exactly one child modulo \\(p^{L+1}\\). Iterating gives a unique \\(p\\) adic zero above the initial class.\n\\end{histcorollary}","\\begin{proof}\nThe reduced Jacobian is invertible, so every residual has one correction. The resulting compatible sequence is Cauchy in \\(\\Zp^d\\), and continuity of \\(F\\) gives a zero in the limit. Uniqueness holds at every depth.\n\\end{proof}","\\begin{histexample}[A singular polynomial obstruction]\\label{hist:ex:singular_polynomial_obstruction}\nLet \\(p=3\\) and consider the class \\(x_1=0\\) modulo three. For\n\\[\nF_0(x)=x^2,\n\\]\nthe residual and the reduced derivative both vanish. Every correction \\(y\\in\\FF_3\\) gives\n\\[\nF_0(3y)\\equiv0\\pmod9,\n\\]\nso the class has three children. For\n\\[\nF_1(x)=x^2+3,\n\\]\nthe reduced derivative is still zero, but the reduced residual is one. Its class in the cokernel is nonzero, so no lift modulo nine exists.\n\\end{histexample}","\\begin{histtheorem}[Packet decomposition of a lift obstruction]\\label{hist:thm:packet_lift_obstruction}\nSuppose\n\\[\n\\FF_p^d\n=\n\\bigoplus_{a\\in A}U_a,\n\\qquad\n\\FF_p^r\n=\n\\bigoplus_{a\\in A}V_a,\n\\]\nand suppose a linear correction map \\(D:\\FF_p^d\\to\\FF_p^r\\) satisfies\n\\[\nD(U_a)\\subseteq V_a\n\\]\nfor every \\(a\\). Write \\(D_a=D|_{U_a}\\). Then\n\\[\n\\boxed{\n\\coker D\n\\cong\n\\bigoplus_{a\\in A}\\coker D_a.\n}\n\\]\nA residual lifts if and only if every packet obstruction vanishes.\n\\end{histtheorem}","\\begin{proof}\nThe image of \\(D\\) is the direct sum of the images of the maps \\(D_a\\). Taking the quotient of the codomain packet decomposition gives the stated cokernel decomposition.\n\\end{proof}","Let \\(R\\) be a commutative ring. Let\n\\[\nv_{n+1}=Cv_n,\n\\qquad\nC\\in M_d(R).\n\\]\nFor a positive integer \\(N\\) and an address \\(j\\in\\{0,\\ldots,N-1\\}\\), define the residue state packet\n\\[\nu_k^{(j)}=v_{Nk+j}.\n\\]","\\begin{histtheorem}[Companion jump recurrence]\\label{hist:thm:companion_jump}\nEvery residue state packet satisfies\n\\[\n\\boxed{\nu_{k+1}^{(j)}=C^Nu_k^{(j)}.}\n\\]\nMore explicitly,\n\\[\n\\boxed{\nu_k^{(j)}=C^j(C^N)^kv_0.}\n\\]\n\\end{histtheorem}","\\begin{proof}\nThe state relation gives \\(v_n=C^nv_0\\). Substitute \\(n=Nk+j\\).\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/recurrence_packets.pdf}\n\\caption{New reconstruction. Fibonacci residues $F_{3k+j}$, with $F_0=0$ and $F_1=1$. Each packet evolves by the same jump matrix \\(C^3\\) and has a different initial address.}\n\\label{hist:fig:recurrence_packets}\n\\end{figure}","We use the resultant convention\n\\[\n\\Res_y(f,g)\n=\n\\prod_{f(\\alpha)=0}g(\\alpha)\n\\]\nwhen \\(f\\) is monic, with roots counted with multiplicity.","\\begin{histtheorem}[Characteristic polynomial of the jump]\\label{hist:thm:jump_characteristic}\nLet\n\\[\n\\chi_C(y)=\\det(yI-C).\n\\]\nThen\n\\[\n\\boxed{\n\\chi_{C^N}(x)\n=\n\\Res_y\\bigl(\\chi_C(y),x-y^N\\bigr).\n}\n\\]\nThe identity holds over the coefficient ring after interpreting the resultant universally.\n\\end{histtheorem}","\\begin{proof}\nAfter passage to an algebraic closure, let \\(\\alpha_1,\\ldots,\\alpha_d\\) be the eigenvalues of \\(C\\), counted with algebraic multiplicity. The resultant is\n\\[\n\\prod_{i=1}^{d}(x-\\alpha_i^N),\n\\]\nwhich is the characteristic polynomial of \\(C^N\\). Both sides are polynomial expressions with integer coefficients in the entries of \\(C\\), so the identity descends to every commutative coefficient ring.\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/recurrence_root_power.pdf}\n\\caption{New reconstruction. Passing to every \\(N\\)th term raises each characteristic root to its \\(N\\)th power.}\n\\label{hist:fig:recurrence_root_power}\n\\end{figure}","\\begin{histtheorem}[Residue subsequences of a second order recurrence]\\label{hist:thm:second_order_jump}\nSuppose\n\\[\nx_{n+2}=Px_{n+1}-Qx_n\n\\]\nover a commutative ring. Define \\(V_0=2\\), \\(V_1=P\\), and\n\\[\nV_{n+2}=PV_{n+1}-QV_n.\n\\]\nFor every positive integer \\(N\\) and every address \\(j\\), the subsequence\n\\[\nu_k=x_{Nk+j}\n\\]\nsatisfies\n\\[\n\\boxed{\nu_{k+2}=V_Nu_{k+1}-Q^Nu_k.}\n\\]\n\\end{histtheorem}","\\begin{proof}\nThe companion matrix\n\\[\nC=\n\\begin{pmatrix}\nP&-Q\\\\\n1&0\n\\end{pmatrix}\n\\]\nhas trace \\(P\\) and determinant \\(Q\\). The trace of \\(C^N\\) is \\(V_N\\), while its determinant is \\(Q^N\\). Cayley Hamilton applied to \\(C^N\\) gives\n\\[\n(C^N)^2-V_NC^N+Q^NI=0.\n\\]\nApply this identity to the residue state packet and take its scalar coordinate.\n\\end{proof}","1&0\n\\end{pmatrix}\n\\]\nhas trace \\(P\\) and determinant \\(Q\\). The trace of \\(C^N\\) is \\(V_N\\), while its determinant is \\(Q^N\\). Cayley Hamilton applied to \\(C^N\\) gives\n\\[\n(C^N)^2-V_NC^N+Q^NI=0.\n\\]\nApply this identity to the residue state packet and take its scalar coordinate.\n\\end{proof}","\\begin{histcorollary}[Fibonacci residue packets]\\label{hist:cor:fibonacci_packets}\nLet \\(F_0=0\\), \\(F_1=1\\), and \\(F_{n+2}=F_{n+1}+F_n\\). Let \\(L_N\\) denote the Lucas number. For every \\(N\\geq1\\) and every address \\(j\\),\n\\[\n\\boxed{\nF_{N(k+2)+j}\n=\nL_NF_{N(k+1)+j}\n-(-1)^NF_{Nk+j}.\n}\n\\]\n\\end{histcorollary}","\\begin{proof}\nApply \\cref{hist:thm:second_order_jump} with \\(P=1\\) and \\(Q=-1\\). Then \\(V_N=L_N\\).\n\\end{proof}","\\begin{histtheorem}[Periodicity of an invertible recurrence modulo a prime power]\\label{hist:thm:recurrence_periodicity}\nLet\n\\[\nC\\in M_d(\\ZZ),\n\\qquad\np\\nmid\\det C.\n\\]\nFor every \\(L\\geq1\\), every state orbit modulo \\(p^L\\) is purely periodic. Its period divides the order of \\(C\\) in\n\\[\n\\operatorname{GL}_d(\\ZZ/p^L\\ZZ).\n\\]\nIf \\(p\\mid\\det C\\), every orbit is still eventually periodic.\n\\end{histtheorem}","\\begin{proof}\nWhen \\(p\\nmid\\det C\\), multiplication by \\(C\\) is a permutation of the finite state set \\((\\ZZ/p^L\\ZZ)^d\\). Every orbit of a permutation is periodic, and applying the group order relation for \\(C\\) gives the divisibility statement. Without invertibility, repeated application of a map on a finite set eventually repeats a state, after which the orbit is periodic.\n\\end{proof}","\\begin{histcorollary}[Residue packet periods]\\label{hist:cor:residue_packet_periods}\nUnder the invertibility hypothesis of \\cref{hist:thm:recurrence_periodicity}, every residue packet with jump \\(N\\) has period dividing the order of \\(C^N\\) modulo \\(p^L\\).\n\\end{histcorollary}","\\begin{histtheorem}[Base change of an integral recurrence]\\label{hist:thm:integral_recurrence_spine}\nLet\n\\[\nC\\in M_d(\\ZZ),\n\\qquad\nv_0\\in\\ZZ^d.\n\\]\nThe integral orbit\n\\[\nv_n=C^nv_0\n\\]\ndetermines, by base change, compatible orbits in every ring \\(R\\) equipped with a homomorphism \\(\\ZZ\\to R\\). In particular, the reductions modulo every positive integer and the orbits in every \\(\\Zp^d\\) share the same companion matrix, characteristic polynomial, and integer recurrence relations.\n\\end{histtheorem}","\\begin{proof}\nEvery coordinate of \\(C^nv_0\\) is an integer polynomial in the entries of \\(C\\) and \\(v_0\\). Applying a ring homomorphism commutes with matrix multiplication and preserves every polynomial relation with integer coefficients.\n\\end{proof}","\\begin{histdefinition}[Residue projector]\\label{hist:def:residue_projector}\nFor a sequence \\(a=(a_n)_{n\\geq0}\\), a positive integer \\(N\\), and an address \\(r\\) modulo \\(N\\), define\n\\[\n(P_{N,r}a)_n\n=\n\\begin{cases}\na_n,&n\\equiv r\\pmod N,","\\begin{histdefinition}[Residue projector]\\label{hist:def:residue_projector}\nFor a sequence \\(a=(a_n)_{n\\geq0}\\), a positive integer \\(N\\), and an address \\(r\\) modulo \\(N\\), define\n\\[\n(P_{N,r}a)_n\n=\n\\begin{cases}\na_n,&n\\equiv r\\pmod N,\\\\\n0,&n\\not\\equiv r\\pmod N.\n\\end{cases}\n\\]\n\\end{histdefinition}","\\begin{histproposition}[Complete residue packet system]\\label{hist:prop:residue_packet_system}\nFor fixed \\(N\\), the projectors \\(P_{N,r}\\) satisfy\n\\[\nP_{N,r}P_{N,s}\n=\n\\delta_{rs}P_{N,r}\n\\]\nand\n\\[\n\\boxed{\n\\sum_{r\\bmod N}P_{N,r}=I.\n}\n\\]\n\\end{histproposition}","\\begin{proof}\nEvery nonnegative integer belongs to exactly one residue class modulo \\(N\\).\n\\end{proof}","\\begin{histtheorem}[Coprime packet factorization]\\label{hist:thm:crt_packet_factorization}\nLet \\(A\\) and \\(B\\) be coprime positive integers. Let \\(r_A\\) be an address modulo \\(A\\) and \\(r_B\\) an address modulo \\(B\\). Let \\(r\\) be the unique address modulo \\(AB\\) satisfying\n\\[\nr\\equiv r_A\\pmod A,\n\\qquad\nr\\equiv r_B\\pmod B.\n\\]\nThen\n\\[\n\\boxed{\nP_{AB,r}\n=\nP_{A,r_A}P_{B,r_B}\n=\nP_{B,r_B}P_{A,r_A}.\n}\n\\]\n\\end{histtheorem}","\\begin{proof}\nThe product retains exactly the indices satisfying both congruences. By the Chinese remainder theorem, those indices form one residue class modulo \\(AB\\) \\cite{IrelandRosen1990}.\n\\end{proof}","\\begin{histcorollary}[Several coprime moduli]\\label{hist:cor:many_crt_packets}\nLet \\(N_1,\\ldots,N_s\\) be pairwise coprime and put\n\\[\nN=\\prod_{j=1}^{s}N_j.\n\\]\nFor compatible addresses \\(r_j\\),\n\\[\n\\boxed{\nP_{N,r}\n=\n\\prod_{j=1}^{s}P_{N_j,r_j},\n}\n\\]\nwhere \\(r\\) is the unique combined address modulo \\(N\\).\n\\end{histcorollary}","\\begin{histstandardfact}[Finite local reconstruction]\\label{hist:fact:finite_local_reconstruction}\nLet\n\\[\nN=\\prod_{j=1}^{s}p_j^{a_j}\n\\]\nbe the prime power factorization of a positive integer. The map\n\\[\n\\ZZ/N\\ZZ\n\\longrightarrow\n\\prod_{j=1}^{s}\\ZZ/p_j^{a_j}\\ZZ\n\\]\nis a ring isomorphism. Thus a compatible collection of prime power residues reconstructs one residue modulo \\(N\\) \\cite{IrelandRosen1990}.\n\\end{histstandardfact}","\\begin{histproposition}[Dyadic and odd residue refinement]\\label{hist:prop:dyadic_odd_refinement}\nLet \\(N=2^am\\) with \\(m\\) odd. Every residue packet modulo \\(N\\) factors uniquely as the product of one dyadic packet modulo \\(2^a\\) and one odd packet modulo \\(m\\).\n\\end{histproposition}","\\begin{proof}\nThe moduli \\(2^a\\) and \\(m\\) are coprime. Apply \\cref{hist:thm:crt_packet_factorization}.\n\\end{proof}","Let \\(C_{mp,j}(n)\\) be a congruential reciprocal coefficient and let\n\\[\nh=\\frac{p-1}{m}.\n\\]\nThe unit mode calculations suggest bounds for\n\\[\n\\vp\\left(\\Delta_h^sC_{mp,j}(n)\\right).\n\\]\nThe theorems of \\cref{hist:sec:mahler,hist:sec:polyphase} determine the gain before the congruential triangular inverse is applied. They do not yet determine every valuation loss introduced by that inverse. The full family bound therefore remains open.","The residual formula\n\\[\n\\Theta_{p^a,s}(r,k)\n\\equiv\n\\binom{s}{r-p^ak}\n\\pmod p\n\\]\nis exact. A closed formula for\n\\[\n\\vp\\left(\\Theta_{p^a,s}(r,k)\\right)\n\\]\nin terms of base \\(p\\) carries and the three digit addresses is not proved here. Such a formula would sharpen the min plus valuation ledger and could distinguish optimal refinement trees.","The odd and dyadic cases agree at the level of convolution and differ at the level of integral geometry. At odd primes the source must be depleted before it becomes a unit mode interpolation. At the dyadic prime the source extends directly to an analytic integral function. The dyadic kernel has exact weight \\(4r-1\\), giving the optimal uniform modulus \\(2^{4M+3}\\) after depth \\(M\\).","\\par\\smallskip\\noindent\\textbf{Current status.} This dated statement is superseded for the stated divisor-branch family by Theorem~\\ref{thm:arith-divisor-base}, with the $j!$ normalization and factorial valuation loss explicitly retained. It makes no claim for unrestricted families.\\par","Let \\[\nF=(F_1,\\ldots,F_e)\\in\\mathbb Z[X_1,\\ldots,X_d]^e,\n\\quad q\\ge1,\\quad L\\ge1,\n\\quad x\\in\\mathbb Z^d,\n\\quad F(x)\\equiv0\\pmod{q^L}.\n\\] Write \\(R_q=\\mathbb Z/q\\mathbb Z\\), let \\(\\rho_q\\) denote\ncoordinatewise reduction, and put \\[\nr_x=F(x)/q^L\\in\\mathbb Z^e,\n\\qquad A_x=DF(x):\\mathbb Z^d\\longrightarrow\\mathbb Z^e,\n\\qquad J_x=\\rho_q\\circ A_x.\n\\] Here \\(A_x\\) is computed from the actual formal partial derivatives\nof the input polynomials. The Lean map \\(J_x\\) has domain\n\\(\\mathbb Z^d\\) and codomain \\(R_q^e\\), viewed as integer modules. It\nfactors through reduction to \\(R_q^d\\). The cokernel below is presented\ndirectly using its integer-domain image.","\\quad q\\ge1,\\quad L\\ge1,\n\\quad x\\in\\mathbb Z^d,\n\\quad F(x)\\equiv0\\pmod{q^L}.\n\\] Write \\(R_q=\\mathbb Z/q\\mathbb Z\\), let \\(\\rho_q\\) denote\ncoordinatewise reduction, and put \\[\nr_x=F(x)/q^L\\in\\mathbb Z^e,\n\\qquad A_x=DF(x):\\mathbb Z^d\\longrightarrow\\mathbb Z^e,\n\\qquad J_x=\\rho_q\\circ A_x.\n\\] Here \\(A_x\\) is computed from the actual formal partial derivatives\nof the input polynomials. The Lean map \\(J_x\\) has domain\n\\(\\mathbb Z^d\\) and codomain \\(R_q^e\\), viewed as integer modules. It\nfactors through reduction to \\(R_q^d\\). The cokernel below is presented\ndirectly using its integer-domain image.","L\\ge1,\n\\quad x\\in\\mathbb Z^d,\n\\quad F(x)\\equiv0\\pmod{q^L}.\n\\] Write \\(R_q=\\mathbb Z/q\\mathbb Z\\), let \\(\\rho_q\\) denote\ncoordinatewise reduction, and put \\[\nr_x=F(x)/q^L\\in\\mathbb Z^e,\n\\qquad A_x=DF(x):\\mathbb Z^d\\longrightarrow\\mathbb Z^e,\n\\qquad J_x=\\rho_q\\circ A_x.\n\\] Here \\(A_x\\) is computed from the actual formal partial derivatives\nof the input polynomials. The Lean map \\(J_x\\) has domain\n\\(\\mathbb Z^d\\) and codomain \\(R_q^e\\), viewed as integer modules. It\nfactors through reduction to \\(R_q^d\\). The cokernel below is presented\ndirectly using its integer-domain image.","\\quad F(x)\\equiv0\\pmod{q^L}.\n\\] Write \\(R_q=\\mathbb Z/q\\mathbb Z\\), let \\(\\rho_q\\) denote\ncoordinatewise reduction, and put \\[\nr_x=F(x)/q^L\\in\\mathbb Z^e,\n\\qquad A_x=DF(x):\\mathbb Z^d\\longrightarrow\\mathbb Z^e,\n\\qquad J_x=\\rho_q\\circ A_x.\n\\] Here \\(A_x\\) is computed from the actual formal partial derivatives\nof the input polynomials. The Lean map \\(J_x\\) has domain\n\\(\\mathbb Z^d\\) and codomain \\(R_q^e\\), viewed as integer modules. It\nfactors through reduction to \\(R_q^d\\). The cokernel below is presented\ndirectly using its integer-domain image.","et \\(\\rho_q\\) denote\ncoordinatewise reduction, and put \\[\nr_x=F(x)/q^L\\in\\mathbb Z^e,\n\\qquad A_x=DF(x):\\mathbb Z^d\\longrightarrow\\mathbb Z^e,\n\\qquad J_x=\\rho_q\\circ A_x.\n\\] Here \\(A_x\\) is computed from the actual formal partial derivatives\nof the input polynomials. The Lean map \\(J_x\\) has domain\n\\(\\mathbb Z^d\\) and codomain \\(R_q^e\\), viewed as integer modules. It\nfactors through reduction to \\(R_q^d\\). The cokernel below is presented\ndirectly using its integer-domain image.","Define the finite obstruction module and its distinguished class by \\[\n\\mathcal O_x=R_q^e/\\operatorname{im}J_x,\n\\qquad o_x=[\\rho_q(r_x)]\\in\\mathcal O_x.\n\\] The checked criterion is \\[\n\\boxed{\n o_x=0\n \\iff\n \\exists y\\in\\mathbb Z^d:\\quad\n y\\equiv x\\pmod{q^L},\\qquad F(y)\\equiv0\\pmod{q^{L+1}}.\n}\n\\] The residual exists by the lower congruence and is unique because\n\\(q>0\\). The quotient is finite since its ambient module \\(R_q^e\\) is\nfinite. The theorem permits zero variables or zero equations as well.","truction module and its distinguished class by \\[\n\\mathcal O_x=R_q^e/\\operatorname{im}J_x,\n\\qquad o_x=[\\rho_q(r_x)]\\in\\mathcal O_x.\n\\] The checked criterion is \\[\n\\boxed{\n o_x=0\n \\iff\n \\exists y\\in\\mathbb Z^d:\\quad\n y\\equiv x\\pmod{q^L},\\qquad F(y)\\equiv0\\pmod{q^{L+1}}.\n}\n\\] The residual exists by the lower congruence and is unique because\n\\(q>0\\). The quotient is finite since its ambient module \\(R_q^e\\) is\nfinite. The theorem permits zero variables or zero equations as well.","For every \\(P\\in\\mathbb Z[X_1,\\ldots,X_d]\\), integer \\(a\\), and\n\\(x,t\\in\\mathbb Z^d\\), Lean proves \\[\n\\boxed{\n a^2\\mid P(x+at)-P(x)-a\\sum_{i=1}^{d}(\\partial_iP)(x)t_i.\n}\n\\] The proof uses polynomial induction: constants, addition, and\nmultiplication by a variable. The product rule for the actual partial\nderivatives controls the multiplication step. It does not assume a\nTaylor identity or a liftability condition.","Taking \\(a=q^L\\), the inequality \\(2L\\ge L+1\\) gives \\[\nF(x+q^Lt)\\equiv q^L(r_x+A_xt)\\pmod{q^{L+1}}.\n\\] Since \\(q^L\\ne0\\) in the integers, cancellation in divisibility\nyields \\[\nF(x+q^Lt)\\equiv0\\pmod{q^{L+1}}\n\\iff \\rho_q(r_x)+J_xt=0.\n\\] Every integer representative of the lower residue has the form\n\\(x+q^Lt\\), and this correction equation has a solution exactly when\n\\(o_x=0\\).","The positive-depth hypothesis is necessary for the general nonlinear\ntheorem. For \\(F(X)=X^2-1\\), \\(q=2\\), \\(L=0\\), \\(x=0\\), the Jacobian is\nzero and the residual class is nonzero, yet \\(y=1\\) is a solution modulo\ntwo in the same residue modulo one. Affine systems retain the previously\nchecked \\(L\\ge0\\) theorem.","Suppose \\(x'=x+q^Ls\\) represents the same lower residue, with exact\nresidual \\(r_{x'}\\). Polynomial evaluation respects coordinatewise\ncongruences, including when applied to each partial derivative. Hence \\[\nJ_{x'}=J_x,\n\\qquad\n\\rho_q(r_{x'})=\\rho_q(r_x)+J_xs.\n\\] The two cokernels therefore have the same ambient module and the same\ndefining submodule, and \\[\n[\\rho_q(r_{x'})]=[\\rho_q(r_x)].\n\\] Lean states this using one explicit common reduced map, avoiding an\nimplicit comparison between differently indexed quotient types. This\nestablishes independence of the integer representative within the fixed\npolynomial presentation.","Changing the polynomial presentation is a separate question. Appending a\nzero equation can change the raw cokernel by adding a copy of \\(R_q\\),\neven though it does not change the solutions.","Let \\(G\\in\\mathbb Z[X_1,\\ldots,X_d]^f\\) be a second fixed system with\nthe same variables and the same lower root \\(x\\). Let\n\\(K_x=\\rho_q\\circ DG(x)\\), and define \\[\n\\mathcal C_x=\\mathbb Z^d/(\\ker J_x+\\ker K_x).\n\\] This quotient is finite: it is a quotient of the finite image of\n\\(J_x\\), or equivalently of the finite image of \\(K_x\\).","Suppose \\(u,v\\in\\mathbb Z^d\\) are separately valid corrections: \\[\nF(x+q^Lu)\\equiv0\\pmod{q^{L+1}},\n\\qquad\nG(x+q^Lv)\\equiv0\\pmod{q^{L+1}}.\n\\] Then \\[\n\\boxed{\n\\begin{gathered}\n[u]=[v]\\text{ in }\\mathcal C_x\n\\iff\n\\exists y\\equiv x\\pmod{q^L}:","Suppose \\(u,v\\in\\mathbb Z^d\\) are separately valid corrections: \\[\nF(x+q^Lu)\\equiv0\\pmod{q^{L+1}},\n\\qquad\nG(x+q^Lv)\\equiv0\\pmod{q^{L+1}}.\n\\] Then \\[\n\\boxed{\n\\begin{gathered}\n[u]=[v]\\text{ in }\\mathcal C_x\n\\iff\n\\exists y\\equiv x\\pmod{q^L}:\\\\\nF(y)\\equiv0\\pmod{q^{L+1}},\\quad G(y)\\equiv0\\pmod{q^{L+1}}.\n\\end{gathered}\n}\n\\] The labels are independent of the choices of valid \\(u\\) and \\(v\\).\nAn unconditional version asks for separately valid corrections with\nequal labels. It does not assume that each block is liftable.","F(y)\\equiv0\\pmod{q^{L+1}},\\quad G(y)\\equiv0\\pmod{q^{L+1}}.\n\\end{gathered}\n}\n\\] The labels are independent of the choices of valid \\(u\\) and \\(v\\).\nAn unconditional version asks for separately valid corrections with\nequal labels. It does not assume that each block is liftable.","Taylor transport turns a common correction \\(t\\) into the simultaneous\nequations \\(J_xt=J_xu\\) and \\(K_xt=K_xv\\). The previously checked\ncommon-quotient theorem gives precisely the displayed equivalence.\nSeparate solvability alone does not imply joint solvability. The\ncontrols include \\(F(X)=X\\), \\(G(X)=X-2\\), starting at \\(x=0\\) modulo\ntwo and lifting to modulus four.","Let \\(p\\) be prime, \\(L\\ge1\\), and \\[\nF\\in\\mathbb Z[X_1,\\ldots,X_d]^e,\n\\qquad x\\in\\mathbb Z^d,\n\\qquad F(x)\\equiv0\\pmod{p^L}.\n\\] Write \\[\nr_x=F(x)/p^L,\\qquad b_x=r_x\\bmod p,\n\\qquad A_x=DF(x)\\bmod p:\\mathbb F_p^d\\longrightarrow\\mathbb F_p^e,\n\\qquad \\rho_x=\\operatorname{rank}_{\\mathbb F_p}A_x.\n\\] The actual lift set is \\[\n\\mathcal L_{x,L}=\n\\left\\{y\\in(\\mathbb Z/p^{L+1}\\mathbb Z)^d:\n y\\equiv x\\pmod{p^L},\\quad F(y)=0\\right\\},\n\\] where polynomial evaluation in the displayed set is modulo\n\\(p^{L+1}\\). Define \\[\n\\mathcal O_x=\\mathbb F_p^e/\\operatorname{im}A_x,\n\\qquad o_x=[b_x].\n\\] Lean proves \\[\n\\boxed{\n\\#\\mathcal L_{x,L}=\n\\begin{cases}\np^{d-\\rho_x},&o_x=0,","Z^d,\n\\qquad F(x)\\equiv0\\pmod{p^L}.\n\\] Write \\[\nr_x=F(x)/p^L,\\qquad b_x=r_x\\bmod p,\n\\qquad A_x=DF(x)\\bmod p:\\mathbb F_p^d\\longrightarrow\\mathbb F_p^e,\n\\qquad \\rho_x=\\operatorname{rank}_{\\mathbb F_p}A_x.\n\\] The actual lift set is \\[\n\\mathcal L_{x,L}=\n\\left\\{y\\in(\\mathbb Z/p^{L+1}\\mathbb Z)^d:\n y\\equiv x\\pmod{p^L},\\quad F(y)=0\\right\\},\n\\] where polynomial evaluation in the displayed set is modulo\n\\(p^{L+1}\\). Define \\[\n\\mathcal O_x=\\mathbb F_p^e/\\operatorname{im}A_x,\n\\qquad o_x=[b_x].\n\\] Lean proves \\[\n\\boxed{\n\\#\\mathcal L_{x,L}=\n\\begin{cases}\np^{d-\\rho_x},&o_x=0,","uiv0\\pmod{p^L}.\n\\] Write \\[\nr_x=F(x)/p^L,\\qquad b_x=r_x\\bmod p,\n\\qquad A_x=DF(x)\\bmod p:\\mathbb F_p^d\\longrightarrow\\mathbb F_p^e,\n\\qquad \\rho_x=\\operatorname{rank}_{\\mathbb F_p}A_x.\n\\] The actual lift set is \\[\n\\mathcal L_{x,L}=\n\\left\\{y\\in(\\mathbb Z/p^{L+1}\\mathbb Z)^d:\n y\\equiv x\\pmod{p^L},\\quad F(y)=0\\right\\},\n\\] where polynomial evaluation in the displayed set is modulo\n\\(p^{L+1}\\). Define \\[\n\\mathcal O_x=\\mathbb F_p^e/\\operatorname{im}A_x,\n\\qquad o_x=[b_x].\n\\] Lean proves \\[\n\\boxed{\n\\#\\mathcal L_{x,L}=\n\\begin{cases}\np^{d-\\rho_x},&o_x=0,","p:\\mathbb F_p^d\\longrightarrow\\mathbb F_p^e,\n\\qquad \\rho_x=\\operatorname{rank}_{\\mathbb F_p}A_x.\n\\] The actual lift set is \\[\n\\mathcal L_{x,L}=\n\\left\\{y\\in(\\mathbb Z/p^{L+1}\\mathbb Z)^d:\n y\\equiv x\\pmod{p^L},\\quad F(y)=0\\right\\},\n\\] where polynomial evaluation in the displayed set is modulo\n\\(p^{L+1}\\). Define \\[\n\\mathcal O_x=\\mathbb F_p^e/\\operatorname{im}A_x,\n\\qquad o_x=[b_x].\n\\] Lean proves \\[\n\\boxed{\n\\#\\mathcal L_{x,L}=\n\\begin{cases}\np^{d-\\rho_x},&o_x=0,\\\\\n0,&o_x\\ne0,\n\\end{cases}\n\\qquad\n\\#\\mathcal O_x=p^{e-\\rho_x}.\n}\n\\] These formulas permit singular Jacobians and zero variables or zero\nequations. Nonsingularity is not assumed. In particular, rank alone\ngives the size of a nonempty lift fiber. The residual obstruction\ndetermines whether it is empty.","For every positive integer \\(q\\), including composite moduli and modulus\none, and every \\(L\\ge0\\), the checked map \\[\n\\Phi_{x,L}:(\\mathbb Z/q\\mathbb Z)^d\\longrightarrow\n\\{y\\bmod q^{L+1}:y\\equiv x\\pmod{q^L}\\},\n\\qquad t\\longmapsto x+q^L\\widetilde t\n\\] is a bijection. Here \\(\\widetilde t\\) is the canonical integer\nrepresentative of the digit vector. The entire residue fiber therefore\nhas \\(q^d\\) elements. Injectivity is proved by cancelling \\(q^L\\) in the\ncorresponding integer divisibility statement. Surjectivity comes from\nthe lower-residue congruence.","composite moduli and modulus\none, and every \\(L\\ge0\\), the checked map \\[\n\\Phi_{x,L}:(\\mathbb Z/q\\mathbb Z)^d\\longrightarrow\n\\{y\\bmod q^{L+1}:y\\equiv x\\pmod{q^L}\\},\n\\qquad t\\longmapsto x+q^L\\widetilde t\n\\] is a bijection. Here \\(\\widetilde t\\) is the canonical integer\nrepresentative of the digit vector. The entire residue fiber therefore\nhas \\(q^d\\) elements. Injectivity is proved by cancelling \\(q^L\\) in the\ncorresponding integer divisibility statement. Surjectivity comes from\nthe lower-residue congruence.","The formal type \\mcode{ActualLifts} is a subtype of these genuine\n\\texttt{ZMod\\ (q\\^{}(L+1))} residue vectors. Its polynomial predicate is\nevaluated on canonical integer representatives and tested for\ndivisibility by \\(q^{L+1}\\). The checked polynomial congruence theorem\nmakes that predicate independent of the integer representative. Thus\nusing canonical representatives does not discard or duplicate residue\nclasses.","At positive depth, the preceding actual multivariable Taylor theorem\ngives \\[\nF(x+p^L\\widetilde t)\\equiv\np^L(r_x+DF(x)\\widetilde t)\\pmod{p^{L+1}}.\n\\] The new bridge proves that the evaluated integer Jacobian reduces to\nthe actual \\(\\mathbb F_p\\)-linear map \\(A_x\\). Restricting\n\\(\\Phi_{x,L}\\) therefore gives an explicit equivalence \\[\n\\boxed{\\{t\\in\\mathbb F_p^d:b_x+A_xt=0\\}\\simeq\\mathcal L_{x,L}.}\n\\] The proof uses the actual formal partial derivatives of the original\npolynomial list.","Let \\(\\mathcal T_x=\\{t:b_x+A_xt=0\\}\\). Its natural kernel action is \\[\nk\\cdot t=k+t,\\qquad k\\in\\ker A_x.\n\\] For every pair \\(u,t\\in\\mathcal T_x\\), exactly one kernel vector\nsends \\(u\\) to \\(t\\), namely \\(t-u\\). Thus every nonempty correction\nfiber is a torsor for \\(\\ker A_x\\). Choosing one valid digit \\(u\\) gives\n\\[\n\\mathcal T_x\\simeq\\ker A_x,\n\\qquad t\\longmapsto t-u.\n\\] Lean constructs an explicit \\mcode{AddTorsor} structure for the\ndigit fiber, proves the unique translation property, and supplies a\nchosen-point equivalence from the actual residue lift set to the kernel.\nA basepoint is required for this equivalence, but the translation action\nitself does not depend on choosing one.","The earlier obstruction used the integer-domain map \\[\nJ_x=\\rho_p\\circ DF(x):\\mathbb Z^d\\to\\mathbb F_p^e.\n\\] The new count uses \\(A_x:\\mathbb F_p^d\\to\\mathbb F_p^e\\). Reduction\nof integer vectors onto finite vectors is surjective, so the two maps\nhave the same image after restricting scalars. Lean constructs a\nclass-preserving integer-linear equivalence \\[\n\\mathbb F_p^e/\\operatorname{im}J_x\n\\simeq\n\\mathbb F_p^e/\\operatorname{im}A_x.\n\\] Consequently the cardinality theorem counts the original obstruction\ngroup, preserving its distinguished residual class. It is not merely a\nformula for a separately introduced comparison object.","Rank-nullity gives \\(\\dim\\ker A_x=d-\\rho_x\\), and the quotient dimension\nis \\(e-\\rho_x\\). Cardinalities of finite vector spaces complete both\nformulas.","Let \\(p\\) be prime, \\(L\\ge1\\), and \\[\nF\\in\\mathbb Z[X_1,\\ldots,X_d]^e,\\qquad x\\in\\mathbb Z^d,\n\\qquad F(x)\\equiv0\\pmod{p^L}.\n\\] Assume the actual reduced Jacobian \\[\nA_x=DF(x)\\bmod p:\\mathbb F_p^d\\longrightarrow\\mathbb F_p^e\n\\] is surjective. Then Lean proves that there is a point \\[\n\\boxed{a\\in\\mathbb Z_p^d,\\qquad F(a)=0,\\qquad\n a\\equiv x\\pmod{p^L}.}\n\\] Here the coefficients of the original integer polynomials are cast\ninto \\(\\mathbb Z_p\\), and their multivariable evaluation is exactly\nzero. This is an existence theorem for a fixed polynomial presentation\nwith the stated Jacobian hypothesis.","and \\[\nF\\in\\mathbb Z[X_1,\\ldots,X_d]^e,\\qquad x\\in\\mathbb Z^d,\n\\qquad F(x)\\equiv0\\pmod{p^L}.\n\\] Assume the actual reduced Jacobian \\[\nA_x=DF(x)\\bmod p:\\mathbb F_p^d\\longrightarrow\\mathbb F_p^e\n\\] is surjective. Then Lean proves that there is a point \\[\n\\boxed{a\\in\\mathbb Z_p^d,\\qquad F(a)=0,\\qquad\n a\\equiv x\\pmod{p^L}.}\n\\] Here the coefficients of the original integer polynomials are cast\ninto \\(\\mathbb Z_p\\), and their multivariable evaluation is exactly\nzero. This is an existence theorem for a fixed polynomial presentation\nwith the stated Jacobian hypothesis.","The stronger checked statement constructs \\(z_n\\in\\mathbb Z^d\\),\n\\(n\\ge0\\), with \\[\n\\begin{aligned}\nz_0&=x,","The stronger checked statement constructs \\(z_n\\in\\mathbb Z^d\\),\n\\(n\\ge0\\), with \\[\n\\begin{aligned}\nz_0&=x,\\\\\nF(z_n)&\\equiv0\\pmod{p^{L+n}},\\\\\nz_{n+1}&\\equiv z_n\\pmod{p^{L+n}},\\\\\nDF(z_n)\\bmod p&=A_x,\\\\\nz_n&\\longrightarrow a\\quad\\text{in }\\mathbb Z_p^d,\\\\\na-z_n&\\in p^{L+n}\\mathbb Z_p^d.\n\\end{aligned}\n\\] The last inclusion retains every prescribed precision. It is a\ndivisibility lower bound, not an assertion that the error has exactly\nthat valuation. An approximation can already equal the limiting root.","The one-step Taylor theorem already supplies the exact correction\nequation \\[\nb_n+A_{z_n}t=0,\\qquad\nb_n=F(z_n)/p^{L+n}\\bmod p.\n\\] Surjectivity supplies a correcting digit for every residual \\(b_n\\).\nThe chosen correction gives an actual new integer point \\(z_{n+1}\\),\nusing the proved nonlinear Taylor remainder.","Every coordinate change is divisible by \\(p^{L+n}\\), hence by \\(p\\).\nEach formal partial derivative is itself an integer polynomial, so its\nvalue is unchanged modulo \\(p\\). Therefore \\[\nA_{z_{n+1}}=A_{z_n}=A_x.\n\\] The proof assumes no separate surjectivity or solvability statement\nat later levels. A dependent recursion chooses each next point from the\nproved one-step existence theorem. It produces one compatible sequence,\nrather than unrelated witnesses at different precisions.","The integer tower construction works more generally for every positive\nmodulus \\(q\\), including composite \\(q\\) and \\(q=1\\), with a surjective\nmap over \\(\\mathbb Z/q\\mathbb Z\\). The p-adic completion and the\nrank-based branching formulas in this block use prime \\(p\\).","Compatibility implies, for \\(m\\ge n\\), \\[\np^{L+n}\\mid z_m-z_n.\n\\] After casting to \\(\\mathbb Z_p\\), this bounds each coordinate\ndistance by \\(p^{-(L+n)}\\), so the sequence is Cauchy. Completeness\nsupplies its limit. Closed norm bounds show that the limit retains the\ncongruence at every stage.","The generic convergence theorem is proved independently of the\npolynomial application. It also proves \\[\np^{L+n}\\mid v_n\\ \\text{for all }n\n\\quad\\Longrightarrow\\quad v_n\\longrightarrow0\\text{ in }\\mathbb Z_p.\n\\] Apply this to each integer sequence \\(F_i(z_n)\\). Continuity of\nmultivariable polynomial evaluation also gives \\(F_i(z_n)\\to F_i(a)\\).\nUniqueness of limits then yields \\(F_i(a)=0\\).","Surjectivity implies \\[\n\\operatorname{rank}_{\\mathbb F_p}A_x=e,\\qquad e\\le d.\n\\] Combining this with the preceding exact residue-lift count gives \\[\n\\boxed{\\#\\mathcal L_{z_n,L+n}=p^{d-e}\\quad\\text{for every }n.}\n\\] These are distinct solutions modulo \\(p^{L+n+1}\\) lying above the\nselected point modulo \\(p^{L+n}\\). The original finite obstruction group\nat each smooth point has cardinality one.","The theorem constructs one path through these finite choices. When\n\\(d>e\\), it does not claim that all paths have the same limit. The\ngeneric uniqueness theorem concerns the limit of a fixed sequence only.","\\begin{enumerate}\n\\def\\labelenumi{\\arabic{enumi}.}\n\\tightlist\n\\item\n  \\textbf{An initial lift does not guarantee an infinite tower without\n  the Jacobian condition.} \\(X^2-5\\) has the odd lower root modulo two\n  and a lift modulo four, but no root modulo eight. Lean proves that no\n  integer tower can continue through all precisions in this example.\n\\item\n  \\textbf{Surjective rectangular Jacobians do not give uniqueness of all\n  roots.} The system \\(F(X,Y)=X\\) has a surjective Jacobian and distinct\n  dyadic roots \\((0,0)\\) and \\((0,2)\\), both in the same residue modulo\n  two.\n\\item\n  \\textbf{The retained initial precision cannot be increased without\n  justification.} For \\(F(X)=X-2\\), zero is a lower root modulo two with\n  surjective derivative, but its exact root two does not retain zero\n  modulo four.\n\\end{enumerate}","ightlist\n\\item\n  \\textbf{An initial lift does not guarantee an infinite tower without\n  the Jacobian condition.} \\(X^2-5\\) has the odd lower root modulo two\n  and a lift modulo four, but no root modulo eight. Lean proves that no\n  integer tower can continue through all precisions in this example.\n\\item\n  \\textbf{Surjective rectangular Jacobians do not give uniqueness of all\n  roots.} The system \\(F(X,Y)=X\\) has a surjective Jacobian and distinct\n  dyadic roots \\((0,0)\\) and \\((0,2)\\), both in the same residue modulo\n  two.\n\\item\n  \\textbf{The retained initial precision cannot be increased without\n  justification.} For \\(F(X)=X-2\\), zero is a lower root modulo two with\n  surjective derivative, but its exact root two does not retain zero\n  modulo four.\n\\end{enumerate}","nite tower without\n  the Jacobian condition.} \\(X^2-5\\) has the odd lower root modulo two\n  and a lift modulo four, but no root modulo eight. Lean proves that no\n  integer tower can continue through all precisions in this example.\n\\item\n  \\textbf{Surjective rectangular Jacobians do not give uniqueness of all\n  roots.} The system \\(F(X,Y)=X\\) has a surjective Jacobian and distinct\n  dyadic roots \\((0,0)\\) and \\((0,2)\\), both in the same residue modulo\n  two.\n\\item\n  \\textbf{The retained initial precision cannot be increased without\n  justification.} For \\(F(X)=X-2\\), zero is a lower root modulo two with\n  surjective derivative, but its exact root two does not retain zero\n  modulo four.\n\\end{enumerate}","hout\n  the Jacobian condition.} \\(X^2-5\\) has the odd lower root modulo two\n  and a lift modulo four, but no root modulo eight. Lean proves that no\n  integer tower can continue through all precisions in this example.\n\\item\n  \\textbf{Surjective rectangular Jacobians do not give uniqueness of all\n  roots.} The system \\(F(X,Y)=X\\) has a surjective Jacobian and distinct\n  dyadic roots \\((0,0)\\) and \\((0,2)\\), both in the same residue modulo\n  two.\n\\item\n  \\textbf{The retained initial precision cannot be increased without\n  justification.} For \\(F(X)=X-2\\), zero is a lower root modulo two with\n  surjective derivative, but its exact root two does not retain zero\n  modulo four.\n\\end{enumerate}","integer tower can continue through all precisions in this example.\n\\item\n  \\textbf{Surjective rectangular Jacobians do not give uniqueness of all\n  roots.} The system \\(F(X,Y)=X\\) has a surjective Jacobian and distinct\n  dyadic roots \\((0,0)\\) and \\((0,2)\\), both in the same residue modulo\n  two.\n\\item\n  \\textbf{The retained initial precision cannot be increased without\n  justification.} For \\(F(X)=X-2\\), zero is a lower root modulo two with\n  surjective derivative, but its exact root two does not retain zero\n  modulo four.\n\\end{enumerate}","Let \\(p\\) be prime, \\(L\\ge1\\), and \\[\nF,H\\in\\mathbb Z[X_1,\\ldots,X_d]^e,\n\\qquad x\\in\\mathbb Z^d.\n\\] Write \\[\nA_x=DF(x)\\bmod p:\\mathbb F_p^d\\longrightarrow\\mathbb F_p^e,\n\\qquad B_L(x)=x+p^L\\mathbb Z_p^d.\n\\] For an integer \\(c\\), the perturbed system is the actual polynomial\nlist \\[\nG=F+cH.\n\\] There is no restriction on the sign of \\(c\\), and \\(c=0\\) is allowed.\nDivisibility conclusions give lower bounds on precision and allow zero\ndifferences.","If \\(d=e\\), \\(A_x\\) is surjective, and \\(F(x)\\equiv0\\pmod{p^L}\\), then\n\\[\n\\boxed{\\exists!\\,a\\in B_L(x):F(a)=0.}\n\\] The stronger comparison lemma proves that any two exact roots of\n\\(F\\) in \\(B_L(x)\\) coincide, without needing a separate lower-root\nhypothesis.","\\textbf{Proof implemented.} Map every arbitrary p-adic root into the\nactual residue rings \\(\\mathbb Z/p^n\\mathbb Z\\), and choose its\ncanonical integer representative. Evaluation commutes with reduction, so\nthese representatives are genuine finite polynomial roots. The preceding\ncount theorem gives exactly \\(p^{d-d}=1\\) next-level residue lift.\nInductively any two roots agree at every precision. The p-adic residue\nmaps separate points, so the roots are equal. Existence is the preceding\nchecked tower theorem.","If \\(p^L\\mid c\\), the actual identities give \\[\nG(x)=F(x)+cH(x),\\qquad DG(x)\\equiv DF(x)\\pmod p.\n\\] Thus if \\(F(x)\\equiv0\\pmod{p^L}\\) and \\(A_x\\) is surjective, then \\[\n\\exists b\\in B_L(x):G(b)=0.\n\\] This theorem allows rectangular systems. The evaluation, derivative\nand Jacobian identities are proved from the polynomial definition\n\\(G_i=F_i+C(c)H_i\\), rather than supplied as hypotheses. Algebraic\ncongruence helpers also permit composite moduli. Exact p-adic root\nexistence uses prime \\(p\\).","od p.\n\\] Thus if \\(F(x)\\equiv0\\pmod{p^L}\\) and \\(A_x\\) is surjective, then \\[\n\\exists b\\in B_L(x):G(b)=0.\n\\] This theorem allows rectangular systems. The evaluation, derivative\nand Jacobian identities are proved from the polynomial definition\n\\(G_i=F_i+C(c)H_i\\), rather than supplied as hypotheses. Algebraic\ncongruence helpers also permit composite moduli. Exact p-adic root\nexistence uses prime \\(p\\).","Let \\(M\\ge1\\) and \\(p^M\\mid c\\). Suppose \\(A_x\\) is surjective and\n\\(a\\in\\mathbb Z_p^d\\) is an exact root of \\(F\\) with\n\\(a\\equiv x\\pmod p\\). Then \\[\n\\boxed{\\exists b\\in\\mathbb Z_p^d:\\quad G(b)=0,\n\\qquad b\\equiv a\\pmod{p^M}.}\n\\] No uniqueness is claimed when \\(d>e\\).","\\textbf{Proof implemented.} Truncate \\(a\\) to an integer vector \\(y\\)\nmodulo \\(p^M\\). It is an actual lower root of \\(F\\), and its reduced\nJacobian agrees with \\(A_x\\). Apply the coefficient-perturbed existence\ntheorem at precision \\(M\\), then combine the two congruences through\n\\(y\\).","Suppose \\(d=e\\), \\(M\\ge L\\ge1\\), \\(p^M\\mid c\\), and \\(A_x\\) is\nsurjective. If \\[\na,b\\in B_L(x),\\qquad F(a)=0,\\qquad G(b)=0,\n\\] then \\[\n\\boxed{b-a\\in p^M\\mathbb Z_p^d.}\n\\] In particular, for \\(G=F+p^MH\\), no p-adic digits below depth \\(M\\)\nare changed in the selected root.","The formalization also proves a combined existence theorem. Starting\nonly with \\(F(x)\\equiv0\\pmod{p^L}\\), it constructs an original root\n\\(a\\in B_L(x)\\) and proves that there is exactly one \\(b\\in B_L(x)\\)\nsatisfying \\(G(b)=0\\) and \\(b\\equiv a\\pmod{p^M}\\). The comparison\ntheorem shows that the precision bound holds for every root of \\(G\\) in\nthat ball, so uniqueness is not obtained merely by adding a restrictive\ndisplacement condition.","\\textbf{Proof implemented.} Section 3 produces a nearby root \\(b'\\).\nSince \\(M\\ge L\\), it belongs to \\(B_L(x)\\). The perturbed Jacobian\nremains surjective, and square uniqueness identifies \\(b'\\) with the\narbitrary root \\(b\\).","Let \\(p\\) be prime, \\(L\\ge1\\), \\(e,q\\ge0\\), and \\[\nF\\in\\mathbb Z[X_1,\\ldots,X_{e+q}]^e,\\qquad\nx\\in\\mathbb Z^{e+q},\\qquad F(x)\\equiv0\\pmod{p^L}.\n\\] Assume the actual reduced Jacobian \\[\nA_x=DF(x)\\bmod p:\\mathbb F_p^{e+q}\\longrightarrow\\mathbb F_p^e\n\\] is surjective. All vector congruences are coordinatewise.","There is a coordinate permutation \\(\\sigma\\), with\n\\((\\sigma z)_i=z_{\\sigma(i)}\\). Write \\(\\sigma z=(y,t)\\), using \\(e\\)\nsolved coordinates and \\(q\\) free coordinates. Define \\[\n(y_0,t_0)=\\sigma x,\\qquad\nB=x+p^L\\mathbb Z_p^{e+q},\\qquad\nT=t_0+p^L\\mathbb Z_p^q,\\qquad\n\\mathcal R=\\{z\\in B:F(z)=0\\}.\n\\] There is a homeomorphism \\[\nh:T\\xrightarrow{\\ \\sim\\ }\\mathcal R,\\qquad\nh^{-1}(z)=\\operatorname{free}(\\sigma z).\n\\] Equivalently, a continuous function\n\\(\\phi:T\\to y_0+p^L\\mathbb Z_p^e\\) gives the entire root locus through\n\\(\\sigma h(t)=(\\phi(t),t)\\). Every parameter has exactly one\ncorresponding root in \\(B\\). For every \\(N\\ge0\\), \\[\n\\boxed{h(t)\\equiv h(u)\\pmod{p^N}\\iff t\\equiv u\\pmod{p^N}.}\n\\]","((\\sigma z)_i=z_{\\sigma(i)}\\). Write \\(\\sigma z=(y,t)\\), using \\(e\\)\nsolved coordinates and \\(q\\) free coordinates. Define \\[\n(y_0,t_0)=\\sigma x,\\qquad\nB=x+p^L\\mathbb Z_p^{e+q},\\qquad\nT=t_0+p^L\\mathbb Z_p^q,\\qquad\n\\mathcal R=\\{z\\in B:F(z)=0\\}.\n\\] There is a homeomorphism \\[\nh:T\\xrightarrow{\\ \\sim\\ }\\mathcal R,\\qquad\nh^{-1}(z)=\\operatorname{free}(\\sigma z).\n\\] Equivalently, a continuous function\n\\(\\phi:T\\to y_0+p^L\\mathbb Z_p^e\\) gives the entire root locus through\n\\(\\sigma h(t)=(\\phi(t),t)\\). Every parameter has exactly one\ncorresponding root in \\(B\\). For every \\(N\\ge0\\), \\[\n\\boxed{h(t)\\equiv h(u)\\pmod{p^N}\\iff t\\equiv u\\pmod{p^N}.}\n\\]","lved coordinates and \\(q\\) free coordinates. Define \\[\n(y_0,t_0)=\\sigma x,\\qquad\nB=x+p^L\\mathbb Z_p^{e+q},\\qquad\nT=t_0+p^L\\mathbb Z_p^q,\\qquad\n\\mathcal R=\\{z\\in B:F(z)=0\\}.\n\\] There is a homeomorphism \\[\nh:T\\xrightarrow{\\ \\sim\\ }\\mathcal R,\\qquad\nh^{-1}(z)=\\operatorname{free}(\\sigma z).\n\\] Equivalently, a continuous function\n\\(\\phi:T\\to y_0+p^L\\mathbb Z_p^e\\) gives the entire root locus through\n\\(\\sigma h(t)=(\\phi(t),t)\\). Every parameter has exactly one\ncorresponding root in \\(B\\). For every \\(N\\ge0\\), \\[\n\\boxed{h(t)\\equiv h(u)\\pmod{p^N}\\iff t\\equiv u\\pmod{p^N}.}\n\\]","x+p^L\\mathbb Z_p^{e+q},\\qquad\nT=t_0+p^L\\mathbb Z_p^q,\\qquad\n\\mathcal R=\\{z\\in B:F(z)=0\\}.\n\\] There is a homeomorphism \\[\nh:T\\xrightarrow{\\ \\sim\\ }\\mathcal R,\\qquad\nh^{-1}(z)=\\operatorname{free}(\\sigma z).\n\\] Equivalently, a continuous function\n\\(\\phi:T\\to y_0+p^L\\mathbb Z_p^e\\) gives the entire root locus through\n\\(\\sigma h(t)=(\\phi(t),t)\\). Every parameter has exactly one\ncorresponding root in \\(B\\). For every \\(N\\ge0\\), \\[\n\\boxed{h(t)\\equiv h(u)\\pmod{p^N}\\iff t\\equiv u\\pmod{p^N}.}\n\\]","{e+q},\\qquad\nT=t_0+p^L\\mathbb Z_p^q,\\qquad\n\\mathcal R=\\{z\\in B:F(z)=0\\}.\n\\] There is a homeomorphism \\[\nh:T\\xrightarrow{\\ \\sim\\ }\\mathcal R,\\qquad\nh^{-1}(z)=\\operatorname{free}(\\sigma z).\n\\] Equivalently, a continuous function\n\\(\\phi:T\\to y_0+p^L\\mathbb Z_p^e\\) gives the entire root locus through\n\\(\\sigma h(t)=(\\phi(t),t)\\). Every parameter has exactly one\ncorresponding root in \\(B\\). For every \\(N\\ge0\\), \\[\n\\boxed{h(t)\\equiv h(u)\\pmod{p^N}\\iff t\\equiv u\\pmod{p^N}.}\n\\]","d\nT=t_0+p^L\\mathbb Z_p^q,\\qquad\n\\mathcal R=\\{z\\in B:F(z)=0\\}.\n\\] There is a homeomorphism \\[\nh:T\\xrightarrow{\\ \\sim\\ }\\mathcal R,\\qquad\nh^{-1}(z)=\\operatorname{free}(\\sigma z).\n\\] Equivalently, a continuous function\n\\(\\phi:T\\to y_0+p^L\\mathbb Z_p^e\\) gives the entire root locus through\n\\(\\sigma h(t)=(\\phi(t),t)\\). Every parameter has exactly one\ncorresponding root in \\(B\\). For every \\(N\\ge0\\), \\[\n\\boxed{h(t)\\equiv h(u)\\pmod{p^N}\\iff t\\equiv u\\pmod{p^N}.}\n\\]","The coordinate choice is a conclusion. A suitable minor and a\npreexisting chart are not hypotheses. The theorem covers the original\nroot locus after undoing the coordinate reordering. Since \\(x\\) is an\napproximate root, it does not assert \\(\\phi(t_0)=y_0\\) unless \\(F(x)=0\\)\nexactly. The cases \\(p=2\\), \\(L=1\\), \\(e=0\\), and \\(q=0\\) are included.","For a rectangular integer polynomial system with a surjective reduced\nJacobian at \\(x\\), and \\(L\\ge1\\), \\[\nF(x+p^L\\mathbb Z_p^d)=F(x)+p^L\\mathbb Z_p^e.\n\\] This covers arbitrary p-adic output targets. No lower-root hypothesis\nis needed.","When \\(d=e\\), the actual restricted polynomial map is a homeomorphism.\nMoreover, for arbitrary \\(a,b\\in x+p\\mathbb Z_p^d\\) and every \\(N\\ge0\\),\n\\[\nF(a)\\equiv F(b)\\pmod{p^N}\\iff a\\equiv b\\pmod{p^N}.\n\\] Its inverse therefore preserves every congruence depth.","}.}\n\\tightlist\n\\item\n  The actual integer Taylor formula gives one additional input digit\n  from one additional output digit when the reduced square Jacobian is\n  surjective. Its finite domain and codomain have the same size, so it\n  is injective. Induction gives every integer precision.\n\\item\n  Canonical integer truncations of arbitrary p-adic vectors preserve\n  evaluation modulo each power of \\(p\\). Applying the integer result to\n  truncations proves the p-adic congruence equivalence.\n\\item\n  Subtracting an integer target from the constant terms leaves the\n  actual Jacobian unchanged. The previous root-lifting theorem supplies\n  a preimage. The image of the compact input ball is closed and contains\n  the convergent truncations of every admissible p-adic target, hence\n  contains the target. No compatibility of arbitrarily chosen preimages\n  is assumed.\n\\item\n  In the square case, the resulting continuous bijection from a compact\n  ball to a Hausdorff ball is a homeomorphism. Its forward map is actual\n  polynomial evaluation, and its inverse solves the output equation.\n\\item\n  For a …","compatibility of arbitrarily chosen preimages\n  is assumed.\n\\item\n  In the square case, the resulting continuous bijection from a compact\n  ball to a Hausdorff ball is a homeomorphism. Its forward map is actual\n  polynomial evaluation, and its inverse solves the output equation.\n\\item\n  For a specified surjective solved-variable block, apply this inverse\n  to the actual augmented polynomial map \\[\n  \\Psi(y,t)=(F(y,t),t).\n  \\] Its derivative has an identity free-coordinate block. The inverse\n  at \\((0,t)\\) gives the root. The lower-root congruence places this\n  target in the output ball. Injectivity proves uniqueness and full\n  graph coverage. Projection identifies the inverse chart and reflects\n  congruences.\n\\item\n  The columns of a surjective finite-dimensional linear map contain a\n  basis. Select those columns and extend their injection to a coordinate\n  permutation. Polynomial renaming by \\(\\sigma^{-1}\\) transports actual\n  evaluations and partial derivatives. The fixed-coordinate chart then\n  gives the main theorem on the entire original root locus.\n\\end{enumerate}","ontinuous bijection from a compact\n  ball to a Hausdorff ball is a homeomorphism. Its forward map is actual\n  polynomial evaluation, and its inverse solves the output equation.\n\\item\n  For a specified surjective solved-variable block, apply this inverse\n  to the actual augmented polynomial map \\[\n  \\Psi(y,t)=(F(y,t),t).\n  \\] Its derivative has an identity free-coordinate block. The inverse\n  at \\((0,t)\\) gives the root. The lower-root congruence places this\n  target in the output ball. Injectivity proves uniqueness and full\n  graph coverage. Projection identifies the inverse chart and reflects\n  congruences.\n\\item\n  The columns of a surjective finite-dimensional linear map contain a\n  basis. Select those columns and extend their injection to a coordinate\n  permutation. Polynomial renaming by \\(\\sigma^{-1}\\) transports actual\n  evaluations and partial derivatives. The fixed-coordinate chart then\n  gives the main theorem on the entire original root locus.\n\\end{enumerate}","\\) gives the root. The lower-root congruence places this\n  target in the output ball. Injectivity proves uniqueness and full\n  graph coverage. Projection identifies the inverse chart and reflects\n  congruences.\n\\item\n  The columns of a surjective finite-dimensional linear map contain a\n  basis. Select those columns and extend their injection to a coordinate\n  permutation. Polynomial renaming by \\(\\sigma^{-1}\\) transports actual\n  evaluations and partial derivatives. The fixed-coordinate chart then\n  gives the main theorem on the entire original root locus.\n\\end{enumerate}","Full Jacobian surjectivity need not allow an arbitrary prescribed\nparameter choice. At the dyadic origin, \\(F(X,T)=X^2-T\\) has a\nsurjective full Jacobian but zero \\(X\\)-derivative. Prescribing \\(T=2\\)\ngives no root even modulo \\(4\\). Suitable coordinates instead solve for\n\\(T\\).","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). Th","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\)","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed in Lean by existence of an integral matrix\nwith exactly tha","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,r\\in\\mathbb N\\). Let\n\\(F=(F_1,\\ldots,F_d)\\in R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed in Lean by existence of an integral matrix\nwith exactly that image under the coefficient inclusion\n\\(R\\hookrightarrow K\\).","n R[X_1,\\ldots,X_d]^d\\), \\(a\\in R^d\\), and let \\[\nJ=DF(a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed in Lean by existence of an integral matrix\nwith exactly that image under the coefficient inclusion\n\\(R\\hookrightarrow K\\).","a)=\\bigl(\\partial_jF_i(a)\\bigr)_{i,j}.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad\nF(a)\\in p^rR^d,\\qquad r>2s.\n\\] Then \\[\n\\boxed{\\exists!\\,b\\in a+p^{s+1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed in Lean by existence of an integral matrix\nwith exactly that image under the coefficient inclusion\n\\(R\\hookrightarrow K\\).","1}R^d:\\ F(b)=0,}\n\\qquad\n\\boxed{b-a\\in p^{r-s}R^d.}\n\\] All vector divisibility assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed in Lean by existence of an integral matrix\nwith exactly that image under the coefficient inclusion\n\\(R\\hookrightarrow K\\).","ity assertions are coordinatewise. The uniqueness\npredicate is the entire ball of depth \\(s+1\\): competitors need not be\nassumed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed in Lean by existence of an integral matrix\nwith exactly that image under the coefficient inclusion\n\\(R\\hookrightarrow K\\).","ed to have depth \\(r-s\\). The theorem permits arbitrary integral\np-adic coefficients and centers, every prime including two, and\ndimension zero. It retains the supplied inverse-denominator exponent\n\\(s\\). There is no replacement by a determinant valuation. Integrality\nof \\(p^sJ^{-1}\\) is expressed in Lean by existence of an integral matrix\nwith exactly that image under the coefficient inclusion\n\\(R\\hookrightarrow K\\).","Write \\(A=p^sJ^{-1}\\in M_d(R)\\). Then \\[\nAJ=JA=p^sI.\n\\] The formal matrix bridge proves the equivalence with the stated\nfraction-field hypotheses, including nonsingularity. In particular,\nmultiplication by \\(A\\) is injective, since multiplication by \\(J\\)\nsends \\(Av=0\\) to \\(p^sv=0\\).","Put \\(G=AF\\), \\(t=p^{s+1}\\), and \\(k=r-(2s+1)\\). Taylor expansion of the\nactual preconditioned polynomials gives one polynomial system \\(Q\\) such\nthat \\[\nG(a+tu)=G(a)+p^{2s+1}u+p^{2s+2}Q(u),\\qquad Q(0)=0.\n\\] This expansion is proved first over every commutative ring. Its\nremainder is a polynomial valid for every \\(u\\), not a pointwise\nquotient depending on \\(u\\). Since \\(G(a)\\in p^rR^d\\), it can be written\n\\(p^{2s+1}C\\) with \\(C\\in p^kR^d\\). Thus \\[\nG(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr).\n\\]",", \\(t=p^{s+1}\\), and \\(k=r-(2s+1)\\). Taylor expansion of the\nactual preconditioned polynomials gives one polynomial system \\(Q\\) such\nthat \\[\nG(a+tu)=G(a)+p^{2s+1}u+p^{2s+2}Q(u),\\qquad Q(0)=0.\n\\] This expansion is proved first over every commutative ring. Its\nremainder is a polynomial valid for every \\(u\\), not a pointwise\nquotient depending on \\(u\\). Since \\(G(a)\\in p^rR^d\\), it can be written\n\\(p^{2s+1}C\\) with \\(C\\in p^kR^d\\). Thus \\[\nG(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr).\n\\]","d \\(k=r-(2s+1)\\). Taylor expansion of the\nactual preconditioned polynomials gives one polynomial system \\(Q\\) such\nthat \\[\nG(a+tu)=G(a)+p^{2s+1}u+p^{2s+2}Q(u),\\qquad Q(0)=0.\n\\] This expansion is proved first over every commutative ring. Its\nremainder is a polynomial valid for every \\(u\\), not a pointwise\nquotient depending on \\(u\\). Since \\(G(a)\\in p^rR^d\\), it can be written\n\\(p^{2s+1}C\\) with \\(C\\in p^kR^d\\). Thus \\[\nG(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr).\n\\]","or expansion of the\nactual preconditioned polynomials gives one polynomial system \\(Q\\) such\nthat \\[\nG(a+tu)=G(a)+p^{2s+1}u+p^{2s+2}Q(u),\\qquad Q(0)=0.\n\\] This expansion is proved first over every commutative ring. Its\nremainder is a polynomial valid for every \\(u\\), not a pointwise\nquotient depending on \\(u\\). Since \\(G(a)\\in p^rR^d\\), it can be written\n\\(p^{2s+1}C\\) with \\(C\\in p^kR^d\\). Thus \\[\nG(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr).\n\\]","Integral polynomial evaluation is 1-Lipschitz in the supremum p-adic\nmetric. Consequently \\(T(u)=-C-pQ(u)\\) has contraction constant\n\\(|p|_p<1\\) on the complete space \\(R^d\\), and has exactly one fixed\npoint \\(u\\).","For completeness, the sharp precision is additional to this fixed-point\nexistence. Inductively, if \\(u\\in p^nR^d\\), then polynomial congruence\nand \\(Q(0)=0\\) imply \\(Q(u)\\in p^nR^d\\). For \\(n<k\\), the equation\n\\(u=-C-pQ(u)\\) therefore gives \\(u\\in p^{n+1}R^d\\). Starting at \\(n=0\\),\nthis proves \\(u\\in p^kR^d\\). Hence \\(b=a+tu\\) satisfies \\[\nb-a\\in p^{s+1+k}R^d=p^{r-s}R^d.\n\\] The normalization and injectivity of \\(A\\) give \\(F(b)=0\\).",", the sharp precision is additional to this fixed-point\nexistence. Inductively, if \\(u\\in p^nR^d\\), then polynomial congruence\nand \\(Q(0)=0\\) imply \\(Q(u)\\in p^nR^d\\). For \\(n<k\\), the equation\n\\(u=-C-pQ(u)\\) therefore gives \\(u\\in p^{n+1}R^d\\). Starting at \\(n=0\\),\nthis proves \\(u\\in p^kR^d\\). Hence \\(b=a+tu\\) satisfies \\[\nb-a\\in p^{s+1+k}R^d=p^{r-s}R^d.\n\\] The normalization and injectivity of \\(A\\) give \\(F(b)=0\\).","precision is additional to this fixed-point\nexistence. Inductively, if \\(u\\in p^nR^d\\), then polynomial congruence\nand \\(Q(0)=0\\) imply \\(Q(u)\\in p^nR^d\\). For \\(n<k\\), the equation\n\\(u=-C-pQ(u)\\) therefore gives \\(u\\in p^{n+1}R^d\\). Starting at \\(n=0\\),\nthis proves \\(u\\in p^kR^d\\). Hence \\(b=a+tu\\) satisfies \\[\nb-a\\in p^{s+1+k}R^d=p^{r-s}R^d.\n\\] The normalization and injectivity of \\(A\\) give \\(F(b)=0\\).","Every other \\(z\\in a+p^{s+1}R^d\\) has the form \\(a+tw\\). If \\(F(z)=0\\),\nthe same normalization, with cancellation of the nonzero scalar\n\\(p^{2s+1}\\), gives \\(w=T(w)\\). Fixed-point uniqueness implies \\(w=u\\),\nhence \\(z=b\\).","The strict inequality cannot in general be weakened to \\(r\\ge2s\\): for\n\\(p=2\\), \\(F(X)=X^2+3\\), and \\(a=1\\), we have \\(F(a)=4\\), \\(DF(a)=2\\),\nand may take \\(s=1,r=2\\). Yet no root exists in \\(\\mathbb Z_2\\), since\nthere is none modulo eight. The auxiliary Lean controls prove this\nobstruction and the exact residual depth.","The displacement exponent cannot uniformly be increased by one. For\n\\(F(X)=p^sX+p^r\\) at \\(a=0\\), with \\(r>2s\\), its root is \\(b=-p^{r-s}\\),\nwhich is not divisible by \\(p^{r-s+1}\\). The controls prove the linear\nroot and the absence of that extra digit for all primes.","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Ja","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis o","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and c","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\np","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite squ","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p","Let \\(p\\) be prime, \\(R=\\mathbb Z_p\\), \\(K=\\mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p=2\\) and\ndimension zero.","mathbb Q_p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p=2\\) and\ndimension zero.","p\\), and\n\\(d,s,M\\in\\mathbb N\\). Let \\(F,H\\in R[X_1,\\ldots,X_d]^d\\), let\n\\(a\\in R^d\\) satisfy \\(F(a)=0\\), and set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p=2\\) and\ndimension zero.","set \\[\nJ=DF(a),\\qquad G=F+p^M H.\n\\] Assume \\[\nJ\\in\\operatorname{GL}_d(K),\\qquad p^sJ^{-1}\\in M_d(R),\\qquad M>2s.\n\\] Then there is exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p=2\\) and\ndimension zero.","exactly one root \\(b\\) of \\(G\\) in \\(a+p^{s+1}R^d\\). It\nsatisfies \\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p=2\\) and\ndimension zero.","\\[\nb-a\\in p^{M-s}R^d,\n\\] and there is \\(\\varepsilon\\in p^{2M-3s}R^d\\) such that, in \\(K^d\\),\n\\[\n\\boxed{b-a=-p^M J^{-1}H(a)+\\varepsilon.}\n\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p=2\\) and\ndimension zero.","\\] The inverse in this formula belongs to the original Jacobian\n\\(DF(a)\\). The theorem constructs the perturbed root from these\nhypotheses and proves uniqueness throughout the full ball of depth\n\\(s+1\\). It does not require an additional Jacobian hypothesis on \\(G\\).\nCoefficients and centers are arbitrary integral p-adic elements. Every\nprime and finite square dimension are included, including \\(p=2\\) and\ndimension zero.","The error depth exceeds the displacement depth by \\[\n(2M-3s)-(M-s)=M-2s\\ge1.\n\\] Thus the first-order term determines strictly more digits than the\ndisplacement bound alone. These are guaranteed divisibility bounds.\nEquality of valuations is not asserted for every system.","Write \\(A=p^sJ^{-1}\\in M_d(R)\\), so \\(AJ=JA=p^sI\\). Let \\(t=p^{s+1}\\)\nand \\(k=M-(2s+1)\\ge0\\). The actual polynomial Taylor and difference\nidentities give polynomial systems \\(Q_F,Q_H\\), both vanishing at zero,\nsuch that \\[\n(AF)(a+tu)=p^{2s+1}u+p^{2s+2}Q_F(u),\n\\] \\[\n(AH)(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s","Write \\(A=p^sJ^{-1}\\in M_d(R)\\), so \\(AJ=JA=p^sI\\). Let \\(t=p^{s+1}\\)\nand \\(k=M-(2s+1)\\ge0\\). The actual polynomial Taylor and difference\nidentities give polynomial systems \\(Q_F,Q_H\\), both vanishing at zero,\nsuch that \\[\n(AF)(a+tu)=p^{2s+1}u+p^{2s+2}Q_F(u),\n\\] \\[\n(AH)(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","p^{s+1}\\)\nand \\(k=M-(2s+1)\\ge0\\). The actual polynomial Taylor and difference\nidentities give polynomial systems \\(Q_F,Q_H\\), both vanishing at zero,\nsuch that \\[\n(AF)(a+tu)=p^{2s+1}u+p^{2s+2}Q_F(u),\n\\] \\[\n(AH)(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","difference\nidentities give polynomial systems \\(Q_F,Q_H\\), both vanishing at zero,\nsuch that \\[\n(AF)(a+tu)=p^{2s+1}u+p^{2s+2}Q_F(u),\n\\] \\[\n(AH)(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","ishing at zero,\nsuch that \\[\n(AF)(a+tu)=p^{2s+1}u+p^{2s+2}Q_F(u),\n\\] \\[\n(AH)(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","^{2s+2}Q_F(u),\n\\] \\[\n(AH)(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","F(u),\n\\] \\[\n(AH)(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).",")(a+tu)=(AH)(a)+tQ_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","Q_H(u).\n\\] Consequently \\[\n(AG)(a+tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","tu)=p^{2s+1}\\bigl(C+u+pQ(u)\\bigr),\n\\] where \\[\nC=p^kAH(a),\\qquad\nQ=Q_F+p^{M-s-1}Q_H,\\qquad Q(0)=0.\n\\] All exponents are nonnegative under \\(M>2s\\). Actual integral\npolynomial evaluation is 1-Lipschitz, so \\(u\\mapsto-C-pQ(u)\\) is a\nstrict contraction on the complete space \\(R^d\\). Its unique fixed point\ninherits the depth \\(k\\) of \\(C\\), because \\(Q(0)=0\\). Thus \\(b=a+tu\\)\nhas displacement depth \\(s+1+k=M-s\\).","The certificate \\(JA=p^sI\\) makes multiplication by \\(A\\) injective, so\nthe normalized zero is an actual root of \\(G\\). Every root in \\(a+tR^d\\)\nhas the form \\(a+tw\\) and satisfies the same fixed-point equation. This\nproves uniqueness on the entire stated ball using the original matrix\ncertificate.","Put \\(\\delta=b-a\\) and \\(L=M-s\\). The displacement form of Taylor's\ntheorem, proved over any commutative ring, gives \\[\nE_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2L}R^d.\n\\] Polynomial congruence gives \\[\nE_H=(AH)(b)-(AH)(a)\\in p^LR^d.\n\\] Since \\(F(a)=0\\) and \\(G(b)=0\\), \\[\np^s\\delta+p^MAH(a)=-E_F-p^ME_H.\n\\] We have \\(M+L\\ge2L\\), so the right side lies in \\(p^{2L}R^d\\).\nCancelling the nonzero scalar \\(p^s\\) proves the integral estimate \\[\n\\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\] Define this integral vector to be \\(\\varepsilon\\). Under\n\\(R\\hookrightarrow K\\), the exact identity \\(p^{M-s}A=p^MJ^{-1}\\) gives\nthe displayed first-order formula. The Lean bridge checks matrix\nevaluation and all coefficient inclusions expl","Put \\(\\delta=b-a\\) and \\(L=M-s\\). The displacement form of Taylor's\ntheorem, proved over any commutative ring, gives \\[\nE_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2L}R^d.\n\\] Polynomial congruence gives \\[\nE_H=(AH)(b)-(AH)(a)\\in p^LR^d.\n\\] Since \\(F(a)=0\\) and \\(G(b)=0\\), \\[\np^s\\delta+p^MAH(a)=-E_F-p^ME_H.\n\\] We have \\(M+L\\ge2L\\), so the right side lies in \\(p^{2L}R^d\\).\nCancelling the nonzero scalar \\(p^s\\) proves the integral estimate \\[\n\\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\] Define this integral vector to be \\(\\varepsilon\\). Under\n\\(R\\hookrightarrow K\\), the exact identity \\(p^{M-s}A=p^MJ^{-1}\\) gives\nthe displayed first-order formula. The Lean bridge checks matrix\nevaluation and all coefficient inclusions explicitly.","he displacement form of Taylor's\ntheorem, proved over any commutative ring, gives \\[\nE_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2L}R^d.\n\\] Polynomial congruence gives \\[\nE_H=(AH)(b)-(AH)(a)\\in p^LR^d.\n\\] Since \\(F(a)=0\\) and \\(G(b)=0\\), \\[\np^s\\delta+p^MAH(a)=-E_F-p^ME_H.\n\\] We have \\(M+L\\ge2L\\), so the right side lies in \\(p^{2L}R^d\\).\nCancelling the nonzero scalar \\(p^s\\) proves the integral estimate \\[\n\\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\] Define this integral vector to be \\(\\varepsilon\\). Under\n\\(R\\hookrightarrow K\\), the exact identity \\(p^{M-s}A=p^MJ^{-1}\\) gives\nthe displayed first-order formula. The Lean bridge checks matrix\nevaluation and all coefficient inclusions explicitly.","ives \\[\nE_F=(AF)(b)-(AF)(a)-p^s\\delta\\in p^{2L}R^d.\n\\] Polynomial congruence gives \\[\nE_H=(AH)(b)-(AH)(a)\\in p^LR^d.\n\\] Since \\(F(a)=0\\) and \\(G(b)=0\\), \\[\np^s\\delta+p^MAH(a)=-E_F-p^ME_H.\n\\] We have \\(M+L\\ge2L\\), so the right side lies in \\(p^{2L}R^d\\).\nCancelling the nonzero scalar \\(p^s\\) proves the integral estimate \\[\n\\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\] Define this integral vector to be \\(\\varepsilon\\). Under\n\\(R\\hookrightarrow K\\), the exact identity \\(p^{M-s}A=p^MJ^{-1}\\) gives\nthe displayed first-order formula. The Lean bridge checks matrix\nevaluation and all coefficient inclusions explicitly.",")(a)-p^s\\delta\\in p^{2L}R^d.\n\\] Polynomial congruence gives \\[\nE_H=(AH)(b)-(AH)(a)\\in p^LR^d.\n\\] Since \\(F(a)=0\\) and \\(G(b)=0\\), \\[\np^s\\delta+p^MAH(a)=-E_F-p^ME_H.\n\\] We have \\(M+L\\ge2L\\), so the right side lies in \\(p^{2L}R^d\\).\nCancelling the nonzero scalar \\(p^s\\) proves the integral estimate \\[\n\\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\] Define this integral vector to be \\(\\varepsilon\\). Under\n\\(R\\hookrightarrow K\\), the exact identity \\(p^{M-s}A=p^MJ^{-1}\\) gives\nthe displayed first-order formula. The Lean bridge checks matrix\nevaluation and all coefficient inclusions explicitly.","congruence gives \\[\nE_H=(AH)(b)-(AH)(a)\\in p^LR^d.\n\\] Since \\(F(a)=0\\) and \\(G(b)=0\\), \\[\np^s\\delta+p^MAH(a)=-E_F-p^ME_H.\n\\] We have \\(M+L\\ge2L\\), so the right side lies in \\(p^{2L}R^d\\).\nCancelling the nonzero scalar \\(p^s\\) proves the integral estimate \\[\n\\delta+p^{M-s}AH(a)\\in p^{2M-3s}R^d.\n\\] Define this integral vector to be \\(\\varepsilon\\). Under\n\\(R\\hookrightarrow K\\), the exact identity \\(p^{M-s}A=p^MJ^{-1}\\) gives\nthe displayed first-order formula. The Lean bridge checks matrix\nevaluation and all coefficient inclusions explicitly.","For \\(p=2\\), take \\(F(X)=X+X^2\\), \\(H(X)=1\\), \\(a=0\\), \\(s=0\\), \\(M=1\\).\nThere is a unique root \\(b\\in2\\mathbb Z_2\\) of \\(X+X^2+2\\). Its equation\ngives \\(b+2=-b^2\\). It also forces \\(b\\equiv2\\pmod4\\). Therefore \\[\nv_2(b+2)=2=2M-3s.\n\\] The accompanying Lean controls show that the error is divisible by\nfour and is not divisible by eight. This refutes a uniform replacement\nof \\(2M-3s\\) by \\(2M-3s+1\\), even for a one-variable unit Jacobian.","The prior strict-threshold example also applies to perturbations: take\n\\(F(X)=X^2-1\\), \\(H(X)=1\\), \\(a=1\\), \\(p=2\\), \\(s=1\\), \\(M=2\\). Then\n\\(F(a)=0\\), \\(DF(a)=2\\), and \\(F+2^2H=X^2+3\\) has no root modulo eight,\nhence none in \\(\\mathbb Z_2\\). The new explanatory identification uses\nthe previously checked no-root polynomial. The strict inequality\n\\(M>2s\\) cannot uniformly be replaced by \\(M\\ge2s\\).","Let \\(q>0\\), \\(L\\ge1\\), and \\(m,k,e\\ge0\\). Write\n\\(S=\\mathbb Z/q\\mathbb Z\\). Let \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equi","Let \\(q>0\\), \\(L\\ge1\\), and \\(m,k,e\\ge0\\). Write\n\\(S=\\mathbb Z/q\\mathbb Z\\). Let \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{","Let \\(q>0\\), \\(L\\ge1\\), and \\(m,k,e\\ge0\\). Write\n\\(S=\\mathbb Z/q\\mathbb Z\\). Let \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\eq","Let \\(q>0\\), \\(L\\ge1\\), and \\(m,k,e\\ge0\\). Write\n\\(S=\\mathbb Z/q\\mathbb Z\\). Let \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}","Let \\(q>0\\), \\(L\\ge1\\), and \\(m,k,e\\ge0\\). Write\n\\(S=\\mathbb Z/q\\mathbb Z\\). Let \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously check","Let \\(q>0\\), \\(L\\ge1\\), and \\(m,k,e\\ge0\\). Write\n\\(S=\\mathbb Z/q\\mathbb Z\\). Let \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changin","Let \\(q>0\\), \\(L\\ge1\\), and \\(m,k,e\\ge0\\). Write\n\\(S=\\mathbb Z/q\\mathbb Z\\). Let \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) doe…","t \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) does not change its prescribed\nresidue modulo \\(q^{L+1}\\).","x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) does not change its prescribed\nresidue modulo \\(q^{L+1}\\).","Z^{m+k},\n\\qquad F(x)\\equiv0\\pmod{q^L}.\n\\] The actual lower root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) does not change its prescribed\nresidue modulo \\(q^{L+1}\\).","r root supplies the unique integral residual \\[\nr=q^{-L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) does not change its prescribed\nresidue modulo \\(q^{L+1}\\).","L}F(x)\\in\\mathbb Z^e,\\qquad \\bar r=r\\bmod q\\in S^e.\n\\] Let the actual reduced Jacobian blocks be \\[\nJ=(D_YF)(x)\\bmod q:S^m\\to S^e,\n\\qquad K=(D_TF)(x)\\bmod q:S^k\\to S^e.\n\\] For a prescribed parameter digit \\(v\\in S^k\\), choose its canonical\nintegral representative \\(\\widehat v\\). Define \\(\\mathcal L_v\\) to be\nthe set of actual residue vectors\n\\(z\\in(\\mathbb Z/q^{L+1}\\mathbb Z)^{m+k}\\) satisfying \\[\nz\\equiv x\\pmod{q^L},\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) does not change its prescribed\nresidue modulo \\(q^{L+1}\\).","\\qquad F(z)\\equiv0\\pmod{q^{L+1}},\\qquad\nz_T\\equiv t_0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) does not change its prescribed\nresidue modulo \\(q^{L+1}\\).","0+q^L\\widehat v\\pmod{q^{L+1}}.\n\\] The formal predicate uses canonical integer representatives for\nevaluation and the lower congruence. The previously checked polynomial\ncongruence theorem and digit-to-residue equivalence establish the actual\nresidue-class interpretation. Changing an integral representative of a\nparameter digit by a multiple of \\(q\\) does not change its prescribed\nresidue modulo \\(q^{L+1}\\).","Define the finite solved-block cokernel and its parameter-dependent\nclass by \\[\n\\mathcal C_x=S^e/\\operatorname{im}J,\n\\qquad o_x(v)=[\\bar r+Kv]\\in\\mathcal C_x.\n\\] Then \\[\n\\boxed{\\mathcal L_v\\ne\\varnothing\\iff o_x(v)=0.}\n\\] More precisely, the checked digit-to-lift equivalence is \\[\n\\boxed{\\mathcal L_v\\cong\\{u\\in S^m:Ju=-\\bar r-Kv\\}.}\n\\] It sends a solved digit \\(u\\) to the actual next residue represented\nby \\(x+q^L(\\widehat u,\\widehat v)\\). The obstruction depends on the\nsolved block \\(J\\), not on the range of the full map \\((J\\ K)\\).","inite solved-block cokernel and its parameter-dependent\nclass by \\[\n\\mathcal C_x=S^e/\\operatorname{im}J,\n\\qquad o_x(v)=[\\bar r+Kv]\\in\\mathcal C_x.\n\\] Then \\[\n\\boxed{\\mathcal L_v\\ne\\varnothing\\iff o_x(v)=0.}\n\\] More precisely, the checked digit-to-lift equivalence is \\[\n\\boxed{\\mathcal L_v\\cong\\{u\\in S^m:Ju=-\\bar r-Kv\\}.}\n\\] It sends a solved digit \\(u\\) to the actual next residue represented\nby \\(x+q^L(\\widehat u,\\widehat v)\\). The obstruction depends on the\nsolved block \\(J\\), not on the range of the full map \\((J\\ K)\\).","_x=S^e/\\operatorname{im}J,\n\\qquad o_x(v)=[\\bar r+Kv]\\in\\mathcal C_x.\n\\] Then \\[\n\\boxed{\\mathcal L_v\\ne\\varnothing\\iff o_x(v)=0.}\n\\] More precisely, the checked digit-to-lift equivalence is \\[\n\\boxed{\\mathcal L_v\\cong\\{u\\in S^m:Ju=-\\bar r-Kv\\}.}\n\\] It sends a solved digit \\(u\\) to the actual next residue represented\nby \\(x+q^L(\\widehat u,\\widehat v)\\). The obstruction depends on the\nsolved block \\(J\\), not on the range of the full map \\((J\\ K)\\).","=[\\bar r+Kv]\\in\\mathcal C_x.\n\\] Then \\[\n\\boxed{\\mathcal L_v\\ne\\varnothing\\iff o_x(v)=0.}\n\\] More precisely, the checked digit-to-lift equivalence is \\[\n\\boxed{\\mathcal L_v\\cong\\{u\\in S^m:Ju=-\\bar r-Kv\\}.}\n\\] It sends a solved digit \\(u\\) to the actual next residue represented\nby \\(x+q^L(\\widehat u,\\widehat v)\\). The obstruction depends on the\nsolved block \\(J\\), not on the range of the full map \\((J\\ K)\\).","Every nonempty \\(\\mathcal L_v\\) has a checked free and transitive action\nof \\(\\ker J\\). The action is defined without choosing a base lift.\nChoosing one valid solved digit then gives an explicit bijection\n\\(\\mathcal L_v\\cong\\ker J\\). In particular, for every positive modulus,\n\\[\no_x(v)=0\\quad\\Longrightarrow\\quad\\#\\mathcal L_v=\\#\\ker J.\n\\] These results permit composite \\(q\\), the degenerate modulus \\(q=1\\),\nand zero dimensions. Neither \\(J\\) nor the full reduced Jacobian is\nassumed surjective.","y nonempty \\(\\mathcal L_v\\) has a checked free and transitive action\nof \\(\\ker J\\). The action is defined without choosing a base lift.\nChoosing one valid solved digit then gives an explicit bijection\n\\(\\mathcal L_v\\cong\\ker J\\). In particular, for every positive modulus,\n\\[\no_x(v)=0\\quad\\Longrightarrow\\quad\\#\\mathcal L_v=\\#\\ker J.\n\\] These results permit composite \\(q\\), the degenerate modulus \\(q=1\\),\nand zero dimensions. Neither \\(J\\) nor the full reduced Jacobian is\nassumed surjective.","ked free and transitive action\nof \\(\\ker J\\). The action is defined without choosing a base lift.\nChoosing one valid solved digit then gives an explicit bijection\n\\(\\mathcal L_v\\cong\\ker J\\). In particular, for every positive modulus,\n\\[\no_x(v)=0\\quad\\Longrightarrow\\quad\\#\\mathcal L_v=\\#\\ker J.\n\\] These results permit composite \\(q\\), the degenerate modulus \\(q=1\\),\nand zero dimensions. Neither \\(J\\) nor the full reduced Jacobian is\nassumed surjective.","When \\(q=p\\) is prime, put \\(\\rho=\\operatorname{rank}_{\\mathbb F_p}J\\).\nThen \\[\n\\boxed{\\#\\mathcal L_v=\n\\begin{cases}\np^{m-\\rho},&o_x(v)=0,","When \\(q=p\\) is prime, put \\(\\rho=\\operatorname{rank}_{\\mathbb F_p}J\\).\nThen \\[\n\\boxed{\\#\\mathcal L_v=\n\\begin{cases}\np^{m-\\rho},&o_x(v)=0,\\\\\n0,&o_x(v)\\ne0,\n\\end{cases}}\n\\qquad\n\\boxed{\\#\\mathcal C_x=p^{e-\\rho}.}\n\\] The count is of distinct actual residue classes at level \\(L+1\\). The\nsolved dimension \\(m\\), rather than the total number \\(m+k\\) of\nvariables, appears in the exponent because the parameter digit is fixed.\nThese rank formulas are asserted only over prime fields.","0,&o_x(v)\\ne0,\n\\end{cases}}\n\\qquad\n\\boxed{\\#\\mathcal C_x=p^{e-\\rho}.}\n\\] The count is of distinct actual residue classes at level \\(L+1\\). The\nsolved dimension \\(m\\), rather than the total number \\(m+k\\) of\nvariables, appears in the exponent because the parameter digit is fixed.\nThese rank formulas are asserted only over prime fields.","The main wrapper assumes only the original lower polynomial congruence.\nIt constructs one residual \\(r\\) and supplies the criterion and count\nuniformly for every \\(v\\), together with the exact obstruction\ncardinality.","The actual derivative calculation gives \\[\n(DF(x)\\bmod q)(u,v)=Ju+Kv.\n\\] Taylor expansion at depth \\(L\\ge1\\) makes the quadratic remainder\ndivisible by \\(q^{L+1}\\). Therefore the digit \\((u,v)\\) is a polynomial\nlifting correction exactly when \\[\n\\bar r+Ju+Kv=0.\n\\] The quotient class \\([\\bar r+Kv]\\) vanishes exactly when this\nequation admits \\(u\\). This uses additive inverses in the image of\n\\(J\\), so the sign agrees with the actual Taylor equation.","The earlier unrestricted correction-digit equivalence is then restricted\nto the predicate that the parameter digit equals \\(v\\). Taking free\ncoordinates commutes with forming digit residues, and digit residues are\ninjective in the digit. Consequently the restricted equivalence enforces\nexactly the actual parameter residue condition on \\(\\mathcal L_v\\).","Translation by \\(\\ker J\\) acts freely and transitively on the\nsolved-digit fiber. Transporting this explicit action through the\nresidue equivalence yields the torsor on \\(\\mathcal L_v\\). Rank-nullity\nover \\(\\mathbb F_p\\) then gives the two displayed cardinalities. No\nnumerical enumeration is used to establish the general theorem.","Take \\(F(Y,T)=T\\), \\(q=2\\), \\(L=1\\), and \\(x=(0,0)\\). The full Jacobian\n\\((0\\ 1)\\) is surjective, but \\(J=0\\). Prescribing \\(v=1\\) forces\n\\(T\\equiv2\\pmod4\\), which cannot satisfy \\(F=0\\pmod4\\). Prescribing\n\\(v=0\\) gives exactly two lifts, corresponding to \\(Y\\equiv0,2\\pmod4\\).","Let \\(q>0\\), \\(L\\ge1\\), and \\[\nF\\in\\mathbb Z[Y_1,\\ldots,Y_m,T_1,\\ldots,T_k]^e,\n\\qquad x=(y_0,t_0)\\in\\mathbb Z^{m+k},\n\\qquad F(x)\\in q^L\\mathbb Z^e.\n\\] Write \\(S_q=\\mathbb Z/q\\mathbb Z\\), and let the actual reduced\nJacobian blocks be \\[\nJ_x=D_YF(x)\\bmod q:S_q^m\\longrightarrow S_q^e,\n\\qquad K_x=D_TF(x)\\bmod q:S_q^k\\longrightarrow S_q^e.\n\\] Assume \\(J_x\\) is surjective. A vector divisibility condition below\nis coordinatewise.","\\textbf{Integer prescribed-path theorem.} For every supplied path\n\\(t:\\mathbb N\\to\\mathbb Z^k\\) satisfying \\[\nt(0)=t_0,\\qquad t(n+1)-t(n)\\in q^{L+n}\\mathbb Z^k,\n\\] there is a single tower \\(z:\\mathbb N\\to\\mathbb Z^{m+k}\\) with \\[\nz(0)=x,\\qquad \\operatorname{free}(z(n))=t(n),\n\\] \\[\nF(z(n))\\in q^{L+n}\\mathbb Z^e,\\qquad\nz(n+1)-z(n)\\in q^{L+n}\\mathbb Z^{m+k}.\n\\] The parameter equality is exact in integers. For \\(a\\le b\\), \\[\nz(b)-z(a)\\in q^{L+a}\\mathbb Z^{m+k},\\qquad\nDF(z(n))\\bmod q=DF(x)\\bmod q.\n\\] Consequently \\(J_{z(n)}=J_x\\) at every level. No later surjectivity\nor per-step solvability is assumed. This theorem includes positive\ncomposite bases and \\(q=1\\).","ger prescribed-path theorem.} For every supplied path\n\\(t:\\mathbb N\\to\\mathbb Z^k\\) satisfying \\[\nt(0)=t_0,\\qquad t(n+1)-t(n)\\in q^{L+n}\\mathbb Z^k,\n\\] there is a single tower \\(z:\\mathbb N\\to\\mathbb Z^{m+k}\\) with \\[\nz(0)=x,\\qquad \\operatorname{free}(z(n))=t(n),\n\\] \\[\nF(z(n))\\in q^{L+n}\\mathbb Z^e,\\qquad\nz(n+1)-z(n)\\in q^{L+n}\\mathbb Z^{m+k}.\n\\] The parameter equality is exact in integers. For \\(a\\le b\\), \\[\nz(b)-z(a)\\in q^{L+a}\\mathbb Z^{m+k},\\qquad\nDF(z(n))\\bmod q=DF(x)\\bmod q.\n\\] Consequently \\(J_{z(n)}=J_x\\) at every level. No later surjectivity\nor per-step solvability is assumed. This theorem includes positive\ncomposite bases and \\(q=1\\).","ed-path theorem.} For every supplied path\n\\(t:\\mathbb N\\to\\mathbb Z^k\\) satisfying \\[\nt(0)=t_0,\\qquad t(n+1)-t(n)\\in q^{L+n}\\mathbb Z^k,\n\\] there is a single tower \\(z:\\mathbb N\\to\\mathbb Z^{m+k}\\) with \\[\nz(0)=x,\\qquad \\operatorname{free}(z(n))=t(n),\n\\] \\[\nF(z(n))\\in q^{L+n}\\mathbb Z^e,\\qquad\nz(n+1)-z(n)\\in q^{L+n}\\mathbb Z^{m+k}.\n\\] The parameter equality is exact in integers. For \\(a\\le b\\), \\[\nz(b)-z(a)\\in q^{L+a}\\mathbb Z^{m+k},\\qquad\nDF(z(n))\\bmod q=DF(x)\\bmod q.\n\\] Consequently \\(J_{z(n)}=J_x\\) at every level. No later surjectivity\nor per-step solvability is assumed. This theorem includes positive\ncomposite bases and \\(q=1\\).","_0,\\qquad t(n+1)-t(n)\\in q^{L+n}\\mathbb Z^k,\n\\] there is a single tower \\(z:\\mathbb N\\to\\mathbb Z^{m+k}\\) with \\[\nz(0)=x,\\qquad \\operatorname{free}(z(n))=t(n),\n\\] \\[\nF(z(n))\\in q^{L+n}\\mathbb Z^e,\\qquad\nz(n+1)-z(n)\\in q^{L+n}\\mathbb Z^{m+k}.\n\\] The parameter equality is exact in integers. For \\(a\\le b\\), \\[\nz(b)-z(a)\\in q^{L+a}\\mathbb Z^{m+k},\\qquad\nDF(z(n))\\bmod q=DF(x)\\bmod q.\n\\] Consequently \\(J_{z(n)}=J_x\\) at every level. No later surjectivity\nor per-step solvability is assumed. This theorem includes positive\ncomposite bases and \\(q=1\\).","free}(z(n))=t(n),\n\\] \\[\nF(z(n))\\in q^{L+n}\\mathbb Z^e,\\qquad\nz(n+1)-z(n)\\in q^{L+n}\\mathbb Z^{m+k}.\n\\] The parameter equality is exact in integers. For \\(a\\le b\\), \\[\nz(b)-z(a)\\in q^{L+a}\\mathbb Z^{m+k},\\qquad\nDF(z(n))\\bmod q=DF(x)\\bmod q.\n\\] Consequently \\(J_{z(n)}=J_x\\) at every level. No later surjectivity\nor per-step solvability is assumed. This theorem includes positive\ncomposite bases and \\(q=1\\).","\\textbf{Arbitrary p-adic target theorem.} Now let \\(q=p\\) be prime. For\nevery \\[\n\\tau\\in t_0+p^L\\mathbb Z_p^k,\n\\] there are a compatible integer tower \\(z\\) as above and\n\\(b\\in\\mathbb Z_p^{m+k}\\) such that \\[\nz(n)\\longrightarrow b,\\qquad F(b)=0,\\qquad\n\\operatorname{free}(b)=\\tau,\n\\qquad b-z(n)\\in p^{L+n}\\mathbb Z_p^{m+k}\\quad(n\\ge0).\n\\] A compatible integer parameter path converging to \\(\\tau\\) is\nconstructed in the proof, with its initial representative fixed exactly.\nA separate theorem accepts every supplied compatible integer path with\nlimit \\(\\tau\\) and retains its exact coordinates in the constructed\ntower.","z\\) as above and\n\\(b\\in\\mathbb Z_p^{m+k}\\) such that \\[\nz(n)\\longrightarrow b,\\qquad F(b)=0,\\qquad\n\\operatorname{free}(b)=\\tau,\n\\qquad b-z(n)\\in p^{L+n}\\mathbb Z_p^{m+k}\\quad(n\\ge0).\n\\] A compatible integer parameter path converging to \\(\\tau\\) is\nconstructed in the proof, with its initial representative fixed exactly.\nA separate theorem accepts every supplied compatible integer path with\nlimit \\(\\tau\\) and retains its exact coordinates in the constructed\ntower.","\\textbf{Exact branching.} At any stage \\(n\\), put \\(N=L+n\\). For every\nparameter digit \\(v\\in\\mathbb F_p^k\\), let \\[\n\\mathcal L_{n,v}=\\left\\{\\bar w\\in(\\mathbb Z/p^{N+1}\\mathbb Z)^{m+k}:\n\\begin{array}{l}\n\\bar w\\equiv z(n)\\pmod{p^N},","\\textbf{Exact branching.} At any stage \\(n\\), put \\(N=L+n\\). For every\nparameter digit \\(v\\in\\mathbb F_p^k\\), let \\[\n\\mathcal L_{n,v}=\\left\\{\\bar w\\in(\\mathbb Z/p^{N+1}\\mathbb Z)^{m+k}:\n\\begin{array}{l}\n\\bar w\\equiv z(n)\\pmod{p^N},\\\\\nF(\\bar w)=0,\\\\\n\\operatorname{free}(\\bar w)=t(n)+p^Nv\\pmod{p^{N+1}}\n\\end{array}\\right\\}.\n\\] Here \\(p^Nv\\) means any integer representative of \\(v\\), multiplied\nby \\(p^N\\). The residue class does not depend on the representative.\nThen \\[\n\\#\\mathcal L_{n,v}=p^{m-e}\\qquad\\text{for every }n\\text{ and every }v.\n\\] Surjectivity implies \\(e\\le m\\). This is a count of distinct actual\nnext-level residue classes. It does not count integer representatives or\nentire infinite towers.","\\operatorname{free}(\\bar w)=t(n)+p^Nv\\pmod{p^{N+1}}\n\\end{array}\\right\\}.\n\\] Here \\(p^Nv\\) means any integer representative of \\(v\\), multiplied\nby \\(p^N\\). The residue class does not depend on the representative.\nThen \\[\n\\#\\mathcal L_{n,v}=p^{m-e}\\qquad\\text{for every }n\\text{ and every }v.\n\\] Surjectivity implies \\(e\\le m\\). This is a count of distinct actual\nnext-level residue classes. It does not count integer representatives or\nentire infinite towers.","\\textbf{Square solved-block uniqueness.} If \\(m=e\\), every prescribed\n\\(\\tau\\) has exactly one root in the original ball: \\[\n\\exists!\\,b\\in x+p^L\\mathbb Z_p^{e+k}:\\quad\nF(b)=0,\\qquad\\operatorname{free}(b)=\\tau.\n\\] Thus all valid integer parameter approximations and all lifting\nchoices with the same target have the same endpoint. This does not\nassert uniqueness of the integer representatives in the tower. All prime\nstatements include \\(p=2\\).","} If \\(m=e\\), every prescribed\n\\(\\tau\\) has exactly one root in the original ball: \\[\n\\exists!\\,b\\in x+p^L\\mathbb Z_p^{e+k}:\\quad\nF(b)=0,\\qquad\\operatorname{free}(b)=\\tau.\n\\] Thus all valid integer parameter approximations and all lifting\nchoices with the same target have the same endpoint. This does not\nassert uniqueness of the integer representatives in the tower. All prime\nstatements include \\(p=2\\).","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative g","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced J","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJa","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, const","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof u","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not supply an\noptimized computational lifting algorit","At depth \\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not supply an\noptimized computational lifting algorithm.","\\(N\\ge1\\), suppose \\(F(a)=q^Nr\\) and a desired next integer\nparameter target satisfies \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not supply an\noptimized computational lifting algorithm.","s \\(t'-\\operatorname{free}(a)=q^Nw\\), with\n\\(w\\in\\mathbb Z^k\\). Surjectivity supplies a solved digit \\(u\\in S_q^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not supply an\noptimized computational lifting algorithm.","^m\\)\nsatisfying \\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not supply an\noptimized computational lifting algorithm.","\\[\nJ_a u=-\\bar r-K_a\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not supply an\noptimized computational lifting algorithm.","\\bar w.\n\\] Choose an integer representative \\(\\widetilde u\\), and set \\[\na'=a+q^N(\\widetilde u,w).\n\\] The parameter quotient is the supplied integer \\(w\\). Therefore\n\\(\\operatorname{free}(a')=t'\\) exactly. The checked polynomial Taylor\nidentity gives \\[\nF(a')\\equiv F(a)+q^NDF(a)(\\widetilde u,w)\n\\equiv0\\pmod{q^{N+1}}.\n\\] Its quadratic remainder is divisible by \\(q^{2N}\\), and\n\\(2N\\ge N+1\\). Also \\(a'\\equiv a\\pmod q\\), so evaluation of every\npolynomial partial derivative gives the same reduced Jacobian. Induction\non the level, carrying the root, exact parameters, and fixed solved\nJacobian as its state, constructs the entire integer tower. The\nexistence proof uses ordinary classical choice. It does not supply an\noptimized computational lifting algorithm.","For a p-adic target \\(\\tau\\), use \\(t(0)=t_0\\) and, for \\(n\\ge1\\), the\ncanonical integer representatives of \\(\\tau\\bmod p^{L+n}\\). The initial\ncongruence hypothesis and the truncation precision imply \\[\n\\tau-t(n)\\in p^{L+n}\\mathbb Z_p^k.\n\\] Subtracting successive precision relations gives compatibility in\n\\(\\mathbb Z_p\\). The checked faithful integer-divisibility theorem gives\ncompatibility in \\(\\mathbb Z\\). These precision relations also give\nconvergence to \\(\\tau\\).",", the\ncanonical integer representatives of \\(\\tau\\bmod p^{L+n}\\). The initial\ncongruence hypothesis and the truncation precision imply \\[\n\\tau-t(n)\\in p^{L+n}\\mathbb Z_p^k.\n\\] Subtracting successive precision relations gives compatibility in\n\\(\\mathbb Z_p\\). The checked faithful integer-divisibility theorem gives\ncompatibility in \\(\\mathbb Z\\). These precision relations also give\nconvergence to \\(\\tau\\).","The compatible vector-tower completion theorem constructs \\(b\\), proves\nconvergence, and retains every stage precision. Continuous polynomial\nevaluation and the increasing root precision give \\(F(b)=0\\). Each\nparameter coordinate has both the corresponding coordinate of \\(b\\) and\n\\(\\tau\\) as its limit, so uniqueness of limits gives exact parameter\nequality.","For every proposed next parameter digit, the previously checked\nobstruction is \\([\\bar r+K_av]\\in S_p^e/\\operatorname{im}J_a\\). The\nfixed solved Jacobian is surjective, so this class vanishes for every\ndigit at every stage. The actual lift fiber is a torsor for\n\\(\\ker J_a\\). Rank-nullity gives its cardinality \\(p^{m-e}\\).","When \\(m=e\\), the new solved-Jacobian definition is definitionally the\nexisting square block used by the checked continuous parameter chart.\nEvery root in the initial ball with the given parameters is that chart\npoint. Applying this to any two roots proves equality, including\nendpoints obtained by different constructions.","The inherited checked example \\(F(Y,T)=T\\), \\(p=2\\), \\(x=0\\), \\(L=1\\)\nhas a surjective full Jacobian but zero solved block. Prescribing\nparameter digit \\(1\\) forces \\(T=2\\pmod4\\), so no next root exists.\nFull-Jacobian surjectivity cannot replace the hypothesis on the\nprescribed solved block.","The new checked example \\(F(Y_1,Y_2,T)=Y_1\\) has a surjective solved\nblock \\((1,0)\\). Nevertheless \\((0,0,0)\\) and \\((0,2,0)\\) are distinct\nexact roots in the same modulo-two ball with identical parameters.\nUniqueness therefore requires the square solved case. The corresponding\nnegative control deliberately attempts the stronger false conclusion and\nmust be rejected by Lean.","Depth \\(L\\ge1\\) is essential for the Taylor step used here. The general\nsingular case can have a nonzero next obstruction after an earlier zero\nobstruction. Neither the finite one-step criterion nor this smooth\ntheorem establishes unrestricted singular persistence.","Let \\(p\\) be any prime, \\(R=\\mathbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral v","Let \\(p\\) be any prime, \\(R=\\mathbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}","Let \\(p\\) be any prime, \\(R=\\mathbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q","Let \\(p\\) be any prime, \\(R=\\mathbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same origi","Let \\(p\\) be any prime, \\(R=\\mathbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\","Let \\(p\\) be any prime, \\(R=\\mathbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","rime, \\(R=\\mathbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","hbb Z_p\\), and \\(Q=\\mathbb Q_p\\). Let \\[\nF,H\\in R[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","\\qquad a=(y_0,t_0)\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.",")\\in R^{e+k},\\qquad F(a)=0.\n\\] Here \\(e,k\\ge0\\) are arbitrary finite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","inite dimensions. There are as many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","s many\nsolved coordinates as equations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","ations. Define the actual derivative blocks at\nthe original root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","l root: \\[\nJ=D_YF(a)\\in M_e(R),\\qquad K=D_TF(a)\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.",")\\in M_{e\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","\\times k}(R).\n\\] Suppose, for natural numbers \\(s,M\\), that \\[\nJ\\in\\operatorname{GL}_e(Q),\\qquad p^sJ^{-1}\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","\\in M_e(R),\\qquad M>2s.\n\\] For every \\(v\\in R^k\\), set \\[\n\\tau=t_0+p^Mv,\\qquad G=F+p^MH.\n\\] Then there is exactly one \\(y\\in y_0+p^{s+1}R^e\\) with \\[\nG(y,\\tau)=0.\n\\] It satisfies \\[\ny-y_0\\in p^{M-s}R^e\n\\] and there is an integral vector \\(\\varepsilon\\in p^{2M-3s}R^e\\) such\nthat, in \\(Q^e\\), \\[\n\\boxed{\\displaystyle\n y-y_0=-p^MJ^{-1}\\bigl(H(a)+Kv\\bigr)+\\varepsilon.\n}\n\\] The same original exponent \\(s\\) appears throughout. No inverse or\nrank assumption on the perturbed Jacobian is imposed. The statement\nincludes arbitrary integral p-adic coefficients and centers, the dyadic\nprime, and zero dimensions.","The full-point theorem constructs \\(b=(y,\\tau)\\) and also proves \\[\nb-a\\in p^{M-s}R^{e+k}\\cap p^{s+1}R^{e+k}.\n\\] Parameter equality \\(b_T=\\tau\\) is exact. Uniqueness concerns every\nroot in the stated solved-coordinate ball with these parameters, not\nonly roots obtained by the construction. No global uniqueness is\nclaimed.","Define polynomial systems with \\(e+k\\) equations and variables by \\[\n\\mathcal F(Y,T)=\\bigl(F(Y,T),T-t_0\\bigr),\\qquad\n\\mathcal H(Y,T)=\\bigl(H(Y,T),-v\\bigr).\n\\] Both are actual polynomials over \\(R\\). Direct evaluation proves \\[\n\\mathcal F(a)=0,\n\\qquad\n(\\mathcal F+p^M\\mathcal H)(b)=0\n\\iff F(b)+p^MH(b)=0\\ \\text{and}\\ b_T=t_0+p^Mv.\n\\] Actual polynomial partial differentiation gives \\[\nD\\mathcal F(a)=B=\\begin{pmatrix}J&K","Define polynomial systems with \\(e+k\\) equations and variables by \\[\n\\mathcal F(Y,T)=\\bigl(F(Y,T),T-t_0\\bigr),\\qquad\n\\mathcal H(Y,T)=\\bigl(H(Y,T),-v\\bigr).\n\\] Both are actual polynomials over \\(R\\). Direct evaluation proves \\[\n\\mathcal F(a)=0,\n\\qquad\n(\\mathcal F+p^M\\mathcal H)(b)=0\n\\iff F(b)+p^MH(b)=0\\ \\text{and}\\ b_T=t_0+p^Mv.\n\\] Actual polynomial partial differentiation gives \\[\nD\\mathcal F(a)=B=\\begin{pmatrix}J&K\\\\0&I_k\\end{pmatrix}.\n\\] These identities are proved from evaluation and partial derivatives,\nrather than supplied as additional hypotheses.","Choose the integral matrix \\(A=p^sJ^{-1}\\). The existing\ninverse-certificate equivalence gives \\[\nAJ=JA=p^sI_e.\n\\] Put \\(c=p^s\\) and define an integral block matrix \\[\nC=\\begin{pmatrix}A&-AK","Choose the integral matrix \\(A=p^sJ^{-1}\\). The existing\ninverse-certificate equivalence gives \\[\nAJ=JA=p^sI_e.\n\\] Put \\(c=p^s\\) and define an integral block matrix \\[\nC=\\begin{pmatrix}A&-AK\\\\0&cI_k\\end{pmatrix}.\n\\] Then \\[\nCB=\\begin{pmatrix}AJ&AK-AK","0&cI_k\\end{pmatrix}.\n\\] Then \\[\nCB=\\begin{pmatrix}AJ&AK-AK\\\\0&cI_k\\end{pmatrix}=cI_{e+k},\n\\] \\[\nBC=\\begin{pmatrix}JA&-JAK+cK","0&cI_k\\end{pmatrix}=cI_{e+k},\n\\] \\[\nBC=\\begin{pmatrix}JA&-JAK+cK\\\\0&cI_k\\end{pmatrix}=cI_{e+k}.\n\\] The second equality uses \\(JA=cI_e\\). The generic block proof works\nover any commutative ring. The p-adic specialization also explicitly\nproves that \\(B\\) is invertible over \\(Q\\) and that \\(p^sB^{-1}\\) is\nintegral. There is no denominator loss from appending the parameter\nequations.","0&cI_k\\end{pmatrix}=cI_{e+k}.\n\\] The second equality uses \\(JA=cI_e\\). The generic block proof works\nover any commutative ring. The p-adic specialization also explicitly\nproves that \\(B\\) is invertible over \\(Q\\) and that \\(p^sB^{-1}\\) is\nintegral. There is no denominator loss from appending the parameter\nequations.","Apply the previously checked weighted coefficient-perturbation\nconstruction to \\(\\mathcal F,\\mathcal H,a,C\\). Its hypotheses follow\nfrom the actual augmented identities and the two-sided certificate. It\nconstructs \\(b\\), proves the full displacement bound \\(p^{M-s}\\), and\nproves uniqueness throughout \\(a+p^{s+1}R^{e+k}\\). The augmented root\nequivalence gives the required original equations and exact parameter\nvalues.","The checked integral first-order estimate gives \\[\nb-a+p^{M-s}C\\mathcal H(a)\\in p^{2M-3s}R^{e+k}.\n\\] The decisive block action is \\[\nC\\binom{H(a)}{-v}\n=\n\\binom{A\\bigl(H(a)+Kv\\bigr)}{-p^sv}.\n\\] Projecting the solved coordinates therefore gives \\[\ny-y_0+p^{M-s}A\\bigl(H(a)+Kv\\bigr)\n\\in p^{2M-3s}R^e.\n\\] The existing rational-conversion theorem turns \\(p^{M-s}A\\) into\n\\(p^MJ^{-1}\\) and retains an integral error vector. This proves the\ndisplayed first-order formula with the original derivative blocks.","For any competing solved-coordinate root \\(z\\in y_0+p^{s+1}R^e\\), the\njoined point \\((z,\\tau)\\) lies in the augmented uniqueness ball:\n\\(M>2s\\) implies \\(M\\ge s+1\\), so \\(\\tau-t_0=p^Mv\\in p^{s+1}R^k\\).\nFull-point uniqueness gives \\((z,\\tau)=b\\), hence \\(z=y\\). This is the\nadditional comparison needed to obtain uniqueness in exactly the ball\nstated in the solved-coordinate theorem.","For \\(p=2\\), \\(s=0\\), \\(F(Y,T)=Y+T\\), \\(H=1\\), \\(a=0\\), and \\(v=1\\),\nboth original blocks satisfy \\(J=K=1\\). At \\(M=1\\), the exact parameter\nis \\(\\tau=2\\), and the unique solved value is \\(y=-4\\). The combined\ncorrection is \\(-2(1+1)=-4\\), with zero error.","The sign controls use \\(M=2\\): the actual augmented root is\n\\((y,\\tau)=(-8,4)\\), and the required error ideal is \\(16\\mathbb Z_2\\).\nOmitting \\(Kv\\) predicts \\(-4\\), with error \\(-4\\notin16\\mathbb Z_2\\).\nReversing its sign predicts \\(0\\), with error \\(-8\\notin16\\mathbb Z_2\\).\nBoth failures are checked. At binary depth \\(M=1\\), the reversed-sign\ndifference would lie in the allowed error ideal, so that shallower\nexample alone would not distinguish the sign. This control-design\nlimitation was identified and corrected before the release gate.","For parameter-only motion, use \\[\nF(Y,T)=Y+Y^2+T,\\quad H=0,\\quad a=0,\\quad v=1,\\quad p=2,\\ s=0,\\ M=1.\n\\] A checked transfer from the prior scalar quadratic root constructs an\neven \\(y\\) with \\[\ny+y^2+2=0,\\qquad 4\\mid y+2,\\qquad 8\\nmid y+2.\n\\] Thus the error exponent \\(2M-3s=2\\) is attained even when only\nparameters move. No uniform extra error digit can be added to the\ngeneral theorem. This example establishes that limitation. It does not\nassert optimality at every choice of \\(p,s,M\\).","ad p=2,\\ s=0,\\ M=1.\n\\] A checked transfer from the prior scalar quadratic root constructs an\neven \\(y\\) with \\[\ny+y^2+2=0,\\qquad 4\\mid y+2,\\qquad 8\\nmid y+2.\n\\] Thus the error exponent \\(2M-3s=2\\) is attained even when only\nparameters move. No uniform extra error digit can be added to the\ngeneral theorem. This example establishes that limitation. It does not\nassert optimality at every choice of \\(p,s,M\\).","The argument establishes both analyticity and differentiability of the\nactual local polynomial solution map at a root with an invertible solved\nJacobian. The statements apply to every prime, including \\(p=2\\). They\nalso hold over any complete nontrivially normed field.","Let \\(\\mathbb K\\) be such a field, let \\(e,k\\in\\mathbb N\\), and let \\[\nF\\in\\mathbb K[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in\\mathbb K^e\\times\\mathbb K^k.\n\\] Evaluation of \\(F\\) means coordinatewise evaluation of its actual\npolynomials. Define the actual derivative blocks \\[\nJ=D_YF(a)\\in M_e(\\mathbb K),\n\\qquad K=D_TF(a)\\in M_{e\\times k}(\\mathbb K),\n\\] where each matrix entry is the corresponding formal partial\nderivative evaluated at \\(a\\). The hypotheses are exactly \\[\nF(a)=0,\n\\qquad J\\in\\operatorname{GL}_e(\\mathbb K).\n\\] Neither analyticity of a proposed solution nor a derivative formula\nis assumed.","N\\), and let \\[\nF\\in\\mathbb K[Y_1,\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in\\mathbb K^e\\times\\mathbb K^k.\n\\] Evaluation of \\(F\\) means coordinatewise evaluation of its actual\npolynomials. Define the actual derivative blocks \\[\nJ=D_YF(a)\\in M_e(\\mathbb K),\n\\qquad K=D_TF(a)\\in M_{e\\times k}(\\mathbb K),\n\\] where each matrix entry is the corresponding formal partial\nderivative evaluated at \\(a\\). The hypotheses are exactly \\[\nF(a)=0,\n\\qquad J\\in\\operatorname{GL}_e(\\mathbb K).\n\\] Neither analyticity of a proposed solution nor a derivative formula\nis assumed.","\\ldots,Y_e,T_1,\\ldots,T_k]^e,\n\\qquad a=(y_0,t_0)\\in\\mathbb K^e\\times\\mathbb K^k.\n\\] Evaluation of \\(F\\) means coordinatewise evaluation of its actual\npolynomials. Define the actual derivative blocks \\[\nJ=D_YF(a)\\in M_e(\\mathbb K),\n\\qquad K=D_TF(a)\\in M_{e\\times k}(\\mathbb K),\n\\] where each matrix entry is the corresponding formal partial\nderivative evaluated at \\(a\\). The hypotheses are exactly \\[\nF(a)=0,\n\\qquad J\\in\\operatorname{GL}_e(\\mathbb K).\n\\] Neither analyticity of a proposed solution nor a derivative formula\nis assumed.","\\textbf{Local analytic graph theorem.} There exist a map \\[\n\\psi:\\mathbb K^k\\longrightarrow\\mathbb K^e\n\\] and open sets \\(U\\subseteq\\mathbb K^k\\), \\(V\\subseteq\\mathbb K^e\\)\nsuch that \\[\nt_0\\in U,\\qquad y_0\\in V,\\qquad \\psi(t_0)=y_0,\n\\qquad \\psi(U)\\subseteq V.\n\\] The map is analytic at every point of \\(U\\), and its graph describes\nall roots in the product neighborhood: \\[\n\\forall t\\in U\\;\\forall y\\in V,\n\\qquad F(y,t)=0\\quad\\Longleftrightarrow\\quad y=\\psi(t).\n\\] At the base parameter it has the strict Fréchet derivative \\[\n\\boxed{D\\psi(t_0)=-J^{-1}K.}\n\\] Consequently it is differentiable on \\(U\\). The map is defined on the\nambient parameter space, but its root, analyticity and uniqueness\nconclusions are restricted to the specified neighborhood. Uniqueness is\namong solved coordinates in \\(V\\).","bseteq\\mathbb K^k\\), \\(V\\subseteq\\mathbb K^e\\)\nsuch that \\[\nt_0\\in U,\\qquad y_0\\in V,\\qquad \\psi(t_0)=y_0,\n\\qquad \\psi(U)\\subseteq V.\n\\] The map is analytic at every point of \\(U\\), and its graph describes\nall roots in the product neighborhood: \\[\n\\forall t\\in U\\;\\forall y\\in V,\n\\qquad F(y,t)=0\\quad\\Longleftrightarrow\\quad y=\\psi(t).\n\\] At the base parameter it has the strict Fréchet derivative \\[\n\\boxed{D\\psi(t_0)=-J^{-1}K.}\n\\] Consequently it is differentiable on \\(U\\). The map is defined on the\nambient parameter space, but its root, analyticity and uniqueness\nconclusions are restricted to the specified neighborhood. Uniqueness is\namong solved coordinates in \\(V\\).","uad y_0\\in V,\\qquad \\psi(t_0)=y_0,\n\\qquad \\psi(U)\\subseteq V.\n\\] The map is analytic at every point of \\(U\\), and its graph describes\nall roots in the product neighborhood: \\[\n\\forall t\\in U\\;\\forall y\\in V,\n\\qquad F(y,t)=0\\quad\\Longleftrightarrow\\quad y=\\psi(t).\n\\] At the base parameter it has the strict Fréchet derivative \\[\n\\boxed{D\\psi(t_0)=-J^{-1}K.}\n\\] Consequently it is differentiable on \\(U\\). The map is defined on the\nambient parameter space, but its root, analyticity and uniqueness\nconclusions are restricted to the specified neighborhood. Uniqueness is\namong solved coordinates in \\(V\\).","y\\in V,\n\\qquad F(y,t)=0\\quad\\Longleftrightarrow\\quad y=\\psi(t).\n\\] At the base parameter it has the strict Fréchet derivative \\[\n\\boxed{D\\psi(t_0)=-J^{-1}K.}\n\\] Consequently it is differentiable on \\(U\\). The map is defined on the\nambient parameter space, but its root, analyticity and uniqueness\nconclusions are restricted to the specified neighborhood. Uniqueness is\namong solved coordinates in \\(V\\).","\\textbf{Proof chain.} The scalar polynomial derivative bridge is proved\nby induction on actual multivariate polynomials: constants, sums, and\nmultiplication by a variable. For a scalar polynomial \\(P\\), its strict\nderivative is \\[\nv\\longmapsto\\sum_j \\frac{\\partial P}{\\partial x_j}(a)v_j.\n\\] Assembling the coordinates gives the strict derivative of the\nevaluated system as its actual Jacobian matrix. Polynomial analyticity\nis supplied by the pinned Mathlib theorem for multivariate polynomial\nevaluation, followed by finite coordinate assembly.","Consider the actual augmented polynomial map \\[\n\\Phi(y,t)=(F(y,t),t).\n\\] Its derivative and inverse derivative at \\(a\\) are \\[\nD\\Phi(a)=\n\\begin{pmatrix}J&K","Consider the actual augmented polynomial map \\[\n\\Phi(y,t)=(F(y,t),t).\n\\] Its derivative and inverse derivative at \\(a\\) are \\[\nD\\Phi(a)=\n\\begin{pmatrix}J&K\\\\0&I_k\\end{pmatrix},\n\\qquad\nD\\Phi(a)^{-1}=\n\\begin{pmatrix}J^{-1}&-J^{-1}K\\\\0&I_k\\end{pmatrix}.\n\\] The proof constructs the continuous linear equivalence from these\nexplicit matrices and verifies both inverse identities. The strict\ninverse theorem gives a local inverse \\(g\\) of \\(\\Phi\\), and the\nanalytic inverse result shows that this same inverse is analytic near\n\\(\\Phi(a)=(0,t_0)\\). Set \\[\n\\psi(t)=\\pi_Y g(0,t).\n\\] The two inverse identities and a sufficiently small product\nneighborhood give the exact local graph statement. Composing the\nderivative of \\(g\\) with parameter insertion and solved-coordinate\nprojection yields \\(-J^{-1}K\\). The chosen parameter neighborhood is\nalso restricted so that the inverse identities, analyticity and\ninclusion \\(\\psi(U)\\subseteq V\\) hold there simulta","0&I_k\\end{pmatrix}.\n\\] The proof constructs the continuous linear equivalence from these\nexplicit matrices and verifies both inverse identities. The strict\ninverse theorem gives a local inverse \\(g\\) of \\(\\Phi\\), and the\nanalytic inverse result shows that this same inverse is analytic near\n\\(\\Phi(a)=(0,t_0)\\). Set \\[\n\\psi(t)=\\pi_Y g(0,t).\n\\] The two inverse identities and a sufficiently small product\nneighborhood give the exact local graph statement. Composing the\nderivative of \\(g\\) with parameter insertion and solved-coordinate\nprojection yields \\(-J^{-1}K\\). The chosen parameter neighborhood is\nalso restricted so that the inverse identities, analyticity and\ninclusion \\(\\psi(U)\\subseteq V\\) hold there simultaneously.","ese\nexplicit matrices and verifies both inverse identities. The strict\ninverse theorem gives a local inverse \\(g\\) of \\(\\Phi\\), and the\nanalytic inverse result shows that this same inverse is analytic near\n\\(\\Phi(a)=(0,t_0)\\). Set \\[\n\\psi(t)=\\pi_Y g(0,t).\n\\] The two inverse identities and a sufficiently small product\nneighborhood give the exact local graph statement. Composing the\nderivative of \\(g\\) with parameter insertion and solved-coordinate\nprojection yields \\(-J^{-1}K\\). The chosen parameter neighborhood is\nalso restricted so that the inverse identities, analyticity and\ninclusion \\(\\psi(U)\\subseteq V\\) hold there simultaneously.","inverse theorem gives a local inverse \\(g\\) of \\(\\Phi\\), and the\nanalytic inverse result shows that this same inverse is analytic near\n\\(\\Phi(a)=(0,t_0)\\). Set \\[\n\\psi(t)=\\pi_Y g(0,t).\n\\] The two inverse identities and a sufficiently small product\nneighborhood give the exact local graph statement. Composing the\nderivative of \\(g\\) with parameter insertion and solved-coordinate\nprojection yields \\(-J^{-1}K\\). The chosen parameter neighborhood is\nalso restricted so that the inverse identities, analyticity and\ninclusion \\(\\psi(U)\\subseteq V\\) hold there simultaneously.","ytic near\n\\(\\Phi(a)=(0,t_0)\\). Set \\[\n\\psi(t)=\\pi_Y g(0,t).\n\\] The two inverse identities and a sufficiently small product\nneighborhood give the exact local graph statement. Composing the\nderivative of \\(g\\) with parameter insertion and solved-coordinate\nprojection yields \\(-J^{-1}K\\). The chosen parameter neighborhood is\nalso restricted so that the inverse identities, analyticity and\ninclusion \\(\\psi(U)\\subseteq V\\) hold there simultaneously.","\\textbf{Transfer to a supplied continuous solution.} Suppose\n\\(g:\\mathbb K^k\\to\\mathbb K^e\\) is continuous at \\(t_0\\),\n\\(g(t_0)=y_0\\), and \\[\nF(g(t),t)=0\n\\] for all \\(t\\) in some neighborhood of \\(t_0\\). Under the same\ninvertibility hypothesis at \\(a\\), continuity places \\(g(t)\\) in the\nuniqueness neighborhood \\(V\\) for all sufficiently close parameters.\nHence \\(g\\) and \\(\\psi\\) agree on a neighborhood of \\(t_0\\). The theorem\n\\mcode{continuous_selection_regularity} transfers both analyticity\nand the strict derivative formula to \\(g\\).","nsfer to a supplied continuous solution.} Suppose\n\\(g:\\mathbb K^k\\to\\mathbb K^e\\) is continuous at \\(t_0\\),\n\\(g(t_0)=y_0\\), and \\[\nF(g(t),t)=0\n\\] for all \\(t\\) in some neighborhood of \\(t_0\\). Under the same\ninvertibility hypothesis at \\(a\\), continuity places \\(g(t)\\) in the\nuniqueness neighborhood \\(V\\) for all sufficiently close parameters.\nHence \\(g\\) and \\(\\psi\\) agree on a neighborhood of \\(t_0\\). The theorem\n\\mcode{continuous_selection_regularity} transfers both analyticity\nand the strict derivative formula to \\(g\\).","nd \\[\nF(g(t),t)=0\n\\] for all \\(t\\) in some neighborhood of \\(t_0\\). Under the same\ninvertibility hypothesis at \\(a\\), continuity places \\(g(t)\\) in the\nuniqueness neighborhood \\(V\\) for all sufficiently close parameters.\nHence \\(g\\) and \\(\\psi\\) agree on a neighborhood of \\(t_0\\). The theorem\n\\mcode{continuous_selection_regularity} transfers both analyticity\nand the strict derivative formula to \\(g\\).","There is also an explicit open-set version. Let \\(U\\) be open, let \\(g\\)\nbe continuous on \\(U\\), and suppose for every \\(t\\in U\\) that \\[\nF(g(t),t)=0,\n\\qquad\nJ_t:=D_YF(g(t),t)\\in\\operatorname{GL}_e(\\mathbb K).\n\\] Writing \\(K_t=D_TF(g(t),t)\\), the theorem\n\\mcode{continuous_selection_analyticOnNhd} proves \\[\ng\\text{ is analytic at every }t\\in U,\n\\qquad Dg(t)=-J_t^{-1}K_t\\quad(t\\in U),\n\\] with a strict Fréchet derivative at each point. The invertibility hypothesis at each point is part of this theorem. This is a transfer\nstatement for continuous local root selections satisfying its\nhypotheses. It does not assert equality of arbitrary global branch\nchoices.","\\textbf{Integral p-adic specialization.} For any prime \\(p\\), let \\[\nF\\in\\mathbb Z_p[Y,T]^e,\n\\qquad a\\in\\mathbb Z_p^{e+k},\n\\qquad F(a)=0.\n\\] Let \\(\\iota:\\mathbb Z_p\\hookrightarrow\\mathbb Q_p\\) be the\ncoefficient inclusion, applied entrywise to points and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferenti","\\textbf{Integral p-adic specialization.} For any prime \\(p\\), let \\[\nF\\in\\mathbb Z_p[Y,T]^e,\n\\qquad a\\in\\mathbb Z_p^{e+k},\n\\qquad F(a)=0.\n\\] Let \\(\\iota:\\mathbb Z_p\\hookrightarrow\\mathbb Q_p\\) be the\ncoefficient inclusion, applied entrywise to points and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility mod","\\textbf{Integral p-adic specialization.} For any prime \\(p\\), let \\[\nF\\in\\mathbb Z_p[Y,T]^e,\n\\qquad a\\in\\mathbb Z_p^{e+k},\n\\qquad F(a)=0.\n\\] Let \\(\\iota:\\mathbb Z_p\\hookrightarrow\\mathbb Q_p\\) be the\ncoefficient inclusion, applied entrywise to points and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This perm","\\textbf{Integral p-adic specialization.} For any prime \\(p\\), let \\[\nF\\in\\mathbb Z_p[Y,T]^e,\n\\qquad a\\in\\mathbb Z_p^{e+k},\n\\qquad F(a)=0.\n\\] Let \\(\\iota:\\mathbb Z_p\\hookrightarrow\\mathbb Q_p\\) be the\ncoefficient inclusion, applied entrywise to points and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field th",",T]^e,\n\\qquad a\\in\\mathbb Z_p^{e+k},\n\\qquad F(a)=0.\n\\] Let \\(\\iota:\\mathbb Z_p\\hookrightarrow\\mathbb Q_p\\) be the\ncoefficient inclusion, applied entrywise to points and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","hbb Z_p^{e+k},\n\\qquad F(a)=0.\n\\] Let \\(\\iota:\\mathbb Z_p\\hookrightarrow\\mathbb Q_p\\) be the\ncoefficient inclusion, applied entrywise to points and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","ient inclusion, applied entrywise to points and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","oints and matrices. Assume\nonly that \\[\n\\iota(D_YF(a))\\in\\operatorname{GL}_e(\\mathbb Q_p).\n\\] The theorem\n\\mcode{PolynomialRegularityIntegral.exists_analytic_integral_implicit}\nconstructs the preceding analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","ng analytic graph in the ambient spaces\n\\(\\mathbb Q_p^k\\) and \\(\\mathbb Q_p^e\\). Its root equation evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","tion evaluates the\noriginal integral polynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","olynomial coefficients using \\(\\iota\\), and its base\nderivative is \\[\n-\\iota(D_YF(a))^{-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","-1}\\,\\iota(D_TF(a)).\n\\] The source proves that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","es that polynomial evaluation, formal partial\ndifferentiation and both Jacobian blocks commute with this inclusion. No\ninvertibility modulo \\(p\\) or integral inverse is required. This permits\na solved Jacobian that is singular modulo \\(p\\) while remaining\ninvertible over \\(\\mathbb Q_p\\). The new specialization is an\nambient-field theorem. It does not treat \\(\\mathbb Z_p\\) as a vector\nspace over \\(\\mathbb Q_p\\) or assert a new explicit integral radius.","2.2 & Filtered finite coefficient reconstruction, including $\\Z_p$","\\subsubsection*{The correction is preserved}\nThe normalized packet is not the same object as the packet multiplied by $j!$. The checked recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad C_{6,3}(2)=\\frac{703}{5460},\n\\]\n\\[\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455},\\qquad\n v_3\\!\\left(\\frac{64}{455}\\right)=0.\n\\]\nThus the unrestricted normalized no-loss claim fails already at a valid one-step parameter instance. Multiplication by $3!$ restores one factor of three in this instance. The general corrected finite-difference theorem is \\textbf{not} inferred from this calculation and is not yet formalized here.","ion is preserved}\nThe normalized packet is not the same object as the packet multiplied by $j!$. The checked recurrence gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\qquad C_{6,3}(2)=\\frac{703}{5460},\n\\]\n\\[\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455},\\qquad\n v_3\\!\\left(\\frac{64}{455}\\right)=0.\n\\]\nThus the unrestricted normalized no-loss claim fails already at a valid one-step parameter instance. Multiplication by $3!$ restores one factor of three in this instance. The general corrected finite-difference theorem is \\textbf{not} inferred from this calculation and is not yet formalized here.","theorems and all 22 explicit definitions. Their transitive axiom dependencies are confined to \\mcode{propext}, \\mcode{Classical.choice} and \\mcode{Quot.sound}, or subsets. Three invalid-proof controls were rejected. The proof, scope, and numerical-regression records remain distinct.\n\\par\\medskip\n\\pageheading{1. Residue packets and actual EGF coordinates}{Source items 1.2 and 2.1}\nFor a sequence $a:\\N\\to R$, define the actual selector\n\\[\n (P_{N,j}a)(n)=\\begin{cases}a(n),&n\\bmod N=j,",". Their transitive axiom dependencies are confined to \\mcode{propext}, \\mcode{Classical.choice} and \\mcode{Quot.sound}, or subsets. Three invalid-proof controls were rejected. The proof, scope, and numerical-regression records remain distinct.\n\\par\\medskip\n\\pageheading{1. Residue packets and actual EGF coordinates}{Source items 1.2 and 2.1}\nFor a sequence $a:\\N\\to R$, define the actual selector\n\\[\n (P_{N,j}a)(n)=\\begin{cases}a(n),&n\\bmod N=j,\\\\0,&\\text{otherwise.}\\end{cases}\n\\]\nThe source proves idempotence for every selector, orthogonality for distinct standard residue labels, and, when $N>0$,\n\\[\n \\sum_{j=0}^{N-1}P_{N,j}a=a.\n\\]\nIf $N\\mid M$ and $M>0$, it also proves\n\\[\n P_{N,j}a=\\sum_{\\substack{0\\le r<M","0,&\\text{otherwise.}\\end{cases}\n\\]\nThe source proves idempotence for every selector, orthogonality for distinct standard residue labels, and, when $N>0$,\n\\[\n \\sum_{j=0}^{N-1}P_{N,j}a=a.\n\\]\nIf $N\\mid M$ and $M>0$, it also proves\n\\[\n P_{N,j}a=\\sum_{\\substack{0\\le r<M","0,&\\text{otherwise.}\\end{cases}\n\\]\nThe source proves idempotence for every selector, orthogonality for distinct standard residue labels, and, when $N>0$,\n\\[\n \\sum_{j=0}^{N-1}P_{N,j}a=a.\n\\]\nIf $N\\mid M$ and $M>0$, it also proves\n\\[\n P_{N,j}a=\\sum_{\\substack{0\\le r<M\\\\r\\bmod N=j}}P_{M,r}a.\n\\]\nThe formal power-series versions are proved coefficientwise for actual \\mcode{PowerSeries} objects. There is no finite truncation of the sequence domain in these theorems.","\\subsection*{A constructed inverse, not an inverse hypothesis}\nLet $K$ be a characteristic-zero field. Define\n\\[\n \\mathcal E(a)=\\sum_{n\\ge0}\\frac{a_n}{n!}t^n,\\qquad\n \\mathcal D(A)_n=n![t^n]A.\n\\]\nBoth inverse laws $\\mathcal D\\mathcal E=I$ and $\\mathcal E\\mathcal D=I$ are proved. The ordinary power-series product then yields the binomial convolution\n\\[\n \\mathcal D\\bigl(\\mathcal E(h)\\mathcal E(a)\\bigr)_n\n =\\sum_{r=0}^{n}\\binom nr h_r a_{n-r}.\n\\]\nFor $h_0=1$, the formal inverse of $H=\\mathcal E(h)$ is used to construct a sequence equivalence\n\\[\n U_H a=\\mathcal D(H\\mathcal E(a)),\\qquad\n U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a)).\n\\]\nThe equivalence has proved left and right inverse laws. The inverse's constant coefficient is proved to be one. The associated lower-t","\\subsection*{A constructed inverse, not an inverse hypothesis}\nLet $K$ be a characteristic-zero field. Define\n\\[\n \\mathcal E(a)=\\sum_{n\\ge0}\\frac{a_n}{n!}t^n,\\qquad\n \\mathcal D(A)_n=n![t^n]A.\n\\]\nBoth inverse laws $\\mathcal D\\mathcal E=I$ and $\\mathcal E\\mathcal D=I$ are proved. The ordinary power-series product then yields the binomial convolution\n\\[\n \\mathcal D\\bigl(\\mathcal E(h)\\mathcal E(a)\\bigr)_n\n =\\sum_{r=0}^{n}\\binom nr h_r a_{n-r}.\n\\]\nFor $h_0=1$, the formal inverse of $H=\\mathcal E(h)$ is used to construct a sequence equivalence\n\\[\n U_H a=\\mathcal D(H\\mathcal E(a)),\\qquad\n U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a)).\n\\]\nThe equivalence has proved left and right inverse laws. The inverse's constant coefficient is proved to be one. The associated lower-triangular entries have diagonal one and vanish above the diagonal.","ypothesis}\nLet $K$ be a characteristic-zero field. Define\n\\[\n \\mathcal E(a)=\\sum_{n\\ge0}\\frac{a_n}{n!}t^n,\\qquad\n \\mathcal D(A)_n=n![t^n]A.\n\\]\nBoth inverse laws $\\mathcal D\\mathcal E=I$ and $\\mathcal E\\mathcal D=I$ are proved. The ordinary power-series product then yields the binomial convolution\n\\[\n \\mathcal D\\bigl(\\mathcal E(h)\\mathcal E(a)\\bigr)_n\n =\\sum_{r=0}^{n}\\binom nr h_r a_{n-r}.\n\\]\nFor $h_0=1$, the formal inverse of $H=\\mathcal E(h)$ is used to construct a sequence equivalence\n\\[\n U_H a=\\mathcal D(H\\mathcal E(a)),\\qquad\n U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a)).\n\\]\nThe equivalence has proved left and right inverse laws. The inverse's constant coefficient is proved to be one. The associated lower-triangular entries have diagonal one and vanish above the diagonal.","eristic-zero field. Define\n\\[\n \\mathcal E(a)=\\sum_{n\\ge0}\\frac{a_n}{n!}t^n,\\qquad\n \\mathcal D(A)_n=n![t^n]A.\n\\]\nBoth inverse laws $\\mathcal D\\mathcal E=I$ and $\\mathcal E\\mathcal D=I$ are proved. The ordinary power-series product then yields the binomial convolution\n\\[\n \\mathcal D\\bigl(\\mathcal E(h)\\mathcal E(a)\\bigr)_n\n =\\sum_{r=0}^{n}\\binom nr h_r a_{n-r}.\n\\]\nFor $h_0=1$, the formal inverse of $H=\\mathcal E(h)$ is used to construct a sequence equivalence\n\\[\n U_H a=\\mathcal D(H\\mathcal E(a)),\\qquad\n U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a)).\n\\]\nThe equivalence has proved left and right inverse laws. The inverse's constant coefficient is proved to be one. The associated lower-triangular entries have diagonal one and vanish above the diagonal.","a_n}{n!}t^n,\\qquad\n \\mathcal D(A)_n=n![t^n]A.\n\\]\nBoth inverse laws $\\mathcal D\\mathcal E=I$ and $\\mathcal E\\mathcal D=I$ are proved. The ordinary power-series product then yields the binomial convolution\n\\[\n \\mathcal D\\bigl(\\mathcal E(h)\\mathcal E(a)\\bigr)_n\n =\\sum_{r=0}^{n}\\binom nr h_r a_{n-r}.\n\\]\nFor $h_0=1$, the formal inverse of $H=\\mathcal E(h)$ is used to construct a sequence equivalence\n\\[\n U_H a=\\mathcal D(H\\mathcal E(a)),\\qquad\n U_H^{-1}a=\\mathcal D(H^{-1}\\mathcal E(a)).\n\\]\nThe equivalence has proved left and right inverse laws. The inverse's constant coefficient is proved to be one. The associated lower-triangular entries have diagonal one and vanish above the diagonal.","For the actual conjugated selectors\n\\[\n \\widehat P_{N,j}=U_HP_{N,j}U_H^{-1},\n\\]\nidempotence, orthogonality, and finite reconstruction are proved from the constructed maps. No projector identity for the conjugates is inserted as an assumption.","\\textbf{Principal targets.}\\par\n\\target{ResiduePackets.packet_reconstruction}\\par\n\\target{ResiduePackets.packet_refinement}\\par\n\\target{EGF.coordinateEquiv, EGF.coordinate_forward, EGF.coordinate_inverse}\\par\n\\target{EGF.transported_reconstruction}\n\\par\\medskip\n\\pageheading{2. Triangular inversion and denominator control}{Source items 3.2 and 3.3}\nFor a general lower-triangular coefficient array $T$ over a field with nonzero diagonal, the code defines forward substitution by well-founded recursion:\n\\[\n a_n=\\frac{b_n-\\sum_{q<n}T_{nq}a_q}{T_{nn}}.\n\\]\nIt proves that this constructed sequence solves every row and that every sequence solving those rows equals it. No existence or uniqueness premise is used.","acket_refinement}\\par\n\\target{EGF.coordinateEquiv, EGF.coordinate_forward, EGF.coordinate_inverse}\\par\n\\target{EGF.transported_reconstruction}\n\\par\\medskip\n\\pageheading{2. Triangular inversion and denominator control}{Source items 3.2 and 3.3}\nFor a general lower-triangular coefficient array $T$ over a field with nonzero diagonal, the code defines forward substitution by well-founded recursion:\n\\[\n a_n=\\frac{b_n-\\sum_{q<n}T_{nq}a_q}{T_{nn}}.\n\\]\nIt proves that this constructed sequence solves every row and that every sequence solving those rows equals it. No existence or uniqueness premise is used.","\\subsection*{The congruential coefficients}\nOver $\\Q$, specialize to\n\\[\n T_{nq}=\\binom{Nn+j}{Nq},\\qquad b_n=\\delta_{n0}.\n\\]\nThe diagonal is positive as an integer, hence nonzero. The resulting sequence $C_{N,j}(n)$ satisfies\n\\[\n C_{N,j}(0)=1,\\qquad\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0}.\n\\]\nThe displayed recurrence in the manuscript and the unique-solution theorem are consequences of this definition.","The formalization does not stop at a recurrence. In the variable $z=t^N$, construct\n\\[\n K_{N,j}(z)=\\sum_{n\\ge0}\\frac{j!}{(Nn+j)!}z^n,\\qquad\n F_{N,j}(z)=\\sum_{n\\ge0}\\frac{C_{N,j}(n)}{(Nn)!}z^n.\n\\]\nIt proves the factorial coefficient identity, then $F_{N,j}K_{N,j}=1$ and therefore $F_{N,j}=K_{N,j}^{-1}$.","For $N>0$ and $0\\le j<N$, the independently defined original exponential section is\n\\[\n E_{N,j}(t)=\\sum_{d\\equiv j\\ ({\\rm mod}\\ N)}\\frac{t^d}{d!}.\n\\]\nUsing the actual power-series expansion homomorphism $z\\mapsto t^N$, the code proves\n\\[\n E_{N,j}(t)=\\frac{t^j}{j!}K_{N,j}(t^N),\\qquad\n \\boxed{F_{N,j}(t^N)E_{N,j}(t)=\\frac{t^j}{j!}.}\n\\]\nThus the recurrence is connected back to the original normalized EGF definition. The section vanishes at zero when $j>0$. No inverse of that nonunit section is assumed.","original exponential section is\n\\[\n E_{N,j}(t)=\\sum_{d\\equiv j\\ ({\\rm mod}\\ N)}\\frac{t^d}{d!}.\n\\]\nUsing the actual power-series expansion homomorphism $z\\mapsto t^N$, the code proves\n\\[\n E_{N,j}(t)=\\frac{t^j}{j!}K_{N,j}(t^N),\\qquad\n \\boxed{F_{N,j}(t^N)E_{N,j}(t)=\\frac{t^j}{j!}.}\n\\]\nThus the recurrence is connected back to the original normalized EGF definition. The section vanishes at zero when $j>0$. No inverse of that nonunit section is assumed.","\\subsection*{Universal clearing}\nDefine\n\\[\n D_{N,j}(n)=\\prod_{r=1}^{n}\\binom{Nr+j}{Nr},\\qquad D_{N,j}(0)=1.\n\\]\nFor every $n$, the code proves $D_{N,j}(n)C_{N,j}(n)\\in\\Z$. The general proof promotes each earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral.\n\\par\\medskip\n\\pageheading{3. Filtered tails and the standard window}{Source items 2.2 and 8.2}\n\\subsection*{Filtration at a finite coefficient row}\nLet $R$ be any commutative ring. For $w\\in\\N\\cup\\{\\infty\\}$ define\n\\[\n p^{[w]}=\\begin{cases}p^w,&w<\\infty,","proves $D_{N,j}(n)C_{N,j}(n)\\in\\Z$. The general proof promotes each earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral.\n\\par\\medskip\n\\pageheading{3. Filtered tails and the standard window}{Source items 2.2 and 8.2}\n\\subsection*{Filtration at a finite coefficient row}\nLet $R$ be any commutative ring. For $w\\in\\N\\cup\\{\\infty\\}$ define\n\\[\n p^{[w]}=\\begin{cases}p^w,&w<\\infty,","$. The general proof promotes each earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral.\n\\par\\medskip\n\\pageheading{3. Filtered tails and the standard window}{Source items 2.2 and 8.2}\n\\subsection*{Filtration at a finite coefficient row}\nLet $R$ be any commutative ring. For $w\\in\\N\\cup\\{\\infty\\}$ define\n\\[\n p^{[w]}=\\begin{cases}p^w,&w<\\infty,","ach earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral.\n\\par\\medskip\n\\pageheading{3. Filtered tails and the standard window}{Source items 2.2 and 8.2}\n\\subsection*{Filtration at a finite coefficient row}\nLet $R$ be any commutative ring. For $w\\in\\N\\cup\\{\\infty\\}$ define\n\\[\n p^{[w]}=\\begin{cases}p^w,&w<\\infty,\\\\0,&w=\\infty.\\end{cases}\n\\]\nThe infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least $L$ is divisible by $p^L$. A finite-sum proof yields\n\\[\n \\boxed{p^L\\ \\bigm|\\ \n \\sum_{r=0}^{n}\\binom nr c_rp^{[w(r)]}S_{n-r}\n -\\sum_{\\substack{0\\le r\\le n","0,&w=\\infty.\\end{cases}\n\\]\nThe infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least $L$ is divisible by $p^L$. A finite-sum proof yields\n\\[\n \\boxed{p^L\\ \\bigm|\\ \n \\sum_{r=0}^{n}\\binom nr c_rp^{[w(r)]}S_{n-r}\n -\\sum_{\\substack{0\\le r\\le n","0,&w=\\infty.\\end{cases}\n\\]\nThe infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least $L$ is divisible by $p^L$. A finite-sum proof yields\n\\[\n \\boxed{p^L\\ \\bigm|\\ \n \\sum_{r=0}^{n}\\binom nr c_rp^{[w(r)]}S_{n-r}\n -\\sum_{\\substack{0\\le r\\le n\\\\w(r)<L}}\n     \\binom nr c_rp^{[w(r)]}S_{n-r}.}\n\\]\nThe result is specialized to actual $p$-adic integers $R=\\Z_p$ for prime $p$. Divisibility there is precisely the asserted membership in $p^L\\Z_p$.","w(r)<L}}\n     \\binom nr c_rp^{[w(r)]}S_{n-r}.}\n\\]\nThe result is specialized to actual $p$-adic integers $R=\\Z_p$ for prime $p$. Divisibility there is precisely the asserted membership in $p^L\\Z_p$.","For an individual row the sum is finite, so the proof does not require $w(r)\\to\\infty$. This is a stronger coefficient-row scope, not a theorem about an infinite limiting process. A supporting lemma also proves a $p^{2M+2}$ tail bound for a sum with visible factors $p^{2m}$. It is not counted as either Euler supercongruence theorem: their generating-function reconstruction identities remain pending.","\\subsection*{Actual Lucas reduction}\nFor prime $p$, all $m,n,q\\in\\N$, and $0\\le j<p$, the checked binomial congruence is\n\\[\n \\boxed{\\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p.}\n\\]\nThe proof applies Mathlib's Lucas low-digit reduction to the actual natural binomial coefficients. It does not assume a pre-reduced row. The statement also permits $q>n$. If $m>0$, both binomial coefficients then vanish.","Taking $q=n$ gives\n\\[\n \\binom{mpn+j}{mpn}\\equiv1\\pmod p.\n\\]\nNondivisibility by $p$ and invertibility of that diagonal in $\\Z/p\\Z$ are proved. The separate supporting lemma $p\\nmid j!$ for $j<p$ establishes the standard-window factorial-unit condition.","\\textbf{Principal targets.}\\par\n\\target{Filtered.padic_filtered_reconstruction}\\par\n\\target{StandardWindow.lucas_row_reduction}\\par\n\\target{StandardWindow.standard_diagonal_unit}\\par\n\\target{StandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","ed.padic_filtered_reconstruction}\\par\n\\target{StandardWindow.lucas_row_reduction}\\par\n\\target{StandardWindow.standard_diagonal_unit}\\par\n\\target{StandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","c_filtered_reconstruction}\\par\n\\target{StandardWindow.lucas_row_reduction}\\par\n\\target{StandardWindow.standard_diagonal_unit}\\par\n\\target{StandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","red_reconstruction}\\par\n\\target{StandardWindow.lucas_row_reduction}\\par\n\\target{StandardWindow.standard_diagonal_unit}\\par\n\\target{StandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","truction}\\par\n\\target{StandardWindow.lucas_row_reduction}\\par\n\\target{StandardWindow.standard_diagonal_unit}\\par\n\\target{StandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","n}\\par\n\\target{StandardWindow.lucas_row_reduction}\\par\n\\target{StandardWindow.standard_diagonal_unit}\\par\n\\target{StandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","dow.lucas_row_reduction}\\par\n\\target{StandardWindow.standard_diagonal_unit}\\par\n\\target{StandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","tandardWindow.factorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","ctorial_valuation_zero}\n\\par\\medskip\n\\pageheading{4. Sharp normalization boundary}{Source item 13.1}\nThe construction fixes the two coefficient conventions:\n\\[\n C_{N,j}(n),\\qquad \\widetilde C_{N,j}(n)=j!\\,C_{N,j}(n).\n\\]\nThe witness uses $p=3$, $m=2$, $N=mp=6$, $j=3$, $n=1$ and difference step $h=(p-1)/m=1$. The parameter conditions are checked explicitly, including odd primality, $m\\mid p-1$ and $j<N$.","The first row gives $84C_{6,3}(1)+1=0$. The second row, evaluated by the same recursive definition, gives $C_{6,3}(2)=703/5460$. Therefore\n\\[\n \\Delta_1 C_{6,3}(1)=\\frac{64}{455}\\ne0.\n\\]\nNeither numerator nor denominator is divisible by three, so its actual rational 3-adic valuation is zero. A further theorem negates the universal one-step no-loss statement by applying it to this valid tuple and deriving $1\\le0$.","t row gives $84C_{6,3}(1)+1=0$. The second row, evaluated by the same recursive definition, gives $C_{6,3}(2)=703/5460$. Therefore\n\\[\n \\Delta_1 C_{6,3}(1)=\\frac{64}{455}\\ne0.\n\\]\nNeither numerator nor denominator is divisible by three, so its actual rational 3-adic valuation is zero. A further theorem negates the universal one-step no-loss statement by applying it to this valid tuple and deriving $1\\le0$.","For the unnormalized convention,\n\\[\n \\Delta_1\\widetilde C_{6,3}(1)=3!\\,\\frac{64}{455}=\\frac{384}{455},\n \\qquad v_3\\!\\left(\\frac{384}{455}\\right)=1.\n\\]\nThe factorial's valuation is also proved:\n\\[\n v_3(3!)=1,\\qquad\n v_3\\bigl(\\Delta_1C_{6,3}(1)\\bigr)\n =1-v_3(3!)=0.\n\\]\nThis proves sharpness at the displayed normalization boundary. It does not establish the corrected global inequality for all parameters, and does not assert that every nonunit-factorial parameter tuple must violate every no-loss specialization. All valuation calculations here concern nonzero rationals.","\\subsubsection*{A useful scope strengthening}\nThe algebraic proof needs \\emph{oddness}, not primality. The exact reconstruction and the integer divisibility statement are therefore checked for every positive odd length $p$, including $p=1$ and odd composites. The manuscript's odd-prime statement follows as a specialization. This is a sharpening of assumptions in this development, not a historical-priority claim. Even lengths are not included. A checked row-zero counterexample shows why.","-prime finite-difference theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment.\n\\par\\medskip\n\\pageheading{1. The actual odd-power generating function}{Lemma 4.1; manuscript page 10}\nWork in the rational formal power-series ring $\\Q[[t]]$. Define\n\\[\n S_n(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\\in\\Z,\n \\qquad F_p(t)=\\sum_{n\\ge0}S_n(p)\\frac{t^n}{n!}.\n\\]\nThe formal exponential is Mathlib's \\mcode{PowerSeries.exp}. Scalar substitution is its actual \\mcode{rescale} homomorphism. We define\n\\[\n \\exp_a(t)=e^{at},\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2,\\qquad\n \\sech(t)=\\cosh_1(t)^{-1}.\n\\]\nThe constant term of $\\cosh_a$ is proved to be one, so the reciprocal used here is a unit inverse.","rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment.\n\\par\\medskip\n\\pageheading{1. The actual odd-power generating function}{Lemma 4.1; manuscript page 10}\nWork in the rational formal power-series ring $\\Q[[t]]$. Define\n\\[\n S_n(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\\in\\Z,\n \\qquad F_p(t)=\\sum_{n\\ge0}S_n(p)\\frac{t^n}{n!}.\n\\]\nThe formal exponential is Mathlib's \\mcode{PowerSeries.exp}. Scalar substitution is its actual \\mcode{rescale} homomorphism. We define\n\\[\n \\exp_a(t)=e^{at},\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2,\\qquad\n \\sech(t)=\\cosh_1(t)^{-1}.\n\\]\nThe constant term of $\\cosh_a$ is proved to be one, so the reciprocal used here is a unit inverse.","r\\medskip\n\\pageheading{1. The actual odd-power generating function}{Lemma 4.1; manuscript page 10}\nWork in the rational formal power-series ring $\\Q[[t]]$. Define\n\\[\n S_n(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\\in\\Z,\n \\qquad F_p(t)=\\sum_{n\\ge0}S_n(p)\\frac{t^n}{n!}.\n\\]\nThe formal exponential is Mathlib's \\mcode{PowerSeries.exp}. Scalar substitution is its actual \\mcode{rescale} homomorphism. We define\n\\[\n \\exp_a(t)=e^{at},\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2,\\qquad\n \\sech(t)=\\cosh_1(t)^{-1}.\n\\]\nThe constant term of $\\cosh_a$ is proved to be one, so the reciprocal used here is a unit inverse.","formal power-series ring $\\Q[[t]]$. Define\n\\[\n S_n(p)=\\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\\in\\Z,\n \\qquad F_p(t)=\\sum_{n\\ge0}S_n(p)\\frac{t^n}{n!}.\n\\]\nThe formal exponential is Mathlib's \\mcode{PowerSeries.exp}. Scalar substitution is its actual \\mcode{rescale} homomorphism. We define\n\\[\n \\exp_a(t)=e^{at},\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2,\\qquad\n \\sech(t)=\\cosh_1(t)^{-1}.\n\\]\nThe constant term of $\\cosh_a$ is proved to be one, so the reciprocal used here is a unit inverse.","\\subsection*{From finite sums to a series identity}\nCoefficient extraction proves\n\\[\n F_p(t)=\\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.\n\\]\nAn induction on the actual finite summation length establishes the telescoping identity\n\\[\n F_p(t)\\cosh t=\\frac{1-(-1)^p e^{2pt}}2.\n\\]\nFor odd $p$ this gives, without a geometric-series convergence assumption,\n\\[\n \\boxed{F_p(t)=\\frac{1+e^{2pt}}2\\,\\sech t.}\n\\]\nThis is an equality of complete formal series, not a fixed-degree polynomial check.","\\subsection*{Even projection is identified coefficientwise}\nFor any formal series $f$, let $\\operatorname{EvenPart}(f)=(f(t)+f(-t))/2$. Its $n$th coefficient is proved to be the coefficient of $f$ when $n$ is even and zero otherwise. Rescaling the inverse law proves $\\sech(-t)=\\sech(t)$. The exponential multiplication identity then gives\n\\[\n \\cosh^2(pt)=\\frac{2+e^{2pt}+e^{-2pt}}4,\n\\]\nand hence\n\\[\n \\boxed{\\sum_{r\\ge0}S_{2r}(p)\\frac{t^{2r}}{(2r)!}\n       =\\cosh^2(pt)\\,\\sech t.}\n\\]\nBoth the full and even EGF formulas are derived from the original integer sum. Neither is supplied as a hypothesis.","\\textbf{Principal targets.}\\par\n\\target{OddPower.source_as_exponentials}\\par\n\\target{OddPower.source_times_cosh}\\par\n\\target{OddPower.odd_source_egf, OddPower.odd_source_even_egf}\n\\par\\medskip\n\\pageheading{2. Integral Euler coefficients and the zigzag bridge}{Coefficient foundations for Theorem 4.2}\nThe first block already constructed $C_{N,j}(r)$ by the manuscript's triangular recurrence. At $(N,j)=(2,0)$, every diagonal binomial coefficient equals one. Its denominator-clearing product is therefore one, and $C_{2,0}(r)$ is an integer.","r\n\\target{OddPower.odd_source_egf, OddPower.odd_source_even_egf}\n\\par\\medskip\n\\pageheading{2. Integral Euler coefficients and the zigzag bridge}{Coefficient foundations for Theorem 4.2}\nThe first block already constructed $C_{N,j}(r)$ by the manuscript's triangular recurrence. At $(N,j)=(2,0)$, every diagonal binomial coefficient equals one. Its denominator-clearing product is therefore one, and $C_{2,0}(r)$ is an integer.","The source defines $E_{2r}$ as the integer numerator of that rational coefficient and proves that its rational cast equals $C_{2,0}(r)$. Set $E_{2r+1}=0$. The actual exponential-section identity from the first block proves\n\\[\n \\boxed{\\sum_{n\\ge0}E_n\\frac{t^n}{n!}=\\sech t.}\n\\]\nThus the integer sequence is connected to the intended signed Euler family, rather than merely assigned the name Euler.","\\subsection*{The inverse coefficients are integral by construction}\nDefine the integer binomial convolution\n\\[\n B_n=\\sum_{r=0}^{n}\\binom nr E_rE_{n-r},\\qquad b_m=B_{2m}.\n\\]\nThe product formula for actual EGFs gives\n\\[\n \\sum_{n\\ge0}B_n\\frac{t^n}{n!}=\\sech^2 t.\n\\]\nOdd coefficients vanish by parity. In particular, every $b_m$ is an integer. This is the integrality required by the later congruence.","\\subsection*{Connecting to the chapter's $Z_n$ convention}\nThe chapter defines $Z_n$ through $\\sec t+\\tan t$. The Lean construction uses Mathlib's actual formal sine and cosine series and defines\n\\[\n \\sec t=(\\cos t)^{-1},\\quad \\tan t=(\\sin t)\\sec t,\n \\quad Z_n=n![t^n](\\sec t+\\tan t).\n\\]\nIt proves the sine and cosine derivative identities, $\\sin^2t+\\cos^2t=1$, and\n\\[\n (\\tan t)'=\\sec^2t.\n\\]\nSign substitution in the squared variable carries the hyperbolic cosine kernel to the actual cosine kernel. The same homomorphism carries their inverses and their squares. Extracting the even coefficient in the derivative of tangent yields\n\\[\n \\boxed{b_m=(-1)^m Z_{2m+1}.}\n\\]\nThis formula is \\emph{proved}, not assumed as an input to reconstruction. The","\\subsection*{Connecting to the chapter's $Z_n$ convention}\nThe chapter defines $Z_n$ through $\\sec t+\\tan t$. The Lean construction uses Mathlib's actual formal sine and cosine series and defines\n\\[\n \\sec t=(\\cos t)^{-1},\\quad \\tan t=(\\sin t)\\sec t,\n \\quad Z_n=n![t^n](\\sec t+\\tan t).\n\\]\nIt proves the sine and cosine derivative identities, $\\sin^2t+\\cos^2t=1$, and\n\\[\n (\\tan t)'=\\sec^2t.\n\\]\nSign substitution in the squared variable carries the hyperbolic cosine kernel to the actual cosine kernel. The same homomorphism carries their inverses and their squares. Extracting the even coefficient in the derivative of tangent yields\n\\[\n \\boxed{b_m=(-1)^m Z_{2m+1}.}\n\\]\nThis formula is \\emph{proved}, not assumed as an input to reconstruction. The formalization follows the generating-","\\subsection*{Connecting to the chapter's $Z_n$ convention}\nThe chapter defines $Z_n$ through $\\sec t+\\tan t$. The Lean construction uses Mathlib's actual formal sine and cosine series and defines\n\\[\n \\sec t=(\\cos t)^{-1},\\quad \\tan t=(\\sin t)\\sec t,\n \\quad Z_n=n![t^n](\\sec t+\\tan t).\n\\]\nIt proves the sine and cosine derivative identities, $\\sin^2t+\\cos^2t=1$, and\n\\[\n (\\tan t)'=\\sec^2t.\n\\]\nSign substitution in the squared variable carries the hyperbolic cosine kernel to the actual cosine kernel. The same homomorphism carries their inverses and their squares. Extracting the even coefficient in the derivative of tangent yields\n\\[\n \\boxed{b_m=(-1)^m Z_{2m+1}.}\n\\]\nThis formula is \\emph{proved}, not assumed as an input to reconstruction. The formalization follows the generating-function definition of z","\\subsection*{Connecting to the chapter's $Z_n$ convention}\nThe chapter defines $Z_n$ through $\\sec t+\\tan t$. The Lean construction uses Mathlib's actual formal sine and cosine series and defines\n\\[\n \\sec t=(\\cos t)^{-1},\\quad \\tan t=(\\sin t)\\sec t,\n \\quad Z_n=n![t^n](\\sec t+\\tan t).\n\\]\nIt proves the sine and cosine derivative identities, $\\sin^2t+\\cos^2t=1$, and\n\\[\n (\\tan t)'=\\sec^2t.\n\\]\nSign substitution in the squared variable carries the hyperbolic cosine kernel to the actual cosine kernel. The same homomorphism carries their inverses and their squares. Extracting the even coefficient in the derivative of tangent yields\n\\[\n \\boxed{b_m=(-1)^m Z_{2m+1}.}\n\\]\nThis formula is \\emph{proved}, not assumed as an input to reconstruction. The formalization follows the generating-function definition of zigzag numbers. It does not additionally formalize their interpretation as counts of alternating permutations.","(\\sin t)\\sec t,\n \\quad Z_n=n![t^n](\\sec t+\\tan t).\n\\]\nIt proves the sine and cosine derivative identities, $\\sin^2t+\\cos^2t=1$, and\n\\[\n (\\tan t)'=\\sec^2t.\n\\]\nSign substitution in the squared variable carries the hyperbolic cosine kernel to the actual cosine kernel. The same homomorphism carries their inverses and their squares. Extracting the even coefficient in the derivative of tangent yields\n\\[\n \\boxed{b_m=(-1)^m Z_{2m+1}.}\n\\]\nThis formula is \\emph{proved}, not assumed as an input to reconstruction. The formalization follows the generating-function definition of zigzag numbers. It does not additionally formalize their interpretation as counts of alternating permutations.","\\textbf{Principal targets.}\\par\n\\target{EulerCoefficients.euler_egf, EulerCoefficients.inverse_coeff_cast}\\par\n\\target{Zigzag.derivative_tan, Zigzag.inverse_is_signed_zigzag}\n\\par\\medskip\n\\pageheading{3. Exact reconstruction and the triangular matrix}{Theorem 4.2; manuscript page 11}\nFor odd $p$, rescaling the already proved inverse law gives\n\\[\n \\sech(pt)\\cosh(pt)=1.\n\\]\nMultiplying the even-source identity by $\\sech^2(pt)$ therefore proves\n\\[\n \\sech^2(pt)\\left(\\sum_{r\\ge0}S_{2r}(p)\\frac{t^{2r}}{(2r)!}\\right)=\\sech t.\n\\]\nActual EGF multiplication now gives the integer coefficient formula\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).}\n\\]\nThe formal proof first extracts the sum over all indices and then proves that the odd summands vanish. A checked even-index summation identity supplies the displayed change of variables.","al targets.}\\par\n\\target{EulerCoefficients.euler_egf, EulerCoefficients.inverse_coeff_cast}\\par\n\\target{Zigzag.derivative_tan, Zigzag.inverse_is_signed_zigzag}\n\\par\\medskip\n\\pageheading{3. Exact reconstruction and the triangular matrix}{Theorem 4.2; manuscript page 11}\nFor odd $p$, rescaling the already proved inverse law gives\n\\[\n \\sech(pt)\\cosh(pt)=1.\n\\]\nMultiplying the even-source identity by $\\sech^2(pt)$ therefore proves\n\\[\n \\sech^2(pt)\\left(\\sum_{r\\ge0}S_{2r}(p)\\frac{t^{2r}}{(2r)!}\\right)=\\sech t.\n\\]\nActual EGF multiplication now gives the integer coefficient formula\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).}\n\\]\nThe formal proof first extracts the sum over all indices and then proves that the odd summands vanish. A checked even-index summation identity supplies the displayed change of variables.","efficients.euler_egf, EulerCoefficients.inverse_coeff_cast}\\par\n\\target{Zigzag.derivative_tan, Zigzag.inverse_is_signed_zigzag}\n\\par\\medskip\n\\pageheading{3. Exact reconstruction and the triangular matrix}{Theorem 4.2; manuscript page 11}\nFor odd $p$, rescaling the already proved inverse law gives\n\\[\n \\sech(pt)\\cosh(pt)=1.\n\\]\nMultiplying the even-source identity by $\\sech^2(pt)$ therefore proves\n\\[\n \\sech^2(pt)\\left(\\sum_{r\\ge0}S_{2r}(p)\\frac{t^{2r}}{(2r)!}\\right)=\\sech t.\n\\]\nActual EGF multiplication now gives the integer coefficient formula\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).}\n\\]\nThe formal proof first extracts the sum over all indices and then proves that the odd summands vanish. A checked even-index summation identity supplies the displayed change of variables.","ing{3. Exact reconstruction and the triangular matrix}{Theorem 4.2; manuscript page 11}\nFor odd $p$, rescaling the already proved inverse law gives\n\\[\n \\sech(pt)\\cosh(pt)=1.\n\\]\nMultiplying the even-source identity by $\\sech^2(pt)$ therefore proves\n\\[\n \\sech^2(pt)\\left(\\sum_{r\\ge0}S_{2r}(p)\\frac{t^{2r}}{(2r)!}\\right)=\\sech t.\n\\]\nActual EGF multiplication now gives the integer coefficient formula\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).}\n\\]\nThe formal proof first extracts the sum over all indices and then proves that the odd summands vanish. A checked even-index summation identity supplies the displayed change of variables.","Substituting the independently proved signed-zigzag identity recovers the manuscript's notation. For every even degree $n$,\n\\[\n \\boxed{E_n=\\sum_{m=0}^{n/2}(-1)^m Z_{2m+1}\\binom n{2m}p^{2m}S_{n-2m}(p).}\n\\]\nThe Lean version uses $n=2r$ to avoid ambiguous natural-number division. Every even natural degree has that form. Primality is not used in this equality.","\\subsection*{The row action is connected to the matrix entries}\nReversing the finite sum and using binomial symmetry gives the actual source-index row action\n\\[\n E_{2r}=\\sum_{q=0}^{r}U_{r,q}(p)S_{2q}(p),\n\\]\nwhere\n\\[\n U_{r,q}(p)=\n \\begin{cases}\n \\displaystyle\\binom{2r}{2q}b_{r-q}p^{2(r-q)},&q\\le r,","\\subsection*{The row action is connected to the matrix entries}\nReversing the finite sum and using binomial symmetry gives the actual source-index row action\n\\[\n E_{2r}=\\sum_{q=0}^{r}U_{r,q}(p)S_{2q}(p),\n\\]\nwhere\n\\[\n U_{r,q}(p)=\n \\begin{cases}\n \\displaystyle\\binom{2r}{2q}b_{r-q}p^{2(r-q)},&q\\le r,\\\\\n 0,&q>r.\n \\end{cases}\n\\]\nThe row-action equality, the diagonal identity $U_{r,r}=1$, and vanishing above the diagonal are all checked. The arithmetic weight is also explicit:\n\\[\n p^{2(r-q)}\\mid U_{r,q}(p)\\quad\\text{in }\\Z,\\qquad 0\\le q\\le r.\n\\]\nNo inverse-coordinate identity, unit diagonal or weight bound is assumed for these rows.","0,&q>r.\n \\end{cases}\n\\]\nThe row-action equality, the diagonal identity $U_{r,r}=1$, and vanishing above the diagonal are all checked. The arithmetic weight is also explicit:\n\\[\n p^{2(r-q)}\\mid U_{r,q}(p)\\quad\\text{in }\\Z,\\qquad 0\\le q\\le r.\n\\]\nNo inverse-coordinate identity, unit diagonal or weight bound is assumed for these rows.","\\textbf{Principal targets.}\\par\n\\target{OddEuler.odd_reconstruction_series}\\par\n\\target{OddEuler.odd_euler_reconstruction}\\par\n\\target{OddEuler.odd_euler_reconstruction_zigzag}\\par\n\\target{OddEuler.row_action, OddEuler.row_diagonal, OddEuler.row_weight}\n\\par\\medskip\n\\pageheading{4. Uniform truncation and a sharp exact example}{Corollary 4.3 and Example 4.4}\nFor row $2r$ and depth $M$, the actual truncation is\n\\[\n T_M(p,2r)=\\sum_{m=0}^{\\min(M,r)}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).\n\\]\nEvery omitted term has $m\\ge M+1$. Its other factors are integers, so the first-block omitted-tail theorem, now applied to the \\emph{proved Euler identity}, gives\n\\[\n \\boxed{p^{2M+2}\\mid E_{2r}-T_M(p,2r)\\quad\\text{in }\\Z.}\n\\]\nThis is not divisibility in $\\Q$, where a nonzero divisor would make the assertion vacuous. The statement holds for every row and depth. For an odd prime it is precisely the claimed supercongruence. If $M\\ge r$, complete retention is proved exact.","al targets.}\\par\n\\target{OddEuler.odd_reconstruction_series}\\par\n\\target{OddEuler.odd_euler_reconstruction}\\par\n\\target{OddEuler.odd_euler_reconstruction_zigzag}\\par\n\\target{OddEuler.row_action, OddEuler.row_diagonal, OddEuler.row_weight}\n\\par\\medskip\n\\pageheading{4. Uniform truncation and a sharp exact example}{Corollary 4.3 and Example 4.4}\nFor row $2r$ and depth $M$, the actual truncation is\n\\[\n T_M(p,2r)=\\sum_{m=0}^{\\min(M,r)}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).\n\\]\nEvery omitted term has $m\\ge M+1$. Its other factors are integers, so the first-block omitted-tail theorem, now applied to the \\emph{proved Euler identity}, gives\n\\[\n \\boxed{p^{2M+2}\\mid E_{2r}-T_M(p,2r)\\quad\\text{in }\\Z.}\n\\]\nThis is not divisibility in $\\Q$, where a nonzero divisor would make the assertion vacuous. The statement holds for every row and depth. For an odd prime it is precisely the claimed supercongruence. If $M\\ge r$, complete retention is proved exact.","uction}\\par\n\\target{OddEuler.odd_euler_reconstruction_zigzag}\\par\n\\target{OddEuler.row_action, OddEuler.row_diagonal, OddEuler.row_weight}\n\\par\\medskip\n\\pageheading{4. Uniform truncation and a sharp exact example}{Corollary 4.3 and Example 4.4}\nFor row $2r$ and depth $M$, the actual truncation is\n\\[\n T_M(p,2r)=\\sum_{m=0}^{\\min(M,r)}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).\n\\]\nEvery omitted term has $m\\ge M+1$. Its other factors are integers, so the first-block omitted-tail theorem, now applied to the \\emph{proved Euler identity}, gives\n\\[\n \\boxed{p^{2M+2}\\mid E_{2r}-T_M(p,2r)\\quad\\text{in }\\Z.}\n\\]\nThis is not divisibility in $\\Q$, where a nonzero divisor would make the assertion vacuous. The statement holds for every row and depth. For an odd prime it is precisely the claimed supercongruence. If $M\\ge r$, complete retention is proved exact.","ar\\medskip\n\\pageheading{4. Uniform truncation and a sharp exact example}{Corollary 4.3 and Example 4.4}\nFor row $2r$ and depth $M$, the actual truncation is\n\\[\n T_M(p,2r)=\\sum_{m=0}^{\\min(M,r)}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).\n\\]\nEvery omitted term has $m\\ge M+1$. Its other factors are integers, so the first-block omitted-tail theorem, now applied to the \\emph{proved Euler identity}, gives\n\\[\n \\boxed{p^{2M+2}\\mid E_{2r}-T_M(p,2r)\\quad\\text{in }\\Z.}\n\\]\nThis is not divisibility in $\\Q$, where a nonzero divisor would make the assertion vacuous. The statement holds for every row and depth. For an odd prime it is precisely the claimed supercongruence. If $M\\ge r$, complete retention is proved exact.","Example 4.4}\nFor row $2r$ and depth $M$, the actual truncation is\n\\[\n T_M(p,2r)=\\sum_{m=0}^{\\min(M,r)}\\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).\n\\]\nEvery omitted term has $m\\ge M+1$. Its other factors are integers, so the first-block omitted-tail theorem, now applied to the \\emph{proved Euler identity}, gives\n\\[\n \\boxed{p^{2M+2}\\mid E_{2r}-T_M(p,2r)\\quad\\text{in }\\Z.}\n\\]\nThis is not divisibility in $\\Q$, where a nonzero divisor would make the assertion vacuous. The statement holds for every row and depth. For an odd prime it is precisely the claimed supercongruence. If $M\\ge r$, complete retention is proved exact.","M+1$. Its other factors are integers, so the first-block omitted-tail theorem, now applied to the \\emph{proved Euler identity}, gives\n\\[\n \\boxed{p^{2M+2}\\mid E_{2r}-T_M(p,2r)\\quad\\text{in }\\Z.}\n\\]\nThis is not divisibility in $\\Q$, where a nonzero divisor would make the assertion vacuous. The statement holds for every row and depth. For an odd prime it is precisely the claimed supercongruence. If $M\\ge r$, complete retention is proved exact.","\\subsection*{The displayed manuscript row is kernel-checked}\nThe scalar anchors are derived from the same general coefficient recurrence:\n\\[\n (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385),\n\\]\n\\[\n (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936).\n\\]\nThe literal source sums at $p=5$ give\n\\[\n (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985).\n\\]\nConsequently,\n\\[\n T_2(5,8)=2\\,731\\,001\\,385,\\qquad\n E_8-T_2(5,8)=-2\\,731\\,000\\,000.\n\\]\nThe source proves both\n\\[\n 5^6\\mid E_8-T_2(5,8),\\qquad 5^7\\nmid E_8-T_2(5,8).\n\\]\nThus the guaranteed power is attained in this example. No stronger uniform exponent is inferred.","kernel-checked}\nThe scalar anchors are derived from the same general coefficient recurrence:\n\\[\n (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385),\n\\]\n\\[\n (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936).\n\\]\nThe literal source sums at $p=5$ give\n\\[\n (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985).\n\\]\nConsequently,\n\\[\n T_2(5,8)=2\\,731\\,001\\,385,\\qquad\n E_8-T_2(5,8)=-2\\,731\\,000\\,000.\n\\]\nThe source proves both\n\\[\n 5^6\\mid E_8-T_2(5,8),\\qquad 5^7\\nmid E_8-T_2(5,8).\n\\]\nThus the guaranteed power is attained in this example. No stronger uniform exponent is inferred.","\\subsection*{Oddness cannot be silently dropped}\nFor length two, $S_0(2)=1-1=0$, whereas $E_0=1$. The row-zero reconstruction already fails. The source proves this failure, and the release verifier separately attempts and rejects an invalid proof of the even-length identity.","\\subsubsection*{The completed mathematical chain}\nThe inverse coefficients are defined from a genuine unit power series and connected to $z^2/\\sinh^2z$. The source is the chapter's explicit rational formula, not an abstract integral input. Its valuation calculation proves the stronger statement that \\emph{every even source coefficient is a $2$-adic unit}. Formal-series multiplication and coefficient extraction then yield the Euler identity with the correct falling factorial. The omitted-tail theorem gives divisibility in the actual completed integer ring $\\Z_2$.","e is the chapter's explicit rational formula, not an abstract integral input. Its valuation calculation proves the stronger statement that \\emph{every even source coefficient is a $2$-adic unit}. Formal-series multiplication and coefficient extraction then yield the Euler identity with the correct falling factorial. The omitted-tail theorem gives divisibility in the actual completed integer ring $\\Z_2$.","cients is not universally formalized here. Formal-series identities do not establish analytic convergence. The corrected global odd-prime finite-difference theorem remains pending. Historical numerical evidence is not presented as a fresh reproduction.\n\\par\\medskip\n\\pageheading{1. Integral coefficients and an exact unit source}{Lemmas 5.1 and 5.2; manuscript page 14}\nDefine the rational subring\n\\[\n \\mathcal O_2=\\{q\\in\\Q:|q|_2\\leq1\\}.\n\\]\nIts ring laws follow from the actual nonarchimedean norm. For every $q\\in\\mathcal O_2$, the code constructs its element in Mathlib's completed ring $\\Z_2$, and proves that the underlying value in $\\Q_2$ is the rational cast of $q$. No integrality claim is hidden in a coercion.","corrected global odd-prime finite-difference theorem remains pending. Historical numerical evidence is not presented as a fresh reproduction.\n\\par\\medskip\n\\pageheading{1. Integral coefficients and an exact unit source}{Lemmas 5.1 and 5.2; manuscript page 14}\nDefine the rational subring\n\\[\n \\mathcal O_2=\\{q\\in\\Q:|q|_2\\leq1\\}.\n\\]\nIts ring laws follow from the actual nonarchimedean norm. For every $q\\in\\mathcal O_2$, the code constructs its element in Mathlib's completed ring $\\Z_2$, and proves that the underlying value in $\\Q_2$ is the rational cast of $q$. No integrality claim is hidden in a coercion.","l numerical evidence is not presented as a fresh reproduction.\n\\par\\medskip\n\\pageheading{1. Integral coefficients and an exact unit source}{Lemmas 5.1 and 5.2; manuscript page 14}\nDefine the rational subring\n\\[\n \\mathcal O_2=\\{q\\in\\Q:|q|_2\\leq1\\}.\n\\]\nIts ring laws follow from the actual nonarchimedean norm. For every $q\\in\\mathcal O_2$, the code constructs its element in Mathlib's completed ring $\\Z_2$, and proves that the underlying value in $\\Q_2$ is the rational cast of $q$. No integrality claim is hidden in a coercion.","reproduction.\n\\par\\medskip\n\\pageheading{1. Integral coefficients and an exact unit source}{Lemmas 5.1 and 5.2; manuscript page 14}\nDefine the rational subring\n\\[\n \\mathcal O_2=\\{q\\in\\Q:|q|_2\\leq1\\}.\n\\]\nIts ring laws follow from the actual nonarchimedean norm. For every $q\\in\\mathcal O_2$, the code constructs its element in Mathlib's completed ring $\\Z_2$, and proves that the underlying value in $\\Q_2$ is the rational cast of $q$. No integrality claim is hidden in a coercion.","geheading{1. Integral coefficients and an exact unit source}{Lemmas 5.1 and 5.2; manuscript page 14}\nDefine the rational subring\n\\[\n \\mathcal O_2=\\{q\\in\\Q:|q|_2\\leq1\\}.\n\\]\nIts ring laws follow from the actual nonarchimedean norm. For every $q\\in\\mathcal O_2$, the code constructs its element in Mathlib's completed ring $\\Z_2$, and proves that the underlying value in $\\Q_2$ is the rational cast of $q$. No integrality claim is hidden in a coercion.","\\subsection*{The unit kernel and its reciprocal}\nUse a squared formal variable $u$ and define\n\\[\n k_m=\\frac{2^{2m+1}}{(2m+2)!},\\qquad\n K(u)=\\sum_{m\\geq0}k_mu^m,\\qquad\n \\beta_m=[u^m]K(u)^{-1}.\n\\]\nThe constant term is proved to be one. The factorial valuation bound gives\n\\[\n v_2(k_m)=2m+1-v_2((2m+2)!)\\geq0.\n\\]\nThus every $k_m$ belongs to $\\mathcal O_2$. A general coefficientwise subring theorem is proved by strong induction using the actual unit-inverse coefficient formula. Applied to $K$, it establishes\n\\[\n \\boxed{\\beta_m\\in\\mathcal O_2\\quad\\text{for every }m\\geq0.}\n\\]\nThe reciprocal's coefficients are therefore proved integral. The proof does not assume them integral in order to obtain the final congruence. Their connection to the intended hyperbolic function is proved o","\\subsection*{The unit kernel and its reciprocal}\nUse a squared formal variable $u$ and define\n\\[\n k_m=\\frac{2^{2m+1}}{(2m+2)!},\\qquad\n K(u)=\\sum_{m\\geq0}k_mu^m,\\qquad\n \\beta_m=[u^m]K(u)^{-1}.\n\\]\nThe constant term is proved to be one. The factorial valuation bound gives\n\\[\n v_2(k_m)=2m+1-v_2((2m+2)!)\\geq0.\n\\]\nThus every $k_m$ belongs to $\\mathcal O_2$. A general coefficientwise subring theorem is proved by strong induction using the actual unit-inverse coefficient formula. Applied to $K$, it establishes\n\\[\n \\boxed{\\beta_m\\in\\mathcal O_2\\quad\\text{for every }m\\geq0.}\n\\]\nThe reciprocal's coefficients are therefore proved integral. The proof does not assume them integral in order to obtain the final congruence. Their connection to the intended hyperbolic function is proved on the next page.","fine\n\\[\n k_m=\\frac{2^{2m+1}}{(2m+2)!},\\qquad\n K(u)=\\sum_{m\\geq0}k_mu^m,\\qquad\n \\beta_m=[u^m]K(u)^{-1}.\n\\]\nThe constant term is proved to be one. The factorial valuation bound gives\n\\[\n v_2(k_m)=2m+1-v_2((2m+2)!)\\geq0.\n\\]\nThus every $k_m$ belongs to $\\mathcal O_2$. A general coefficientwise subring theorem is proved by strong induction using the actual unit-inverse coefficient formula. Applied to $K$, it establishes\n\\[\n \\boxed{\\beta_m\\in\\mathcal O_2\\quad\\text{for every }m\\geq0.}\n\\]\nThe reciprocal's coefficients are therefore proved integral. The proof does not assume them integral in order to obtain the final congruence. Their connection to the intended hyperbolic function is proved on the next page.","{2m+1}}{(2m+2)!},\\qquad\n K(u)=\\sum_{m\\geq0}k_mu^m,\\qquad\n \\beta_m=[u^m]K(u)^{-1}.\n\\]\nThe constant term is proved to be one. The factorial valuation bound gives\n\\[\n v_2(k_m)=2m+1-v_2((2m+2)!)\\geq0.\n\\]\nThus every $k_m$ belongs to $\\mathcal O_2$. A general coefficientwise subring theorem is proved by strong induction using the actual unit-inverse coefficient formula. Applied to $K$, it establishes\n\\[\n \\boxed{\\beta_m\\in\\mathcal O_2\\quad\\text{for every }m\\geq0.}\n\\]\nThe reciprocal's coefficients are therefore proved integral. The proof does not assume them integral in order to obtain the final congruence. Their connection to the intended hyperbolic function is proved on the next page.","\\subsection*{The source has valuation exactly zero}\nAt even degree $2r$, define the literal rational source\n\\[\n R_{2r}=\\frac{3^{2r+2}-1}{4(2r+1)(2r+2)}.\n\\]\nBoth numerator and denominator are nonzero. The pinned Mathlib lifting-the-exponent theorem gives\n\\[\n v_2(3^{2r+2}-1)=2+v_2(2r+2).\n\\]\nBecause $2r+1$ is odd, the denominator has the same valuation. Consequently,\n\\[\n \\boxed{v_2(R_{2r})=0,\\qquad |R_{2r}|_2=1.}\n\\]\nThe code proves actual \\mcode{IsUnit} status for the constructed $\\Z_2$ element. This strengthens the manuscript's integrality conclusion without changing its source. The argument applies at $r=0$, where $R_0=1$, as well as at every later row.","literal rational source\n\\[\n R_{2r}=\\frac{3^{2r+2}-1}{4(2r+1)(2r+2)}.\n\\]\nBoth numerator and denominator are nonzero. The pinned Mathlib lifting-the-exponent theorem gives\n\\[\n v_2(3^{2r+2}-1)=2+v_2(2r+2).\n\\]\nBecause $2r+1$ is odd, the denominator has the same valuation. Consequently,\n\\[\n \\boxed{v_2(R_{2r})=0,\\qquad |R_{2r}|_2=1.}\n\\]\nThe code proves actual \\mcode{IsUnit} status for the constructed $\\Z_2$ element. This strengthens the manuscript's integrality conclusion without changing its source. The argument applies at $r=0$, where $R_0=1$, as well as at every later row.","d Mathlib lifting-the-exponent theorem gives\n\\[\n v_2(3^{2r+2}-1)=2+v_2(2r+2).\n\\]\nBecause $2r+1$ is odd, the denominator has the same valuation. Consequently,\n\\[\n \\boxed{v_2(R_{2r})=0,\\qquad |R_{2r}|_2=1.}\n\\]\nThe code proves actual \\mcode{IsUnit} status for the constructed $\\Z_2$ element. This strengthens the manuscript's integrality conclusion without changing its source. The argument applies at $r=0$, where $R_0=1$, as well as at every later row.","ting-the-exponent theorem gives\n\\[\n v_2(3^{2r+2}-1)=2+v_2(2r+2).\n\\]\nBecause $2r+1$ is odd, the denominator has the same valuation. Consequently,\n\\[\n \\boxed{v_2(R_{2r})=0,\\qquad |R_{2r}|_2=1.}\n\\]\nThe code proves actual \\mcode{IsUnit} status for the constructed $\\Z_2$ element. This strengthens the manuscript's integrality conclusion without changing its source. The argument applies at $r=0$, where $R_0=1$, as well as at every later row.","\\textbf{Principal targets.}\\par\n\\target{Dyadic.kernel_coefficient_integral, Dyadic.beta_integral}\\par\n\\target{Dyadic.source_valuation, Dyadic.sourcePadic_isUnit}\n\\par\\medskip\n\\pageheading{2. Connecting the definitions to actual formal series}{Generating-function foundations for Theorem 5.3}\nWork in $\\Q[[t]]$. The formal exponential and rescaling homomorphism are the same checked objects used in the second block. Define\n\\[\n \\sinh_a(t)=\\frac{e^{at}-e^{-at}}2,\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2.\n\\]\nLet $B(t)=K(t^2)^{-1}=\\sum_{m\\geq0}\\beta_mt^{2m}$ and let\n\\[\n F(t)=\\sum_{r\\geq0}R_{2r}\\frac{t^{2r}}{(2r)!}.\n\\]\nThese series are constructed independently from the definitions on the preceding page.",".}\\par\n\\target{Dyadic.kernel_coefficient_integral, Dyadic.beta_integral}\\par\n\\target{Dyadic.source_valuation, Dyadic.sourcePadic_isUnit}\n\\par\\medskip\n\\pageheading{2. Connecting the definitions to actual formal series}{Generating-function foundations for Theorem 5.3}\nWork in $\\Q[[t]]$. The formal exponential and rescaling homomorphism are the same checked objects used in the second block. Define\n\\[\n \\sinh_a(t)=\\frac{e^{at}-e^{-at}}2,\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2.\n\\]\nLet $B(t)=K(t^2)^{-1}=\\sum_{m\\geq0}\\beta_mt^{2m}$ and let\n\\[\n F(t)=\\sum_{r\\geq0}R_{2r}\\frac{t^{2r}}{(2r)!}.\n\\]\nThese series are constructed independently from the definitions on the preceding page.","adic.source_valuation, Dyadic.sourcePadic_isUnit}\n\\par\\medskip\n\\pageheading{2. Connecting the definitions to actual formal series}{Generating-function foundations for Theorem 5.3}\nWork in $\\Q[[t]]$. The formal exponential and rescaling homomorphism are the same checked objects used in the second block. Define\n\\[\n \\sinh_a(t)=\\frac{e^{at}-e^{-at}}2,\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2.\n\\]\nLet $B(t)=K(t^2)^{-1}=\\sum_{m\\geq0}\\beta_mt^{2m}$ and let\n\\[\n F(t)=\\sum_{r\\geq0}R_{2r}\\frac{t^{2r}}{(2r)!}.\n\\]\nThese series are constructed independently from the definitions on the preceding page.","medskip\n\\pageheading{2. Connecting the definitions to actual formal series}{Generating-function foundations for Theorem 5.3}\nWork in $\\Q[[t]]$. The formal exponential and rescaling homomorphism are the same checked objects used in the second block. Define\n\\[\n \\sinh_a(t)=\\frac{e^{at}-e^{-at}}2,\\qquad\n \\cosh_a(t)=\\frac{e^{at}+e^{-at}}2.\n\\]\nLet $B(t)=K(t^2)^{-1}=\\sum_{m\\geq0}\\beta_mt^{2m}$ and let\n\\[\n F(t)=\\sum_{r\\geq0}R_{2r}\\frac{t^{2r}}{(2r)!}.\n\\]\nThese series are constructed independently from the definitions on the preceding page.","\\subsection*{No inverse of a zero-constant series is assumed}\nCoefficient extraction proves the cleared identity\n\\[\n t^2K(t^2)=\\frac{\\cosh(2t)-1}{2}.\n\\]\nA separate identity from the formal exponential gives\n\\[\n \\sinh^2t=\\frac{\\cosh(2t)-1}{2}.\n\\]\nSince $K(t^2)$ has constant term one, its reciprocal exists as a unit. Multiplication then proves\n\\[\n \\boxed{B(t)\\sinh^2t=t^2.}\n\\]\nThis is the precise formal meaning of $B(t)=t^2/\\sinh^2t$. The forced quadratic zero is removed first. The proof does not claim that $\\sinh t$ itself is an invertible formal power series.","inverse of a zero-constant series is assumed}\nCoefficient extraction proves the cleared identity\n\\[\n t^2K(t^2)=\\frac{\\cosh(2t)-1}{2}.\n\\]\nA separate identity from the formal exponential gives\n\\[\n \\sinh^2t=\\frac{\\cosh(2t)-1}{2}.\n\\]\nSince $K(t^2)$ has constant term one, its reciprocal exists as a unit. Multiplication then proves\n\\[\n \\boxed{B(t)\\sinh^2t=t^2.}\n\\]\nThis is the precise formal meaning of $B(t)=t^2/\\sinh^2t$. The forced quadratic zero is removed first. The proof does not claim that $\\sinh t$ itself is an invertible formal power series.","y\n\\[\n t^2K(t^2)=\\frac{\\cosh(2t)-1}{2}.\n\\]\nA separate identity from the formal exponential gives\n\\[\n \\sinh^2t=\\frac{\\cosh(2t)-1}{2}.\n\\]\nSince $K(t^2)$ has constant term one, its reciprocal exists as a unit. Multiplication then proves\n\\[\n \\boxed{B(t)\\sinh^2t=t^2.}\n\\]\nThis is the precise formal meaning of $B(t)=t^2/\\sinh^2t$. The forced quadratic zero is removed first. The proof does not claim that $\\sinh t$ itself is an invertible formal power series.","\\subsection*{The explicit rational source supplies its own identity}\nFrom the formula for $R_{2r}$, coefficient arithmetic proves\n\\[\n \\boxed{t^2 F(t)=\\frac{\\cosh(3t)-\\cosh t}{4}.}\n\\]\nThe product relation for the formal hyperbolic cosines yields\n\\[\n F(t)\\cosh t=K(4t^2).\n\\]\nThis equality is proved after multiplication by $t^2$ and cancellation of that monomial in the formal-series ring. It is not an assumed source-transform relation.","Rescale the reciprocal by $t\\mapsto2t$. The unit inverse law and the preceding identity now give\n\\[\n \\boxed{B(2t)F(t)=\\sech t.}\n\\]\nThe right side is connected to the same integral signed Euler sequence constructed in the previous release. These are equalities of full formal series, not fixed-degree polynomial experiments.","\\textbf{Principal targets.}\\par\n\\target{DyadicSeries.beta_original_generating_function}\\par\n\\target{DyadicSeries.source_shift_identity, DyadicSeries.source_times_cosh}\\par\n\\target{DyadicSeries.dyadic_reconstruction_series}\n\\par\\medskip\n\\pageheading{3. Exact reconstruction and genuine dyadic congruence}{Theorem 5.3 and Corollary 5.4; manuscript pages 14--15}\nThe reciprocal coefficients $\\beta_m$ use an \\emph{ordinary} power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree $2r$ from $B(2t)F(t)=\\sech t$ therefore gives\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\beta_m2^{2m}\\fall{2r}{2m}R_{2(r-m)},}\n \\qquad\n \\fall{2r}{2m}=\\frac{(2r)!}{(2r-2m)!}.\n\\]\nThe formal proof uses the actual natural-number descending factorial and proves its factorial-quotient identity for $2m\\leq2r$. The odd coefficient terms vanish by a proved parity argument. The binomial EGF product then supplies the falling factorial, rather than treating it as an extra assumed factor.","entity, DyadicSeries.source_times_cosh}\\par\n\\target{DyadicSeries.dyadic_reconstruction_series}\n\\par\\medskip\n\\pageheading{3. Exact reconstruction and genuine dyadic congruence}{Theorem 5.3 and Corollary 5.4; manuscript pages 14--15}\nThe reciprocal coefficients $\\beta_m$ use an \\emph{ordinary} power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree $2r$ from $B(2t)F(t)=\\sech t$ therefore gives\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\beta_m2^{2m}\\fall{2r}{2m}R_{2(r-m)},}\n \\qquad\n \\fall{2r}{2m}=\\frac{(2r)!}{(2r-2m)!}.\n\\]\nThe formal proof uses the actual natural-number descending factorial and proves its factorial-quotient identity for $2m\\leq2r$. The odd coefficient terms vanish by a proved parity argument. The binomial EGF product then supplies the falling factorial, rather than treating it as an extra assumed factor.","adicSeries.source_times_cosh}\\par\n\\target{DyadicSeries.dyadic_reconstruction_series}\n\\par\\medskip\n\\pageheading{3. Exact reconstruction and genuine dyadic congruence}{Theorem 5.3 and Corollary 5.4; manuscript pages 14--15}\nThe reciprocal coefficients $\\beta_m$ use an \\emph{ordinary} power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree $2r$ from $B(2t)F(t)=\\sech t$ therefore gives\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\beta_m2^{2m}\\fall{2r}{2m}R_{2(r-m)},}\n \\qquad\n \\fall{2r}{2m}=\\frac{(2r)!}{(2r-2m)!}.\n\\]\nThe formal proof uses the actual natural-number descending factorial and proves its factorial-quotient identity for $2m\\leq2r$. The odd coefficient terms vanish by a proved parity argument. The binomial EGF product then supplies the falling factorial, rather than treating it as an extra assumed factor.","rget{DyadicSeries.dyadic_reconstruction_series}\n\\par\\medskip\n\\pageheading{3. Exact reconstruction and genuine dyadic congruence}{Theorem 5.3 and Corollary 5.4; manuscript pages 14--15}\nThe reciprocal coefficients $\\beta_m$ use an \\emph{ordinary} power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree $2r$ from $B(2t)F(t)=\\sech t$ therefore gives\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\beta_m2^{2m}\\fall{2r}{2m}R_{2(r-m)},}\n \\qquad\n \\fall{2r}{2m}=\\frac{(2r)!}{(2r-2m)!}.\n\\]\nThe formal proof uses the actual natural-number descending factorial and proves its factorial-quotient identity for $2m\\leq2r$. The odd coefficient terms vanish by a proved parity argument. The binomial EGF product then supplies the falling factorial, rather than treating it as an extra assumed factor.","ordinary} power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree $2r$ from $B(2t)F(t)=\\sech t$ therefore gives\n\\[\n \\boxed{E_{2r}=\\sum_{m=0}^{r}\\beta_m2^{2m}\\fall{2r}{2m}R_{2(r-m)},}\n \\qquad\n \\fall{2r}{2m}=\\frac{(2r)!}{(2r-2m)!}.\n\\]\nThe formal proof uses the actual natural-number descending factorial and proves its factorial-quotient identity for $2m\\leq2r$. The odd coefficient terms vanish by a proved parity argument. The binomial EGF product then supplies the falling factorial, rather than treating it as an extra assumed factor.","Both row indexing by $r$ and the literal even-degree version with hypothesis $\\operatorname{Even}(n)$ are checked. All rows include degree zero. The exact identity is a rational equality with the Euler coefficient's integer-to-rational cast made explicit.","\\subsection*{The finite truncation}\nFor $M\\geq0$, define\n\\[\n T_M(2r)=\\sum_{m=0}^{\\min(M,r)}\\beta_m2^{2m}\\fall{2r}{2m}R_{2(r-m)}.\n\\]\nThe integral rational factors are embedded into $\\Z_2$ using their proved witnesses. The cast of this $\\Z_2$ truncation into $\\Q_2$ is proved to equal the cast of the literal rational truncation.","Each omitted term has $m\\geq M+1$. Its remaining factors lie in $\\Z_2$, so the checked weighted-tail theorem proves\n\\[\n \\boxed{2^{2M+2}\\mid E_{2r}-T_M(2r)\\quad\\text{in }\\Z_2.}\n\\]\nThis is not divisibility in $\\Q$ or $\\Q_2$, where division by any nonzero scalar would make the claim vacuous. A second theorem supplies the equivalent rational statement\n\\[\n \\boxed{\\frac{E_{2r}-T_M(2r)}{2^{2M+2}}\\in\\mathcal O_2.}\n\\]\nIt follows from actual $\\Z_2$ divisibility and the proved rational/completed-field bridge. No precision assertion is passed in as a hypothesis.","s remaining factors lie in $\\Z_2$, so the checked weighted-tail theorem proves\n\\[\n \\boxed{2^{2M+2}\\mid E_{2r}-T_M(2r)\\quad\\text{in }\\Z_2.}\n\\]\nThis is not divisibility in $\\Q$ or $\\Q_2$, where division by any nonzero scalar would make the claim vacuous. A second theorem supplies the equivalent rational statement\n\\[\n \\boxed{\\frac{E_{2r}-T_M(2r)}{2^{2M+2}}\\in\\mathcal O_2.}\n\\]\nIt follows from actual $\\Z_2$ divisibility and the proved rational/completed-field bridge. No precision assertion is passed in as a hypothesis.","\\subsection*{Endpoints and scope}\nWhen $M\\geq r$, the sum retains every term and $T_M(2r)=E_{2r}$ exactly, in both rational and dyadic coordinates. The zero-error case is included. No finite valuation of zero is used. This is a separate dyadic construction, not substitution of $p=2$ into the odd-length theorem.","\\textbf{Principal targets.}\\par\n\\target{DyadicReconstruction.dyadic_reconstruction}\\par\n\\target{DyadicReconstruction.dyadic_supercongruence}\\par\n\\target{DyadicReconstruction.normalized_error_integral}\n\\par\\medskip\n\\pageheading{4. Exact examples and adversarial checks}{Example 5.5 and controls against incorrect normalizations}\nThe actual reciprocal recurrence gives\n\\[\n (\\beta_0,\\beta_1,\\beta_2,\\beta_3,\\beta_4)\n =\\left(1,-\\frac13,\\frac1{15},-\\frac2{189},\\frac1{675}\\right).\n\\]\nThese values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives\n\\[\n (R_0,R_2,R_4,R_6,R_8)\n =\\left(1,\\frac53,\\frac{91}{15},\\frac{205}{7},\\frac{7381}{45}\\right).\n\\]\nUsing the retained Euler anchor $E_8=1385$, Lean checks the literal decomposition\n\\[\n E_8=\\frac{7381}{45}-\\frac{6560}{3}+\\frac{163072}{15}\n       -\\frac{204800}{9}+\\frac{229376}{15}=1385.\n\\]","al checks}{Example 5.5 and controls against incorrect normalizations}\nThe actual reciprocal recurrence gives\n\\[\n (\\beta_0,\\beta_1,\\beta_2,\\beta_3,\\beta_4)\n =\\left(1,-\\frac13,\\frac1{15},-\\frac2{189},\\frac1{675}\\right).\n\\]\nThese values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives\n\\[\n (R_0,R_2,R_4,R_6,R_8)\n =\\left(1,\\frac53,\\frac{91}{15},\\frac{205}{7},\\frac{7381}{45}\\right).\n\\]\nUsing the retained Euler anchor $E_8=1385$, Lean checks the literal decomposition\n\\[\n E_8=\\frac{7381}{45}-\\frac{6560}{3}+\\frac{163072}{15}\n       -\\frac{204800}{9}+\\frac{229376}{15}=1385.\n\\]","\\beta_2,\\beta_3,\\beta_4)\n =\\left(1,-\\frac13,\\frac1{15},-\\frac2{189},\\frac1{675}\\right).\n\\]\nThese values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives\n\\[\n (R_0,R_2,R_4,R_6,R_8)\n =\\left(1,\\frac53,\\frac{91}{15},\\frac{205}{7},\\frac{7381}{45}\\right).\n\\]\nUsing the retained Euler anchor $E_8=1385$, Lean checks the literal decomposition\n\\[\n E_8=\\frac{7381}{45}-\\frac{6560}{3}+\\frac{163072}{15}\n       -\\frac{204800}{9}+\\frac{229376}{15}=1385.\n\\]","15},-\\frac2{189},\\frac1{675}\\right).\n\\]\nThese values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives\n\\[\n (R_0,R_2,R_4,R_6,R_8)\n =\\left(1,\\frac53,\\frac{91}{15},\\frac{205}{7},\\frac{7381}{45}\\right).\n\\]\nUsing the retained Euler anchor $E_8=1385$, Lean checks the literal decomposition\n\\[\n E_8=\\frac{7381}{45}-\\frac{6560}{3}+\\frac{163072}{15}\n       -\\frac{204800}{9}+\\frac{229376}{15}=1385.\n\\]","\\subsection*{Truncations and actual valuations}\n\\begin{center}\n\\begin{tabular}{@{}crr@{}}\n\\toprule\nDepth $M$ & $T_M(8)$ & Guaranteed exponent $2M+2$","midrule\n0 & $7381/45$ & 2","[3pt]\n1 & $-91019/45$ & 4\\\\[3pt]\n2 & $398197/45$ & 6\\\\[3pt]\n3 & $-208601/15$ & 8\\\\[3pt]\n4 & $1385$ & 10\\\\\\bottomrule\n\\end{tabular}\n\\end{center}\nThe checked nonzero error valuations are\n\\[\n E_8-T_0(8)=\\frac{54944}{45},\\qquad v_2(E_8-T_0(8))=5,\n\\]\n\\[\n E_8-T_3(8)=\\frac{229376}{15},\\qquad v_2(E_8-T_3(8))=15.\n\\]\nThey exceed the uniform guarantees. These two examples do not establish a universally stronger exponent.","bottomrule\n\\end{tabular}\n\\end{center}\nThe checked nonzero error valuations are\n\\[\n E_8-T_0(8)=\\frac{54944}{45},\\qquad v_2(E_8-T_0(8))=5,\n\\]\n\\[\n E_8-T_3(8)=\\frac{229376}{15},\\qquad v_2(E_8-T_3(8))=15.\n\\]\nThey exceed the uniform guarantees. These two examples do not establish a universally stronger exponent.","\\subsection*{What the negative controls rule out}\nOmitting the rescaling $2^{2m}$ already breaks the degree-two identity: the altered sum is $1$, whereas $E_2=-1$. Confusing ordinary beta coefficients with EGF coefficients would replace $\\beta_1=-1/3$ by $-2/3$. A checked inequality excludes that normalization. A deliberately false claim that the depth-zero degree-eight error has valuation at least six is rejected.","\\subsubsection*{Two explicit specification boundaries}\nA finite sample through $N$ determines $\\Delta^s f(n)$ only when $n+s\\leq N$. For arbitrary orders, the finite-coefficient identity describes the constructed interpolant, not every unknown continuation of the sample. All three versions are separately checked.","The additive valuation is extended to $\\ord_p(0)=\\infty$. This avoids incorrectly applying Mathlib's finite-valued default at zero to an inequality that must also cover cancellation and identically zero differences.","d rejected 12 invalid-proof controls. All 74 infrastructure and release tests passed. Independent exact-rational mirrors are recorded separately from the universal proofs. The archive is also subjected to a fresh-extraction rebuild and integrity audit. Its separate report records that outcome.\n\\par\\medskip\n\\pageheading{1. Reconstruction and finite-data scope}{Theorem 7.2; manuscript page 17}\nLet $R$ be any commutative ring. The implementation uses the actual forward-difference operator\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad c_r=\\Delta_1^r f(0).\n\\]\nFor a full sequence $f:\\N\\to R$, pinned Mathlib's Gregory--Newton formula gives\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparen…","dent exact-rational mirrors are recorded separately from the universal proofs. The archive is also subjected to a fresh-extraction rebuild and integrity audit. Its separate report records that outcome.\n\\par\\medskip\n\\pageheading{1. Reconstruction and finite-data scope}{Theorem 7.2; manuscript page 17}\nLet $R$ be any commutative ring. The implementation uses the actual forward-difference operator\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad c_r=\\Delta_1^r f(0).\n\\]\nFor a full sequence $f:\\N\\to R$, pinned Mathlib's Gregory--Newton formula gives\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparently infinite coefficient sum is finite at each natural argument. Neither continuity nor conve…","he archive is also subjected to a fresh-extraction rebuild and integrity audit. Its separate report records that outcome.\n\\par\\medskip\n\\pageheading{1. Reconstruction and finite-data scope}{Theorem 7.2; manuscript page 17}\nLet $R$ be any commutative ring. The implementation uses the actual forward-difference operator\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad c_r=\\Delta_1^r f(0).\n\\]\nFor a full sequence $f:\\N\\to R$, pinned Mathlib's Gregory--Newton formula gives\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparently infinite coefficient sum is finite at each natural argument. Neither continuity nor convergence is needed for these identities.","d integrity audit. Its separate report records that outcome.\n\\par\\medskip\n\\pageheading{1. Reconstruction and finite-data scope}{Theorem 7.2; manuscript page 17}\nLet $R$ be any commutative ring. The implementation uses the actual forward-difference operator\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad c_r=\\Delta_1^r f(0).\n\\]\nFor a full sequence $f:\\N\\to R$, pinned Mathlib's Gregory--Newton formula gives\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparently infinite coefficient sum is finite at each natural argument. Neither continuity nor convergence is needed for these identities.","script page 17}\nLet $R$ be any commutative ring. The implementation uses the actual forward-difference operator\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad c_r=\\Delta_1^r f(0).\n\\]\nFor a full sequence $f:\\N\\to R$, pinned Mathlib's Gregory--Newton formula gives\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparently infinite coefficient sum is finite at each natural argument. Neither continuity nor convergence is needed for these identities.","d-difference operator\n\\[\n \\Delta_h f(n)=f(n+h)-f(n),\\qquad c_r=\\Delta_1^r f(0).\n\\]\nFor a full sequence $f:\\N\\to R$, pinned Mathlib's Gregory--Newton formula gives\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparently infinite coefficient sum is finite at each natural argument. Neither continuity nor convergence is needed for these identities.","nce $f:\\N\\to R$, pinned Mathlib's Gregory--Newton formula gives\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparently infinite coefficient sum is finite at each natural argument. Neither continuity nor convergence is needed for these identities.","ves\n\\[\n \\boxed{f(n)=\\sum_{r=0}^{n}c_r\\binom nr.}\n\\]\nThe source applies the theorem to these exact definitions. A separate explicit formula relates an $s$-th difference to its original sampled values, so the data dependence is inspectable:\n\\[\n \\Delta_h^s f(n)=\\sum_{j=0}^{s}(-1)^{s-j}\\binom sj f(n+jh).\n\\]\nApplying Newton reconstruction to $\\Delta_1^s f$ and composing the iterated operators proves\n\\[\n \\boxed{\\Delta_1^s f(n)=\\sum_{j=0}^{n}c_{s+j}\\binom nj.}\n\\]\nThe apparently infinite coefficient sum is finite at each natural argument. Neither continuity nor convergence is needed for these identities.","\\subsection*{An independently defined interpolant}\nFor a finite coefficient list, define\n\\[\n I_N(c)(n)=\\sum_{r=0}^{N}c_r\\binom nr.\n\\]\nThe source proves exact coefficient extraction:\n\\[\n \\Delta_1^s I_N(c)(0)=\n \\begin{cases}c_s,&s\\leq N,","\\subsection*{An independently defined interpolant}\nFor a finite coefficient list, define\n\\[\n I_N(c)(n)=\\sum_{r=0}^{N}c_r\\binom nr.\n\\]\nThe source proves exact coefficient extraction:\n\\[\n \\Delta_1^s I_N(c)(0)=\n \\begin{cases}c_s,&s\\leq N,\\\\0,&s>N.\\end{cases}\n\\]\nIt follows that $I_N(c)$ has the stated shifted-coefficient formula at every order, with zero coefficients beyond $N$. Every difference of order greater than $N$ vanishes. Its coefficients are uniquely determined by its values at $0,\\ldots,N$.","0,&s>N.\\end{cases}\n\\]\nIt follows that $I_N(c)$ has the stated shifted-coefficient formula at every order, with zero coefficients beyond $N$. Every difference of order greater than $N$ vanishes. Its coefficients are uniquely determined by its values at $0,\\ldots,N$.","\\subsection*{What the finite sample does and does not determine}\nFor $c_r=\\Delta_1^r f(0)$ with $r\\leq N$, the source proves\n\\[\n I_N(c)(n)=f(n)\\quad(n\\leq N),\\qquad\n \\Delta_1^s I_N(c)(n)=\\Delta_1^s f(n)\\quad(n+s\\leq N).\n\\]\nThe condition is essential. The sequences $f(n)=n$ and $g(n)=0$ agree on the sample $\\{0\\}$, but $\\Delta f(0)=1$ and $\\Delta g(0)=0$. This is a checked counterexample to unrestricted inference outside the observed stencil.","\\textbf{Principal targets.}\\par\n\\target{Mahler.finite_mahler_reconstruction, Mahler.coefficient_interpolant}\\par\n\\target{Mahler.difference_from_coefficients, Mahler.finite_sample_difference}\n\\par\\medskip\n\\pageheading{2. Weighted bounds in the actual ring $\\Z_p$}{Corollary 7.3; manuscript page 18}\nLet $p$ be prime. For $x\\in\\Z_p$, define\n\\[\n \\ord_p(x)=\\begin{cases}\\infty,&x=0,","\\textbf{Principal targets.}\\par\n\\target{Mahler.finite_mahler_reconstruction, Mahler.coefficient_interpolant}\\par\n\\target{Mahler.difference_from_coefficients, Mahler.finite_sample_difference}\n\\par\\medskip\n\\pageheading{2. Weighted bounds in the actual ring $\\Z_p$}{Corollary 7.3; manuscript page 18}\nLet $p$ be prime. For $x\\in\\Z_p$, define\n\\[\n \\ord_p(x)=\\begin{cases}\\infty,&x=0,\\\\v_p(x),&x\\neq0.\\end{cases}\n\\]\nThe nonzero branch uses Mathlib's actual $p$-adic valuation. The implementation proves its connection to genuine integral divisibility:\n\\[\n \\boxed{L\\leq\\ord_p(x)\\quad\\Longleftrightarrow\\quad p^L\\mid x\n \\ \\text{in }\\Z_p,\\qquad L\\in\\N.}\n\\]\nThis is not divisibility in $\\Q_p$, where a nonzero scalar can divide any field element. The zero case is proved separately, so zero does not spuriously violate a positive lower bound.","v_p(x),&x\\neq0.\\end{cases}\n\\]\nThe nonzero branch uses Mathlib's actual $p$-adic valuation. The implementation proves its connection to genuine integral divisibility:\n\\[\n \\boxed{L\\leq\\ord_p(x)\\quad\\Longleftrightarrow\\quad p^L\\mid x\n \\ \\text{in }\\Z_p,\\qquad L\\in\\N.}\n\\]\nThis is not divisibility in $\\Q_p$, where a nonzero scalar can divide any field element. The zero case is proved separately, so zero does not spuriously violate a positive lower bound.","\\subsection*{The binomial valuation is retained}\nLet $f:\\N\\to\\Z_p$, let $c_r=\\Delta_1^r f(0)$, and suppose\n\\[\n w(s+j)\\leq\\ord_p(c_{s+j})\\quad(0\\leq j\\leq n),\n \\qquad w:\\N\\to\\N\\cup\\{\\infty\\}.\n\\]\nSet\n\\[\n B_p(w;n,s)=\\min_{0\\leq j\\leq n}\n \\left\\{w(s+j)+v_p\\!\\binom nj\\right\\}.\n\\]\nEvery binomial in this range is nonzero. The code proves\n\\[\n \\boxed{B_p(w;n,s)\\leq\\ord_p(\\Delta_1^s f(n)).}\n\\]\nThis is the manuscript minimum after the substitution $r=s+j$. Coefficients outside this range have zero binomial multiplier. Weights equal to infinity are allowed, and a completely vanishing difference has order infinity.","ained}\nLet $f:\\N\\to\\Z_p$, let $c_r=\\Delta_1^r f(0)$, and suppose\n\\[\n w(s+j)\\leq\\ord_p(c_{s+j})\\quad(0\\leq j\\leq n),\n \\qquad w:\\N\\to\\N\\cup\\{\\infty\\}.\n\\]\nSet\n\\[\n B_p(w;n,s)=\\min_{0\\leq j\\leq n}\n \\left\\{w(s+j)+v_p\\!\\binom nj\\right\\}.\n\\]\nEvery binomial in this range is nonzero. The code proves\n\\[\n \\boxed{B_p(w;n,s)\\leq\\ord_p(\\Delta_1^s f(n)).}\n\\]\nThis is the manuscript minimum after the substitution $r=s+j$. Coefficients outside this range have zero binomial multiplier. Weights equal to infinity are allowed, and a completely vanishing difference has order infinity.","For a uniform coefficient hypothesis, the simpler conclusion follows:\n\\[\n \\boxed{\\inf_{j\\geq0} w(s+j)\\leq\\ord_p(\\Delta_1^s f(n)).}\n\\]\nFor each finite $L\\leq B_p(w;n,s)$, a separate theorem yields $p^L\\mid\\Delta_1^s f(n)$ inside $\\Z_p$. Another specialization handles arbitrary finite interpolants with infinite weights beyond their degree.","\\textbf{Principal targets.}\\par\n\\target{PadicBounds.nat_le_order_iff, PadicBounds.weighted_difference_bound}\\par\n\\target{PadicBounds.weighted_difference_divisibility}\\par\n\\target{PadicBounds.coefficient_tail_bound, PadicBounds.interpolant_weighted_bound}\n\\par\\medskip\n\\pageheading{3. Unit modes and residue-field reduction}{Theorem 7.4; manuscript page 18}\nFor $u$ in any commutative ring, and every natural step $h$, iteration proves\n\\[\n \\boxed{\\Delta_h^s(u^n)=u^n(u^h-1)^s.}\n\\]\nThe proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility $p^k\\mid u^h-1$ implies\n\\[\n p^{ks}\\mid\\Delta_h^s(u^n).\n\\]","pal targets.}\\par\n\\target{PadicBounds.nat_le_order_iff, PadicBounds.weighted_difference_bound}\\par\n\\target{PadicBounds.weighted_difference_divisibility}\\par\n\\target{PadicBounds.coefficient_tail_bound, PadicBounds.interpolant_weighted_bound}\n\\par\\medskip\n\\pageheading{3. Unit modes and residue-field reduction}{Theorem 7.4; manuscript page 18}\nFor $u$ in any commutative ring, and every natural step $h$, iteration proves\n\\[\n \\boxed{\\Delta_h^s(u^n)=u^n(u^h-1)^s.}\n\\]\nThe proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility $p^k\\mid u^h-1$ implies\n\\[\n p^{ks}\\mid\\Delta_h^s(u^n).\n\\]","et{PadicBounds.nat_le_order_iff, PadicBounds.weighted_difference_bound}\\par\n\\target{PadicBounds.weighted_difference_divisibility}\\par\n\\target{PadicBounds.coefficient_tail_bound, PadicBounds.interpolant_weighted_bound}\n\\par\\medskip\n\\pageheading{3. Unit modes and residue-field reduction}{Theorem 7.4; manuscript page 18}\nFor $u$ in any commutative ring, and every natural step $h$, iteration proves\n\\[\n \\boxed{\\Delta_h^s(u^n)=u^n(u^h-1)^s.}\n\\]\nThe proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility $p^k\\mid u^h-1$ implies\n\\[\n p^{ks}\\mid\\Delta_h^s(u^n).\n\\]","3. Unit modes and residue-field reduction}{Theorem 7.4; manuscript page 18}\nFor $u$ in any commutative ring, and every natural step $h$, iteration proves\n\\[\n \\boxed{\\Delta_h^s(u^n)=u^n(u^h-1)^s.}\n\\]\nThe proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility $p^k\\mid u^h-1$ implies\n\\[\n p^{ks}\\mid\\Delta_h^s(u^n).\n\\]","-field reduction}{Theorem 7.4; manuscript page 18}\nFor $u$ in any commutative ring, and every natural step $h$, iteration proves\n\\[\n \\boxed{\\Delta_h^s(u^n)=u^n(u^h-1)^s.}\n\\]\nThe proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility $p^k\\mid u^h-1$ implies\n\\[\n p^{ks}\\mid\\Delta_h^s(u^n).\n\\]","\\subsection*{Why a $p$-adic unit supplies one factor of $p$}\nNow let $u\\in\\Z_p^\\times$. The actual ring homomorphism\n\\[\n \\pi:\\Z_p\\longrightarrow\\mathbb F_p\n\\]\n sends $u$ to a nonzero unit. Fermat's finite-field theorem therefore gives\n\\[\n \\pi(u)^{p-1}=1,\\qquad \\pi(u^{p-1}-1)=0.\n\\]\nUsing the proved kernel identity $\\ker\\pi=p\\Z_p$, the source derives\n\\[\n \\boxed{p\\mid u^{p-1}-1\\quad\\text{inside }\\Z_p.}\n\\]\nThe conclusion is not supplied as an abstract certificate. It follows from the unit hypothesis and the standard residue-field construction.","Combining it with the geometric identity establishes\n\\[\n \\boxed{p^s\\mid\\Delta_{p-1}^s(u^n),\\qquad\n s\\leq\\ord_p(\\Delta_{p-1}^s(u^n)).}\n\\]\nThese statements hold for every prime, including $p=2$ with step one. Zero differences and order zero are included. Every multiple of the Fermat step satisfies the same lower bound.","\\subsection*{Finite integral linear combinations}\nFor any finite index set $S$, coefficients $a_i\\in\\Z_p$ and units $u_i\\in\\Z_p^\\times$, linearity of the actual iterated operator gives\n\\[\n \\Delta_h^s\\!\\left(\\sum_{i\\in S}a_i u_i^n\\right)\n =\\sum_{i\\in S}a_i u_i^n(u_i^h-1)^s.\n\\]\nAt $h=p-1$, the result is divisible by $p^s$. The empty sum is permitted. This does not prove that the general congruential Euler packets of Theorem 7.5 admit such a representation.","rincipal targets.}\\par\n\\target{UnitModes.geometric_difference, UnitModes.unit_fermat_divisibility}\\par\n\\target{UnitModes.unit_mode_valuation_gain}\\par\n\\target{UnitModes.finite_unit_modes_valuation_gain}\n\\par\\medskip\n\\pageheading{4. Exact examples and adversarial checks}{The examples use the same definitions as the general results}\n\\subsection*{A polynomial anchor}\nThe actual difference table for $f(n)=n^2$ yields\n\\[\n (c_0,c_1,c_2,c_3)=(0,1,2,0).\n\\]\nThe finite interpolant therefore reconstructs $I_2(c)(7)=49$. This checks the sign and indexing conventions, not just the names of the defined objects.","par\n\\target{UnitModes.geometric_difference, UnitModes.unit_fermat_divisibility}\\par\n\\target{UnitModes.unit_mode_valuation_gain}\\par\n\\target{UnitModes.finite_unit_modes_valuation_gain}\n\\par\\medskip\n\\pageheading{4. Exact examples and adversarial checks}{The examples use the same definitions as the general results}\n\\subsection*{A polynomial anchor}\nThe actual difference table for $f(n)=n^2$ yields\n\\[\n (c_0,c_1,c_2,c_3)=(0,1,2,0).\n\\]\nThe finite interpolant therefore reconstructs $I_2(c)(7)=49$. This checks the sign and indexing conventions, not just the names of the defined objects.","ar\n\\target{UnitModes.unit_mode_valuation_gain}\\par\n\\target{UnitModes.finite_unit_modes_valuation_gain}\n\\par\\medskip\n\\pageheading{4. Exact examples and adversarial checks}{The examples use the same definitions as the general results}\n\\subsection*{A polynomial anchor}\nThe actual difference table for $f(n)=n^2$ yields\n\\[\n (c_0,c_1,c_2,c_3)=(0,1,2,0).\n\\]\nThe finite interpolant therefore reconstructs $I_2(c)(7)=49$. This checks the sign and indexing conventions, not just the names of the defined objects.","\\subsection*{A sharp binomial-weight improvement}\nLet\n\\[\n F(n)=81\\binom n1+3\\binom n2+3\\binom n3+81\\binom n4.\n\\]\nAt $n=3$ and $s=1$, the four coefficient valuations are $(4,1,1,4)$, while the binomial valuations are $(0,1,1,0)$. Thus the refined bound is\n\\[\n B_3(w;3,1)=\\min(4,2,2,4)=2,\n\\]\nstronger than the coefficient-only lower bound one. The source checks both the minimum and the actual result:\n\\[\n \\boxed{\\Delta F(3)=180,\\qquad\\ord_3(180)=2.}\n\\]\nThe valuation equality is proved in the actual completed ring $\\Z_3$, with an independent exact integer divisibility check as well.","ubsection*{A sharp binomial-weight improvement}\nLet\n\\[\n F(n)=81\\binom n1+3\\binom n2+3\\binom n3+81\\binom n4.\n\\]\nAt $n=3$ and $s=1$, the four coefficient valuations are $(4,1,1,4)$, while the binomial valuations are $(0,1,1,0)$. Thus the refined bound is\n\\[\n B_3(w;3,1)=\\min(4,2,2,4)=2,\n\\]\nstronger than the coefficient-only lower bound one. The source checks both the minimum and the actual result:\n\\[\n \\boxed{\\Delta F(3)=180,\\qquad\\ord_3(180)=2.}\n\\]\nThe valuation equality is proved in the actual completed ring $\\Z_3$, with an independent exact integer divisibility check as well.","$n=3$ and $s=1$, the four coefficient valuations are $(4,1,1,4)$, while the binomial valuations are $(0,1,1,0)$. Thus the refined bound is\n\\[\n B_3(w;3,1)=\\min(4,2,2,4)=2,\n\\]\nstronger than the coefficient-only lower bound one. The source checks both the minimum and the actual result:\n\\[\n \\boxed{\\Delta F(3)=180,\\qquad\\ord_3(180)=2.}\n\\]\nThe valuation equality is proved in the actual completed ring $\\Z_3$, with an independent exact integer divisibility check as well.","\\subsection*{A sharp unit-mode gain}\nFor $p=5$ and $u=2$,\n\\[\n \\Delta_4^2(u^n)\\big|_{n=0}=(2^4-1)^2=225.\n\\]\nThe source proves $5^2\\mid225$ and $5^3\\nmid225$. This example attains the guaranteed gain. No additional uniform power of $p$ is claimed.","\\subsection*{What fails when hypotheses or conventions are dropped}\nThe nonunit $u=3$ at $p=3$ gives $\\Delta_2(u^n)|_{n=0}=8$, with no factor of three. A constant unit has every positive-order difference zero, hence infinite extended valuation. The finite-sample counterexample on page 2 distinguishes a true interpolant identity from an unsupported claim about unknown continuation.","bottomrule\n\\end{tabular}\n\\end{center}\nThe 61 new theorems include supporting lemmas and examples, not 61 independent discoveries. Pinned Mathlib supplies finite-field, unit-group, ideal-power, $p$-adic residue and Chinese-remainder results. The new modules connect these to the actual packet coefficients, explicit finite lifts and existing selectors.","\\subsubsection*{The specification matters}\nRational coefficients are not reduced through an imaginary map $\\Q\\to\\F_p$. The proof first constructs their integral lifts in $\\Z_p$ and establishes equality in $\\Q_p$. Root representatives are actual units at the requested precision, not unchanged residues that need not satisfy the root equation modulo $p^\\ell$.","reports, and rejected all 17 invalid-proof controls. All 103 infrastructure and release tests passed. Independent finite regressions are recorded separately from the proofs. A separate release audit records the archive's fresh-extraction rebuild and integrity checks.\n\\par\\medskip\n\\pageheading{1. Integral reduction of the original coefficients}{Corollary 8.3; manuscript pages 19--20}\nFix a prime $p$, an integer $m\\geq1$ and a window address $0\\leq j<p$. The earlier project defines the rational packet coefficients by the actual recurrence\n\\[\n \\sum_{q=0}^{n}\\binom{mpn+j}{mpq}C_{mp,j}(q)=\\delta_{n0}.\n\\]\nThe previous Lucas theorem proves\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p,\\qquad\n \\binom{mpn+j}{mpn}\\equiv1\\pmod p.\n\\]\nThe last congruence makes each diagonal entry a unit in the actual completed ring $\\Z_p$.","jected all 17 invalid-proof controls. All 103 infrastructure and release tests passed. Independent finite regressions are recorded separately from the proofs. A separate release audit records the archive's fresh-extraction rebuild and integrity checks.\n\\par\\medskip\n\\pageheading{1. Integral reduction of the original coefficients}{Corollary 8.3; manuscript pages 19--20}\nFix a prime $p$, an integer $m\\geq1$ and a window address $0\\leq j<p$. The earlier project defines the rational packet coefficients by the actual recurrence\n\\[\n \\sum_{q=0}^{n}\\binom{mpn+j}{mpq}C_{mp,j}(q)=\\delta_{n0}.\n\\]\nThe previous Lucas theorem proves\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p,\\qquad\n \\binom{mpn+j}{mpn}\\equiv1\\pmod p.\n\\]\nThe last congruence makes each diagonal entry a unit in the actual completed ring $\\Z_p$.","ntrols. All 103 infrastructure and release tests passed. Independent finite regressions are recorded separately from the proofs. A separate release audit records the archive's fresh-extraction rebuild and integrity checks.\n\\par\\medskip\n\\pageheading{1. Integral reduction of the original coefficients}{Corollary 8.3; manuscript pages 19--20}\nFix a prime $p$, an integer $m\\geq1$ and a window address $0\\leq j<p$. The earlier project defines the rational packet coefficients by the actual recurrence\n\\[\n \\sum_{q=0}^{n}\\binom{mpn+j}{mpq}C_{mp,j}(q)=\\delta_{n0}.\n\\]\nThe previous Lucas theorem proves\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p,\\qquad\n \\binom{mpn+j}{mpn}\\equiv1\\pmod p.\n\\]\nThe last congruence makes each diagonal entry a unit in the actual completed ring $\\Z_p$.","separately from the proofs. A separate release audit records the archive's fresh-extraction rebuild and integrity checks.\n\\par\\medskip\n\\pageheading{1. Integral reduction of the original coefficients}{Corollary 8.3; manuscript pages 19--20}\nFix a prime $p$, an integer $m\\geq1$ and a window address $0\\leq j<p$. The earlier project defines the rational packet coefficients by the actual recurrence\n\\[\n \\sum_{q=0}^{n}\\binom{mpn+j}{mpq}C_{mp,j}(q)=\\delta_{n0}.\n\\]\nThe previous Lucas theorem proves\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p,\\qquad\n \\binom{mpn+j}{mpn}\\equiv1\\pmod p.\n\\]\nThe last congruence makes each diagonal entry a unit in the actual completed ring $\\Z_p$.","ld and integrity checks.\n\\par\\medskip\n\\pageheading{1. Integral reduction of the original coefficients}{Corollary 8.3; manuscript pages 19--20}\nFix a prime $p$, an integer $m\\geq1$ and a window address $0\\leq j<p$. The earlier project defines the rational packet coefficients by the actual recurrence\n\\[\n \\sum_{q=0}^{n}\\binom{mpn+j}{mpq}C_{mp,j}(q)=\\delta_{n0}.\n\\]\nThe previous Lucas theorem proves\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p,\\qquad\n \\binom{mpn+j}{mpn}\\equiv1\\pmod p.\n\\]\nThe last congruence makes each diagonal entry a unit in the actual completed ring $\\Z_p$.","q1$ and a window address $0\\leq j<p$. The earlier project defines the rational packet coefficients by the actual recurrence\n\\[\n \\sum_{q=0}^{n}\\binom{mpn+j}{mpq}C_{mp,j}(q)=\\delta_{n0}.\n\\]\nThe previous Lucas theorem proves\n\\[\n \\binom{mpn+j}{mpq}\\equiv\\binom{mn}{mq}\\pmod p,\\qquad\n \\binom{mpn+j}{mpn}\\equiv1\\pmod p.\n\\]\nThe last congruence makes each diagonal entry a unit in the actual completed ring $\\Z_p$.","\\subsection*{First construct the integral solution}\nFor a ring $R$, a triangular row system $T$, right-hand side $b$, and proved unit pivots $d_n=T_{nn}$, define\n\\[\n s_n=d_n^{-1}\\left(b_n-\\sum_{q<n}T_{nq}s_q\\right).\n\\]\nThe new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homomorphisms into fields.","Specializing to $R=\\Z_p$ constructs $c^{(p)}_{m,j}(n)$. Mapping to $\\Q_p$ and comparing with the original rational forward substitution gives\n\\[\n \\boxed{c^{(p)}_{m,j}(n)=C_{mp,j}(n)\\quad\\text{in }\\Q_p.}\n\\]\nThus every original standard-window coefficient is $p$-integral, for every row. Integrality is a conclusion, not an additional hypothesis on the coefficients.","\\subsection*{Then apply the residue map}\nIndependently define $e^{(m,p)}$ over $\\F_p$ by the unit-diagonal system\n\\[\n \\sum_{q=0}^{n}\\binom{mn}{mq}e^{(m,p)}_q=\\delta_{n0}.\n\\]\nIts recurrence and uniqueness are checked. Applying the actual homomorphism $\\pi:\\Z_p\\to\\F_p$ to the integral row, and then Lucas reduction, proves\n\\[\n \\boxed{\\pi(c^{(p)}_{m,j}(n))=e^{(m,p)}_n.}\n\\]\nFor $j,k<p$, a separate ideal statement gives\n\\[\n p\\mid c^{(p)}_{m,j}(n)-c^{(p)}_{m,k}(n)\\quad\\text{inside }\\Z_p.\n\\]\nThe formal solver also covers $m=0$. The manuscript uses $m\\geq1$. No global finite-difference theorem or period assertion enters this proof.","ply the residue map}\nIndependently define $e^{(m,p)}$ over $\\F_p$ by the unit-diagonal system\n\\[\n \\sum_{q=0}^{n}\\binom{mn}{mq}e^{(m,p)}_q=\\delta_{n0}.\n\\]\nIts recurrence and uniqueness are checked. Applying the actual homomorphism $\\pi:\\Z_p\\to\\F_p$ to the integral row, and then Lucas reduction, proves\n\\[\n \\boxed{\\pi(c^{(p)}_{m,j}(n))=e^{(m,p)}_n.}\n\\]\nFor $j,k<p$, a separate ideal statement gives\n\\[\n p\\mid c^{(p)}_{m,j}(n)-c^{(p)}_{m,k}(n)\\quad\\text{inside }\\Z_p.\n\\]\nThe formal solver also covers $m=0$. The manuscript uses $m\\geq1$. No global finite-difference theorem or period assertion enters this proof.","um_{q=0}^{n}\\binom{mn}{mq}e^{(m,p)}_q=\\delta_{n0}.\n\\]\nIts recurrence and uniqueness are checked. Applying the actual homomorphism $\\pi:\\Z_p\\to\\F_p$ to the integral row, and then Lucas reduction, proves\n\\[\n \\boxed{\\pi(c^{(p)}_{m,j}(n))=e^{(m,p)}_n.}\n\\]\nFor $j,k<p$, a separate ideal statement gives\n\\[\n p\\mid c^{(p)}_{m,j}(n)-c^{(p)}_{m,k}(n)\\quad\\text{inside }\\Z_p.\n\\]\nThe formal solver also covers $m=0$. The manuscript uses $m\\geq1$. No global finite-difference theorem or period assertion enters this proof.","^{(m,p)}_q=\\delta_{n0}.\n\\]\nIts recurrence and uniqueness are checked. Applying the actual homomorphism $\\pi:\\Z_p\\to\\F_p$ to the integral row, and then Lucas reduction, proves\n\\[\n \\boxed{\\pi(c^{(p)}_{m,j}(n))=e^{(m,p)}_n.}\n\\]\nFor $j,k<p$, a separate ideal statement gives\n\\[\n p\\mid c^{(p)}_{m,j}(n)-c^{(p)}_{m,k}(n)\\quad\\text{inside }\\Z_p.\n\\]\nThe formal solver also covers $m=0$. The manuscript uses $m\\geq1$. No global finite-difference theorem or period assertion enters this proof.","\\textbf{Principal targets.}\\par\n\\target{WindowIndependence.coeffPadic_coe, .standard_window_integral}\\par\n\\target{WindowIndependence.residue_independence, .reduced_unique}\\par\n\\target{WindowIndependence.two_window_addresses_congruent}\n\\par\\medskip\n\\pageheading{2. Explicit finite prime-power decomposition}{Proposition 6.2; odd-prime branch}\nLet $p$ be prime, $\\ell\\geq1$, and\n\\[\n a\\in\\bigl(\\Z/p^\\ell\\Z\\bigr)^\\times.\n\\]\nDefine actual units in the same ring by\n\\[\n \\boxed{\\omega_\\ell(a)=a^{p^{\\ell-1}},\\qquad\n \\langle a\\rangle_\\ell=a\\,\\omega_\\ell(a)^{-1}.}\n\\]\nThe Lean source writes $\\ell=L+1$, so zero precision is excluded without truncated natural-number subtraction.","ffPadic_coe, .standard_window_integral}\\par\n\\target{WindowIndependence.residue_independence, .reduced_unique}\\par\n\\target{WindowIndependence.two_window_addresses_congruent}\n\\par\\medskip\n\\pageheading{2. Explicit finite prime-power decomposition}{Proposition 6.2; odd-prime branch}\nLet $p$ be prime, $\\ell\\geq1$, and\n\\[\n a\\in\\bigl(\\Z/p^\\ell\\Z\\bigr)^\\times.\n\\]\nDefine actual units in the same ring by\n\\[\n \\boxed{\\omega_\\ell(a)=a^{p^{\\ell-1}},\\qquad\n \\langle a\\rangle_\\ell=a\\,\\omega_\\ell(a)^{-1}.}\n\\]\nThe Lean source writes $\\ell=L+1$, so zero precision is excluded without truncated natural-number subtraction.","{WindowIndependence.two_window_addresses_congruent}\n\\par\\medskip\n\\pageheading{2. Explicit finite prime-power decomposition}{Proposition 6.2; odd-prime branch}\nLet $p$ be prime, $\\ell\\geq1$, and\n\\[\n a\\in\\bigl(\\Z/p^\\ell\\Z\\bigr)^\\times.\n\\]\nDefine actual units in the same ring by\n\\[\n \\boxed{\\omega_\\ell(a)=a^{p^{\\ell-1}},\\qquad\n \\langle a\\rangle_\\ell=a\\,\\omega_\\ell(a)^{-1}.}\n\\]\nThe Lean source writes $\\ell=L+1$, so zero precision is excluded without truncated natural-number subtraction.","\\subsection*{The root equation and residue are derived}\nEuler's unit-group exponent gives\n\\[\n \\omega_\\ell(a)^{p-1}\n =a^{p^{\\ell-1}(p-1)}=a^{\\varphi(p^\\ell)}=1.\n\\]\nReduction to the actual field $\\F_p$ commutes with powers. Its Frobenius identity gives\n\\[\n \\overline{\\omega_\\ell(a)}\n =\\bar a^{p^{\\ell-1}}=\\bar a.\n\\]\nConsequently,\n\\[\n \\boxed{a=\\omega_\\ell(a)\\langle a\\rangle_\\ell,\\quad\n \\omega_\\ell(a)^{p-1}=1,\\quad\n \\overline{\\langle a\\rangle_\\ell}=1.}\n\\]\nThe manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.","\\subsection*{Uniqueness requires a lifting argument}\nThe source proves the actual kernel statement\n\\[\n \\bar x=0\\quad\\Longleftrightarrow\\quad p\\mid x\n \\quad\\text{in }\\Z/p^\\ell\\Z.\n\\]\nFor $x\\equiv y\\pmod p$, the checked ideal-power lifting theorem gives\n\\[\n x^{p^{\\ell-1}}-y^{p^{\\ell-1}}\\in(p)^\\ell=(0).\n\\]\nIf $w^{p-1}=1$, then $w^p=w$ and hence $w^{p^k}=w$ for every $k$. Therefore two roots with the same residue are equal. The decomposition is unique whenever its second factor reduces to one. Multiplicativity of both factors is also proved.","For $p=3,\\ell=2,a=2$, the checked values are\n\\[\n \\omega_2(2)=8,\\qquad \\langle2\\rangle_2=7,\\qquad\n 8^2=1,\\quad8\\cdot7=2\\quad\\text{in }\\Z/9\\Z.\n\\]\nThe unlifted representative $2$ is not a square root of one modulo nine.","\\textbf{Principal targets.}\\par\n\\target{FiniteUnits.finite_unit_decomposition, .power_lift_congruence}\\par\n\\target{FiniteUnits.omega_unique, .decomposition_unique}\\par\n\\target{FiniteUnits.omega_mul, .principal_mul}\n\\par\\medskip\n\\pageheading{3. Precision compatibility and the dyadic sign}{Finite refinements and Proposition 6.2's dyadic branch}\nFor $1\\leq k\\leq\\ell$, let\n\\[\n \\rho_{\\ell,k}:\\Z/p^\\ell\\Z\\longrightarrow\\Z/p^k\\Z\n\\]\nbe the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies\n\\[\n \\boxed{\\rho_{\\ell,k}(\\omega_\\ell(a))\n =\\omega_k(\\rho_{\\ell,k}(a)),}\n\\]\n\\[\n \\boxed{\\rho_{\\ell,k}(\\langle a\\rangle_\\ell)\n =\\langle\\rho_{\\ell,k}(a)\\rangle_k.}\n\\]\nThe proof uses the exponent factorization\n$p^{\\ell-1}=p^{k-1}p^{\\ell-k}$ and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy\n\\[\n \\langle a\\rangle_\\ell^{\\,p^{\\ell-1}}=1.\n\\]\nAt precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or topological unit-group equivalence is","\\textbf{Principal targets.}\\par\n\\target{FiniteUnits.finite_unit_decomposition, .power_lift_congruence}\\par\n\\target{FiniteUnits.omega_unique, .decomposition_unique}\\par\n\\target{FiniteUnits.omega_mul, .principal_mul}\n\\par\\medskip\n\\pageheading{3. Precision compatibility and the dyadic sign}{Finite refinements and Proposition 6.2's dyadic branch}\nFor $1\\leq k\\leq\\ell$, let\n\\[\n \\rho_{\\ell,k}:\\Z/p^\\ell\\Z\\longrightarrow\\Z/p^k\\Z\n\\]\nbe the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies\n\\[\n \\boxed{\\rho_{\\ell,k}(\\omega_\\ell(a))\n =\\omega_k(\\rho_{\\ell,k}(a)),}\n\\]\n\\[\n \\boxed{\\rho_{\\ell,k}(\\langle a\\rangle_\\ell)\n =\\langle\\rho_{\\ell,k}(a)\\rangle_k.}\n\\]\nThe proof uses the exponent factorization\n$p^{\\ell-1}=p^{k-1}p^{\\ell-k}$ and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy\n\\[\n \\langle a\\rangle_\\ell^{\\,p^{\\ell-1}}=1.\n\\]\nAt precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or topological unit-group equivalence is constructed in this release.","a_unique, .decomposition_unique}\\par\n\\target{FiniteUnits.omega_mul, .principal_mul}\n\\par\\medskip\n\\pageheading{3. Precision compatibility and the dyadic sign}{Finite refinements and Proposition 6.2's dyadic branch}\nFor $1\\leq k\\leq\\ell$, let\n\\[\n \\rho_{\\ell,k}:\\Z/p^\\ell\\Z\\longrightarrow\\Z/p^k\\Z\n\\]\nbe the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies\n\\[\n \\boxed{\\rho_{\\ell,k}(\\omega_\\ell(a))\n =\\omega_k(\\rho_{\\ell,k}(a)),}\n\\]\n\\[\n \\boxed{\\rho_{\\ell,k}(\\langle a\\rangle_\\ell)\n =\\langle\\rho_{\\ell,k}(a)\\rangle_k.}\n\\]\nThe proof uses the exponent factorization\n$p^{\\ell-1}=p^{k-1}p^{\\ell-k}$ and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy\n\\[\n \\langle a\\rangle_\\ell^{\\,p^{\\ell-1}}=1.\n\\]\nAt precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or topological unit-group equivalence is constructed in this release.","pal_mul}\n\\par\\medskip\n\\pageheading{3. Precision compatibility and the dyadic sign}{Finite refinements and Proposition 6.2's dyadic branch}\nFor $1\\leq k\\leq\\ell$, let\n\\[\n \\rho_{\\ell,k}:\\Z/p^\\ell\\Z\\longrightarrow\\Z/p^k\\Z\n\\]\nbe the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies\n\\[\n \\boxed{\\rho_{\\ell,k}(\\omega_\\ell(a))\n =\\omega_k(\\rho_{\\ell,k}(a)),}\n\\]\n\\[\n \\boxed{\\rho_{\\ell,k}(\\langle a\\rangle_\\ell)\n =\\langle\\rho_{\\ell,k}(a)\\rangle_k.}\n\\]\nThe proof uses the exponent factorization\n$p^{\\ell-1}=p^{k-1}p^{\\ell-k}$ and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy\n\\[\n \\langle a\\rangle_\\ell^{\\,p^{\\ell-1}}=1.\n\\]\nAt precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or topological unit-group equivalence is constructed in this release.","c branch}\nFor $1\\leq k\\leq\\ell$, let\n\\[\n \\rho_{\\ell,k}:\\Z/p^\\ell\\Z\\longrightarrow\\Z/p^k\\Z\n\\]\nbe the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies\n\\[\n \\boxed{\\rho_{\\ell,k}(\\omega_\\ell(a))\n =\\omega_k(\\rho_{\\ell,k}(a)),}\n\\]\n\\[\n \\boxed{\\rho_{\\ell,k}(\\langle a\\rangle_\\ell)\n =\\langle\\rho_{\\ell,k}(a)\\rangle_k.}\n\\]\nThe proof uses the exponent factorization\n$p^{\\ell-1}=p^{k-1}p^{\\ell-k}$ and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy\n\\[\n \\langle a\\rangle_\\ell^{\\,p^{\\ell-1}}=1.\n\\]\nAt precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or topological unit-group equivalence is constructed in this release.","hism. It induces a map on units. The explicit construction satisfies\n\\[\n \\boxed{\\rho_{\\ell,k}(\\omega_\\ell(a))\n =\\omega_k(\\rho_{\\ell,k}(a)),}\n\\]\n\\[\n \\boxed{\\rho_{\\ell,k}(\\langle a\\rangle_\\ell)\n =\\langle\\rho_{\\ell,k}(a)\\rangle_k.}\n\\]\nThe proof uses the exponent factorization\n$p^{\\ell-1}=p^{k-1}p^{\\ell-k}$ and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy\n\\[\n \\langle a\\rangle_\\ell^{\\,p^{\\ell-1}}=1.\n\\]\nAt precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or topological unit-group equivalence is constructed in this release.","\\subsection*{Why the mod-four branch is separate}\nNow let $\\ell\\geq2$ and $a\\in(\\Z/2^\\ell\\Z)^\\times$. The actual units of $\\Z/4\\Z$ are exhausted by $1$ and $-1$. Define\n\\[\n \\varepsilon_\\ell(a)=\n \\begin{cases}1,&a\\equiv1\\pmod4,","\\subsection*{Why the mod-four branch is separate}\nNow let $\\ell\\geq2$ and $a\\in(\\Z/2^\\ell\\Z)^\\times$. The actual units of $\\Z/4\\Z$ are exhausted by $1$ and $-1$. Define\n\\[\n \\varepsilon_\\ell(a)=\n \\begin{cases}1,&a\\equiv1\\pmod4,\\\\-1,&a\\equiv3\\pmod4,\\end{cases}\n \\qquad\n \\langle a\\rangle_{2,\\ell}=\\varepsilon_\\ell(a)a.\n\\]\nThe source uses $\\ell=L+2$. It proves\n\\[\n \\boxed{a=\\varepsilon_\\ell(a)\\langle a\\rangle_{2,\\ell},\\quad\n \\varepsilon_\\ell(a)\\in\\{1,-1\\},\\quad\n \\langle a\\rangle_{2,\\ell}\\equiv1\\pmod4.}\n\\]\nThe sign is uniquely determined by the residue modulo four, and cancellation gives uniqueness of the principal factor.","-1,&a\\equiv3\\pmod4,\\end{cases}\n \\qquad\n \\langle a\\rangle_{2,\\ell}=\\varepsilon_\\ell(a)a.\n\\]\nThe source uses $\\ell=L+2$. It proves\n\\[\n \\boxed{a=\\varepsilon_\\ell(a)\\langle a\\rangle_{2,\\ell},\\quad\n \\varepsilon_\\ell(a)\\in\\{1,-1\\},\\quad\n \\langle a\\rangle_{2,\\ell}\\equiv1\\pmod4.}\n\\]\nThe sign is uniquely determined by the residue modulo four, and cancellation gives uniqueness of the principal factor.","The checked example $a=3$ modulo eight has sign $7=-1$ and principal factor $5$:\n\\[\n 7\\cdot5=3\\pmod8,\\qquad5=1\\pmod4.\n\\]\nThe code separately proves $1=-1$ in $\\Z/2\\Z$ but $1\\neq-1$ in $\\Z/4\\Z$. Modulo two alone cannot identify the sign. The generic prime-power construction at $p=2$ only asks for a principal residue modulo two and is not silently substituted for this stronger mod-four factorization.","\\textbf{Principal targets.}\\par\n\\target{UnitRefinement.omega_compatible, .principal_compatible, .principal_power}\\par\n\\target{DyadicUnits.dyadic_unit_decomposition, .decomposition_unique}\n\\par\\medskip\n\\pageheading{4. CRT addresses and the actual packet selectors}{Theorem 10.1; manuscript page 22}\nLet $a\\geq0$, let $M$ be odd, and put $N=2^aM$. Coprimality is derived from oddness:\n\\[\n \\gcd(2^a,M)=1.\n\\]\nThe construction uses the actual Chinese-remainder ring equivalence\n\\[\n \\boxed{\\Z/N\\Z\\ \\simeq\\ \\Z/2^a\\Z\\times\\Z/M\\Z.}\n\\]\nIts bijectivity and component formula are checked. On a natural representative $r$, the components are precisely its reductions modulo the two factors, not arbitrary coordinate labels.","rincipal_compatible, .principal_power}\\par\n\\target{DyadicUnits.dyadic_unit_decomposition, .decomposition_unique}\n\\par\\medskip\n\\pageheading{4. CRT addresses and the actual packet selectors}{Theorem 10.1; manuscript page 22}\nLet $a\\geq0$, let $M$ be odd, and put $N=2^aM$. Coprimality is derived from oddness:\n\\[\n \\gcd(2^a,M)=1.\n\\]\nThe construction uses the actual Chinese-remainder ring equivalence\n\\[\n \\boxed{\\Z/N\\Z\\ \\simeq\\ \\Z/2^a\\Z\\times\\Z/M\\Z.}\n\\]\nIts bijectivity and component formula are checked. On a natural representative $r$, the components are precisely its reductions modulo the two factors, not arbitrary coordinate labels.","heading{4. CRT addresses and the actual packet selectors}{Theorem 10.1; manuscript page 22}\nLet $a\\geq0$, let $M$ be odd, and put $N=2^aM$. Coprimality is derived from oddness:\n\\[\n \\gcd(2^a,M)=1.\n\\]\nThe construction uses the actual Chinese-remainder ring equivalence\n\\[\n \\boxed{\\Z/N\\Z\\ \\simeq\\ \\Z/2^a\\Z\\times\\Z/M\\Z.}\n\\]\nIts bijectivity and component formula are checked. On a natural representative $r$, the components are precisely its reductions modulo the two factors, not arbitrary coordinate labels.","\\subsection*{The algebraic bridge to projectors}\nFor natural indices $n,r$, the source proves\n\\[\n n\\equiv r\\pmod{2^aM}\n \\quad\\Longleftrightarrow\\quad\n \\bigl(n\\equiv r\\pmod{2^a}\\text{ and }n\\equiv r\\pmod M\\bigr).\n\\]\nFor the original coefficient selector\n\\[\n (P_{N,r}f)(n)=\n \\begin{cases}f(n),&n\\bmod N=r,","\\subsection*{The algebraic bridge to projectors}\nFor natural indices $n,r$, the source proves\n\\[\n n\\equiv r\\pmod{2^aM}\n \\quad\\Longleftrightarrow\\quad\n \\bigl(n\\equiv r\\pmod{2^a}\\text{ and }n\\equiv r\\pmod M\\bigr).\n\\]\nFor the original coefficient selector\n\\[\n (P_{N,r}f)(n)=\n \\begin{cases}f(n),&n\\bmod N=r,\\\\0,&\\text{otherwise},\\end{cases}\n\\]\ntake the canonical label $0\\leq r<N$. The logical equivalence yields the actual equality of sequence operators\n\\[\n \\boxed{P_{N,r}f=\n P_{2^a,\\,r\\bmod2^a}\\bigl(P_{M,\\,r\\bmod M}f\\bigr).}\n\\]\nThis statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.","0,&\\text{otherwise},\\end{cases}\n\\]\ntake the canonical label $0\\leq r<N$. The logical equivalence yields the actual equality of sequence operators\n\\[\n \\boxed{P_{N,r}f=\n P_{2^a,\\,r\\bmod2^a}\\bigl(P_{M,\\,r\\bmod M}f\\bigr).}\n\\]\nThis statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.","The cases $a=0$ and $M=1$ are included. The corresponding one-element residue ring is not excluded. Oddness of $M$ already implies positivity.","Coprimality cannot be discarded. The distinct classes $0$ and $2$ modulo four both map to $(0,0)$ when both components are reduced modulo two. The source checks this counterexample. It also checks a composite-modulus failure of the Fermat exponent and a coefficient outside the standard window that is not $3$-adically integral.","_bijective, .address_components, .residue_equivalence}\\par\n\\target{CRTPackets.packet_factorization, .series_packet_factorization}\\par\n\\target{FiniteArithmeticExamples.outside_window_not_integral}\n\\par\\medskip\n\\pageheading{5. Verification, regressions and remaining work}{Cumulative release evidence}\nThe fresh build started with an empty output directory under Lean 4.34.0-rc2 and Mathlib revision\n\\[\n \\mcode{aeb72f1e9981b6d35b426b16dca7759be3e7ba06}.\n\\]\nAll nine dependency revisions matched the runtime manifest and repository heads.","\\subsubsection*{The main mathematical connection}\nA sampled state $u_k=v_{Nk+j}$ evolves by the actual jump matrix $C^N$. Its recurrence is derived from the characteristic polynomial of that matrix. No scalar recurrence or spectral decomposition is supplied as a substitute for the desired conclusion.","For Fibonacci numbers, the state is independently defined from Mathlib's actual sequence. The Lucas sequence is independently defined by its initial values and recurrence. The trace and determinant of $C^N$ are then proved to supply the sampled recurrence coefficients.","\\subsubsection*{Important qualification}\nA $d$-dimensional state system supplies a recurrence of \\textbf{order at most $d$}. Its minimal recurrence need not have order $d$. The original statement is read as the characteristic-polynomial recurrence, not a minimal-order claim. The formal equations also handle singular matrices, zero-dimensional spaces and zero step.","\\par\\medskip\n\\pageheading{1. Actual matrix orbits and arithmetic jumps}{Theorem 9.1; manuscript page 22}\nLet $R$ be a commutative ring, $I$ a finite coordinate type and $d=|I|$. For $C\\in M_I(R)$ and $v_0\\in R^I$, define\n\\[\n \\operatorname{orbit}(C,v_0,n)=C^nv_0,\n \\qquad u_k=\\operatorname{packet}(C,v_0,N,j,k)=C^{Nk+j}v_0.\n\\]\nThe implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence $v$ with\n\\[\n v_{n+1}=Cv_n\n\\]\nis proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.","trix orbits and arithmetic jumps}{Theorem 9.1; manuscript page 22}\nLet $R$ be a commutative ring, $I$ a finite coordinate type and $d=|I|$. For $C\\in M_I(R)$ and $v_0\\in R^I$, define\n\\[\n \\operatorname{orbit}(C,v_0,n)=C^nv_0,\n \\qquad u_k=\\operatorname{packet}(C,v_0,N,j,k)=C^{Nk+j}v_0.\n\\]\nThe implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence $v$ with\n\\[\n v_{n+1}=Cv_n\n\\]\nis proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.","s and arithmetic jumps}{Theorem 9.1; manuscript page 22}\nLet $R$ be a commutative ring, $I$ a finite coordinate type and $d=|I|$. For $C\\in M_I(R)$ and $v_0\\in R^I$, define\n\\[\n \\operatorname{orbit}(C,v_0,n)=C^nv_0,\n \\qquad u_k=\\operatorname{packet}(C,v_0,N,j,k)=C^{Nk+j}v_0.\n\\]\nThe implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence $v$ with\n\\[\n v_{n+1}=Cv_n\n\\]\nis proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.","\\subsection*{The jump formula}\nAssociativity and the laws for powers of one matrix give\n\\[\n \\boxed{u_k=C^j(C^N)^kv_0=(C^N)^k(C^jv_0).}\n\\]\nOnly powers of the same matrix are interchanged. The argument does not assume that arbitrary matrices commute. In particular,\n\\[\n u_{k+1}=C^Nu_k,\\qquad u_{k+a}=(C^N)^a u_k.\n\\]\nThe step matrix is $C^N$, not $C$. A checked example with the Fibonacci matrix and $N=2$ rejects replacing the jump with a single original step.","\\subsection*{Refinement of an arithmetic address}\nTaking another subsequence with step $M$ and offset $s$ yields\n\\[\n u_{Mk+s}=v_{(NM)k+(Ns+j)}.\n\\]\nThe formal proof establishes this identity for the actual packet function by checking the index arithmetic. It is not inferred from a diagram or a naming convention.","\\subsection*{Endpoints and interpretation}\nThe algebraic identities allow $N=0$, in which case $u_k=v_j$ is constant. Positive-modulus residue packets are the special case $N>0$ and $0\\le j<N$. The formula also permits offsets beyond that standard range. No inverse matrix, diagonalization, eigenvalue, or characteristic-zero assumption is needed.","\\textbf{Principal targets.}\\par\n\\target{Companion.orbit_of_recurrence}\\par\n\\target{Companion.companion_jump, Companion.source_companion_jump}\\par\n\\target{Companion.packet_step, Companion.packet_advance}\\par\n\\target{Companion.packet_refinement, Companion.packet_zero_step}\n\\par\\medskip\n\\pageheading{2. From the characteristic polynomial to a recurrence}{Theorem 9.1, continued}\nPut $D=C^N$ and let\n\\[\n \\chi_D(X)=\\det(XI-D)=X^d+\\sum_{a=0}^{d-1}c_aX^a\n\\]\nbe its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves\n\\[\n \\sum_{a=0}^{d}\\chi_D[a]D^a=0,\n \\qquad \\chi_D[d]=1.\n\\]\nHere $\\chi_D[a]$ denotes the coefficient at index $a$. The identities hold over any commutative coefficient ring. The zero-ring case is handled explicitly by the proof's subsingleton branch.","packet_refinement, Companion.packet_zero_step}\n\\par\\medskip\n\\pageheading{2. From the characteristic polynomial to a recurrence}{Theorem 9.1, continued}\nPut $D=C^N$ and let\n\\[\n \\chi_D(X)=\\det(XI-D)=X^d+\\sum_{a=0}^{d-1}c_aX^a\n\\]\nbe its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves\n\\[\n \\sum_{a=0}^{d}\\chi_D[a]D^a=0,\n \\qquad \\chi_D[d]=1.\n\\]\nHere $\\chi_D[a]$ denotes the coefficient at index $a$. The identities hold over any commutative coefficient ring. The zero-ring case is handled explicitly by the proof's subsingleton branch.","ageheading{2. From the characteristic polynomial to a recurrence}{Theorem 9.1, continued}\nPut $D=C^N$ and let\n\\[\n \\chi_D(X)=\\det(XI-D)=X^d+\\sum_{a=0}^{d-1}c_aX^a\n\\]\nbe its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves\n\\[\n \\sum_{a=0}^{d}\\chi_D[a]D^a=0,\n \\qquad \\chi_D[d]=1.\n\\]\nHere $\\chi_D[a]$ denotes the coefficient at index $a$. The identities hold over any commutative coefficient ring. The zero-ring case is handled explicitly by the proof's subsingleton branch.","ial to a recurrence}{Theorem 9.1, continued}\nPut $D=C^N$ and let\n\\[\n \\chi_D(X)=\\det(XI-D)=X^d+\\sum_{a=0}^{d-1}c_aX^a\n\\]\nbe its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves\n\\[\n \\sum_{a=0}^{d}\\chi_D[a]D^a=0,\n \\qquad \\chi_D[d]=1.\n\\]\nHere $\\chi_D[a]$ denotes the coefficient at index $a$. The identities hold over any commutative coefficient ring. The zero-ring case is handled explicitly by the proof's subsingleton branch.","\\subsection*{The complete vector recurrence}\nApply the matrix identity to the actual state $u_k$. The already proved advance identity identifies each $D^a u_k$ with $u_{k+a}$. Consequently,\n\\[\n \\sum_{a=0}^{d}\\chi_D[a]u_{k+a}=0,\n \\qquad\n \\boxed{u_{k+d}=-\\sum_{a=0}^{d-1}c_a u_{k+a}.}\n\\]\nThis is the desired recurrence, with its monic leading coefficient and all earlier coefficients supplied by $\\chi_D$. It is not an assumption on the packet sequence.","Every coordinate and every $R$-linear scalar observation $\\lambda:R^I\\to R$ satisfies the corresponding equation:\n\\[\n \\lambda(u_{k+d})=-\\sum_{a=0}^{d-1}c_a\\lambda(u_{k+a}).\n\\]\nA separate source-sequence theorem transports the recurrence back to the original $v_{Nk+j}$ notation.","\\subsection*{Order versus minimal order}\nThe characteristic polynomial gives a recurrence of order at most $d$. For example, $C=I_2$ gives a degree-two characteristic polynomial, but every orbit is constant and obeys $u_{k+1}=u_k$. Both statements are checked. No minimal-polynomial classification is claimed.","For $d=0$, the unique empty vector is zero and the formula remains valid. For singular matrices, Cayley--Hamilton still applies: invertibility is irrelevant. The nilpotent example $C=\\left(\\begin{smallmatrix}0&1","For $d=0$, the unique empty vector is zero and the formula remains valid. For singular matrices, Cayley--Hamilton still applies: invertibility is irrelevant. The nilpotent example $C=\\left(\\begin{smallmatrix}0&1\\\\0&0\\end{smallmatrix}\\right)$ has $C^2=0$ and $\\chi_C=X^2$.","0&0\\end{smallmatrix}\\right)$ has $C^2=0$ and $\\chi_C=X^2$.","s, Companion.charpoly_coeff_dimension}\\par\n\\target{Companion.packet_charpoly_annihilation}\\par\n\\target{Companion.packet_charpoly_recurrence}\\par\n\\target{Companion.packet_coordinate_recurrence}\\par\n\\target{Companion.packet_observable_recurrence}\n\\par\\medskip\n\\pageheading{3. Fibonacci and Lucas coefficients are derived}{Corollary 9.2; manuscript page 22}\nDefine the actual integer matrix and state\n\\[\n C=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix},\n \\qquad v_n=\\begin{pmatrix}F_{n+1}\\\\F_n\\end{pmatrix},\n \\qquad v_0=\\begin{pmatrix}1\\\\0\\end{pmatrix}.\n\\]\nThe values $F_n$ are Mathlib's \\mcode{Nat.fib}, cast into $\\Z$. Its basic recurrence proves $v_{n+1}=Cv_n$. The second coordinate of the packet state is therefore proved to be the intended $F_{Nk+j}$, not a newly assumed sequence.","0\\end{pmatrix}.\n\\]\nThe values $F_n$ are Mathlib's \\mcode{Nat.fib}, cast into $\\Z$. Its basic recurrence proves $v_{n+1}=Cv_n$. The second coordinate of the packet state is therefore proved to be the intended $F_{Nk+j}$, not a newly assumed sequence.","\\subsection*{An independent Lucas sequence}\nDefine\n\\[\n L_0=2,\\quad L_1=1,\\quad L_{n+2}=L_n+L_{n+1}.\n\\]\nDirect finite matrix arithmetic proves $C^2=C+I$. Multiplication by $C^n$ gives $C^{n+2}=C^n+C^{n+1}$. Taking traces shows that $\\operatorname{tr}(C^n)$ satisfies the same recurrence and initial values as $L_n$. Induction yields\n\\[\n \\boxed{\\operatorname{tr}(C^N)=L_N.}\n\\]\nThe implementation separately proves\n\\[\n \\det C=-1,\\qquad \\det(C^N)=(-1)^N,\n \\qquad L_N=2F_{N+1}-F_N.\n\\]\nNo Binet roots or diagonalizability argument is needed.","n independent Lucas sequence}\nDefine\n\\[\n L_0=2,\\quad L_1=1,\\quad L_{n+2}=L_n+L_{n+1}.\n\\]\nDirect finite matrix arithmetic proves $C^2=C+I$. Multiplication by $C^n$ gives $C^{n+2}=C^n+C^{n+1}$. Taking traces shows that $\\operatorname{tr}(C^n)$ satisfies the same recurrence and initial values as $L_n$. Induction yields\n\\[\n \\boxed{\\operatorname{tr}(C^N)=L_N.}\n\\]\nThe implementation separately proves\n\\[\n \\det C=-1,\\qquad \\det(C^N)=(-1)^N,\n \\qquad L_N=2F_{N+1}-F_N.\n\\]\nNo Binet roots or diagonalizability argument is needed.","\\subsection*{The sampled Fibonacci recurrence}\nThe actual two-by-two characteristic-polynomial formula gives\n\\[\n \\chi_{C^N}(X)=X^2-L_NX+(-1)^N.\n\\]\nInsert these coefficients into the general packet recurrence and take the second coordinate:\n\\[\n \\boxed{F_{N(k+2)+j}=L_NF_{N(k+1)+j}-(-1)^NF_{Nk+j}.}\n\\]\nThe full two-coordinate state satisfies the same equation. All subtraction is in $\\Z$, not truncated natural-number subtraction. The result holds for all natural $N,j,k$, including zero step and zero initial values.","y-two characteristic-polynomial formula gives\n\\[\n \\chi_{C^N}(X)=X^2-L_NX+(-1)^N.\n\\]\nInsert these coefficients into the general packet recurrence and take the second coordinate:\n\\[\n \\boxed{F_{N(k+2)+j}=L_NF_{N(k+1)+j}-(-1)^NF_{Nk+j}.}\n\\]\nThe full two-coordinate state satisfies the same equation. All subtraction is in $\\Z$, not truncated natural-number subtraction. The result holds for all natural $N,j,k$, including zero step and zero initial values.","s.state_is_orbit, FibonacciPackets.packet_is_fibonacci}\\par\n\\target{FibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power}\\par\n\\target{FibonacciPackets.jump_charpoly}\\par\n\\target{FibonacciPackets.fibonacci_packet_recurrence}\n\\par\\medskip\n\\pageheading{4. Exact examples and meaningful boundary checks}{Example 9.3 and supporting results}\nThe independently defined Lucas recurrence gives\n\\[\n (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).\n\\]\nFor $N=3$, the general theorem therefore specializes to\n\\[\n u_{k+2}=4u_{k+1}+u_k.\n\\]\nThe implementation proves this for every offset $j$ and every $k$, not only the displayed sample. At $j=2$, direct evaluation of the actual Fibonacci sequence gives\n\\[\n (F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).\n\\]\nThis is the exact manuscript example.","ci}\\par\n\\target{FibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power}\\par\n\\target{FibonacciPackets.jump_charpoly}\\par\n\\target{FibonacciPackets.fibonacci_packet_recurrence}\n\\par\\medskip\n\\pageheading{4. Exact examples and meaningful boundary checks}{Example 9.3 and supporting results}\nThe independently defined Lucas recurrence gives\n\\[\n (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).\n\\]\nFor $N=3$, the general theorem therefore specializes to\n\\[\n u_{k+2}=4u_{k+1}+u_k.\n\\]\nThe implementation proves this for every offset $j$ and every $k$, not only the displayed sample. At $j=2$, direct evaluation of the actual Fibonacci sequence gives\n\\[\n (F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).\n\\]\nThis is the exact manuscript example.",", FibonacciPackets.det_power}\\par\n\\target{FibonacciPackets.jump_charpoly}\\par\n\\target{FibonacciPackets.fibonacci_packet_recurrence}\n\\par\\medskip\n\\pageheading{4. Exact examples and meaningful boundary checks}{Example 9.3 and supporting results}\nThe independently defined Lucas recurrence gives\n\\[\n (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).\n\\]\nFor $N=3$, the general theorem therefore specializes to\n\\[\n u_{k+2}=4u_{k+1}+u_k.\n\\]\nThe implementation proves this for every offset $j$ and every $k$, not only the displayed sample. At $j=2$, direct evaluation of the actual Fibonacci sequence gives\n\\[\n (F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).\n\\]\nThis is the exact manuscript example.","r\n\\target{FibonacciPackets.fibonacci_packet_recurrence}\n\\par\\medskip\n\\pageheading{4. Exact examples and meaningful boundary checks}{Example 9.3 and supporting results}\nThe independently defined Lucas recurrence gives\n\\[\n (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).\n\\]\nFor $N=3$, the general theorem therefore specializes to\n\\[\n u_{k+2}=4u_{k+1}+u_k.\n\\]\nThe implementation proves this for every offset $j$ and every $k$, not only the displayed sample. At $j=2$, direct evaluation of the actual Fibonacci sequence gives\n\\[\n (F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).\n\\]\nThis is the exact manuscript example.","nacciPackets.fibonacci_packet_recurrence}\n\\par\\medskip\n\\pageheading{4. Exact examples and meaningful boundary checks}{Example 9.3 and supporting results}\nThe independently defined Lucas recurrence gives\n\\[\n (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).\n\\]\nFor $N=3$, the general theorem therefore specializes to\n\\[\n u_{k+2}=4u_{k+1}+u_k.\n\\]\nThe implementation proves this for every offset $j$ and every $k$, not only the displayed sample. At $j=2$, direct evaluation of the actual Fibonacci sequence gives\n\\[\n (F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).\n\\]\nThis is the exact manuscript example.","e}\n\\par\\medskip\n\\pageheading{4. Exact examples and meaningful boundary checks}{Example 9.3 and supporting results}\nThe independently defined Lucas recurrence gives\n\\[\n (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).\n\\]\nFor $N=3$, the general theorem therefore specializes to\n\\[\n u_{k+2}=4u_{k+1}+u_k.\n\\]\nThe implementation proves this for every offset $j$ and every $k$, not only the displayed sample. At $j=2$, direct evaluation of the actual Fibonacci sequence gives\n\\[\n (F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).\n\\]\nThis is the exact manuscript example.","ndary checks}{Example 9.3 and supporting results}\nThe independently defined Lucas recurrence gives\n\\[\n (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).\n\\]\nFor $N=3$, the general theorem therefore specializes to\n\\[\n u_{k+2}=4u_{k+1}+u_k.\n\\]\nThe implementation proves this for every offset $j$ and every $k$, not only the displayed sample. At $j=2$, direct evaluation of the actual Fibonacci sequence gives\n\\[\n (F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).\n\\]\nThis is the exact manuscript example.","\\subsection*{The parity sign cannot be removed}\nFor $N=2$, the recurrence is\n\\[\n u_{k+2}=3u_{k+1}-u_k.\n\\]\nAt $j=1,k=0$, the correct equation is $5=3\\cdot2-1$. Replacing the minus sign with a plus would assert $5=7$. The source proves the inequality, and a separate invalid-proof control attempts and rejects the false equation.","At $N=1$ the ordinary Fibonacci recurrence is recovered. At $N=0$, $L_0=2$ and the packet is constant, so the same formula becomes $u=2u-u$.","\\subsection*{Uniqueness from the first two packet values}\nFor a fixed $N,j$, any integer sequence satisfying the proved sampled recurrence and starting at $F_j,F_{N+j}$ agrees with $F_{Nk+j}$ for all $k$. This is proved by two-step induction. It is a consequence of the established coefficients and recurrence, not a premise used to identify Fibonacci values initially.","\\subsubsection*{The independent route}\nIn characteristic $p$, both the first and $p$th coefficient shifts obey a Leibniz rule for Hurwitz convolution. A section supported at multiples of $m$, with $m\\mid p-1$, has equal first and $p$th shifts. Its constructed reciprocal inherits this equality. Identifying that reciprocal with the existing reduced Euler sequence gives its period on the positive-index tail.","\\subsubsection*{What remains}\nTheorem 7.5 asks for powers as high as $p^{\\min(k,mn)}$ and covers nonstandard residue classes. A period modulo $p$ does not establish that stronger result. This release proves a genuine standard-window mod-$p$ fragment and precise conditional interfaces for the remaining argument, while keeping item 7.5 pending.","\\par\\medskip\n\\pageheading{1. Coefficient products and reciprocal differentiation}{A characteristic-$p$ argument without factorial division}\nFor sequences $f,g:\\N\\to K$, where $K$ is a field, define\n\\[\n (f\\star g)(d)=\\sum_{i=0}^{d}\\binom di f(i)g(d-i),\n \\qquad (U^a f)(d)=f(d+a).\n\\]\nThese are Hurwitz coefficients. The notation does not divide by $d!$ in $\\F_p$. In particular, the implementation is not ordinary power-series multiplication with undefined inverse factorials in positive characteristic.","The identity sequence is $\\mathbf1(d)=\\delta_{d0}$. The new source proves the actual higher product rule\n\\[\n U^a(f\\star g)=\\sum_{i=0}^{a}\\binom ai\n       (U^i f)\\star(U^{a-i}g).\n\\]\nThe argument starts with Pascal's identity for the first shift, then inducts on $a$. All sums are finite at each coefficient.","\\subsection*{The prime-shift cancellation}\nFor prime $p$ and characteristic $p$, every interior binomial coefficient in row $p$ vanishes. Hence\n\\[\n U^p(f\\star g)=(U^pf)\\star g+f\\star(U^pg).\n\\]\nThe first-shift rule has exactly the same form. Both are checked for the literal coefficient product.","Suppose $g(0)=1$. The source constructs $f$ by the full triangular recurrence\n\\[\n f(d)=\\delta_{d0}-\\sum_{i<d}\\binom di f(i)g(d-i)\n\\]\nand proves $f\\star g=\\mathbf1$. It also proves cancellation: if $a\\star g=0$, then $a=0$, by induction on the first nonzero coefficient.","If $U^pg=Ug$, differentiate the inverse equation with both operators and subtract. The terms involving $f\\star(Ug)$ cancel, leaving\n\\[\n (U^pf-Uf)\\star g=0.\n\\]\nThe proved cancellation law yields\n\\[\n \\boxed{U^pf=Uf.}\n\\]\nThe section identity is an explicit premise of this general reciprocal lemma. The next page proves that premise for the specific section used by the Euler sequence.","\\par\\medskip\n\\pageheading{2. Identification with the reduced Euler sequence}{Corollary 8.5; manuscript page 21}\nWork over $\\F_p=\\Z/p\\Z$. For positive $m\\mid p-1$, take the literal section\n\\[\n g_m(d)=\\begin{cases}1,&m\\mid d,","\\par\\medskip\n\\pageheading{2. Identification with the reduced Euler sequence}{Corollary 8.5; manuscript page 21}\nWork over $\\F_p=\\Z/p\\Z$. For positive $m\\mid p-1$, take the literal section\n\\[\n g_m(d)=\\begin{cases}1,&m\\mid d,\\\\0,&m\\nmid d.\\end{cases}\n\\]\nIt satisfies $g_m(0)=1$. Since $p\\equiv1\\pmod m$, the conditions $m\\mid d+p$ and $m\\mid d+1$ are equivalent. The source therefore proves $U^pg_m=Ug_m$.","0,&m\\nmid d.\\end{cases}\n\\]\nIt satisfies $g_m(0)=1$. Since $p\\equiv1\\pmod m$, the conditions $m\\mid d+p$ and $m\\mid d+1$ are equivalent. The source therefore proves $U^pg_m=Ug_m$.","Let $f_m$ be the reciprocal constructed on the previous page. A separate strong-induction proof establishes its actual support:\n\\[\n m\\nmid d\\quad\\Longrightarrow\\quad f_m(d)=0.\n\\]\nThis support is not included as an assumption of the construction.","\\subsection*{The same residual sequence, not a renamed substitute}\nEvaluate $f_m\\star g_m=\\mathbf1$ at $d=mn$. Reindex the surviving summation indices by $i=mq$ using the proved support. The result is\n\\[\n \\sum_{q=0}^{n}\\binom{mn}{mq}f_m(mq)=\\delta_{n0}.\n\\]\nThe fifth block independently constructed the unique sequence $e_n^{(m,p)}$ satisfying this triangular system. Applying that uniqueness theorem gives\n\\[\n \\boxed{f_m(mn)=e_n^{(m,p)}.}\n\\]\nThus the reciprocal really is connected to the manuscript's Lucas-reduced inverse.","on*{The same residual sequence, not a renamed substitute}\nEvaluate $f_m\\star g_m=\\mathbf1$ at $d=mn$. Reindex the surviving summation indices by $i=mq$ using the proved support. The result is\n\\[\n \\sum_{q=0}^{n}\\binom{mn}{mq}f_m(mq)=\\delta_{n0}.\n\\]\nThe fifth block independently constructed the unique sequence $e_n^{(m,p)}$ satisfying this triangular system. Applying that uniqueness theorem gives\n\\[\n \\boxed{f_m(mn)=e_n^{(m,p)}.}\n\\]\nThus the reciprocal really is connected to the manuscript's Lucas-reduced inverse.","\\subsection*{The positive-tail period}\nWrite $h=(p-1)/m$. For $n\\ge1$, evaluate $U^pf_m=Uf_m$ at the nonnegative degree $mn-1$:\n\\[\n f_m(mn+p-1)=f_m(mn).\n\\]\nSince $mn+p-1=m(n+h)$, the identified sequence satisfies\n\\[\n \\boxed{e_{n+h}^{(m,p)}=e_n^{(m,p)}\\quad(n\\ge1).}\n\\]\nInduction gives $e_{n+uh}^{(m,p)}=e_n^{(m,p)}$ for every $u\\ge0$. This proves a period, not necessarily the least positive period. The proof also includes $p=2$, where $m=1$ and $h=1$.","-tail period}\nWrite $h=(p-1)/m$. For $n\\ge1$, evaluate $U^pf_m=Uf_m$ at the nonnegative degree $mn-1$:\n\\[\n f_m(mn+p-1)=f_m(mn).\n\\]\nSince $mn+p-1=m(n+h)$, the identified sequence satisfies\n\\[\n \\boxed{e_{n+h}^{(m,p)}=e_n^{(m,p)}\\quad(n\\ge1).}\n\\]\nInduction gives $e_{n+uh}^{(m,p)}=e_n^{(m,p)}$ for every $u\\ge0$. This proves a period, not necessarily the least positive period. The proof also includes $p=2$, where $m=1$ and $h=1$.","Write $h=(p-1)/m$. For $n\\ge1$, evaluate $U^pf_m=Uf_m$ at the nonnegative degree $mn-1$:\n\\[\n f_m(mn+p-1)=f_m(mn).\n\\]\nSince $mn+p-1=m(n+h)$, the identified sequence satisfies\n\\[\n \\boxed{e_{n+h}^{(m,p)}=e_n^{(m,p)}\\quad(n\\ge1).}\n\\]\nInduction gives $e_{n+uh}^{(m,p)}=e_n^{(m,p)}$ for every $u\\ge0$. This proves a period, not necessarily the least positive period. The proof also includes $p=2$, where $m=1$ and $h=1$.","(p-1)/m$. For $n\\ge1$, evaluate $U^pf_m=Uf_m$ at the nonnegative degree $mn-1$:\n\\[\n f_m(mn+p-1)=f_m(mn).\n\\]\nSince $mn+p-1=m(n+h)$, the identified sequence satisfies\n\\[\n \\boxed{e_{n+h}^{(m,p)}=e_n^{(m,p)}\\quad(n\\ge1).}\n\\]\nInduction gives $e_{n+uh}^{(m,p)}=e_n^{(m,p)}$ for every $u\\ge0$. This proves a period, not necessarily the least positive period. The proof also includes $p=2$, where $m=1$ and $h=1$.","\\par\\medskip\n\\pageheading{3. Actual packet congruences and sharp boundaries}{Consequences for the standard window; not the full Theorem 7.5}\nThe fifth block constructed integral lifts $\\widehat C_{mp,j}(n)\\in\\Z_p$ of the original rational coefficients when $0\\le j<p$, and proved that their residues equal $e_n^{(m,p)}$.","Applying the actual residue map and its kernel ideal to the new period gives\n\\[\n \\boxed{p\\mid\\widehat C_{mp,j}(n+uh)-\\widehat C_{mp,j}(n)\n       \\quad\\hbox{in }\\Z_p}\n\\]\nfor $n\\ge1$ and $u\\ge0$. Every higher positive difference is a difference of terms already divisible by $p$. Consequently,\n\\[\n p\\mid\\Delta_{uh}^{k}\\widehat C_{mp,j}(n)\\qquad(k\\ge1,\\ n\\ge1).\n\\]\nA ring-homomorphism lemma identifies the image of these differences in $\\Q_p$ with the differences of the original rational coefficient function. This is not vacuous field divisibility. The valuation version uses order infinity at zero.","d its kernel ideal to the new period gives\n\\[\n \\boxed{p\\mid\\widehat C_{mp,j}(n+uh)-\\widehat C_{mp,j}(n)\n       \\quad\\hbox{in }\\Z_p}\n\\]\nfor $n\\ge1$ and $u\\ge0$. Every higher positive difference is a difference of terms already divisible by $p$. Consequently,\n\\[\n p\\mid\\Delta_{uh}^{k}\\widehat C_{mp,j}(n)\\qquad(k\\ge1,\\ n\\ge1).\n\\]\nA ring-homomorphism lemma identifies the image of these differences in $\\Q_p$ with the differences of the original rational coefficient function. This is not vacuous field divisibility. The valuation version uses order infinity at zero.","\\subsection*{Three checked boundaries}\n\\textbf{The initial row.} For $(p,m)=(3,2)$, the same reduced recurrence gives $e_0=1$ and $e_1=-1=2$ in $\\F_3$. Its proposed period is one, so the positive-tail result cannot in general include row zero.","\\textbf{The divisor hypothesis.} At $(p,m)=(5,3)$, where $3\\nmid4$, the independently computed reduced row is\n\\[\n (e_0,e_1,e_2,e_3)=(1,4,4,2).\n\\]\nReplacing $h$ by the integer quotient $\\lfloor4/3\\rfloor=1$ does not give a period: $e_3\\ne e_2$.","\\textbf{No automatic extra power.} The original rational triangular coefficients satisfy\n\\[\n C_{6,0}(1)=-1,\\quad C_{6,0}(2)=923,\\quad\n v_3\\bigl(C_{6,0}(2)-C_{6,0}(1)\\bigr)=v_3(924)=1.\n\\]\nA period modulo $3$ therefore cannot simply be reported as a period modulo $9$. This is consistent with the global theorem's proposed first-difference bound. It is not a counterexample to Theorem 7.5.","\\par\\medskip\n\\pageheading{4. Exact interfaces for the remaining global argument}{Conditional results are not counted as completion of item 7.5}\nLet $p$ be prime and $m$ a positive divisor of $p-1$. For a sequence $e$ in a commutative coefficient ring, define its actual packed Hurwitz sequence\n\\[\n X(d)=\\begin{cases}e(d/N),&N\\mid d,","\\par\\medskip\n\\pageheading{4. Exact interfaces for the remaining global argument}{Conditional results are not counted as completion of item 7.5}\nLet $p$ be prime and $m$ a positive divisor of $p-1$. For a sequence $e$ in a commutative coefficient ring, define its actual packed Hurwitz sequence\n\\[\n X(d)=\\begin{cases}e(d/N),&N\\mid d,\\\\0,&N\\nmid d,\\end{cases}\n \\qquad N=mp.\n\\]\nLet $L=p(p-1)$ and $\\delta=U^{p+L}-U^p=U^{p^2}-U^p$. The source proves the operator dictionary\n\\[\n (\\delta^sX)(d)=\\Delta_L^s X(d+ps).\n\\]\nAt the valid nonnegative coefficient degree $d=mpn-ps$, with $s\\le mn$, this becomes\n\\[\n (\\delta^sX)(mpn-ps)=\\Delta_h^s e(n),\\qquad h=(p-1)/m.\n\\]\nThe packed sequence and difference operations are definitions. Their coefficient correspondence is proved before any divisibility assumption is introduced.","0,&N\\nmid d,\\end{cases}\n \\qquad N=mp.\n\\]\nLet $L=p(p-1)$ and $\\delta=U^{p+L}-U^p=U^{p^2}-U^p$. The source proves the operator dictionary\n\\[\n (\\delta^sX)(d)=\\Delta_L^s X(d+ps).\n\\]\nAt the valid nonnegative coefficient degree $d=mpn-ps$, with $s\\le mn$, this becomes\n\\[\n (\\delta^sX)(mpn-ps)=\\Delta_h^s e(n),\\qquad h=(p-1)/m.\n\\]\nThe packed sequence and difference operations are definitions. Their coefficient correspondence is proved before any divisibility assumption is introduced.","\\subsection*{A sufficient condition, explicitly conditional}\nDefine flatness by the actual coefficientwise divisibility assertion\n\\[\n \\delta^sX(d)\\in p^sR\\qquad(s,d\\ge0).\n\\]\nThe new theorem proves that this hypothesis implies\n\\[\n \\boxed{p^{\\min(k,mn)}\\mid\\Delta_h^k e(n).}\n\\]\nFirst, coefficient extraction proves the triangular region $k\\le mn$. A separate induction on difference order proves that this region is equivalent to the complete saturated estimate. This implication is useful over $\\Z_p$. Interpreting it over a field would not supply the intended integral content.","fficientwise divisibility assertion\n\\[\n \\delta^sX(d)\\in p^sR\\qquad(s,d\\ge0).\n\\]\nThe new theorem proves that this hypothesis implies\n\\[\n \\boxed{p^{\\min(k,mn)}\\mid\\Delta_h^k e(n).}\n\\]\nFirst, coefficient extraction proves the triangular region $k\\le mn$. A separate induction on difference order proves that this region is equivalent to the complete saturated estimate. This implication is useful over $\\Z_p$. Interpreting it over a field would not supply the intended integral content.","\\subsection*{Normalization is also an explicit implication}\nUsing the existing definition $E^{(N,j)}_{Nn}=j!C_{N,j}(n)$, the source proves that multiplication by $j!$ commutes with every iterated difference. If the unnormalized difference has precision $A$, the normalized difference has precision $A-v_p(j!)$. Precision is defined as\n\\[\n x=0\\quad\\hbox{or}\\quad v_p(x)\\ge A,\n\\]\nso zero and negative factorial-loss thresholds are handled correctly. For $j<p$, the loss is zero.","ation}\nUsing the existing definition $E^{(N,j)}_{Nn}=j!C_{N,j}(n)$, the source proves that multiplication by $j!$ commutes with every iterated difference. If the unnormalized difference has precision $A$, the normalized difference has precision $A-v_p(j!)$. Precision is defined as\n\\[\n x=0\\quad\\hbox{or}\\quad v_p(x)\\ge A,\n\\]\nso zero and negative factorial-loss thresholds are handled correctly. For $j<p$, the loss is zero.","\\subsection*{Independent exact finite checks}\nThe new rational and finite-field mirrors passed 4,848 reciprocal coefficient rows, 2,592 higher-Leibniz cases, 4,520 reciprocal prime-shift cases, 3,720 period comparisons, and 1,476 standard-window rational rows. They also checked 1,104 rational period differences, 620 higher differences modulo $p$, and 586 Dwork extraction interfaces. Eleven invalid input configurations were rejected.","omplete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nThe all-residue Theorem 7.5 remains pending. This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","etains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nThe all-residue Theorem 7.5 remains pending. This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","tus of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nThe all-residue Theorem 7.5 remains pending. This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","ted original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nThe all-residue Theorem 7.5 remains pending. This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nThe all-residue Theorem 7.5 remains pending. This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","ing or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nThe all-residue Theorem 7.5 remains pending. This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","rage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nThe all-residue Theorem 7.5 remains pending. This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.",". This release proves its \\textbf{full saturated higher-power estimate throughout the standard window}, rather than only a period modulo $p$. For every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","every prime $p$, positive $m\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","\\mid p-1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","1$, $0\\le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","le j<p$, and natural $n,k,u$, put $h=(p-1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","1)/m$. Then\n\\[\n\\boxed{\\quad\n\\Delta_{uh}^{k}C_{mp,j}(n)=0\\quad\\text{or}\\quad\nv_p\\!\\left(\\Delta_{uh}^{k}C_{mp,j}(n)\\right)\\ge\\min\\{k,mn\\}.\n\\quad}\n\\]\nThe unnormalized family $j!C_{mp,j}(n)$ satisfies the same bound. Both statements concern the original rational coefficient function, identified with its actual integral $p$-adic lift. They include $p=2$, $k=0$, $n=0$, and the additional harmless case $u=0$.","\\textbf{Coverage remains 23 of 24 named items proved in the manuscript.} Three classical inputs and two conjectures remain separate. The new result covers every permissible residue when $m=1$, but not the range $p\\le j<mp$ when $m>1$.","\\par\\medskip\n\\pageheading{1. The characteristic-zero product rule}{The normalized defect is not an ordinary derivation}\nLet $A$ be a commutative algebra over a characteristic-zero field $K$, and let $U$ be an actual $K$-derivation. The source defines\n\\[\n B=\\frac{U^{p^2}-U^p}{p},\\qquad\n M_q(f,g)=\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coe","\\par\\medskip\n\\pageheading{1. The characteristic-zero product rule}{The normalized defect is not an ordinary derivation}\nLet $A$ be a commutative algebra over a characteristic-zero field $K$, and let $U$ be an actual $K$-derivation. The source defines\n\\[\n B=\\frac{U^{p^2}-U^p}{p},\\qquad\n M_q(f,g)=\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is","\\par\\medskip\n\\pageheading{1. The characteristic-zero product rule}{The normalized defect is not an ordinary derivation}\nLet $A$ be a commutative algebra over a characteristic-zero field $K$, and let $U$ be an actual $K$-derivation. The source defines\n\\[\n B=\\frac{U^{p^2}-U^p}{p},\\qquad\n M_q(f,g)=\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This in","\\par\\medskip\n\\pageheading{1. The characteristic-zero product rule}{The normalized defect is not an ordinary derivation}\nLet $A$ be a commutative algebra over a characteristic-zero field $K$, and let $U$ be an actual $K$-derivation. The source defines\n\\[\n B=\\frac{U^{p^2}-U^p}{p},\\qquad\n M_q(f,g)=\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itsel","\\par\\medskip\n\\pageheading{1. The characteristic-zero product rule}{The normalized defect is not an ordinary derivation}\nLet $A$ be a commutative algebra over a characteristic-zero field $K$, and let $U$ be an actual $K$-derivation. The source defines\n\\[\n B=\\frac{U^{p^2}-U^p}{p},\\qquad\n M_q(f,g)=\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","medskip\n\\pageheading{1. The characteristic-zero product rule}{The normalized defect is not an ordinary derivation}\nLet $A$ be a commutative algebra over a characteristic-zero field $K$, and let $U$ be an actual $K$-derivation. The source defines\n\\[\n B=\\frac{U^{p^2}-U^p}{p},\\qquad\n M_q(f,g)=\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","ordinary derivation}\nLet $A$ be a commutative algebra over a characteristic-zero field $K$, and let $U$ be an actual $K$-derivation. The source defines\n\\[\n B=\\frac{U^{p^2}-U^p}{p},\\qquad\n M_q(f,g)=\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","\\sum_{a=1}^{q-1}\\frac{1}{p}\\binom qa\n          (U^a f)(U^{q-a}g),\\quad q=p,p^2.\n\\]\nThe coefficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","efficients of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","nts of each $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","h $M_q$ are proved integral: the middle entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","le entries of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","of a prime-power binomial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","ial row are divisible by $p$. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence\n\\[\n\\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}\n\\]\nThe middle terms are essential. At $p=3$, $f=t$ and $g=t^2$, the constant coefficient of $B(fg)$ is $-2$, whereas that of $B(f)g+fB(g)$ is zero. This inequality is itself Lean-checked.","\\subsection*{All orders, without a circular closure assumption}\nLet $I\\subset A$ be a subring stable under $U$. Define\n\\[\n \\operatorname{Bounded}_r(f)\\iff B^s f\\in I\\ (0\\le s\\le r),\n \\qquad \\operatorname{Flat}(f)\\iff B^s f\\in I\\ (s\\ge0).\n\\]\nThe definitions contain no product or inverse closure requirement. Since $B$ commutes with $U$, derivatives of a bounded element remain bounded. Induction on $r$ in the displayed product rule then proves closure under multiplication. The finite middle sums use the lower-order induction hypothesis, not the desired next-order conclusion.","For an actual inverse $fg=1$, applying the product rule and multiplying by $g$ gives\n\\[\n B(g)=-g^2B(f)-gM_{p^2}(f,g)+gM_p(f,g).\n\\]\nIf $g\\in I$, induction on $r$ proves boundedness of $g$ from boundedness of $f$. At each successor stage the preceding stage supplies boundedness of $g$ needed on the right. Passing to all finite orders proves reciprocal closure for an integral inverse.","\\par\\medskip\n\\pageheading{2. The actual integral Hurwitz lattice}{Connecting differential expressions to coefficient divisibility}\nThe specialization works inside $\\Q_p[[t]]$ and uses factorial-scaled coefficients\n\\[\n a_f(d)=d![t^d]f,\\qquad\n I_p=\\{f\\in\\Q_p[[t]]:a_f(d)\\in\\Z_p\\text{ for every }d\\}.\n\\]\nThe source constructs $I_p$ as a subring. Actual exponential-series multiplication gives\n\\[\n a_{fg}(d)=\\sum_{r=0}^{d}\\binom dr a_f(r)a_g(d-r),\n \\qquad a_{Uf}(d)=a_f(d+1).\n\\]\nThese identities prove multiplication and derivative stability of the lattice. Membership does not require ordinary power-series coefficients to lie in $\\Z_p$.","t^d]f,\\qquad\n I_p=\\{f\\in\\Q_p[[t]]:a_f(d)\\in\\Z_p\\text{ for every }d\\}.\n\\]\nThe source constructs $I_p$ as a subring. Actual exponential-series multiplication gives\n\\[\n a_{fg}(d)=\\sum_{r=0}^{d}\\binom dr a_f(r)a_g(d-r),\n \\qquad a_{Uf}(d)=a_f(d+1).\n\\]\nThese identities prove multiplication and derivative stability of the lattice. Membership does not require ordinary power-series coefficients to lie in $\\Z_p$.","\\subsection*{The reciprocal is integral}\nFor $f\\in I_p$ with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence\n\\[\n a_{f^{-1}}(d)=\\delta_{d0}-\\sum_{r=0}^{d-1}\n     \\binom dr a_{f^{-1}}(r)a_f(d-r).\n\\]\nStrong induction proves that all of these coefficients are integral. Combining this construction with the preceding all-order theorem proves flatness of the \\emph{actual reciprocal}. No flatness or integrality of that reciprocal is assumed in the normalized-series specialization.","\\subsection*{The earlier coefficient criterion is recovered exactly}\nDefine the literal coefficient defect\n\\[\n (\\delta a)(d)=a(d+p^2)-a(d+p).\n\\]\nThe new source proves, for every $s,d$,\n\\[\n a_{B^s f}(d)=p^{-s}(\\delta^s a_f)(d).\n\\]\nFor an integral coefficient $x$, it proves the bridge\n\\[\n\\boxed{p^{-s}x\\in\\Z_p\\quad\\Longleftrightarrow\\quad p^s\\mid x\\text{ in }\\Z_p.}\n\\]\nThus the differential definition is equivalent to the existing coefficient-level Dwork criterion. Division occurs in $\\Q_p$, while the assertion after division is membership in $\\Z_p$. It is not divisibility in the field. Zero coefficients satisfy every finite precision condition.","l coefficient defect\n\\[\n (\\delta a)(d)=a(d+p^2)-a(d+p).\n\\]\nThe new source proves, for every $s,d$,\n\\[\n a_{B^s f}(d)=p^{-s}(\\delta^s a_f)(d).\n\\]\nFor an integral coefficient $x$, it proves the bridge\n\\[\n\\boxed{p^{-s}x\\in\\Z_p\\quad\\Longleftrightarrow\\quad p^s\\mid x\\text{ in }\\Z_p.}\n\\]\nThus the differential definition is equivalent to the existing coefficient-level Dwork criterion. Division occurs in $\\Q_p$, while the assertion after division is membership in $\\Z_p$. It is not divisibility in the field. Zero coefficients satisfy every finite precision condition.","he new source proves, for every $s,d$,\n\\[\n a_{B^s f}(d)=p^{-s}(\\delta^s a_f)(d).\n\\]\nFor an integral coefficient $x$, it proves the bridge\n\\[\n\\boxed{p^{-s}x\\in\\Z_p\\quad\\Longleftrightarrow\\quad p^s\\mid x\\text{ in }\\Z_p.}\n\\]\nThus the differential definition is equivalent to the existing coefficient-level Dwork criterion. Division occurs in $\\Q_p$, while the assertion after division is membership in $\\Z_p$. It is not divisibility in the field. Zero coefficients satisfy every finite precision condition.","\\par\\medskip\n\\pageheading{3. A nonsingular Euler-operator argument}{The standard interval without an analytic power-series-in-the-index premise}\nOn Hurwitz coefficients, the actual Euler operator plus $c$ is\n\\[\n (\\Lambda_c a)(d)=(d+c)a(d).\n\\]\nDirect induction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\","\\par\\medskip\n\\pageheading{3. A nonsingular Euler-operator argument}{The standard interval without an analytic power-series-in-the-index premise}\nOn Hurwitz coefficients, the actual Euler operator plus $c$ is\n\\[\n (\\Lambda_c a)(d)=(d+c)a(d).\n\\]\nDirect induction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","3. A nonsingular Euler-operator argument}{The standard interval without an analytic power-series-in-the-index premise}\nOn Hurwitz coefficients, the actual Euler operator plus $c$ is\n\\[\n (\\Lambda_c a)(d)=(d+c)a(d).\n\\]\nDirect induction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","interval without an analytic power-series-in-the-index premise}\nOn Hurwitz coefficients, the actual Euler operator plus $c$ is\n\\[\n (\\Lambda_c a)(d)=(d+c)a(d).\n\\]\nDirect induction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","without an analytic power-series-in-the-index premise}\nOn Hurwitz coefficients, the actual Euler operator plus $c$ is\n\\[\n (\\Lambda_c a)(d)=(d+c)a(d).\n\\]\nDirect induction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","in-the-index premise}\nOn Hurwitz coefficients, the actual Euler operator plus $c$ is\n\\[\n (\\Lambda_c a)(d)=(d+c)a(d).\n\\]\nDirect induction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","coefficients, the actual Euler operator plus $c$ is\n\\[\n (\\Lambda_c a)(d)=(d+c)a(d).\n\\]\nDirect induction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","uction proves its commutator with every defect iterate:\n\\[\n \\delta^s\\Lambda_c a\n =\\Lambda_c\\delta^s a\n  +sp\\bigl(pU^{p^2}-U^p\\bigr)\\delta^{s-1}a,\n \\qquad s\\ge1.\n\\]\nSuppose $a$ is integral and supported at degrees divisible by $p$, and $p\\nmid c$. Every supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","very supported pivot $d+c$ is proved to be a unit of $\\Z_p$. The defect preserves this support. Induction in the commutator identity therefore proves\n\\[\n \\operatorname{Flat}(\\Lambda_c a)\\Longrightarrow\\operatorname{Flat}(a).\n\\]\nThis is a reflection theorem on the integral supported domain. It does not assume flatness of the proposed preimage, and does not assert invertibility at a singular pivot $p\\mid c$.","\\subsection*{Apply it to the explicit normalized sections}\nSet $N=mp$ and, for $0\\le j<p$, define integral Hurwitz coefficients\n\\[\n g_j(d)=\n \\begin{cases}\n \\displaystyle\\binom{d+j}{d}^{-1},&N\\mid d,","\\subsection*{Apply it to the explicit normalized sections}\nSet $N=mp$ and, for $0\\le j<p$, define integral Hurwitz coefficients\n\\[\n g_j(d)=\n \\begin{cases}\n \\displaystyle\\binom{d+j}{d}^{-1},&N\\mid d,\\\\[1mm]\n 0,&N\\nmid d.\n \\end{cases}\n\\]\nThe inverse binomial is constructed using the actual unit pivots already proved by Lucas reduction. Its image in $\\Q_p$ is proved equal to the displayed reciprocal.","[1mm]\n 0,&N\\nmid d.\n \\end{cases}\n\\]\nThe inverse binomial is constructed using the actual unit pivots already proved by Lucas reduction. Its image in $\\Q_p$ is proved equal to the displayed reciprocal.","For $j=0$, $g_0$ is the literal $N$-divisibility indicator. Because $N\\mid p(p-1)$, the two shifts in $\\delta g_0$ agree and $\\delta g_0=0$. It is flat at every order. The actual binomial identity gives\n\\[\n \\Lambda_{j+1}g_{j+1}=(j+1)g_j.\n\\]\nAs $1\\le j+1<p$, the nonsingular reflection theorem inductively proves flatness of every $g_j$ in the standard window. There is no uniform-convergence assumption in this argument.","\\par\\medskip\n\\pageheading{4. From these sections to the original Euler packets}{Actual reciprocal, actual rational sequence, every order and step multiple}\nThe factorial-ratio identity is proved coefficient by coefficient:\n\\[\n \\sum_{d\\ge0}g_j(d)\\frac{t^d}{d!}\n =j!\\sum_{r\\ge0}\\frac{t^{Nr}}{(Nr+j)!}.\n\\]\nThe right side is precisely the normalized source kernel from the first block, not a replacement family. The earlier rational identity for its reciprocal is transported to $\\Q_p[[t]]$. A separate encoding theorem identifies the reciprocal's coefficients with the already constructed $\\Z_p$ lifts of the original $C_{N,j}(r)$.","ts}{Actual reciprocal, actual rational sequence, every order and step multiple}\nThe factorial-ratio identity is proved coefficient by coefficient:\n\\[\n \\sum_{d\\ge0}g_j(d)\\frac{t^d}{d!}\n =j!\\sum_{r\\ge0}\\frac{t^{Nr}}{(Nr+j)!}.\n\\]\nThe right side is precisely the normalized source kernel from the first block, not a replacement family. The earlier rational identity for its reciprocal is transported to $\\Q_p[[t]]$. A separate encoding theorem identifies the reciprocal's coefficients with the already constructed $\\Z_p$ lifts of the original $C_{N,j}(r)$.","proved coefficient by coefficient:\n\\[\n \\sum_{d\\ge0}g_j(d)\\frac{t^d}{d!}\n =j!\\sum_{r\\ge0}\\frac{t^{Nr}}{(Nr+j)!}.\n\\]\nThe right side is precisely the normalized source kernel from the first block, not a replacement family. The earlier rational identity for its reciprocal is transported to $\\Q_p[[t]]$. A separate encoding theorem identifies the reciprocal's coefficients with the already constructed $\\Z_p$ lifts of the original $C_{N,j}(r)$.","cient:\n\\[\n \\sum_{d\\ge0}g_j(d)\\frac{t^d}{d!}\n =j!\\sum_{r\\ge0}\\frac{t^{Nr}}{(Nr+j)!}.\n\\]\nThe right side is precisely the normalized source kernel from the first block, not a replacement family. The earlier rational identity for its reciprocal is transported to $\\Q_p[[t]]$. A separate encoding theorem identifies the reciprocal's coefficients with the already constructed $\\Z_p$ lifts of the original $C_{N,j}(r)$.","All-order reciprocal closure therefore gives flatness of the actual packed Euler coefficient sequence. The earlier coefficient-extraction theorem now has a \\emph{proved antecedent} for every $j<p$, and yields\n\\[\n p^{\\min(k,mn)}\\mid\\Delta_h^k C_{mp,j}(n)\\quad\\text{in }\\Z_p,\n \\qquad h=(p-1)/m.\n\\]\nA new finite telescoping argument proves the extension to arbitrary natural multiples of $h$. Its first-step identity is\n\\[\n f(n+uh)-f(n)=\\sum_{a=0}^{u-1}\\Delta_h f(n+ah).\n\\]\nCommutation of differences and induction on their order transfer any fixed ideal bound valid on a forward tail. The required precision $\\min(k,mn)$ is nondecreasing as the starting row advances. Hence\n\\[\n \\boxed{p^{\\min(k,mn)}\\mid\\Delta_{uh}^{k}C_{mp,j}(n)\\quad(0\\le j<p).}\n\\]","l closure therefore gives flatness of the actual packed Euler coefficient sequence. The earlier coefficient-extraction theorem now has a \\emph{proved antecedent} for every $j<p$, and yields\n\\[\n p^{\\min(k,mn)}\\mid\\Delta_h^k C_{mp,j}(n)\\quad\\text{in }\\Z_p,\n \\qquad h=(p-1)/m.\n\\]\nA new finite telescoping argument proves the extension to arbitrary natural multiples of $h$. Its first-step identity is\n\\[\n f(n+uh)-f(n)=\\sum_{a=0}^{u-1}\\Delta_h f(n+ah).\n\\]\nCommutation of differences and induction on their order transfer any fixed ideal bound valid on a forward tail. The required precision $\\min(k,mn)$ is nondecreasing as the starting row advances. Hence\n\\[\n \\boxed{p^{\\min(k,mn)}\\mid\\Delta_{uh}^{k}C_{mp,j}(n)\\quad(0\\le j<p).}\n\\]","s\n\\[\n p^{\\min(k,mn)}\\mid\\Delta_h^k C_{mp,j}(n)\\quad\\text{in }\\Z_p,\n \\qquad h=(p-1)/m.\n\\]\nA new finite telescoping argument proves the extension to arbitrary natural multiples of $h$. Its first-step identity is\n\\[\n f(n+uh)-f(n)=\\sum_{a=0}^{u-1}\\Delta_h f(n+ah).\n\\]\nCommutation of differences and induction on their order transfer any fixed ideal bound valid on a forward tail. The required precision $\\min(k,mn)$ is nondecreasing as the starting row advances. Hence\n\\[\n \\boxed{p^{\\min(k,mn)}\\mid\\Delta_{uh}^{k}C_{mp,j}(n)\\quad(0\\le j<p).}\n\\]","\\qquad h=(p-1)/m.\n\\]\nA new finite telescoping argument proves the extension to arbitrary natural multiples of $h$. Its first-step identity is\n\\[\n f(n+uh)-f(n)=\\sum_{a=0}^{u-1}\\Delta_h f(n+ah).\n\\]\nCommutation of differences and induction on their order transfer any fixed ideal bound valid on a forward tail. The required precision $\\min(k,mn)$ is nondecreasing as the starting row advances. Hence\n\\[\n \\boxed{p^{\\min(k,mn)}\\mid\\Delta_{uh}^{k}C_{mp,j}(n)\\quad(0\\le j<p).}\n\\]","The scalar-extension bridge identifies these differences with the original rational coefficient function. The rational theorem explicitly allows zero differences, avoiding the library's finite valuation convention at zero. Since $j!$ is a unit for $j<p$, both normalized and unnormalized versions have the same precision.","\\textbf{Scope.} The proof includes all primes, all rows and orders, and all natural step multipliers. For $m=1$, the standard window is the complete allowed residue range. For larger $m$, residues $p\\le j<mp$ are not covered by this theorem. The earlier one-power period is not the source of the new higher-power estimate.","\\par\\medskip\n\\pageheading{5. Examples, verification and the remaining gap}{The coverage count is unchanged; the unresolved theorem now has a stronger proved part}\nThe original rational recurrence gives\n\\[\n C_{6,2}(1)=-\\frac1{28},\\quad C_{6,2}(2)=\\frac{425}{364},\\quad\n C_{6,2}(3)=-\\frac{2039759}{2660}.\n\\]\nConsequently, the new source proves\n\\[\n \\Delta_1^2 C_{6,2}(1)=-\\frac{949959}{1235},\\qquad\n v_3\\!\\left(\\Delta_1^2 C_{6,2}(1)\\right)=2.\n\\]\nThis attains the bound $\\min(2,2\\cdot1)=2$. A claim of a third power is rejected. The separate cubic example proves that the normalized differential defect is not an ordinary derivation. Controls also reject an asserted unit pivot at $c=p$ and the prime-power binomial-divisibility formula at a composite base.","ng gap}{The coverage count is unchanged; the unresolved theorem now has a stronger proved part}\nThe original rational recurrence gives\n\\[\n C_{6,2}(1)=-\\frac1{28},\\quad C_{6,2}(2)=\\frac{425}{364},\\quad\n C_{6,2}(3)=-\\frac{2039759}{2660}.\n\\]\nConsequently, the new source proves\n\\[\n \\Delta_1^2 C_{6,2}(1)=-\\frac{949959}{1235},\\qquad\n v_3\\!\\left(\\Delta_1^2 C_{6,2}(1)\\right)=2.\n\\]\nThis attains the bound $\\min(2,2\\cdot1)=2$. A claim of a third power is rejected. The separate cubic example proves that the normalized differential defect is not an ordinary derivation. Controls also reject an asserted unit pivot at $c=p$ and the prime-power binomial-divisibility formula at a composite base.",")=-\\frac{2039759}{2660}.\n\\]\nConsequently, the new source proves\n\\[\n \\Delta_1^2 C_{6,2}(1)=-\\frac{949959}{1235},\\qquad\n v_3\\!\\left(\\Delta_1^2 C_{6,2}(1)\\right)=2.\n\\]\nThis attains the bound $\\min(2,2\\cdot1)=2$. A claim of a third power is rejected. The separate cubic example proves that the normalized differential defect is not an ordinary derivation. Controls also reject an asserted unit pivot at $c=p$ and the prime-power binomial-divisibility formula at a composite base.","\\subsection*{What is still unproved}\nThe all-residue Theorem 7.5 requires the additional range $p\\le j<mp$. Its supporting prose proof treats singular ratios at $c=p,2p,\\ldots,(m-1)p$ through weighted Amice estimates and a dual Mahler pairing, then propagates across the intervening indices. Those actual-family estimates and the full remaining propagation are not supplied by this release. A unit argument at $p\\nmid c$ cannot be used at $p\\mid c$.","*{What is still unproved}\nThe all-residue Theorem 7.5 requires the additional range $p\\le j<mp$. Its supporting prose proof treats singular ratios at $c=p,2p,\\ldots,(m-1)p$ through weighted Amice estimates and a dual Mahler pairing, then propagates across the intervening indices. Those actual-family estimates and the full remaining propagation are not supplied by this release. A unit argument at $p\\nmid c$ cannot be used at $p\\mid c$.","all-residue Theorem 7.5 requires the additional range $p\\le j<mp$. Its supporting prose proof treats singular ratios at $c=p,2p,\\ldots,(m-1)p$ through weighted Amice estimates and a dual Mahler pairing, then propagates across the intervening indices. Those actual-family estimates and the full remaining propagation are not supplied by this release. A unit argument at $p\\nmid c$ cannot be used at $p\\mid c$.","\\clearpage\\subsection{Singular Amice Supplement}\\label{app:sup9}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nLet $p$ be prime, let $m\\mid p-1$, and put $N=mp$. For $1\\le a<m$, put $c=ap$ and define the original exponential section and ratio by\n\\[\n H_{N,c}(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\qquad\n R_c(t)=\\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.\n\\]\nThe quotient is interpreted after removing its forced zero. This block proves, for the actual rational coefficients $\\alpha_d$,\n\\[\n R_{ap}(\\log(1+T))=\\sum_{d\\ge0}\\alpha_dT^d,\n \\qquad\n \\boxed{\\alpha_d=0\\ \\text{or}\\ v_p(\\alpha_d)\\ge-\\left\\lfloor d/p\\right\\rfloor.}\n\\]\nEquivalently, every $p^{\\lfloor d/p\\rfloor}\\alpha_d$ belongs to the actual completed ring $\\Z_p$. The zero-safe alternative is explicit in the formal rational statement.","}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and result}\nLet $p$ be prime, let $m\\mid p-1$, and put $N=mp$. For $1\\le a<m$, put $c=ap$ and define the original exponential section and ratio by\n\\[\n H_{N,c}(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\qquad\n R_c(t)=\\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.\n\\]\nThe quotient is interpreted after removing its forced zero. This block proves, for the actual rational coefficients $\\alpha_d$,\n\\[\n R_{ap}(\\log(1+T))=\\sum_{d\\ge0}\\alpha_dT^d,\n \\qquad\n \\boxed{\\alpha_d=0\\ \\text{or}\\ v_p(\\alpha_d)\\ge-\\left\\lfloor d/p\\right\\rfloor.}\n\\]\nEquivalently, every $p^{\\lfloor d/p\\rfloor}\\alpha_d$ belongs to the actual completed ring $\\Z_p$. The zero-safe alternative is explicit in the formal rational statement.","r\\medskip\n\\subsubsection*{Scope and result}\nLet $p$ be prime, let $m\\mid p-1$, and put $N=mp$. For $1\\le a<m$, put $c=ap$ and define the original exponential section and ratio by\n\\[\n H_{N,c}(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\qquad\n R_c(t)=\\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.\n\\]\nThe quotient is interpreted after removing its forced zero. This block proves, for the actual rational coefficients $\\alpha_d$,\n\\[\n R_{ap}(\\log(1+T))=\\sum_{d\\ge0}\\alpha_dT^d,\n \\qquad\n \\boxed{\\alpha_d=0\\ \\text{or}\\ v_p(\\alpha_d)\\ge-\\left\\lfloor d/p\\right\\rfloor.}\n\\]\nEquivalently, every $p^{\\lfloor d/p\\rfloor}\\alpha_d$ belongs to the actual completed ring $\\Z_p$. The zero-safe alternative is explicit in the formal rational statement.","bsubsection*{Scope and result}\nLet $p$ be prime, let $m\\mid p-1$, and put $N=mp$. For $1\\le a<m$, put $c=ap$ and define the original exponential section and ratio by\n\\[\n H_{N,c}(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\qquad\n R_c(t)=\\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.\n\\]\nThe quotient is interpreted after removing its forced zero. This block proves, for the actual rational coefficients $\\alpha_d$,\n\\[\n R_{ap}(\\log(1+T))=\\sum_{d\\ge0}\\alpha_dT^d,\n \\qquad\n \\boxed{\\alpha_d=0\\ \\text{or}\\ v_p(\\alpha_d)\\ge-\\left\\lfloor d/p\\right\\rfloor.}\n\\]\nEquivalently, every $p^{\\lfloor d/p\\rfloor}\\alpha_d$ belongs to the actual completed ring $\\Z_p$. The zero-safe alternative is explicit in the formal rational statement.","ratio by\n\\[\n H_{N,c}(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\qquad\n R_c(t)=\\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.\n\\]\nThe quotient is interpreted after removing its forced zero. This block proves, for the actual rational coefficients $\\alpha_d$,\n\\[\n R_{ap}(\\log(1+T))=\\sum_{d\\ge0}\\alpha_dT^d,\n \\qquad\n \\boxed{\\alpha_d=0\\ \\text{or}\\ v_p(\\alpha_d)\\ge-\\left\\lfloor d/p\\right\\rfloor.}\n\\]\nEquivalently, every $p^{\\lfloor d/p\\rfloor}\\alpha_d$ belongs to the actual completed ring $\\Z_p$. The zero-safe alternative is explicit in the formal rational statement.","_c(t)=\\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.\n\\]\nThe quotient is interpreted after removing its forced zero. This block proves, for the actual rational coefficients $\\alpha_d$,\n\\[\n R_{ap}(\\log(1+T))=\\sum_{d\\ge0}\\alpha_dT^d,\n \\qquad\n \\boxed{\\alpha_d=0\\ \\text{or}\\ v_p(\\alpha_d)\\ge-\\left\\lfloor d/p\\right\\rfloor.}\n\\]\nEquivalently, every $p^{\\lfloor d/p\\rfloor}\\alpha_d$ belongs to the actual completed ring $\\Z_p$. The zero-safe alternative is explicit in the formal rational statement.","eorem assumes primality, the modulus divisibility and the specified residue bounds. It does not assume weighted section membership, reciprocal closure, the ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","sumes primality, the modulus divisibility and the specified residue bounds. It does not assume weighted section membership, reciprocal closure, the ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","y, the modulus divisibility and the specified residue bounds. It does not assume weighted section membership, reciprocal closure, the ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","ified residue bounds. It does not assume weighted section membership, reciprocal closure, the ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","he ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","entity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","escribe the constructed objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","ted objects and proved bridges.\n\\par\\medskip\n\\subsubsection*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","section*{1. Integral cyclic coefficients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","ients}\nWork first in an arbitrary commutative ring $R$. If $z^p=1$, set\n\\[\n I=(p,z-1),\\qquad P=(p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","p).\n\\]\nExpanding $((z-1)+1)^p=1$ and using divisibility of every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","f every interior binomial coefficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","ficient in prime row $p$ gives\n\\[\n (z-1)^p\\in PI.\n\\]\nEvery other degree-$p$ product of the two generators of $I$ also belongs to $PI$. The formal ideal proof therefore establishes\n\\[\n \\boxed{I^p\\subseteq PI.}\n\\]\nFor $q=p\\lfloor q/p\\rfloor+(q\\bmod p)$, strong induction and the factorial valuation identity give\n\\[\n \\boxed{I^q\\subseteq P^{v_p(q!)}}.\n\\]\nIn detail, write $b=\\lfloor q/p\\rfloor$. Then\n\\[\n I^q\\subseteq P^b I^b\\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.\n\\]\nNo integral-domain hypothesis or division in a field is used here.","Now take the integral cyclic group ring\n\\[\n R_N=\\Z[\\Z/N\\Z],\\qquad x=[1],\\qquad x^N=1,\n\\]\nand assume $N\\mid p(p-1)$. Thus $(x^{p-1})^p=1$. Each genuine consecutive block\n\\[\n \\prod_{i=0}^{p-1}\\bigl(x-(pb+i)\\bigr)\n\\]\nbelongs to $(p,x^{p-1}-1)$. This follows from the actual polynomial identity\n\\[\n \\prod_{i\\in\\F_p}(X-i)=X^p-X\n\\]\nmodulo $p$, not from a hypothesized valuation of the product.","For the falling factorial $(x)_d=\\prod_{i=0}^{d-1}(x-i)$, grouping into consecutive blocks gives\n\\[\n (x)_d\\in I^{\\lfloor d/p\\rfloor}\n \\subseteq\\bigl(p^{v_p(\\lfloor d/p\\rfloor!)}\\bigr).\n\\]\nEvery integral residue coordinate therefore satisfies\n\\[\n \\boxed{p^{v_p(\\lfloor d/p\\rfloor!)}\\mid [(r)](x)_d\\quad\\text{in }\\Z.}\n\\]\nDividing a coordinate by $d!$ is done only afterward, using\n$v_p(d!)=\\lfloor d/p\\rfloor+v_p(\\lfloor d/p\\rfloor!)$. This establishes the weighted denominator estimate before any field-valued divisibility could make it vacuous.","\\textbf{Modules.} \\mcode{CyclotomicFiltration}, \\mcode{PrimeBlockPolynomial}, and \\mcode{CyclicAmiceCoefficients}.\n\\par\\medskip\n\\subsubsection*{2. Connection to the actual section}\nAn integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue $r\\in\\Z/N\\Z$, define\n\\[\n H_r(t)=\\sum_{d\\ge0}\\mathbf1_{d\\equiv r\\ (N)}\\frac{t^d}{d!},\n \\qquad F_r(T)=H_r(\\log(1+T)).\n\\]\nThe implementation uses the actual formal logarithm and substitution in $\\Q[[T]]$. It proves\n\\[\n H_r'=H_{r-1},\\qquad (1+T)F_r'=F_{r-1},\n\\]\nand hence the coefficient recurrence\n\\[\n (d+1)\\coef{T}{d+1}F_r\n =\\coef{T}{d}F_{r-1}-d\\coef{T}{d}F_r.\n\\]\nThe initial values are $F_r(0)=\\mathbf1_{r=0}$.","fficients}.\n\\par\\medskip\n\\subsubsection*{2. Connection to the actual section}\nAn integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue $r\\in\\Z/N\\Z$, define\n\\[\n H_r(t)=\\sum_{d\\ge0}\\mathbf1_{d\\equiv r\\ (N)}\\frac{t^d}{d!},\n \\qquad F_r(T)=H_r(\\log(1+T)).\n\\]\nThe implementation uses the actual formal logarithm and substitution in $\\Q[[T]]$. It proves\n\\[\n H_r'=H_{r-1},\\qquad (1+T)F_r'=F_{r-1},\n\\]\nand hence the coefficient recurrence\n\\[\n (d+1)\\coef{T}{d+1}F_r\n =\\coef{T}{d}F_{r-1}-d\\coef{T}{d}F_r.\n\\]\nThe initial values are $F_r(0)=\\mathbf1_{r=0}$.","ip\n\\subsubsection*{2. Connection to the actual section}\nAn integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue $r\\in\\Z/N\\Z$, define\n\\[\n H_r(t)=\\sum_{d\\ge0}\\mathbf1_{d\\equiv r\\ (N)}\\frac{t^d}{d!},\n \\qquad F_r(T)=H_r(\\log(1+T)).\n\\]\nThe implementation uses the actual formal logarithm and substitution in $\\Q[[T]]$. It proves\n\\[\n H_r'=H_{r-1},\\qquad (1+T)F_r'=F_{r-1},\n\\]\nand hence the coefficient recurrence\n\\[\n (d+1)\\coef{T}{d+1}F_r\n =\\coef{T}{d}F_{r-1}-d\\coef{T}{d}F_r.\n\\]\nThe initial values are $F_r(0)=\\mathbf1_{r=0}$.","mate becomes relevant only after connecting it to the manuscript's generating functions. For a residue $r\\in\\Z/N\\Z$, define\n\\[\n H_r(t)=\\sum_{d\\ge0}\\mathbf1_{d\\equiv r\\ (N)}\\frac{t^d}{d!},\n \\qquad F_r(T)=H_r(\\log(1+T)).\n\\]\nThe implementation uses the actual formal logarithm and substitution in $\\Q[[T]]$. It proves\n\\[\n H_r'=H_{r-1},\\qquad (1+T)F_r'=F_{r-1},\n\\]\nand hence the coefficient recurrence\n\\[\n (d+1)\\coef{T}{d+1}F_r\n =\\coef{T}{d}F_{r-1}-d\\coef{T}{d}F_r.\n\\]\nThe initial values are $F_r(0)=\\mathbf1_{r=0}$.","For a residue $r\\in\\Z/N\\Z$, define\n\\[\n H_r(t)=\\sum_{d\\ge0}\\mathbf1_{d\\equiv r\\ (N)}\\frac{t^d}{d!},\n \\qquad F_r(T)=H_r(\\log(1+T)).\n\\]\nThe implementation uses the actual formal logarithm and substitution in $\\Q[[T]]$. It proves\n\\[\n H_r'=H_{r-1},\\qquad (1+T)F_r'=F_{r-1},\n\\]\nand hence the coefficient recurrence\n\\[\n (d+1)\\coef{T}{d+1}F_r\n =\\coef{T}{d}F_{r-1}-d\\coef{T}{d}F_r.\n\\]\nThe initial values are $F_r(0)=\\mathbf1_{r=0}$.","Independently, let $b_{d,r}$ be the residue-$r$ coordinate of $(x)_d$ in $R_N$. Actual group-ring multiplication gives\n\\[\n b_{0,r}=\\mathbf1_{r=0},\\qquad\n b_{d+1,r}=b_{d,r-1}-d\\,b_{d,r}.\n\\]\nInduction comparing these two recurrences proves\n\\[\n \\boxed{d!\\coef{T}{d}H_r(\\log(1+T))=b_{d,r}.}\n\\]\nThus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label $0\\le c<N$, the formal section is also proved equal to the original \\mcode{ExponentialSection.}\\allowbreak\\mcode{originalSection} used in the existing Euler project.","\\subsection*{Weighted membership}\nThe class used in the formalization is\n\\[\n \\Wp=\\left\\{F\\in\\Q_p[[T]]:\n p^{\\lfloor d/p\\rfloor}\\coef{T}{d}F\\in\\Z_p\\ \\text{for every }d\\right\\}.\n\\]\nThe integral-coordinate estimate from page 2 now proves\n\\[\n \\boxed{H_r(\\log(1+T))\\in\\Wp}\n\\]\nfor every residue $r$, including the singular ones. Rational coefficients are mapped into $\\Q_p$ by the actual coefficient ring homomorphism.","The same source package retains the earlier independent formal identity\n\\[\n d!\\coef{T}{d}\\exp(x\\log(1+T))=(x)_d\n\\]\nover a commutative $\\Q$-algebra. The section bridge above does not rely on selecting or averaging roots in a cyclotomic extension.","\\textbf{Principal targets.}\\par \\mcode{factorial_coeff_section}; \\mcode{section_weighted_integral};\\par \\mcode{weighted_section}.\n\\par\\medskip\n\\subsubsection*{3. Weighted closure and the normalized denominator}\nLet $w(d)=\\lfloor d/p\\rfloor$. Since $w(i)+w(d-i)\\le w(d)$, the coefficient formula for a product proves closure of $\\Wp$ under multiplication. Additive closure is immediate. For $U\\in\\Wp$ with $U(0)=1$, strong induction in the actual reciprocal coefficient recurrence proves\n\\[\n \\boxed{U^{-1}\\in\\Wp.}\n\\]\nThis inverse property is a theorem, not an additional field in a structure encoding weighted series.","f{Principal targets.}\\par \\mcode{factorial_coeff_section}; \\mcode{section_weighted_integral};\\par \\mcode{weighted_section}.\n\\par\\medskip\n\\subsubsection*{3. Weighted closure and the normalized denominator}\nLet $w(d)=\\lfloor d/p\\rfloor$. Since $w(i)+w(d-i)\\le w(d)$, the coefficient formula for a product proves closure of $\\Wp$ under multiplication. Additive closure is immediate. For $U\\in\\Wp$ with $U(0)=1$, strong induction in the actual reciprocal coefficient recurrence proves\n\\[\n \\boxed{U^{-1}\\in\\Wp.}\n\\]\nThis inverse property is a theorem, not an additional field in a structure encoding weighted series.","integral};\\par \\mcode{weighted_section}.\n\\par\\medskip\n\\subsubsection*{3. Weighted closure and the normalized denominator}\nLet $w(d)=\\lfloor d/p\\rfloor$. Since $w(i)+w(d-i)\\le w(d)$, the coefficient formula for a product proves closure of $\\Wp$ under multiplication. Additive closure is immediate. For $U\\in\\Wp$ with $U(0)=1$, strong induction in the actual reciprocal coefficient recurrence proves\n\\[\n \\boxed{U^{-1}\\in\\Wp.}\n\\]\nThis inverse property is a theorem, not an additional field in a structure encoding weighted series.","\\subsection*{Differentiation and the Euler field}\nThe derivative coefficient is $(d+1)a_{d+1}$. If $p\\nmid d+1$, the adjacent weights agree. If $p\\mid d+1$, the integer factor $d+1$ compensates for the one-unit weight increase. Therefore ordinary differentiation preserves $\\Wp$.","For the actual logarithm, its coefficient denominator is $d$, and the proved inequality\n\\[\n v_p(d)\\le\\lfloor d/p\\rfloor\\qquad(d>0)\n\\]\nimplies $\\log(1+T)\\in\\Wp$. Consequently\n\\[\n L_TF=\\log(1+T)(1+T)F'\n\\]\nalso preserves $\\Wp$.","\\subsection*{Removing a singular forced zero}\nPut $c=ap<N$ and define the actual normalized kernel\n\\[\n K_c(t)=c!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+c)!},\\qquad K_c(0)=1.\n\\]\nLet $\\ell(T)=\\log(1+T)/T$, defined by shifting logarithm coefficients, not by inverting $T$. Set\n\\[\n U(T)=\\ell(T)^cK_c(\\log(1+T)).\n\\]\nThe formal proof establishes $U(0)=1$ and the exact factorization\n\\[\n H_{N,c}(\\log(1+T))=\\frac{T^c}{c!}U(T).\n\\]\nIt follows coefficientwise that\n\\[\n \\coef{T}{d}U=c!\\coef{T}{d+c}H_{N,c}(\\log(1+T)).\n\\]\nBecause $p^a\\mid(ap)!$ and $\\lfloor(d+ap)/p\\rfloor=a+\\lfloor d/p\\rfloor$, the already proved section estimate gives $U\\in\\Wp$. Thus $U^{-1}\\in\\Wp$ as well. The factorial divisibility is derived from the pinned Legendre valuation theorem.","ut $c=ap<N$ and define the actual normalized kernel\n\\[\n K_c(t)=c!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+c)!},\\qquad K_c(0)=1.\n\\]\nLet $\\ell(T)=\\log(1+T)/T$, defined by shifting logarithm coefficients, not by inverting $T$. Set\n\\[\n U(T)=\\ell(T)^cK_c(\\log(1+T)).\n\\]\nThe formal proof establishes $U(0)=1$ and the exact factorization\n\\[\n H_{N,c}(\\log(1+T))=\\frac{T^c}{c!}U(T).\n\\]\nIt follows coefficientwise that\n\\[\n \\coef{T}{d}U=c!\\coef{T}{d+c}H_{N,c}(\\log(1+T)).\n\\]\nBecause $p^a\\mid(ap)!$ and $\\lfloor(d+ap)/p\\rfloor=a+\\lfloor d/p\\rfloor$, the already proved section estimate gives $U\\in\\Wp$. Thus $U^{-1}\\in\\Wp$ as well. The factorial divisibility is derived from the pinned Legendre valuation theorem.","=c!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+c)!},\\qquad K_c(0)=1.\n\\]\nLet $\\ell(T)=\\log(1+T)/T$, defined by shifting logarithm coefficients, not by inverting $T$. Set\n\\[\n U(T)=\\ell(T)^cK_c(\\log(1+T)).\n\\]\nThe formal proof establishes $U(0)=1$ and the exact factorization\n\\[\n H_{N,c}(\\log(1+T))=\\frac{T^c}{c!}U(T).\n\\]\nIt follows coefficientwise that\n\\[\n \\coef{T}{d}U=c!\\coef{T}{d+c}H_{N,c}(\\log(1+T)).\n\\]\nBecause $p^a\\mid(ap)!$ and $\\lfloor(d+ap)/p\\rfloor=a+\\lfloor d/p\\rfloor$, the already proved section estimate gives $U\\in\\Wp$. Thus $U^{-1}\\in\\Wp$ as well. The factorial divisibility is derived from the pinned Legendre valuation theorem.","c{t^{Nn}}{(Nn+c)!},\\qquad K_c(0)=1.\n\\]\nLet $\\ell(T)=\\log(1+T)/T$, defined by shifting logarithm coefficients, not by inverting $T$. Set\n\\[\n U(T)=\\ell(T)^cK_c(\\log(1+T)).\n\\]\nThe formal proof establishes $U(0)=1$ and the exact factorization\n\\[\n H_{N,c}(\\log(1+T))=\\frac{T^c}{c!}U(T).\n\\]\nIt follows coefficientwise that\n\\[\n \\coef{T}{d}U=c!\\coef{T}{d+c}H_{N,c}(\\log(1+T)).\n\\]\nBecause $p^a\\mid(ap)!$ and $\\lfloor(d+ap)/p\\rfloor=a+\\lfloor d/p\\rfloor$, the already proved section estimate gives $U\\in\\Wp$. Thus $U^{-1}\\in\\Wp$ as well. The factorial divisibility is derived from the pinned Legendre valuation theorem.","arithm coefficients, not by inverting $T$. Set\n\\[\n U(T)=\\ell(T)^cK_c(\\log(1+T)).\n\\]\nThe formal proof establishes $U(0)=1$ and the exact factorization\n\\[\n H_{N,c}(\\log(1+T))=\\frac{T^c}{c!}U(T).\n\\]\nIt follows coefficientwise that\n\\[\n \\coef{T}{d}U=c!\\coef{T}{d+c}H_{N,c}(\\log(1+T)).\n\\]\nBecause $p^a\\mid(ap)!$ and $\\lfloor(d+ap)/p\\rfloor=a+\\lfloor d/p\\rfloor$, the already proved section estimate gives $U\\in\\Wp$. Thus $U^{-1}\\in\\Wp$ as well. The factorial divisibility is derived from the pinned Legendre valuation theorem.","nts, not by inverting $T$. Set\n\\[\n U(T)=\\ell(T)^cK_c(\\log(1+T)).\n\\]\nThe formal proof establishes $U(0)=1$ and the exact factorization\n\\[\n H_{N,c}(\\log(1+T))=\\frac{T^c}{c!}U(T).\n\\]\nIt follows coefficientwise that\n\\[\n \\coef{T}{d}U=c!\\coef{T}{d+c}H_{N,c}(\\log(1+T)).\n\\]\nBecause $p^a\\mid(ap)!$ and $\\lfloor(d+ap)/p\\rfloor=a+\\lfloor d/p\\rfloor$, the already proved section estimate gives $U\\in\\Wp$. Thus $U^{-1}\\in\\Wp$ as well. The factorial divisibility is derived from the pinned Legendre valuation theorem.","\\textbf{Boundary.} The original positive-address section has zero constant coefficient. It is not the series to which unit inverse closure is applied.\n\\par\\medskip\n\\subsubsection*{4. The singular ratio estimate}\nFor a positive address $c$, the rational ratio is independently constructed from the original kernels:\n\\[\n R_c(t)=c\\,K_{c-1}(t)K_c(t)^{-1}.\n\\]\nThe scalar factor $c$ is necessary. The source proves the cleared identity\n\\[\n \\boxed{R_c(t)H_{N,c}(t)=tH_{N,c-1}(t).}\n\\]\nThis connects the constructed object to the original unnormalized ratio $A_{c-1}/A_c$. It does not assign the desired transformed formula as its definition.","positive-address section has zero constant coefficient. It is not the series to which unit inverse closure is applied.\n\\par\\medskip\n\\subsubsection*{4. The singular ratio estimate}\nFor a positive address $c$, the rational ratio is independently constructed from the original kernels:\n\\[\n R_c(t)=c\\,K_{c-1}(t)K_c(t)^{-1}.\n\\]\nThe scalar factor $c$ is necessary. The source proves the cleared identity\n\\[\n \\boxed{R_c(t)H_{N,c}(t)=tH_{N,c-1}(t).}\n\\]\nThis connects the constructed object to the original unnormalized ratio $A_{c-1}/A_c$. It does not assign the desired transformed formula as its definition.","\\par\\medskip\n\\subsubsection*{4. The singular ratio estimate}\nFor a positive address $c$, the rational ratio is independently constructed from the original kernels:\n\\[\n R_c(t)=c\\,K_{c-1}(t)K_c(t)^{-1}.\n\\]\nThe scalar factor $c$ is necessary. The source proves the cleared identity\n\\[\n \\boxed{R_c(t)H_{N,c}(t)=tH_{N,c-1}(t).}\n\\]\nThis connects the constructed object to the original unnormalized ratio $A_{c-1}/A_c$. It does not assign the desired transformed formula as its definition.","After actual logarithmic substitution and the chain rule,\n\\[\n R_c(\\log(1+T))H_{N,c}(\\log(1+T))\n =L_TH_{N,c}(\\log(1+T)).\n\\]\nInsert the proved factorization $H_{N,c}(\\log(1+T))=T^cU/c!$. Leibniz differentiation and cancellation of that nonzero section give\n\\[\n \\boxed{R_c(\\log(1+T))\n =c(1+T)\\ell(T)+(L_TU)U^{-1}.}\n\\]\nAll differentiation, substitution, nonzeroness and cancellation obligations are proved in the source.","For $c=ap$, the second summand belongs to $\\Wp$ by the closure results on page 4. The first does too: shifting the logarithm coefficient loses at most one unit of the floor weight, so $p\\ell(T)\\in\\Wp$, and the integer $a$ supplies the remaining factor. Therefore\n\\[\n \\boxed{R_{ap}(\\log(1+T))\\in\\Wp.}\n\\]\nThe final rational coefficient theorem states the bound with an explicit zero alternative. No finite-valued default valuation at zero is used as the meaning of unlimited precision.","\\subsection*{Checks that prevent a normalization mistake}\nThe source proves that the actual ratio has constant coefficient $c$, not one. In the concrete case $N=6,c=3$, the universal theorem gives weighted membership and the constant coefficient is proved to be $3$.","It also derives $\\coef{T}{3}H_{3,0}(\\log(1+T))=1/6$ from the actual differential recurrence and checks its valuation $v_3(1/6)=-1$. Thus weighted membership does not mean ordinary coefficient integrality. This value attains the allowed floor weight at degree three.","bottomrule\n\\end{tabular}\\end{center}\nFor $p=3,N=6,c=3$, this independent computation obtains the degree-six ratio coefficient $1/10080$, with valuation $-2$, attaining the allowed weight. This particular coefficient calculation is recorded as a finite regression, not as a new Lean theorem. A violating modulus $p=3,N=4$ yields a section coefficient of valuation $-4$ at degree nine, showing that the modulus divisibility hypothesis cannot simply be discarded.","\\clearpage\\subsection{All Residue Completion Supplement}\\label{app:sup10}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_","\\clearpage\\subsection{All Residue Completion Supplement}\\label{app:sup10}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\big","\\clearpage\\subsection{All Residue Completion Supplement}\\label{app:sup10}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,m","\\clearpage\\subsection{All Residue Completion Supplement}\\label{app:sup10}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}","arpage\\subsection{All Residue Completion Supplement}\\label{app:sup10}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe…","Residue Completion Supplement}\\label{app:sup10}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prim…","nt\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal…","ement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal statement also includes zero row, zero order, zero multiplier, and the $p=2,m=1$ subcase. This does …","are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal statement also includes zero row, zero order, zero multiplier, and the $p=2,m=1$ subcase. This does not assert a different or broader external dyadic …","Scope and final statement}\nLet $p$ be a prime, let $m>0$ divide $p-1$, put $N=mp$ and $h=(p-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal statement also includes zero row, zero order, zero multiplier, and the $p=2,m=1$ subcase. This does not assert a different or broader external dyadic conjecture.","-1)/m$, and let $0\\le j<N$. The original rational packet is defined by\n\\[\n \\sum_{q=0}^{n}\\binom{Nn+j}{Nq}C_{N,j}(q)=\\delta_{n0},\n \\qquad E_j(n)=j!\\,C_{N,j}(n).\n\\]\nFor every $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal statement also includes zero row, zero order, zero multiplier, and the $p=2,m=1$ subcase. This does not assert a different or broader external dyadic conjecture.","ery $n,k,u\\in\\N$, the new formal result establishes\n\\[\n \\boxed{E_j(n)\\in\\Z_p,\\qquad\n \\Delta_{uh}^{k}E_j(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nHere $\\Delta_h f(n)=f(n+h)-f(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal statement also includes zero row, zero order, zero multiplier, and the $p=2,m=1$ subcase. This does not assert a different or broader external dyadic conjecture.","(n)$. The corresponding rational valuation statement is zero-safe:\n\\[\n \\Delta_{uh}^{k}E_j(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal statement also includes zero row, zero order, zero multiplier, and the $p=2,m=1$ subcase. This does not assert a different or broader external dyadic conjecture.","v_p\\bigl(\\Delta_{uh}^{k}E_j(n)\\bigr)\\ge\\min(k,mn).\n\\]\nFor the normalized coefficients, the exact correction is retained:\n\\[\n \\boxed{\\Delta_{uh}^{k}C_{N,j}(n)=0\\quad\\text{or}\\quad\n v_p\\bigl(\\Delta_{uh}^{k}C_{N,j}(n)\\bigr)\\ge\\min(k,mn)-v_p(j!).}\n\\]\nThe manuscript's odd-prime, positive-$u$ theorem is a specialization. The formal statement also includes zero row, zero order, zero multiplier, and the $p=2,m=1$ subcase. This does not assert a different or broader external dyadic conjecture.","\\textbf{Principal declaration.} \\path{KomorebiPaperIX.Theorem75.global_finite_difference} states the integrality and both precision bounds together, for the original coefficient functions. Its hypotheses are primality, $m>0$, $m\\mid p-1$ and $j<mp$, not an assumed flatness or weightedness certificate.\n\\par\\medskip\n\\subsubsection*{1. Stability of the weighted differential ring}\nThe recovered checkpoint used the actual weighted subring\n\\[\n \\Wp=\\left\\{F(T)=\\sum_{d\\ge0}a_dT^d:\\\n p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\ \\text{for every }d\\right\\}.\n\\]\nIts coefficient condition is identified with the one in the previously checked singular-Amice development. The failed \\mcode{ThetaWeighted} draft is retained separately. The repaired active source checks the complete formal-series statement below.","morebiPaperIX.Theorem75.global_finite_difference} states the integrality and both precision bounds together, for the original coefficient functions. Its hypotheses are primality, $m>0$, $m\\mid p-1$ and $j<mp$, not an assumed flatness or weightedness certificate.\n\\par\\medskip\n\\subsubsection*{1. Stability of the weighted differential ring}\nThe recovered checkpoint used the actual weighted subring\n\\[\n \\Wp=\\left\\{F(T)=\\sum_{d\\ge0}a_dT^d:\\\n p^{\\lfloor d/p\\rfloor}a_d\\in\\Z_p\\ \\text{for every }d\\right\\}.\n\\]\nIts coefficient condition is identified with the one in the previously checked singular-Amice development. The failed \\mcode{ThetaWeighted} draft is retained separately. The repaired active source checks the complete formal-series statement below.","Put\n\\[\n \\Theta=(1+T)\\frac{d}{dT},\\qquad\n B_\\Theta=\\frac{\\Theta^{p^2}-\\Theta^p}{p}.\n\\]\nThe coefficient formula\n\\[\n [T^d]\\Theta F=(d+1)a_{d+1}+da_d\n\\]\nproves $\\Theta\\Wp\\subseteq\\Wp$. If the floor weight increases between $d$ and $d+1$, then $p\\mid d+1$, which compensates for that increase.","\\textbf{The normalized defect is not assumed to be a derivation.}\nIts proved product rule contains the two middle Leibniz sums:\n\\[\n B_\\Theta(FG)=(B_\\Theta F)G+F(B_\\Theta G)+\\mathcal B_p(F,G).\n\\]\nThe coefficients of $\\mathcal B_p$ are the actual integral quotients\n$\\binom{p^2}{a}/p$ and $\\binom pa/p$. Derivative stability and this correction formula show that the elements $F\\in\\Wp$ satisfying $B_\\Theta F\\in\\Wp$ are closed under the finite ring operations.","The scalar input is proved for every natural $b$, including nonunits:\n\\[\n \\boxed{p^2\\mid b^{p^2}-b^p\\quad\\text{in }\\Z.}\n\\]\nBecause $\\Theta(1+T)^b=b(1+T)^b$, the binomial expansion of $T^p=((1+T)-1)^p$ then proves stability of the divided generator $T^p/p$. For $d=pq+r$, $0\\le r<p$, every weighted monomial has the exact factorization\n\\[\n a_dT^d=(p^qa_d)\\left(\\frac{T^p}{p}\\right)^qT^r.\n\\]\nThe scalar $p^qa_d$ is integral by the actual weight condition. Thus all weighted monomials, and all finite weighted polynomials, satisfy the defect condition.","e scalar input is proved for every natural $b$, including nonunits:\n\\[\n \\boxed{p^2\\mid b^{p^2}-b^p\\quad\\text{in }\\Z.}\n\\]\nBecause $\\Theta(1+T)^b=b(1+T)^b$, the binomial expansion of $T^p=((1+T)-1)^p$ then proves stability of the divided generator $T^p/p$. For $d=pq+r$, $0\\le r<p$, every weighted monomial has the exact factorization\n\\[\n a_dT^d=(p^qa_d)\\left(\\frac{T^p}{p}\\right)^qT^r.\n\\]\nThe scalar $p^qa_d$ is integral by the actual weight condition. Thus all weighted monomials, and all finite weighted polynomials, satisfy the defect condition.","Finally, $[T^d]B_\\Theta F$ depends only on coefficients through $d+p^2$. Replacing $F$ by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption:\n\\[\n \\boxed{B_\\Theta\\Wp\\subseteq\\Wp,\\qquad B_\\Theta^s\\Wp\\subseteq\\Wp\\ (s\\ge0).}\n\\]\n\\textbf{Modules.} \\mcode{WeightedAmiceRing}, \\mcode{FrobeniusSquare}, \\mcode{ThetaWeighted}.\n\\par\\medskip\n\\subsubsection*{2. Actual substitution and singular higher differences}\nLet\n\\[\n \\mathcal E(F)(t)=F(e^t-1),\\qquad D=\\frac{d}{dt},\\qquad\n B_D=\\frac{D^{p^2}-D^p}{p}.\n\\]\nAll substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives\n\\[\n D\\mathcal E(F)=\\mathcal E(\\Theta F),\\qquad\n B_D^s\\mat","Finally, $[T^d]B_\\Theta F$ depends only on coefficients through $d+p^2$. Replacing $F$ by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption:\n\\[\n \\boxed{B_\\Theta\\Wp\\subseteq\\Wp,\\qquad B_\\Theta^s\\Wp\\subseteq\\Wp\\ (s\\ge0).}\n\\]\n\\textbf{Modules.} \\mcode{WeightedAmiceRing}, \\mcode{FrobeniusSquare}, \\mcode{ThetaWeighted}.\n\\par\\medskip\n\\subsubsection*{2. Actual substitution and singular higher differences}\nLet\n\\[\n \\mathcal E(F)(t)=F(e^t-1),\\qquad D=\\frac{d}{dt},\\qquad\n B_D=\\frac{D^{p^2}-D^p}{p}.\n\\]\nAll substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives\n\\[\n D\\mathcal E(F)=\\mathcal E(\\Theta F),\\qquad\n B_D^s\\mathcal E(F)=\\mathcal E(B_\\Theta^sF).\n\\]\nThe imp","Finally, $[T^d]B_\\Theta F$ depends only on coefficients through $d+p^2$. Replacing $F$ by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption:\n\\[\n \\boxed{B_\\Theta\\Wp\\subseteq\\Wp,\\qquad B_\\Theta^s\\Wp\\subseteq\\Wp\\ (s\\ge0).}\n\\]\n\\textbf{Modules.} \\mcode{WeightedAmiceRing}, \\mcode{FrobeniusSquare}, \\mcode{ThetaWeighted}.\n\\par\\medskip\n\\subsubsection*{2. Actual substitution and singular higher differences}\nLet\n\\[\n \\mathcal E(F)(t)=F(e^t-1),\\qquad D=\\frac{d}{dt},\\qquad\n B_D=\\frac{D^{p^2}-D^p}{p}.\n\\]\nAll substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives\n\\[\n D\\mathcal E(F)=\\mathcal E(\\Theta F),\\qquad\n B_D^s\\mathcal E(F)=\\mathcal E(B_\\Theta^sF).\n\\]\nThe implementation als","Finally, $[T^d]B_\\Theta F$ depends only on coefficients through $d+p^2$. Replacing $F$ by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption:\n\\[\n \\boxed{B_\\Theta\\Wp\\subseteq\\Wp,\\qquad B_\\Theta^s\\Wp\\subseteq\\Wp\\ (s\\ge0).}\n\\]\n\\textbf{Modules.} \\mcode{WeightedAmiceRing}, \\mcode{FrobeniusSquare}, \\mcode{ThetaWeighted}.\n\\par\\medskip\n\\subsubsection*{2. Actual substitution and singular higher differences}\nLet\n\\[\n \\mathcal E(F)(t)=F(e^t-1),\\qquad D=\\frac{d}{dt},\\qquad\n B_D=\\frac{D^{p^2}-D^p}{p}.\n\\]\nAll substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives\n\\[\n D\\mathcal E(F)=\\mathcal E(\\Theta F),\\qquad\n B_D^s\\mathcal E(F)=\\mathcal E(B_\\Theta^sF).\n\\]\nThe implementation also proves the inverse-substitution identity\n\\[\n \\mathcal E\\bigl(f(\\log(1+T))\\bigr)=f(t).\n\\]","p^2$. Replacing $F$ by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption:\n\\[\n \\boxed{B_\\Theta\\Wp\\subseteq\\Wp,\\qquad B_\\Theta^s\\Wp\\subseteq\\Wp\\ (s\\ge0).}\n\\]\n\\textbf{Modules.} \\mcode{WeightedAmiceRing}, \\mcode{FrobeniusSquare}, \\mcode{ThetaWeighted}.\n\\par\\medskip\n\\subsubsection*{2. Actual substitution and singular higher differences}\nLet\n\\[\n \\mathcal E(F)(t)=F(e^t-1),\\qquad D=\\frac{d}{dt},\\qquad\n B_D=\\frac{D^{p^2}-D^p}{p}.\n\\]\nAll substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives\n\\[\n D\\mathcal E(F)=\\mathcal E(\\Theta F),\\qquad\n B_D^s\\mathcal E(F)=\\mathcal E(B_\\Theta^sF).\n\\]\nThe implementation also proves the inverse-substitution identity\n\\[\n \\mathcal E\\bigl(f(\\log(1+T))\\bigr)=f(t).\n\\]","(s\\ge0).}\n\\]\n\\textbf{Modules.} \\mcode{WeightedAmiceRing}, \\mcode{FrobeniusSquare}, \\mcode{ThetaWeighted}.\n\\par\\medskip\n\\subsubsection*{2. Actual substitution and singular higher differences}\nLet\n\\[\n \\mathcal E(F)(t)=F(e^t-1),\\qquad D=\\frac{d}{dt},\\qquad\n B_D=\\frac{D^{p^2}-D^p}{p}.\n\\]\nAll substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives\n\\[\n D\\mathcal E(F)=\\mathcal E(\\Theta F),\\qquad\n B_D^s\\mathcal E(F)=\\mathcal E(B_\\Theta^sF).\n\\]\nThe implementation also proves the inverse-substitution identity\n\\[\n \\mathcal E\\bigl(f(\\log(1+T))\\bigr)=f(t).\n\\]","d singular higher differences}\nLet\n\\[\n \\mathcal E(F)(t)=F(e^t-1),\\qquad D=\\frac{d}{dt},\\qquad\n B_D=\\frac{D^{p^2}-D^p}{p}.\n\\]\nAll substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives\n\\[\n D\\mathcal E(F)=\\mathcal E(\\Theta F),\\qquad\n B_D^s\\mathcal E(F)=\\mathcal E(B_\\Theta^sF).\n\\]\nThe implementation also proves the inverse-substitution identity\n\\[\n \\mathcal E\\bigl(f(\\log(1+T))\\bigr)=f(t).\n\\]","The relevant integral subring is the \\emph{Hurwitz} subring\n\\[\n \\Hp=\\{f\\in\\Q_p[[t]]:\\ n![t^n]f\\in\\Z_p\\ \\text{for every }n\\}.\n\\]\nIt is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is\n\\[\n n![t^n]\\mathcal E(F)=[T^0]\\Theta^nF.\n\\]\nSince every iterate of $\\Theta$ preserves $\\Wp$, the right side is integral. Therefore $\\mathcal E(\\Wp)\\subseteq\\Hp$. Applying the same statement to every $B_\\Theta^sF$ proves\n\\[\n \\boxed{f(\\log(1+T))\\in\\Wp\\quad\\Longrightarrow\\quad\n B_D^sf\\in\\Hp\\text{ for every }s.}\n\\]\nThis is the proved transfer that was missing in the earlier attempt. No weighted Mahler pairing is assumed. The formal differential route supplies an alternative bridge.","s the \\emph{Hurwitz} subring\n\\[\n \\Hp=\\{f\\in\\Q_p[[t]]:\\ n![t^n]f\\in\\Z_p\\ \\text{for every }n\\}.\n\\]\nIt is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is\n\\[\n n![t^n]\\mathcal E(F)=[T^0]\\Theta^nF.\n\\]\nSince every iterate of $\\Theta$ preserves $\\Wp$, the right side is integral. Therefore $\\mathcal E(\\Wp)\\subseteq\\Hp$. Applying the same statement to every $B_\\Theta^sF$ proves\n\\[\n \\boxed{f(\\log(1+T))\\in\\Wp\\quad\\Longrightarrow\\quad\n B_D^sf\\in\\Hp\\text{ for every }s.}\n\\]\nThis is the proved transfer that was missing in the earlier attempt. No weighted Mahler pairing is assumed. The formal differential route supplies an alternative bridge.","ubring\n\\[\n \\Hp=\\{f\\in\\Q_p[[t]]:\\ n![t^n]f\\in\\Z_p\\ \\text{for every }n\\}.\n\\]\nIt is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is\n\\[\n n![t^n]\\mathcal E(F)=[T^0]\\Theta^nF.\n\\]\nSince every iterate of $\\Theta$ preserves $\\Wp$, the right side is integral. Therefore $\\mathcal E(\\Wp)\\subseteq\\Hp$. Applying the same statement to every $B_\\Theta^sF$ proves\n\\[\n \\boxed{f(\\log(1+T))\\in\\Wp\\quad\\Longrightarrow\\quad\n B_D^sf\\in\\Hp\\text{ for every }s.}\n\\]\nThis is the proved transfer that was missing in the earlier attempt. No weighted Mahler pairing is assumed. The formal differential route supplies an alternative bridge.","\\subsection*{Application to the actual singular ratios}\nThe preceding v0.9.0 source independently constructs\n\\[\n H_{N,c}(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\quad\n A_c(t)=t^{-c}H_{N,c}(t),\\quad R_c=\\frac{A_{c-1}}{A_c}.\n\\]\nThe notation $t^{-c}H_{N,c}$ means removing its proved initial zero coefficients. For each $c=ap$, $1\\le a<m$, that source proves\n$R_{ap}(\\log(1+T))\\in\\Wp$. The new transfer consequently proves flatness of the \\emph{same} rational ratio after scalar extension to $\\Q_p$.","ndently constructs\n\\[\n H_{N,c}(t)=\\sum_{n\\ge0}\\frac{t^{Nn+c}}{(Nn+c)!},\\quad\n A_c(t)=t^{-c}H_{N,c}(t),\\quad R_c=\\frac{A_{c-1}}{A_c}.\n\\]\nThe notation $t^{-c}H_{N,c}$ means removing its proved initial zero coefficients. For each $c=ap$, $1\\le a<m$, that source proves\n$R_{ap}(\\log(1+T))\\in\\Wp$. The new transfer consequently proves flatness of the \\emph{same} rational ratio after scalar extension to $\\Q_p$.","The actual ratio is also proved to be an expansion in $t^N$, so its coefficients vanish outside multiples of $N$. Constructed $\\Z_p$ lifts and the established coefficient criterion now give, for its literal Hurwitz coefficients $r_a(n)$,\n\\[\n \\boxed{\\Delta_{uh}^{k}r_a(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nThe separate rational precision theorem is obtained through the coefficient-lift equality. Divisibility in the field $\\Q_p$ is never substituted for this assertion.","is also proved to be an expansion in $t^N$, so its coefficients vanish outside multiples of $N$. Constructed $\\Z_p$ lifts and the established coefficient criterion now give, for its literal Hurwitz coefficients $r_a(n)$,\n\\[\n \\boxed{\\Delta_{uh}^{k}r_a(n)\\in p^{\\min(k,mn)}\\Z_p.}\n\\]\nThe separate rational precision theorem is obtained through the coefficient-lift equality. Divisibility in the field $\\Q_p$ is never substituted for this assertion.","\\textbf{Modules.} \\mcode{WeightedDworkTransfer}, \\mcode{SeriesDifferenceBridge}, \\mcode{RatioSeriesSupport}.\n\\par\\medskip\n\\subsubsection*{3. Nonsingular propagation without an assumed limit}\nFor consecutive original sections, the already defined ratios satisfy\n\\[\n R_cH_{N,c}=tH_{N,c-1},\\qquad H_{N,c}'=H_{N,c-1}.\n\\]\nDifferentiating the first relation, using the relation for $R_{c-1}$, and cancelling a section whose leading coefficient is proved nonzero gives the actual Riccati equation\n\\[\n \\boxed{tR_c'=R_c(1+R_{c-1}-R_c),\\qquad 2\\le c<N.}\n\\]\nIt is not an equation imposed on a replacement ratio. Write\n\\[\n q=R_{c-1}-(c-1),\\qquad y=R_c-c.\n\\]\nBoth have zero constant coefficient. With $\\Lambda_c=tD+c$, the equation becomes\n\\[\n \\boxed{\\Lambda_c y=cq+yq-y^2.}\n\\]","oSeriesSupport}.\n\\par\\medskip\n\\subsubsection*{3. Nonsingular propagation without an assumed limit}\nFor consecutive original sections, the already defined ratios satisfy\n\\[\n R_cH_{N,c}=tH_{N,c-1},\\qquad H_{N,c}'=H_{N,c-1}.\n\\]\nDifferentiating the first relation, using the relation for $R_{c-1}$, and cancelling a section whose leading coefficient is proved nonzero gives the actual Riccati equation\n\\[\n \\boxed{tR_c'=R_c(1+R_{c-1}-R_c),\\qquad 2\\le c<N.}\n\\]\nIt is not an equation imposed on a replacement ratio. Write\n\\[\n q=R_{c-1}-(c-1),\\qquad y=R_c-c.\n\\]\nBoth have zero constant coefficient. With $\\Lambda_c=tD+c$, the equation becomes\n\\[\n \\boxed{\\Lambda_c y=cq+yq-y^2.}\n\\]","iginal sections, the already defined ratios satisfy\n\\[\n R_cH_{N,c}=tH_{N,c-1},\\qquad H_{N,c}'=H_{N,c-1}.\n\\]\nDifferentiating the first relation, using the relation for $R_{c-1}$, and cancelling a section whose leading coefficient is proved nonzero gives the actual Riccati equation\n\\[\n \\boxed{tR_c'=R_c(1+R_{c-1}-R_c),\\qquad 2\\le c<N.}\n\\]\nIt is not an equation imposed on a replacement ratio. Write\n\\[\n q=R_{c-1}-(c-1),\\qquad y=R_c-c.\n\\]\nBoth have zero constant coefficient. With $\\Lambda_c=tD+c$, the equation becomes\n\\[\n \\boxed{\\Lambda_c y=cq+yq-y^2.}\n\\]","quad H_{N,c}'=H_{N,c-1}.\n\\]\nDifferentiating the first relation, using the relation for $R_{c-1}$, and cancelling a section whose leading coefficient is proved nonzero gives the actual Riccati equation\n\\[\n \\boxed{tR_c'=R_c(1+R_{c-1}-R_c),\\qquad 2\\le c<N.}\n\\]\nIt is not an equation imposed on a replacement ratio. Write\n\\[\n q=R_{c-1}-(c-1),\\qquad y=R_c-c.\n\\]\nBoth have zero constant coefficient. With $\\Lambda_c=tD+c$, the equation becomes\n\\[\n \\boxed{\\Lambda_c y=cq+yq-y^2.}\n\\]","ing the first relation, using the relation for $R_{c-1}$, and cancelling a section whose leading coefficient is proved nonzero gives the actual Riccati equation\n\\[\n \\boxed{tR_c'=R_c(1+R_{c-1}-R_c),\\qquad 2\\le c<N.}\n\\]\nIt is not an equation imposed on a replacement ratio. Write\n\\[\n q=R_{c-1}-(c-1),\\qquad y=R_c-c.\n\\]\nBoth have zero constant coefficient. With $\\Lambda_c=tD+c$, the equation becomes\n\\[\n \\boxed{\\Lambda_c y=cq+yq-y^2.}\n\\]","When $p\\nmid c$, define the actual inverse coefficientwise:\n\\[\n [t^d]\\Lambda_c^{-1}f=\\frac{[t^d]f}{d+c}.\n\\]\nOn $p$-supported degrees, $d+c$ is a genuine $p$-adic unit. The proof establishes Hurwitz integrality of this solution, its support, both inverse identities, and preservation of full flatness using the earlier all-order Euler commutator theorem. The singular pivots $c=ap$ are never declared to be units.","Now define explicit iterations\n\\[\n y_0=0,\\qquad y_{r+1}=\\Lambda_c^{-1}(cq+y_rq-y_r^2).\n\\]\nIf $q$ is flat and $p$-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem:\n\\[\n \\boxed{[t^d]y_r=[t^d]y\\qquad(d\\le r).}\n\\]\nIndeed,\n\\[\n (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y).\n\\]\nThe second factor has zero constant term. Multiplication therefore raises the established agreement prefix by one. Coefficientwise division by $d+c$ preserves that prefix.","_rq-y_r^2).\n\\]\nIf $q$ is flat and $p$-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem:\n\\[\n \\boxed{[t^d]y_r=[t^d]y\\qquad(d\\le r).}\n\\]\nIndeed,\n\\[\n (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y).\n\\]\nThe second factor has zero constant term. Multiplication therefore raises the established agreement prefix by one. Coefficientwise division by $d+c$ preserves that prefix.","For fixed $s,d$, the coefficient $d![t^d]B_D^sy$ depends only on the coefficients of $y$ through $d+p^2s$. Choosing the matching finite iterate proves its integrality. This establishes flatness of the actual $y$, and hence of $R_c$, without postulating flatness of the solution or invoking an unproved analytic fixed-point theorem.","\\textbf{Modules.} \\mcode{EulerSeriesInverse}, \\mcode{RiccatiFlatPropagation}, \\mcode{RatioRiccati}.\n\\par\\medskip\n\\subsubsection*{4. All-residue assembly and original coefficients}\nThe standard-window source and inverse results supply $R_1$ and $F_{N,0}$. Induction over $1\\le c<N$ now has two proved cases. At $p\\mid c$, the singular weighted estimate and the new substitution theorem give flatness. At $p\\nmid c$, the preceding Riccati propagation applies. Thus every actual ratio $R_c$ is flat.","tbf{Modules.} \\mcode{EulerSeriesInverse}, \\mcode{RiccatiFlatPropagation}, \\mcode{RatioRiccati}.\n\\par\\medskip\n\\subsubsection*{4. All-residue assembly and original coefficients}\nThe standard-window source and inverse results supply $R_1$ and $F_{N,0}$. Induction over $1\\le c<N$ now has two proved cases. At $p\\mid c$, the singular weighted estimate and the new substitution theorem give flatness. At $p\\nmid c$, the preceding Riccati propagation applies. Thus every actual ratio $R_c$ is flat.","e{RatioRiccati}.\n\\par\\medskip\n\\subsubsection*{4. All-residue assembly and original coefficients}\nThe standard-window source and inverse results supply $R_1$ and $F_{N,0}$. Induction over $1\\le c<N$ now has two proved cases. At $p\\mid c$, the singular weighted estimate and the new substitution theorem give flatness. At $p\\nmid c$, the preceding Riccati propagation applies. Thus every actual ratio $R_c$ is flat.","Let $K_j(t)$ be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is\n\\[\n F_{N,j}(t)=j!\\,K_j(t)^{-1}.\n\\]\nThe source/inverse identities and the factorial successor identity prove\n\\[\n F_{N,j+1}=F_{N,j}R_{j+1},\\qquad\n F_{N,j}=F_{N,0}\\prod_{c=1}^{j}R_c.\n\\]\nThe product formula is not an assembly hypothesis. Proved flatness of all factors and the original product-closure theorem establish flatness of the full family.","Coefficient extraction identifies, literally,\n\\[\n (Nn)![t^{Nn}]F_{N,j}=j!\\,C_{N,j}(n).\n\\]\nThe support outside multiples of $N$ is proved separately. The constructed integral lift therefore represents the original rational coefficient function. The existing saturation criterion and the proved step-multiple argument give the ideal bound at every order, row and multiplier. Multiplication by $j!$ commutes with finite differences, giving precisely the displayed normalization loss.","\\subsection*{Exact examples outside the standard window}\nFor $(p,m,j)=(3,2,3)$, the source derives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\quad C_{6,3}(2)=\\frac{703}{5460},\\quad\n C_{6,3}(3)=-\\frac{223199}{7980}.\n\\]\nConsequently,\n\\[\n \\Delta_1^2\\bigl(3!C_{6,3}\\bigr)(1)=-\\frac{1464768}{8645},\\qquad\n v_3\\!\\left(-\\frac{1464768}{8645}\\right)=2.\n\\]\nThis attains $\\min(2,2\\cdot1)=2$. At the regular address $j=4$ beyond the singular boundary, the source likewise proves\n\\[\n \\Delta_1^2\\bigl(4!C_{6,4}\\bigr)(1)=-\\frac{102745512}{2377375},\n\\]\nand applies the universal exponent-two bound to this actual coefficient function.","w}\nFor $(p,m,j)=(3,2,3)$, the source derives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\quad C_{6,3}(2)=\\frac{703}{5460},\\quad\n C_{6,3}(3)=-\\frac{223199}{7980}.\n\\]\nConsequently,\n\\[\n \\Delta_1^2\\bigl(3!C_{6,3}\\bigr)(1)=-\\frac{1464768}{8645},\\qquad\n v_3\\!\\left(-\\frac{1464768}{8645}\\right)=2.\n\\]\nThis attains $\\min(2,2\\cdot1)=2$. At the regular address $j=4$ beyond the singular boundary, the source likewise proves\n\\[\n \\Delta_1^2\\bigl(4!C_{6,4}\\bigr)(1)=-\\frac{102745512}{2377375},\n\\]\nand applies the universal exponent-two bound to this actual coefficient function.","The earlier no-loss counterexample is retained: $v_3(C_{6,3}(2)-C_{6,3}(1))=0$. Proving the corrected theorem does not make that false normalized claim true.","\\subsection*{Independent finite checks}\nThe standard-library Python mirror checks 328 complete residue families for primes $2,3,5,7,11$, including 233 higher-residue families. It checks 4,264 integral unnormalized coefficients, 22,137 normalized and 22,137 unnormalized difference bounds, and 1,640 comparisons with a separately computed ordinary reciprocal. Further checks cover 108 weighted differential iterations, 205 integer Frobenius-square cases and 180 exact Picard agreement prefixes.","\\clearpage\\subsection{Nishibuchi conjecture completion}\\label{app:sup11}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{1. The complete external statement}\nFor $N>0$ and $0\\le j<N$, use the original unnormalized convention\n\\[\n \\sum_{d\\ge0}\\EE_d^{(N,j)}\\frac{t^d}{d!}\n =\\left(\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!}\\right)^{-1}.\n\\]\nThe existing packet convention satisfies\n$\\EE_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The formalization defines the degree-indexed coefficients through the actual factorial-scaled series coefficients, proves their packet identification, and checks the cleared generating identity with the original exponential section.","{Nishibuchi conjecture completion}\\label{app:sup11}\n\\noindent\\textbf{Checkpoint scope.} This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{1. The complete external statement}\nFor $N>0$ and $0\\le j<N$, use the original unnormalized convention\n\\[\n \\sum_{d\\ge0}\\EE_d^{(N,j)}\\frac{t^d}{d!}\n =\\left(\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!}\\right)^{-1}.\n\\]\nThe existing packet convention satisfies\n$\\EE_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The formalization defines the degree-indexed coefficients through the actual factorial-scaled series coefficients, proves their packet identification, and checks the cleared generating identity with the original exponential section.","ed original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution.\\par\\medskip\n\\subsubsection*{1. The complete external statement}\nFor $N>0$ and $0\\le j<N$, use the original unnormalized convention\n\\[\n \\sum_{d\\ge0}\\EE_d^{(N,j)}\\frac{t^d}{d!}\n =\\left(\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!}\\right)^{-1}.\n\\]\nThe existing packet convention satisfies\n$\\EE_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The formalization defines the degree-indexed coefficients through the actual factorial-scaled series coefficients, proves their packet identification, and checks the cleared generating identity with the original exponential section.","\\begin{theorem}\nLet $p$ be prime, $m>0$, and assume $m\\mid p-1$ or $m=2$. Put $N=mp$ and $q=\\operatorname{lcm}(2,p-1)$. Every $\\EE_{Nn}^{(N,j)}$ is $p$-adically integral. For every $r\\ge1$,\n\\[\n\\boxed{\\EE_{Nn+qp^r}^{(N,j)}\\equiv\\EE_{Nn}^{(N,j)}\\pmod{p^r}\n       \\qquad(0\\le j<N,\\ n\\ge r).}\n\\]\nIn particular, the positive sufficient threshold $n_0=r$ works uniformly in all allowed parameters.\n\\end{theorem}","This is the parameter range and displacement of Nishibuchi's Conjecture 3.9 in arXiv:2605.10677v2. The explicit threshold also establishes the suggested bound $n_0\\le r$. The original paper already proves the $p=2,m=2,j=0$ period as a special case, with all starting rows. This extension does not claim that special case as a new discovery.","The theorem type includes only the stated arithmetic hypotheses. There is no assumed congruence, weighted estimate, or flatness certificate. Precision is the zero-safe predicate $x=0$ or $v_p(x)\\ge r$. Its connection to genuine ideal divisibility in $\\Z_p$ is used in the proof.","\\par\\medskip\n\\subsubsection*{2. The missing dyadic branch}\nFor odd primes, the extra option $m=2$ already divides $p-1$. Therefore the only allowed family outside the preceding all-residue theorem is $p=2,m=2$, with $N=4$.","The new proof establishes a higher-difference estimate for every one of its four addresses:\n\\[\n\\boxed{v_2\\!\\left(\\Delta_u^k C_{4,j}(n)\\right)\\ge\\min(k,n)\n       \\qquad(0\\le j<4).}\n\\]\nAll natural $u,k,n$ are allowed, including zero step, zero order and zero differences. The same bound holds for $j!C_{4,j}(n)$. Both a genuine $\\Z_2$ divisibility theorem and a valuation theorem for the original rational sequence are included.","\\subsection*{An explicit source instead of an assumed regularity condition}\nLet $g_j(n)=\\binom{4n+j}{4n}^{-1}$. Its four formulas are\n\\[\n\\begin{aligned}\n g_0(n)&=1,","\\subsection*{An explicit source instead of an assumed regularity condition}\nLet $g_j(n)=\\binom{4n+j}{4n}^{-1}$. Its four formulas are\n\\[\n\\begin{aligned}\n g_0(n)&=1,\\\\\n g_1(n)&=(4n+1)^{-1},\\\\\n g_2(n)&=((4n+1)(2n+1))^{-1},\\\\\n g_3(n)&=3((4n+1)(2n+1)(4n+3))^{-1}.\n\\end{aligned}\n\\]\nEvery denominator is odd. The code constructs the actual unit inverses in $\\Z_2$ and proves these identities against the original binomial coefficients, including $n=0$.","g_3(n)&=3((4n+1)(2n+1)(4n+3))^{-1}.\n\\end{aligned}\n\\]\nEvery denominator is odd. The code constructs the actual unit inverses in $\\Z_2$ and proves these identities against the original binomial coefficients, including $n=0$.","Define a Kummer sequence over a commutative ring by\n\\[\n\\PP_a(f)\\quad\\Longleftrightarrow\\quad\n a^k\\mid\\Delta_1^k f(n)\\ \\text{for all }k,n.\n\\]\nThe exact discrete product rule\n\\[\n\\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n     \\Delta_1^i f(n)\\,\\Delta_1^{s-i}g(n+i)\n\\]\nis proved by induction. It proves closure under products. Applied to $fg=1$, the same formula isolates\n$\\Delta_1^s g(n)f(n+s)$. Induction and multiplication by $g(n+s)$ prove closure under actual pointwise inverses.","e over a commutative ring by\n\\[\n\\PP_a(f)\\quad\\Longleftrightarrow\\quad\n a^k\\mid\\Delta_1^k f(n)\\ \\text{for all }k,n.\n\\]\nThe exact discrete product rule\n\\[\n\\Delta_1^s(fg)(n)=\\sum_{i=0}^s\\binom si\n     \\Delta_1^i f(n)\\,\\Delta_1^{s-i}g(n+i)\n\\]\nis proved by induction. It proves closure under products. Applied to $fg=1$, the same formula isolates\n$\\Delta_1^s g(n)f(n+s)$. Induction and multiplication by $g(n+s)$ prove closure under actual pointwise inverses.","An affine sequence whose slope is divisible by $a$ satisfies $\\PP_a$. Consequently the four displayed source sequences satisfy $\\PP_2$. No numerical interpolation or finite set of samples is used in that deduction.","\\par\\medskip\n\\subsubsection*{3. The dyadic differential operator and reciprocity}\nThe operator required here is\n\\[\n\\boxed{B_4=\\frac{D^8-D^4}{2},\\qquad D=\\frac{d}{dt}.}\n\\]\nIt is not the preceding $p=2$ specialization $(D^4-D^2)/2$. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are $\\binom 8i/2$ and $\\binom 4i/2$.","dskip\n\\subsubsection*{3. The dyadic differential operator and reciprocity}\nThe operator required here is\n\\[\n\\boxed{B_4=\\frac{D^8-D^4}{2},\\qquad D=\\frac{d}{dt}.}\n\\]\nIt is not the preceding $p=2$ specialization $(D^4-D^2)/2$. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are $\\binom 8i/2$ and $\\binom 4i/2$.","Write $\\mathcal H_{\\mathrm{int}}$ for the subring of $\\Q_2[[t]]$ whose factorial-scaled coefficients belong to $\\Z_2$. Flatness means $B_4^sF\\in\\mathcal H_{\\mathrm{int}}$ for every $s$. The proof of multiplicative and reciprocal closure is induction on $s$, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when $F(0)=1$.","Pack $g_j$ into degrees divisible by four and form its actual EGF. The formal coefficient identity is\n\\[\nG_j(t)=\\sum_{n\\ge0}g_j(n)\\frac{t^{4n}}{(4n)!}\n       =j!\\sum_{n\\ge0}\\frac{t^{4n}}{(4n+j)!}.\n\\]\nIts inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of $G_j$. The proved reciprocal closure gives flatness of $G_j^{-1}$.","For a packed coefficient sequence $e$, the exact extraction formula is\n\\[\n\\bigl(U^8-U^4\\bigr)^k\\operatorname{pack}_4(e)(4n-4k)\n =\\Delta_1^k e(n),\\qquad k\\le n.\n\\]\nHere $U$ is the coefficient shift. The nonnegative extraction index is why the first estimate covers $k\\le n$. The existing saturation lemma extends it to all $k$ with exponent $\\min(k,n)$, and the step-multiple lemma extends step one to every natural $u$.","fficient sequence $e$, the exact extraction formula is\n\\[\n\\bigl(U^8-U^4\\bigr)^k\\operatorname{pack}_4(e)(4n-4k)\n =\\Delta_1^k e(n),\\qquad k\\le n.\n\\]\nHere $U$ is the coefficient shift. The nonnegative extraction index is why the first estimate covers $k\\le n$. The existing saturation lemma extends it to all $k$ with exponent $\\min(k,n)$, and the step-multiple lemma extends step one to every natural $u$.","\\subsection*{Identity with the original rational coefficients}\nThe integral lift is constructed from the decoded coefficients of the actual reciprocal. The code proves\n\\[\n \\operatorname{coe}(e_j(n))=C_{4,j}(n)\\quad\\text{in }\\Q_2,\n\\]\nand proves that the coefficient map commutes with every difference stencil. Multiplying by $j!$ yields the unnormalized family. The valuation result is not divisibility in a field.","\\par\\medskip\n\\subsubsection*{4. From differences to the exact external period}\nLet $f$ take values in a commutative ring and suppose\n\\[\n p^{\\min(k,r)}\\mid\\Delta_h^k f(n)\\qquad\\text{for every }k.\n\\]\nThe proved finite Newton expansion is\n\\[\n f(n+p^{r-1}h)-f(n)=\n \\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nFor $k\\ge r$, the difference already supplies $p^r$. For $1\\le k<r$, the code proves the integer divisibility\n\\[\n p^r\\mid\\binom{p^{r-1}}k p^k.\n\\]\nIt follows from $k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}$ and $v_p(k)<k$. Thus every summand is divisible by $p^r$, including the endpoint $r=1$.","4. From differences to the exact external period}\nLet $f$ take values in a commutative ring and suppose\n\\[\n p^{\\min(k,r)}\\mid\\Delta_h^k f(n)\\qquad\\text{for every }k.\n\\]\nThe proved finite Newton expansion is\n\\[\n f(n+p^{r-1}h)-f(n)=\n \\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nFor $k\\ge r$, the difference already supplies $p^r$. For $1\\le k<r$, the code proves the integer divisibility\n\\[\n p^r\\mid\\binom{p^{r-1}}k p^k.\n\\]\nIt follows from $k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}$ and $v_p(k)<k$. Thus every summand is divisible by $p^r$, including the endpoint $r=1$.","et $f$ take values in a commutative ring and suppose\n\\[\n p^{\\min(k,r)}\\mid\\Delta_h^k f(n)\\qquad\\text{for every }k.\n\\]\nThe proved finite Newton expansion is\n\\[\n f(n+p^{r-1}h)-f(n)=\n \\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nFor $k\\ge r$, the difference already supplies $p^r$. For $1\\le k<r$, the code proves the integer divisibility\n\\[\n p^r\\mid\\binom{p^{r-1}}k p^k.\n\\]\nIt follows from $k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}$ and $v_p(k)<k$. Thus every summand is divisible by $p^r$, including the endpoint $r=1$.","[\n p^{\\min(k,r)}\\mid\\Delta_h^k f(n)\\qquad\\text{for every }k.\n\\]\nThe proved finite Newton expansion is\n\\[\n f(n+p^{r-1}h)-f(n)=\n \\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nFor $k\\ge r$, the difference already supplies $p^r$. For $1\\le k<r$, the code proves the integer divisibility\n\\[\n p^r\\mid\\binom{p^{r-1}}k p^k.\n\\]\nIt follows from $k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}$ and $v_p(k)<k$. Thus every summand is divisible by $p^r$, including the endpoint $r=1$.","for every }k.\n\\]\nThe proved finite Newton expansion is\n\\[\n f(n+p^{r-1}h)-f(n)=\n \\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nFor $k\\ge r$, the difference already supplies $p^r$. For $1\\le k<r$, the code proves the integer divisibility\n\\[\n p^r\\mid\\binom{p^{r-1}}k p^k.\n\\]\nIt follows from $k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}$ and $v_p(k)<k$. Thus every summand is divisible by $p^r$, including the endpoint $r=1$.","nite Newton expansion is\n\\[\n f(n+p^{r-1}h)-f(n)=\n \\sum_{k=1}^{p^{r-1}}\\binom{p^{r-1}}k\\Delta_h^k f(n).\n\\]\nFor $k\\ge r$, the difference already supplies $p^r$. For $1\\le k<r$, the code proves the integer divisibility\n\\[\n p^r\\mid\\binom{p^{r-1}}k p^k.\n\\]\nIt follows from $k\\binom{p^{r-1}}k=p^{r-1}\\binom{p^{r-1}-1}{k-1}$ and $v_p(k)<k$. Thus every summand is divisible by $p^r$, including the endpoint $r=1$.","\\subsection*{Divisor branch}\nFor $m\\mid p-1$, the preceding all-residue theorem provides the needed differences as soon as $r\\le mn$. With $q=\\operatorname{lcm}(2,p-1)$, the required packet step $q/m$ is proved to be an integer multiple of $(p-1)/m$. The degree conversion is checked exactly:\n\\[\n mp\\left(n+p^{r-1}\\frac qm\\right)=mpn+qp^r.\n\\]\nThis retains the stronger sufficient condition $r\\le mn$ on the divisor branch. The uniform choice $n\\ge r$ implies it for every positive $m$.","$, the preceding all-residue theorem provides the needed differences as soon as $r\\le mn$. With $q=\\operatorname{lcm}(2,p-1)$, the required packet step $q/m$ is proved to be an integer multiple of $(p-1)/m$. The degree conversion is checked exactly:\n\\[\n mp\\left(n+p^{r-1}\\frac qm\\right)=mpn+qp^r.\n\\]\nThis retains the stronger sufficient condition $r\\le mn$ on the divisor branch. The uniform choice $n\\ge r$ implies it for every positive $m$.","ovides the needed differences as soon as $r\\le mn$. With $q=\\operatorname{lcm}(2,p-1)$, the required packet step $q/m$ is proved to be an integer multiple of $(p-1)/m$. The degree conversion is checked exactly:\n\\[\n mp\\left(n+p^{r-1}\\frac qm\\right)=mpn+qp^r.\n\\]\nThis retains the stronger sufficient condition $r\\le mn$ on the divisor branch. The uniform choice $n\\ge r$ implies it for every positive $m$.","\\subsection*{Exceptional branch}\nFor $p=2,m=2$, $q=2$ and the packet displacement is $2^{r-1}$. The new dyadic bound supplies the hypothesis when $n\\ge r$, and\n\\[\n 4\\bigl(n+2^{r-1}\\bigr)=4n+2^{r+1}.\n\\]\nThe $p=2,m=1$ branch is also checked using $q=2$, not an incorrectly substituted $q=p-1=1$.","\\subsection*{An exact arithmetic check, not an optimality claim}\nThe independent rational recurrence gives\n$C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. Their difference is $26/9$, of $2$-adic valuation one. This agrees with the new bound at $n=k=u=1$. The source also contains universal instantiations of the degree-indexed theorem at all four dyadic addresses and at an odd singular address.","\\subsection*{Reference}\nYuta Nishibuchi. \\emph{$p$-adic Congruences of Generalized Euler Numbers and Relations to Even Zeta Values}. arXiv:2605.10677v2, revised 1 June 2026. Definition 2.1, Conjecture 3.9 and Proposition 3.10. The version identifier is fixed for the statement comparison.","To distinguish ordinary signed Euler numbers from the generalized numbers\nof the main text, write\n\\[\n \\operatorname{sech}t=\\sum_{n\\ge0}\\mathcal E_n\\frac{t^n}{n!},\n \\qquad \\frac{t}{e^t-1}=\\sum_{n\\ge0}B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\l","To distinguish ordinary signed Euler numbers from the generalized numbers\nof the main text, write\n\\[\n \\operatorname{sech}t=\\sum_{n\\ge0}\\mathcal E_n\\frac{t^n}{n!},\n \\qquad \\frac{t}{e^t-1}=\\sum_{n\\ge0}B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nT","To distinguish ordinary signed Euler numbers from the generalized numbers\nof the main text, write\n\\[\n \\operatorname{sech}t=\\sum_{n\\ge0}\\mathcal E_n\\frac{t^n}{n!},\n \\qquad \\frac{t}{e^t-1}=\\sum_{n\\ge0}B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\","To distinguish ordinary signed Euler numbers from the generalized numbers\nof the main text, write\n\\[\n \\operatorname{sech}t=\\sum_{n\\ge0}\\mathcal E_n\\frac{t^n}{n!},\n \\qquad \\frac{t}{e^t-1}=\\sum_{n\\ge0}B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0","To distinguish ordinary signed Euler numbers from the generalized numbers\nof the main text, write\n\\[\n \\operatorname{sech}t=\\sum_{n\\ge0}\\mathcal E_n\\frac{t^n}{n!},\n \\qquad \\frac{t}{e^t-1}=\\sum_{n\\ge0}B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after m","o distinguish ordinary signed Euler numbers from the generalized numbers\nof the main text, write\n\\[\n \\operatorname{sech}t=\\sum_{n\\ge0}\\mathcal E_n\\frac{t^n}{n!},\n \\qquad \\frac{t}{e^t-1}=\\sum_{n\\ge0}B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplicatio…","rac{t}{e^t-1}=\\sum_{n\\ge0}B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0…","B_n\\frac{t^n}{n!},\n \\qquad B_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)…","_1=-\\frac12.\n\\]\nThroughout this appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m…","appendix, $v_2(2)=1$ and $v_2(0)=\\infty$.\nLet $s_2(n)$ be the number of ones in the binary expansion of the\nnonnegative integer $n$. Define\n\\[\n A(N)=\\frac{3^{2N+2}-1}{4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N…","4(2N+1)(2N+2)},\\qquad\n \\left(\\frac z{\\sinh z}\\right)^2=\\sum_{m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","m\\ge0}\\beta_m z^{2m},\n \\qquad \\kappa_m=2^{2m}(2m)!\\beta_m.\n\\]\nThe initial values are $\\beta_0=\\kappa_0=1$. For $m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","m\\ge1$, differentiation\nof $z\\coth z$ gives\n\\begin{equation}\n \\beta_m=-\\frac{2^{2m}(2m-1)B_{2m}}{(2m)!},\\qquad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","ad\n \\kappa_m=-2^{4m}(2m-1)B_{2m}.\n \\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","\\label{eq:root:beta}\n\\end{equation}\nThe classical dyadic Bernoulli denominator law\n$v_2(B_{2m})=-1$ and Legendre's formula yield\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","ld\n\\begin{equation}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","tion}\n v_2(\\beta_m)=s_2(m)-1,\\qquad\n v_2(\\kappa_m)=4m-1.\\label{eq:root:weights}\n\\end{equation}\nThe source identity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","ntity\n\\[\n \\sum_{N\\ge0}A(N)\\frac{t^{2N}}{(2N)!}\n =\\frac{\\sinh(2t)\\sinh t}{2t^2}\n\\]\nimplies, after multiplication by $4t^2/\\sinh^2(2t)$,\n\\begin{equation}\n \\mathcal E_{2N}=\\sum_{m=0}^N\n    \\kappa_m\\binom{2N}{2m}A(N-m).\n \\label{eq:root:packet}\n\\end{equation}\nFor a retained depth $M\\ge0$, put\n\\[\n \\mathcal T_M(N)=\\mathcal E_{2N}\n   -\\sum_{m=0}^{\\min(M,N)}\\kappa_m\\binom{2N}{2m}A(N-m).\n\\]\nAn empty tail, $N\\le M$, is exactly zero.","\\begin{proposition}[Sharp uniform classical truncation bound]\nFor every $M,N\\ge0$,\n\\[\n v_2\\bigl(\\mathcal T_M(N)\\bigr)\\ge4M+3.\n\\]\nFor each fixed $M$, equality holds at $N=M+1$.\n\\end{proposition}\n\\begin{proof}\nThe source $A(n)$ is a dyadic unit for every nonnegative integer $n$.\nEach nonempty summand has $m\\ge M+1$, so\n\\eqref{eq:root:weights} and integrality of the binomial coefficient give\nthe asserted bound. At $N=M+1$, only the first omitted summand remains.\nIts binomial coefficient and $A(0)$ both equal one.\n\\end{proof}","on}[Sharp uniform classical truncation bound]\nFor every $M,N\\ge0$,\n\\[\n v_2\\bigl(\\mathcal T_M(N)\\bigr)\\ge4M+3.\n\\]\nFor each fixed $M$, equality holds at $N=M+1$.\n\\end{proposition}\n\\begin{proof}\nThe source $A(n)$ is a dyadic unit for every nonnegative integer $n$.\nEach nonempty summand has $m\\ge M+1$, so\n\\eqref{eq:root:weights} and integrality of the binomial coefficient give\nthe asserted bound. At $N=M+1$, only the first omitted summand remains.\nIts binomial coefficient and $A(0)$ both equal one.\n\\end{proof}","l(\\mathcal T_M(N)\\bigr)\\ge4M+3.\n\\]\nFor each fixed $M$, equality holds at $N=M+1$.\n\\end{proposition}\n\\begin{proof}\nThe source $A(n)$ is a dyadic unit for every nonnegative integer $n$.\nEach nonempty summand has $m\\ge M+1$, so\n\\eqref{eq:root:weights} and integrality of the binomial coefficient give\nthe asserted bound. At $N=M+1$, only the first omitted summand remains.\nIts binomial coefficient and $A(0)$ both equal one.\n\\end{proof}","Let $\\mathbb Z_2\\langle x\\rangle$ denote the restricted power series:\ntheir integral coefficients tend to zero dyadically. Extend $A$ by\n\\[\n \\mathscr A(x)=\\frac{9^{x+1}-1}{8(x+1)(2x+1)},\n \\qquad \\mathscr A(-1)=-\\frac{\\log9}{8}.\n\\]\nThe apparent singularity at $x=-1$ is removable. Since\n$v_2(\\log9)=3$, its convergent expansion is\n\\begin{equation}\n \\mathscr A(x)=\\frac{\\log9/8}{1+2x}\n   \\sum_{h\\ge0}\\frac{(\\log9)^h}{(h+1)!}(x+1)^h.\n \\label{eq:root:source}\n\\end{equation}\nIt follows coefficientwise that\n$\\mathscr A\\in\\mathbb Z_2\\langle x\\rangle^\\times$,\n$\\mathscr A\\equiv1\\pmod2$, and $\\mathscr A(N)=A(N)$.","to zero dyadically. Extend $A$ by\n\\[\n \\mathscr A(x)=\\frac{9^{x+1}-1}{8(x+1)(2x+1)},\n \\qquad \\mathscr A(-1)=-\\frac{\\log9}{8}.\n\\]\nThe apparent singularity at $x=-1$ is removable. Since\n$v_2(\\log9)=3$, its convergent expansion is\n\\begin{equation}\n \\mathscr A(x)=\\frac{\\log9/8}{1+2x}\n   \\sum_{h\\ge0}\\frac{(\\log9)^h}{(h+1)!}(x+1)^h.\n \\label{eq:root:source}\n\\end{equation}\nIt follows coefficientwise that\n$\\mathscr A\\in\\mathbb Z_2\\langle x\\rangle^\\times$,\n$\\mathscr A\\equiv1\\pmod2$, and $\\mathscr A(N)=A(N)$.","=\\frac{9^{x+1}-1}{8(x+1)(2x+1)},\n \\qquad \\mathscr A(-1)=-\\frac{\\log9}{8}.\n\\]\nThe apparent singularity at $x=-1$ is removable. Since\n$v_2(\\log9)=3$, its convergent expansion is\n\\begin{equation}\n \\mathscr A(x)=\\frac{\\log9/8}{1+2x}\n   \\sum_{h\\ge0}\\frac{(\\log9)^h}{(h+1)!}(x+1)^h.\n \\label{eq:root:source}\n\\end{equation}\nIt follows coefficientwise that\n$\\mathscr A\\in\\mathbb Z_2\\langle x\\rangle^\\times$,\n$\\mathscr A\\equiv1\\pmod2$, and $\\mathscr A(N)=A(N)$.",",\n \\qquad \\mathscr A(-1)=-\\frac{\\log9}{8}.\n\\]\nThe apparent singularity at $x=-1$ is removable. Since\n$v_2(\\log9)=3$, its convergent expansion is\n\\begin{equation}\n \\mathscr A(x)=\\frac{\\log9/8}{1+2x}\n   \\sum_{h\\ge0}\\frac{(\\log9)^h}{(h+1)!}(x+1)^h.\n \\label{eq:root:source}\n\\end{equation}\nIt follows coefficientwise that\n$\\mathscr A\\in\\mathbb Z_2\\langle x\\rangle^\\times$,\n$\\mathscr A\\equiv1\\pmod2$, and $\\mathscr A(N)=A(N)$.","Set $r=M+1\\ge1$ and define\n\\[\n P_q(x)=\\prod_{i=0}^{q-1}(x-i)(2x-(2i+1)),\\qquad P_0=1,\n\\]\n\\[\n H_r(x)=\\sum_{q\\ge0}8^q\\frac{\\beta_{r+q}}{\\beta_r}\n                    P_q(x)\\mathscr A(x-q).\n\\]\nThe coefficient valuations tend to infinity. At any nonnegative integer\n$x$, terms with $q>x$ vanish. Cancelling the factorials in\n\\eqref{eq:root:packet} therefore gives the exact identity\n\\begin{equation}\n \\mathcal T_{r-1}(N)=\\kappa_r\\binom{2N}{2r}H_r(N-r)\n \\qquad(N\\ge r).\\label{eq:root:factor}\n\\end{equation}\nThis is a finite rational identity at every original Euler row.","The scalar packet weights have exact valuations\n\\[\n \\lambda_r(q)=3q+s_2(r+q)-s_2(r).\n\\]\nPut\n\\[\n \\mu_r=\\min_{q\\ge0}\\lambda_r(q),\\qquad\n J_r=\\{q:\\lambda_r(q)=\\mu_r\\},\\qquad G_r=2^{-\\mu_r}H_r.\n\\]\nSince $\\lambda_r(0)=0$ and\n$\\lambda_r(q)\\ge3q+1-s_2(r)$, every minimizing address satisfies\n\\begin{equation}\n 0\\le q\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n \\label{eq:root:cutoff}\n\\end{equation}\nIn particular $\\mu_r\\le0$, and the minimum is found by a finite carry\ncalculation. Reduction of the whole series gives\n\\begin{equation}\n \\overline G_r(X)=\\sum_{q\\in J_r}X^{\\underline q}\n       \\quad\\text{in }\\mathbb F_2[X].\n \\label{eq:root:reduction}\n\\end{equation}\nIts highest degree is $\\max J_r$, with leading coefficient one.\nThus the normalization is the actual Gauss normalization.","packet weights have exact valuations\n\\[\n \\lambda_r(q)=3q+s_2(r+q)-s_2(r).\n\\]\nPut\n\\[\n \\mu_r=\\min_{q\\ge0}\\lambda_r(q),\\qquad\n J_r=\\{q:\\lambda_r(q)=\\mu_r\\},\\qquad G_r=2^{-\\mu_r}H_r.\n\\]\nSince $\\lambda_r(0)=0$ and\n$\\lambda_r(q)\\ge3q+1-s_2(r)$, every minimizing address satisfies\n\\begin{equation}\n 0\\le q\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n \\label{eq:root:cutoff}\n\\end{equation}\nIn particular $\\mu_r\\le0$, and the minimum is found by a finite carry\ncalculation. Reduction of the whole series gives\n\\begin{equation}\n \\overline G_r(X)=\\sum_{q\\in J_r}X^{\\underline q}\n       \\quad\\text{in }\\mathbb F_2[X].\n \\label{eq:root:reduction}\n\\end{equation}\nIts highest degree is $\\max J_r$, with leading coefficient one.\nThus the normalization is the actual Gauss normalization.","ad\n J_r=\\{q:\\lambda_r(q)=\\mu_r\\},\\qquad G_r=2^{-\\mu_r}H_r.\n\\]\nSince $\\lambda_r(0)=0$ and\n$\\lambda_r(q)\\ge3q+1-s_2(r)$, every minimizing address satisfies\n\\begin{equation}\n 0\\le q\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n \\label{eq:root:cutoff}\n\\end{equation}\nIn particular $\\mu_r\\le0$, and the minimum is found by a finite carry\ncalculation. Reduction of the whole series gives\n\\begin{equation}\n \\overline G_r(X)=\\sum_{q\\in J_r}X^{\\underline q}\n       \\quad\\text{in }\\mathbb F_2[X].\n \\label{eq:root:reduction}\n\\end{equation}\nIts highest degree is $\\max J_r$, with leading coefficient one.\nThus the normalization is the actual Gauss normalization.","\\begin{equation}\n 0\\le q\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n \\label{eq:root:cutoff}\n\\end{equation}\nIn particular $\\mu_r\\le0$, and the minimum is found by a finite carry\ncalculation. Reduction of the whole series gives\n\\begin{equation}\n \\overline G_r(X)=\\sum_{q\\in J_r}X^{\\underline q}\n       \\quad\\text{in }\\mathbb F_2[X].\n \\label{eq:root:reduction}\n\\end{equation}\nIts highest degree is $\\max J_r$, with leading coefficient one.\nThus the normalization is the actual Gauss normalization.","\\begin{theorem}[Exact classical tail valuation]\nFor every $r\\ge1$ and $N\\ge r$,\n\\begin{equation}\n v_2\\bigl(\\mathcal T_{r-1}(N)\\bigr)\n =4r-1+v_2\\binom Nr+\\mu_r+v_2G_r(N-r).\n \\label{eq:root:exactvaluation}\n\\end{equation}\nThere are unique factors\n\\[\n G_r(X)=W_r(X)U_r(X),\n\\]\nwhere $W_r\\in\\mathbb Z_2[X]$ is monic of degree $d_r=\\max J_r$ and\n$U_r\\in\\mathbb Z_2\\langle X\\rangle^\\times$ satisfies $U_r\\equiv1\\pmod2$.\nConsequently $v_2G_r(x)=v_2W_r(x)$ for every $x\\in\\mathbb Z_2$, and\n\\[\n \\deg W_r\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n\\]\n\\label{thm:root:finitepreparation}\n\\end{theorem}\n\\begin{proof}\nEquation~\\eqref{eq:root:factor}, the exact weight in\n\\eqref{eq:root:weights}, and the digit-sum identity\n$v_2\\binom{2N}{2r}=v_2\\binom Nr$ prove\n\\eqref{eq:root","\\begin{theorem}[Exact classical tail valuation]\nFor every $r\\ge1$ and $N\\ge r$,\n\\begin{equation}\n v_2\\bigl(\\mathcal T_{r-1}(N)\\bigr)\n =4r-1+v_2\\binom Nr+\\mu_r+v_2G_r(N-r).\n \\label{eq:root:exactvaluation}\n\\end{equation}\nThere are unique factors\n\\[\n G_r(X)=W_r(X)U_r(X),\n\\]\nwhere $W_r\\in\\mathbb Z_2[X]$ is monic of degree $d_r=\\max J_r$ and\n$U_r\\in\\mathbb Z_2\\langle X\\rangle^\\times$ satisfies $U_r\\equiv1\\pmod2$.\nConsequently $v_2G_r(x)=v_2W_r(x)$ for every $x\\in\\mathbb Z_2$, and\n\\[\n \\deg W_r\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n\\]\n\\label{thm:root:finitepreparation}\n\\end{theorem}\n\\begin{proof}\nEquation~\\eqref{eq:root:factor}, the exact weight in\n\\eqref{eq:root:weights}, and the digit-sum identity\n$v_2\\binom{2N}{2r}=v_2\\binom Nr$ prove\n\\eqref{eq:root:exactvaluati","\\begin{theorem}[Exact classical tail valuation]\nFor every $r\\ge1$ and $N\\ge r$,\n\\begin{equation}\n v_2\\bigl(\\mathcal T_{r-1}(N)\\bigr)\n =4r-1+v_2\\binom Nr+\\mu_r+v_2G_r(N-r).\n \\label{eq:root:exactvaluation}\n\\end{equation}\nThere are unique factors\n\\[\n G_r(X)=W_r(X)U_r(X),\n\\]\nwhere $W_r\\in\\mathbb Z_2[X]$ is monic of degree $d_r=\\max J_r$ and\n$U_r\\in\\mathbb Z_2\\langle X\\rangle^\\times$ satisfies $U_r\\equiv1\\pmod2$.\nConsequently $v_2G_r(x)=v_2W_r(x)$ for every $x\\in\\mathbb Z_2$, and\n\\[\n \\deg W_r\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n\\]\n\\label{thm:root:finitepreparation}\n\\end{theorem}\n\\begin{proof}\nEquation~\\eqref{eq:root:factor}, the exact weight in\n\\eqref{eq:root:weights}, and the digit-sum identity\n$v_2\\binom{2N}{2r}=v_2\\binom Nr$ prove\n\\eqref{eq:root:exactvaluation}. The binomial coefficients themselves\nneed not be equal.","sical tail valuation]\nFor every $r\\ge1$ and $N\\ge r$,\n\\begin{equation}\n v_2\\bigl(\\mathcal T_{r-1}(N)\\bigr)\n =4r-1+v_2\\binom Nr+\\mu_r+v_2G_r(N-r).\n \\label{eq:root:exactvaluation}\n\\end{equation}\nThere are unique factors\n\\[\n G_r(X)=W_r(X)U_r(X),\n\\]\nwhere $W_r\\in\\mathbb Z_2[X]$ is monic of degree $d_r=\\max J_r$ and\n$U_r\\in\\mathbb Z_2\\langle X\\rangle^\\times$ satisfies $U_r\\equiv1\\pmod2$.\nConsequently $v_2G_r(x)=v_2W_r(x)$ for every $x\\in\\mathbb Z_2$, and\n\\[\n \\deg W_r\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n\\]\n\\label{thm:root:finitepreparation}\n\\end{theorem}\n\\begin{proof}\nEquation~\\eqref{eq:root:factor}, the exact weight in\n\\eqref{eq:root:weights}, and the digit-sum identity\n$v_2\\binom{2N}{2r}=v_2\\binom Nr$ prove\n\\eqref{eq:root:exactvaluation}. The binomial coefficients themselves\nneed not be equal.","$r\\ge1$ and $N\\ge r$,\n\\begin{equation}\n v_2\\bigl(\\mathcal T_{r-1}(N)\\bigr)\n =4r-1+v_2\\binom Nr+\\mu_r+v_2G_r(N-r).\n \\label{eq:root:exactvaluation}\n\\end{equation}\nThere are unique factors\n\\[\n G_r(X)=W_r(X)U_r(X),\n\\]\nwhere $W_r\\in\\mathbb Z_2[X]$ is monic of degree $d_r=\\max J_r$ and\n$U_r\\in\\mathbb Z_2\\langle X\\rangle^\\times$ satisfies $U_r\\equiv1\\pmod2$.\nConsequently $v_2G_r(x)=v_2W_r(x)$ for every $x\\in\\mathbb Z_2$, and\n\\[\n \\deg W_r\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n\\]\n\\label{thm:root:finitepreparation}\n\\end{theorem}\n\\begin{proof}\nEquation~\\eqref{eq:root:factor}, the exact weight in\n\\eqref{eq:root:weights}, and the digit-sum identity\n$v_2\\binom{2N}{2r}=v_2\\binom Nr$ prove\n\\eqref{eq:root:exactvaluation}. The binomial coefficients themselves\nneed not be equal.","egin{equation}\n v_2\\bigl(\\mathcal T_{r-1}(N)\\bigr)\n =4r-1+v_2\\binom Nr+\\mu_r+v_2G_r(N-r).\n \\label{eq:root:exactvaluation}\n\\end{equation}\nThere are unique factors\n\\[\n G_r(X)=W_r(X)U_r(X),\n\\]\nwhere $W_r\\in\\mathbb Z_2[X]$ is monic of degree $d_r=\\max J_r$ and\n$U_r\\in\\mathbb Z_2\\langle X\\rangle^\\times$ satisfies $U_r\\equiv1\\pmod2$.\nConsequently $v_2G_r(x)=v_2W_r(x)$ for every $x\\in\\mathbb Z_2$, and\n\\[\n \\deg W_r\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n\\]\n\\label{thm:root:finitepreparation}\n\\end{theorem}\n\\begin{proof}\nEquation~\\eqref{eq:root:factor}, the exact weight in\n\\eqref{eq:root:weights}, and the digit-sum identity\n$v_2\\binom{2N}{2r}=v_2\\binom Nr$ prove\n\\eqref{eq:root:exactvaluation}. The binomial coefficients themselves\nneed not be equal.","degree $d_r=\\max J_r$ and\n$U_r\\in\\mathbb Z_2\\langle X\\rangle^\\times$ satisfies $U_r\\equiv1\\pmod2$.\nConsequently $v_2G_r(x)=v_2W_r(x)$ for every $x\\in\\mathbb Z_2$, and\n\\[\n \\deg W_r\\le\\left\\lfloor\\frac{s_2(r)-1}{3}\\right\\rfloor.\n\\]\n\\label{thm:root:finitepreparation}\n\\end{theorem}\n\\begin{proof}\nEquation~\\eqref{eq:root:factor}, the exact weight in\n\\eqref{eq:root:weights}, and the digit-sum identity\n$v_2\\binom{2N}{2r}=v_2\\binom Nr$ prove\n\\eqref{eq:root:exactvaluation}. The binomial coefficients themselves\nneed not be equal.","For preparation, start from the monic polynomial\n$g=\\overline G_r$ and its zero-one lift $W_1$, with $U_1=1$.\nSuppose $G_r-W_kU_k$ is coefficientwise divisible by $2^k$.\nIts divided reduction is a finite polynomial because $G_r$ is restricted.\nDivide it by $g$ in $\\mathbb F_2[X]$:\n\\[\n \\overline{(G_r-W_kU_k)/2^k}=gq+s,\\qquad\\deg s<d_r.\n\\]\nWith zero-one coefficient lifts, set\n$W_{k+1}=W_k+2^ks$ and $U_{k+1}=U_k+2^kq$.\nThe error is now divisible by $2^{k+1}$, the polynomial remains monic of\ndegree $d_r$, and the unit reduces to one. The sequences converge in the\nGauss norm. Uniqueness of quotient and remainder at successive binary\ndigits proves uniqueness of the factors. For $d_r=0$, take $W_r=1$.\nThe degree bound follows from \\eqref{eq:root:cutoff}.\n\\end{pr","For preparation, start from the monic polynomial\n$g=\\overline G_r$ and its zero-one lift $W_1$, with $U_1=1$.\nSuppose $G_r-W_kU_k$ is coefficientwise divisible by $2^k$.\nIts divided reduction is a finite polynomial because $G_r$ is restricted.\nDivide it by $g$ in $\\mathbb F_2[X]$:\n\\[\n \\overline{(G_r-W_kU_k)/2^k}=gq+s,\\qquad\\deg s<d_r.\n\\]\nWith zero-one coefficient lifts, set\n$W_{k+1}=W_k+2^ks$ and $U_{k+1}=U_k+2^kq$.\nThe error is now divisible by $2^{k+1}$, the polynomial remains monic of\ndegree $d_r$, and the unit reduces to one. The sequences converge in the\nGauss norm. Uniqueness of quotient and remainder at successive binary\ndigits proves uniqueness of the factors. For $d_r=0$, take $W_r=1$.\nThe degree bound follows from \\eqref{eq:root:cutoff}.\n\\end{proof}","or preparation, start from the monic polynomial\n$g=\\overline G_r$ and its zero-one lift $W_1$, with $U_1=1$.\nSuppose $G_r-W_kU_k$ is coefficientwise divisible by $2^k$.\nIts divided reduction is a finite polynomial because $G_r$ is restricted.\nDivide it by $g$ in $\\mathbb F_2[X]$:\n\\[\n \\overline{(G_r-W_kU_k)/2^k}=gq+s,\\qquad\\deg s<d_r.\n\\]\nWith zero-one coefficient lifts, set\n$W_{k+1}=W_k+2^ks$ and $U_{k+1}=U_k+2^kq$.\nThe error is now divisible by $2^{k+1}$, the polynomial remains monic of\ndegree $d_r$, and the unit reduces to one. The sequences converge in the\nGauss norm. Uniqueness of quotient and remainder at successive binary\ndigits proves uniqueness of the factors. For $d_r=0$, take $W_r=1$.\nThe degree bound follows from \\eqref{eq:root:cutoff}.\n\\end{proof}","=\\overline G_r$ and its zero-one lift $W_1$, with $U_1=1$.\nSuppose $G_r-W_kU_k$ is coefficientwise divisible by $2^k$.\nIts divided reduction is a finite polynomial because $G_r$ is restricted.\nDivide it by $g$ in $\\mathbb F_2[X]$:\n\\[\n \\overline{(G_r-W_kU_k)/2^k}=gq+s,\\qquad\\deg s<d_r.\n\\]\nWith zero-one coefficient lifts, set\n$W_{k+1}=W_k+2^ks$ and $U_{k+1}=U_k+2^kq$.\nThe error is now divisible by $2^{k+1}$, the polynomial remains monic of\ndegree $d_r$, and the unit reduces to one. The sequences converge in the\nGauss norm. Uniqueness of quotient and remainder at successive binary\ndigits proves uniqueness of the factors. For $d_r=0$, take $W_r=1$.\nThe degree bound follows from \\eqref{eq:root:cutoff}.\n\\end{proof}","$U_1=1$.\nSuppose $G_r-W_kU_k$ is coefficientwise divisible by $2^k$.\nIts divided reduction is a finite polynomial because $G_r$ is restricted.\nDivide it by $g$ in $\\mathbb F_2[X]$:\n\\[\n \\overline{(G_r-W_kU_k)/2^k}=gq+s,\\qquad\\deg s<d_r.\n\\]\nWith zero-one coefficient lifts, set\n$W_{k+1}=W_k+2^ks$ and $U_{k+1}=U_k+2^kq$.\nThe error is now divisible by $2^{k+1}$, the polynomial remains monic of\ndegree $d_r$, and the unit reduces to one. The sequences converge in the\nGauss norm. Uniqueness of quotient and remainder at successive binary\ndigits proves uniqueness of the factors. For $d_r=0$, take $W_r=1$.\nThe degree bound follows from \\eqref{eq:root:cutoff}.\n\\end{proof}","\\[\n \\overline{(G_r-W_kU_k)/2^k}=gq+s,\\qquad\\deg s<d_r.\n\\]\nWith zero-one coefficient lifts, set\n$W_{k+1}=W_k+2^ks$ and $U_{k+1}=U_k+2^kq$.\nThe error is now divisible by $2^{k+1}$, the polynomial remains monic of\ndegree $d_r$, and the unit reduces to one. The sequences converge in the\nGauss norm. Uniqueness of quotient and remainder at successive binary\ndigits proves uniqueness of the factors. For $d_r=0$, take $W_r=1$.\nThe degree bound follows from \\eqref{eq:root:cutoff}.\n\\end{proof}","(G_r-W_kU_k)/2^k}=gq+s,\\qquad\\deg s<d_r.\n\\]\nWith zero-one coefficient lifts, set\n$W_{k+1}=W_k+2^ks$ and $U_{k+1}=U_k+2^kq$.\nThe error is now divisible by $2^{k+1}$, the polynomial remains monic of\ndegree $d_r$, and the unit reduces to one. The sequences converge in the\nGauss norm. Uniqueness of quotient and remainder at successive binary\ndigits proves uniqueness of the factors. For $d_r=0$, take $W_r=1$.\nThe degree bound follows from \\eqref{eq:root:cutoff}.\n\\end{proof}","This theorem reduces every depth to a finite polynomial without discarding\nthe infinite analytic tail. It does not explicitly classify every\n$W_r$, decide every repeated root from a fixed precision, or prove that\nall its dyadic roots are simple.","For a wall label $j\\ge1$ and $0\\le q<j$, define\n\\[\n e_j(q)=3q+s_2(j-1)-s_2(j-q-1),\\qquad\n E_j=\\min_{q<j}e_j(q),\n\\]\n\\[\n \\Theta(j)=3j+1+s_2(j-1),\\qquad L_j=\\Theta(j)-E_j.\n\\]\nHere $E_j$ is a carry minimum, not an Euler number. A wall label is\n\\emph{regular} when $e_j(q)>0$ for $1\\le q<j$.\nFor $r=2^L-j$ with $L>L_j$, put $\\delta=L-L_j$.\nThe carry identities give\n\\[\n \\mu_r=E_j-\\delta,\\qquad J_r=\\{j\\},\\qquad\n G_r(x)=P_j(x)U(x-j)+2^\\delta V(x),\n\\]\nwhere $U\\equiv1\\pmod2$. For a regular label, also\n$E_j=0$ and $V\\equiv1\\pmod2$.","e_j(q)=3q+s_2(j-1)-s_2(j-q-1),\\qquad\n E_j=\\min_{q<j}e_j(q),\n\\]\n\\[\n \\Theta(j)=3j+1+s_2(j-1),\\qquad L_j=\\Theta(j)-E_j.\n\\]\nHere $E_j$ is a carry minimum, not an Euler number. A wall label is\n\\emph{regular} when $e_j(q)>0$ for $1\\le q<j$.\nFor $r=2^L-j$ with $L>L_j$, put $\\delta=L-L_j$.\nThe carry identities give\n\\[\n \\mu_r=E_j-\\delta,\\qquad J_r=\\{j\\},\\qquad\n G_r(x)=P_j(x)U(x-j)+2^\\delta V(x),\n\\]\nwhere $U\\equiv1\\pmod2$. For a regular label, also\n$E_j=0$ and $V\\equiv1\\pmod2$.","in_{q<j}e_j(q),\n\\]\n\\[\n \\Theta(j)=3j+1+s_2(j-1),\\qquad L_j=\\Theta(j)-E_j.\n\\]\nHere $E_j$ is a carry minimum, not an Euler number. A wall label is\n\\emph{regular} when $e_j(q)>0$ for $1\\le q<j$.\nFor $r=2^L-j$ with $L>L_j$, put $\\delta=L-L_j$.\nThe carry identities give\n\\[\n \\mu_r=E_j-\\delta,\\qquad J_r=\\{j\\},\\qquad\n G_r(x)=P_j(x)U(x-j)+2^\\delta V(x),\n\\]\nwhere $U\\equiv1\\pmod2$. For a regular label, also\n$E_j=0$ and $V\\equiv1\\pmod2$.","q),\n\\]\n\\[\n \\Theta(j)=3j+1+s_2(j-1),\\qquad L_j=\\Theta(j)-E_j.\n\\]\nHere $E_j$ is a carry minimum, not an Euler number. A wall label is\n\\emph{regular} when $e_j(q)>0$ for $1\\le q<j$.\nFor $r=2^L-j$ with $L>L_j$, put $\\delta=L-L_j$.\nThe carry identities give\n\\[\n \\mu_r=E_j-\\delta,\\qquad J_r=\\{j\\},\\qquad\n G_r(x)=P_j(x)U(x-j)+2^\\delta V(x),\n\\]\nwhere $U\\equiv1\\pmod2$. For a regular label, also\n$E_j=0$ and $V\\equiv1\\pmod2$.","For $0\\le i<j$, let\n\\[\n d_i=v_2(i!)+v_2((j-1-i)!),\\qquad\n k_i=1+\\left\\lfloor\\log_2\\max(i,j-1-i)\\right\\rfloor,\n\\]\n\\[\n B_i=i+2^{k_i}\\mathbb Z_2,\\qquad t_i=d_i+k_i.\n\\]\nAt $j=1$ use $d_0=k_0=t_0=0$. The balls $B_i$ partition\n$\\mathbb Z_2$. Indeed a residue represented twice among $0,\\ldots,j-1$\nsplits into its two children. Every other represented residue remains\none ball. On each such ball,\n\\[\n P_j(i+2^{k_i}y)=2^{t_i}yQ_i(y),\\qquad\n Q_i\\in\\mathbb Z_2\\langle y\\rangle^\\times,\n \\qquad Q_i\\equiv1\\pmod2.\n\\]","\\begin{theorem}[Regular-wall root law]\nFor every regular label $j$ and every integer $\\delta\\ge1$,\n\\[\n v_2G_r(x)=\n \\begin{cases}\n  \\delta,&x\\in B_i,\\ \\delta<t_i,","\\begin{theorem}[Regular-wall root law]\nFor every regular label $j$ and every integer $\\delta\\ge1$,\n\\[\n v_2G_r(x)=\n \\begin{cases}\n  \\delta,&x\\in B_i,\\ \\delta<t_i,\\\\\n  d_i+v_2(x-\\alpha_i),&x\\in B_i,\\ \\delta\\ge t_i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contribution","d_i+v_2(x-\\alpha_i),&x\\in B_i,\\ \\delta\\ge t_i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exa","d_i+v_2(x-\\alpha_i),&x\\in B_i,\\ \\delta\\ge t_i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distan","d_i+v_2(x-\\alpha_i),&x\\in B_i,\\ \\delta\\ge t_i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated dis","d_i+v_2(x-\\alpha_i),&x\\in B_i,\\ \\delta\\ge t_i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition","d_i+v_2(x-\\alpha_i),&x\\in B_i,\\ \\delta\\ge t_i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","-\\alpha_i),&x\\in B_i,\\ \\delta\\ge t_i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","i.\n \\end{cases}\n\\]\nIn the second case there is exactly one root $\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","\\alpha_i$ in $B_i$,\nit is simple, and $v_2(\\alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","alpha_i-i)=\\delta-d_i$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","$.\nThus the number of dyadic roots is exactly\n\\[\n \\#\\{i:0\\le i<j,\\ t_i\\le\\delta\\}.\n\\]\n\\label{thm:root:regular}\n\\end{theorem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","rem}\n\\begin{proof}\nIf $\\delta<t_i$, the unit perturbation has smaller valuation than\nthe skeleton term throughout $B_i$, so the valuation is exactly\n$\\delta$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","a$. Otherwise, after the ball substitution and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","and division by\n$2^{t_i}$, the coefficientwise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","ise reduction is $y+c$ for a constant\n$c\\in\\mathbb F_2$. A restricted integral series with this reduction\nis a dyadic isometry: division of its difference by $y-z$ gives an\nodd linear coefficient plus even higher contributions. Binary lifting\ntherefore gives its unique root and exact distance law. Evaluating at\n$x=i$ gives the stated displacement, since\n$G_r(i)=2^\\delta V(i)$ has exact valuation $\\delta$.\nThe partition proves exhaustion of the roots.\n\\end{proof}","The largest emergence threshold is\n\\[\n \\max_i t_i=v_2((j-1)!)+\\lceil\\log_2 j\\rceil.\n\\]\nFor $j=2^a$, all thresholds equal $j-1$. There are no roots before\nthat distance and exactly $j$ simple roots at and after it, within the\npositive-distance range. The equality and pre-wall cases are also\nresolved: regular labels $j\\ge2$ have no roots there. $j=1$ has one\nsimple odd root at equality.","For example, $j=5$ has $(t_0,\\ldots,t_4)=(6,3,4,3,6)$.\nThe exact root counts at distances $1,2,3,4,5,\\ge6$ are\n$0,0,2,3,3,5$. The older sufficient stages $3,5,7$ are not the\nsharp emergence stages, which are $3,4,6$.\nThe first irregular label, $j=17$, has $V\\equiv1+X\\pmod2$.\nThe regular-wall theorem does not apply. Complete coefficient\ncertificates close distances $1$ through $19$, and the general separation\ntheorem gives all seventeen simple roots for $\\delta\\ge20$.\nThose two kinds of evidence together resolve that family.","exact root counts at distances $1,2,3,4,5,\\ge6$ are\n$0,0,2,3,3,5$. The older sufficient stages $3,5,7$ are not the\nsharp emergence stages, which are $3,4,6$.\nThe first irregular label, $j=17$, has $V\\equiv1+X\\pmod2$.\nThe regular-wall theorem does not apply. Complete coefficient\ncertificates close distances $1$ through $19$, and the general separation\ntheorem gives all seventeen simple roots for $\\delta\\ge20$.\nThose two kinds of evidence together resolve that family.","For a general depth $r$, call $j_r=\\max J_r$ its \\emph{true terminal}.\nThis definition depends on the actual carry minimum. Rewriting the same\ninteger as another wall expression does not change $j_r$.\nThe universal simplicity target is\n\\begin{equation}\n \\text{for every integer }r\\ge1\\text{ and }x\\in\\mathbb Z_2,\n \\qquad G_r(x)=0\\ \\Longrightarrow\\ G_r'(x)\\ne0.\n \\label{eq:root:simplicitytarget}\n\\end{equation}\nThe following are completed restricted results of the later continuations.\nTheir family statements include a proof that the indicated label is the\ntrue terminal throughout the displayed ranges.","\\begin{center}\n\\begin{tabular}{lll}\n\\toprule\nDepth $r$ & Odd multiplier range & Outer scales","\\midrule\n$B2^L-129$ & $0<B<2^{89}$ & $L\\ge392$","$B2^L-161$ & $0<B<2^{932}$ & $L\\ge487$","\\bottomrule\n\\end{tabular}\n\\end{center}\nAll dyadic roots are simple in both infinite integer families.\nFor arbitrary larger admissible multipliers, a multiple root with true\nterminal $129$ can occur only at $L=515$ and in two necessary multiplier\nclasses modulo $2^{90}$. With true terminal $161$, it can occur only at\n$L=640$ and in one necessary multiplier class modulo $2^{935}$.\nThese classes are necessary conditions. The records do not assert\nexistence of a singular ordinary Euler index in them.","For terminal $161$, the neighboring phases are resolved uniformly:\n$L=639$ has exactly one simple root and $L=641$ exactly three simple\nroots, for every positive odd multiplier with that true terminal.\nThe middle phase is reduced, before the deeper certificate is applied,\nto\n\\[\n B\\equiv17659\\pmod{2^{18}},\\qquad\n x\\equiv10202\\pmod{2^{15}}.\n\\]\nThe $935$-bit multiplier condition composes this whole-phase reduction\nwith an inherited $917$-bit local parameter certificate. It is neither\na new $935$-bit reconstruction nor an all-precision interpolation of\nthe undepleted Euler family.","g phases are resolved uniformly:\n$L=639$ has exactly one simple root and $L=641$ exactly three simple\nroots, for every positive odd multiplier with that true terminal.\nThe middle phase is reduced, before the deeper certificate is applied,\nto\n\\[\n B\\equiv17659\\pmod{2^{18}},\\qquad\n x\\equiv10202\\pmod{2^{15}}.\n\\]\nThe $935$-bit multiplier condition composes this whole-phase reduction\nwith an inherited $917$-bit local parameter certificate. It is neither\na new $935$-bit reconstruction nor an all-precision interpolation of\nthe undepleted Euler family.","mple root and $L=641$ exactly three simple\nroots, for every positive odd multiplier with that true terminal.\nThe middle phase is reduced, before the deeper certificate is applied,\nto\n\\[\n B\\equiv17659\\pmod{2^{18}},\\qquad\n x\\equiv10202\\pmod{2^{15}}.\n\\]\nThe $935$-bit multiplier condition composes this whole-phase reduction\nwith an inherited $917$-bit local parameter certificate. It is neither\na new $935$-bit reconstruction nor an all-precision interpolation of\nthe undepleted Euler family.","The structural argument also extends beyond lower address one.\nFor a certified true-terminal wall, let $a$ be its last lower minimizing\naddress, $k=j_r-a$ the regular complement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","ll, let $a$ be its last lower minimizing\naddress, $k=j_r-a$ the regular complement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","s last lower minimizing\naddress, $k=j_r-a$ the regular complement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","lower minimizing\naddress, $k=j_r-a$ the regular complement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","ing\naddress, $k=j_r-a$ the regular complement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","=j_r-a$ the regular complement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","the regular complement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","mplement, and $t=v_2(k)$.\nUnder the regular lower-address hypotheses of the source theorem,\nall roots above emerged complement balls are simple. Every root is\nsimple at every wall distance in the band\n\\[\n \\Theta(a)\\le t<\\Theta(a)+v_2(a!).\n\\]\nEquality walls are simple even outside that band. This includes\ncertified terminal-$258$ walls with $a=2$, $k=256$, and $L\\ge777$.\nFor $a=3$, $k=4096$, only $L=16385$ remains a possible critical scale.\nOutside the band, the theorem gives necessary local valuation conditions.\nIt does not exclude the remaining unit cancellation. Irregular lower\naddresses need their own prefix analysis.","The current focused obstruction is the admitted ordinary-index family\n\\begin{equation}\n r=(17659+2^{18}T)2^{640}-161,\n \\qquad T\\in\\mathbb Z_{\\ge0},\\qquad j_r=161.\n \\label{eq:root:focusedfamily}\n\\end{equation}\nOn its surviving chart define the \\emph{actual}, undepleted function\n\\[\n K_r(Z)=2^{-178}G_r(10202+2^{15}Z).\n\\]\nThe complete coefficient theorem gives\n\\begin{equation}\n K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z),\n \\quad a_r,b_r\\in\\mathbb Z_2,\\quad\n u_r\\in\\mathbb Z_2^\\times,\\quad\n R_r\\in\\mathbb Z_2\\langle Z\\rangle.\n \\label{eq:root:quadraticclass}\n\\end{equation}\nIts half derivative is a dyadic isometry. It therefore has a unique\ncritical point $c_r\\in\\mathbb Z_2$. Set\n$\\Phi_r^{\\mathrm{act}}=K_r(c_r)$.\nThe earlier independent-undepletion notation for this same scalar is\n$\\Phi_r(2^{2r-25})$.\nExcluding a multiple root on the surviving chart is exactly the assertion\n$\\Phi_r^{\\mathrm{act}}\\","The current focused obstruction is the admitted ordinary-index family\n\\begin{equation}\n r=(17659+2^{18}T)2^{640}-161,\n \\qquad T\\in\\mathbb Z_{\\ge0},\\qquad j_r=161.\n \\label{eq:root:focusedfamily}\n\\end{equation}\nOn its surviving chart define the \\emph{actual}, undepleted function\n\\[\n K_r(Z)=2^{-178}G_r(10202+2^{15}Z).\n\\]\nThe complete coefficient theorem gives\n\\begin{equation}\n K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z),\n \\quad a_r,b_r\\in\\mathbb Z_2,\\quad\n u_r\\in\\mathbb Z_2^\\times,\\quad\n R_r\\in\\mathbb Z_2\\langle Z\\rangle.\n \\label{eq:root:quadraticclass}\n\\end{equation}\nIts half derivative is a dyadic isometry. It therefore has a unique\ncritical point $c_r\\in\\mathbb Z_2$. Set\n$\\Phi_r^{\\mathrm{act}}=K_r(c_r)$.\nThe earlier independent-undepletion notation for this same scalar is\n$\\Phi_r(2^{2r-25})$.\nExcluding a multiple root on the surviving chart is exactly the assertion\n$\\Phi_r^{\\mathrm{act}}\\ne0$.","actual}, undepleted function\n\\[\n K_r(Z)=2^{-178}G_r(10202+2^{15}Z).\n\\]\nThe complete coefficient theorem gives\n\\begin{equation}\n K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z),\n \\quad a_r,b_r\\in\\mathbb Z_2,\\quad\n u_r\\in\\mathbb Z_2^\\times,\\quad\n R_r\\in\\mathbb Z_2\\langle Z\\rangle.\n \\label{eq:root:quadraticclass}\n\\end{equation}\nIts half derivative is a dyadic isometry. It therefore has a unique\ncritical point $c_r\\in\\mathbb Z_2$. Set\n$\\Phi_r^{\\mathrm{act}}=K_r(c_r)$.\nThe earlier independent-undepletion notation for this same scalar is\n$\\Phi_r(2^{2r-25})$.\nExcluding a multiple root on the surviving chart is exactly the assertion\n$\\Phi_r^{\\mathrm{act}}\\ne0$.","ion\n\\[\n K_r(Z)=2^{-178}G_r(10202+2^{15}Z).\n\\]\nThe complete coefficient theorem gives\n\\begin{equation}\n K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z),\n \\quad a_r,b_r\\in\\mathbb Z_2,\\quad\n u_r\\in\\mathbb Z_2^\\times,\\quad\n R_r\\in\\mathbb Z_2\\langle Z\\rangle.\n \\label{eq:root:quadraticclass}\n\\end{equation}\nIts half derivative is a dyadic isometry. It therefore has a unique\ncritical point $c_r\\in\\mathbb Z_2$. Set\n$\\Phi_r^{\\mathrm{act}}=K_r(c_r)$.\nThe earlier independent-undepletion notation for this same scalar is\n$\\Phi_r(2^{2r-25})$.\nExcluding a multiple root on the surviving chart is exactly the assertion\n$\\Phi_r^{\\mathrm{act}}\\ne0$.","K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z),\n \\quad a_r,b_r\\in\\mathbb Z_2,\\quad\n u_r\\in\\mathbb Z_2^\\times,\\quad\n R_r\\in\\mathbb Z_2\\langle Z\\rangle.\n \\label{eq:root:quadraticclass}\n\\end{equation}\nIts half derivative is a dyadic isometry. It therefore has a unique\ncritical point $c_r\\in\\mathbb Z_2$. Set\n$\\Phi_r^{\\mathrm{act}}=K_r(c_r)$.\nThe earlier independent-undepletion notation for this same scalar is\n$\\Phi_r(2^{2r-25})$.\nExcluding a multiple root on the surviving chart is exactly the assertion\n$\\Phi_r^{\\mathrm{act}}\\ne0$.","mes,\\quad\n R_r\\in\\mathbb Z_2\\langle Z\\rangle.\n \\label{eq:root:quadraticclass}\n\\end{equation}\nIts half derivative is a dyadic isometry. It therefore has a unique\ncritical point $c_r\\in\\mathbb Z_2$. Set\n$\\Phi_r^{\\mathrm{act}}=K_r(c_r)$.\nThe earlier independent-undepletion notation for this same scalar is\n$\\Phi_r(2^{2r-25})$.\nExcluding a multiple root on the surviving chart is exactly the assertion\n$\\Phi_r^{\\mathrm{act}}\\ne0$.","\\begin{proposition}[Exact quadratic compression]\nThere is a unique preparation\n\\[\n K_r(Z)=\\mathcal U_r(Z)\\mathcal W_r(Z),\\qquad\n \\mathcal W_r(Z)=Z^2+2\\beta_rZ+\\alpha_r,\\qquad\n \\mathcal U_r\\equiv u_r\\pmod{256}.\n\\]\nWith $\\delta_r=\\beta_r^2-\\alpha_r$ and\n$\\tau_r=\\mathcal U_r'(c_r)/\\mathcal U_r(c_r)$,\n\\begin{equation}\n \\delta_r=-\\frac{\\Phi_r^{\\mathrm{act}}}{\\mathcal U_r(c_r)}\n  \\left(1-\\frac{\\tau_r^2\\Phi_r^{\\mathrm{act}}}{4\\mathcal U_r(c_r)}\\right),\n \\qquad v_2(\\delta_r)=v_2(\\Phi_r^{\\mathrm{act}}).\n \\label{eq:root:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while","\\begin{proposition}[Exact quadratic compression]\nThere is a unique preparation\n\\[\n K_r(Z)=\\mathcal U_r(Z)\\mathcal W_r(Z),\\qquad\n \\mathcal W_r(Z)=Z^2+2\\beta_rZ+\\alpha_r,\\qquad\n \\mathcal U_r\\equiv u_r\\pmod{256}.\n\\]\nWith $\\delta_r=\\beta_r^2-\\alpha_r$ and\n$\\tau_r=\\mathcal U_r'(c_r)/\\mathcal U_r(c_r)$,\n\\begin{equation}\n \\delta_r=-\\frac{\\Phi_r^{\\mathrm{act}}}{\\mathcal U_r(c_r)}\n  \\left(1-\\frac{\\tau_r^2\\Phi_r^{\\mathrm{act}}}{4\\mathcal U_r(c_r)}\\right),\n \\qquad v_2(\\delta_r)=v_2(\\Phi_r^{\\mathrm{act}}).\n \\label{eq:root:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U","\\begin{proposition}[Exact quadratic compression]\nThere is a unique preparation\n\\[\n K_r(Z)=\\mathcal U_r(Z)\\mathcal W_r(Z),\\qquad\n \\mathcal W_r(Z)=Z^2+2\\beta_rZ+\\alpha_r,\\qquad\n \\mathcal U_r\\equiv u_r\\pmod{256}.\n\\]\nWith $\\delta_r=\\beta_r^2-\\alpha_r$ and\n$\\tau_r=\\mathcal U_r'(c_r)/\\mathcal U_r(c_r)$,\n\\begin{equation}\n \\delta_r=-\\frac{\\Phi_r^{\\mathrm{act}}}{\\mathcal U_r(c_r)}\n  \\left(1-\\frac{\\tau_r^2\\Phi_r^{\\mathrm{act}}}{4\\mathcal U_r(c_r)}\\right),\n \\qquad v_2(\\delta_r)=v_2(\\Phi_r^{\\mathrm{act}}).\n \\label{eq:root:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha","\\begin{proposition}[Exact quadratic compression]\nThere is a unique preparation\n\\[\n K_r(Z)=\\mathcal U_r(Z)\\mathcal W_r(Z),\\qquad\n \\mathcal W_r(Z)=Z^2+2\\beta_rZ+\\alpha_r,\\qquad\n \\mathcal U_r\\equiv u_r\\pmod{256}.\n\\]\nWith $\\delta_r=\\beta_r^2-\\alpha_r$ and\n$\\tau_r=\\mathcal U_r'(c_r)/\\mathcal U_r(c_r)$,\n\\begin{equation}\n \\delta_r=-\\frac{\\Phi_r^{\\mathrm{act}}}{\\mathcal U_r(c_r)}\n  \\left(1-\\frac{\\tau_r^2\\Phi_r^{\\mathrm{act}}}{4\\mathcal U_r(c_r)}\\right),\n \\qquad v_2(\\delta_r)=v_2(\\Phi_r^{\\mathrm{act}}).\n \\label{eq:root:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha_r=W'(c_r)^2/4-W(c_r)$ proves the formula.\nSince $\\tau_r\\in256\\mathbb Z_2$, its bracket is a unit.\n\\end{…","mathcal U_r(c_r)}\\right),\n \\qquad v_2(\\delta_r)=v_2(\\Phi_r^{\\mathrm{act}}).\n \\label{eq:root:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha_r=W'(c_r)^2/4-W(c_r)$ proves the formula.\nSince $\\tau_r\\in256\\mathbb Z_2$, its bracket is a unit.\n\\end{proof}","ad v_2(\\delta_r)=v_2(\\Phi_r^{\\mathrm{act}}).\n \\label{eq:root:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha_r=W'(c_r)^2/4-W(c_r)$ proves the formula.\nSince $\\tau_r\\in256\\mathbb Z_2$, its bracket is a unit.\n\\end{proof}","athrm{act}}).\n \\label{eq:root:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha_r=W'(c_r)^2/4-W(c_r)$ proves the formula.\nSince $\\tau_r\\in256\\mathbb Z_2$, its bracket is a unit.\n\\end{proof}","oot:discriminant}\n\\end{equation}\nThe equality includes the zero case.\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha_r=W'(c_r)^2/4-W(c_r)$ proves the formula.\nSince $\\tau_r\\in256\\mathbb Z_2$, its bracket is a unit.\n\\end{proof}",".\n\\end{proposition}\n\\begin{proof}\nThe same successive monic polynomial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha_r=W'(c_r)^2/4-W(c_r)$ proves the formula.\nSince $\\tau_r\\in256\\mathbb Z_2$, its bracket is a unit.\n\\end{proof}","mial division used in\nTheorem~\\ref{thm:root:finitepreparation}, started at precision eight,\npreserves the quadratic and unit reductions of\n\\eqref{eq:root:quadraticclass}. Write\n$W=Z^2+2\\beta_rZ+\\alpha_r$ and $w=W(c_r)$.\nStationarity gives $W'(c_r)=-\\tau_rw$, while\n$w=\\Phi_r^{\\mathrm{act}}/\\mathcal U_r(c_r)$.\nThe quadratic identity\n$\\beta_r^2-\\alpha_r=W'(c_r)^2/4-W(c_r)$ proves the formula.\nSince $\\tau_r\\in256\\mathbb Z_2$, its bracket is a unit.\n\\end{proof}","This compression retains the unit-cofactor derivative. It does not\nassert that $c_r$, $\\alpha_r$, or $\\beta_r$ is rational or algebraic\nover $\\mathbb Q$. The remaining arithmetic obligation is therefore\n\\begin{equation}\n \\delta_r\\ne0\\quad\\text{for every admitted ordinary index in\n \\eqref{eq:root:focusedfamily}.}\n \\label{eq:root:remaining}\n\\end{equation}\nEven this assertion would still leave the terminal-$129$ and higher-address\nobligations in the universal target \\eqref{eq:root:simplicitytarget}.","pression retains the unit-cofactor derivative. It does not\nassert that $c_r$, $\\alpha_r$, or $\\beta_r$ is rational or algebraic\nover $\\mathbb Q$. The remaining arithmetic obligation is therefore\n\\begin{equation}\n \\delta_r\\ne0\\quad\\text{for every admitted ordinary index in\n \\eqref{eq:root:focusedfamily}.}\n \\label{eq:root:remaining}\n\\end{equation}\nEven this assertion would still leave the terminal-$129$ and higher-address\nobligations in the universal target \\eqref{eq:root:simplicitytarget}.","The latest continuation retains the exact Bernoulli undepletion factor\n\\[\n e_{r,q}=\\frac{1-2^{2r-1}}{1-2^{2r-1}4^q},\\qquad q\\ge0.\n\\]\nIt is a dyadic unit, including $e_{r,0}=1$. Multiplication by this\nfactor preserves the all-$q$ error of the shifted Jacobi moment model.\nConsequently the actual model critical values satisfy, for every $m\\ge25$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\","The latest continuation retains the exact Bernoulli undepletion factor\n\\[\n e_{r,q}=\\frac{1-2^{2r-1}}{1-2^{2r-1}4^q},\\qquad q\\ge0.\n\\]\nIt is a dyadic unit, including $e_{r,0}=1$. Multiplication by this\nfactor preserves the all-$q$ error of the shifted Jacobi moment model.\nConsequently the actual model critical values satisfy, for every $m\\ge25$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{a","The latest continuation retains the exact Bernoulli undepletion factor\n\\[\n e_{r,q}=\\frac{1-2^{2r-1}}{1-2^{2r-1}4^q},\\qquad q\\ge0.\n\\]\nIt is a dyadic unit, including $e_{r,0}=1$. Multiplication by this\nfactor preserves the all-$q$ error of the shifted Jacobi moment model.\nConsequently the actual model critical values satisfy, for every $m\\ge25$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{act}})<L_m.\n \\label{eq:root:stricttest}\n\\end{equation}","The latest continuation retains the exact Bernoulli undepletion factor\n\\[\n e_{r,q}=\\frac{1-2^{2r-1}}{1-2^{2r-1}4^q},\\qquad q\\ge0.\n\\]\nIt is a dyadic unit, including $e_{r,0}=1$. Multiplication by this\nfactor preserves the all-$q$ error of the shifted Jacobi moment model.\nConsequently the actual model critical values satisfy, for every $m\\ge25$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{act}})<L_m.\n \\label{eq:root:stricttest}\n\\end{equation}\nIndeed a passing strict test has the same valuation as the limit. Conversely\nany finite limit valuation is eventual","The latest continuation retains the exact Bernoulli undepletion factor\n\\[\n e_{r,q}=\\frac{1-2^{2r-1}}{1-2^{2r-1}4^q},\\qquad q\\ge0.\n\\]\nIt is a dyadic unit, including $e_{r,0}=1$. Multiplication by this\nfactor preserves the all-$q$ error of the shifted Jacobi moment model.\nConsequently the actual model critical values satisfy, for every $m\\ge25$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{act}})<L_m.\n \\label{eq:root:stricttest}\n\\end{equation}\nIndeed a passing strict test has the same valuation as the limit. Conversely\nany finite limit valuation is eventually smaller than $L_m$.\nThe existence of a passing depth f…","oment model.\nConsequently the actual model critical values satisfy, for every $m\\ge25$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{act}})<L_m.\n \\label{eq:root:stricttest}\n\\end{equation}\nIndeed a passing strict test has the same valuation as the limit. Conversely\nany finite limit valuation is eventually smaller than $L_m$.\nThe existence of a passing depth for every ordinary index is unproved.\nNonzero models at every depth are compatible with an exactly zero limit.","ual model critical values satisfy, for every $m\\ge25$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{act}})<L_m.\n \\label{eq:root:stricttest}\n\\end{equation}\nIndeed a passing strict test has the same valuation as the limit. Conversely\nany finite limit valuation is eventually smaller than $L_m$.\nThe existence of a passing depth for every ordinary index is unproved.\nNonzero models at every depth are compatible with an exactly zero limit.","$,\n\\begin{equation}\n v_2\\bigl(\\Phi_{r,m}^{\\mathrm{act}}-\\Phi_r^{\\mathrm{act}}\\bigr)\n \\ge L_m=8m-4s_2(m)-178,\n \\qquad L_m\\longrightarrow\\infty.\n \\label{eq:root:actualapprox}\n\\end{equation}\nThe estimate includes every packet and controls motion of the critical\npoint. There is no ceiling at the undepletion exponent $2r-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{act}})<L_m.\n \\label{eq:root:stricttest}\n\\end{equation}\nIndeed a passing strict test has the same valuation as the limit. Conversely\nany finite limit valuation is eventually smaller than $L_m$.\nThe existence of a passing depth for every ordinary index is unproved.\nNonzero models at every depth are compatible with an exactly zero limit.","-25$.\nFor each fixed admitted $r$ it implies the exact equivalence\n\\begin{equation}\n \\Phi_r^{\\mathrm{act}}\\ne0\n \\quad\\Longleftrightarrow\\quad\n \\text{there exists }m\\ge25\\text{ such that }\n v_2(\\Phi_{r,m}^{\\mathrm{act}})<L_m.\n \\label{eq:root:stricttest}\n\\end{equation}\nIndeed a passing strict test has the same valuation as the limit. Conversely\nany finite limit valuation is eventually smaller than $L_m$.\nThe existence of a passing depth for every ordinary index is unproved.\nNonzero models at every depth are compatible with an exactly zero limit.","The degree-two preparation also reduces the old representative-height\ncost to a linear cost, about $2P$ bits at precision $P$. This remains\nlarger than the $P$ bits needed for a strict nonzero residue certificate.\nThe fixed rational control $(Z-1/3)^2$ has zero critical value while\nits least-residue integer discriminant representatives are nonzero at\nevery even $P\\ge6$. Thus a nonzero integer representative alone cannot\nprove a nonzero dyadic discriminant.","One sufficient new arithmetic input would be an odd-denominator\nprojective quadratic\n\\[\n A Z^2+2BZ+C\\equiv A\\mathcal W_r(Z)\\pmod{2^P},\\qquad A\\text{ odd},\n \\qquad 0<|B^2-AC|<2^P,\n\\]\nat a proved precision for every admitted ordinary $r$.\nThen $(B^2-AC)/A^2\\equiv\\delta_r\\pmod{2^P}$ cannot vanish\nmodulo $2^P$. No depth-independent theorem supplying this inequality\nhas been established. The tested recurrence and exterior-contact\nidentities retain balancing terms and do not force a contradiction.\nFinite exponential models approximate the actual target at every depth,\nbut their order grows with precision, their clearing polynomial has\nzeros on the actual chart, and their proved construction-specific height\ncost does not give the required discriminant bound.","Sharp uniform ordinary Euler tail exponent $4M+3$\n & Proved, with earlier formal coverage","Universal simplicity of every $G_r$\n & Open","Throughout these sections, the normalized source and its reciprocal are\n\\begin{equation}\n G_{N,j}(t)=j!\\sum_{n\\geq0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n F_{N,j}(t)=G_{N,j}(t)^{-1}\n   =\\sum_{n\\geq0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!},\n \\label{eq:final-normalization}\n\\end{equation}\nwhere $N\\geq1$ and $0\\leq j<N$. In particular, $C_{N,j}(0)=1$.\nThe coefficients in the original convention satisfy\n\\begin{equation}\n E_{Nn}^{(N,j)}=j!C_{N,j}(n),\\qquad\n E_d^{(N,j)}=0\\quad(N\\nmid d).\n \\label{eq:final-original-normalization}\n\\end{equation}\nThus every valuation of a nonzero difference gains $v_p(j!)$ on returning\nto the original coefficients. This factor must be retained when stating\noptimal periods or the final congruence.","hroughout these sections, the normalized source and its reciprocal are\n\\begin{equation}\n G_{N,j}(t)=j!\\sum_{n\\geq0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n F_{N,j}(t)=G_{N,j}(t)^{-1}\n   =\\sum_{n\\geq0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!},\n \\label{eq:final-normalization}\n\\end{equation}\nwhere $N\\geq1$ and $0\\leq j<N$. In particular, $C_{N,j}(0)=1$.\nThe coefficients in the original convention satisfy\n\\begin{equation}\n E_{Nn}^{(N,j)}=j!C_{N,j}(n),\\qquad\n E_d^{(N,j)}=0\\quad(N\\nmid d).\n \\label{eq:final-original-normalization}\n\\end{equation}\nThus every valuation of a nonzero difference gains $v_p(j!)$ on returning\nto the original coefficients. This factor must be retained when stating\noptimal periods or the final congruence.","=\\sum_{n\\geq0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!},\n \\label{eq:final-normalization}\n\\end{equation}\nwhere $N\\geq1$ and $0\\leq j<N$. In particular, $C_{N,j}(0)=1$.\nThe coefficients in the original convention satisfy\n\\begin{equation}\n E_{Nn}^{(N,j)}=j!C_{N,j}(n),\\qquad\n E_d^{(N,j)}=0\\quad(N\\nmid d).\n \\label{eq:final-original-normalization}\n\\end{equation}\nThus every valuation of a nonzero difference gains $v_p(j!)$ on returning\nto the original coefficients. This factor must be retained when stating\noptimal periods or the final congruence.","Write $S f(n)=f(n+1)$ and $\\Delta_H=S^H-1$. A mixed difference\n$\\Delta_{H_1}\\cdots\\Delta_{H_k}$ acts on the row variable $n$, rather than\non the full Euler subscript. We use $v_p(0)=+\\infty$. Orders and starting\nrows are nonnegative integers. Steps are positive integers unless stated\notherwise. An empty mixed difference is the original value, and a zero\nordinary step makes every nonempty product containing it zero.","\\begin{definition}[Hurwitz integral series]\nFor a prime $p$, let\n\\[\n \\mathcal H_p=\\left\\{\\sum_{d\\geq0}f_d\\frac{t^d}{d!}:\n                    f_d\\in\\mathbb Z_p\\right\\}\n       \\subset\\mathbb Q_p[[t]].\n\\]\nFor $N=p^a$ and $\\lambda\\in\\mathbb Z_p$, put\n\\[\n D=\\frac{d}{dt},\\qquad L_\\lambda=\\frac{D^N-\\lambda}{p}.\n\\]\nA series $A$ is $\\lambda$-flat if $L_\\lambda^k A\\in\\mathcal H_p$ for every\n$k\\geq0$.\n\\end{definition}","Products preserve $\\mathcal H_p$ because the coefficient at factorial\ndegree $d$ is\n\\[\n (AB)_d=\\sum_{i=0}^d\\binom di a_i b_{d-i}.\n\\]\nThe derivative $D$ acts by the coefficient shift $(DA)_d=a_{d+1}$.\nConsequently every nonnegative power of $D$ preserves the lattice.\nIf $A\\in\\mathcal H_p$ has unit constant term, its reciprocal belongs to\nthe same lattice: with $B=A^{-1}$,\n\\begin{equation}\n b_0=a_0^{-1},\\qquad\n b_d=-a_0^{-1}\\sum_{i=1}^d\\binom di a_i b_{d-i}.\n \\label{eq:hurwitz-reciprocal-recurrence}\n\\end{equation}\nIf $A$ is supported at degrees divisible by $N$, this recurrence also\nproves that $B$ has the same support.","_{d-i}.\n\\]\nThe derivative $D$ acts by the coefficient shift $(DA)_d=a_{d+1}$.\nConsequently every nonnegative power of $D$ preserves the lattice.\nIf $A\\in\\mathcal H_p$ has unit constant term, its reciprocal belongs to\nthe same lattice: with $B=A^{-1}$,\n\\begin{equation}\n b_0=a_0^{-1},\\qquad\n b_d=-a_0^{-1}\\sum_{i=1}^d\\binom di a_i b_{d-i}.\n \\label{eq:hurwitz-reciprocal-recurrence}\n\\end{equation}\nIf $A$ is supported at degrees divisible by $N$, this recurrence also\nproves that $B$ has the same support.","coefficient shift $(DA)_d=a_{d+1}$.\nConsequently every nonnegative power of $D$ preserves the lattice.\nIf $A\\in\\mathcal H_p$ has unit constant term, its reciprocal belongs to\nthe same lattice: with $B=A^{-1}$,\n\\begin{equation}\n b_0=a_0^{-1},\\qquad\n b_d=-a_0^{-1}\\sum_{i=1}^d\\binom di a_i b_{d-i}.\n \\label{eq:hurwitz-reciprocal-recurrence}\n\\end{equation}\nIf $A$ is supported at degrees divisible by $N$, this recurrence also\nproves that $B$ has the same support.","onsequently every nonnegative power of $D$ preserves the lattice.\nIf $A\\in\\mathcal H_p$ has unit constant term, its reciprocal belongs to\nthe same lattice: with $B=A^{-1}$,\n\\begin{equation}\n b_0=a_0^{-1},\\qquad\n b_d=-a_0^{-1}\\sum_{i=1}^d\\binom di a_i b_{d-i}.\n \\label{eq:hurwitz-reciprocal-recurrence}\n\\end{equation}\nIf $A$ is supported at degrees divisible by $N$, this recurrence also\nproves that $B$ has the same support.","\\begin{lemma}[Translated product formula]\n\\label{lem:translated-hurwitz-product}\nFor $N=p^a$, $a\\geq1$, and $\\lambda,\\mu\\in\\mathbb Z_p$,\n\\begin{equation}\n L_{\\lambda+\\mu}(AB)\n  =(L_\\lambda A)B+A(L_\\mu B)\n       +\\sum_{i=1}^{N-1}c_i(D^iA)(D^{N-i}B),\n \\qquad c_i=\\frac1p\\binom Ni\\in\\mathbb Z.\n \\label{eq:translated-product}\n\\end{equation}\nFlatness types add under multiplication.\n\\end{lemma}","\\begin{proof}\nThe identity $(1+X)^{p^a}=1+X^{p^a}$ over $\\mathbb F_p$ shows that every\ninterior binomial coefficient $\\binom Ni$ is divisible by $p$.\nSubtract $(\\lambda+\\mu)AB$ from the ordinary $N$th derivative product\nformula and divide by $p$ to obtain \\eqref{eq:translated-product}.","For completeness, the iterated formula is an integral multinomial\nformula, rather than a product rule with unspecified remainders. On the\ntensor of the two factors use the mutually commuting operators\n\\[\n L_\\lambda\\otimes1,\\quad 1\\otimes L_\\mu,\\quad\n c_iD^i\\otimes D^{N-i}\\quad(1\\leq i<N).\n\\]\nApplying multiplication after the $k$th power of their sum gives exactly\n$L_{\\lambda+\\mu}^k(AB)$. Expanding it yields terms indexed by\n$a,b,e_1,\\ldots,e_{N-1}\\geq0$, with $a+b+\\sum_i e_i=k$:\n\\begin{equation}\n \\frac{k!}{a!b!\\prod_i e_i!}\\prod_i c_i^{e_i}\n (D^\\alpha L_\\lambda^a A)(D^\\beta L_\\mu^b B),\n \\quad\n \\alpha=\\sum_i i e_i,\\quad\n \\beta=\\sum_i(N-i)e_i.\n \\label{eq:translated-multinomial}\n\\end{equation}\nEvery scalar is integral. If $A$ and $B$ have the stated flatness types,\nboth factors in every term belong to $\\mathcal H_p$, as does their product.\n\\end{proof}","d formula is an integral multinomial\nformula, rather than a product rule with unspecified remainders. On the\ntensor of the two factors use the mutually commuting operators\n\\[\n L_\\lambda\\otimes1,\\quad 1\\otimes L_\\mu,\\quad\n c_iD^i\\otimes D^{N-i}\\quad(1\\leq i<N).\n\\]\nApplying multiplication after the $k$th power of their sum gives exactly\n$L_{\\lambda+\\mu}^k(AB)$. Expanding it yields terms indexed by\n$a,b,e_1,\\ldots,e_{N-1}\\geq0$, with $a+b+\\sum_i e_i=k$:\n\\begin{equation}\n \\frac{k!}{a!b!\\prod_i e_i!}\\prod_i c_i^{e_i}\n (D^\\alpha L_\\lambda^a A)(D^\\beta L_\\mu^b B),\n \\quad\n \\alpha=\\sum_i i e_i,\\quad\n \\beta=\\sum_i(N-i)e_i.\n \\label{eq:translated-multinomial}\n\\end{equation}\nEvery scalar is integral. If $A$ and $B$ have the stated flatness types,\nboth factors in every term belong to $\\mathcal H_p$, as does their product.\n\\end{proof}","formula, rather than a product rule with unspecified remainders. On the\ntensor of the two factors use the mutually commuting operators\n\\[\n L_\\lambda\\otimes1,\\quad 1\\otimes L_\\mu,\\quad\n c_iD^i\\otimes D^{N-i}\\quad(1\\leq i<N).\n\\]\nApplying multiplication after the $k$th power of their sum gives exactly\n$L_{\\lambda+\\mu}^k(AB)$. Expanding it yields terms indexed by\n$a,b,e_1,\\ldots,e_{N-1}\\geq0$, with $a+b+\\sum_i e_i=k$:\n\\begin{equation}\n \\frac{k!}{a!b!\\prod_i e_i!}\\prod_i c_i^{e_i}\n (D^\\alpha L_\\lambda^a A)(D^\\beta L_\\mu^b B),\n \\quad\n \\alpha=\\sum_i i e_i,\\quad\n \\beta=\\sum_i(N-i)e_i.\n \\label{eq:translated-multinomial}\n\\end{equation}\nEvery scalar is integral. If $A$ and $B$ have the stated flatness types,\nboth factors in every term belong to $\\mathcal H_p$, as does their product.\n\\end{proof}","e with unspecified remainders. On the\ntensor of the two factors use the mutually commuting operators\n\\[\n L_\\lambda\\otimes1,\\quad 1\\otimes L_\\mu,\\quad\n c_iD^i\\otimes D^{N-i}\\quad(1\\leq i<N).\n\\]\nApplying multiplication after the $k$th power of their sum gives exactly\n$L_{\\lambda+\\mu}^k(AB)$. Expanding it yields terms indexed by\n$a,b,e_1,\\ldots,e_{N-1}\\geq0$, with $a+b+\\sum_i e_i=k$:\n\\begin{equation}\n \\frac{k!}{a!b!\\prod_i e_i!}\\prod_i c_i^{e_i}\n (D^\\alpha L_\\lambda^a A)(D^\\beta L_\\mu^b B),\n \\quad\n \\alpha=\\sum_i i e_i,\\quad\n \\beta=\\sum_i(N-i)e_i.\n \\label{eq:translated-multinomial}\n\\end{equation}\nEvery scalar is integral. If $A$ and $B$ have the stated flatness types,\nboth factors in every term belong to $\\mathcal H_p$, as does their product.\n\\end{proof}","th power of their sum gives exactly\n$L_{\\lambda+\\mu}^k(AB)$. Expanding it yields terms indexed by\n$a,b,e_1,\\ldots,e_{N-1}\\geq0$, with $a+b+\\sum_i e_i=k$:\n\\begin{equation}\n \\frac{k!}{a!b!\\prod_i e_i!}\\prod_i c_i^{e_i}\n (D^\\alpha L_\\lambda^a A)(D^\\beta L_\\mu^b B),\n \\quad\n \\alpha=\\sum_i i e_i,\\quad\n \\beta=\\sum_i(N-i)e_i.\n \\label{eq:translated-multinomial}\n\\end{equation}\nEvery scalar is integral. If $A$ and $B$ have the stated flatness types,\nboth factors in every term belong to $\\mathcal H_p$, as does their product.\n\\end{proof}","of their sum gives exactly\n$L_{\\lambda+\\mu}^k(AB)$. Expanding it yields terms indexed by\n$a,b,e_1,\\ldots,e_{N-1}\\geq0$, with $a+b+\\sum_i e_i=k$:\n\\begin{equation}\n \\frac{k!}{a!b!\\prod_i e_i!}\\prod_i c_i^{e_i}\n (D^\\alpha L_\\lambda^a A)(D^\\beta L_\\mu^b B),\n \\quad\n \\alpha=\\sum_i i e_i,\\quad\n \\beta=\\sum_i(N-i)e_i.\n \\label{eq:translated-multinomial}\n\\end{equation}\nEvery scalar is integral. If $A$ and $B$ have the stated flatness types,\nboth factors in every term belong to $\\mathcal H_p$, as does their product.\n\\end{proof}","terms indexed by\n$a,b,e_1,\\ldots,e_{N-1}\\geq0$, with $a+b+\\sum_i e_i=k$:\n\\begin{equation}\n \\frac{k!}{a!b!\\prod_i e_i!}\\prod_i c_i^{e_i}\n (D^\\alpha L_\\lambda^a A)(D^\\beta L_\\mu^b B),\n \\quad\n \\alpha=\\sum_i i e_i,\\quad\n \\beta=\\sum_i(N-i)e_i.\n \\label{eq:translated-multinomial}\n\\end{equation}\nEvery scalar is integral. If $A$ and $B$ have the stated flatness types,\nboth factors in every term belong to $\\mathcal H_p$, as does their product.\n\\end{proof}","\\begin{lemma}[Reciprocal transfer]\n\\label{lem:translated-hurwitz-inverse}\nIf $A$ is $\\lambda$-flat and has unit constant term, then $A^{-1}$ is\n$(-\\lambda)$-flat.\n\\end{lemma}","\\begin{proof}\nSet $B=A^{-1}$. Its order-zero integrality was established in\n\\eqref{eq:hurwitz-reciprocal-recurrence}. Assume by induction that\n$L_{-\\lambda}^bB\\in\\mathcal H_p$ for $b<k$. Since $AB=1$,\n\\[\n L_0^k(AB)=\\left(\\frac{D^N}{p}\\right)^k1=0\\qquad(k>0).\n\\]\nUse \\eqref{eq:translated-multinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathca","\\begin{proof}\nSet $B=A^{-1}$. Its order-zero integrality was established in\n\\eqref{eq:hurwitz-reciprocal-recurrence}. Assume by induction that\n$L_{-\\lambda}^bB\\in\\mathcal H_p$ for $b<k$. Since $AB=1$,\n\\[\n L_0^k(AB)=\\left(\\frac{D^N}{p}\\right)^k1=0\\qquad(k>0).\n\\]\nUse \\eqref{eq:translated-multinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proo","\\begin{proof}\nSet $B=A^{-1}$. Its order-zero integrality was established in\n\\eqref{eq:hurwitz-reciprocal-recurrence}. Assume by induction that\n$L_{-\\lambda}^bB\\in\\mathcal H_p$ for $b<k$. Since $AB=1$,\n\\[\n L_0^k(AB)=\\left(\\frac{D^N}{p}\\right)^k1=0\\qquad(k>0).\n\\]\nUse \\eqref{eq:translated-multinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proof}","begin{proof}\nSet $B=A^{-1}$. Its order-zero integrality was established in\n\\eqref{eq:hurwitz-reciprocal-recurrence}. Assume by induction that\n$L_{-\\lambda}^bB\\in\\mathcal H_p$ for $b<k$. Since $AB=1$,\n\\[\n L_0^k(AB)=\\left(\\frac{D^N}{p}\\right)^k1=0\\qquad(k>0).\n\\]\nUse \\eqref{eq:translated-multinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proof}","o integrality was established in\n\\eqref{eq:hurwitz-reciprocal-recurrence}. Assume by induction that\n$L_{-\\lambda}^bB\\in\\mathcal H_p$ for $b<k$. Since $AB=1$,\n\\[\n L_0^k(AB)=\\left(\\frac{D^N}{p}\\right)^k1=0\\qquad(k>0).\n\\]\nUse \\eqref{eq:translated-multinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proof}","r $b<k$. Since $AB=1$,\n\\[\n L_0^k(AB)=\\left(\\frac{D^N}{p}\\right)^k1=0\\qquad(k>0).\n\\]\nUse \\eqref{eq:translated-multinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proof}","eft(\\frac{D^N}{p}\\right)^k1=0\\qquad(k>0).\n\\]\nUse \\eqref{eq:translated-multinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proof}","tinomial} with $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proof}","ith $\\mu=-\\lambda$.\nExactly one term has inverse order $b=k$: it has $a=0$ and all $e_i=0$,\nand is $A L_{-\\lambda}^k B$. In every other term $b<k$, so its second\nfactor is integral by induction and derivative-shift integrality. Its\nfirst factor is integral by the flatness of $A$, and its scalar is\nintegral. Hence $A L_{-\\lambda}^k B\\in\\mathcal H_p$. Multiplying by the\nalready integral $B$ proves $L_{-\\lambda}^k B\\in\\mathcal H_p$.\nThis induction takes place in the entire Hurwitz lattice. It places no\nrestriction on the coefficient degree or the starting row.\n\\end{proof}","The packed factorial coefficient of $G_{N,j}$ is\n\\begin{equation}\n g_{N,j}(n)=\\frac{j!(Nn)!}{(Nn+j)!}\n   =\\prod_{i=1}^j\\left(1+\\frac Ni n\\right)^{-1}.\n \\label{eq:actual-prime-power-kernel}\n\\end{equation}\nThe empty product is one. If $N=p^a$ and $1\\leq i<N$, then\n$v_p(i)<a$, so $N/i\\in p\\mathbb Z_p$. Every affine denominator in\n\\eqref{eq:actual-prime-power-kernel} is therefore a unit on $\\mathbb Z_p$.","For $s\\in p\\mathbb Z_p$, an exact finite-difference identity is\n\\begin{equation}\n \\Delta_1^k\\frac1{1+sn}\n  =\\frac{(-s)^k k!}{\\prod_{h=0}^k(1+s(n+h))}.\n \\label{eq:affine-reciprocal-exact}\n\\end{equation}\nIt follows by induction, subtracting adjacent rational functions. Each\nsuccessive subtraction adds the factor $-(k+1)s$ in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$","For $s\\in p\\mathbb Z_p$, an exact finite-difference identity is\n\\begin{equation}\n \\Delta_1^k\\frac1{1+sn}\n  =\\frac{(-s)^k k!}{\\prod_{h=0}^k(1+s(n+h))}.\n \\label{eq:affine-reciprocal-exact}\n\\end{equation}\nIt follows by induction, subtracting adjacent rational functions. Each\nsuccessive subtraction adds the factor $-(k+1)s$ in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","ence identity is\n\\begin{equation}\n \\Delta_1^k\\frac1{1+sn}\n  =\\frac{(-s)^k k!}{\\prod_{h=0}^k(1+s(n+h))}.\n \\label{eq:affine-reciprocal-exact}\n\\end{equation}\nIt follows by induction, subtracting adjacent rational functions. Each\nsuccessive subtraction adds the factor $-(k+1)s$ in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","It follows by induction, subtracting adjacent rational functions. Each\nsuccessive subtraction adds the factor $-(k+1)s$ in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","nal functions. Each\nsuccessive subtraction adds the factor $-(k+1)s$ in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","btraction adds the factor $-(k+1)s$ in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","n adds the factor $-(k+1)s$ in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","in the numerator.\nIts denominator is a unit, so it belongs to $p^k\\mathbb Z_p$.\nThe discrete Leibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","ibniz formula\n\\begin{equation}\n \\Delta_1^k(fg)(n)\n =\\sum_{r=0}^k\\binom kr\n       (\\Delta_1^r f)(n)(\\Delta_1^{k-r}g)(n+r)\n \\label{eq:discrete-leibniz-final}\n\\end{equation}\nthen proves $\\Delta_1^k g_{N,j}(n)\\in p^k\\mathbb Z_p$ for every $k,n$.\nThe coefficient at degree $Nn$ of $L_1^kG_{N,j}$ is precisely\n$p^{-k}\\Delta_1^k g_{N,j}(n)$, and the other residue classes vanish.\nThus the actual source is $1$-flat.","\\begin{theorem}[All-row prime-power transfer]\n\\label{thm:all-prime-power-transfer}\nLet $p$ be prime, $a\\geq1$, $N=p^a$, and $0\\leq j<N$.\nThe coefficients $C_{N,j}(n)$ belong to $\\mathbb Z_p$ for all $n\\geq0$.\nSet $A_{N,j}(n)=(-1)^n C_{N,j}(n)$ and\n$T_H=S^H-(-1)^H$. For $k\\geq1$ and positive $H_1,\\ldots,H_k$,\n\\begin{align}\n v_p(\\Delta_{H_1}\\cdots\\Delta_{H_k}A_{N,j}(n))\n  &\\geq k+\\sum_{\\nu=1}^k v_p(H_\\nu),\n \\label{eq:prime-power-signed-bound}","\\begin{theorem}[All-row prime-power transfer]\n\\label{thm:all-prime-power-transfer}\nLet $p$ be prime, $a\\geq1$, $N=p^a$, and $0\\leq j<N$.\nThe coefficients $C_{N,j}(n)$ belong to $\\mathbb Z_p$ for all $n\\geq0$.\nSet $A_{N,j}(n)=(-1)^n C_{N,j}(n)$ and\n$T_H=S^H-(-1)^H$. For $k\\geq1$ and positive $H_1,\\ldots,H_k$,\n\\begin{align}\n v_p(\\Delta_{H_1}\\cdots\\Delta_{H_k}A_{N,j}(n))\n  &\\geq k+\\sum_{\\nu=1}^k v_p(H_\\nu),\n \\label{eq:prime-power-signed-bound}\\\\\n v_p(T_{H_1}\\cdots T_{H_k}E_{Nn}^{(N,j)})\n  &\\geq v_p(j!)+k+\\sum_{\\nu=1}^k v_p(H_\\nu).\n \\label{eq:prime-power-original-twist}\n\\end{align}\nAt $p=2$, the ordinary differences of $C_{N,j}$ and of the original\ncoefficients satisfy the same bounds. These dyadic statements also hold\nfor $N=1$.\n\\end{theorem}","v_p(T_{H_1}\\cdots T_{H_k}E_{Nn}^{(N,j)})\n  &\\geq v_p(j!)+k+\\sum_{\\nu=1}^k v_p(H_\\nu).\n \\label{eq:prime-power-original-twist}\n\\end{align}\nAt $p=2$, the ordinary differences of $C_{N,j}$ and of the original\ncoefficients satisfy the same bounds. These dyadic statements also hold\nfor $N=1$.\n\\end{theorem}","\\begin{proof}\nReciprocal transfer gives $L_{-1}^kF_{N,j}\\in\\mathcal H_p$.\nExtraction of the packed coefficient gives\n$(S+1)^k C_{N,j}(n)\\in p^k\\mathbb Z_p$. Moreover,\n\\[\n \\Delta_1^k A_{N,j}(n)=(-1)^{n+k}(S+1)^k C_{N,j}(n),\n\\]\nso $A_{N,j}$ has the ordinary $p^k$ difference property.\nAt $p=2$, $L_1=L_{-1}-1$. Its integral binomial expansion on\n$F_{N,j}$ gives $\\Delta_1^k C_{N,j}(n)\\in2^k\\mathbb Z_2$ directly.\nFor $N=1$ the interior sum in the product formula is empty, so the same\nargument applies without change. Equivalently, $F_{1,0}(t)=e^{-t}$.","ciprocal transfer gives $L_{-1}^kF_{N,j}\\in\\mathcal H_p$.\nExtraction of the packed coefficient gives\n$(S+1)^k C_{N,j}(n)\\in p^k\\mathbb Z_p$. Moreover,\n\\[\n \\Delta_1^k A_{N,j}(n)=(-1)^{n+k}(S+1)^k C_{N,j}(n),\n\\]\nso $A_{N,j}$ has the ordinary $p^k$ difference property.\nAt $p=2$, $L_1=L_{-1}-1$. Its integral binomial expansion on\n$F_{N,j}$ gives $\\Delta_1^k C_{N,j}(n)\\in2^k\\mathbb Z_2$ directly.\nFor $N=1$ the interior sum in the product formula is empty, so the same\nargument applies without change. Equivalently, $F_{1,0}(t)=e^{-t}$.","j}(n)\\in p^k\\mathbb Z_p$. Moreover,\n\\[\n \\Delta_1^k A_{N,j}(n)=(-1)^{n+k}(S+1)^k C_{N,j}(n),\n\\]\nso $A_{N,j}$ has the ordinary $p^k$ difference property.\nAt $p=2$, $L_1=L_{-1}-1$. Its integral binomial expansion on\n$F_{N,j}$ gives $\\Delta_1^k C_{N,j}(n)\\in2^k\\mathbb Z_2$ directly.\nFor $N=1$ the interior sum in the product formula is empty, so the same\nargument applies without change. Equivalently, $F_{1,0}(t)=e^{-t}$.","To retain the valuation of a step, let $B$ be any sequence satisfying\n$\\Delta_1^m B\\in p^m\\mathbb Z_p$ at every order and row. Use the finite\noperator identity\n\\[\n \\Delta_H=\\sum_{i=1}^H\\binom Hi\\Delta_1^i.\n\\]\nThe relation $i\\binom Hi=H\\binom{H-1}{i-1}$ implies\n\\[\n v_p\\left(p^i\\binom Hi\\right)\n \\geq i+v_p(H)-v_p(i)\\geq1+v_p(H).\n\\]\nMultiplying these identities for the $k$ steps, every term of total\norder $M$ has its scalar multiplied by $p^M$, and is divisible by\n$p^{k+\\sum_\\nu v_p(H_\\nu)}$. This proves the mixed-step bound.\nFinally,\n\\[\n \\Delta_H A_{N,j}(n)=(-1)^{n+H}T_H C_{N,j}(n).\n\\]\nIteration contributes only a global sign. Multiplying by $j!$ proves\n\\eqref{eq:prime-power-original-twist}.\n\\end{proof}","in the valuation of a step, let $B$ be any sequence satisfying\n$\\Delta_1^m B\\in p^m\\mathbb Z_p$ at every order and row. Use the finite\noperator identity\n\\[\n \\Delta_H=\\sum_{i=1}^H\\binom Hi\\Delta_1^i.\n\\]\nThe relation $i\\binom Hi=H\\binom{H-1}{i-1}$ implies\n\\[\n v_p\\left(p^i\\binom Hi\\right)\n \\geq i+v_p(H)-v_p(i)\\geq1+v_p(H).\n\\]\nMultiplying these identities for the $k$ steps, every term of total\norder $M$ has its scalar multiplied by $p^M$, and is divisible by\n$p^{k+\\sum_\\nu v_p(H_\\nu)}$. This proves the mixed-step bound.\nFinally,\n\\[\n \\Delta_H A_{N,j}(n)=(-1)^{n+H}T_H C_{N,j}(n).\n\\]\nIteration contributes only a global sign. Multiplying by $j!$ proves\n\\eqref{eq:prime-power-original-twist}.\n\\end{proof}","B$ be any sequence satisfying\n$\\Delta_1^m B\\in p^m\\mathbb Z_p$ at every order and row. Use the finite\noperator identity\n\\[\n \\Delta_H=\\sum_{i=1}^H\\binom Hi\\Delta_1^i.\n\\]\nThe relation $i\\binom Hi=H\\binom{H-1}{i-1}$ implies\n\\[\n v_p\\left(p^i\\binom Hi\\right)\n \\geq i+v_p(H)-v_p(i)\\geq1+v_p(H).\n\\]\nMultiplying these identities for the $k$ steps, every term of total\norder $M$ has its scalar multiplied by $p^M$, and is divisible by\n$p^{k+\\sum_\\nu v_p(H_\\nu)}$. This proves the mixed-step bound.\nFinally,\n\\[\n \\Delta_H A_{N,j}(n)=(-1)^{n+H}T_H C_{N,j}(n).\n\\]\nIteration contributes only a global sign. Multiplying by $j!$ proves\n\\eqref{eq:prime-power-original-twist}.\n\\end{proof}","ng\n$\\Delta_1^m B\\in p^m\\mathbb Z_p$ at every order and row. Use the finite\noperator identity\n\\[\n \\Delta_H=\\sum_{i=1}^H\\binom Hi\\Delta_1^i.\n\\]\nThe relation $i\\binom Hi=H\\binom{H-1}{i-1}$ implies\n\\[\n v_p\\left(p^i\\binom Hi\\right)\n \\geq i+v_p(H)-v_p(i)\\geq1+v_p(H).\n\\]\nMultiplying these identities for the $k$ steps, every term of total\norder $M$ has its scalar multiplied by $p^M$, and is divisible by\n$p^{k+\\sum_\\nu v_p(H_\\nu)}$. This proves the mixed-step bound.\nFinally,\n\\[\n \\Delta_H A_{N,j}(n)=(-1)^{n+H}T_H C_{N,j}(n).\n\\]\nIteration contributes only a global sign. Multiplying by $j!$ proves\n\\eqref{eq:prime-power-original-twist}.\n\\end{proof}","e\noperator identity\n\\[\n \\Delta_H=\\sum_{i=1}^H\\binom Hi\\Delta_1^i.\n\\]\nThe relation $i\\binom Hi=H\\binom{H-1}{i-1}$ implies\n\\[\n v_p\\left(p^i\\binom Hi\\right)\n \\geq i+v_p(H)-v_p(i)\\geq1+v_p(H).\n\\]\nMultiplying these identities for the $k$ steps, every term of total\norder $M$ has its scalar multiplied by $p^M$, and is divisible by\n$p^{k+\\sum_\\nu v_p(H_\\nu)}$. This proves the mixed-step bound.\nFinally,\n\\[\n \\Delta_H A_{N,j}(n)=(-1)^{n+H}T_H C_{N,j}(n).\n\\]\nIteration contributes only a global sign. Multiplying by $j!$ proves\n\\eqref{eq:prime-power-original-twist}.\n\\end{proof}","1}$ implies\n\\[\n v_p\\left(p^i\\binom Hi\\right)\n \\geq i+v_p(H)-v_p(i)\\geq1+v_p(H).\n\\]\nMultiplying these identities for the $k$ steps, every term of total\norder $M$ has its scalar multiplied by $p^M$, and is divisible by\n$p^{k+\\sum_\\nu v_p(H_\\nu)}$. This proves the mixed-step bound.\nFinally,\n\\[\n \\Delta_H A_{N,j}(n)=(-1)^{n+H}T_H C_{N,j}(n).\n\\]\nIteration contributes only a global sign. Multiplying by $j!$ proves\n\\eqref{eq:prime-power-original-twist}.\n\\end{proof}","\\begin{corollary}[Threshold-free dyadic congruence]\n\\label{cor:all-dyadic-final}\nFor $N=2^a$, $a\\geq0$, all permitted $j$, and $n\\geq0$,\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}E_{Nn}^{(N,j)})\n  \\geq v_2(j!)+k+\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:all-dyadic-final-bound}\n\\end{equation}\nIn particular, $\\Delta_H^k$ gains $k(1+v_2(H))$ bits before normalization.\nFor $r>v_2(j!)$, the row displacement\n$H=2^{\\max(0,r-v_2(j!)-1)}$ is a sufficient period modulo $2^r$ from\nrow zero. If $r\\leq v_2(j!)$, all original coefficients vanish modulo\n$2^r$.\n\\end{corollary}","ary}[Threshold-free dyadic congruence]\n\\label{cor:all-dyadic-final}\nFor $N=2^a$, $a\\geq0$, all permitted $j$, and $n\\geq0$,\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}E_{Nn}^{(N,j)})\n  \\geq v_2(j!)+k+\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:all-dyadic-final-bound}\n\\end{equation}\nIn particular, $\\Delta_H^k$ gains $k(1+v_2(H))$ bits before normalization.\nFor $r>v_2(j!)$, the row displacement\n$H=2^{\\max(0,r-v_2(j!)-1)}$ is a sufficient period modulo $2^r$ from\nrow zero. If $r\\leq v_2(j!)$, all original coefficients vanish modulo\n$2^r$.\n\\end{corollary}","ic-final}\nFor $N=2^a$, $a\\geq0$, all permitted $j$, and $n\\geq0$,\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}E_{Nn}^{(N,j)})\n  \\geq v_2(j!)+k+\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:all-dyadic-final-bound}\n\\end{equation}\nIn particular, $\\Delta_H^k$ gains $k(1+v_2(H))$ bits before normalization.\nFor $r>v_2(j!)$, the row displacement\n$H=2^{\\max(0,r-v_2(j!)-1)}$ is a sufficient period modulo $2^r$ from\nrow zero. If $r\\leq v_2(j!)$, all original coefficients vanish modulo\n$2^r$.\n\\end{corollary}","a\\geq0$, all permitted $j$, and $n\\geq0$,\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}E_{Nn}^{(N,j)})\n  \\geq v_2(j!)+k+\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:all-dyadic-final-bound}\n\\end{equation}\nIn particular, $\\Delta_H^k$ gains $k(1+v_2(H))$ bits before normalization.\nFor $r>v_2(j!)$, the row displacement\n$H=2^{\\max(0,r-v_2(j!)-1)}$ is a sufficient period modulo $2^r$ from\nrow zero. If $r\\leq v_2(j!)$, all original coefficients vanish modulo\n$2^r$.\n\\end{corollary}","}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}E_{Nn}^{(N,j)})\n  \\geq v_2(j!)+k+\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:all-dyadic-final-bound}\n\\end{equation}\nIn particular, $\\Delta_H^k$ gains $k(1+v_2(H))$ bits before normalization.\nFor $r>v_2(j!)$, the row displacement\n$H=2^{\\max(0,r-v_2(j!)-1)}$ is a sufficient period modulo $2^r$ from\nrow zero. If $r\\leq v_2(j!)$, all original coefficients vanish modulo\n$2^r$.\n\\end{corollary}","\\begin{remark}[Necessary boundaries]\nAt an odd prime the ordinary step-one difference cannot replace the\nsign-twisted operator in the general theorem. For $j=0$,\n$C_{N,0}(0)=1$ and $C_{N,0}(1)=-1$, so their ordinary difference is\n$-2$, a unit at every odd prime. The proof also does not extend\nunchanged to non-prime-power $N$, whose interior binomial coefficients\nneed not be divisible by the chosen prime. Neither observation is a\ncounterexample to Nishibuchi's original eventual congruence.\n\\end{remark}","Write $C_j=C_{4,j}$ and $F_j=F_{4,j}$. The all-row theorem removes the\nthreshold, but at $N=4$ it leaves a further factorial gain unused.\nThe exact algebra below proves that gain for all four addresses and\nidentifies several infinite families where it is sharp.","\\begin{theorem}[Four-address factorial bound]\n\\label{thm:four-factorial-bound}\nFor $j\\in\\{0,1,2,3\\}$, $n\\geq0$, $k\\geq1$, and positive mixed steps,\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}C_j(n))\n \\geq k+v_2(k!)+\\sum_{\\nu=1}^k v_2(H_\\nu)+\\beta_j,\n \\quad (\\beta_0,\\beta_1,\\beta_2,\\beta_3)=(0,0,3,1).\n \\label{eq:factorial-four-lower}\n\\end{equation}\nAddress two also satisfies\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}C_2(n))\n \\geq 2k+v_2(k!)+\\left\\lfloor\\frac{k}{2}\\right\\rfloor\n             +\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:factorial-four-j2-filtration}\n\\end{equation}\nFor the original coefficients, the offsets in\n\\eqref{eq:factorial-four-lower} are $(0,0,4,2)$.\nNo positive-order offset is asserted for the empty difference.\n\\end{theorem}","egin{theorem}[Four-address factorial bound]\n\\label{thm:four-factorial-bound}\nFor $j\\in\\{0,1,2,3\\}$, $n\\geq0$, $k\\geq1$, and positive mixed steps,\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}C_j(n))\n \\geq k+v_2(k!)+\\sum_{\\nu=1}^k v_2(H_\\nu)+\\beta_j,\n \\quad (\\beta_0,\\beta_1,\\beta_2,\\beta_3)=(0,0,3,1).\n \\label{eq:factorial-four-lower}\n\\end{equation}\nAddress two also satisfies\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}C_2(n))\n \\geq 2k+v_2(k!)+\\left\\lfloor\\frac{k}{2}\\right\\rfloor\n             +\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:factorial-four-j2-filtration}\n\\end{equation}\nFor the original coefficients, the offsets in\n\\eqref{eq:factorial-four-lower} are $(0,0,4,2)$.\nNo positive-order offset is asserted for the empty difference.\n\\end{theorem}","_1,\\beta_2,\\beta_3)=(0,0,3,1).\n \\label{eq:factorial-four-lower}\n\\end{equation}\nAddress two also satisfies\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}C_2(n))\n \\geq 2k+v_2(k!)+\\left\\lfloor\\frac{k}{2}\\right\\rfloor\n             +\\sum_{\\nu=1}^k v_2(H_\\nu).\n \\label{eq:factorial-four-j2-filtration}\n\\end{equation}\nFor the original coefficients, the offsets in\n\\eqref{eq:factorial-four-lower} are $(0,0,4,2)$.\nNo positive-order offset is asserted for the empty difference.\n\\end{theorem}","The proof of the first two addresses uses one classical arithmetic\ninput: Frobenius's exact congruence for the ordinary Euler numbers in\nthe $\\operatorname{sech}$ convention, as stated and derived by\nGessel~\\cite{gessel}, Section~9. The address-two and address-three\nbounds below are finite Bernoulli calculations and do not require that\ninput. A divided-power argument is necessary: ordinary integral moments\nalone do not justify division by $k!$.","e classical arithmetic\ninput: Frobenius's exact congruence for the ordinary Euler numbers in\nthe $\\operatorname{sech}$ convention, as stated and derived by\nGessel~\\cite{gessel}, Section~9. The address-two and address-three\nbounds below are finite Bernoulli calculations and do not require that\ninput. A divided-power argument is necessary: ordinary integral moments\nalone do not justify division by $k!$.","Let $\\mathcal L$ be the ordinary Euler evaluation functional on a\nvariable $\\mathcal E$, defined by\n\\[\n \\mathcal L(\\mathcal E^d)=\\operatorname{Euler}_d,\n \\qquad \\sum_{d\\geq0}\\operatorname{Euler}_d\\frac{t^d}{d!}\n                  =\\operatorname{sech}t.\n\\]\nThe functional is linear. It is not a multiplicative evaluation.\nIndependent variables mean that moments of polynomials in separate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.","Let $\\mathcal L$ be the ordinary Euler evaluation functional on a\nvariable $\\mathcal E$, defined by\n\\[\n \\mathcal L(\\mathcal E^d)=\\operatorname{Euler}_d,\n \\qquad \\sum_{d\\geq0}\\operatorname{Euler}_d\\frac{t^d}{d!}\n                  =\\operatorname{sech}t.\n\\]\nThe functional is linear. It is not a multiplicative evaluation.\nIndependent variables mean that moments of polynomials in separate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ s","Let $\\mathcal L$ be the ordinary Euler evaluation functional on a\nvariable $\\mathcal E$, defined by\n\\[\n \\mathcal L(\\mathcal E^d)=\\operatorname{Euler}_d,\n \\qquad \\sum_{d\\geq0}\\operatorname{Euler}_d\\frac{t^d}{d!}\n                  =\\operatorname{sech}t.\n\\]\nThe functional is linear. It is not a multiplicative evaluation.\nIndependent variables mean that moments of polynomials in separate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an ind","ariable $\\mathcal E$, defined by\n\\[\n \\mathcal L(\\mathcal E^d)=\\operatorname{Euler}_d,\n \\qquad \\sum_{d\\geq0}\\operatorname{Euler}_d\\frac{t^d}{d!}\n                  =\\operatorname{sech}t.\n\\]\nThe functional is linear. It is not a multiplicative evaluation.\nIndependent variables mean that moments of polynomials in separate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","he functional is linear. It is not a multiplicative evaluation.\nIndependent variables mean that moments of polynomials in separate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","dependent variables mean that moments of polynomials in separate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","at moments of polynomials in separate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","parate\nvariables factor. Odd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","dd moments vanish. Frobenius's congruence is\n\\begin{equation}\n v_2\\left(\\sum_{r=0}^m(-1)^{m-r}\\binom mr\n                 \\operatorname{Euler}_{2n+2r}\\right)\n       =m+v_2(m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","m!)\\qquad(n,m\\geq0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","0).\n \\label{eq:classical-frobenius-exact}\n\\end{equation}\nFor $m=0$, the assertion is that every even Euler number is odd.\nWith $P=(\\mathcal E^2-1)/2$ and\n$\\gamma_m(P)=P^m/m!$, the case $n=0$ says\n\\begin{equation}\n \\mathcal L(\\gamma_m(P))\\in\\mathbb Z_2^{\\times}\n                 \\qquad(m\\geq0).\n \\label{eq:euler-pd-odd-moments}\n\\end{equation}\nIntroduce an independent Euler variable $\\mathcal F$ and\n$Q=(\\mathcal F^2-1)/2$.","Put $K=\\mathbb Q_2(i)$, $R=\\mathbb Z_2[i]$,\n$a=(1+i)/2$, and $b=(1-i)/2$. The exact identity\n\\begin{equation}\n F_0(t)=\\frac{2}{\\cosh t+\\cos t}\n       =\\operatorname{sech}(at)\\operatorname{sech}(bt)\n \\label{eq:four-euler-factorization}\n\\end{equation}\ngives, for $U=a\\mathcal E+b\\mathcal F$,\n$\\mathcal L(U^{4n})=C_0(n)$. Although $a$ and $b$ have negative\nvaluation, their denominators disappear from the following exact\npolynomial identities:\n\\begin{align}\n U^2&=\\mathcal E\\mathcal F+i(P-Q),\\nonumber",", and $b=(1-i)/2$. The exact identity\n\\begin{equation}\n F_0(t)=\\frac{2}{\\cosh t+\\cos t}\n       =\\operatorname{sech}(at)\\operatorname{sech}(bt)\n \\label{eq:four-euler-factorization}\n\\end{equation}\ngives, for $U=a\\mathcal E+b\\mathcal F$,\n$\\mathcal L(U^{4n})=C_0(n)$. Although $a$ and $b$ have negative\nvaluation, their denominators disappear from the following exact\npolynomial identities:\n\\begin{align}\n U^2&=\\mathcal E\\mathcal F+i(P-Q),\\nonumber\\\\\n H_0:=\\frac{U^4-1}{2}\n   &=P+Q-\\frac{P^2}{2}-\\frac{Q^2}{2}\n                 +3PQ+i\\mathcal E\\mathcal F(P-Q).\n \\label{eq:four-euler-H}\n\\end{align}","Let $\\Gamma_R(P,Q)$ be the $R$-subalgebra generated by all\n$\\gamma_m(P)$ and $\\gamma_r(Q)$. Adjoin $\\mathcal E,\\mathcal F$ with\n\\[\n \\mathcal E^2=1+2P,\\qquad \\mathcal F^2=1+2Q,\n\\]\nand call the resulting algebra $\\mathcal A_0$. Its elements are finite\nexpressions. No completion is being presumed. It is generated as a\n$\\Gamma_R(P,Q)$-module by $1,\\mathcal E,\\mathcal F,\\mathcal E\\mathcal F$.\nThe functional maps this module to $R$. For an even basis term this is\n\\eqref{eq:euler-pd-odd-moments} and independence. For a basis term with\nan odd Euler factor the moment is zero, since $P$ and $Q$ are even\npolynomials in their respective variables.","$R$-subalgebra generated by all\n$\\gamma_m(P)$ and $\\gamma_r(Q)$. Adjoin $\\mathcal E,\\mathcal F$ with\n\\[\n \\mathcal E^2=1+2P,\\qquad \\mathcal F^2=1+2Q,\n\\]\nand call the resulting algebra $\\mathcal A_0$. Its elements are finite\nexpressions. No completion is being presumed. It is generated as a\n$\\Gamma_R(P,Q)$-module by $1,\\mathcal E,\\mathcal F,\\mathcal E\\mathcal F$.\nThe functional maps this module to $R$. For an even basis term this is\n\\eqref{eq:euler-pd-odd-moments} and independence. For a basis term with\nan odd Euler factor the moment is zero, since $P$ and $Q$ are even\npolynomials in their respective variables.","d call the resulting algebra $\\mathcal A_0$. Its elements are finite\nexpressions. No completion is being presumed. It is generated as a\n$\\Gamma_R(P,Q)$-module by $1,\\mathcal E,\\mathcal F,\\mathcal E\\mathcal F$.\nThe functional maps this module to $R$. For an even basis term this is\n\\eqref{eq:euler-pd-odd-moments} and independence. For a basis term with\nan odd Euler factor the moment is zero, since $P$ and $Q$ are even\npolynomials in their respective variables.","he resulting algebra $\\mathcal A_0$. Its elements are finite\nexpressions. No completion is being presumed. It is generated as a\n$\\Gamma_R(P,Q)$-module by $1,\\mathcal E,\\mathcal F,\\mathcal E\\mathcal F$.\nThe functional maps this module to $R$. For an even basis term this is\n\\eqref{eq:euler-pd-odd-moments} and independence. For a basis term with\nan odd Euler factor the moment is zero, since $P$ and $Q$ are even\npolynomials in their respective variables.","Every displayed component of $H_0$ has all its divided powers in\n$\\mathcal A_0$. Besides the generators, the needed identities are\n\\begin{align}\n \\gamma_m(P^2/2)\n     &=\\frac{(2m)!}{2^m m!}\\gamma_{2m}(P),\\nonumber","Every displayed component of $H_0$ has all its divided powers in\n$\\mathcal A_0$. Besides the generators, the needed identities are\n\\begin{align}\n \\gamma_m(P^2/2)\n     &=\\frac{(2m)!}{2^m m!}\\gamma_{2m}(P),\\nonumber\\\\\n \\gamma_m(3PQ)&=3^m m!\\gamma_m(P)\\gamma_m(Q),\\nonumber\\\\\n \\gamma_m(P-Q)&=\\sum_{r=0}^m(-1)^r\n                   \\gamma_{m-r}(P)\\gamma_r(Q).\n \\label{eq:components-divided-powers}\n\\end{align}\nThe first scalar counts partitions into pairs and is an integer.\nFor the last term of $H_0$, multiply the third identity by\n$(i\\mathcal E\\mathcal F)^m$ and reduce the Euler powers using the\nintegral quadratic relations. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$","\\gamma_m(P-Q)&=\\sum_{r=0}^m(-1)^r\n                   \\gamma_{m-r}(P)\\gamma_r(Q).\n \\label{eq:components-divided-powers}\n\\end{align}\nThe first scalar counts partitions into pairs and is an integer.\nFor the last term of $H_0$, multiply the third identity by\n$(i\\mathcal E\\mathcal F)^m$ and reduce the Euler powers using the\nintegral quadratic relations. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","-1)^r\n                   \\gamma_{m-r}(P)\\gamma_r(Q).\n \\label{eq:components-divided-powers}\n\\end{align}\nThe first scalar counts partitions into pairs and is an integer.\nFor the last term of $H_0$, multiply the third identity by\n$(i\\mathcal E\\mathcal F)^m$ and reduce the Euler powers using the\nintegral quadratic relations. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","onents-divided-powers}\n\\end{align}\nThe first scalar counts partitions into pairs and is an integer.\nFor the last term of $H_0$, multiply the third identity by\n$(i\\mathcal E\\mathcal F)^m$ and reduce the Euler powers using the\nintegral quadratic relations. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","he first scalar counts partitions into pairs and is an integer.\nFor the last term of $H_0$, multiply the third identity by\n$(i\\mathcal E\\mathcal F)^m$ and reduce the Euler powers using the\nintegral quadratic relations. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","hcal E\\mathcal F)^m$ and reduce the Euler powers using the\nintegral quadratic relations. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","atic relations. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","ons. Finally,\n\\[\n \\gamma_k(X+Y)=\\sum_{r=0}^k\\gamma_r(X)\\gamma_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","_{k-r}(Y)\n\\]\nproves $\\gamma_k(H_0)\\in\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","n\\mathcal A_0$. Applying the functional and\nputting $W=U^4-1=2H_0$ gives\n\\begin{equation}\n \\mathcal L(W^m)\\in2^m m!R\\qquad(m\\geq0).\n \\label{eq:euler-W-factorial}\n\\end{equation}\nThese moments are rational. The identity\n$R\\cap\\mathbb Q_2=\\mathbb Z_2$, obtained from the linear independence\nof $1,i$ over $\\mathbb Q_2$, descends every such divisibility statement\nto $\\mathbb Z_2$. No field property of $R/2R$ is needed.","For every starting row,\n\\begin{equation}\n \\Delta_1^k C_0(n)\n   =\\mathcal L(U^{4n}W^k)\n   =\\sum_{s=0}^n\\binom ns\\mathcal L(W^{k+s}).\n \\label{eq:base-row-finite-expansion}\n\\end{equation}\nEach summand lies in $2^k k!\\mathbb Z_2$, proving the factorial\nbound for address zero from row zero onward.","The conversion from unit steps to arbitrary mixed steps is used several\ntimes, so it is useful to make it explicit. Write $\\Delta=\\Delta_1$.\nFor $H>0$,\n\\begin{equation}\n \\Delta_H=\\sum_{m=1}^H\\binom Hm\\Delta^m.\n \\label{eq:finite-step-expansion}\n\\end{equation}\nA mixed term with local orders $m_1,\\ldots,m_k\\geq1$ and\n$M=\\sum_\\nu m_\\nu$ therefore has valuation at least\n\\[\n M+v_2(M!)+\\beta\n    +\\sum_\\nu\\bigl(v_2(H_\\nu)-v_2(m_\\nu)\\bigr)\n\\]\nif the unit-order estimate has offset $\\beta$ for every positive order.\nThe inequalities\n\\[\n m-1\\geq v_2(m),\\qquad M\\geq k,\\qquad v_2(M!)\\geq v_2(k!)\n\\]\nmake this at least $k+v_2(k!)+\\beta+\\sum_\\nu v_2(H_\\nu)$.\nEvery expansion is finite. This proves the desired mixed-step transfer\nwithout an exchange of limits.",", so it is useful to make it explicit. Write $\\Delta=\\Delta_1$.\nFor $H>0$,\n\\begin{equation}\n \\Delta_H=\\sum_{m=1}^H\\binom Hm\\Delta^m.\n \\label{eq:finite-step-expansion}\n\\end{equation}\nA mixed term with local orders $m_1,\\ldots,m_k\\geq1$ and\n$M=\\sum_\\nu m_\\nu$ therefore has valuation at least\n\\[\n M+v_2(M!)+\\beta\n    +\\sum_\\nu\\bigl(v_2(H_\\nu)-v_2(m_\\nu)\\bigr)\n\\]\nif the unit-order estimate has offset $\\beta$ for every positive order.\nThe inequalities\n\\[\n m-1\\geq v_2(m),\\qquad M\\geq k,\\qquad v_2(M!)\\geq v_2(k!)\n\\]\nmake this at least $k+v_2(k!)+\\beta+\\sum_\\nu v_2(H_\\nu)$.\nEvery expansion is finite. This proves the desired mixed-step transfer\nwithout an exchange of limits.","=\\Delta_1$.\nFor $H>0$,\n\\begin{equation}\n \\Delta_H=\\sum_{m=1}^H\\binom Hm\\Delta^m.\n \\label{eq:finite-step-expansion}\n\\end{equation}\nA mixed term with local orders $m_1,\\ldots,m_k\\geq1$ and\n$M=\\sum_\\nu m_\\nu$ therefore has valuation at least\n\\[\n M+v_2(M!)+\\beta\n    +\\sum_\\nu\\bigl(v_2(H_\\nu)-v_2(m_\\nu)\\bigr)\n\\]\nif the unit-order estimate has offset $\\beta$ for every positive order.\nThe inequalities\n\\[\n m-1\\geq v_2(m),\\qquad M\\geq k,\\qquad v_2(M!)\\geq v_2(k!)\n\\]\nmake this at least $k+v_2(k!)+\\beta+\\sum_\\nu v_2(H_\\nu)$.\nEvery expansion is finite. This proves the desired mixed-step transfer\nwithout an exchange of limits.","\\label{eq:finite-step-expansion}\n\\end{equation}\nA mixed term with local orders $m_1,\\ldots,m_k\\geq1$ and\n$M=\\sum_\\nu m_\\nu$ therefore has valuation at least\n\\[\n M+v_2(M!)+\\beta\n    +\\sum_\\nu\\bigl(v_2(H_\\nu)-v_2(m_\\nu)\\bigr)\n\\]\nif the unit-order estimate has offset $\\beta$ for every positive order.\nThe inequalities\n\\[\n m-1\\geq v_2(m),\\qquad M\\geq k,\\qquad v_2(M!)\\geq v_2(k!)\n\\]\nmake this at least $k+v_2(k!)+\\beta+\\sum_\\nu v_2(H_\\nu)$.\nEvery expansion is finite. This proves the desired mixed-step transfer\nwithout an exchange of limits.","Let $B_m$ denote the Bernoulli numbers with $B_1=-1/2$.\nThe identity obtained by substituting $u=e^t-1$ into\n$\\log(1+u)/u$ is\n\\begin{equation}\n B_m=\\sum_{r=0}^m(-1)^r\\frac{r!}{r+1}\n                                  \\left\\{\\begin{matrix}m","Let $B_m$ denote the Bernoulli numbers with $B_1=-1/2$.\nThe identity obtained by substituting $u=e^t-1$ into\n$\\log(1+u)/u$ is\n\\begin{equation}\n B_m=\\sum_{r=0}^m(-1)^r\\frac{r!}{r+1}\n                                  \\left\\{\\begin{matrix}m\\\\r\\end{matrix}\\right\\},\n \\label{eq:bernoulli-stirling-denominator}\n\\end{equation}\nwhere the braces are Stirling numbers of the second kind. Each\n$2r!/(r+1)$ is dyadically integral. Indeed, if $s=v_2(r+1)>0$, then\n$r\\geq2^s-1$ and\n\\[\n v_2(r!)\\geq v_2((2^s-1)!)=2^s-1-s\\geq s-1.\n\\]\nThe last inequality is $2^s\\geq2s$. The case $s=0$ is immediate.\nConsequently\n\\begin{equation}\n 2B_m\\in\\mathbb Z_2\\qquad(m\\geq0).\n \\label{eq:elementary-bernoulli-bound}\n\\end{equation}\nThus the transfer factors below require no appeal to the\nvon Staudt--Clausen theorem.","r\\end{matrix}\\right\\},\n \\label{eq:bernoulli-stirling-denominator}\n\\end{equation}\nwhere the braces are Stirling numbers of the second kind. Each\n$2r!/(r+1)$ is dyadically integral. Indeed, if $s=v_2(r+1)>0$, then\n$r\\geq2^s-1$ and\n\\[\n v_2(r!)\\geq v_2((2^s-1)!)=2^s-1-s\\geq s-1.\n\\]\nThe last inequality is $2^s\\geq2s$. The case $s=0$ is immediate.\nConsequently\n\\begin{equation}\n 2B_m\\in\\mathbb Z_2\\qquad(m\\geq0).\n \\label{eq:elementary-bernoulli-bound}\n\\end{equation}\nThus the transfer factors below require no appeal to the\nvon Staudt--Clausen theorem.","The address-one source is\n$G_{4,1}(t)=(\\sinh t+\\sin t)/(2t)$. The identities $a+b=1$,\n$a-b=i$, and $ab=1/2$ give\n\\begin{equation}\n D_1(t):=G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1","The address-one source is\n$G_{4,1}(t)=(\\sinh t+\\sin t)/(2t)$. The identities $a+b=1$,\n$a-b=i$, and $ab=1/2$ give\n\\begin{equation}\n D_1(t):=G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernou","The address-one source is\n$G_{4,1}(t)=(\\sinh t+\\sin t)/(2t)$. The identities $a+b=1$,\n$a-b=i$, and $ab=1/2$ give\n\\begin{equation}\n D_1(t):=G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound","The address-one source is\n$G_{4,1}(t)=(\\sinh t+\\sin t)/(2t)$. The identities $a+b=1$,\n$a-b=i$, and $ab=1/2$ give\n\\begin{equation}\n D_1(t):=G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_","The address-one source is\n$G_{4,1}(t)=(\\sinh t+\\sin t)/(2t)$. The identities $a+b=1$,\n$a-b=i$, and $ab=1/2$ give\n\\begin{equation}\n D_1(t):=G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse","The address-one source is\n$G_{4,1}(t)=(\\sinh t+\\sin t)/(2t)$. The identities $a+b=1$,\n$a-b=i$, and $ab=1/2$ give\n\\begin{equation}\n D_1(t):=G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multip…","source is\n$G_{4,1}(t)=(\\sinh t+\\sin t)/(2t)$. The identities $a+b=1$,\n$a-b=i$, and $ab=1/2$ give\n\\begin{equation}\n D_1(t):=G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral…","G_{4,1}(t)F_0(t)\n =\\frac12\\left(\\frac{\\tanh(at)}{at}\n                         +\\frac{\\tanh(bt)}{bt}\\right).\n \\label{eq:address-one-tangent-factor}\n\\end{equation}\nFor example, its numerator follows from\n\\[\n \\sinh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:addre…","nh t+\\sin t\n =2b\\sinh(at)\\cosh(bt)+2a\\cosh(at)\\sinh(bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","bt),\n\\]\nwhile $\\cosh t+\\cos t=2\\cosh(at)\\cosh(bt)$.\nOnly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","nly degrees divisible by four survive. Its actual Hurwitz packets are\n\\begin{equation}\n D_1(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=\\left(-\\frac14\\right)^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.",")^i\n \\frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","\\label{eq:address-one-packets}\n\\end{equation}\nThe constant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","stant is one. Since $v_2(4i+2)=1$, the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","the elementary Bernoulli bound\nshows $v_2(d_i)\\geq2i$. Hurwitz inversion therefore gives\n\\[\n D_1(t)^{-1}=\\sum_{i\\geq0}h_i\\frac{t^{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","{4i}}{(4i)!},\n \\qquad h_0=1,\\quad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","ad v_2(h_i)\\geq2i.\n\\]\nThis follows by induction from the inverse recurrence: for packet index\n$i$, every product $d_rh_{i-r}$ has valuation at least $2i$, and its\nbinomial multiplier is integral.\nBecause $F_1=F_0D_1^{-1}$,\n\\begin{equation}\n C_1(n)-C_0(n)\n     =\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_0(n-i).\n \\label{eq:address-one-exact-transfer}\n\\end{equation}\nAt a natural $n$ this is a finite sum, with $i\\leq n$.","Here is the polynomial estimate that supplies one additional bit.\nFor $i\\geq1$, put $P_i(x)=4^i\\binom{4x}{4i}$. Its constant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nF","Here is the polynomial estimate that supplies one additional bit.\nFor $i\\geq1$, put $P_i(x)=4^i\\binom{4x}{4i}$. Its constant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ va","Here is the polynomial estimate that supplies one additional bit.\nFor $i\\geq1$, put $P_i(x)=4^i\\binom{4x}{4i}$. Its constant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polyno","Here is the polynomial estimate that supplies one additional bit.\nFor $i\\geq1$, put $P_i(x)=4^i\\binom{4x}{4i}$. Its constant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","polynomial estimate that supplies one additional bit.\nFor $i\\geq1$, put $P_i(x)=4^i\\binom{4x}{4i}$. Its constant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","t supplies one additional bit.\nFor $i\\geq1$, put $P_i(x)=4^i\\binom{4x}{4i}$. Its constant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","geq1$, put $P_i(x)=4^i\\binom{4x}{4i}$. Its constant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","onstant is zero,\nand its degree-$d$ coefficient is\n\\begin{equation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","quation}\n [x^d]P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","P_i(x)=\\frac{4^i4^d s(4i,d)}{(4i)!},\n \\label{eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","eq:scaled-binomial-coefficients}\n\\end{equation}\nwhere $s(m,d)$ is a signed Stirling number of the first kind.\nIn each product contributing to $s(4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","4i,d)$, at most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","most $d-1$ of the\n$2i-1$ even integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","integers in $1,\\ldots,4i-1$ can be omitted. Thus\n\\[\n v_2(s(4i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","i,d))\\geq\\max(0,2i-d).\n\\]\nWriting $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d","g $s_2(i)$ for the number of ones in the binary expansion of $i$,\nand using $v_2((4i)!)=4i-s_2(i)$, gives\n\\begin{equation}\n v_2([x^d]P_i)\n \\geq2d-2i+s_2(i)+\\max(0,2i-d)\n \\geq d+s_2(i)\\geq d+1\\quad(d\\geq1).\n \\label{eq:scaled-binomial-degree-bound}\n\\end{equation}\nFor $k\\geq1$, $\\Delta_1^k x^d$ vanishes when $d<k$ and is an\ninteger polynomial with values divisible by $k!$. To see the latter,\nexpand $x^d=\\sum_r\\left\\{\\begin{smallmatrix}d\\\\r\\end{smallmatrix}\\right\\}\nr!\\binom xr$ and apply\n$\\Delta_1^k\\binom xr=\\binom{x}{r-k}$.\nIt follows that\n\\begin{equation}\n \\Delta_1^kP_i(x)\\in2^{k+1}k!\\mathbb Z_2\n                      \\qquad(k\\geq0,\\ x\\in\\mathbb Z_2).\n \\label{eq:scaled-binomial-difference-bound}\n\\end{equation}\nFor $k=0$ the bound follows directly from the coefficient estimate.","r\\end{smallmatrix}\\right\\}\nr!\\binom xr$ and apply\n$\\Delta_1^k\\binom xr=\\binom{x}{r-k}$.\nIt follows that\n\\begin{equation}\n \\Delta_1^kP_i(x)\\in2^{k+1}k!\\mathbb Z_2\n                      \\qquad(k\\geq0,\\ x\\in\\mathbb Z_2).\n \\label{eq:scaled-binomial-difference-bound}\n\\end{equation}\nFor $k=0$ the bound follows directly from the coefficient estimate.","To apply discrete products in \\eqref{eq:address-one-exact-transfer}\nwithout a moving truncation, first extend $C_0$ to $\\mathbb Z_2$ by its\nuniformly convergent Newton series. The all-row bound\n$v_2(\\Delta_1^m C_0(0))\\geq m$ proves this convergence. Its factorial\nbounds extend to every dyadic starting point by continuity and\napproximation with natural rows. The sum in\n\\eqref{eq:address-one-exact-transfer} is then uniformly convergent on\n$\\mathbb Z_2$: both the binomial polynomial and $C_0(x-i)$ are\nintegral-valued and $v_2(h_i)\\geq2i$.","dress-one-exact-transfer}\nwithout a moving truncation, first extend $C_0$ to $\\mathbb Z_2$ by its\nuniformly convergent Newton series. The all-row bound\n$v_2(\\Delta_1^m C_0(0))\\geq m$ proves this convergence. Its factorial\nbounds extend to every dyadic starting point by continuity and\napproximation with natural rows. The sum in\n\\eqref{eq:address-one-exact-transfer} is then uniformly convergent on\n$\\mathbb Z_2$: both the binomial polynomial and $C_0(x-i)$ are\nintegral-valued and $v_2(h_i)\\geq2i$.","ation, first extend $C_0$ to $\\mathbb Z_2$ by its\nuniformly convergent Newton series. The all-row bound\n$v_2(\\Delta_1^m C_0(0))\\geq m$ proves this convergence. Its factorial\nbounds extend to every dyadic starting point by continuity and\napproximation with natural rows. The sum in\n\\eqref{eq:address-one-exact-transfer} is then uniformly convergent on\n$\\mathbb Z_2$: both the binomial polynomial and $C_0(x-i)$ are\nintegral-valued and $v_2(h_i)\\geq2i$.","bb Z_2$ by its\nuniformly convergent Newton series. The all-row bound\n$v_2(\\Delta_1^m C_0(0))\\geq m$ proves this convergence. Its factorial\nbounds extend to every dyadic starting point by continuity and\napproximation with natural rows. The sum in\n\\eqref{eq:address-one-exact-transfer} is then uniformly convergent on\n$\\mathbb Z_2$: both the binomial polynomial and $C_0(x-i)$ are\nintegral-valued and $v_2(h_i)\\geq2i$.","For the pointwise product $P_i(x)C_0(x-i)$, the discrete Leibniz formula\nhas scalar divisibility\n\\[\n \\binom kr(2^{r+1}r!)(2^{k-r}(k-r)!)=2^{k+1}k!.\n\\]\nMultiplication by $h_i/4^i\\in\\mathbb Z_2$ and uniform summation now give\n\\begin{equation}\n v_2(\\Delta_1^k(C_1-C_0)(n))\n               \\geq k+v_2(k!)+1\\qquad(k,n\\geq0).\n \\label{eq:address-one-extra-bit}\n\\end{equation}\nThe same extra bit holds for positive mixed steps by\n\\eqref{eq:finite-step-expansion}. Combining this with the address-zero\nbound proves Theorem~\\ref{thm:four-factorial-bound} for address one.","\\begin{lemma}[Bernoulli divided-power moments]\n\\label{lem:bernoulli-central-pd}\nLet $Z$ be the ordinary Bernoulli variable,\n$\\mathcal L(Z^r)=B_r$, with exponential transform $t/(e^t-1)$.\nFor $P=Z(Z+1)$ and $\\mu_m=\\mathcal L(P^m/m!)$,\n\\begin{equation}\n \\mu_m\\in\\mathbb Z_2,\\qquad\n v_2(\\mu_m)\\geq s_2(m)-1\\quad(m\\geq1).\n \\label{eq:bernoulli-central-moment-bound}\n\\end{equation}\nMoreover, $\\mu_m$ is odd exactly when $m=0$ or $m$ is a positive power\nof two.\n\\end{lemma}","ernoulli divided-power moments]\n\\label{lem:bernoulli-central-pd}\nLet $Z$ be the ordinary Bernoulli variable,\n$\\mathcal L(Z^r)=B_r$, with exponential transform $t/(e^t-1)$.\nFor $P=Z(Z+1)$ and $\\mu_m=\\mathcal L(P^m/m!)$,\n\\begin{equation}\n \\mu_m\\in\\mathbb Z_2,\\qquad\n v_2(\\mu_m)\\geq s_2(m)-1\\quad(m\\geq1).\n \\label{eq:bernoulli-central-moment-bound}\n\\end{equation}\nMoreover, $\\mu_m$ is odd exactly when $m=0$ or $m$ is a positive power\nof two.\n\\end{lemma}","divided-power moments]\n\\label{lem:bernoulli-central-pd}\nLet $Z$ be the ordinary Bernoulli variable,\n$\\mathcal L(Z^r)=B_r$, with exponential transform $t/(e^t-1)$.\nFor $P=Z(Z+1)$ and $\\mu_m=\\mathcal L(P^m/m!)$,\n\\begin{equation}\n \\mu_m\\in\\mathbb Z_2,\\qquad\n v_2(\\mu_m)\\geq s_2(m)-1\\quad(m\\geq1).\n \\label{eq:bernoulli-central-moment-bound}\n\\end{equation}\nMoreover, $\\mu_m$ is odd exactly when $m=0$ or $m$ is a positive power\nof two.\n\\end{lemma}","\\begin{proof}\nDefine the central-factorial basis polynomials\n\\[\n P_r(X)=\\prod_{a=0}^{r-1}(X-a(a+1)).\n\\]\nAt $X=Z(Z+1)$, they have the exact factorization\n\\begin{equation}\n P_r(Z(Z+1))\n   =\\prod_{a=0}^{r-1}(Z-a)(Z+a+1)\n   =(2r)!\\binom{Z+r}{2r}.\n \\label{eq:central-factorial-substitution}\n\\end{equation}\nThe Bernoulli transform gives\n$\\mathcal L((1+T)^Z)=\\log(1+T)/T$, whence\n$\\mathcal L(\\binom Zj)=(-1)^j/(j+1)$. Finite Vandermonde expansion\ntherefore gives\n\\begin{align}\n \\mathcal L\\left(\\binom{Z+r}{2r}\\right)\n &=(-1)^r\\sum_{q=0}^r(-1)^q\\binom rq\\frac1{r+q+1}\\nonumber","X)=\\prod_{a=0}^{r-1}(X-a(a+1)).\n\\]\nAt $X=Z(Z+1)$, they have the exact factorization\n\\begin{equation}\n P_r(Z(Z+1))\n   =\\prod_{a=0}^{r-1}(Z-a)(Z+a+1)\n   =(2r)!\\binom{Z+r}{2r}.\n \\label{eq:central-factorial-substitution}\n\\end{equation}\nThe Bernoulli transform gives\n$\\mathcal L((1+T)^Z)=\\log(1+T)/T$, whence\n$\\mathcal L(\\binom Zj)=(-1)^j/(j+1)$. Finite Vandermonde expansion\ntherefore gives\n\\begin{align}\n \\mathcal L\\left(\\binom{Z+r}{2r}\\right)\n &=(-1)^r\\sum_{q=0}^r(-1)^q\\binom rq\\frac1{r+q+1}\\nonumber\\\\\n &=(-1)^r\\frac{(r!)^2}{(2r+1)!}.\n \\label{eq:central-factorial-evaluation}\n\\end{align}\nThe last equality is the rational identity obtained by expanding\n$\\int_0^1t^r(1-t)^r\\,dt$. This use of an ordinary finite integral makes\nno assertion about a positive or bounded $p$-adic measure. In\nparticular,\n\\[\n \\mathcal L(P_r(P))=(-1)^r\\frac{(r!)^2}{2r+1}.\n\\]","&=(-1)^r\\frac{(r!)^2}{(2r+1)!}.\n \\label{eq:central-factorial-evaluation}\n\\end{align}\nThe last equality is the rational identity obtained by expanding\n$\\int_0^1t^r(1-t)^r\\,dt$. This use of an ordinary finite integral makes\nno assertion about a positive or bounded $p$-adic measure. In\nparticular,\n\\[\n \\mathcal L(P_r(P))=(-1)^r\\frac{(r!)^2}{2r+1}.\n\\]","Expand $X^m=\\sum_{r=0}^mT(m,r)P_r(X)$. Since\n$XP_r(X)=P_{r+1}(X)+r(r+1)P_r(X)$, these integer coefficients satisfy\n\\begin{equation}\n T(m+1,r)=T(m,r-1)+r(r+1)T(m,r),\n \\quad T(0,0)=1.\n \\label{eq:central-factorial-recurrence}\n\\end{equation}\nEvery $r(r+1)$ is even. Induction, including the boundary indices,\nproves $2^{m-r}\\mid T(m,r)$ for $0\\leq r\\leq m$.\nFor $m>0$ the $r=0$ term vanishes. Each remaining summand of\n$\\mathcal L(P^m/m!)$ has valuation at least\n\\begin{equation}\n m-r+2v_2(r!)-v_2(m!)\n   =r+s_2(m)-2s_2(r)\\geq s_2(m)-1\\geq0.\n \\label{eq:central-factorial-term-bound}\n\\end{equation}\nHere $r\\geq2s_2(r)-1$ follows from\n$r\\geq2^{s_2(r)}-1$ and $2^u\\geq2u$ for $u\\geq1$.\nThis proves integrality and the stronger bound.","_{r=0}^mT(m,r)P_r(X)$. Since\n$XP_r(X)=P_{r+1}(X)+r(r+1)P_r(X)$, these integer coefficients satisfy\n\\begin{equation}\n T(m+1,r)=T(m,r-1)+r(r+1)T(m,r),\n \\quad T(0,0)=1.\n \\label{eq:central-factorial-recurrence}\n\\end{equation}\nEvery $r(r+1)$ is even. Induction, including the boundary indices,\nproves $2^{m-r}\\mid T(m,r)$ for $0\\leq r\\leq m$.\nFor $m>0$ the $r=0$ term vanishes. Each remaining summand of\n$\\mathcal L(P^m/m!)$ has valuation at least\n\\begin{equation}\n m-r+2v_2(r!)-v_2(m!)\n   =r+s_2(m)-2s_2(r)\\geq s_2(m)-1\\geq0.\n \\label{eq:central-factorial-term-bound}\n\\end{equation}\nHere $r\\geq2s_2(r)-1$ follows from\n$r\\geq2^{s_2(r)}-1$ and $2^u\\geq2u$ for $u\\geq1$.\nThis proves integrality and the stronger bound.","X)$, these integer coefficients satisfy\n\\begin{equation}\n T(m+1,r)=T(m,r-1)+r(r+1)T(m,r),\n \\quad T(0,0)=1.\n \\label{eq:central-factorial-recurrence}\n\\end{equation}\nEvery $r(r+1)$ is even. Induction, including the boundary indices,\nproves $2^{m-r}\\mid T(m,r)$ for $0\\leq r\\leq m$.\nFor $m>0$ the $r=0$ term vanishes. Each remaining summand of\n$\\mathcal L(P^m/m!)$ has valuation at least\n\\begin{equation}\n m-r+2v_2(r!)-v_2(m!)\n   =r+s_2(m)-2s_2(r)\\geq s_2(m)-1\\geq0.\n \\label{eq:central-factorial-term-bound}\n\\end{equation}\nHere $r\\geq2s_2(r)-1$ follows from\n$r\\geq2^{s_2(r)}-1$ and $2^u\\geq2u$ for $u\\geq1$.\nThis proves integrality and the stronger bound.","-factorial-recurrence}\n\\end{equation}\nEvery $r(r+1)$ is even. Induction, including the boundary indices,\nproves $2^{m-r}\\mid T(m,r)$ for $0\\leq r\\leq m$.\nFor $m>0$ the $r=0$ term vanishes. Each remaining summand of\n$\\mathcal L(P^m/m!)$ has valuation at least\n\\begin{equation}\n m-r+2v_2(r!)-v_2(m!)\n   =r+s_2(m)-2s_2(r)\\geq s_2(m)-1\\geq0.\n \\label{eq:central-factorial-term-bound}\n\\end{equation}\nHere $r\\geq2s_2(r)-1$ follows from\n$r\\geq2^{s_2(r)}-1$ and $2^u\\geq2u$ for $u\\geq1$.\nThis proves integrality and the stronger bound.","uation}\nEvery $r(r+1)$ is even. Induction, including the boundary indices,\nproves $2^{m-r}\\mid T(m,r)$ for $0\\leq r\\leq m$.\nFor $m>0$ the $r=0$ term vanishes. Each remaining summand of\n$\\mathcal L(P^m/m!)$ has valuation at least\n\\begin{equation}\n m-r+2v_2(r!)-v_2(m!)\n   =r+s_2(m)-2s_2(r)\\geq s_2(m)-1\\geq0.\n \\label{eq:central-factorial-term-bound}\n\\end{equation}\nHere $r\\geq2s_2(r)-1$ follows from\n$r\\geq2^{s_2(r)}-1$ and $2^u\\geq2u$ for $u\\geq1$.\nThis proves integrality and the stronger bound.","s even. Induction, including the boundary indices,\nproves $2^{m-r}\\mid T(m,r)$ for $0\\leq r\\leq m$.\nFor $m>0$ the $r=0$ term vanishes. Each remaining summand of\n$\\mathcal L(P^m/m!)$ has valuation at least\n\\begin{equation}\n m-r+2v_2(r!)-v_2(m!)\n   =r+s_2(m)-2s_2(r)\\geq s_2(m)-1\\geq0.\n \\label{eq:central-factorial-term-bound}\n\\end{equation}\nHere $r\\geq2s_2(r)-1$ follows from\n$r\\geq2^{s_2(r)}-1$ and $2^u\\geq2u$ for $u\\geq1$.\nThis proves integrality and the stronger bound.","n, including the boundary indices,\nproves $2^{m-r}\\mid T(m,r)$ for $0\\leq r\\leq m$.\nFor $m>0$ the $r=0$ term vanishes. Each remaining summand of\n$\\mathcal L(P^m/m!)$ has valuation at least\n\\begin{equation}\n m-r+2v_2(r!)-v_2(m!)\n   =r+s_2(m)-2s_2(r)\\geq s_2(m)-1\\geq0.\n \\label{eq:central-factorial-term-bound}\n\\end{equation}\nHere $r\\geq2s_2(r)-1$ follows from\n$r\\geq2^{s_2(r)}-1$ and $2^u\\geq2u$ for $u\\geq1$.\nThis proves integrality and the stronger bound.","If $m$ is not a power of two, $s_2(m)\\geq2$ makes $\\mu_m$ even.\nIf $m$ is a positive power of two, the only central indices which can\ncontribute an odd residue are $r=1$ and $r=3$, since\n$r-2s_2(r)=-1$ only at these two positive integers. The $r=1$ term is\nodd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","al indices which can\ncontribute an odd residue are $r=1$ and $r=3$, since\n$r-2s_2(r)=-1$ only at these two positive integers. The $r=1$ term is\nodd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","bute an odd residue are $r=1$ and $r=3$, since\n$r-2s_2(r)=-1$ only at these two positive integers. The $r=1$ term is\nodd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","and $r=3$, since\n$r-2s_2(r)=-1$ only at these two positive integers. The $r=1$ term is\nodd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","since\n$r-2s_2(r)=-1$ only at these two positive integers. The $r=1$ term is\nodd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","2(r)=-1$ only at these two positive integers. The $r=1$ term is\nodd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","itive integers. The $r=1$ term is\nodd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","odd: $T(m,1)=2^{m-1}$ and $v_2(m!)=m-1$. For $m\\geq4$ a power of\ntwo, the $r=3$ term is even. More explicitly,\n\\[\n \\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6),\n \\qquad\n \\sum_{d\\geq0}h_d(1,3,6)t^d\n       =\\frac1{(1-t)(1-3t)(1-6t)}.\n\\]\nReduction modulo two gives $(1-t)^{-2}$, whose degree $m-3$\ncoefficient has parity $m-2=0$. At $m=1,2$ the $r=3$ term is absent,\nand the $r=1$ term is again odd. Finally $\\mu_0=1$.\n\\end{proof}","The variable $\\mathcal B=2Z+1$ has transform\n\\[\n e^t\\frac{2t}{e^{2t}-1}=\\frac{t}{\\sinh t}.\n\\]\nIts odd moments vanish, and\n$P=(\\mathcal B^2-1)/4=Z(Z+1)$ has the divided-power moments of\nLemma~\\ref{lem:bernoulli-central-pd}. Introduce an independent copy\n$\\mathcal D$ and $Q=(\\mathcal D^2-1)/4$. The exact factorization\n\\begin{equation}\n F_2(t)=\\frac{t^2}{\\cosh t-\\cos t}\n       =\\frac{at}{\\sinh(at)}\\frac{bt}{\\sinh(bt)}\n \\label{eq:address-two-factorization}\n\\end{equation}\nshows that $U=a\\mathcal B+b\\mathcal D$ satisfies\n$\\mathcal L(U^{4n})=C_2(n)$. Direct expansion gives\n\\begin{equation}\n H_2:=\\frac{U^4-1}{4}\n     =P+Q-P^2-Q^2+6PQ+i\\mathcal B\\mathcal D(P-Q).\n \\label{eq:address-two-H}\n\\end{equation}","nh t}.\n\\]\nIts odd moments vanish, and\n$P=(\\mathcal B^2-1)/4=Z(Z+1)$ has the divided-power moments of\nLemma~\\ref{lem:bernoulli-central-pd}. Introduce an independent copy\n$\\mathcal D$ and $Q=(\\mathcal D^2-1)/4$. The exact factorization\n\\begin{equation}\n F_2(t)=\\frac{t^2}{\\cosh t-\\cos t}\n       =\\frac{at}{\\sinh(at)}\\frac{bt}{\\sinh(bt)}\n \\label{eq:address-two-factorization}\n\\end{equation}\nshows that $U=a\\mathcal B+b\\mathcal D$ satisfies\n$\\mathcal L(U^{4n})=C_2(n)$. Direct expansion gives\n\\begin{equation}\n H_2:=\\frac{U^4-1}{4}\n     =P+Q-P^2-Q^2+6PQ+i\\mathcal B\\mathcal D(P-Q).\n \\label{eq:address-two-H}\n\\end{equation}","$P=(\\mathcal B^2-1)/4=Z(Z+1)$ has the divided-power moments of\nLemma~\\ref{lem:bernoulli-central-pd}. Introduce an independent copy\n$\\mathcal D$ and $Q=(\\mathcal D^2-1)/4$. The exact factorization\n\\begin{equation}\n F_2(t)=\\frac{t^2}{\\cosh t-\\cos t}\n       =\\frac{at}{\\sinh(at)}\\frac{bt}{\\sinh(bt)}\n \\label{eq:address-two-factorization}\n\\end{equation}\nshows that $U=a\\mathcal B+b\\mathcal D$ satisfies\n$\\mathcal L(U^{4n})=C_2(n)$. Direct expansion gives\n\\begin{equation}\n H_2:=\\frac{U^4-1}{4}\n     =P+Q-P^2-Q^2+6PQ+i\\mathcal B\\mathcal D(P-Q).\n \\label{eq:address-two-H}\n\\end{equation}","a~\\ref{lem:bernoulli-central-pd}. Introduce an independent copy\n$\\mathcal D$ and $Q=(\\mathcal D^2-1)/4$. The exact factorization\n\\begin{equation}\n F_2(t)=\\frac{t^2}{\\cosh t-\\cos t}\n       =\\frac{at}{\\sinh(at)}\\frac{bt}{\\sinh(bt)}\n \\label{eq:address-two-factorization}\n\\end{equation}\nshows that $U=a\\mathcal B+b\\mathcal D$ satisfies\n$\\mathcal L(U^{4n})=C_2(n)$. Direct expansion gives\n\\begin{equation}\n H_2:=\\frac{U^4-1}{4}\n     =P+Q-P^2-Q^2+6PQ+i\\mathcal B\\mathcal D(P-Q).\n \\label{eq:address-two-H}\n\\end{equation}","Let $\\mathcal A_2$ be the divided-power algebra over $R$ generated by\n$P,Q$, adjoining $\\mathcal B,\\mathcal D$ with\n$\\mathcal B^2=1+4P$, $\\mathcal D^2=1+4Q$.\nThe functional is integral on its module basis\n$1,\\mathcal B,\\mathcal D,\\mathcal B\\mathcal D$ by the Bernoulli lemma,\nindependence, and vanishing of odd moments. Every component in\n\\eqref{eq:address-two-H} has integral divided powers, by the same\nfinite identities as before. $P^2$ is even easier than $P^2/2$.\nConsequently\n\\begin{equation}\n \\mathcal L((U^4-1)^m)\\in4^m m!\\mathbb Z_2,\n \\qquad\n v_2(\\Delta_1^kC_2(n))\\geq2k+v_2(k!).\n \\label{eq:address-two-basic-factorial}\n\\end{equation}\nThe starting-row conclusion follows from the finite base expansion\n\\eqref","Let $\\mathcal A_2$ be the divided-power algebra over $R$ generated by\n$P,Q$, adjoining $\\mathcal B,\\mathcal D$ with\n$\\mathcal B^2=1+4P$, $\\mathcal D^2=1+4Q$.\nThe functional is integral on its module basis\n$1,\\mathcal B,\\mathcal D,\\mathcal B\\mathcal D$ by the Bernoulli lemma,\nindependence, and vanishing of odd moments. Every component in\n\\eqref{eq:address-two-H} has integral divided powers, by the same\nfinite identities as before. $P^2$ is even easier than $P^2/2$.\nConsequently\n\\begin{equation}\n \\mathcal L((U^4-1)^m)\\in4^m m!\\mathbb Z_2,\n \\qquad\n v_2(\\Delta_1^kC_2(n))\\geq2k+v_2(k!).\n \\label{eq:address-two-basic-factorial}\n\\end{equation}\nThe starting-row conclusion follows from the finite base expansion\n\\eqref{eq:base-row-finite-expansion}, now wi","Let $\\mathcal A_2$ be the divided-power algebra over $R$ generated by\n$P,Q$, adjoining $\\mathcal B,\\mathcal D$ with\n$\\mathcal B^2=1+4P$, $\\mathcal D^2=1+4Q$.\nThe functional is integral on its module basis\n$1,\\mathcal B,\\mathcal D,\\mathcal B\\mathcal D$ by the Bernoulli lemma,\nindependence, and vanishing of odd moments. Every component in\n\\eqref{eq:address-two-H} has integral divided powers, by the same\nfinite identities as before. $P^2$ is even easier than $P^2/2$.\nConsequently\n\\begin{equation}\n \\mathcal L((U^4-1)^m)\\in4^m m!\\mathbb Z_2,\n \\qquad\n v_2(\\Delta_1^kC_2(n))\\geq2k+v_2(k!).\n \\label{eq:address-two-basic-factorial}\n\\end{equation}\nThe starting-row conclusion follows from the finite base expansion\n\\eqref{eq:base-row-finite-expansion}, now with $W=4H_2$.","ower algebra over $R$ generated by\n$P,Q$, adjoining $\\mathcal B,\\mathcal D$ with\n$\\mathcal B^2=1+4P$, $\\mathcal D^2=1+4Q$.\nThe functional is integral on its module basis\n$1,\\mathcal B,\\mathcal D,\\mathcal B\\mathcal D$ by the Bernoulli lemma,\nindependence, and vanishing of odd moments. Every component in\n\\eqref{eq:address-two-H} has integral divided powers, by the same\nfinite identities as before. $P^2$ is even easier than $P^2/2$.\nConsequently\n\\begin{equation}\n \\mathcal L((U^4-1)^m)\\in4^m m!\\mathbb Z_2,\n \\qquad\n v_2(\\Delta_1^kC_2(n))\\geq2k+v_2(k!).\n \\label{eq:address-two-basic-factorial}\n\\end{equation}\nThe starting-row conclusion follows from the finite base expansion\n\\eqref{eq:base-row-finite-expansion}, now with $W=4H_2$.","rated by\n$P,Q$, adjoining $\\mathcal B,\\mathcal D$ with\n$\\mathcal B^2=1+4P$, $\\mathcal D^2=1+4Q$.\nThe functional is integral on its module basis\n$1,\\mathcal B,\\mathcal D,\\mathcal B\\mathcal D$ by the Bernoulli lemma,\nindependence, and vanishing of odd moments. Every component in\n\\eqref{eq:address-two-H} has integral divided powers, by the same\nfinite identities as before. $P^2$ is even easier than $P^2/2$.\nConsequently\n\\begin{equation}\n \\mathcal L((U^4-1)^m)\\in4^m m!\\mathbb Z_2,\n \\qquad\n v_2(\\Delta_1^kC_2(n))\\geq2k+v_2(k!).\n \\label{eq:address-two-basic-factorial}\n\\end{equation}\nThe starting-row conclusion follows from the finite base expansion\n\\eqref{eq:base-row-finite-expansion}, now with $W=4H_2$.",",\\mathcal D$ with\n$\\mathcal B^2=1+4P$, $\\mathcal D^2=1+4Q$.\nThe functional is integral on its module basis\n$1,\\mathcal B,\\mathcal D,\\mathcal B\\mathcal D$ by the Bernoulli lemma,\nindependence, and vanishing of odd moments. Every component in\n\\eqref{eq:address-two-H} has integral divided powers, by the same\nfinite identities as before. $P^2$ is even easier than $P^2/2$.\nConsequently\n\\begin{equation}\n \\mathcal L((U^4-1)^m)\\in4^m m!\\mathbb Z_2,\n \\qquad\n v_2(\\Delta_1^kC_2(n))\\geq2k+v_2(k!).\n \\label{eq:address-two-basic-factorial}\n\\end{equation}\nThe starting-row conclusion follows from the finite base expansion\n\\eqref{eq:base-row-finite-expansion}, now with $W=4H_2$.","o-H} has integral divided powers, by the same\nfinite identities as before. $P^2$ is even easier than $P^2/2$.\nConsequently\n\\begin{equation}\n \\mathcal L((U^4-1)^m)\\in4^m m!\\mathbb Z_2,\n \\qquad\n v_2(\\Delta_1^kC_2(n))\\geq2k+v_2(k!).\n \\label{eq:address-two-basic-factorial}\n\\end{equation}\nThe starting-row conclusion follows from the finite base expansion\n\\eqref{eq:base-row-finite-expansion}, now with $W=4H_2$.","The low Newton coefficients of the actual reciprocal are\n\\begin{equation}\n C_2(0)=1,\\quad C_2(1)=-\\frac1{15},\\quad C_2(2)=\\frac{13}{45},\n \\qquad a_1=-\\frac{16}{15},\\quad a_2=\\frac{64}{45},\n \\label{eq:address-two-initial-newton}\n\\end{equation}\nwhere $a_m=\\Delta_1^m C_2(0)$. Thus $v_2(a_1)=4$ and $v_2(a_2)=6$.\nFor $m\\geq3$, \\eqref{eq:address-two-basic-factorial} bounds $a_m$ by\n$2m+v_2(m!)$. The finite shifted Newton formula\n\\begin{equation}\n \\Delta_1^kC_2(n)=\\sum_{m=k}^{n+k}a_m\\binom n{m-k}\n \\label{eq:shifted-newton-finite}\n\\end{equation}\ntherefore gives\n$v_2(\\Delta_1^kC_2(n))\\geq k+v_2(k!)+3$ for every $k\\geq1$.\nFor $k\\geq3$ this is $2k\\geq k+3$. For $k=1,2$ it uses the exact\ninitial coefficients and the higher tail bound. The finite step\nexpansion proves the offset three in\n\\eqref{eq:factorial-four-lower}.","eciprocal are\n\\begin{equation}\n C_2(0)=1,\\quad C_2(1)=-\\frac1{15},\\quad C_2(2)=\\frac{13}{45},\n \\qquad a_1=-\\frac{16}{15},\\quad a_2=\\frac{64}{45},\n \\label{eq:address-two-initial-newton}\n\\end{equation}\nwhere $a_m=\\Delta_1^m C_2(0)$. Thus $v_2(a_1)=4$ and $v_2(a_2)=6$.\nFor $m\\geq3$, \\eqref{eq:address-two-basic-factorial} bounds $a_m$ by\n$2m+v_2(m!)$. The finite shifted Newton formula\n\\begin{equation}\n \\Delta_1^kC_2(n)=\\sum_{m=k}^{n+k}a_m\\binom n{m-k}\n \\label{eq:shifted-newton-finite}\n\\end{equation}\ntherefore gives\n$v_2(\\Delta_1^kC_2(n))\\geq k+v_2(k!)+3$ for every $k\\geq1$.\nFor $k\\geq3$ this is $2k\\geq k+3$. For $k=1,2$ it uses the exact\ninitial coefficients and the higher tail bound. The finite step\nexpansion proves the offset three in\n\\eqref{eq:factorial-four-lower}.","5},\\quad a_2=\\frac{64}{45},\n \\label{eq:address-two-initial-newton}\n\\end{equation}\nwhere $a_m=\\Delta_1^m C_2(0)$. Thus $v_2(a_1)=4$ and $v_2(a_2)=6$.\nFor $m\\geq3$, \\eqref{eq:address-two-basic-factorial} bounds $a_m$ by\n$2m+v_2(m!)$. The finite shifted Newton formula\n\\begin{equation}\n \\Delta_1^kC_2(n)=\\sum_{m=k}^{n+k}a_m\\binom n{m-k}\n \\label{eq:shifted-newton-finite}\n\\end{equation}\ntherefore gives\n$v_2(\\Delta_1^kC_2(n))\\geq k+v_2(k!)+3$ for every $k\\geq1$.\nFor $k\\geq3$ this is $2k\\geq k+3$. For $k=1,2$ it uses the exact\ninitial coefficients and the higher tail bound. The finite step\nexpansion proves the offset three in\n\\eqref{eq:factorial-four-lower}.","-initial-newton}\n\\end{equation}\nwhere $a_m=\\Delta_1^m C_2(0)$. Thus $v_2(a_1)=4$ and $v_2(a_2)=6$.\nFor $m\\geq3$, \\eqref{eq:address-two-basic-factorial} bounds $a_m$ by\n$2m+v_2(m!)$. The finite shifted Newton formula\n\\begin{equation}\n \\Delta_1^kC_2(n)=\\sum_{m=k}^{n+k}a_m\\binom n{m-k}\n \\label{eq:shifted-newton-finite}\n\\end{equation}\ntherefore gives\n$v_2(\\Delta_1^kC_2(n))\\geq k+v_2(k!)+3$ for every $k\\geq1$.\nFor $k\\geq3$ this is $2k\\geq k+3$. For $k=1,2$ it uses the exact\ninitial coefficients and the higher tail bound. The finite step\nexpansion proves the offset three in\n\\eqref{eq:factorial-four-lower}.","n}\n\\end{equation}\nwhere $a_m=\\Delta_1^m C_2(0)$. Thus $v_2(a_1)=4$ and $v_2(a_2)=6$.\nFor $m\\geq3$, \\eqref{eq:address-two-basic-factorial} bounds $a_m$ by\n$2m+v_2(m!)$. The finite shifted Newton formula\n\\begin{equation}\n \\Delta_1^kC_2(n)=\\sum_{m=k}^{n+k}a_m\\binom n{m-k}\n \\label{eq:shifted-newton-finite}\n\\end{equation}\ntherefore gives\n$v_2(\\Delta_1^kC_2(n))\\geq k+v_2(k!)+3$ for every $k\\geq1$.\nFor $k\\geq3$ this is $2k\\geq k+3$. For $k=1,2$ it uses the exact\ninitial coefficients and the higher tail bound. The finite step\nexpansion proves the offset three in\n\\eqref{eq:factorial-four-lower}.","where $a_m=\\Delta_1^m C_2(0)$. Thus $v_2(a_1)=4$ and $v_2(a_2)=6$.\nFor $m\\geq3$, \\eqref{eq:address-two-basic-factorial} bounds $a_m$ by\n$2m+v_2(m!)$. The finite shifted Newton formula\n\\begin{equation}\n \\Delta_1^kC_2(n)=\\sum_{m=k}^{n+k}a_m\\binom n{m-k}\n \\label{eq:shifted-newton-finite}\n\\end{equation}\ntherefore gives\n$v_2(\\Delta_1^kC_2(n))\\geq k+v_2(k!)+3$ for every $k\\geq1$.\nFor $k\\geq3$ this is $2k\\geq k+3$. For $k=1,2$ it uses the exact\ninitial coefficients and the higher tail bound. The finite step\nexpansion proves the offset three in\n\\eqref{eq:factorial-four-lower}.","_1^m C_2(0)$. Thus $v_2(a_1)=4$ and $v_2(a_2)=6$.\nFor $m\\geq3$, \\eqref{eq:address-two-basic-factorial} bounds $a_m$ by\n$2m+v_2(m!)$. The finite shifted Newton formula\n\\begin{equation}\n \\Delta_1^kC_2(n)=\\sum_{m=k}^{n+k}a_m\\binom n{m-k}\n \\label{eq:shifted-newton-finite}\n\\end{equation}\ntherefore gives\n$v_2(\\Delta_1^kC_2(n))\\geq k+v_2(k!)+3$ for every $k\\geq1$.\nFor $k\\geq3$ this is $2k\\geq k+3$. For $k=1,2$ it uses the exact\ninitial coefficients and the higher tail bound. The finite step\nexpansion proves the offset three in\n\\eqref{eq:factorial-four-lower}.","There is a still stronger filtration. Put\n\\[\n X=P+Q,\\qquad \\tau=i\\mathcal B\\mathcal D,\\qquad \\epsilon=1+\\tau.\n\\]\nThen the exact identities are\n\\begin{equation}\n H_2=\\epsilon X+2G,\\quad\n G=-\\gamma_2(P)-\\gamma_2(Q)+3PQ-\\tau Q,\\quad\n \\epsilon^2=2V,\\quad V=\\tau-2X-8PQ.\n \\label{eq:address-two-filtration-algebra}\n\\end{equation}\nEvery divided power of $G$ is integral component by component.\nThe finite addition identity gives\n\\begin{equation}\n \\gamma_k(H_2)=\\sum_{r=0}^k2^r\\gamma_r(G)\n                  \\epsilon^{k-r}\\gamma_{k-r}(X)\n          \\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version o","There is a still stronger filtration. Put\n\\[\n X=P+Q,\\qquad \\tau=i\\mathcal B\\mathcal D,\\qquad \\epsilon=1+\\tau.\n\\]\nThen the exact identities are\n\\begin{equation}\n H_2=\\epsilon X+2G,\\quad\n G=-\\gamma_2(P)-\\gamma_2(Q)+3PQ-\\tau Q,\\quad\n \\epsilon^2=2V,\\quad V=\\tau-2X-8PQ.\n \\label{eq:address-two-filtration-algebra}\n\\end{equation}\nEvery divided power of $G$ is integral component by component.\nThe finite addition identity gives\n\\begin{equation}\n \\gamma_k(H_2)=\\sum_{r=0}^k2^r\\gamma_r(G)\n                  \\epsilon^{k-r}\\gamma_{k-r}(X)\n          \\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\le","There is a still stronger filtration. Put\n\\[\n X=P+Q,\\qquad \\tau=i\\mathcal B\\mathcal D,\\qquad \\epsilon=1+\\tau.\n\\]\nThen the exact identities are\n\\begin{equation}\n H_2=\\epsilon X+2G,\\quad\n G=-\\gamma_2(P)-\\gamma_2(Q)+3PQ-\\tau Q,\\quad\n \\epsilon^2=2V,\\quad V=\\tau-2X-8PQ.\n \\label{eq:address-two-filtration-algebra}\n\\end{equation}\nEvery divided power of $G$ is integral component by component.\nThe finite addition identity gives\n\\begin{equation}\n \\gamma_k(H_2)=\\sum_{r=0}^k2^r\\gamma_r(G)\n                  \\epsilon^{k-r}\\gamma_{k-r}(X)\n          \\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v","Put\n\\[\n X=P+Q,\\qquad \\tau=i\\mathcal B\\mathcal D,\\qquad \\epsilon=1+\\tau.\n\\]\nThen the exact identities are\n\\begin{equation}\n H_2=\\epsilon X+2G,\\quad\n G=-\\gamma_2(P)-\\gamma_2(Q)+3PQ-\\tau Q,\\quad\n \\epsilon^2=2V,\\quad V=\\tau-2X-8PQ.\n \\label{eq:address-two-filtration-algebra}\n\\end{equation}\nEvery divided power of $G$ is integral component by component.\nThe finite addition identity gives\n\\begin{equation}\n \\gamma_k(H_2)=\\sum_{r=0}^k2^r\\gamma_r(G)\n                  \\epsilon^{k-r}\\gamma_{k-r}(X)\n          \\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\\geq0.\n\\]\nIndeed $\\sum_\\nu v_2(m_\\nu)\\leq M-k$. This pr…",",\\quad V=\\tau-2X-8PQ.\n \\label{eq:address-two-filtration-algebra}\n\\end{equation}\nEvery divided power of $G$ is integral component by component.\nThe finite addition identity gives\n\\begin{equation}\n \\gamma_k(H_2)=\\sum_{r=0}^k2^r\\gamma_r(G)\n                  \\epsilon^{k-r}\\gamma_{k-r}(X)\n          \\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\\geq0.\n\\]\nIndeed $\\sum_\\nu v_2(m_\\nu)\\leq M-k$. This proves the stronger mixed\nbound. The earlier offset-three bound is retained, since it is stronger\nat the lowest orders.","\\label{eq:address-two-filtration-algebra}\n\\end{equation}\nEvery divided power of $G$ is integral component by component.\nThe finite addition identity gives\n\\begin{equation}\n \\gamma_k(H_2)=\\sum_{r=0}^k2^r\\gamma_r(G)\n                  \\epsilon^{k-r}\\gamma_{k-r}(X)\n          \\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\\geq0.\n\\]\nIndeed $\\sum_\\nu v_2(m_\\nu)\\leq M-k$. This proves the stronger mixed\nbound. The earlier offset-three bound is retained, since it is stronger\nat the lowest orders.","\\epsilon^{k-r}\\gamma_{k-r}(X)\n          \\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\\geq0.\n\\]\nIndeed $\\sum_\\nu v_2(m_\\nu)\\leq M-k$. This proves the stronger mixed\nbound. The earlier offset-three bound is retained, since it is stronger\nat the lowest orders.","\\in2^{\\lfloor k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\\geq0.\n\\]\nIndeed $\\sum_\\nu v_2(m_\\nu)\\leq M-k$. This proves the stronger mixed\nbound. The earlier offset-three bound is retained, since it is stronger\nat the lowest orders.","k/2\\rfloor}\\mathcal A_2,\n \\label{eq:address-two-filtration-pd}\n\\end{equation}\nbecause the $r$th term belongs to\n$2^{r+\\lfloor(k-r)/2\\rfloor}\\mathcal A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\\geq0.\n\\]\nIndeed $\\sum_\\nu v_2(m_\\nu)\\leq M-k$. This proves the stronger mixed\nbound. The earlier offset-three bound is retained, since it is stronger\nat the lowest orders.","A_2$.\nApplying the functional, rational descent, and the base-index expansion\nproves the unit-step version of\n\\eqref{eq:factorial-four-j2-filtration}.\nFor a mixed term of total order $M$, its lower bound exceeds the target\nat order $k$ by at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor\n          -\\left\\lfloor\\frac k2\\right\\rfloor\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\\geq0.\n\\]\nIndeed $\\sum_\\nu v_2(m_\\nu)\\leq M-k$. This proves the stronger mixed\nbound. The earlier offset-three bound is retained, since it is stronger\nat the lowest orders.","The address-three source is\n$G_{4,3}(t)=3(\\sinh t-\\sin t)/t^3$. Addition identities give\n\\[\n \\sinh t-\\sin t\n =2a\\sinh(at)\\cosh(bt)+2b\\cosh(at)\\sinh(bt),\n \\qquad \\cosh t-\\cos t=2\\sinh(at)\\sinh(bt).\n\\]\nConsequently\n\\begin{equation}\n D_3(t):=G_{4,3}(t)F_2(t)\n =\\frac32\\left(\\frac{\\coth(at)}{at}\n                            +\\frac{\\coth(bt)}{bt}\\right).\n \\label{eq:address-three-cotangent-factor}\n\\end{equation}\nThis expression is a power series: its two polar terms cancel because\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-th","The address-three source is\n$G_{4,3}(t)=3(\\sinh t-\\sin t)/t^3$. Addition identities give\n\\[\n \\sinh t-\\sin t\n =2a\\sinh(at)\\cosh(bt)+2b\\cosh(at)\\sinh(bt),\n \\qquad \\cosh t-\\cos t=2\\sinh(at)\\sinh(bt).\n\\]\nConsequently\n\\begin{equation}\n D_3(t):=G_{4,3}(t)F_2(t)\n =\\frac32\\left(\\frac{\\coth(at)}{at}\n                            +\\frac{\\coth(bt)}{bt}\\right).\n \\label{eq:address-three-cotangent-factor}\n\\end{equation}\nThis expression is a power series: its two polar terms cancel because\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\be","The address-three source is\n$G_{4,3}(t)=3(\\sinh t-\\sin t)/t^3$. Addition identities give\n\\[\n \\sinh t-\\sin t\n =2a\\sinh(at)\\cosh(bt)+2b\\cosh(at)\\sinh(bt),\n \\qquad \\cosh t-\\cos t=2\\sinh(at)\\sinh(bt).\n\\]\nConsequently\n\\begin{equation}\n D_3(t):=G_{4,3}(t)F_2(t)\n =\\frac32\\left(\\frac{\\coth(at)}{at}\n                            +\\frac{\\coth(bt)}{bt}\\right).\n \\label{eq:address-three-cotangent-factor}\n\\end{equation}\nThis expression is a power series: its two polar terms cancel because\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound","ties give\n\\[\n \\sinh t-\\sin t\n =2a\\sinh(at)\\cosh(bt)+2b\\cosh(at)\\sinh(bt),\n \\qquad \\cosh t-\\cos t=2\\sinh(at)\\sinh(bt).\n\\]\nConsequently\n\\begin{equation}\n D_3(t):=G_{4,3}(t)F_2(t)\n =\\frac32\\left(\\frac{\\coth(at)}{at}\n                            +\\frac{\\coth(bt)}{bt}\\right).\n \\label{eq:address-three-cotangent-factor}\n\\end{equation}\nThis expression is a power series: its two polar terms cancel because\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous ext…","ntly\n\\begin{equation}\n D_3(t):=G_{4,3}(t)F_2(t)\n =\\frac32\\left(\\frac{\\coth(at)}{at}\n                            +\\frac{\\coth(bt)}{bt}\\right).\n \\label{eq:address-three-cotangent-factor}\n\\end{equation}\nThis expression is a power series: its two polar terms cancel because\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise…","d{equation}\nThis expression is a power series: its two polar terms cancel because\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation…","its two polar terms cancel because\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra…","e\n$a^{-2}+b^{-2}=0$. Its constant is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra bits, so this gives the\none-bit …","is one, only degrees divisible by four\nsurvive, and the Bernoulli expansion gives\n\\begin{equation}\n D_3(t)=\\sum_{i\\geq0}d_i\\frac{t^{4i}}{(4i)!},\\qquad\n d_i=3\\left(-\\frac14\\right)^i\n          \\frac{2^{4i+2}B_{4i+2}}{(4i+2)(4i+1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra bits, so this gives the\none-bit offset for $C_3$. Finite step expa…","1)}.\n \\label{eq:address-three-packets}\n\\end{equation}\nAgain $v_2(d_i)\\geq2i$, and the inverse coefficients $h_i$ have the same\nbound. Since $F_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra bits, so this gives the\none-bit offset for $C_3$. Finite step expansion finishes the proof of\nTheorem~\\ref{thm:four-factorial-bound}.","_3=F_2D_3^{-1}$,\n\\begin{equation}\n C_3(n)-C_2(n)=\\sum_{i\\geq1}h_i\\binom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra bits, so this gives the\none-bit offset for $C_3$. Finite step expansion finishes the proof of\nTheorem~\\ref{thm:four-factorial-bound}.","nom{4n}{4i}C_2(n-i).\n \\label{eq:address-three-exact-transfer}\n\\end{equation}\nThe scaled-binomial argument\n\\eqref{eq:scaled-binomial-difference-bound} applies unchanged.\nThe basic bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra bits, so this gives the\none-bit offset for $C_3$. Finite step expansion finishes the proof of\nTheorem~\\ref{thm:four-factorial-bound}.","bound for $C_2$ includes order zero. Its continuous extension\nis available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra bits, so this gives the\none-bit offset for $C_3$. Finite step expansion finishes the proof of\nTheorem~\\ref{thm:four-factorial-bound}.","available from the threshold-free theorem. Thus the shifted sums\nconverge uniformly on $\\mathbb Z_2$, and the pointwise product rule\nproves\n\\begin{equation}\n v_2(\\Delta_1^k(C_3-C_2)(n))\\geq k+v_2(k!)+1\n                                      \\qquad(k,n\\geq0).\n \\label{eq:address-three-extra-bit}\n\\end{equation}\nAt every positive order $C_2$ has three extra bits, so this gives the\none-bit offset for $C_3$. Finite step expansion finishes the proof of\nTheorem~\\ref{thm:four-factorial-bound}.","\\begin{theorem}[Exact factorial precision at addresses zero and one]\n\\label{thm:zero-one-exact-mixed}\nLet $j\\in\\{0,1\\}$, $k\\geq1$, $n\\geq0$, and $H_1,\\ldots,H_k>0$.\nPut\n\\[\n L=k+v_2(k!)+\\sum_{\\nu=1}^k v_2(H_\\nu),\\qquad\n e=\\#\\{\\nu:2\\mid H_\\nu\\}.\n\\]\nThe mixed difference has valuation exactly $L$ if $k$ is odd or $e$ is\neven. If $k$ is even and $e$ is odd, it has valuation at least $L+1$.\nIn particular, for every homogeneous positive step,\n\\begin{equation}\n v_2(\\Delta_H^kC_j(n))=k+v_2(k!)+k v_2(H).\n \\label{eq:zero-one-homogeneous-exact}\n\\end{equation}\n\\end{theorem}","ero and one]\n\\label{thm:zero-one-exact-mixed}\nLet $j\\in\\{0,1\\}$, $k\\geq1$, $n\\geq0$, and $H_1,\\ldots,H_k>0$.\nPut\n\\[\n L=k+v_2(k!)+\\sum_{\\nu=1}^k v_2(H_\\nu),\\qquad\n e=\\#\\{\\nu:2\\mid H_\\nu\\}.\n\\]\nThe mixed difference has valuation exactly $L$ if $k$ is odd or $e$ is\neven. If $k$ is even and $e$ is odd, it has valuation at least $L+1$.\nIn particular, for every homogeneous positive step,\n\\begin{equation}\n v_2(\\Delta_H^kC_j(n))=k+v_2(k!)+k v_2(H).\n \\label{eq:zero-one-homogeneous-exact}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nReturn to the Euler variables of address zero and put\n$X=P+Q$, $\\tau=i\\mathcal E\\mathcal F$, $\\epsilon=1+\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equati","\\begin{proof}\nReturn to the Euler variables of address zero and put\n$X=P+Q$, $\\tau=i\\mathcal E\\mathcal F$, $\\epsilon=1+\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv","\\begin{proof}\nReturn to the Euler variables of address zero and put\n$X=P+Q$, $\\tau=i\\mathcal E\\mathcal F$, $\\epsilon=1+\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\e","\\begin{proof}\nReturn to the Euler variables of address zero and put\n$X=P+Q$, $\\tau=i\\mathcal E\\mathcal F$, $\\epsilon=1+\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref","\\begin{proof}\nReturn to the Euler variables of address zero and put\n$X=P+Q$, $\\tau=i\\mathcal E\\mathcal F$, $\\epsilon=1+\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independe","o and put\n$X=P+Q$, $\\tau=i\\mathcal E\\mathcal F$, $\\epsilon=1+\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weigh…",", $\\tau=i\\mathcal E\\mathcal F$, $\\epsilon=1+\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanish…","\\tau$.\nDirect rearrangement of \\eqref{eq:four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach …",":four-euler-H} gives\n\\begin{equation}\n H_0=\\gamma_2(X)+\\epsilon X+2G,\\qquad\n G=-\\gamma_2(P)-\\gamma_2(Q)+PQ-\\tau Q,\n \\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term…","\\qquad \\epsilon^2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k…","2=2(\\tau-P-Q-2PQ).\n \\label{eq:zero-parity-decomposition}\n\\end{equation}\nThe divided powers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd a…","wers of $G$ are integral by the same component formulas\nused above. Thus modulo $2\\mathcal A_0$, $\\gamma_k(H_0)$ equals the\ndivided power of $\\gamma_2(X)+\\epsilon X$. The identity\n\\[\n \\gamma_r(\\gamma_2(X))\n   =\\frac{(2r)!}{2^r r!}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).…","}\\gamma_{2r}(X)\n\\]\nhas an odd integer scalar. Every term involving $\\epsilon^j$ for\n$j\\geq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).\n\\]\nIn the base-row expansion \\eqref{eq:base-row-finite-expansion}, every\n$s>0$ term has at least one extra bit after division by $2^k k!$.\nThus the same exact valuation holds at every starting row.\nThe extra-bit transfer \\eqref{…","eq2$ vanishes modulo two. The surviving zero and first powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).\n\\]\nIn the base-row expansion \\eqref{eq:base-row-finite-expansion}, every\n$s>0$ term has at least one extra bit after division by $2^k k!$.\nThus the same exact valuation holds at every starting row.\nThe extra-bit transfer \\eqref{eq:address-one-extra-bit} proves it for\naddress one as well.","t powers give\n\\begin{equation}\n \\gamma_k(H_0)\\equiv\n       \\gamma_{2k}(X)+\\epsilon\\gamma_{2k-1}(X)\n                                  \\pmod{2\\mathcal A_0}.\n \\label{eq:zero-parity-leading-pd}\n\\end{equation}\nThe multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).\n\\]\nIn the base-row expansion \\eqref{eq:base-row-finite-expansion}, every\n$s>0$ term has at least one extra bit after division by $2^k k!$.\nThus the same exact valuation holds at every starting row.\nThe extra-bit transfer \\eqref{eq:address-one-extra-bit} proves it for\naddress one as well.","multiplication coefficient in the second term is $2k-1$, also odd.\nBy the odd moments \\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).\n\\]\nIn the base-row expansion \\eqref{eq:base-row-finite-expansion}, every\n$s>0$ term has at least one extra bit after division by $2^k k!$.\nThus the same exact valuation holds at every starting row.\nThe extra-bit transfer \\eqref{eq:address-one-extra-bit} proves it for\naddress one as well.","\\eqref{eq:euler-pd-odd-moments} and independence,\n\\[\n \\mathcal L(\\gamma_m(P+Q))\\equiv m+1\\pmod2.\n\\]\nThe $\\tau$-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).\n\\]\nIn the base-row expansion \\eqref{eq:base-row-finite-expansion}, every\n$s>0$ term has at least one extra bit after division by $2^k k!$.\nThus the same exact valuation holds at every starting row.\nThe extra-bit transfer \\eqref{eq:address-one-extra-bit} proves it for\naddress one as well.","-weighted moment vanishes because it contains an odd power of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).\n\\]\nIn the base-row expansion \\eqref{eq:base-row-finite-expansion}, every\n$s>0$ term has at least one extra bit after division by $2^k k!$.\nThus the same exact valuation holds at every starting row.\nThe extra-bit transfer \\eqref{eq:address-one-extra-bit} proves it for\naddress one as well.","er of\neach Euler variable. Therefore the first term of\n\\eqref{eq:zero-parity-leading-pd} evaluates to one modulo two and the\nsecond to zero. Hence $\\mathcal L(\\gamma_k(H_0))$ is odd at every order,\nincluding order zero. This proves\n\\[\n v_2(\\Delta_1^k C_0(0))=k+v_2(k!).\n\\]\nIn the base-row expansion \\eqref{eq:base-row-finite-expansion}, every\n$s>0$ term has at least one extra bit after division by $2^k k!$.\nThus the same exact valuation holds at every starting row.\nThe extra-bit transfer \\eqref{eq:address-one-extra-bit} proves it for\naddress one as well.","It remains to track possible ties in the mixed-step expansion.\nFor local orders $m_\\nu\\geq1$ and total $M$, the excess over $L$ is\nat least\n\\begin{equation}\n \\left(M-k-\\sum_\\nu v_2(m_\\nu)\\right)\n                       +\\left(v_2(M!)-v_2(k!)\\right).\n \\label{eq:zero-one-mixed-excess}\n\\end{equation}\nThe first bracket is nonnegative, and equality requires every\n$m_\\nu$ to be one or two. Two or more occurrences of two make the\nsecond bracket positive, since the intervening factorial interval then\ncontains an even number. Exactly one occurrence of two can tie only\nwhen $k$ is even. Its binomial coefficient can have the minimally\nallowed valuation only if its associated step is even.","m_\\nu v_2(m_\\nu)\\right)\n                       +\\left(v_2(M!)-v_2(k!)\\right).\n \\label{eq:zero-one-mixed-excess}\n\\end{equation}\nThe first bracket is nonnegative, and equality requires every\n$m_\\nu$ to be one or two. Two or more occurrences of two make the\nsecond bracket positive, since the intervening factorial interval then\ncontains an even number. Exactly one occurrence of two can tie only\nwhen $k$ is even. Its binomial coefficient can have the minimally\nallowed valuation only if its associated step is even.","The all-one multi-index contributes one after normalization by $2^L$.\nWhen $k$ is even, each even step supplies exactly one further tied term,\nagain with normalized residue one. For example,\n$\\binom H2=H(H-1)/2$ attains $v_2(H)-1$ exactly when $H$ is even.\nAll remaining factors and the exact normalized order-$k+1$ difference\nare odd. Thus the total normalized parity is one for odd $k$, and\n$1+e$ for even $k$. This proves the classification.\nIn the homogeneous case, $e$ is zero or $k$, so it is even whenever\n$k$ is even, proving \\eqref{eq:zero-one-homogeneous-exact}.\n\\end{proof}","ne multi-index contributes one after normalization by $2^L$.\nWhen $k$ is even, each even step supplies exactly one further tied term,\nagain with normalized residue one. For example,\n$\\binom H2=H(H-1)/2$ attains $v_2(H)-1$ exactly when $H$ is even.\nAll remaining factors and the exact normalized order-$k+1$ difference\nare odd. Thus the total normalized parity is one for odd $k$, and\n$1+e$ for even $k$. This proves the classification.\nIn the homogeneous case, $e$ is zero or $k$, so it is even whenever\n$k$ is even, proving \\eqref{eq:zero-one-homogeneous-exact}.\n\\end{proof}","hen $k$ is even, each even step supplies exactly one further tied term,\nagain with normalized residue one. For example,\n$\\binom H2=H(H-1)/2$ attains $v_2(H)-1$ exactly when $H$ is even.\nAll remaining factors and the exact normalized order-$k+1$ difference\nare odd. Thus the total normalized parity is one for odd $k$, and\n$1+e$ for even $k$. This proves the classification.\nIn the homogeneous case, $e$ is zero or $k$, so it is even whenever\n$k$ is even, proving \\eqref{eq:zero-one-homogeneous-exact}.\n\\end{proof}",", each even step supplies exactly one further tied term,\nagain with normalized residue one. For example,\n$\\binom H2=H(H-1)/2$ attains $v_2(H)-1$ exactly when $H$ is even.\nAll remaining factors and the exact normalized order-$k+1$ difference\nare odd. Thus the total normalized parity is one for odd $k$, and\n$1+e$ for even $k$. This proves the classification.\nIn the homogeneous case, $e$ is zero or $k$, so it is even whenever\n$k$ is even, proving \\eqref{eq:zero-one-homogeneous-exact}.\n\\end{proof}","actly one further tied term,\nagain with normalized residue one. For example,\n$\\binom H2=H(H-1)/2$ attains $v_2(H)-1$ exactly when $H$ is even.\nAll remaining factors and the exact normalized order-$k+1$ difference\nare odd. Thus the total normalized parity is one for odd $k$, and\n$1+e$ for even $k$. This proves the classification.\nIn the homogeneous case, $e$ is zero or $k$, so it is even whenever\n$k$ is even, proving \\eqref{eq:zero-one-homogeneous-exact}.\n\\end{proof}","\\begin{remark}[A mixed cancellation]\nThe exceptional class can gain more than one bit. The exact values\n\\[\n \\Delta_1\\Delta_2C_0(0)=-33728,\\qquad\n \\Delta_1\\Delta_2C_1(0)=-\\frac{150208}{585}\n\\]\nboth have valuation six, whereas their baseline $L$ is four.\nThe theorem classifies attainment of the baseline. It does not give a\ncomplete deeper valuation in the cancellation class.\n\\end{remark}","\\begin{theorem}[Exact address-two power orders]\n\\label{thm:two-exact-power-orders}\nFor every starting row $n\\geq0$ and all positive mixed steps,\n\\begin{equation}\n v_2(\\Delta_{H_1}\\cdots\\Delta_{H_k}C_2(n))\n       =V_2(k)+\\sum_{\\nu=1}^k v_2(H_\\nu)\n \\label{eq:two-power-orders-exact}\n\\end{equation}\nwhenever $k$ is a positive power of two, where\n\\[\n V_2(1)=4,\\qquad V_2(2)=6,\\qquad\n V_2(k)=2k+v_2(k!)+\\frac{k}{2}\\quad(k\\geq4).\n\\]\nFor the original coefficients, the right side gains one bit.\n\\end{theorem}","\\begin{proof}\nFor $k=2^r\\geq4$, put $q=k/2$, which is even. Normalize the exact\nexpansion \\eqref{eq:address-two-filtration-pd} by $2^q$ before reducing\nmodulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\","\\begin{proof}\nFor $k=2^r\\geq4$, put $q=k/2$, which is even. Normalize the exact\nexpansion \\eqref{eq:address-two-filtration-pd} by $2^q$ before reducing\nmodulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter","\\begin{proof}\nFor $k=2^r\\geq4$, put $q=k/2$, which is even. Normalize the exact\nexpansion \\eqref{eq:address-two-filtration-pd} by $2^q$ before reducing\nmodulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to","\\begin{proof}\nFor $k=2^r\\geq4$, put $q=k/2$, which is even. Normalize the exact\nexpansion \\eqref{eq:address-two-filtration-pd} by $2^q$ before reducing\nmodulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)","\\begin{proof}\nFor $k=2^r\\geq4$, put $q=k/2$, which is even. Normalize the exact\nexpansion \\eqref{eq:address-two-filtration-pd} by $2^q$ before reducing\nmodulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}","\\begin{proof}\nFor $k=2^r\\geq4$, put $q=k/2$, which is even. Normalize the exact\nexpansion \\eqref{eq:address-two-filtration-pd} by $2^q$ before reducing\nmodulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\m…","2^q$ before reducing\nmodulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. …","ulo two. Its $u$th auxiliary term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernou…","term has excess\n$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this require…","$u+\\lfloor(k-u)/2\\rfloor-q=\\lfloor u/2\\rfloor$.\nOnly auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen …","auxiliary indices zero and one can therefore survive:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. He…","ve:\n\\begin{equation}\n \\frac{\\gamma_k(H_2)}{2^q}\n \\equiv V^q\\gamma_k(X)\n       +\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","+\\epsilon V^{q-1}G\\gamma_{k-1}(X)\n                                \\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","\\pmod{2\\mathcal A_2}.\n \\label{eq:two-power-leading-normalized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","malized}\n\\end{equation}\nHere $V\\equiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","quiv\\tau\\pmod2$ and $\\tau^2\\equiv1\\pmod2$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","$.\nSince $q$ is even and $\\epsilon\\tau\\equiv\\epsilon\\pmod2$, this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","this reduces\nto $\\gamma_k(X)+\\epsilon G\\gamma_{k-1}(X)$.\nMoreover, $G\\equiv\\gamma_2(X)+\\tau Q\\pmod2$.\nAfter evaluation, every term with a single $\\tau$ vanishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","nishes, so the\nsecond contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","d contribution reduces to\n\\[\n \\mathcal L\\bigl((\\gamma_2(X)+Q)\\gamma_{k-1}(X)\\bigr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","igr).\n\\]\nThe $\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","\\gamma_2(X)$ product has scalar $\\binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","binom{k+1}{2}$, which is even\nbecause $4\\mid k$. The other contribution expands as\n\\begin{equation}\n \\mathcal L(Q\\gamma_{k-1}(P+Q))\n    =\\sum_{a=0}^{k-1}(k-a)\\mu_a\\mu_{k-a}.\n \\label{eq:two-power-Q-evaluation}\n\\end{equation}\nAn odd summand would require $k-a$ to be odd and $\\mu_{k-a}$ to be\nodd. By Lemma~\\ref{lem:bernoulli-central-pd}, this requires $k-a=1$.\nThen $a=k-1$ is not a power of two and $\\mu_a$ is even. Hence the\nsecond contribution is zero modulo two.","For the first contribution, independence gives\n$\\mathcal L(\\gamma_k(P+Q))=\\sum_{a=0}^k\\mu_a\\mu_{k-a}$.\nThe parity generating series of these convolution moments is\n\\[\n (1+t+t^2+t^4+t^8+\\cdots)^2\n        =1+t^2+t^4+t^8+\\cdots\\quad\\text{in }\\mathbb F_2[[t]].\n\\]\nIts coefficient is one at every $k=2^r\\geq4$. Thus the normalized\nmoment in \\eqref{eq:two-power-leading-normalized} is odd and\n\\[\n v_2(\\Delta_1^kC_2(0))=2k+v_2(k!)+k/2.\n\\]\nInserting $U^{4n}=(1+4H_2)^n$ introduces only corrections with strictly\nhigher valuation, by the filtered bound at orders $k+s$ for $s>0$.\nThe equality consequently holds at every starting row.","ves\n$\\mathcal L(\\gamma_k(P+Q))=\\sum_{a=0}^k\\mu_a\\mu_{k-a}$.\nThe parity generating series of these convolution moments is\n\\[\n (1+t+t^2+t^4+t^8+\\cdots)^2\n        =1+t^2+t^4+t^8+\\cdots\\quad\\text{in }\\mathbb F_2[[t]].\n\\]\nIts coefficient is one at every $k=2^r\\geq4$. Thus the normalized\nmoment in \\eqref{eq:two-power-leading-normalized} is odd and\n\\[\n v_2(\\Delta_1^kC_2(0))=2k+v_2(k!)+k/2.\n\\]\nInserting $U^{4n}=(1+4H_2)^n$ introduces only corrections with strictly\nhigher valuation, by the filtered bound at orders $k+s$ for $s>0$.\nThe equality consequently holds at every starting row.","ments is\n\\[\n (1+t+t^2+t^4+t^8+\\cdots)^2\n        =1+t^2+t^4+t^8+\\cdots\\quad\\text{in }\\mathbb F_2[[t]].\n\\]\nIts coefficient is one at every $k=2^r\\geq4$. Thus the normalized\nmoment in \\eqref{eq:two-power-leading-normalized} is odd and\n\\[\n v_2(\\Delta_1^kC_2(0))=2k+v_2(k!)+k/2.\n\\]\nInserting $U^{4n}=(1+4H_2)^n$ introduces only corrections with strictly\nhigher valuation, by the filtered bound at orders $k+s$ for $s>0$.\nThe equality consequently holds at every starting row.","[\n (1+t+t^2+t^4+t^8+\\cdots)^2\n        =1+t^2+t^4+t^8+\\cdots\\quad\\text{in }\\mathbb F_2[[t]].\n\\]\nIts coefficient is one at every $k=2^r\\geq4$. Thus the normalized\nmoment in \\eqref{eq:two-power-leading-normalized} is odd and\n\\[\n v_2(\\Delta_1^kC_2(0))=2k+v_2(k!)+k/2.\n\\]\nInserting $U^{4n}=(1+4H_2)^n$ introduces only corrections with strictly\nhigher valuation, by the filtered bound at orders $k+s$ for $s>0$.\nThe equality consequently holds at every starting row.","For mixed steps the all-one local multi-index gives the claimed leading\nterm. A higher term of total order $M>k$ has excess at least\n\\[\n 2(M-k)+\\left\\lfloor\\frac M2\\right\\rfloor-\\frac k2\n       +v_2(M!)-v_2(k!)-\\sum_\\nu v_2(m_\\nu)\n       \\geq M-k\\geq1.\n\\]\nHence the leading term cannot cancel.","At order one, $a_1$ in \\eqref{eq:address-two-initial-newton} has\nvaluation four and the Newton tail starts at valuation six. Its shifted\nvalue remains exact. In the step expansion a local order two can lose\nat most one bit through its binomial coefficient, so every correction\nis still above $4+v_2(H_1)$. At order two, $a_2$ has valuation six and\nthe next coefficient has lower bound eight from the stronger\nfiltration. A total-order-three correction loses at most one bit,\nso it stays above the leading valuation six, while later orders gain\nstill more. The same argument proves exactness for both mixed steps.\n\\end{proof}","\\begin{lemma}[A first-difference criterion]\n\\label{lem:newton-exact-isometry-criterion}\nLet $C:\\mathbb Z_{\\geq0}\\to\\mathbb Z_2$ have Newton coefficients\n$a_k=\\Delta_1^kC(0)$. Suppose for an integer $b\\geq0$ that\n\\begin{equation}\n v_2(a_1)=b,\\qquad\n v_2(a_k)\\geq b+1+\\lfloor\\log_2k\\rfloor\\quad(k\\geq2).\n \\label{eq:newton-isometry-criterion}\n\\end{equation}\nThen for every $n\\geq0$ and $H>0$,\n\\begin{equation}\n v_2(C(n+H)-C(n))=b+v_2(H).\n \\label{eq:newton-isometry-conclusion}\n\\end{equation}\nThe Newton interpolation extends continuously to $\\mathbb Z_2$ and\nsatisfies the same identity for any two distinct dyadic arguments.\n\\end{lemma}","{lemma}[A first-difference criterion]\n\\label{lem:newton-exact-isometry-criterion}\nLet $C:\\mathbb Z_{\\geq0}\\to\\mathbb Z_2$ have Newton coefficients\n$a_k=\\Delta_1^kC(0)$. Suppose for an integer $b\\geq0$ that\n\\begin{equation}\n v_2(a_1)=b,\\qquad\n v_2(a_k)\\geq b+1+\\lfloor\\log_2k\\rfloor\\quad(k\\geq2).\n \\label{eq:newton-isometry-criterion}\n\\end{equation}\nThen for every $n\\geq0$ and $H>0$,\n\\begin{equation}\n v_2(C(n+H)-C(n))=b+v_2(H).\n \\label{eq:newton-isometry-conclusion}\n\\end{equation}\nThe Newton interpolation extends continuously to $\\mathbb Z_2$ and\nsatisfies the same identity for any two distinct dyadic arguments.\n\\end{lemma}","coefficients\n$a_k=\\Delta_1^kC(0)$. Suppose for an integer $b\\geq0$ that\n\\begin{equation}\n v_2(a_1)=b,\\qquad\n v_2(a_k)\\geq b+1+\\lfloor\\log_2k\\rfloor\\quad(k\\geq2).\n \\label{eq:newton-isometry-criterion}\n\\end{equation}\nThen for every $n\\geq0$ and $H>0$,\n\\begin{equation}\n v_2(C(n+H)-C(n))=b+v_2(H).\n \\label{eq:newton-isometry-conclusion}\n\\end{equation}\nThe Newton interpolation extends continuously to $\\mathbb Z_2$ and\nsatisfies the same identity for any two distinct dyadic arguments.\n\\end{lemma}","\\begin{proof}\nAt a natural argument the exact Newton expansion is finite.\nVandermonde gives\n\\[\n \\binom{n+H}{k}-\\binom nk\n   =\\sum_{i=1}^k\\binom Hi\\binom n{k-i}.\n\\]\nFor $1\\leq i\\leq H$,\n$v_2(\\binom Hi)\\geq v_2(H)-v_2(i)\n \\geq v_2(H)-\\lfloor\\log_2k\\rfloor$. Terms with $i>H$ vanish.\nThus each Newton summand with $k\\geq2$ has valuation at least\n$b+v_2(H)+1$. The $k=1$ summand is $Ha_1$, of exact valuation\n$b+v_2(H)$, proving \\eqref{eq:newton-isometry-conclusion}.","{proof}\nAt a natural argument the exact Newton expansion is finite.\nVandermonde gives\n\\[\n \\binom{n+H}{k}-\\binom nk\n   =\\sum_{i=1}^k\\binom Hi\\binom n{k-i}.\n\\]\nFor $1\\leq i\\leq H$,\n$v_2(\\binom Hi)\\geq v_2(H)-v_2(i)\n \\geq v_2(H)-\\lfloor\\log_2k\\rfloor$. Terms with $i>H$ vanish.\nThus each Newton summand with $k\\geq2$ has valuation at least\n$b+v_2(H)+1$. The $k=1$ summand is $Ha_1$, of exact valuation\n$b+v_2(H)$, proving \\eqref{eq:newton-isometry-conclusion}.","The coefficient hypothesis implies $a_k\\to0$ dyadically, so the\nNewton series converges uniformly on $\\mathbb Z_2$, where\n$\\binom xk$ is integral-valued. For distinct $x,y\\in\\mathbb Z_2$,\napproximate the pair by natural integer pairs with their difference\nvaluation eventually equal to $v_2(x-y)$. The corresponding function\ndifferences have the fixed exact valuation $b+v_2(x-y)$.\nA limit of elements with that fixed valuation has the same valuation:\nthe normalized residues are units, and $\\mathbb Z_2^\\times$ is closed.\n\\end{proof}","ormly on $\\mathbb Z_2$, where\n$\\binom xk$ is integral-valued. For distinct $x,y\\in\\mathbb Z_2$,\napproximate the pair by natural integer pairs with their difference\nvaluation eventually equal to $v_2(x-y)$. The corresponding function\ndifferences have the fixed exact valuation $b+v_2(x-y)$.\nA limit of elements with that fixed valuation has the same valuation:\nthe normalized residues are units, and $\\mathbb Z_2^\\times$ is closed.\n\\end{proof}","\\begin{theorem}[Exact four-address first differences]\n\\label{thm:rank-four-exact-periods}\nFor every $n\\geq0$, $H>0$, and $j\\in\\{0,1,2,3\\}$,\n\\begin{align}\n v_2(C_j(n+H)-C_j(n))&=v_2(H)+b_j,\n      &(b_0,b_1,b_2,b_3)&=(1,1,4,2),\\label{eq:rank-four-normalized-isometry}","\\begin{theorem}[Exact four-address first differences]\n\\label{thm:rank-four-exact-periods}\nFor every $n\\geq0$, $H>0$, and $j\\in\\{0,1,2,3\\}$,\n\\begin{align}\n v_2(C_j(n+H)-C_j(n))&=v_2(H)+b_j,\n      &(b_0,b_1,b_2,b_3)&=(1,1,4,2),\\label{eq:rank-four-normalized-isometry}\\\\\n v_2(E_{4(n+H)}^{(4,j)}-E_{4n}^{(4,j)})&=v_2(H)+B_j,\n      &(B_0,B_1,B_2,B_3)&=(1,1,5,3).\n \\label{eq:rank-four-original-isometry}\n\\end{align}\nConsequently, the least positive row period modulo $2^r$ is exactly\n\\begin{equation}\n P_j(r)=2^{\\max(0,r-B_j)}\\qquad(r\\geq1).\n \\label{eq:rank-four-least-period-final}\n\\end{equation}\nThe corresponding displacement in the original Euler subscript is\n$4P_j(r)$.\n\\end{theorem}","v_2(E_{4(n+H)}^{(4,j)}-E_{4n}^{(4,j)})&=v_2(H)+B_j,\n      &(B_0,B_1,B_2,B_3)&=(1,1,5,3).\n \\label{eq:rank-four-original-isometry}\n\\end{align}\nConsequently, the least positive row period modulo $2^r$ is exactly\n\\begin{equation}\n P_j(r)=2^{\\max(0,r-B_j)}\\qquad(r\\geq1).\n \\label{eq:rank-four-least-period-final}\n\\end{equation}\nThe corresponding displacement in the original Euler subscript is\n$4P_j(r)$.\n\\end{theorem}","\\begin{proof}\nThe actual reciprocal recurrence is\n\\begin{equation}\n C_{N,j}(n)=-\\sum_{q=1}^n\\binom{Nn}{Nq}\n           g_{N,j}(q)C_{N,j}(n-q),\\qquad C_{N,j}(0)=1.\n \\label{eq:exact-reciprocal-row-recurrence}\n\\end{equation}\nFor $N=4$, the four kernels are\n\\[\n 1,\\qquad\\frac1{4q+1},\\qquad\n \\frac1{(4q+1)(2q+1)},\\qquad\n \\frac3{(4q+1)(2q+1)(4q+3)}.\n\\]\nThe threshold-free theorem gives $v_2(a_k)\\geq k$ in every address.\nThe following exact values cover all orders where that bound does not\nalready imply \\eqref{eq:newton-isometry-criterion}:\n\\begin{center}\n\\begin{tabular}{c r l r}\n$j$&$k$&$a_k$&$v_2(a_k)$","t-reciprocal-row-recurrence}\n\\end{equation}\nFor $N=4$, the four kernels are\n\\[\n 1,\\qquad\\frac1{4q+1},\\qquad\n \\frac1{(4q+1)(2q+1)},\\qquad\n \\frac3{(4q+1)(2q+1)(4q+3)}.\n\\]\nThe threshold-free theorem gives $v_2(a_k)\\geq k$ in every address.\nThe following exact values cover all orders where that bound does not\nalready imply \\eqref{eq:newton-isometry-criterion}:\n\\begin{center}\n\\begin{tabular}{c r l r}\n$j$&$k$&$a_k$&$v_2(a_k)$","ciprocal-row-recurrence}\n\\end{equation}\nFor $N=4$, the four kernels are\n\\[\n 1,\\qquad\\frac1{4q+1},\\qquad\n \\frac1{(4q+1)(2q+1)},\\qquad\n \\frac3{(4q+1)(2q+1)(4q+3)}.\n\\]\nThe threshold-free theorem gives $v_2(a_k)\\geq k$ in every address.\nThe following exact values cover all orders where that bound does not\nalready imply \\eqref{eq:newton-isometry-criterion}:\n\\begin{center}\n\\begin{tabular}{c r l r}\n$j$&$k$&$a_k$&$v_2(a_k)$","ocal-row-recurrence}\n\\end{equation}\nFor $N=4$, the four kernels are\n\\[\n 1,\\qquad\\frac1{4q+1},\\qquad\n \\frac1{(4q+1)(2q+1)},\\qquad\n \\frac3{(4q+1)(2q+1)(4q+3)}.\n\\]\nThe threshold-free theorem gives $v_2(a_k)\\geq k$ in every address.\nThe following exact values cover all orders where that bound does not\nalready imply \\eqref{eq:newton-isometry-criterion}:\n\\begin{center}\n\\begin{tabular}{c r l r}\n$j$&$k$&$a_k$&$v_2(a_k)$","ow-recurrence}\n\\end{equation}\nFor $N=4$, the four kernels are\n\\[\n 1,\\qquad\\frac1{4q+1},\\qquad\n \\frac1{(4q+1)(2q+1)},\\qquad\n \\frac3{(4q+1)(2q+1)(4q+3)}.\n\\]\nThe threshold-free theorem gives $v_2(a_k)\\geq k$ in every address.\nThe following exact values cover all orders where that bound does not\nalready imply \\eqref{eq:newton-isometry-criterion}:\n\\begin{center}\n\\begin{tabular}{c r l r}\n$j$&$k$&$a_k$&$v_2(a_k)$","hline\n0&1&$-2$&1","0&2&$72$&3","1&1&$-6/5$&1","1&2&$184/45$&3","2&1&$-16/15$&4","2&2&$64/45$&6","2&3&$-14848/1365$&9","2&4&$71426048/69615$&13","2&5&$-25509691392/85085$&16","2&6&$149113198870528/765765$&22","3&1&$-36/35$&2","3&2&$256/231$&8","3&3&$-65792/35035$&8","3&4&$161644544/4849845$&15\n\\end{tabular}\n\\end{center}\nThey are finite rational evaluations of\n\\eqref{eq:exact-reciprocal-row-recurrence}, followed by\n$a_k=\\sum_{h=0}^k(-1)^{k-h}\\binom kh C_j(h)$.\nFor addresses zero and one, $k\\geq3$ is covered by the generic bound.\nFor address two, $k\\geq7$ is covered. For address three, $k\\geq5$ is\ncovered. In each range the elementary inequality is\n$k-\\lfloor\\log_2k\\rfloor\\geq b_j+1$.\nLemma~\\ref{lem:newton-exact-isometry-criterion} proves the normalized\nidentity. Multiplication by $j!$ proves the original identity.","w-recurrence}, followed by\n$a_k=\\sum_{h=0}^k(-1)^{k-h}\\binom kh C_j(h)$.\nFor addresses zero and one, $k\\geq3$ is covered by the generic bound.\nFor address two, $k\\geq7$ is covered. For address three, $k\\geq5$ is\ncovered. In each range the elementary inequality is\n$k-\\lfloor\\log_2k\\rfloor\\geq b_j+1$.\nLemma~\\ref{lem:newton-exact-isometry-criterion} proves the normalized\nidentity. Multiplication by $j!$ proves the original identity.","A row displacement $H$ preserves the original coefficient modulo\n$2^r$ at even one row exactly when $v_2(H)+B_j\\geq r$.\nThe least positive such integer is\n$2^{\\max(0,r-B_j)}$, and the same displacement works at every row.\nThis proves both optimality and the absence of a starting threshold.\n\\end{proof}","\\begin{corollary}[Exact image and residue enumeration]\n\\label{cor:four-exact-images}\nLet $\\widetilde C_j$ denote the dyadic Newton interpolation.\nThe map\n\\[\n T_j(x)=\\frac{\\widetilde C_j(x)-1}{2^{b_j}}\n\\]\nis a bijective isometry $\\mathbb Z_2\\to\\mathbb Z_2$, and\n\\begin{equation}\n \\widetilde C_j(\\mathbb Z_2)=1+2^{b_j}\\mathbb Z_2,\n \\qquad j!\\widetilde C_j(\\mathbb Z_2)=j!+2^{B_j}\\mathbb Z_2.\n \\label{eq:four-interpolation-images-final}\n\\end{equation}\nFor $r>B_j$, one least period modulo $2^r$ visits every residue in\n$j!+2^{B_j}\\mathbb Z/2^r\\mathbb Z$ exactly once. For $r\\leq B_j$,\nthere is one residue and the least row period is one.\n\\end{corollary}","\\label{cor:four-exact-images}\nLet $\\widetilde C_j$ denote the dyadic Newton interpolation.\nThe map\n\\[\n T_j(x)=\\frac{\\widetilde C_j(x)-1}{2^{b_j}}\n\\]\nis a bijective isometry $\\mathbb Z_2\\to\\mathbb Z_2$, and\n\\begin{equation}\n \\widetilde C_j(\\mathbb Z_2)=1+2^{b_j}\\mathbb Z_2,\n \\qquad j!\\widetilde C_j(\\mathbb Z_2)=j!+2^{B_j}\\mathbb Z_2.\n \\label{eq:four-interpolation-images-final}\n\\end{equation}\nFor $r>B_j$, one least period modulo $2^r$ visits every residue in\n$j!+2^{B_j}\\mathbb Z/2^r\\mathbb Z$ exactly once. For $r\\leq B_j$,\nthere is one residue and the least row period is one.\n\\end{corollary}","t $\\widetilde C_j$ denote the dyadic Newton interpolation.\nThe map\n\\[\n T_j(x)=\\frac{\\widetilde C_j(x)-1}{2^{b_j}}\n\\]\nis a bijective isometry $\\mathbb Z_2\\to\\mathbb Z_2$, and\n\\begin{equation}\n \\widetilde C_j(\\mathbb Z_2)=1+2^{b_j}\\mathbb Z_2,\n \\qquad j!\\widetilde C_j(\\mathbb Z_2)=j!+2^{B_j}\\mathbb Z_2.\n \\label{eq:four-interpolation-images-final}\n\\end{equation}\nFor $r>B_j$, one least period modulo $2^r$ visits every residue in\n$j!+2^{B_j}\\mathbb Z/2^r\\mathbb Z$ exactly once. For $r\\leq B_j$,\nthere is one residue and the least row period is one.\n\\end{corollary}","dyadic Newton interpolation.\nThe map\n\\[\n T_j(x)=\\frac{\\widetilde C_j(x)-1}{2^{b_j}}\n\\]\nis a bijective isometry $\\mathbb Z_2\\to\\mathbb Z_2$, and\n\\begin{equation}\n \\widetilde C_j(\\mathbb Z_2)=1+2^{b_j}\\mathbb Z_2,\n \\qquad j!\\widetilde C_j(\\mathbb Z_2)=j!+2^{B_j}\\mathbb Z_2.\n \\label{eq:four-interpolation-images-final}\n\\end{equation}\nFor $r>B_j$, one least period modulo $2^r$ visits every residue in\n$j!+2^{B_j}\\mathbb Z/2^r\\mathbb Z$ exactly once. For $r\\leq B_j$,\nthere is one residue and the least row period is one.\n\\end{corollary}","{\\widetilde C_j(x)-1}{2^{b_j}}\n\\]\nis a bijective isometry $\\mathbb Z_2\\to\\mathbb Z_2$, and\n\\begin{equation}\n \\widetilde C_j(\\mathbb Z_2)=1+2^{b_j}\\mathbb Z_2,\n \\qquad j!\\widetilde C_j(\\mathbb Z_2)=j!+2^{B_j}\\mathbb Z_2.\n \\label{eq:four-interpolation-images-final}\n\\end{equation}\nFor $r>B_j$, one least period modulo $2^r$ visits every residue in\n$j!+2^{B_j}\\mathbb Z/2^r\\mathbb Z$ exactly once. For $r\\leq B_j$,\nthere is one residue and the least row period is one.\n\\end{corollary}","\\begin{proof}\nThe exact difference identity gives\n$v_2(T_j(x)-T_j(y))=v_2(x-y)$ and $T_j(0)=0$.\nFor every $s\\geq1$, the map induces an injection of\n$\\mathbb Z_2/2^s\\mathbb Z_2$ into itself. It is a bijection because\nboth sets have $2^s$ elements. These quotient bijections are compatible\nwith reduction. Inverting them gives a compatible preimage system for\nevery dyadic integer, proving surjectivity. Multiplication by $j!$\nchanges the image ideal by exactly $v_2(j!)$.\nThe finite quotient bijection is also precisely the residue enumeration\nin a least row period.\n\\end{proof}","ct difference identity gives\n$v_2(T_j(x)-T_j(y))=v_2(x-y)$ and $T_j(0)=0$.\nFor every $s\\geq1$, the map induces an injection of\n$\\mathbb Z_2/2^s\\mathbb Z_2$ into itself. It is a bijection because\nboth sets have $2^s$ elements. These quotient bijections are compatible\nwith reduction. Inverting them gives a compatible preimage system for\nevery dyadic integer, proving surjectivity. Multiplication by $j!$\nchanges the image ideal by exactly $v_2(j!)$.\nThe finite quotient bijection is also precisely the residue enumeration\nin a least row period.\n\\end{proof}","-T_j(y))=v_2(x-y)$ and $T_j(0)=0$.\nFor every $s\\geq1$, the map induces an injection of\n$\\mathbb Z_2/2^s\\mathbb Z_2$ into itself. It is a bijection because\nboth sets have $2^s$ elements. These quotient bijections are compatible\nwith reduction. Inverting them gives a compatible preimage system for\nevery dyadic integer, proving surjectivity. Multiplication by $j!$\nchanges the image ideal by exactly $v_2(j!)$.\nThe finite quotient bijection is also precisely the residue enumeration\nin a least row period.\n\\end{proof}","\\begin{theorem}[Lower-half exact laws]\n\\label{thm:lower-half-all-dyadic-isometry}\nLet $N=2^a\\geq4$ and $0\\leq j<N/2$. Then for all $n\\geq0$ and $H>0$,\n\\begin{equation}\n v_2(C_{N,j}(n+H)-C_{N,j}(n))=1+v_2(H).\n \\label{eq:lower-half-all-dyadic-exact}\n\\end{equation}\nIts normalized interpolation is a bijection onto $1+2\\mathbb Z_2$.\nFor the original coefficients, the least row period modulo $2^r$ is\n$2^{\\max(0,r-1-v_2(j!))}$.\n\\end{theorem}","\\begin{proof}\nEvery slope $N/i$ in the source has valuation at least two, since\n$i\\leq j<N/2$. Hence $g(1)\\equiv1\\pmod4$ and\n$g(2)\\equiv1\\pmod8$. The first two Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\e","\\begin{proof}\nEvery slope $N/i$ in the source has valuation at least two, since\n$i\\leq j<N/2$. Hence $g(1)\\equiv1\\pmod4$ and\n$g(2)\\equiv1\\pmod8$. The first two Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\ge","\\begin{proof}\nEvery slope $N/i$ in the source has valuation at least two, since\n$i\\leq j<N/2$. Hence $g(1)\\equiv1\\pmod4$ and\n$g(2)\\equiv1\\pmod8$. The first two Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criteri","\\begin{proof}\nEvery slope $N/i$ in the source has valuation at least two, since\n$i\\leq j<N/2$. Hence $g(1)\\equiv1\\pmod4$ and\n$g(2)\\equiv1\\pmod8$. The first two Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The perio","\\begin{proof}\nEvery slope $N/i$ in the source has valuation at least two, since\n$i\\leq j<N/2$. Hence $g(1)\\equiv1\\pmod4$ and\n$g(2)\\equiv1\\pmod8$. The first two Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","$N/i$ in the source has valuation at least two, since\n$i\\leq j<N/2$. Hence $g(1)\\equiv1\\pmod4$ and\n$g(2)\\equiv1\\pmod8$. The first two Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","at least two, since\n$i\\leq j<N/2$. Hence $g(1)\\equiv1\\pmod4$ and\n$g(2)\\equiv1\\pmod8$. The first two Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","o Newton coefficients from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","ents from the\nreciprocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","rocal recurrence are\n\\begin{equation}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","}\n a_1=-g(1)-1,\\qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","qquad\n a_2=\\binom{2N}{N}g(1)^2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","2-g(2)+2g(1)+1.\n \\label{eq:lower-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","r-half-initial-newton}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","on}\n\\end{equation}\nThe first has valuation exactly one. Also\n$\\binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","binom{2N}{N}\\equiv6\\pmod8$ for every $N=2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","2^a\\geq2$.\nTo see this, write\n\\[\n B_a=\\binom{2^{a+1}}{2^a}\n       =2\\prod_{h=1}^{2^a-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","-1}\\left(1+\\frac{2^a}{h}\\right).\n\\]\nThe even-$h$ factors give $B_{a-1}$. The odd-$h$ factors are one\nmodulo four for $a\\geq2$. Since $B_{a-1}$ has valuation one,\n$B_a\\equiv B_{a-1}\\pmod8$, and $B_1=6$.\nTherefore $a_2\\equiv6-1+2+1=0\\pmod8$.\nThe global Mahler bound covers all $k\\geq3$, completing the criterion\nwith $b=1$. The period and image deductions are the same as above.\nThe proof also covers $N=2,j=0$, whose kernel is identically one.\n\\end{proof}","\\begin{proposition}[Small-rank complete isometry table]\n\\label{prop:two-four-eight-isometry-table}\nEvery address at $N\\in\\{2,4,8\\}$ satisfies\n\\[\n v_2(C_{N,j}(n+H)-C_{N,j}(n))=b_{N,j}+v_2(H).\n\\]\nThe following entries are ordered by $j=0,\\ldots,N-1$:\n\\begin{center}\n\\begin{tabular}{c l l}\n$N$&normalized shifts $b_{N,j}$&original shifts $B_{N,j}=b_{N,j}+v_2(j!)$","hline\n2&$1,2$&$1,2$","4&$1,1,4,2$&$1,1,5,3$","8&$1,1,1,1,4,3,2,2$&$1,1,2,2,7,6,6,6$\n\\end{tabular}\n\\end{center}\nFor each entry, the least original row period is\n$2^{\\max(0,r-B_{N,j})}$, its Euler subscript displacement is $N$ times\nthat period, and the exact-image and residue-enumeration conclusions\nhold with the corresponding shifts.\n\\end{proposition}","\\begin{proof}\nRank four was proved above. At ranks two and eight the finite\ncriterion is verified by the actual rational recurrence\n\\eqref{eq:exact-reciprocal-row-recurrence}. The following valuation\nlists record every initial Newton coefficient needed in addition to\nthe generic tail $v_2(a_k)\\geq k$:\n\\begin{center}\n\\begin{tabular}{c c c l}\n$N$&$j$&$b_{N,j}$&$v_2(a_1),v_2(a_2),\\ldots$ through the finite cutoff","hline\n2&0&1&$1,3$","2&1&2&$2,5,9,11$","8&0&1&$1,3$","8&1&1&$1,3$","8&2&1&$1,3$","8&3&1&$1,3$","8&4&4&$4,6,9,13,16,20$","8&5&3&$3,6,10,13,17$","8&6&2&$2,6,8,13$","8&7&2&$2,11,8,15$\n\\end{tabular}\n\\end{center}\nThe first entries can also be read directly from\n$a_1=-1-\\binom{N+j}{j}^{-1}$. At rank eight they are respectively\n\\[\n -2,\\ -\\frac{10}{9},\\ -\\frac{46}{45},\\ -\\frac{166}{165},\\\n -\\frac{496}{495},\\ -\\frac{1288}{1287},\\\n -\\frac{3004}{3003},\\ -\\frac{6436}{6435}.\n\\]\nAll later entries in the table are finite rational arithmetic, with\nfull numerator and denominator records retained in the accompanying\nexact certificate \\texttt{all\\_rank\\_extensions\\_verified.json}.\nEvery entry with $k\\geq2$ meets\n$b_{N,j}+1+\\lfloor\\log_2k\\rfloor$. Beyond the displayed cutoff, the\ninequality $k\\geq b_{N,j}+1+\\lfloor\\log_2k\\rfloor$ holds at every\norder. Thus the finite table and the prove","8&7&2&$2,11,8,15$\n\\end{tabular}\n\\end{center}\nThe first entries can also be read directly from\n$a_1=-1-\\binom{N+j}{j}^{-1}$. At rank eight they are respectively\n\\[\n -2,\\ -\\frac{10}{9},\\ -\\frac{46}{45},\\ -\\frac{166}{165},\\\n -\\frac{496}{495},\\ -\\frac{1288}{1287},\\\n -\\frac{3004}{3003},\\ -\\frac{6436}{6435}.\n\\]\nAll later entries in the table are finite rational arithmetic, with\nfull numerator and denominator records retained in the accompanying\nexact certificate \\texttt{all\\_rank\\_extensions\\_verified.json}.\nEvery entry with $k\\geq2$ meets\n$b_{N,j}+1+\\lfloor\\log_2k\\rfloor$. Beyond the displayed cutoff, the\ninequality $k\\geq b_{N,j}+1+\\lfloor\\log_2k\\rfloor$ holds at every\norder. Thus the finite table and the proved infinite tail establish\nthe full hypothesis of Lemma~\\ref{lem:newton-exact-isometry-criterion}.\nThis is","8&7&2&$2,11,8,15$\n\\end{tabular}\n\\end{center}\nThe first entries can also be read directly from\n$a_1=-1-\\binom{N+j}{j}^{-1}$. At rank eight they are respectively\n\\[\n -2,\\ -\\frac{10}{9},\\ -\\frac{46}{45},\\ -\\frac{166}{165},\\\n -\\frac{496}{495},\\ -\\frac{1288}{1287},\\\n -\\frac{3004}{3003},\\ -\\frac{6436}{6435}.\n\\]\nAll later entries in the table are finite rational arithmetic, with\nfull numerator and denominator records retained in the accompanying\nexact certificate \\texttt{all\\_rank\\_extensions\\_verified.json}.\nEvery entry with $k\\geq2$ meets\n$b_{N,j}+1+\\lfloor\\log_2k\\rfloor$. Beyond the displayed cutoff, the\ninequality $k\\geq b_{N,j}+1+\\lfloor\\log_2k\\rfloor$ holds at every\norder. Thus the finite table and the proved infinite tail establish\nthe full hypothesis of Lemma~\\ref{lem:newton-exact-isometry-criterion}.\nThis is a finite coefficient proof followed by a universal tail\nargument, rather than extrapolation from sampled rows or steps.\n\\end{proof}","rank eight they are respectively\n\\[\n -2,\\ -\\frac{10}{9},\\ -\\frac{46}{45},\\ -\\frac{166}{165},\\\n -\\frac{496}{495},\\ -\\frac{1288}{1287},\\\n -\\frac{3004}{3003},\\ -\\frac{6436}{6435}.\n\\]\nAll later entries in the table are finite rational arithmetic, with\nfull numerator and denominator records retained in the accompanying\nexact certificate \\texttt{all\\_rank\\_extensions\\_verified.json}.\nEvery entry with $k\\geq2$ meets\n$b_{N,j}+1+\\lfloor\\log_2k\\rfloor$. Beyond the displayed cutoff, the\ninequality $k\\geq b_{N,j}+1+\\lfloor\\log_2k\\rfloor$ holds at every\norder. Thus the finite table and the proved infinite tail establish\nthe full hypothesis of Lemma~\\ref{lem:newton-exact-isometry-criterion}.\nThis is a finite coefficient proof followed by a universal tail\nargument, rather than extrapolation from sampled rows or steps.\n\\end{proof}","y are respectively\n\\[\n -2,\\ -\\frac{10}{9},\\ -\\frac{46}{45},\\ -\\frac{166}{165},\\\n -\\frac{496}{495},\\ -\\frac{1288}{1287},\\\n -\\frac{3004}{3003},\\ -\\frac{6436}{6435}.\n\\]\nAll later entries in the table are finite rational arithmetic, with\nfull numerator and denominator records retained in the accompanying\nexact certificate \\texttt{all\\_rank\\_extensions\\_verified.json}.\nEvery entry with $k\\geq2$ meets\n$b_{N,j}+1+\\lfloor\\log_2k\\rfloor$. Beyond the displayed cutoff, the\ninequality $k\\geq b_{N,j}+1+\\lfloor\\log_2k\\rfloor$ holds at every\norder. Thus the finite table and the proved infinite tail establish\nthe full hypothesis of Lemma~\\ref{lem:newton-exact-isometry-criterion}.\nThis is a finite coefficient proof followed by a universal tail\nargument, rather than extrapolation from sampled rows or steps.\n\\end{proof}","-\\frac{496}{495},\\ -\\frac{1288}{1287},\\\n -\\frac{3004}{3003},\\ -\\frac{6436}{6435}.\n\\]\nAll later entries in the table are finite rational arithmetic, with\nfull numerator and denominator records retained in the accompanying\nexact certificate \\texttt{all\\_rank\\_extensions\\_verified.json}.\nEvery entry with $k\\geq2$ meets\n$b_{N,j}+1+\\lfloor\\log_2k\\rfloor$. Beyond the displayed cutoff, the\ninequality $k\\geq b_{N,j}+1+\\lfloor\\log_2k\\rfloor$ holds at every\norder. Thus the finite table and the proved infinite tail establish\nthe full hypothesis of Lemma~\\ref{lem:newton-exact-isometry-criterion}.\nThis is a finite coefficient proof followed by a universal tail\nargument, rather than extrapolation from sampled rows or steps.\n\\end{proof}","\\begin{remark}[Failure of a universal upper-half exact law]\nThe all-rank lower bound remains valid in every address. The exact\nfirst-difference law does not automatically extend to all upper-half\naddresses. Exact rational recurrence calculations give\n\\begin{center}\n\\begin{tabular}{c c c c}\n$(N,j)$&$v_2(C(1)-C(0))$&$v_2(C(2)-C(0))$&naive $b+v_2(2)$","hline\n$(16,9)$&5&7&6","$(32,18)$&6&6&7","$(32,19)$&5&7&6\n\\end{tabular}\n\\end{center}\nFor example, at $(N,j)=(16,9)$ the first difference is\n$-2042976/2042975$, whereas the two-row difference is\n$-2215603929728/2215923048625$. These exact counterexamples refute\nthe additional universal isometry proposal. They preserve the\nthreshold-free lower bound and the original conjecture.\n\\end{remark}","\\begin{theorem}[Taylor coefficient bounds and radii]\n\\label{thm:four-taylor-radii}\nFor $j=0,1,2,3$, the dyadic interpolation has an ordinary power series\n\\[\n \\widetilde C_j(x)=1+\\sum_{\\ell\\geq1}d_{j,\\ell}x^\\ell,\n \\qquad v_2(d_{j,\\ell})\\geq\\ell+\\beta_j.\n\\]\nIts Taylor series converges for $|x|_2<2$.\nAt addresses zero and one,\n\\begin{equation}\n v_2(d_{j,\\ell})=\\ell\\quad(\\ell\\geq1),\n \\qquad R_0=R_1=2.\n \\label{eq:zero-one-exact-taylor-radius}\n\\end{equation}\nAt address two,\n\\begin{equation}\n v_2(d_{2,\\ell})\\geq\n       \\max\\left(2\\ell+\\left\\lfloor\\frac\\ell2\\right\\rfloor,\n                          \\ell+3\\right),\n \\qquad R_2=2^{5/2}.\n \\label{eq:two-exact-taylor-radius}\n\\end{equation}\nThe radius at address three is at least two. Its exact radius is not\nasserted here. The stated exact radii at add","\\begin{theorem}[Taylor coefficient bounds and radii]\n\\label{thm:four-taylor-radii}\nFor $j=0,1,2,3$, the dyadic interpolation has an ordinary power series\n\\[\n \\widetilde C_j(x)=1+\\sum_{\\ell\\geq1}d_{j,\\ell}x^\\ell,\n \\qquad v_2(d_{j,\\ell})\\geq\\ell+\\beta_j.\n\\]\nIts Taylor series converges for $|x|_2<2$.\nAt addresses zero and one,\n\\begin{equation}\n v_2(d_{j,\\ell})=\\ell\\quad(\\ell\\geq1),\n \\qquad R_0=R_1=2.\n \\label{eq:zero-one-exact-taylor-radius}\n\\end{equation}\nAt address two,\n\\begin{equation}\n v_2(d_{2,\\ell})\\geq\n       \\max\\left(2\\ell+\\left\\lfloor\\frac\\ell2\\right\\rfloor,\n                          \\ell+3\\right),\n \\qquad R_2=2^{5/2}.\n \\label{eq:two-exact-taylor-radius}\n\\end{equation}\nThe radius at address three is at least two. Its exact radius is not\nasserted here. The stated exact radii at addresses zero, one, and two\nhold at every center in $\\mathbb Z_2$.\n\\end{theorem}","=0,1,2,3$, the dyadic interpolation has an ordinary power series\n\\[\n \\widetilde C_j(x)=1+\\sum_{\\ell\\geq1}d_{j,\\ell}x^\\ell,\n \\qquad v_2(d_{j,\\ell})\\geq\\ell+\\beta_j.\n\\]\nIts Taylor series converges for $|x|_2<2$.\nAt addresses zero and one,\n\\begin{equation}\n v_2(d_{j,\\ell})=\\ell\\quad(\\ell\\geq1),\n \\qquad R_0=R_1=2.\n \\label{eq:zero-one-exact-taylor-radius}\n\\end{equation}\nAt address two,\n\\begin{equation}\n v_2(d_{2,\\ell})\\geq\n       \\max\\left(2\\ell+\\left\\lfloor\\frac\\ell2\\right\\rfloor,\n                          \\ell+3\\right),\n \\qquad R_2=2^{5/2}.\n \\label{eq:two-exact-taylor-radius}\n\\end{equation}\nThe radius at address three is at least two. Its exact radius is not\nasserted here. The stated exact radii at addresses zero, one, and two\nhold at every center in $\\mathbb Z_2$.\n\\end{theorem}","quation}\nAt address two,\n\\begin{equation}\n v_2(d_{2,\\ell})\\geq\n       \\max\\left(2\\ell+\\left\\lfloor\\frac\\ell2\\right\\rfloor,\n                          \\ell+3\\right),\n \\qquad R_2=2^{5/2}.\n \\label{eq:two-exact-taylor-radius}\n\\end{equation}\nThe radius at address three is at least two. Its exact radius is not\nasserted here. The stated exact radii at addresses zero, one, and two\nhold at every center in $\\mathbb Z_2$.\n\\end{theorem}","\\begin{proof}\nPut $a_m=\\Delta_1^mC_j(0)$ and expand the falling factorial in the\nNewton series. The ordinary coefficients are\n\\begin{equation}\n d_{j,\\ell}=\\sum_{m\\geq\\ell}\\frac{a_m}{m!}s(m,\\ell).\n \\label{eq:newton-taylor-stirling}\n\\end{equation}\nSince $s(m,\\ell)$ is integral and\n$v_2(a_m/m!)\\geq m+\\beta_j$, the coefficient sum converges and has\nvaluation at least $\\ell+\\beta_j$. The sum exchange is justified on\n$\\mathbb Z_2$ by these uniform geometric coefficient bounds.\nThe ordinary power series agrees there with the Newton interpolation\nand converges whenever $|x|_2<2$.","t $a_m=\\Delta_1^mC_j(0)$ and expand the falling factorial in the\nNewton series. The ordinary coefficients are\n\\begin{equation}\n d_{j,\\ell}=\\sum_{m\\geq\\ell}\\frac{a_m}{m!}s(m,\\ell).\n \\label{eq:newton-taylor-stirling}\n\\end{equation}\nSince $s(m,\\ell)$ is integral and\n$v_2(a_m/m!)\\geq m+\\beta_j$, the coefficient sum converges and has\nvaluation at least $\\ell+\\beta_j$. The sum exchange is justified on\n$\\mathbb Z_2$ by these uniform geometric coefficient bounds.\nThe ordinary power series agrees there with the Newton interpolation\nand converges whenever $|x|_2<2$.","l}\\frac{a_m}{m!}s(m,\\ell).\n \\label{eq:newton-taylor-stirling}\n\\end{equation}\nSince $s(m,\\ell)$ is integral and\n$v_2(a_m/m!)\\geq m+\\beta_j$, the coefficient sum converges and has\nvaluation at least $\\ell+\\beta_j$. The sum exchange is justified on\n$\\mathbb Z_2$ by these uniform geometric coefficient bounds.\nThe ordinary power series agrees there with the Newton interpolation\nand converges whenever $|x|_2<2$.","For $j=0,1$, the term $m=\\ell$ in\n\\eqref{eq:newton-taylor-stirling} has exact valuation $\\ell$, since\n$s(\\ell,\\ell)=1$ and the Newton valuation is exact. Every later term\nhas valuation at least $\\ell+1$. Thus\n$v_2(d_{j,\\ell})=\\ell$, and the coefficient radius is exactly two.\nAt a center $x_0\\in\\mathbb Z_2$, the coefficient of $y^\\ell$ in the\ntranslated series is\n\\begin{equation}\n \\sum_{m\\geq\\ell}d_{j,m}\\binom m\\ell x_0^{m-\\ell}.\n \\label{eq:taylor-translation-coefficient}\n\\end{equation}\nIts initial term again has exact valuation $\\ell$, and all later\nterms have at least one more bit. The exact radius is therefore two\nat every integral center.","aluation is exact. Every later term\nhas valuation at least $\\ell+1$. Thus\n$v_2(d_{j,\\ell})=\\ell$, and the coefficient radius is exactly two.\nAt a center $x_0\\in\\mathbb Z_2$, the coefficient of $y^\\ell$ in the\ntranslated series is\n\\begin{equation}\n \\sum_{m\\geq\\ell}d_{j,m}\\binom m\\ell x_0^{m-\\ell}.\n \\label{eq:taylor-translation-coefficient}\n\\end{equation}\nIts initial term again has exact valuation $\\ell$, and all later\nterms have at least one more bit. The exact radius is therefore two\nat every integral center.","For address two, the stronger factorial filtration gives the lower\nbound in \\eqref{eq:two-exact-taylor-radius}. At each\n$\\ell=2^r\\geq4$, Theorem~\\ref{thm:two-exact-power-orders} makes the\n$m=\\ell$ term in \\eqref{eq:newton-taylor-stirling} have valuation\n$2\\ell+\\ell/2=5\\ell/2$. Every $m>\\ell$ term has strictly greater\nlower bound $2m+\\lfloor m/2\\rfloor$. Hence\n\\[\n v_2(d_{2,\\ell})=5\\ell/2\\qquad(\\ell=2^r\\geq4).\n\\]\nThe universal lower bound and this unbounded exact subsequence prove\nthat the coefficient radius is exactly $2^{5/2}$. Translation by an\nintegral center preserves the same unique leading coefficients at\nthese degrees in \\eqref{eq:taylor-translation-coefficient}, proving\nthe exact radius at every such center.\n\\end{proof}","l=2^r\\geq4$, Theorem~\\ref{thm:two-exact-power-orders} makes the\n$m=\\ell$ term in \\eqref{eq:newton-taylor-stirling} have valuation\n$2\\ell+\\ell/2=5\\ell/2$. Every $m>\\ell$ term has strictly greater\nlower bound $2m+\\lfloor m/2\\rfloor$. Hence\n\\[\n v_2(d_{2,\\ell})=5\\ell/2\\qquad(\\ell=2^r\\geq4).\n\\]\nThe universal lower bound and this unbounded exact subsequence prove\nthat the coefficient radius is exactly $2^{5/2}$. Translation by an\nintegral center preserves the same unique leading coefficients at\nthese degrees in \\eqref{eq:taylor-translation-coefficient}, proving\nthe exact radius at every such center.\n\\end{proof}","\\begin{remark}[Boundary interpretation]\nFor addresses zero and one, at an argument of absolute value two every\npositive-degree Taylor term has valuation zero, so that ordinary Taylor\nseries diverges. For address two, over a complete extension admitting\narguments of absolute value $2^{5/2}$, the power-of-two-degree terms\nhave valuation zero and fail to tend to zero. These statements concern\nthe Taylor expansions just constructed. They do not prohibit a\ndifferent analytic continuation. On $\\mathbb Q_2$-valued arguments,\nopen-disc notation must be read with its discrete value group. The\ncoefficient radius is the precise invariant in the theorem.\n\\end{remark}","e value two every\npositive-degree Taylor term has valuation zero, so that ordinary Taylor\nseries diverges. For address two, over a complete extension admitting\narguments of absolute value $2^{5/2}$, the power-of-two-degree terms\nhave valuation zero and fail to tend to zero. These statements concern\nthe Taylor expansions just constructed. They do not prohibit a\ndifferent analytic continuation. On $\\mathbb Q_2$-valued arguments,\nopen-disc notation must be read with its discrete value group. The\ncoefficient radius is the precise invariant in the theorem.\n\\end{remark}","\\begin{corollary}[Final common-step congruence]\n\\label{cor:final-euler-congruence}\nLet $j\\in\\{0,1,2,3\\}$, $n\\geq0$, and $k,H\\geq1$. Define\n\\[\n A_0(k)=A_1(k)=k,\\qquad\n A_2(k)=\\max\\left(k+4,\\ 2k+\\left\\lfloor\\frac k2\\right\\rfloor+1\\right),\n \\qquad A_3(k)=k+2.\n\\]\nThen\n\\begin{equation}\n \\boxed{\\displaystyle\n \\sum_{i=0}^k(-1)^{k-i}\\binom ki E_{4(n+iH)}^{(4,j)}\n       \\equiv0\\pmod{2^{\\,v_2(k!)+k v_2(H)+A_j(k)}}.}\n \\label{eq:boxed-final-euler-congruence}\n\\end{equation}\nFor $j=0,1$, this displayed exponent is the exact valuation for every\n$n,k,H$ in the stated range. For $j=2$, the exact positive\npower-of-two-order valuations are given by\nTheorem~\\ref{thm:two-exact-power-orders}, with one added bit.\n\\end{corollary}","congruence}\nLet $j\\in\\{0,1,2,3\\}$, $n\\geq0$, and $k,H\\geq1$. Define\n\\[\n A_0(k)=A_1(k)=k,\\qquad\n A_2(k)=\\max\\left(k+4,\\ 2k+\\left\\lfloor\\frac k2\\right\\rfloor+1\\right),\n \\qquad A_3(k)=k+2.\n\\]\nThen\n\\begin{equation}\n \\boxed{\\displaystyle\n \\sum_{i=0}^k(-1)^{k-i}\\binom ki E_{4(n+iH)}^{(4,j)}\n       \\equiv0\\pmod{2^{\\,v_2(k!)+k v_2(H)+A_j(k)}}.}\n \\label{eq:boxed-final-euler-congruence}\n\\end{equation}\nFor $j=0,1$, this displayed exponent is the exact valuation for every\n$n,k,H$ in the stated range. For $j=2$, the exact positive\npower-of-two-order valuations are given by\nTheorem~\\ref{thm:two-exact-power-orders}, with one added bit.\n\\end{corollary}","e\n\\[\n A_0(k)=A_1(k)=k,\\qquad\n A_2(k)=\\max\\left(k+4,\\ 2k+\\left\\lfloor\\frac k2\\right\\rfloor+1\\right),\n \\qquad A_3(k)=k+2.\n\\]\nThen\n\\begin{equation}\n \\boxed{\\displaystyle\n \\sum_{i=0}^k(-1)^{k-i}\\binom ki E_{4(n+iH)}^{(4,j)}\n       \\equiv0\\pmod{2^{\\,v_2(k!)+k v_2(H)+A_j(k)}}.}\n \\label{eq:boxed-final-euler-congruence}\n\\end{equation}\nFor $j=0,1$, this displayed exponent is the exact valuation for every\n$n,k,H$ in the stated range. For $j=2$, the exact positive\npower-of-two-order valuations are given by\nTheorem~\\ref{thm:two-exact-power-orders}, with one added bit.\n\\end{corollary}","(k)=\\max\\left(k+4,\\ 2k+\\left\\lfloor\\frac k2\\right\\rfloor+1\\right),\n \\qquad A_3(k)=k+2.\n\\]\nThen\n\\begin{equation}\n \\boxed{\\displaystyle\n \\sum_{i=0}^k(-1)^{k-i}\\binom ki E_{4(n+iH)}^{(4,j)}\n       \\equiv0\\pmod{2^{\\,v_2(k!)+k v_2(H)+A_j(k)}}.}\n \\label{eq:boxed-final-euler-congruence}\n\\end{equation}\nFor $j=0,1$, this displayed exponent is the exact valuation for every\n$n,k,H$ in the stated range. For $j=2$, the exact positive\npower-of-two-order valuations are given by\nTheorem~\\ref{thm:two-exact-power-orders}, with one added bit.\n\\end{corollary}","\\begin{proof}\nThe left side is $j!\\Delta_H^kC_j(n)$.\nAt addresses zero and one, $v_2(j!)=0$ and the exact homogeneous\nformula gives the exponent. At address two, $v_2(2!)=1$. The maximum\nof \\eqref{eq:factorial-four-lower} and\n\\eqref{eq:factorial-four-j2-filtration} yields\n$\\max(k+4,2k+\\lfloor k/2\\rfloor+1)$ after removing the common\n$v_2(k!)+k v_2(H)$. At address three, the one-bit normalized offset\nand $v_2(3!)=1$ give $k+2$.\n\\end{proof}","roof}\nThe left side is $j!\\Delta_H^kC_j(n)$.\nAt addresses zero and one, $v_2(j!)=0$ and the exact homogeneous\nformula gives the exponent. At address two, $v_2(2!)=1$. The maximum\nof \\eqref{eq:factorial-four-lower} and\n\\eqref{eq:factorial-four-j2-filtration} yields\n$\\max(k+4,2k+\\lfloor k/2\\rfloor+1)$ after removing the common\n$v_2(k!)+k v_2(H)$. At address three, the one-bit normalized offset\nand $v_2(3!)=1$ give $k+2$.\n\\end{proof}","de is $j!\\Delta_H^kC_j(n)$.\nAt addresses zero and one, $v_2(j!)=0$ and the exact homogeneous\nformula gives the exponent. At address two, $v_2(2!)=1$. The maximum\nof \\eqref{eq:factorial-four-lower} and\n\\eqref{eq:factorial-four-j2-filtration} yields\n$\\max(k+4,2k+\\lfloor k/2\\rfloor+1)$ after removing the common\n$v_2(k!)+k v_2(H)$. At address three, the one-bit normalized offset\nand $v_2(3!)=1$ give $k+2$.\n\\end{proof}","For mixed steps the same corollary holds with $k v_2(H)$ replaced by\n$\\sum_\\nu v_2(H_\\nu)$ and the binomial stencil replaced by\n$j!\\Delta_{H_1}\\cdots\\Delta_{H_k}C_j(n)$.\nAt addresses zero and one, exactness is then governed by the parity\nclassification in Theorem~\\ref{thm:zero-one-exact-mixed}.\nAt first order these bounds are exact in every address, with original\nvaluation offsets $(1,1,5,3)$. The least periods are those in\n\\eqref{eq:rank-four-least-period-final}.","is checked by the accompanying exact script.\n\\fi\n\\newcommand{\\AtlasHead}[2]{\\ifdefined\\AtlasStarted\\clearpage\\fi\\ifdefined\\IXMainAtlas\\subsection{#1}\\else\\section{#1}\\fi\\label{#2}\\def\\AtlasStarted{1}}\n\\newcommand{\\AtlasImage}[1]{\\begin{center}\\includegraphics[width=.91\\linewidth]{\\AtlasFigDir/#1.pdf}\\end{center}}\n\\AtlasHead{Compute the reciprocal before estimating it}{atlas:reciprocal}\nFor width $N$ and address $j$, the normalized source and its reciprocal are\n\\[\n G_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n G_{N,j}(t)^{-1}=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\]\nMultiplying the two series and clearing the factorials gives an exact triangular recurrence:\n\\[\n C_{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=12…","e accompanying exact script.\n\\fi\n\\newcommand{\\AtlasHead}[2]{\\ifdefined\\AtlasStarted\\clearpage\\fi\\ifdefined\\IXMainAtlas\\subsection{#1}\\else\\section{#1}\\fi\\label{#2}\\def\\AtlasStarted{1}}\n\\newcommand{\\AtlasImage}[1]{\\begin{center}\\includegraphics[width=.91\\linewidth]{\\AtlasFigDir/#1.pdf}\\end{center}}\n\\AtlasHead{Compute the reciprocal before estimating it}{atlas:reciprocal}\nFor width $N$ and address $j$, the normalized source and its reciprocal are\n\\[\n G_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n G_{N,j}(t)^{-1}=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\]\nMultiplying the two series and clearing the factorials gives an exact triangular recurrence:\n\\[\n C_{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are","Head}[2]{\\ifdefined\\AtlasStarted\\clearpage\\fi\\ifdefined\\IXMainAtlas\\subsection{#1}\\else\\section{#1}\\fi\\label{#2}\\def\\AtlasStarted{1}}\n\\newcommand{\\AtlasImage}[1]{\\begin{center}\\includegraphics[width=.91\\linewidth]{\\AtlasFigDir/#1.pdf}\\end{center}}\n\\AtlasHead{Compute the reciprocal before estimating it}{atlas:reciprocal}\nFor width $N$ and address $j$, the normalized source and its reciprocal are\n\\[\n G_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n G_{N,j}(t)^{-1}=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\]\nMultiplying the two series and clearing the factorials gives an exact triangular recurrence:\n\\[\n C_{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are exact rational coefficients, not rounded approxima…","tlasFigDir/#1.pdf}\\end{center}}\n\\AtlasHead{Compute the reciprocal before estimating it}{atlas:reciprocal}\nFor width $N$ and address $j$, the normalized source and its reciprocal are\n\\[\n G_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n G_{N,j}(t)^{-1}=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\]\nMultiplying the two series and clearing the factorials gives an exact triangular recurrence:\n\\[\n C_{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are exact rational coefficients, not rounded approximations. The second implementation in the supplied script inverts the ordinary power series first, then multiplies by $(Nn)!$. Agreement between the two constructions checks the factorial placement.","ciprocal are\n\\[\n G_{N,j}(t)=j!\\sum_{n\\ge0}\\frac{t^{Nn}}{(Nn+j)!},\\qquad\n G_{N,j}(t)^{-1}=\\sum_{n\\ge0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\]\nMultiplying the two series and clearing the factorials gives an exact triangular recurrence:\n\\[\n C_{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are exact rational coefficients, not rounded approximations. The second implementation in the supplied script inverts the ordinary power series first, then multiplies by $(Nn)!$. Agreement between the two constructions checks the factorial placement.","0}C_{N,j}(n)\\frac{t^{Nn}}{(Nn)!}.\n\\]\nMultiplying the two series and clearing the factorials gives an exact triangular recurrence:\n\\[\n C_{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are exact rational coefficients, not rounded approximations. The second implementation in the supplied script inverts the ordinary power series first, then multiplies by $(Nn)!$. Agreement between the two constructions checks the factorial placement.","lar recurrence:\n\\[\n C_{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are exact rational coefficients, not rounded approximations. The second implementation in the supplied script inverts the ordinary power series first, then multiplies by $(Nn)!$. Agreement between the two constructions checks the factorial placement.","{N,j}(0)=1,\\qquad\n C_{N,j}(n)=-\\frac{\\displaystyle\\sum_{r=0}^{n-1}\n \\binom{Nn+j}{Nr}C_{N,j}(r)}{\\binom{Nn+j}{Nn}}.\n\\]\n\\AtlasImage{reciprocal_rows}\nThe table is for $N=4$. The first two address-zero rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are exact rational coefficients, not rounded approximations. The second implementation in the supplied script inverts the ordinary power series first, then multiplies by $(Nn)!$. Agreement between the two constructions checks the factorial placement.","rows illustrate the computation without any valuation estimate:\n\\[\n C_{4,0}(1)=-1,\\qquad\n C_{4,0}(2)=-\\bigl(1+\\binom84(-1)\\bigr)=69.\n\\]\nAt address one, the same recurrence yields $C_{4,1}(1)=-1/5$ and $C_{4,1}(2)=121/45$. These are exact rational coefficients, not rounded approximations. The second implementation in the supplied script inverts the ordinary power series first, then multiplies by $(Nn)!$. Agreement between the two constructions checks the factorial placement.","\\AtlasHead{The normalization loss is a real boundary}{atlas:normalization}\nThe original Euler coefficient is $E_{Nn}^{(N,j)}=j!C_{N,j}(n)$. Multiplication by $j!$ changes the available $p$-adic precision. That change cannot be discarded outside the unit-factorial window.","Take $p=3$, $N=6$, $j=3$, and compare rows one and two. Exact inversion gives\n\\[\n C_{6,3}(1)=-\\frac1{84},\\quad\n C_{6,3}(2)=\\frac{703}{5460},\\quad\n C_{6,3}(2)-C_{6,3}(1)=\\frac{64}{455}.\n\\]\n\\AtlasImage{normalization_loss}\nNeither $64$ nor $455$ is divisible by three, so the normalized difference has valuation zero. The original coefficient difference is $3!\\cdot64/455=384/455$, of valuation one.","This is the counterexample behind the corrected factorial-loss statement. A bound for the original Euler normalization must lose $v_p(j!)$ when translated to $C_{N,j}$. For $0\\le j<p$, that loss is zero. The distinction concerns which sequence the theorem measures, not a numerical weakness of the calculation.","At $N=p=3$ and $j=0$, put $A(n)=(-1)^nC_{3,0}(n)$. The first three rows are\n\\AtlasImage{odd_prime_sign}\nThe ordinary first difference at row zero is $-1-1=-2$. It is a $3$-adic unit, so an asserted lower bound of one for that difference would be false. The twisted first difference is zero, whose mathematical valuation is $+\\infty$. At the next row, the twisted difference is $19-1=18$, of valuation two.","Equivalently, the untwisted sequence is acted on by $T_H=S^H-(-1)^H$, where $S$ advances one row. At odd step one this is $S+1$, not $S-1$. The zero in the example is handled by a zero-safe divisibility statement rather than a finite valuation equality. The explicit sign and the zero convention are both part of the theorem's interface.","\\AtlasHead{Newton coordinates turn precision into a finite computation}{atlas:newton}\nLet $b_k=\\Delta_1^kC_{4,0}(0)$. The first coefficients obtained from the difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ throug","\\AtlasHead{Newton coordinates turn precision into a finite computation}{atlas:newton}\nLet $b_k=\\Delta_1^kC_{4,0}(0)$. The first coefficients obtained from the difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","ates turn precision into a finite computation}{atlas:newton}\nLet $b_k=\\Delta_1^kC_{4,0}(0)$. The first coefficients obtained from the difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","sion into a finite computation}{atlas:newton}\nLet $b_k=\\Delta_1^kC_{4,0}(0)$. The first coefficients obtained from the difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","putation}{atlas:newton}\nLet $b_k=\\Delta_1^kC_{4,0}(0)$. The first coefficients obtained from the difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","}\nLet $b_k=\\Delta_1^kC_{4,0}(0)$. The first coefficients obtained from the difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","ients obtained from the difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","difference triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","e triangle are\n\\input{generated/worked_newton_values.tex}\nThe exact Newton identity recovers each nonnegative integer row:\n\\[\n C_{4,0}(n)=\\sum_{k=0}^{n}b_k\\binom nk.\n\\]\n\\AtlasImage{newton_cutoff}\nFor this address the proved law is $v_2(b_k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","k)=2k-s_2(k)$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","$ for $k\\ge1$. Every $k\\ge5$ therefore has $v_2(b_k)\\ge8$. Since $\\binom nk$ is integral, those terms vanish modulo $256$. Thus, for every $n\\ge0$,\n\\[\n C_{4,0}(n)\\equiv1+254\\binom n1+72\\binom n2+\n 176\\binom n3+128\\binom n4\\pmod{256}.\n\\]\nThe uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows $0$ through $32$. Those finite checks validate the worked arithmetic without replacing the theorem.","\\AtlasHead{A compatible root is built one digit at a time}{atlas:smooth}\nTake $F(x)=x^2-7$ over the $3$-adic integers and start with $x_1=1$ modulo three. If $F(x_L)=3^Lr_L$, the next representative has the form $x_{L+1}=x_L+3^Lt_L$. Its correcting digit satisfies\n\\[\n r_L+2x_Lt_L\\equiv0\\pmod3.\n\\]\n\\AtlasImage{smooth_tower}\nFor example, $F(1)/3=-2$ and $2x_1=2$, so the next digit is one and $x_2=4$. Then $F(4)/9=1$, again giving digit one and $x_3=13$. At the next stage the residual is divisible by three, so digit zero is correct. A zero digit is not a stalled construction.","ead{A compatible root is built one digit at a time}{atlas:smooth}\nTake $F(x)=x^2-7$ over the $3$-adic integers and start with $x_1=1$ modulo three. If $F(x_L)=3^Lr_L$, the next representative has the form $x_{L+1}=x_L+3^Lt_L$. Its correcting digit satisfies\n\\[\n r_L+2x_Lt_L\\equiv0\\pmod3.\n\\]\n\\AtlasImage{smooth_tower}\nFor example, $F(1)/3=-2$ and $2x_1=2$, so the next digit is one and $x_2=4$. Then $F(4)/9=1$, again giving digit one and $x_3=13$. At the next stage the residual is divisible by three, so digit zero is correct. A zero digit is not a stalled construction.","t at a time}{atlas:smooth}\nTake $F(x)=x^2-7$ over the $3$-adic integers and start with $x_1=1$ modulo three. If $F(x_L)=3^Lr_L$, the next representative has the form $x_{L+1}=x_L+3^Lt_L$. Its correcting digit satisfies\n\\[\n r_L+2x_Lt_L\\equiv0\\pmod3.\n\\]\n\\AtlasImage{smooth_tower}\nFor example, $F(1)/3=-2$ and $2x_1=2$, so the next digit is one and $x_2=4$. Then $F(4)/9=1$, again giving digit one and $x_3=13$. At the next stage the residual is divisible by three, so digit zero is correct. A zero digit is not a stalled construction.","Every center remains congruent to one modulo three. Hence $2x_L$ stays a unit, and the correction equation has a unique digit at every stage. The representatives are compatible and their increments are divisible by $3^L$. Completeness gives a limit, and continuity of the polynomial gives a root of $F$. The finite table exhibits the construction. The unit derivative is what supports indefinite continuation.","\\AtlasHead{A singular residual can stop a lift}{atlas:blocked}\nNow take $F(x)=x^2-3$ at the residue $x=0$ modulo three. The residual is $r=F(0)/3=-1$ and the Jacobian is zero modulo three. The next-digit equation would require\n\\[\n -1+0\\cdot t\\equiv0\\pmod3,\n\\]\nwhich has no solution.\n\\AtlasImage{singular_block}\nThe three representatives of the old root class modulo nine all leave residual six modulo nine. Thus the root class exists at precision three but does not lift to precision nine. This is an exact obstruction, not failure of an iterative search.","A singular Jacobian alone does not imply failure. For $F(x)=x^2$ at the same center, the residual is zero and all three candidate digits lift to roots modulo nine. The obstruction is the residual class in the cokernel of the actual Jacobian. It is this class, rather than the rank by itself, that decides whether a next lift exists.","\\AtlasHead{Separate corrections need not be one shared correction}{atlas:shared}\nConsider the two-component system $F(x)=(x,x-3)$ at $x=0$, with $q=3$ and $L=1$. Both components vanish modulo three. After writing a candidate as $y=3t$, its residual vector modulo three is\n\\[\n \\overline r+J\\overline t=(0,2)+(t,t)=(t,t-1).\n\\]\n\\AtlasImage{shared_obstruction}\nThe first coordinate can be corrected by $t=0$. The second can be corrected by $t=1$. No digit corrects both. The available residuals are $(0,2)$, $(1,0)$, and $(2,1)$, none of which is $(0,0)$.","not be one shared correction}{atlas:shared}\nConsider the two-component system $F(x)=(x,x-3)$ at $x=0$, with $q=3$ and $L=1$. Both components vanish modulo three. After writing a candidate as $y=3t$, its residual vector modulo three is\n\\[\n \\overline r+J\\overline t=(0,2)+(t,t)=(t,t-1).\n\\]\n\\AtlasImage{shared_obstruction}\nThe first coordinate can be corrected by $t=0$. The second can be corrected by $t=1$. No digit corrects both. The available residuals are $(0,2)$, $(1,0)$, and $(2,1)$, none of which is $(0,0)$.","o-component system $F(x)=(x,x-3)$ at $x=0$, with $q=3$ and $L=1$. Both components vanish modulo three. After writing a candidate as $y=3t$, its residual vector modulo three is\n\\[\n \\overline r+J\\overline t=(0,2)+(t,t)=(t,t-1).\n\\]\n\\AtlasImage{shared_obstruction}\nThe first coordinate can be corrected by $t=0$. The second can be corrected by $t=1$. No digit corrects both. The available residuals are $(0,2)$, $(1,0)$, and $(2,1)$, none of which is $(0,0)$.","nt system $F(x)=(x,x-3)$ at $x=0$, with $q=3$ and $L=1$. Both components vanish modulo three. After writing a candidate as $y=3t$, its residual vector modulo three is\n\\[\n \\overline r+J\\overline t=(0,2)+(t,t)=(t,t-1).\n\\]\n\\AtlasImage{shared_obstruction}\nThe first coordinate can be corrected by $t=0$. The second can be corrected by $t=1$. No digit corrects both. The available residuals are $(0,2)$, $(1,0)$, and $(2,1)$, none of which is $(0,0)$.","x)=(x,x-3)$ at $x=0$, with $q=3$ and $L=1$. Both components vanish modulo three. After writing a candidate as $y=3t$, its residual vector modulo three is\n\\[\n \\overline r+J\\overline t=(0,2)+(t,t)=(t,t-1).\n\\]\n\\AtlasImage{shared_obstruction}\nThe first coordinate can be corrected by $t=0$. The second can be corrected by $t=1$. No digit corrects both. The available residuals are $(0,2)$, $(1,0)$, and $(2,1)$, none of which is $(0,0)$.","$q=3$ and $L=1$. Both components vanish modulo three. After writing a candidate as $y=3t$, its residual vector modulo three is\n\\[\n \\overline r+J\\overline t=(0,2)+(t,t)=(t,t-1).\n\\]\n\\AtlasImage{shared_obstruction}\nThe first coordinate can be corrected by $t=0$. The second can be corrected by $t=1$. No digit corrects both. The available residuals are $(0,2)$, $(1,0)$, and $(2,1)$, none of which is $(0,0)$.","The combined Jacobian maps $t$ to the diagonal vector $(t,t)$. Its image is smaller than the product of its coordinate images. Checking the two projected obstructions independently therefore loses the compatibility condition. The common-correction theorem retains the full residual vector and the shared digit space, which is exactly the information needed to detect this failure.","\\AtlasHead{A derivative weight can cost one digit of root precision}{atlas:weighted}\nCompare the linear equations $F(x)=3x=0$ and $G_A(x)=3x+3^A=0$, where $A\\ge2$. Their coefficient difference has $3$-adic valuation $A$, but their roots differ by\n\\[\n x_A-0=-3^{A-1},\\qquad v_3(x_A)=A-1.\n\\]\n\\AtlasImage{weighted_motion}\nThe derivative has valuation one. Solving the correction equation divides by three, so one digit of coefficient precision is lost in root displacement. The calculation is exact for every displayed $A$ and also follows immediately from the two linear equations.","x)=3x=0$ and $G_A(x)=3x+3^A=0$, where $A\\ge2$. Their coefficient difference has $3$-adic valuation $A$, but their roots differ by\n\\[\n x_A-0=-3^{A-1},\\qquad v_3(x_A)=A-1.\n\\]\n\\AtlasImage{weighted_motion}\nThe derivative has valuation one. Solving the correction equation divides by three, so one digit of coefficient precision is lost in root displacement. The calculation is exact for every displayed $A$ and also follows immediately from the two linear equations.","This example explains why weighted lifting and deformation estimates track a Jacobian loss rather than silently treating every invertible derivative over $\\mathbb Q_p$ as a unit over $\\mathbb Z_p$. A valuation-zero derivative would not incur this particular loss. Nonlinear systems additionally require their stated neighborhood, residual, and right-inverse hypotheses.","\\[\n\\boxed{\n\\text{residual root}\n\\quad+\\quad\n\\text{positive-depth deformation coordinates}.\n}\n\\]","Let \\(\\Ocal\\) be a complete discrete valuation ring with uniformizer \\(\\varpi\\), maximal ideal \\((\\varpi)\\), residue field\n\\[\nk=\\Ocal/(\\varpi),\\qquad |k|=q,\n\\]\nand fraction field \\(K\\).  When a characteristic-zero conclusion is intended, assume additionally that \\(\\operatorname{char}K=0\\). The general counting statements do not require that restriction.  Let \\(\\mathcal M\\) be a moduli functor on complete local Noetherian \\(\\Ocal\\)-algebras with residue field \\(k\\), equipped with a well-defined reduction map\n\\[\n\\red_A:\\mathcal M(A)\\longrightarrow \\mathcal M(k).\n\\]\nFor a Galois representation this means either that a framed integral model has been fixed or that the residual semisimplified class is used.","discrete valuation ring with uniformizer \\(\\varpi\\), maximal ideal \\((\\varpi)\\), residue field\n\\[\nk=\\Ocal/(\\varpi),\\qquad |k|=q,\n\\]\nand fraction field \\(K\\).  When a characteristic-zero conclusion is intended, assume additionally that \\(\\operatorname{char}K=0\\). The general counting statements do not require that restriction.  Let \\(\\mathcal M\\) be a moduli functor on complete local Noetherian \\(\\Ocal\\)-algebras with residue field \\(k\\), equipped with a well-defined reduction map\n\\[\n\\red_A:\\mathcal M(A)\\longrightarrow \\mathcal M(k).\n\\]\nFor a Galois representation this means either that a framed integral model has been fixed or that the residual semisimplified class is used.","zer \\(\\varpi\\), maximal ideal \\((\\varpi)\\), residue field\n\\[\nk=\\Ocal/(\\varpi),\\qquad |k|=q,\n\\]\nand fraction field \\(K\\).  When a characteristic-zero conclusion is intended, assume additionally that \\(\\operatorname{char}K=0\\). The general counting statements do not require that restriction.  Let \\(\\mathcal M\\) be a moduli functor on complete local Noetherian \\(\\Ocal\\)-algebras with residue field \\(k\\), equipped with a well-defined reduction map\n\\[\n\\red_A:\\mathcal M(A)\\longrightarrow \\mathcal M(k).\n\\]\nFor a Galois representation this means either that a framed integral model has been fixed or that the residual semisimplified class is used.","e field\n\\[\nk=\\Ocal/(\\varpi),\\qquad |k|=q,\n\\]\nand fraction field \\(K\\).  When a characteristic-zero conclusion is intended, assume additionally that \\(\\operatorname{char}K=0\\). The general counting statements do not require that restriction.  Let \\(\\mathcal M\\) be a moduli functor on complete local Noetherian \\(\\Ocal\\)-algebras with residue field \\(k\\), equipped with a well-defined reduction map\n\\[\n\\red_A:\\mathcal M(A)\\longrightarrow \\mathcal M(k).\n\\]\nFor a Galois representation this means either that a framed integral model has been fixed or that the residual semisimplified class is used.","\\begin{definition}[Residual seed system]\\label{px:1.1}\nFor \\(\\bar x\\in\\mathcal M(k)\\), define the residual fiber\n\\[\n\\mathcal M_{\\bar x}(A)\n=\n\\{x\\in\\mathcal M(A):\\red_A(x)=\\bar x\\}.\n\\]\nThe point \\(\\bar x\\) is the residual root.  The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.\n\\end{definition}","For a Galois problem the principal example is\n\\[\n\\bar\\rho:G_{F,S}\\longrightarrow \\GL_n(k),\n\\]\nwith a deformation functor \\(\\Def_{\\bar\\rho}\\) cut out by a determinant and selected local conditions.","\\begin{theorem}[Residual partition]\\label{px:2.1}\nFor every coefficient algebra \\(A\\),\n\\[\n\\boxed{\n\\mathcal M(A)\n=\n\\bigsqcup_{\\bar x\\in\\mathcal M(k)}\n\\mathcal M_{\\bar x}(A).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nEvery \\(A\\)-valued object has one reduced object \\(\\red_A(x)\\in\\mathcal M(k)\\), so the union is exhaustive.  Two distinct fibers cannot intersect because an object cannot have two different reductions.  Hence the union is disjoint.\n\\end{proof}","\\begin{theorem}[No crossing under positive-depth perturbation]\\label{px:2.2}\nSuppose \\(x,y\\in\\mathcal M(\\Ocal)\\) are related by a deformation operation that is congruent to the identity modulo \\(\\varpi\\).  Then\n\\[\n\\red(x)=\\red(y).\n\\]\nEvery finite composition remains in one residual fiber.  If an infinite perturbation stream converges in the \\(\\varpi\\)-adic topology, its limit remains in the same fiber.  The existence of that limit is an explicit hypothesis, not a consequence of positive depth alone.\n\\end{theorem}","\\begin{proof}\nA correction in \\(\\varpi\\Ocal\\) vanishes after reduction modulo \\(\\varpi\\).  Thus one operation preserves the residual root, and induction proves the finite statement.  For a convergent stream, convergence at the first finite precision makes its reduction eventually equal to that of its limit.  Every term already has the initial reduction, so the limit has that reduction too.  Completeness applies to Cauchy streams. It does not make every root-preserving stream Cauchy.  For example, \\(0,3,0,3,\\ldots\\) has constant reduction modulo \\(3\\) but no limit modulo \\(9\\).\n\\end{proof}","ation preserves the residual root, and induction proves the finite statement.  For a convergent stream, convergence at the first finite precision makes its reduction eventually equal to that of its limit.  Every term already has the initial reduction, so the limit has that reduction too.  Completeness applies to Cauchy streams. It does not make every root-preserving stream Cauchy.  For example, \\(0,3,0,3,\\ldots\\) has constant reduction modulo \\(3\\) but no limit modulo \\(9\\).\n\\end{proof}","proves the finite statement.  For a convergent stream, convergence at the first finite precision makes its reduction eventually equal to that of its limit.  Every term already has the initial reduction, so the limit has that reduction too.  Completeness applies to Cauchy streams. It does not make every root-preserving stream Cauchy.  For example, \\(0,3,0,3,\\ldots\\) has constant reduction modulo \\(3\\) but no limit modulo \\(9\\).\n\\end{proof}","t.  For a convergent stream, convergence at the first finite precision makes its reduction eventually equal to that of its limit.  Every term already has the initial reduction, so the limit has that reduction too.  Completeness applies to Cauchy streams. It does not make every root-preserving stream Cauchy.  For example, \\(0,3,0,3,\\ldots\\) has constant reduction modulo \\(3\\) but no limit modulo \\(9\\).\n\\end{proof}","\\begin{numericalcertificate}[Exact residual partition]\\label{px:2.4}\nThe finite implementation partitions the integers \\(0,\\ldots,8\\) into the three classes modulo \\(3\\):\n\\[\n\\{0,3,6\\},\\qquad \\{1,4,7\\},\\qquad \\{2,5,8\\}.\n\\]\nEvery tested correction of the form \\(x\\mapsto x+p h\\) preserves the residual class exactly.\n\\end{numericalcertificate}","Fix \\(\\bar x\\in\\mathcal M(k)\\).  For \\(m\\ge1\\), set\n\\[\nX_m(\\bar x)=\\mathcal M_{\\bar x}(\\Ocal/\\varpi^m).\n\\]\nReduction gives transition maps\n\\[\nr_{m+1,m}:X_{m+1}(\\bar x)\\longrightarrow X_m(\\bar x).\n\\]\nA compatible branch is a sequence \\((x_m)_{m\\ge1}\\) satisfying\n\\[\nr_{m+1,m}(x_{m+1})=x_m.\n\\]","\\begin{classicalinput}[Represented complete deformation problems]\\label{px:3.1}\nWhen a deformation functor is represented by a complete local ring \\(R_{\\bar x}\\), its \\(\\Ocal\\)-valued points are continuous local maps \\(R_{\\bar x}\\to\\Ocal\\).  Completeness identifies such maps with compatible maps to every quotient \\(\\Ocal/\\varpi^m\\).\n\\end{classicalinput}","\\begin{theorem}[Formal inverse-limit reconstruction]\\label{px:3.2}\nAssume \\(\\mathcal M_{\\bar x}\\) is represented by a complete local \\(\\Ocal\\)-algebra \\(R_{\\bar x}\\).  Then reduction induces a natural bijection\n\\[\n\\boxed{\n\\mathcal M_{\\bar x}(\\Ocal)\n\\cong\n\\varprojlim_m\n\\mathcal M_{\\bar x}(\\Ocal/\\varpi^m).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nA continuous local homomorphism \\(R_{\\bar x}\\to\\Ocal\\) reduces to a compatible family of homomorphisms into \\(\\Ocal/\\varpi^m\\).  Conversely, compatible finite-level maps determine a map\n\\[\nR_{\\bar x}\\longrightarrow\\varprojlim_m\\Ocal/\\varpi^m\\cong\\Ocal.\n\\]\nThe two constructions are inverse because equality in a separated complete ring is detected modulo every power of its maximal ideal.\n\\end{proof}","\\begin{theorem}[Finite-branch compactness]\\label{px:3.3}\nSuppose every \\(X_m(\\bar x)\\) is finite and nonempty and every node has finitely many children.  Then the lift tree above \\(\\bar x\\) contains an infinite compatible branch.\n\\end{theorem}","\\begin{proposition}[Smooth formal branch count]\\label{px:3.4}\nIf\n\\[\nR_{\\bar x}\\cong\\Ocal[[t_1,\\ldots,t_d]]\\quad\\text{as complete local }\\Ocal\\text{-algebras},\n\\]\nthen\n\\[\n\\boxed{\n\\#X_m(\\bar x)=q^{d(m-1)}.\n}\n\\]\nFix a uniformizer and a section \\(s:k\\to\\Ocal\\) of reduction, with \\(s(0)=0\\). Every \\(\\Ocal\\)-valued branch is encoded by \\(d\\) positive-depth digit streams\n\\[\nt_i=s(a_{i,1})\\varpi+s(a_{i,2})\\varpi^2+\\cdots,\n\\qquad a_{i,j}\\in k,\n\\]\n\\end{proposition}","\\begin{proof}\nA local map sends each parameter into the image of \\((\\varpi)\\) in \\(\\Ocal/\\varpi^m\\). Conversely, every tuple in that image is nilpotent and defines a coefficient-preserving evaluation homomorphism. This homomorphism is local. To prove uniqueness without assuming continuity of an arbitrary competing homomorphism, observe in one variable that a homomorphism killing \\(X^N\\) only sees the finite polynomial truncation modulo \\(X^N\\). Induction over the finite number of variables gives uniqueness in all dimensions. The actual local maps are therefore in bijection with the full parameter tuple set. Its cardinality is \\(q^{d(m-1)}\\), including the zero-dimensional chart.","eter into the image of \\((\\varpi)\\) in \\(\\Ocal/\\varpi^m\\). Conversely, every tuple in that image is nilpotent and defines a coefficient-preserving evaluation homomorphism. This homomorphism is local. To prove uniqueness without assuming continuity of an arbitrary competing homomorphism, observe in one variable that a homomorphism killing \\(X^N\\) only sees the finite polynomial truncation modulo \\(X^N\\). Induction over the finite number of variables gives uniqueness in all dimensions. The actual local maps are therefore in bijection with the full parameter tuple set. Its cardinality is \\(q^{d(m-1)}\\), including the zero-dimensional chart.","ocal. To prove uniqueness without assuming continuity of an arbitrary competing homomorphism, observe in one variable that a homomorphism killing \\(X^N\\) only sees the finite polynomial truncation modulo \\(X^N\\). Induction over the finite number of variables gives uniqueness in all dimensions. The actual local maps are therefore in bijection with the full parameter tuple set. Its cardinality is \\(q^{d(m-1)}\\), including the zero-dimensional chart.","For integral maps, reduce an arbitrary tuple in \\((\\varpi)\\Ocal\\) at every precision, evaluate there, and use the represented-map reconstruction theorem. The finite evaluations are compatible, the reconstructed algebra map is local, and every competing local map has the same finite reductions. Separatedness gives uniqueness. Local maps are continuous in the maximal-ideal adic topologies. This establishes the integral parameter classification rather than assuming it from the finite count.","For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream \\((a_{i,j})_{j\\ge1}\\) such that\n\\[\n\\varpi^{n+1}\\mid t_i-\\sum_{j=1}^{n}s(a_{i,j})\\varpi^j\\qquad(n\\ge0).\n\\]\nThe section is a set-theoretic section of reduction, not an asserted embedding of residue fields into the coefficient ring. Finally, a supplied local \\(\\Ocal\\)-algebra isomorphism from \\(R_{\\bar x}\\) transports both the map classification and the count.\n\\end{proof}","patible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream \\((a_{i,j})_{j\\ge1}\\) such that\n\\[\n\\varpi^{n+1}\\mid t_i-\\sum_{j=1}^{n}s(a_{i,j})\\varpi^j\\qquad(n\\ge0).\n\\]\nThe section is a set-theoretic section of reduction, not an asserted embedding of residue fields into the coefficient ring. Finally, a supplied local \\(\\Ocal\\)-algebra isomorphism from \\(R_{\\bar x}\\) transports both the map classification and the count.\n\\end{proof}","ct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream \\((a_{i,j})_{j\\ge1}\\) such that\n\\[\n\\varpi^{n+1}\\mid t_i-\\sum_{j=1}^{n}s(a_{i,j})\\varpi^j\\qquad(n\\ge0).\n\\]\nThe section is a set-theoretic section of reduction, not an asserted embedding of residue fields into the coefficient ring. Finally, a supplied local \\(\\Ocal\\)-algebra isomorphism from \\(R_{\\bar x}\\) transports both the map classification and the count.\n\\end{proof}","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.94\\linewidth]{02_residual_digit_forest.pdf}\n\\caption{The first four levels of a full \\(3\\)-adic digit forest.  A path is a compatible finite-depth perturbation stream.}\n\\end{figure}","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.82\\linewidth]{04_inverse_limit_digits.pdf}\n\\caption{A compatible base-five residue tower reconstructs the digit stream \\((2,4,1,3,0,2)\\).}\n\\end{figure}","For a residual representation \\(\\bar\\rho\\), first-order deformations and lifting obstructions are classically controlled by cohomology of the adjoint module.  The projector calculus does not replace that theory.  It resolves the controlling complex into invariant packets when suitable projectors exist.","\\begin{classicalinput}[Tangent and obstruction spaces]\\label{px:4.1}\nFor a framed representation problem, first-order lifts are described by \\(Z^1(G,\\ad\\bar\\rho)\\).  For an unframed Schur problem, tangent directions are described by \\(H^1(G,\\ad\\bar\\rho)\\).  Relations in a presentation of a deformation ring map into \\(H^2(G,\\ad\\bar\\rho)\\), which contains the obstruction classes.\n\\end{classicalinput}","Let\n\\[\nC^0\\xrightarrow{d_0}C^1\\xrightarrow{d_1}C^2\\xrightarrow{}\\cdots\n\\]\nbe a cochain complex over a field.  Suppose a finite family of projectors \\(P_j^r:C^r\\to C^r\\) satisfies\n\\[\n\\sum_jP_j^r=I,\n\\qquad\nP_i^rP_j^r=\\delta_{ij}P_j^r,\n\\qquad\nd_rP_j^r=P_j^{r+1}d_r.\n\\]","\\begin{theorem}[Cohomology packet decomposition]\\label{px:4.2}\nUnder these hypotheses, each packet is a subcomplex and\n\\[\n\\boxed{\nH^r(C^\\bullet)\n\\cong\n\\bigoplus_jH^r(P_jC^\\bullet).\n}\n\\]\nIn particular,\n\\[\nH^1=\\bigoplus_jH^1_j,\n\\qquad\nH^2=\\bigoplus_jH^2_j,\n\\]\nso tangent and obstruction spaces reconstruct exactly from their packet components.\n\\end{theorem}","\\begin{proof}\nFor each packet, use the actual image submodules\n\\(C_j^r=\\operatorname{im}P_j^r\\). Commutation with the differential\nmakes these a subcomplex. Write\n\\[\nZ^r=\\ker d_r,\\qquad B^r=\\operatorname{im}d_{r-1}\\subseteq Z^r,\n\\qquad H^r=Z^r/B^r,\n\\]\nand define \\(Z_j^r,B_j^r,H_j^r\\) using the restricted differentials of\n\\(C_j^\\bullet\\). The inclusion \\(i_j:C_j^\\bullet\\to C^\\bullet\\)\nand the image-valued projection \\(p_j:C^\\bullet\\to C_j^\\bullet\\)\nare cochain maps, so they induce maps on these genuine quotients.","h packet, use the actual image submodules\n\\(C_j^r=\\operatorname{im}P_j^r\\). Commutation with the differential\nmakes these a subcomplex. Write\n\\[\nZ^r=\\ker d_r,\\qquad B^r=\\operatorname{im}d_{r-1}\\subseteq Z^r,\n\\qquad H^r=Z^r/B^r,\n\\]\nand define \\(Z_j^r,B_j^r,H_j^r\\) using the restricted differentials of\n\\(C_j^\\bullet\\). The inclusion \\(i_j:C_j^\\bullet\\to C^\\bullet\\)\nand the image-valued projection \\(p_j:C^\\bullet\\to C_j^\\bullet\\)\nare cochain maps, so they induce maps on these genuine quotients.","For \\(i\\ne j\\), orthogonality gives\n\\(H(p_i)H(i_j)=0\\). Idempotence gives\n\\(H(p_j)H(i_j)=I\\), and completeness gives\n\\(\\sum_j H(i_j)H(p_j)=I\\). Thus the maps\n\\[\n[z]\\longmapsto([P_j^r z])_j,\n\\qquad\n([z_j])_j\\longmapsto\\left[\\sum_j i_j z_j\\right]\n\\]\nare inverse linear maps. The finite product equals the direct sum, proving\nthe assertion. In the formalization, both sides are also identified with\nMathlib's standard homology objects of the genuine graded complexes.\n\\end{proof}","\\paragraph{Scope of the splitting.}\nThe algebraic proof works over any commutative ring and does not divide by\n\\(2\\), so it applies in characteristic two when the stated integral\nchain projectors exist. It does not construct those projectors. The\nidentification of a particular Galois tangent or obstruction problem with\nthis cohomology remains the preceding classical input.\nThe splitting detects vanishing:\n\\[\nH^r(C^\\bullet)=0\\quad\\Longleftrightarrow\\quad\nH^r(C_j^\\bullet)=0\\text{ for every }j.\n\\]\nIt does not force either side to vanish or decouple nonlinear deformation\nequations. Moreover, commutation with the differential is essential:\ncomplementary idempotents on the underlying vector spaces need not\npreserve cocycles or coboundaries.","Scope of the splitting.}\nThe algebraic proof works over any commutative ring and does not divide by\n\\(2\\), so it applies in characteristic two when the stated integral\nchain projectors exist. It does not construct those projectors. The\nidentification of a particular Galois tangent or obstruction problem with\nthis cohomology remains the preceding classical input.\nThe splitting detects vanishing:\n\\[\nH^r(C^\\bullet)=0\\quad\\Longleftrightarrow\\quad\nH^r(C_j^\\bullet)=0\\text{ for every }j.\n\\]\nIt does not force either side to vanish or decouple nonlinear deformation\nequations. Moreover, commutation with the differential is essential:\ncomplementary idempotents on the underlying vector spaces need not\npreserve cocycles or coboundaries.","\\paragraph{Naturality.}\nIf a cochain map \\(F:C^\\bullet\\to D^\\bullet\\) intertwines the supplied\nprojectors, it restricts to maps \\(F_j:C_j^\\bullet\\to D_j^\\bullet\\).\nThe direct-sum decomposition commutes with \\(H^r(F)\\) and the family\n\\(H^r(F_j)\\). This is checked using representatives in the actual\nquotients, independently of any dimension computation.","\\begin{remark}[Odd and dyadic nonlinear equations]\\label{px:4.3}\nAt odd residue characteristic, a differential graded Lie model may be written in Maurer--Cartan form\n\\[\nd\\eta+\\frac12[\\eta,\\eta]=0,\n\\]\nand a cyclic grading converts this into exact packet convolutions.  At residue characteristic two, division by \\(2\\) is not integral.  A quadratic or divided-square refinement must be retained.  This is one reason the dyadic residual forest is weighted rather than obtained by formal substitution into the odd-prime formulas.\n\\end{remark}","\\begin{numericalcertificate}[Finite packet complex]\\label{px:4.4}\nThe exact modular test uses two invariant subcomplexes over \\(\\F_5\\).  Their cohomology dimensions are\n\\[\n(0,0,0),\n\\qquad\n(1,2,1),\n\\]\nand the reconstructed total is\n\\[\n\\dim(H^0,H^1,H^2)=(1,2,1).\n\\]\nThe packet-sum residual is zero.\n\\end{numericalcertificate}","\\paragraph{An explicit realization of the finite pattern.}\nA separate finite-complex implementation uses\n\\[\n\\mathbf F_5^2\\xrightarrow{d_0}\\mathbf F_5^4\\xrightarrow{d_1}\\mathbf F_5^2,\n\\quad d_0(a,b)=(a,0,0,0),\\quad d_1(a,b,c,d)=(b,0).\n\\]\nThe first packet is the exact complex with dimensions \\((1,2,1)\\),\nand the second packet has zero differentials and dimensions \\((1,2,1)\\).\nThe actual middle cohomology is linearly isomorphic to \\(\\mathbf F_5^2\\),\nwhich is checked in Lean. Independent exhaustive coset tests check all\nthree cohomology degrees after changes of basis. This realizes the displayed dimension pattern. Its formal check is\ndistinct from the original companion implementation.","of the finite pattern.}\nA separate finite-complex implementation uses\n\\[\n\\mathbf F_5^2\\xrightarrow{d_0}\\mathbf F_5^4\\xrightarrow{d_1}\\mathbf F_5^2,\n\\quad d_0(a,b)=(a,0,0,0),\\quad d_1(a,b,c,d)=(b,0).\n\\]\nThe first packet is the exact complex with dimensions \\((1,2,1)\\),\nand the second packet has zero differentials and dimensions \\((1,2,1)\\).\nThe actual middle cohomology is linearly isomorphic to \\(\\mathbf F_5^2\\),\nwhich is checked in Lean. Independent exhaustive coset tests check all\nthree cohomology degrees after changes of basis. This realizes the displayed dimension pattern. Its formal check is\ndistinct from the original companion implementation.","At an odd prime, distinct sign characters remain distinct in the residue field.  At the coefficient prime two,\n\\[\n1\\equiv-1\\pmod2.\n\\]\nThus distinct characteristic-zero packets can share one residual root and separate only after more precision is retained.","\\begin{definition}[Visibility weight]\\label{px:5.1}\nFor two integral packet coordinates \\(a,b\\in\\Ocal\\), define their separation depth\n\\[\nw(a,b)=\\min\\{m\\ge1:a\\not\\equiv b\\pmod{\\varpi^m}\\},\n\\]\nwhen such an \\(m\\) exists.  A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.\n\\end{definition}","\\begin{example}[The two sign branches]\\label{px:5.2}\nThe characters \\(\\chi_+(g)=1\\) and \\(\\chi_-(g)=-1\\) have the same reduction modulo \\(2\\), but\n\\[\n1\\not\\equiv-1\\pmod4.\n\\]\nTheir first separation depth is therefore \\(2\\).  One mod-two root contains two branches that become distinct at the next level.\n\\end{example}","A simple weighted formal model already captures delayed visibility.  Let \\(w_i\\ge1\\) and constrain\n\\[\nt_i\\in(\\varpi^{w_i}\\Ocal+\\varpi^m\\Ocal)/\\varpi^m\\Ocal.\n\\]\nThen coordinate \\(i\\) is invisible until \\(m>w_i\\).","\\begin{proposition}[Weighted smooth-chart count]\\label{px:5.3}\nFor weights \\(\\bm w=(w_1,\\ldots,w_d)\\),\n\\[\n\\boxed{\nN_{\\bm w}(m)\n=\nq^{\\sum_{i=1}^d\\max(m-w_i,0)}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe image of \\(\\varpi^{w_i}\\Ocal\\) in \\(\\Ocal/\\varpi^m\\) has one element when \\(m\\le w_i\\), and \\(q^{m-w_i}\\) elements when \\(m>w_i\\).  Multiplying the independent coordinate counts gives the formula.\n\\end{proof}","\\begin{definition}[Seed zeta function]\\label{px:6.1}\nFor a residual root \\(\\bar x\\) with finite levels, define\n\\[\n\\boxed{\nZ_{\\bar x}(T)\n=\n\\sum_{m\\ge1}\\#X_m(\\bar x)T^m.\n}\n\\]\nWhen a weighted chart is specified, write \\(Z_{\\bm w}(T)\\).\n\\end{definition}","\\begin{theorem}[Smooth seed zeta function]\\label{px:6.2}\nFor a smooth formal component of dimension \\(d\\),\n\\[\n\\boxed{\nZ_d(T)=\\frac{T}{1-q^dT}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nBy the proved formal-model map-to-parameter equivalence, \\(\\#X_m=q^{d(m-1)}\\) counts all the actual local coefficient-algebra maps. Summing the geometric series gives the identity. The checked statement is an equality of infinite formal power series, with zero constant coefficient.\n\\end{proof}","\\begin{theorem}[Weighted seed zeta function]\\label{px:6.3}\nLet \\(1\\le w_1\\le\\cdots\\le w_d\\), put \\(W=\\sum_iw_i\\), and let\n\\[\na_m=q^{\\sum_i\\max(m-w_i,0)}.\n\\]\nThen\n\\[\n\\boxed{\nZ_{\\bm w}(T)\n=\n\\sum_{m=1}^{w_d}a_mT^m\n+\nq^{-W}\\frac{(q^dT)^{w_d+1}}{1-q^dT}.\n}\n\\]\nIn particular, every weighted smooth chart has a rational seed zeta function.\n\\end{theorem}","\\begin{proof}\nThe finite sum records the depths before all coordinates are active.  For \\(m\\ge w_d+1\\),\n\\[\na_m=q^{dm-W},\n\\]\nso the remaining terms form one geometric series.\n\\end{proof}","\\begin{theorem}[Singular power seed]\\label{px:6.4}\nFor integers \\(e\\ge1\\) and \\(m\\ge1\\), in \\(\\Ocal/\\varpi^m\\) the equation\n\\[\nx^e=0\n\\]\nhas exactly\n\\[\n\\boxed{\nq^{m-\\lceil m/e\\rceil}\n}\n\\]\nsolutions.  Its seed zeta function is\n\\[\n\\boxed{\nZ_e(T)\n=\n\\frac{\\sum_{r=1}^{e}q^{r-1}T^r}\n{1-q^{e-1}T^e}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe congruence is equivalent to\n\\[\ne\\,v_\\varpi(x)\\ge m,\n\\]\nso \\(x\\) must lie in \\(\\varpi^{\\lceil m/e\\rceil}\\Ocal/\\varpi^m\\), which has the stated cardinality.  Write \\(m=ek+r\\), \\(1\\le r\\le e\\).  The exponent becomes \\((e-1)k+r-1\\).  Summing over \\(k\\ge0\\) and \\(1\\le r\\le e\\) gives the rational expression.\n\\end{proof}","\\begin{definition}[Branching dimension]\\label{px:6.5}\nWhen the limit exists, define the finite-level counting exponent\n\\[\n\\dim_{\\mathrm{seed}}(\\bar x)\n=\n\\lim_{m\\to\\infty}\n\\frac{\\log_q\\#X_m(\\bar x)}{m}.\n\\]\nA limsup may be used in general.\n\\end{definition}","\\begin{corollary}[Fractional singular dimension]\\label{px:6.6}\nFor \\(x^e=0\\),\n\\[\n\\boxed{\n\\dim_{\\mathrm{seed}}=1-\\frac1e.\n}\n\\]\nIn particular, \\(x^2=0\\) has branching dimension \\(1/2\\).","\\paragraph{Counting exponent versus compatible branches.}\nThe preceding exponent counts all finite-level solutions, including nodes which cannot lift indefinitely. It is not the cardinality or dimension of the space of infinite branches. In fact every compatible tower of solutions to \\(x^e=0\\), for \\(e\\ge1\\), is the zero tower. For each \\(m\\), a solution at depth \\(em\\) lies in \\(\\varpi^m\\Ocal/\\varpi^{em}\\). Its reduction at depth \\(m\\) is therefore zero. Compatibility proves the assertion at every depth. The formal companion identifies proofs of both this observation and the real logarithmic limit.","ompatible branches.}\nThe preceding exponent counts all finite-level solutions, including nodes which cannot lift indefinitely. It is not the cardinality or dimension of the space of infinite branches. In fact every compatible tower of solutions to \\(x^e=0\\), for \\(e\\ge1\\), is the zero tower. For each \\(m\\), a solution at depth \\(em\\) lies in \\(\\varpi^m\\Ocal/\\varpi^{em}\\). Its reduction at depth \\(m\\) is therefore zero. Compatibility proves the assertion at every depth. The formal companion identifies proofs of both this observation and the real logarithmic limit.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.83\\linewidth]{06_smooth_singular_counts.pdf}\n\\caption{A smooth two-dimensional formal chart and the singular equation \\(x^2=0\\) have visibly different exponential growth.}\n\\end{figure}","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.78\\linewidth]{08_fractional_seed_dimension.pdf}\n\\caption{The finite-depth branching dimension of \\(x^e=0\\) is \\(1-1/e\\), giving a family of fractional local growth exponents.}\n\\end{figure}","\\begin{definition}[Admissible residual transport]\\label{px:7.1}\nAn admissible transport from \\(\\bar\\rho\\) to \\(\\bar\\rho'\\) consists of\n\\[\n\\mathcal T:\\bar\\rho\\longmapsto\\bar\\rho'\n\\]\ntogether with natural maps\n\\[\n\\mathcal T_A:\\Def_{\\bar\\rho}(A)\\longrightarrow\\Def_{\\bar\\rho'}(A)\n\\]\nfor every coefficient algebra \\(A\\), compatible with base change and reduction.  When the transport is reversible, it is an isomorphism in the residual transport groupoid.  One-way operations give a directed category.\n\\end{definition}","For a finite collection of roots, record a directed edge \\(u\\to v\\) whenever occupancy or the relevant property is known to propagate from \\(u\\) to \\(v\\).","\\begin{definition}[Directed seed rank]\\label{px:7.2}\nFor a directed transport graph \\(G\\), define\n\\[\n\\srank_G\n=\n\\min\\{|S|:\\Cl_G(S)=V(G)\\},\n\\]\nwhere \\(\\Cl_G(S)\\) is the set of vertices reachable from \\(S\\).\n\\end{definition}","\\begin{theorem}[Directed minimum-seed theorem]\\label{px:7.3}\nLet \\(G\\) be a finite directed graph.  Collapse every strongly connected component to one vertex, obtaining the condensation DAG.  Then\n\\[\n\\boxed{\n\\srank_G\n=\n\\#\\{\\text{source strongly connected components of }G\\}.\n}\n\\]\nOne representative from each source component is a minimum seed set.\n\\end{theorem}","If a property $P$ propagates along every supplied edge and holds at all selected\nseeds, it holds at every reached vertex. These are explicit hypotheses. The\nminimum-seed theorem does not construct valid arithmetic edges or supply\ninitial occupancy.","\\paragraph{Why the finite hypothesis cannot be dropped.}\nTake vertices $\\{*\\}\\sqcup\\{v_n:n\\in\\mathbf N\\}$, with $*$ isolated and\nedges $v_{n+1}\\to v_n$. Every strongly connected component is a singleton,\nand $\\{*\\}$ is the unique source component. No finite seed set covers all\nvertices: it can reach only indices no larger than the largest chosen index.\nBoth the unique-source statement and the failure of every finite cover are\nchecked in Lean. Hence one source component in an infinite condensation does\nnot imply that one seed reaches all objects.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.91\\linewidth]{09_directed_transport_graph.pdf}\n\\caption{A finite directed transport model.  The components \\(\\{A,B\\}\\), \\(\\{D,E\\}\\), and \\(\\{G\\}\\) are sources. \\(C\\) and \\(F\\) are reached downstream.}\n\\label{fig:transport-graph}\n\\end{figure}","\\paragraph{An explicit realization of the directed graph.}\nAn independently specified graph has vertices $A,B,C,D,E,F,G$ and the eight edges\n\\[\n A\\to B,\\ B\\to A,\\ B\\to C,\\ C\\to F,\\ D\\to E,\\ E\\to D,\\ E\\to F,\\ G\\to F.\n\\]\nThe source components are $\\{A,B\\}$, $\\{D,E\\}$, and $\\{G\\}$. The other\ncomponents are $\\{C\\}$ and $\\{F\\}$. A bounded path search to length three\nis proved closed under every original edge, which certifies equality with\nunbounded reachability. The resulting label certificate is identified with\nthe actual quotient by mutual reachability. Lean checks seed rank three and\nexactly the four minimum sets\n\\[\n \\{A,D,G\\},\\quad\\{A,E,G\\},\\quad\\{B,D,G\\},\\quad\\{B,E,G\\}.\n\\]\nIndependent exact tests enumerate every directed graph on zero through four\nvertices, including loops, and every seed subset. These finit","\\paragraph{An explicit realization of the directed graph.}\nAn independently specified graph has vertices $A,B,C,D,E,F,G$ and the eight edges\n\\[\n A\\to B,\\ B\\to A,\\ B\\to C,\\ C\\to F,\\ D\\to E,\\ E\\to D,\\ E\\to F,\\ G\\to F.\n\\]\nThe source components are $\\{A,B\\}$, $\\{D,E\\}$, and $\\{G\\}$. The other\ncomponents are $\\{C\\}$ and $\\{F\\}$. A bounded path search to length three\nis proved closed under every original edge, which certifies equality with\nunbounded reachability. The resulting label certificate is identified with\nthe actual quotient by mutual reachability. Lean checks seed rank three and\nexactly the four minimum sets\n\\[\n \\{A,D,G\\},\\quad\\{A,E,G\\},\\quad\\{B,D,G\\},\\quad\\{B,E,G\\}.\n\\]\nIndependent exact tests enumerate every directed graph on zero through four\nvertices, including loops, and every seed subset. These finite tests audit the\nimplementation. The universal theorem is a separate formal proof.","e source components are $\\{A,B\\}$, $\\{D,E\\}$, and $\\{G\\}$. The other\ncomponents are $\\{C\\}$ and $\\{F\\}$. A bounded path search to length three\nis proved closed under every original edge, which certifies equality with\nunbounded reachability. The resulting label certificate is identified with\nthe actual quotient by mutual reachability. Lean checks seed rank three and\nexactly the four minimum sets\n\\[\n \\{A,D,G\\},\\quad\\{A,E,G\\},\\quad\\{B,D,G\\},\\quad\\{B,E,G\\}.\n\\]\nIndependent exact tests enumerate every directed graph on zero through four\nvertices, including loops, and every seed subset. These finite tests audit the\nimplementation. The universal theorem is a separate formal proof.","A compatible system should not be treated as an unrelated collection of \\(\\lambda\\)-adic roots.  Its members share algebraic Frobenius polynomials, Hodge--Tate data, determinant information, and local Weil--Deligne packets.","\\begin{definition}[Common arithmetic spine]\\label{px:8.1}\nLet \\(M\\) be a number field.  A common spine \\(\\Compat\\) consists of:\n\\begin{itemize}\n\\item characteristic polynomials \\(P_v(X)\\in M[X]\\) at almost all places;\n\\item coefficient-prime independent weight and determinant data;\n\\item prescribed local Weil--Deligne or inertial packet labels at the ramified places.\n\\end{itemize}\nFor each finite place \\(\\lambda\\) of \\(M\\), let\n\\[\nX_{\\lambda,m}(\\Compat)\n\\]\nbe the finite set of admissible objects modulo \\(\\varpi_\\lambda^m\\) realizing the reduction of this spine.\n\\end{definition}","efinition}[Common arithmetic spine]\\label{px:8.1}\nLet \\(M\\) be a number field.  A common spine \\(\\Compat\\) consists of:\n\\begin{itemize}\n\\item characteristic polynomials \\(P_v(X)\\in M[X]\\) at almost all places;\n\\item coefficient-prime independent weight and determinant data;\n\\item prescribed local Weil--Deligne or inertial packet labels at the ramified places.\n\\end{itemize}\nFor each finite place \\(\\lambda\\) of \\(M\\), let\n\\[\nX_{\\lambda,m}(\\Compat)\n\\]\nbe the finite set of admissible objects modulo \\(\\varpi_\\lambda^m\\) realizing the reduction of this spine.\n\\end{definition}","on arithmetic spine]\\label{px:8.1}\nLet \\(M\\) be a number field.  A common spine \\(\\Compat\\) consists of:\n\\begin{itemize}\n\\item characteristic polynomials \\(P_v(X)\\in M[X]\\) at almost all places;\n\\item coefficient-prime independent weight and determinant data;\n\\item prescribed local Weil--Deligne or inertial packet labels at the ramified places.\n\\end{itemize}\nFor each finite place \\(\\lambda\\) of \\(M\\), let\n\\[\nX_{\\lambda,m}(\\Compat)\n\\]\nbe the finite set of admissible objects modulo \\(\\varpi_\\lambda^m\\) realizing the reduction of this spine.\n\\end{definition}","ic spine]\\label{px:8.1}\nLet \\(M\\) be a number field.  A common spine \\(\\Compat\\) consists of:\n\\begin{itemize}\n\\item characteristic polynomials \\(P_v(X)\\in M[X]\\) at almost all places;\n\\item coefficient-prime independent weight and determinant data;\n\\item prescribed local Weil--Deligne or inertial packet labels at the ramified places.\n\\end{itemize}\nFor each finite place \\(\\lambda\\) of \\(M\\), let\n\\[\nX_{\\lambda,m}(\\Compat)\n\\]\nbe the finite set of admissible objects modulo \\(\\varpi_\\lambda^m\\) realizing the reduction of this spine.\n\\end{definition}","at\\) consists of:\n\\begin{itemize}\n\\item characteristic polynomials \\(P_v(X)\\in M[X]\\) at almost all places;\n\\item coefficient-prime independent weight and determinant data;\n\\item prescribed local Weil--Deligne or inertial packet labels at the ramified places.\n\\end{itemize}\nFor each finite place \\(\\lambda\\) of \\(M\\), let\n\\[\nX_{\\lambda,m}(\\Compat)\n\\]\nbe the finite set of admissible objects modulo \\(\\varpi_\\lambda^m\\) realizing the reduction of this spine.\n\\end{definition}","The system seed is the fibered object\n\\[\n\\boxed{\n\\Sys_{\\mathrm{sys}}\n=\n\\Sys_2^{\\mathrm{wt}}\n\\times_{\\Compat}\n\\prod_{\\lambda\\nmid2}'\\Sys_\\lambda.\n}\n\\]\nThe fiber product notation states a compatibility requirement.  It does not prove that every fiber is nonempty.","\\begin{theorem}[Adelic finite-depth completion]\\label{px:8.2}\nAssume:\n\\begin{enumerate}\n\\item an algebraic common spine \\(\\Compat\\) over a number field \\(M\\) is fixed;\n\\item for every finite place \\(\\lambda\\) of \\(M\\) and every \\(m\\ge1\\), the companion level \\(X_{\\lambda,m}(\\Compat)\\) is finite and nonempty;\n\\item the reduction forests are finitely branching;\n\\item the selected local conditions are closed under inverse limits.\n\\end{enumerate}\nThen for every \\(\\lambda\\) there is an infinite compatible branch\n\\[\n\\rho_\\lambda\\in\\varprojlim_mX_{\\lambda,m}(\\Compat),\n\\]\nand the collection \\(\\{\\rho_\\lambda\\}_\\lambda\\) realizes the prescribed strictly compatible spine.\n\\end{theorem}","sume:\n\\begin{enumerate}\n\\item an algebraic common spine \\(\\Compat\\) over a number field \\(M\\) is fixed;\n\\item for every finite place \\(\\lambda\\) of \\(M\\) and every \\(m\\ge1\\), the companion level \\(X_{\\lambda,m}(\\Compat)\\) is finite and nonempty;\n\\item the reduction forests are finitely branching;\n\\item the selected local conditions are closed under inverse limits.\n\\end{enumerate}\nThen for every \\(\\lambda\\) there is an infinite compatible branch\n\\[\n\\rho_\\lambda\\in\\varprojlim_mX_{\\lambda,m}(\\Compat),\n\\]\nand the collection \\(\\{\\rho_\\lambda\\}_\\lambda\\) realizes the prescribed strictly compatible spine.\n\\end{theorem}","pine \\(\\Compat\\) over a number field \\(M\\) is fixed;\n\\item for every finite place \\(\\lambda\\) of \\(M\\) and every \\(m\\ge1\\), the companion level \\(X_{\\lambda,m}(\\Compat)\\) is finite and nonempty;\n\\item the reduction forests are finitely branching;\n\\item the selected local conditions are closed under inverse limits.\n\\end{enumerate}\nThen for every \\(\\lambda\\) there is an infinite compatible branch\n\\[\n\\rho_\\lambda\\in\\varprojlim_mX_{\\lambda,m}(\\Compat),\n\\]\nand the collection \\(\\{\\rho_\\lambda\\}_\\lambda\\) realizes the prescribed strictly compatible spine.\n\\end{theorem}","lace \\(\\lambda\\) of \\(M\\) and every \\(m\\ge1\\), the companion level \\(X_{\\lambda,m}(\\Compat)\\) is finite and nonempty;\n\\item the reduction forests are finitely branching;\n\\item the selected local conditions are closed under inverse limits.\n\\end{enumerate}\nThen for every \\(\\lambda\\) there is an infinite compatible branch\n\\[\n\\rho_\\lambda\\in\\varprojlim_mX_{\\lambda,m}(\\Compat),\n\\]\nand the collection \\(\\{\\rho_\\lambda\\}_\\lambda\\) realizes the prescribed strictly compatible spine.\n\\end{theorem}","\\begin{proof}\nFinite-branch compactness gives an infinite branch in every coefficient-prime forest.  Inverse-limit closure gives an admissible \\(\\lambda\\)-adic object.  By definition each finite level has the prescribed reductions of the common Frobenius and local packets.  Equality at every finite precision implies equality in the complete coefficient ring.  Thus all members realize the same spine.\n\\end{proof}","\\begin{corollary}[Residual companion obstruction]\\label{px:8.3}\nIf\n\\[\nX_{\\lambda,1}(\\Compat)=\\varnothing\n\\]\nfor one coefficient prime \\(\\lambda\\), no compatible completion with spine \\(\\Compat\\) exists.  Deeper perturbations at other primes cannot create the missing residual companion.\n\\end{corollary}","\\begin{example}[A scalar companion failure]\\label{px:8.4}\nFor\n\\[\nf(x)=x^2+1,\n\\]\nthere are two simple branches at \\(p=5\\) and \\(p=13\\), no residual roots at \\(p=3\\) or \\(p=7\\), and one root modulo \\(2\\) that has no child modulo \\(4\\).  A complete \\(5\\)-adic branch therefore does not imply an adelic solution.\n\\end{example}","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.79\\linewidth]{12_companion_forest_occupancy.pdf}\n\\caption{Finite companion levels for \\(x^2+1=0\\).  The obstruction at \\(2,3,7\\) is visible before any deep lifting argument begins.}\n\\end{figure}","\\paragraph{Checked representation-level completion and its arithmetic boundary.}\nThe representation-level completion construction works with actual matrix\nrepresentations of a fixed group. For a complete separated ring $R$\nand ideal $I$, compatible maps\n$\\rho_m:G\\to\\operatorname{GL}_d(R/I^m)$ reconstruct a unique integral\nrepresentation for that fixed compatible family. Multiplication and inverses\nare verified after every reduction. If all finite representations are continuous\nfor the discrete quotient topology, the integral representation is continuous\nfor the ideal-adic topology. This does not assert uniqueness among all possible\nbranches of the allowed forest.","All coefficients of the characteristic polynomial commute with reduction.\nConsequently, prescribed integral characteristic polynomials at every precision\nhold exactly in the reconstructed representation. For a common coefficient field\n$E$, local integral polynomials lie in $\\Ocal_\\lambda[X]$ and are compared with\n$P_v\\in E[X]$ only after mapping both into the local coefficient field. No field\nembedding $E\\to\\Ocal_\\lambda$ is assumed. The allowed-place predicate is explicit\nand may depend on $\\lambda$. Trivial action on a supplied subgroup is likewise\ndetected at every precision. Identifying that subgroup with arithmetic inertia\nis a separate input.","fficients of the characteristic polynomial commute with reduction.\nConsequently, prescribed integral characteristic polynomials at every precision\nhold exactly in the reconstructed representation. For a common coefficient field\n$E$, local integral polynomials lie in $\\Ocal_\\lambda[X]$ and are compared with\n$P_v\\in E[X]$ only after mapping both into the local coefficient field. No field\nembedding $E\\to\\Ocal_\\lambda$ is assumed. The allowed-place predicate is explicit\nand may depend on $\\lambda$. Trivial action on a supplied subgroup is likewise\ndetected at every precision. Identifying that subgroup with arithmetic inertia\nis a separate input.","d integral characteristic polynomials at every precision\nhold exactly in the reconstructed representation. For a common coefficient field\n$E$, local integral polynomials lie in $\\Ocal_\\lambda[X]$ and are compared with\n$P_v\\in E[X]$ only after mapping both into the local coefficient field. No field\nembedding $E\\to\\Ocal_\\lambda$ is assumed. The allowed-place predicate is explicit\nand may depend on $\\lambda$. Trivial action on a supplied subgroup is likewise\ndetected at every precision. Identifying that subgroup with arithmetic inertia\nis a separate input.","\\paragraph{Completion from conjugacy classes.}\\phantomsection\\label{book:x-construction-14}\nOver an $I$-adically complete separated commutative coefficient ring $R$,\nfor a fixed finite rank, the formalization now takes finite-level objects to be\nactual matrix representations modulo conjugation by an invertible matrix.\nAssume the coefficient quotients $R/I^k$ are finite.\nConjugation commutes with coefficient reduction, and each conjugacy class has\nfinitely many frames over a finite coefficient ring. Thus finite allowed class\nlevels give finite full preimages of allowed matrix representations. Downward\nadmissibility and positive-depth nonemptiness yield an exactly compatible\nmatrix branch and an actual integral representation. More strongly, every\nprescribed compatible class branch has such re","\\paragraph{Completion from conjugacy classes.}\\phantomsection\\label{book:x-construction-14}\nOver an $I$-adically complete separated commutative coefficient ring $R$,\nfor a fixed finite rank, the formalization now takes finite-level objects to be\nactual matrix representations modulo conjugation by an invertible matrix.\nAssume the coefficient quotients $R/I^k$ are finite.\nConjugation commutes with coefficient reduction, and each conjugacy class has\nfinitely many frames over a finite coefficient ring. Thus finite allowed class\nlevels give finite full preimages of allowed matrix representations. Downward\nadmissibility and positive-depth nonemptiness yield an exactly compatible\nmatrix branch and an actual integral representation. More strongly, every\nprescribed compatible class branch has such representatives, using finite\nnonempty fibers over that branch.","\\paragraph{Completion from conjugacy classes.}\\phantomsection\\label{book:x-construction-14}\nOver an $I$-adically complete separated commutative coefficient ring $R$,\nfor a fixed finite rank, the formalization now takes finite-level objects to be\nactual matrix representations modulo conjugation by an invertible matrix.\nAssume the coefficient quotients $R/I^k$ are finite.\nConjugation commutes with coefficient reduction, and each conjugacy class has\nfinitely many frames over a finite coefficient ring. Thus finite allowed class\nlevels give finite full preimages of allowed matrix representations. Downward\nadmissibility and positive-depth nonemptiness yield an exactly compatible\nmatrix branch and an actual integral representation. More strongly, every\nprescribed compatible class branch has such representatives, using finite\nnonempty fibers over that branch. No surjectivity of reductions or lifting of\nprescribed change-of-basis matrices is assumed. This does not assert uniqueness\nup to integral conjugacy.","For finitely generated $I$ with finite $R/I$, finiteness of $R/I^k$ is derived.\nOne continuous representative per allowed class suffices, because continuity\nis proved invariant under conjugation. Characteristic polynomials, determinants\nand trivial action on a supplied subgroup are likewise actual invariants.\nThe unframed common-spine theorem constructs continuous coefficient-field\nrepresentations with the prescribed whole polynomials and the explicitly\nsupplied inverse-limit-closed local predicate.","An actual $\\mathbb Z_3$ example applies the class-level completion theorem\nfor the cyclic group of order two and polynomial $X^2-1$, producing a nonidentity generator. Another\nchecked example has compatible classes represented by\n$\\operatorname{diag}(1,-1)$ modulo $3$ and\n$\\operatorname{diag}(-1,1)$ modulo $9$: these chosen matrices disagree under\nreduction, and an actual swap of basis repairs them. Arbitrary independent\nrepresentatives therefore do not replace the framing argument.","\\paragraph{Actual local comparisons from fixed-depth certificates.}\\phantomsection\\label{book:x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and","\\paragraph{Actual local comparisons from fixed-depth certificates.}\\phantomsection\\label{book:x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonz","\\paragraph{Actual local comparisons from fixed-depth certificates.}\\phantomsection\\label{book:x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral reali","\\paragraph{Actual local comparisons from fixed-depth certificates.}\\phantomsection\\label{book:x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior cho","\\paragraph{Actual local comparisons from fixed-depth certificates.}\\phantomsection\\label{book:x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior choices of comparisons are not assumed.","local comparisons from fixed-depth certificates.}\\phantomsection\\label{book:x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior choices of comparisons are not assumed.","om fixed-depth certificates.}\\phantomsection\\label{book:x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior choices of comparisons are not assumed.","x-construction-15}\nLet $R$ be an $I$-adically complete separated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior choices of comparisons are not assumed.","eparated commutative ring with finite\ncoefficient quotients and an injective map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior choices of comparisons are not assumed.","e map into a field $K$. Fix a finite matrix\nrank. For finitely many\nlocal group maps $\\phi_s:H_s\\to G$ and integral template representations\n$\\tau_s$, fix detecting depths $d_s$. At precision $m$ retain matrices $P_{s,m}$\nas part of the admissible object and require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior choices of comparisons are not assumed.","d require\n\\[\n\\rho_m(\\phi_s(h))P_{s,m}=P_{s,m}\\tau_{s,m}(h).\n\\]\nFor $m\\ge d_s$ require their determinants to remain nonzero after reduction to\n$R/I^{d_s}$. Earlier levels omit that test. Finite occupied augmented levels\ngive a compatible branch of both representations and comparisons. The integral\nlimits satisfy the actual equations and have nonzero determinants, so the\ncomparisons are units over $K$. A converse supplies all finite nodes from any\nintegral realization with these detectors. Finite-level invertibility and\ncompatible prior choices of comparisons are not assumed.","The formalization specializes to the actual maps\n$G_{\\mathbb Q_p}\\to G_{\\mathbb Q}$ induced by the field embeddings and choices\nof algebraic closures. A nontrivial continuous quadratic character of\n$G_{\\mathbb Q}$, constructed from its action on $i^2=-1$, gives the integral\n$2$-adic actions\n\\[\n\\tau(\\sigma)=\\operatorname{diag}(1,-1),\\quad\n\\rho(\\sigma)=\\begin{pmatrix}1&-1","The formalization specializes to the actual maps\n$G_{\\mathbb Q_p}\\to G_{\\mathbb Q}$ induced by the field embeddings and choices\nof algebraic closures. A nontrivial continuous quadratic character of\n$G_{\\mathbb Q}$, constructed from its action on $i^2=-1$, gives the integral\n$2$-adic actions\n\\[\n\\tau(\\sigma)=\\operatorname{diag}(1,-1),\\quad\n\\rho(\\sigma)=\\begin{pmatrix}1&-1\\\\0&-1\\end{pmatrix},\\quad\nP=\\begin{pmatrix}1&1\\\\0&2\\end{pmatrix}\n\\]\nat a nonidentity quadratic element. The determinant $2$ is detected modulo $4$.\nThe representations are conjugate over $\\mathbb Q_2$ but not integrally\nconjugate. A separate $\\mathbb Z_3$ example invokes simultaneous completion\nand reconstructs a comparison which is an integral nonunit and a field unit.","0&2\\end{pmatrix}\n\\]\nat a nonidentity quadratic element. The determinant $2$ is detected modulo $4$.\nThe representations are conjugate over $\\mathbb Q_2$ but not integrally\nconjugate. A separate $\\mathbb Z_3$ example invokes simultaneous completion\nand reconstructs a comparison which is an integral nonunit and a field unit.","The bound is substantive: for the unipotent actions\n$\\rho_a(g)=\\begin{pmatrix}1&ag\\\\0&1\\end{pmatrix}$, every integral intertwiner\nwith $\\rho_1$ has determinant $aP_{22}^2$. Taking $a=\\pi^n$ for a nonzero\n$\\pi\\in I$ rules out every\nfixed detecting depth, even though all the field-valued approximants are\nisomorphic and their integral limit is trivial. This does not refute Claim 8.2,\nwhich already assumes inverse-limit closure of its chosen local conditions.","0&1\\end{pmatrix}$, every integral intertwiner\nwith $\\rho_1$ has determinant $aP_{22}^2$. Taking $a=\\pi^n$ for a nonzero\n$\\pi\\in I$ rules out every\nfixed detecting depth, even though all the field-valued approximants are\nisomorphic and their integral limit is trivial. This does not refute Claim 8.2,\nwhich already assumes inverse-limit closure of its chosen local conditions.","\\paragraph{Derived comparison bounds.}\\phantomsection\\label{book:x-construction-16}\nFor a finite group $\\Gamma$ whose order is a unit in a commutative local\ncoefficient ring $R$, any prescribed equivariant residual isomorphism lifts to\nan integral isomorphism. Lift its matrix to $A$ and define\n\\[\nP=|\\Gamma|^{-1}\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nReindexing proves equivariance, and reduction recovers the prescribed matrix.\nIts determinant is a unit. Complete local coefficients with finite quotients,\noccupied compatible finite conditions and this residual type therefore give\nactual integral completion with detector depth one. A fixed continuous map\n$G\\to\\Gamma$ permits pullback to infinite source groups. The finite-level\nrepresentations must factor through the same map, and the residual type is","\\paragraph{Derived comparison bounds.}\\phantomsection\\label{book:x-construction-16}\nFor a finite group $\\Gamma$ whose order is a unit in a commutative local\ncoefficient ring $R$, any prescribed equivariant residual isomorphism lifts to\nan integral isomorphism. Lift its matrix to $A$ and define\n\\[\nP=|\\Gamma|^{-1}\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nReindexing proves equivariance, and reduction recovers the prescribed matrix.\nIts determinant is a unit. Complete local coefficients with finite quotients,\noccupied compatible finite conditions and this residual type therefore give\nactual integral completion with detector depth one. A fixed continuous map\n$G\\to\\Gamma$ permits pullback to infinite source groups. The finite-level\nrepresentations must factor through the same map, and the residual type is\nimposed on their $\\Gamma$-actions. No uniform determinant bound o","\\paragraph{Derived comparison bounds.}\\phantomsection\\label{book:x-construction-16}\nFor a finite group $\\Gamma$ whose order is a unit in a commutative local\ncoefficient ring $R$, any prescribed equivariant residual isomorphism lifts to\nan integral isomorphism. Lift its matrix to $A$ and define\n\\[\nP=|\\Gamma|^{-1}\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nReindexing proves equivariance, and reduction recovers the prescribed matrix.\nIts determinant is a unit. Complete local coefficients with finite quotients,\noccupied compatible finite conditions and this residual type therefore give\nactual integral completion with detector depth one. A fixed continuous map\n$G\\to\\Gamma$ permits pullback to infinite source groups. The finite-level\nrepresentations must factor through the same map, and the residual type is\nimposed on their $\\Gamma$-actions. No uniform determinant bound or augmented\ncertificate occupancy is assumed in this sector.","ft its matrix to $A$ and define\n\\[\nP=|\\Gamma|^{-1}\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nReindexing proves equivariance, and reduction recovers the prescribed matrix.\nIts determinant is a unit. Complete local coefficients with finite quotients,\noccupied compatible finite conditions and this residual type therefore give\nactual integral completion with detector depth one. A fixed continuous map\n$G\\to\\Gamma$ permits pullback to infinite source groups. The finite-level\nrepresentations must factor through the same map, and the residual type is\nimposed on their $\\Gamma$-actions. No uniform determinant bound or augmented\ncertificate occupancy is assumed in this sector.","ix.\nIts determinant is a unit. Complete local coefficients with finite quotients,\noccupied compatible finite conditions and this residual type therefore give\nactual integral completion with detector depth one. A fixed continuous map\n$G\\to\\Gamma$ permits pullback to infinite source groups. The finite-level\nrepresentations must factor through the same map, and the residual type is\nimposed on their $\\Gamma$-actions. No uniform determinant bound or augmented\ncertificate occupancy is assumed in this sector.","For a second sector let $R$ be a DVR with uniformizer $\\pi$, fraction field $K$\nand matrix rank $n$. If a fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and","For a second sector let $R$ be a DVR with uniformizer $\\pi$, fraction field $K$\nand matrix rank $n$. If a fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and does\nnot assume the DVR fractio","For a second sector let $R$ be a DVR with uniformizer $\\pi$, fraction field $K$\nand matrix rank $n$. If a fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and does\nnot assume the DVR fraction field is algebraic","For a second sector let $R$ be a DVR with uniformizer $\\pi$, fraction field $K$\nand matrix rank $n$. If a fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and does\nnot assume the DVR fraction field is algebraically closed.","ond sector let $R$ be a DVR with uniformizer $\\pi$, fraction field $K$\nand matrix rank $n$. If a fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and does\nnot assume the DVR fraction field is algebraically closed.","r let $R$ be a DVR with uniformizer $\\pi$, fraction field $K$\nand matrix rank $n$. If a fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and does\nnot assume the DVR fraction field is algebraically closed.","R with uniformizer $\\pi$, fraction field $K$\nand matrix rank $n$. If a fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and does\nnot assume the DVR fraction field is algebraically closed.","fixed template $\\tau$ has full $K$-linear action span,\nfinite generation and denominator clearing give a fixed $c$ such that\n\\[\n\\pi^c M_n(R)\\subseteq\\operatorname{span}_R\\{\\tau(g):g\\in G\\}.\n\\]\nNormalize a field comparison to an integral $P$ with a unit entry. The displayed\nspan constructs an integral $B$ with $BP=\\pi^cI$, so\n\\[\n\\det P\\mid\\pi^{cn},\\qquad \\det P\\not\\equiv0\\pmod{\\pi^{cn+1}}.\n\\]\nOne $c$ works for all integral representations in the fixed field-isomorphism\nclass. Jacobson density derives full span from actual irreducibility and a\nscalar commutant. The formalization states these hypotheses directly and does\nnot assume the DVR fraction field is algebraically closed.","Over $\\mathbb Z_3$, the actual average corrects\n$A=\\left(\\begin{smallmatrix}1&1","Over $\\mathbb Z_3$, the actual average corrects\n$A=\\left(\\begin{smallmatrix}1&1\\\\0&1\\end{smallmatrix}\\right)$ to\n$P=\\left(\\begin{smallmatrix}1&-1/2","0&1\\end{smallmatrix}\\right)$ to\n$P=\\left(\\begin{smallmatrix}1&-1/2\\\\0&1\\end{smallmatrix}\\right)$ for the\ninvolutions $\\rho(g)=\\left(\\begin{smallmatrix}1&1","0&1\\end{smallmatrix}\\right)$ for the\ninvolutions $\\rho(g)=\\left(\\begin{smallmatrix}1&1\\\\0&-1\\end{smallmatrix}\\right)$\nand $\\tau(g)=\\operatorname{diag}(1,-1)$. It preserves the residual comparison\nand has determinant one. Pullback through the quadratic character of the\nactual $G_{\\mathbb Q}$ gives continuous arithmetic examples. In contrast, the\nidentity and $\\operatorname{diag}(1,\\mathrm{sign})$ over $\\mathbb Z/4$ agree\nmodulo two but are not integrally conjugate. Their group order is not a unit.\nThese restricted results do not supply general arithmetic packet attachment,\nall finite-level existence statements, or cross-coefficient compatibility.","0&-1\\end{smallmatrix}\\right)$\nand $\\tau(g)=\\operatorname{diag}(1,-1)$. It preserves the residual comparison\nand has determinant one. Pullback through the quadratic character of the\nactual $G_{\\mathbb Q}$ gives continuous arithmetic examples. In contrast, the\nidentity and $\\operatorname{diag}(1,\\mathrm{sign})$ over $\\mathbb Z/4$ agree\nmodulo two but are not integrally conjugate. Their group order is not a unit.\nThese restricted results do not supply general arithmetic packet attachment,\nall finite-level existence statements, or cross-coefficient compatibility.","\\begin{theorem}[Finite Frobenius separation]\\label{px:9.1}\nLet $\\mathscr W_1,\\ldots,\\mathscr W_r$ be pairwise nonisomorphic strictly\ncompatible systems whose members are absolutely irreducible, with common\nalgebraic unramified trace values $t_i(v)$ in a number field $E$.\nThere are $r$ distinct permitted unramified places $v_1,\\ldots,v_r$ for which\n\\[\n C=(t_i(v_j))_{j,i}\\in M_r(E),\\qquad \\det C\\ne0.\n\\]\nEvery coefficient-field embedding $\\sigma_\\lambda:E\\hookrightarrow E_\\lambda$\ntherefore sends $C$ to an invertible matrix. Its entries agree with the\nordinary representation traces\n$\\operatorname{Tr}(W_{i,\\lambda}(\\operatorname{Frob}_{v_j}))$\nwhen the selected places satisfy the unramified compatibility convention\nfor that coefficient prime. In particular, places above the residue\ncharact","\\begin{theorem}[Finite Frobenius separation]\\label{px:9.1}\nLet $\\mathscr W_1,\\ldots,\\mathscr W_r$ be pairwise nonisomorphic strictly\ncompatible systems whose members are absolutely irreducible, with common\nalgebraic unramified trace values $t_i(v)$ in a number field $E$.\nThere are $r$ distinct permitted unramified places $v_1,\\ldots,v_r$ for which\n\\[\n C=(t_i(v_j))_{j,i}\\in M_r(E),\\qquad \\det C\\ne0.\n\\]\nEvery coefficient-field embedding $\\sigma_\\lambda:E\\hookrightarrow E_\\lambda$\ntherefore sends $C$ to an invertible matrix. Its entries agree with the\nordinary representation traces\n$\\operatorname{Tr}(W_{i,\\lambda}(\\operatorname{Frob}_{v_j}))$\nwhen the selected places satisfy the unramified compatibility convention\nfor that coefficient prime. In particular, places above the residue\ncharacteristic of $\\lambda$ are excluded from this ordinary trace comparison.\n\\end{theorem}","ion]\\label{px:9.1}\nLet $\\mathscr W_1,\\ldots,\\mathscr W_r$ be pairwise nonisomorphic strictly\ncompatible systems whose members are absolutely irreducible, with common\nalgebraic unramified trace values $t_i(v)$ in a number field $E$.\nThere are $r$ distinct permitted unramified places $v_1,\\ldots,v_r$ for which\n\\[\n C=(t_i(v_j))_{j,i}\\in M_r(E),\\qquad \\det C\\ne0.\n\\]\nEvery coefficient-field embedding $\\sigma_\\lambda:E\\hookrightarrow E_\\lambda$\ntherefore sends $C$ to an invertible matrix. Its entries agree with the\nordinary representation traces\n$\\operatorname{Tr}(W_{i,\\lambda}(\\operatorname{Frob}_{v_j}))$\nwhen the selected places satisfy the unramified compatibility convention\nfor that coefficient prime. In particular, places above the residue\ncharacteristic of $\\lambda$ are excluded from this ordinary trace comparison.\n\\end{theorem}","pairwise nonisomorphic strictly\ncompatible systems whose members are absolutely irreducible, with common\nalgebraic unramified trace values $t_i(v)$ in a number field $E$.\nThere are $r$ distinct permitted unramified places $v_1,\\ldots,v_r$ for which\n\\[\n C=(t_i(v_j))_{j,i}\\in M_r(E),\\qquad \\det C\\ne0.\n\\]\nEvery coefficient-field embedding $\\sigma_\\lambda:E\\hookrightarrow E_\\lambda$\ntherefore sends $C$ to an invertible matrix. Its entries agree with the\nordinary representation traces\n$\\operatorname{Tr}(W_{i,\\lambda}(\\operatorname{Frob}_{v_j}))$\nwhen the selected places satisfy the unramified compatibility convention\nfor that coefficient prime. In particular, places above the residue\ncharacteristic of $\\lambda$ are excluded from this ordinary trace comparison.\n\\end{theorem}","cible, with common\nalgebraic unramified trace values $t_i(v)$ in a number field $E$.\nThere are $r$ distinct permitted unramified places $v_1,\\ldots,v_r$ for which\n\\[\n C=(t_i(v_j))_{j,i}\\in M_r(E),\\qquad \\det C\\ne0.\n\\]\nEvery coefficient-field embedding $\\sigma_\\lambda:E\\hookrightarrow E_\\lambda$\ntherefore sends $C$ to an invertible matrix. Its entries agree with the\nordinary representation traces\n$\\operatorname{Tr}(W_{i,\\lambda}(\\operatorname{Frob}_{v_j}))$\nwhen the selected places satisfy the unramified compatibility convention\nfor that coefficient prime. In particular, places above the residue\ncharacteristic of $\\lambda$ are excluded from this ordinary trace comparison.\n\\end{theorem}","ry coefficient-field embedding $\\sigma_\\lambda:E\\hookrightarrow E_\\lambda$\ntherefore sends $C$ to an invertible matrix. Its entries agree with the\nordinary representation traces\n$\\operatorname{Tr}(W_{i,\\lambda}(\\operatorname{Frob}_{v_j}))$\nwhen the selected places satisfy the unramified compatibility convention\nfor that coefficient prime. In particular, places above the residue\ncharacteristic of $\\lambda$ are excluded from this ordinary trace comparison.\n\\end{theorem}","For $r$ independent functions, their evaluation linear forms span the\n$r$-dimensional coefficient dual: a proper span would leave a nonzero\ncoefficient vector annihilated by every evaluation, contradicting independence.\nChoose a basis from the evaluations. Its $r$ distinct sample points give an\ninvertible square matrix. The common algebraic traces put that matrix over\n$E$. An injective field embedding reflects the nonvanishing of its determinant.\nAll further field embeddings preserve the same nonvanishing.\n\\end{proof}","enius coverage remains unformalized. The \\hyperref[book:x-construction-25]{system-isomorphism and finite-cyclotomic comparison} now proves\nthe bridge from componentwise nonisomorphic systems to nonisomorphic members\nat the selected coefficient prime, under density on the required good-place\noverlaps. It also proves the complete restricted separator for a fixed finite\ncyclotomic Galois group over $\\Q$, with arithmetic coverage derived from the\nproved Dirichlet theorem. The full arithmetic theorem remains partial. Selection\nis existential and gives no effective upper bound on the selected primes.\nA common algebraic matrix is still distinct from an ordinary unramified trace\ncomparison at all coefficient primes simultaneously.","\\paragraph{System isomorphism and good-place overlaps.}\\phantomsection\\label{book:x-construction-25}\nThe system equivalence used here is componentwise representation isomorphism:\n\\[\n \\mathscr W_i\\simeq\\mathscr W_j\n \\quad\\Longleftrightarrow\\quad\n \\forall\\lambda,\\ W_{i,\\lambda}\\simeq W_{j,\\lambda}.\n\\]\nFor finite-dimensional irreducible representations of any monoid over a\ncharacteristic-zero field, the new character-rigidity theorem proves that\nequality of characters is equivalent to an actual equivariant linear isomorphism.\nNeither a finite group nor an algebraically closed coefficient field is needed.","Suppose two members are isomorphic at $\\lambda_0$. Injectivity of the common\ncoefficient embedding gives $t_i(v)=t_j(v)$ at every place good for $\\lambda_0$.\nAt another coefficient prime $\\lambda$, this equality compares actual member\ntraces on the intersection of the two good-place sets. Conjugacy density of\nthat intersection and continuity extend the equality to the whole group.\nCharacter rigidity then gives an actual isomorphism at $\\lambda$.\nThus the nonisomorphism of systems supplies the required nonisomorphism of\nmembers. Equality of optional labels or record fields is not used as a\nsubstitute for representation equivalence.","lambda_0$. Injectivity of the common\ncoefficient embedding gives $t_i(v)=t_j(v)$ at every place good for $\\lambda_0$.\nAt another coefficient prime $\\lambda$, this equality compares actual member\ntraces on the intersection of the two good-place sets. Conjugacy density of\nthat intersection and continuity extend the equality to the whole group.\nCharacter rigidity then gives an actual isomorphism at $\\lambda$.\nThus the nonisomorphism of systems supplies the required nonisomorphism of\nmembers. Equality of optional labels or record fields is not used as a\nsubstitute for representation equivalence.","\\paragraph{Arithmetic coverage for a fixed cyclotomic extension.}\\phantomsection\\label{book:x-construction-25-cyclotomic}\nLet $n>0$ and let $L=\\Q(\\zeta_n)$ be a cyclotomic number field. The canonical\nisomorphism\n\\[\n \\operatorname{Gal}(L/\\Q)\\simeq(\\mathbb Z/n\\mathbb Z)^\\times\n\\]\ndefines $\\sigma_p$ by $\\sigma_p(\\zeta_n)=\\zeta_n^p$ when $p\\nmid n$.\nFor every automorphism $\\sigma$, finite set $B\\subset\\mathbb N$ and bound $b$,\nLean proves that there is a rational prime $p>b$, with $p\\notin B$ and\n$p\\nmid n$, such that $\\sigma_p=\\sigma$. The prime-existence step uses the\nproved Dirichlet theorem in the pinned Mathlib dependency.","overage for a fixed cyclotomic extension.}\\phantomsection\\label{book:x-construction-25-cyclotomic}\nLet $n>0$ and let $L=\\Q(\\zeta_n)$ be a cyclotomic number field. The canonical\nisomorphism\n\\[\n \\operatorname{Gal}(L/\\Q)\\simeq(\\mathbb Z/n\\mathbb Z)^\\times\n\\]\ndefines $\\sigma_p$ by $\\sigma_p(\\zeta_n)=\\zeta_n^p$ when $p\\nmid n$.\nFor every automorphism $\\sigma$, finite set $B\\subset\\mathbb N$ and bound $b$,\nLean proves that there is a rational prime $p>b$, with $p\\notin B$ and\n$p\\nmid n$, such that $\\sigma_p=\\sigma$. The prime-existence step uses the\nproved Dirichlet theorem in the pinned Mathlib dependency.","omsection\\label{book:x-construction-25-cyclotomic}\nLet $n>0$ and let $L=\\Q(\\zeta_n)$ be a cyclotomic number field. The canonical\nisomorphism\n\\[\n \\operatorname{Gal}(L/\\Q)\\simeq(\\mathbb Z/n\\mathbb Z)^\\times\n\\]\ndefines $\\sigma_p$ by $\\sigma_p(\\zeta_n)=\\zeta_n^p$ when $p\\nmid n$.\nFor every automorphism $\\sigma$, finite set $B\\subset\\mathbb N$ and bound $b$,\nLean proves that there is a rational prime $p>b$, with $p\\notin B$ and\n$p\\nmid n$, such that $\\sigma_p=\\sigma$. The prime-existence step uses the\nproved Dirichlet theorem in the pinned Mathlib dependency.","bel{book:x-construction-25-cyclotomic}\nLet $n>0$ and let $L=\\Q(\\zeta_n)$ be a cyclotomic number field. The canonical\nisomorphism\n\\[\n \\operatorname{Gal}(L/\\Q)\\simeq(\\mathbb Z/n\\mathbb Z)^\\times\n\\]\ndefines $\\sigma_p$ by $\\sigma_p(\\zeta_n)=\\zeta_n^p$ when $p\\nmid n$.\nFor every automorphism $\\sigma$, finite set $B\\subset\\mathbb N$ and bound $b$,\nLean proves that there is a rational prime $p>b$, with $p\\notin B$ and\n$p\\nmid n$, such that $\\sigma_p=\\sigma$. The prime-existence step uses the\nproved Dirichlet theorem in the pinned Mathlib dependency.","ruction-25-cyclotomic}\nLet $n>0$ and let $L=\\Q(\\zeta_n)$ be a cyclotomic number field. The canonical\nisomorphism\n\\[\n \\operatorname{Gal}(L/\\Q)\\simeq(\\mathbb Z/n\\mathbb Z)^\\times\n\\]\ndefines $\\sigma_p$ by $\\sigma_p(\\zeta_n)=\\zeta_n^p$ when $p\\nmid n$.\nFor every automorphism $\\sigma$, finite set $B\\subset\\mathbb N$ and bound $b$,\nLean proves that there is a rational prime $p>b$, with $p\\notin B$ and\n$p\\nmid n$, such that $\\sigma_p=\\sigma$. The prime-existence step uses the\nproved Dirichlet theorem in the pinned Mathlib dependency.","Let $n>0$ and let $L=\\Q(\\zeta_n)$ be a cyclotomic number field. The canonical\nisomorphism\n\\[\n \\operatorname{Gal}(L/\\Q)\\simeq(\\mathbb Z/n\\mathbb Z)^\\times\n\\]\ndefines $\\sigma_p$ by $\\sigma_p(\\zeta_n)=\\zeta_n^p$ when $p\\nmid n$.\nFor every automorphism $\\sigma$, finite set $B\\subset\\mathbb N$ and bound $b$,\nLean proves that there is a rational prime $p>b$, with $p\\notin B$ and\n$p\\nmid n$, such that $\\sigma_p=\\sigma$. The prime-existence step uses the\nproved Dirichlet theorem in the pinned Mathlib dependency.","The arithmetic attachment is checked in this finite cyclotomic case. There is\nan actual maximal prime $\\mathfrak P$ of $\\mathcal O_L$ above $(p)$, its\nramification index is one, and $\\sigma_p$ belongs to its decomposition group.\nFor every integral $x\\in\\mathcal O_L$,\n\\[\n \\overline{\\sigma_p(x)}=\\overline{x}^{\\,p}\n \\quad\\text{in }\\mathcal O_L/\\mathfrak P.\n\\]\nThe residue identity is proved on all integral elements using\n$\\mathcal O_L=\\mathbb Z[\\zeta_n]$. Existence of $\\mathfrak P$ is obtained by\nlying over, rather than assumed. This specifies arithmetic Frobenius, with\n$p$-power residue action. The action and unramifiedness statements also hold\nfor every maximal prime above the selected $p$.","ent is checked in this finite cyclotomic case. There is\nan actual maximal prime $\\mathfrak P$ of $\\mathcal O_L$ above $(p)$, its\nramification index is one, and $\\sigma_p$ belongs to its decomposition group.\nFor every integral $x\\in\\mathcal O_L$,\n\\[\n \\overline{\\sigma_p(x)}=\\overline{x}^{\\,p}\n \\quad\\text{in }\\mathcal O_L/\\mathfrak P.\n\\]\nThe residue identity is proved on all integral elements using\n$\\mathcal O_L=\\mathbb Z[\\zeta_n]$. Existence of $\\mathfrak P$ is obtained by\nlying over, rather than assumed. This specifies arithmetic Frobenius, with\n$p$-power residue action. The action and unramifiedness statements also hold\nfor every maximal prime above the selected $p$.","here is\nan actual maximal prime $\\mathfrak P$ of $\\mathcal O_L$ above $(p)$, its\nramification index is one, and $\\sigma_p$ belongs to its decomposition group.\nFor every integral $x\\in\\mathcal O_L$,\n\\[\n \\overline{\\sigma_p(x)}=\\overline{x}^{\\,p}\n \\quad\\text{in }\\mathcal O_L/\\mathfrak P.\n\\]\nThe residue identity is proved on all integral elements using\n$\\mathcal O_L=\\mathbb Z[\\zeta_n]$. Existence of $\\mathfrak P$ is obtained by\nlying over, rather than assumed. This specifies arithmetic Frobenius, with\n$p$-power residue action. The action and unramifiedness statements also hold\nfor every maximal prime above the selected $p$.","ne, and $\\sigma_p$ belongs to its decomposition group.\nFor every integral $x\\in\\mathcal O_L$,\n\\[\n \\overline{\\sigma_p(x)}=\\overline{x}^{\\,p}\n \\quad\\text{in }\\mathcal O_L/\\mathfrak P.\n\\]\nThe residue identity is proved on all integral elements using\n$\\mathcal O_L=\\mathbb Z[\\zeta_n]$. Existence of $\\mathfrak P$ is obtained by\nlying over, rather than assumed. This specifies arithmetic Frobenius, with\n$p$-power residue action. The action and unramifiedness statements also hold\nfor every maximal prime above the selected $p$.","=\\overline{x}^{\\,p}\n \\quad\\text{in }\\mathcal O_L/\\mathfrak P.\n\\]\nThe residue identity is proved on all integral elements using\n$\\mathcal O_L=\\mathbb Z[\\zeta_n]$. Existence of $\\mathfrak P$ is obtained by\nlying over, rather than assumed. This specifies arithmetic Frobenius, with\n$p$-power residue action. The action and unramifiedness statements also hold\nfor every maximal prime above the selected $p$.","ite set, with nonzero determinant. Every coefficient embedding preserves its\nnonvanishing. Actual member trace comparisons retain that member's good-prime\ncondition. No coverage or character-continuity hypothesis is added in this\ncyclotomic specialization: the prime coverage is proved, and the finite\nextension's Krull topology is discrete. A concrete conductor-four instance uses\nthe actual field $\\Q(\\zeta_4)$.","\\paragraph{Absolute Galois restriction and arithmetic witnesses.}\\phantomsection\\label{book:x-construction-26}\nFix a cyclotomic number field $L=\\Q(\\zeta_n)$ with $n>0$ and an embedding\n$j:L\\hookrightarrow\\overline{\\Q}$. The new restriction map\n\\[\n q_L:G_{\\Q}\\longrightarrow\\operatorname{Gal}(L/\\Q),\\qquad\n j(q_L(\\gamma)x)=\\gamma(j(x)),\n\\]\nis an actual continuous surjective homomorphism with open kernel. Surjectivity\nis derived by extending automorphisms to the algebraic closure. Pullback along\na surjective homomorphism preserves and reflects equivariant linear\nisomorphisms, invariant subspaces, and irreducibility. These assertions keep\nthe underlying linear maps and submodules and do not assume a finite acting\ngroup. In particular, the equivalence relation for systems is preserved and\nreflected componentwise.","The arithmetic witnesses use the integral closure\n$\\mathcal A$ of $\\mathbb Z$ in $\\overline{\\Q}$. For each rational prime $p$,\nLean constructs a maximal ideal $\\mathfrak Q$ of $\\mathcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group.","The arithmetic witnesses use the integral closure\n$\\mathcal A$ of $\\mathbb Z$ in $\\overline{\\Q}$. For each rational prime $p$,\nLean constructs a maximal ideal $\\mathfrak Q$ of $\\mathcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot ass","The arithmetic witnesses use the integral closure\n$\\mathcal A$ of $\\mathbb Z$ in $\\overline{\\Q}$. For each rational prime $p$,\nLean constructs a maximal ideal $\\mathfrak Q$ of $\\mathcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot assert that an arbitra","The arithmetic witnesses use the integral closure\n$\\mathcal A$ of $\\mathbb Z$ in $\\overline{\\Q}$. For each rational prime $p$,\nLean constructs a maximal ideal $\\mathfrak Q$ of $\\mathcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot assert that an arbitrary global lift belongs to tha","The arithmetic witnesses use the integral closure\n$\\mathcal A$ of $\\mathbb Z$ in $\\overline{\\Q}$. For each rational prime $p$,\nLean constructs a maximal ideal $\\mathfrak Q$ of $\\mathcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot assert that an arbitrary global lift belongs to that group.","$p$,\nLean constructs a maximal ideal $\\mathfrak Q$ of $\\mathcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot assert that an arbitrary global lift belongs to that group.","frak Q$ of $\\mathcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot assert that an arbitrary global lift belongs to that group.","hcal A$ above $(p)$ and\nan actual element $\\gamma_p\\in G_{\\Q}$ such that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot assert that an arbitrary global lift belongs to that group.","ch that\n\\[\n \\gamma_p\\in\\operatorname{Stab}_{G_{\\Q}}(\\mathfrak Q),\\qquad\n \\overline{\\gamma_p(a)}=\\overline a^{\\,p}\n \\quad(a\\in\\mathcal A).\n\\]\nThe existence argument applies Mathlib's proved profinite stabilizer-surjectivity\ntheorem to this actual invariant ring. The residue $p$-power map is proved\nbijective by taking $p$-th roots in $\\overline{\\Q}$ and showing that they\nremain integral when their $p$-th powers are integral. This constructs an\narithmetic Frobenius witness in the prime-ideal decomposition group. It does\nnot assert that an arbitrary global lift belongs to that group.","When $p\\nmid n$, the comparison with the finite cyclotomic sample is proved:\n\\[\n q_L(\\gamma_p)=\\sigma_p,\n \\qquad \\sigma_p(\\zeta_n)=\\zeta_n^p.\n\\]\nReduction distinguishes $n$-th roots of unity at such a prime. Indeed, if\n$x^n=y^n=1$ and their reductions agree, the factorization of $x^n-y^n$ gives\n\\[\n (x-y)\\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.\n\\]\nThe second factor reduces to $n\\overline y^{\\,n-1}\\ne0$, so the domain\nproperty gives $x=y$. This proves both Frobenius recognition on the\ncyclotomic generator and triviality of its inertia action. Consequently,\nevery representation pulled back through $q_L$ is unramified there.","son with the finite cyclotomic sample is proved:\n\\[\n q_L(\\gamma_p)=\\sigma_p,\n \\qquad \\sigma_p(\\zeta_n)=\\zeta_n^p.\n\\]\nReduction distinguishes $n$-th roots of unity at such a prime. Indeed, if\n$x^n=y^n=1$ and their reductions agree, the factorization of $x^n-y^n$ gives\n\\[\n (x-y)\\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.\n\\]\nThe second factor reduces to $n\\overline y^{\\,n-1}\\ne0$, so the domain\nproperty gives $x=y$. This proves both Frobenius recognition on the\ncyclotomic generator and triviality of its inertia action. Consequently,\nevery representation pulled back through $q_L$ is unramified there.","uch a prime. Indeed, if\n$x^n=y^n=1$ and their reductions agree, the factorization of $x^n-y^n$ gives\n\\[\n (x-y)\\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.\n\\]\nThe second factor reduces to $n\\overline y^{\\,n-1}\\ne0$, so the domain\nproperty gives $x=y$. This proves both Frobenius recognition on the\ncyclotomic generator and triviality of its inertia action. Consequently,\nevery representation pulled back through $q_L$ is unramified there.","The valuation formulation is checked separately. For any valuation subring\n$A\\subset\\overline{\\Q}$ with residue characteristic $p\\nmid n$, its actual\nresidue-action kernel fixes every $n$-th root of unity. A supplied element of\nits decomposition subgroup with $p$-power residue action restricts to\n$\\sigma_p$. These statements use Mathlib's valuation inertia and decomposition\nsubgroups. No identification of the valuation and integral-closure prime\nmodels is assumed in the formal comparison.","For a compatible family $\\rho_{i,\\lambda}$ of representations of the fixed\nfinite group, set $V_{i,\\lambda}=\\rho_{i,\\lambda}\\circ q_L$. Suppose the\nactual absolute-Galois members are irreducible and the systems are pairwise\ncomponentwise nonisomorphic. The finite prime separator now yields distinct\npermitted rational primes $p_j$, actual prime ideals $\\mathfrak Q_j$ and\narithmetic Frobenius witnesses $\\gamma_j$ as above, outside any prescribed\nfinite set, such that the common algebraic trace matrix has nonzero\ndeterminant. Every coefficient embedding preserves this nonvanishing, and\n\\[\n \\iota_\\lambda(t_i(p_j))=\n \\operatorname{tr}(V_{i,\\lambda}(\\gamma_j))\n\\]\nwhenever $p_j$ is good for that coefficient member. The operator and trace","For a compatible family $\\rho_{i,\\lambda}$ of representations of the fixed\nfinite group, set $V_{i,\\lambda}=\\rho_{i,\\lambda}\\circ q_L$. Suppose the\nactual absolute-Galois members are irreducible and the systems are pairwise\ncomponentwise nonisomorphic. The finite prime separator now yields distinct\npermitted rational primes $p_j$, actual prime ideals $\\mathfrak Q_j$ and\narithmetic Frobenius witnesses $\\gamma_j$ as above, outside any prescribed\nfinite set, such that the common algebraic trace matrix has nonzero\ndeterminant. Every coefficient embedding preserves this nonvanishing, and\n\\[\n \\iota_\\lambda(t_i(p_j))=\n \\operatorname{tr}(V_{i,\\lambda}(\\gamma_j))\n\\]\nwhenever $p_j$ is good for that coefficient member. The operator and trace\nvalues agree for every global lift with the same cyclotomic restriction.\nNeither Frobenius-w","For a compatible family $\\rho_{i,\\lambda}$ of representations of the fixed\nfinite group, set $V_{i,\\lambda}=\\rho_{i,\\lambda}\\circ q_L$. Suppose the\nactual absolute-Galois members are irreducible and the systems are pairwise\ncomponentwise nonisomorphic. The finite prime separator now yields distinct\npermitted rational primes $p_j$, actual prime ideals $\\mathfrak Q_j$ and\narithmetic Frobenius witnesses $\\gamma_j$ as above, outside any prescribed\nfinite set, such that the common algebraic trace matrix has nonzero\ndeterminant. Every coefficient embedding preserves this nonvanishing, and\n\\[\n \\iota_\\lambda(t_i(p_j))=\n \\operatorname{tr}(V_{i,\\lambda}(\\gamma_j))\n\\]\nwhenever $p_j$ is good for that coefficient member. The operator and trace\nvalues agree for every global lift with the same cyclotomic restriction.\nNeither Frobenius-witness existence nor quotient coverage is a supplied\nhypothesis in this restricted arithmetic theorem. The common cyclotomic\nfactorization and the compatible trace data remain explicit hypotheses.","compatible family $\\rho_{i,\\lambda}$ of representations of the fixed\nfinite group, set $V_{i,\\lambda}=\\rho_{i,\\lambda}\\circ q_L$. Suppose the\nactual absolute-Galois members are irreducible and the systems are pairwise\ncomponentwise nonisomorphic. The finite prime separator now yields distinct\npermitted rational primes $p_j$, actual prime ideals $\\mathfrak Q_j$ and\narithmetic Frobenius witnesses $\\gamma_j$ as above, outside any prescribed\nfinite set, such that the common algebraic trace matrix has nonzero\ndeterminant. Every coefficient embedding preserves this nonvanishing, and\n\\[\n \\iota_\\lambda(t_i(p_j))=\n \\operatorname{tr}(V_{i,\\lambda}(\\gamma_j))\n\\]\nwhenever $p_j$ is good for that coefficient member. The operator and trace\nvalues agree for every global lift with the same cyclotomic restriction.\nNeither Frobenius-witness existence nor quotient coverage is a supplied\nhypothesis in this restricted arithmetic theorem. The common cyclotomic\nfactorization and the compatible trace data remain explicit hypotheses.","le and the systems are pairwise\ncomponentwise nonisomorphic. The finite prime separator now yields distinct\npermitted rational primes $p_j$, actual prime ideals $\\mathfrak Q_j$ and\narithmetic Frobenius witnesses $\\gamma_j$ as above, outside any prescribed\nfinite set, such that the common algebraic trace matrix has nonzero\ndeterminant. Every coefficient embedding preserves this nonvanishing, and\n\\[\n \\iota_\\lambda(t_i(p_j))=\n \\operatorname{tr}(V_{i,\\lambda}(\\gamma_j))\n\\]\nwhenever $p_j$ is good for that coefficient member. The operator and trace\nvalues agree for every global lift with the same cyclotomic restriction.\nNeither Frobenius-witness existence nor quotient coverage is a supplied\nhypothesis in this restricted arithmetic theorem. The common cyclotomic\nfactorization and the compatible trace data remain explicit hypotheses.","tor now yields distinct\npermitted rational primes $p_j$, actual prime ideals $\\mathfrak Q_j$ and\narithmetic Frobenius witnesses $\\gamma_j$ as above, outside any prescribed\nfinite set, such that the common algebraic trace matrix has nonzero\ndeterminant. Every coefficient embedding preserves this nonvanishing, and\n\\[\n \\iota_\\lambda(t_i(p_j))=\n \\operatorname{tr}(V_{i,\\lambda}(\\gamma_j))\n\\]\nwhenever $p_j$ is good for that coefficient member. The operator and trace\nvalues agree for every global lift with the same cyclotomic restriction.\nNeither Frobenius-witness existence nor quotient coverage is a supplied\nhypothesis in this restricted arithmetic theorem. The common cyclotomic\nfactorization and the compatible trace data remain explicit hypotheses.","The boundaries are formalized as well. Pullback along a nonsurjective\nhomomorphism can make inequivalent irreducible representations isomorphic.\nSamples can cover one quotient while missing an open quotient of the full\ngroup. Finally, in $\\mathbb Z_2$, the distinct square roots $1$ and $-1$ have\nthe same residue, so the condition that the root order be a unit cannot be\ndiscarded.","\\paragraph{General finite-field Frobenius comparison.}\\phantomsection\\label{book:x-construction-27}\nLet $L/\\Q$ now be any finite normal number field, with a chosen embedding\n$j:L\\hookrightarrow\\overline{\\Q}$. Write $\\mathcal A$ for the integral\nclosure of $\\mathbb Z$ in $\\overline{\\Q}$. The embedding restricts to an\ninjective map $\\mathcal O_L\\to\\mathcal A$ which intertwines the actual\nGalois actions. If $\\mathfrak Q\\subset\\mathcal A$ is maximal above $(p)$,\nits contraction $\\mathfrak P\\subset\\mathcal O_L$ is a nonzero maximal\nideal above $(p)$, and the induced residue map is injective.","h{General finite-field Frobenius comparison.}\\phantomsection\\label{book:x-construction-27}\nLet $L/\\Q$ now be any finite normal number field, with a chosen embedding\n$j:L\\hookrightarrow\\overline{\\Q}$. Write $\\mathcal A$ for the integral\nclosure of $\\mathbb Z$ in $\\overline{\\Q}$. The embedding restricts to an\ninjective map $\\mathcal O_L\\to\\mathcal A$ which intertwines the actual\nGalois actions. If $\\mathfrak Q\\subset\\mathcal A$ is maximal above $(p)$,\nits contraction $\\mathfrak P\\subset\\mathcal O_L$ is a nonzero maximal\nideal above $(p)$, and the induced residue map is injective.","antomsection\\label{book:x-construction-27}\nLet $L/\\Q$ now be any finite normal number field, with a chosen embedding\n$j:L\\hookrightarrow\\overline{\\Q}$. Write $\\mathcal A$ for the integral\nclosure of $\\mathbb Z$ in $\\overline{\\Q}$. The embedding restricts to an\ninjective map $\\mathcal O_L\\to\\mathcal A$ which intertwines the actual\nGalois actions. If $\\mathfrak Q\\subset\\mathcal A$ is maximal above $(p)$,\nits contraction $\\mathfrak P\\subset\\mathcal O_L$ is a nonzero maximal\nideal above $(p)$, and the induced residue map is injective.",":x-construction-27}\nLet $L/\\Q$ now be any finite normal number field, with a chosen embedding\n$j:L\\hookrightarrow\\overline{\\Q}$. Write $\\mathcal A$ for the integral\nclosure of $\\mathbb Z$ in $\\overline{\\Q}$. The embedding restricts to an\ninjective map $\\mathcal O_L\\to\\mathcal A$ which intertwines the actual\nGalois actions. If $\\mathfrak Q\\subset\\mathcal A$ is maximal above $(p)$,\nits contraction $\\mathfrak P\\subset\\mathcal O_L$ is a nonzero maximal\nideal above $(p)$, and the induced residue map is injective.","ld, with a chosen embedding\n$j:L\\hookrightarrow\\overline{\\Q}$. Write $\\mathcal A$ for the integral\nclosure of $\\mathbb Z$ in $\\overline{\\Q}$. The embedding restricts to an\ninjective map $\\mathcal O_L\\to\\mathcal A$ which intertwines the actual\nGalois actions. If $\\mathfrak Q\\subset\\mathcal A$ is maximal above $(p)$,\nits contraction $\\mathfrak P\\subset\\mathcal O_L$ is a nonzero maximal\nideal above $(p)$, and the induced residue map is injective.","Restriction carries the actual decomposition group at $\\mathfrak Q$ into\nthat at $\\mathfrak P$, carries ideal inertia into ideal inertia, and\npreserves the arithmetic $p$-power residue action. Thus the absolute\nFrobenius constructed in the \\hyperref[book:x-construction-26]{absolute-Galois restriction construction} gives a simultaneous finite Frobenius\nat the contracted prime. This assertion applies to arbitrary finite normal\nnumber fields and does not require a cyclotomic generator.","The exceptional primes are controlled by the actual field discriminant:\n\\[\n S_L=\\{p\\text{ rational prime}:p\\mid\\operatorname{disc}(L)\\}.\n\\]\nThis set is finite. For $p\\notin S_L$, every prime of $\\mathcal O_L$\nabove $(p)$ is unramified. Lean proves that the actual inertia cardinality\nis the ramification index. It follows that inertia is trivial and the\narithmetic Frobenius at a fixed finite prime is unique as a field\nautomorphism. Neither trivial inertia nor uniqueness is assumed in place\nof the arithmetic unramifiedness condition.","The choice of absolute prime and Frobenius lift is also controlled. Let\n$(\\mathfrak Q_u,\\gamma_u)$ and $(\\mathfrak Q_v,\\gamma_v)$ be any two\nactual arithmetic witnesses at the same rational prime. Transitivity on\nprimes of $\\mathcal A$ above $(p)$ gives $a\\in G_{\\Q}$ with\n\\[\n \\mathfrak Q_v=a\\mathfrak Q_u,\\qquad\n \\gamma_v^{-1}a\\gamma_u a^{-1}\\in I_{\\mathfrak Q_v}.\n\\]\nConsequently, any representation trivial on $I_{\\mathfrak Q_v}$ has the\nsame trace on the two witnesses. For representations factoring through\n$L$, triviality on this actual absolute inertia is derived whenever\n$p\\notin S_L$. The restrictions of all absolute witnesses at such a prime\ntherefore form a single conjugacy class. This conclusion requires only\nthe indicated finite factorization and the discriminant exclusion.","absolute prime and Frobenius lift is also controlled. Let\n$(\\mathfrak Q_u,\\gamma_u)$ and $(\\mathfrak Q_v,\\gamma_v)$ be any two\nactual arithmetic witnesses at the same rational prime. Transitivity on\nprimes of $\\mathcal A$ above $(p)$ gives $a\\in G_{\\Q}$ with\n\\[\n \\mathfrak Q_v=a\\mathfrak Q_u,\\qquad\n \\gamma_v^{-1}a\\gamma_u a^{-1}\\in I_{\\mathfrak Q_v}.\n\\]\nConsequently, any representation trivial on $I_{\\mathfrak Q_v}$ has the\nsame trace on the two witnesses. For representations factoring through\n$L$, triviality on this actual absolute inertia is derived whenever\n$p\\notin S_L$. The restrictions of all absolute witnesses at such a prime\ntherefore form a single conjugacy class. This conclusion requires only\nthe indicated finite factorization and the discriminant exclusion.","k Q_v,\\gamma_v)$ be any two\nactual arithmetic witnesses at the same rational prime. Transitivity on\nprimes of $\\mathcal A$ above $(p)$ gives $a\\in G_{\\Q}$ with\n\\[\n \\mathfrak Q_v=a\\mathfrak Q_u,\\qquad\n \\gamma_v^{-1}a\\gamma_u a^{-1}\\in I_{\\mathfrak Q_v}.\n\\]\nConsequently, any representation trivial on $I_{\\mathfrak Q_v}$ has the\nsame trace on the two witnesses. For representations factoring through\n$L$, triviality on this actual absolute inertia is derived whenever\n$p\\notin S_L$. The restrictions of all absolute witnesses at such a prime\ntherefore form a single conjugacy class. This conclusion requires only\nthe indicated finite factorization and the discriminant exclusion.","rational prime. Transitivity on\nprimes of $\\mathcal A$ above $(p)$ gives $a\\in G_{\\Q}$ with\n\\[\n \\mathfrak Q_v=a\\mathfrak Q_u,\\qquad\n \\gamma_v^{-1}a\\gamma_u a^{-1}\\in I_{\\mathfrak Q_v}.\n\\]\nConsequently, any representation trivial on $I_{\\mathfrak Q_v}$ has the\nsame trace on the two witnesses. For representations factoring through\n$L$, triviality on this actual absolute inertia is derived whenever\n$p\\notin S_L$. The restrictions of all absolute witnesses at such a prime\ntherefore form a single conjugacy class. This conclusion requires only\nthe indicated finite factorization and the discriminant exclusion.","There is a final compatibility between embeddings. For a normal subfield\n$L\\subset\\overline{\\Q}$, restriction defined through a chosen embedding\nis conjugate to canonical restriction by one fixed automorphism of $L$.\nThis identifies their conjugacy classes and lets the arithmetic comparison\napply directly to the fixed fields of open normal subgroups of $G_{\\Q}$.\nThose fixed fields are proved to be finite Galois number fields, with the\noriginal open normal subgroup as the kernel of restriction.","The remaining prime-existence obligation now has a precise formulation.\nFor every finite Galois subfield $L$, every finite set $B$ of rational\nprimes, and every $\\sigma\\in\\operatorname{Gal}(L/\\Q)$, require a prime\n$p\\notin B\\cup S_L$ and an actual absolute arithmetic witness $\\gamma_p$\nsuch that\n\\[\n q_L(\\gamma_p)\\sim\\sigma.\n\\]\nLean proves that this assertion is equivalent to the cofinite quotient\ncoverage required by the general separator, for any fixed choice of actual\nFrobenius witnesses. The equivalence derives independence from the witness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unprov","The remaining prime-existence obligation now has a precise formulation.\nFor every finite Galois subfield $L$, every finite set $B$ of rational\nprimes, and every $\\sigma\\in\\operatorname{Gal}(L/\\Q)$, require a prime\n$p\\notin B\\cup S_L$ and an actual absolute arithmetic witness $\\gamma_p$\nsuch that\n\\[\n q_L(\\gamma_p)\\sim\\sigma.\n\\]\nLean proves that this assertion is equivalent to the cofinite quotient\ncoverage required by the general separator, for any fixed choice of actual\nFrobenius witnesses. The equivalence derives independence from the witness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remai","The remaining prime-existence obligation now has a precise formulation.\nFor every finite Galois subfield $L$, every finite set $B$ of rational\nprimes, and every $\\sigma\\in\\operatorname{Gal}(L/\\Q)$, require a prime\n$p\\notin B\\cup S_L$ and an actual absolute arithmetic witness $\\gamma_p$\nsuch that\n\\[\n q_L(\\gamma_p)\\sim\\sigma.\n\\]\nLean proves that this assertion is equivalent to the cofinite quotient\ncoverage required by the general separator, for any fixed choice of actual\nFrobenius witnesses. The equivalence derives independence from the witness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three","The remaining prime-existence obligation now has a precise formulation.\nFor every finite Galois subfield $L$, every finite set $B$ of rational\nprimes, and every $\\sigma\\in\\operatorname{Gal}(L/\\Q)$, require a prime\n$p\\notin B\\cup S_L$ and an actual absolute arithmetic witness $\\gamma_p$\nsuch that\n\\[\n q_L(\\gamma_p)\\sim\\sigma.\n\\]\nLean proves that this assertion is equivalent to the cofinite quotient\ncoverage required by the general separator, for any fixed choice of actual\nFrobenius witnesses. The equivalence derives independence from the witness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three partial.\nThis new arithmetic interfa","The remaining prime-existence obligation now has a precise formulation.\nFor every finite Galois subfield $L$, every finite set $B$ of rational\nprimes, and every $\\sigma\\in\\operatorname{Gal}(L/\\Q)$, require a prime\n$p\\notin B\\cup S_L$ and an actual absolute arithmetic witness $\\gamma_p$\nsuch that\n\\[\n q_L(\\gamma_p)\\sim\\sigma.\n\\]\nLean proves that this assertion is equivalent to the cofinite quotient\ncoverage required by the general separator, for any fixed choice of actual\nFrobenius witnesses. The equivalence derives independence from the witness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three partial.\nThis new arithmetic interface has base $\\Q$. For the source's general-base\ninsta","The remaining prime-existence obligation now has a precise formulation.\nFor every finite Galois subfield $L$, every finite set $B$ of rational\nprimes, and every $\\sigma\\in\\operatorname{Gal}(L/\\Q)$, require a prime\n$p\\notin B\\cup S_L$ and an actual absolute arithmetic witness $\\gamma_p$\nsuch that\n\\[\n q_L(\\gamma_p)\\sim\\sigma.\n\\]\nLean proves that this assertion is equivalent to the cofinite quotient\ncoverage required by the general separator, for any fixed choice of actual\nFrobenius witnesses. The equivalence derives independence from the witness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three partial.\nThis new arithmetic interface has base $\\Q$. For the source's general-base\ninstances involving $G_{F,S}$, the analogous a","e arithmetic witness $\\gamma_p$\nsuch that\n\\[\n q_L(\\gamma_p)\\sim\\sigma.\n\\]\nLean proves that this assertion is equivalent to the cofinite quotient\ncoverage required by the general separator, for any fixed choice of actual\nFrobenius witnesses. The equivalence derives independence from the witness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three partial.\nThis new arithmetic interface has base $\\Q$. For the source's general-base\ninstances involving $G_{F,S}$, the analogous actual-prime and norm-power\nFrobenius attachment and coverage over $F$, or a proved transfer to the\nrational-base result, also remain necessary.","ness\nchoices. It does not supply the displayed prime-existence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three partial.\nThis new arithmetic interface has base $\\Q$. For the source's general-base\ninstances involving $G_{F,S}$, the analogous actual-prime and norm-power\nFrobenius attachment and coverage over $F$, or a proved transfer to the\nrational-base result, also remain necessary.","stence assertion.\nArbitrary finite exclusions absorb $S_L$, and the resulting conditional\ndensity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three partial.\nThis new arithmetic interface has base $\\Q$. For the source's general-base\ninstances involving $G_{F,S}$, the analogous actual-prime and norm-power\nFrobenius attachment and coverage over $F$, or a proved transfer to the\nrational-base result, also remain necessary.","nsity theorem preserves the separate good-prime sets of coefficient\nmembers. General prime coverage remains unproved, so Claim 9.1 remains\npartial. The source ledger is still seventeen checked and three partial.\nThis new arithmetic interface has base $\\Q$. For the source's general-base\ninstances involving $G_{F,S}$, the analogous actual-prime and norm-power\nFrobenius attachment and coverage over $F$, or a proved transfer to the\nrational-base result, also remain necessary.","\\paragraph{Finite deletion can destroy density.}\nIn the actual discrete group $C_2=\\{1,a\\}$, set $s(0)=a$ and $s(m+1)=1$.\nThe original samples are surjective and satisfy quotient coverage. Removing\nindex zero leaves only the identity, so both density and quotient coverage fail.\nThe actual rational sign and trivial characters agree on the remaining samples\nbut differ globally. This checked example explains why ordinary coverage alone\ncannot justify deleting the bad places in the system comparison.","\\begin{theorem}[Compatible-carrier summand completion]\\label{px:9.2}\nSuppose a semisimple strictly compatible carrier decomposes as\n\\[\n\\mathscr W\n\\cong\n\\bigoplus_{i=1}^{r}\n\\mathscr W_i^{\\oplus c_i},\n\\]\nwith the \\(\\mathscr W_i\\) as above.  At one coefficient prime \\(\\lambda_0\\), suppose a semisimple summand has the form\n\\[\nV_{\\lambda_0}\n\\cong\n\\bigoplus_{i=1}^{r}\nW_{i,\\lambda_0}^{\\oplus m_i},\n\\qquad\n0\\le m_i\\le c_i.\n\\]\nThen\n\\[\n\\boxed{\nV_\\lambda\n=\n\\bigoplus_{i=1}^{r}\nW_{i,\\lambda}^{\\oplus m_i}\n}\n\\]\ndefines a strictly compatible summand at every coefficient prime.  The vector \\((m_i)\\) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.\n\\end{theorem}","arrier summand completion]\\label{px:9.2}\nSuppose a semisimple strictly compatible carrier decomposes as\n\\[\n\\mathscr W\n\\cong\n\\bigoplus_{i=1}^{r}\n\\mathscr W_i^{\\oplus c_i},\n\\]\nwith the \\(\\mathscr W_i\\) as above.  At one coefficient prime \\(\\lambda_0\\), suppose a semisimple summand has the form\n\\[\nV_{\\lambda_0}\n\\cong\n\\bigoplus_{i=1}^{r}\nW_{i,\\lambda_0}^{\\oplus m_i},\n\\qquad\n0\\le m_i\\le c_i.\n\\]\nThen\n\\[\n\\boxed{\nV_\\lambda\n=\n\\bigoplus_{i=1}^{r}\nW_{i,\\lambda}^{\\oplus m_i}\n}\n\\]\ndefines a strictly compatible summand at every coefficient prime.  The vector \\((m_i)\\) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.\n\\end{theorem}","W_i^{\\oplus c_i},\n\\]\nwith the \\(\\mathscr W_i\\) as above.  At one coefficient prime \\(\\lambda_0\\), suppose a semisimple summand has the form\n\\[\nV_{\\lambda_0}\n\\cong\n\\bigoplus_{i=1}^{r}\nW_{i,\\lambda_0}^{\\oplus m_i},\n\\qquad\n0\\le m_i\\le c_i.\n\\]\nThen\n\\[\n\\boxed{\nV_\\lambda\n=\n\\bigoplus_{i=1}^{r}\nW_{i,\\lambda}^{\\oplus m_i}\n}\n\\]\ndefines a strictly compatible summand at every coefficient prime.  The vector \\((m_i)\\) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.\n\\end{theorem}","\\begin{proof}\nUse the same multiplicities in every member of the compatible carrier.  Direct sums preserve strict compatibility, and \\(m_i\\le c_i\\) makes \\(V_\\lambda\\) a summand of \\(W_\\lambda\\).  At the separating places, the target trace vector is \\(C\\bm m\\).  Full column rank makes \\(\\bm m\\) the unique solution.\n\\end{proof}","\\paragraph{Recovery and realization are different steps.}\nFor a certified square separator the coefficient vector is $C^{-1}t$.\nThis is a valid carrier multiplicity vector only when every entry is a\nnonnegative integer and satisfies its capacity bound $m_i\\le c_i$.\nThose tests and uniqueness of natural multiplicities in characteristic zero\nare separately Lean-checked, as is transport through a field embedding.\nReduction modulo a residue characteristic is not a field embedding and may\nlose rank. The representation construction and common-polynomial part are now\nLean-checked as described below. The full strictly compatible arithmetic\nstatement, including its local conditions, remains partially formalized.","\\paragraph{Explicit summands inside a supplied decomposition.}\nLet $K$ be a field, let $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the","\\paragraph{Explicit summands inside a supplied decomposition.}\nLet $K$ be a field, let $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unuse","\\paragraph{Explicit summands inside a supplied decomposition.}\nLet $K$ be a field, let $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant id","\\paragraph{Explicit summands inside a supplied decomposition.}\nLet $K$ be a field, let $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the sp","\\paragraph{Explicit summands inside a supplied decomposition.}\nLet $K$ be a field, let $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside","\\paragraph{Explicit summands inside a supplied decomposition.}\nLet $K$ be a field, let $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct","position.}\nLet $K$ be a field, let $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","t $G$ be a monoid, and let\n$\\rho_i:G\\to\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","o\\operatorname{End}_K(W_i)$ be actual representations on\nfinite-dimensional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","ional spaces, indexed by a finite set $I$. For natural numbers\n$m_i\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","\\le c_i$, set\n\\[\n U_m=\\prod_{i\\in I}\\prod_{k<m_i}W_i,\n \\qquad U_c=\\prod_{i\\in I}\\prod_{k<c_i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","i}W_i.\n\\]\nThese finite products realize the corresponding finite direct sums. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","ms. The\nrepresentation on $U_m$ acts componentwise. Define $j:U_m\\to U_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","_c$ by retaining\nthe first $m_i$ copies and filling every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","ing every remaining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","ining coordinate with zero.\nDefine $r:U_c\\to U_m$ by restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","restriction to the selected copies. Lean proves\n\\[\n rj=1_{U_m},\\qquad\n j\\rho_m(g)=\\rho_c(g)j,\\qquad\n r\\rho_c(g)=\\rho_m(g)r.\n\\]\nThe image of $j$ is exactly the subspace whose unused coordinates vanish.\nConsequently $jr$ is an equivariant idempotent. A supplied equivariant\nisomorphism $D:W\\xrightarrow{\\sim}U_c$ gives the split maps\n$D^{-1}j$ and $rD$ inside the original carrier. The decomposition $D$ is an\ninput. These statements do not construct a decomposition of an unknown\narithmetic carrier.","The same formalization proves, for each $g\\in G$,\n\\[\n \\det\\bigl(XI-\\rho_m(g)\\bigr)\n =\\prod_{i\\in I}\\det\\bigl(XI-\\rho_i(g)\\bigr)^{m_i},\n \\qquad\n \\operatorname{Tr}\\bigl(\\rho_m(g)\\bigr)\n =\\sum_{i\\in I}(m_i:K)\\operatorname{Tr}\\bigl(\\rho_i(g)\\bigr).\n\\]\nConstituents may have different dimensions. Zero multiplicities, zero-dimensional\nconstituents, and the empty index set are included. If at a permitted sample\n$g$ the constituent polynomials are the images of $P_{i,g}\\in E[X]$ under a\nspecified map $E\\to K$, the summand polynomial is the image of\n$\\prod_iP_{i,g}^{m_i}$. Applying this independently at each coefficient field\npreserves a supplied common polynomial family. This does not establish the\nHodge--Tate or Weil--Deligne conditions in stric","The same formalization proves, for each $g\\in G$,\n\\[\n \\det\\bigl(XI-\\rho_m(g)\\bigr)\n =\\prod_{i\\in I}\\det\\bigl(XI-\\rho_i(g)\\bigr)^{m_i},\n \\qquad\n \\operatorname{Tr}\\bigl(\\rho_m(g)\\bigr)\n =\\sum_{i\\in I}(m_i:K)\\operatorname{Tr}\\bigl(\\rho_i(g)\\bigr).\n\\]\nConstituents may have different dimensions. Zero multiplicities, zero-dimensional\nconstituents, and the empty index set are included. If at a permitted sample\n$g$ the constituent polynomials are the images of $P_{i,g}\\in E[X]$ under a\nspecified map $E\\to K$, the summand polynomial is the image of\n$\\prod_iP_{i,g}^{m_i}$. Applying this independently at each coefficient field\npreserves a supplied common polynomial family. This does not establish the\nHodge--Tate or Weil--Deligne conditions in strict compatibility.","formalization proves, for each $g\\in G$,\n\\[\n \\det\\bigl(XI-\\rho_m(g)\\bigr)\n =\\prod_{i\\in I}\\det\\bigl(XI-\\rho_i(g)\\bigr)^{m_i},\n \\qquad\n \\operatorname{Tr}\\bigl(\\rho_m(g)\\bigr)\n =\\sum_{i\\in I}(m_i:K)\\operatorname{Tr}\\bigl(\\rho_i(g)\\bigr).\n\\]\nConstituents may have different dimensions. Zero multiplicities, zero-dimensional\nconstituents, and the empty index set are included. If at a permitted sample\n$g$ the constituent polynomials are the images of $P_{i,g}\\in E[X]$ under a\nspecified map $E\\to K$, the summand polynomial is the image of\n$\\prod_iP_{i,g}^{m_i}$. Applying this independently at each coefficient field\npreserves a supplied common polynomial family. This does not establish the\nHodge--Tate or Weil--Deligne conditions in strict compatibility.","\\bigl(XI-\\rho_m(g)\\bigr)\n =\\prod_{i\\in I}\\det\\bigl(XI-\\rho_i(g)\\bigr)^{m_i},\n \\qquad\n \\operatorname{Tr}\\bigl(\\rho_m(g)\\bigr)\n =\\sum_{i\\in I}(m_i:K)\\operatorname{Tr}\\bigl(\\rho_i(g)\\bigr).\n\\]\nConstituents may have different dimensions. Zero multiplicities, zero-dimensional\nconstituents, and the empty index set are included. If at a permitted sample\n$g$ the constituent polynomials are the images of $P_{i,g}\\in E[X]$ under a\nspecified map $E\\to K$, the summand polynomial is the image of\n$\\prod_iP_{i,g}^{m_i}$. Applying this independently at each coefficient field\npreserves a supplied common polynomial family. This does not establish the\nHodge--Tate or Weil--Deligne conditions in strict compatibility.","}\\det\\bigl(XI-\\rho_i(g)\\bigr)^{m_i},\n \\qquad\n \\operatorname{Tr}\\bigl(\\rho_m(g)\\bigr)\n =\\sum_{i\\in I}(m_i:K)\\operatorname{Tr}\\bigl(\\rho_i(g)\\bigr).\n\\]\nConstituents may have different dimensions. Zero multiplicities, zero-dimensional\nconstituents, and the empty index set are included. If at a permitted sample\n$g$ the constituent polynomials are the images of $P_{i,g}\\in E[X]$ under a\nspecified map $E\\to K$, the summand polynomial is the image of\n$\\prod_iP_{i,g}^{m_i}$. Applying this independently at each coefficient field\npreserves a supplied common polynomial family. This does not establish the\nHodge--Tate or Weil--Deligne conditions in strict compatibility.","eratorname{Tr}\\bigl(\\rho_m(g)\\bigr)\n =\\sum_{i\\in I}(m_i:K)\\operatorname{Tr}\\bigl(\\rho_i(g)\\bigr).\n\\]\nConstituents may have different dimensions. Zero multiplicities, zero-dimensional\nconstituents, and the empty index set are included. If at a permitted sample\n$g$ the constituent polynomials are the images of $P_{i,g}\\in E[X]$ under a\nspecified map $E\\to K$, the summand polynomial is the image of\n$\\prod_iP_{i,g}^{m_i}$. Applying this independently at each coefficient field\npreserves a supplied common polynomial family. This does not establish the\nHodge--Tate or Weil--Deligne conditions in strict compatibility.","Joint continuity of the constituent actions implies joint continuity of the\nconstructed action. The inclusion and retraction are continuous for the\nproduct topologies, and triviality on a supplied subgroup is preserved.\nNeither continuity nor triviality is inferred for a carrier whose corresponding\nhypothesis has not been supplied. The maps transported by $D$ are proved\nalgebraically. Their topological interpretation additionally requires the\nappropriate continuity of $D$ and its inverse.","e\nconstructed action. The inclusion and retraction are continuous for the\nproduct topologies, and triviality on a supplied subgroup is preserved.\nNeither continuity nor triviality is inferred for a carrier whose corresponding\nhypothesis has not been supplied. The maps transported by $D$ are proved\nalgebraically. Their topological interpretation additionally requires the\nappropriate continuity of $D$ and its inverse.","For the three-character example below, the construction produces an actual\nthree-dimensional summand inside the carrier of dimension seven. At the sample\n$(0,1)$ its characteristic polynomial is $(X-1)^2(X+1)$. A separate checked\ncounterexample uses the coordinate inclusion $\\mathbf Q\\to\\mathbf Q^2$, its\ncoordinate retraction, and the swap involution: the split law holds, but\ncompression sends the swap to zero and therefore fails to preserve its square.\nThus selecting whole constituent copies, with the proved equivariance, is\nessential to this construction.","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe correspon","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces a","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplici","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon const","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give th","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for t","\\paragraph{Algebraic monodromy data on the selected copies.}\nFix a field $K$ and a nonzero scalar $q\\in K^\\times$. On each constituent\n$W_i$, supply an automorphism $F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for the summand. Whole-space","$F_i$ and a nilpotent endomorphism $N_i$ with\n$F_iN_iF_i^{-1}=qN_i$. The scalar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for the summand. Whole-space Frobenius\npolynomials are not substituted for these kernel polynomials.","alar $q$ is common to this finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for the summand. Whole-space Frobenius\npolynomials are not substituted for these kernel polynomials.","finite family.\nThe componentwise operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for the summand. Whole-space Frobenius\npolynomials are not substituted for these kernel polynomials.","se operators $F_m,N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for the summand. Whole-space Frobenius\npolynomials are not substituted for these kernel polynomials.",",N_m$ on $U_m$ satisfy the same relation.\n$F_m$ is invertible and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for the summand. Whole-space Frobenius\npolynomials are not substituted for these kernel polynomials.","and $N_m$ is nilpotent. The split maps constructed above\nintertwine both operators. The proof first identifies the independently defined\nkernel through an equivariant linear isomorphism\n\\[\n \\ker N_m\\simeq\\prod_{i\\in I}\\prod_{k<m_i}\\ker N_i.\n\\]\nThe relation, with $q\\ne0$, proves that both $F_i$ and its inverse preserve\n$\\ker N_i$. Thus $F_i|_{\\ker N_i}$ is an actual automorphism, and Lean proves\n\\[\n \\det(XI-F_m|_{\\ker N_m})\n   =\\prod_{i\\in I}\\det(XI-F_i|_{\\ker N_i})^{m_i}.\n\\]\nThe corresponding kernel dimensions and traces are additive with multiplicity.\nCommon constituent kernel polynomials, when supplied through a coefficient-ring\nmap, give the mapped product polynomial for the summand. Whole-space Frobenius\npolynomials are not substituted for these kernel polynomials.","The exact special chain has basis $e_0,\\ldots,e_{n-1}$ and operators\n\\[\n Ne_0=0,\\qquad Ne_j=e_{j-1}\\ (j>0),\\qquad Fe_j=aq^{-j}e_j,\n \\qquad a,q\\in K^\\times.\n\\]\nLean proves $N^n=0$, and for $n>0$ also $N^{n-1}\\ne0$ and\n$\\ker N=Ke_0$. Consequently its kernel polynomial is $X-a$, whereas its\nwhole-space polynomial is $\\prod_{j=0}^{n-1}(X-aq^{-j})$.\nThe zero-length chain is included as a zero-dimensional space. The assertion\n$\\ker N=Ke_0$ is made only for positive length.","as basis $e_0,\\ldots,e_{n-1}$ and operators\n\\[\n Ne_0=0,\\qquad Ne_j=e_{j-1}\\ (j>0),\\qquad Fe_j=aq^{-j}e_j,\n \\qquad a,q\\in K^\\times.\n\\]\nLean proves $N^n=0$, and for $n>0$ also $N^{n-1}\\ne0$ and\n$\\ker N=Ke_0$. Consequently its kernel polynomial is $X-a$, whereas its\nwhole-space polynomial is $\\prod_{j=0}^{n-1}(X-aq^{-j})$.\nThe zero-length chain is included as a zero-dimensional space. The assertion\n$\\ker N=Ke_0$ is made only for positive length.","For $n>1$, replacing $N$ by zero retains exactly the same $F$ and the same\nwhole-space characteristic polynomial, while changing the degree of the kernel\npolynomial from one to $n$. Both pairs still satisfy the algebraic monodromy\nrelation. This Lean-checked example shows that whole Frobenius data alone do\nnot determine the monodromy kernel polynomial. It is not asserted to be a\ncounterexample among the intended strictly compatible arithmetic systems.","These statements concern supplied algebraic pairs. No Weil-group or inertia\naction, Hodge--Tate object, arithmetic comparison, or realization of the pair\nfrom a Galois representation is constructed here. The polynomial convention is\n$\\det(XI-F|_{\\ker N})$. No identification with an inertia-dependent local\nEuler factor is asserted. Claim 9.2 remains partial at its full arithmetic scope.","\\paragraph{Checked group, inertia and monodromy finite sums.}\\phantomsection\\label{book:x-construction-12}\nLet $K$ be a field, $G$ a group, and $\\nu:G\\to K^\\times$ a character.\nThe new datum consists of an actual representation $r$ and nilpotent $N$ with\n$r(g)N=\\nu(g)Nr(g)$ for every $g$. For a supplied subgroup $H$, define\n\\[\n E(D,H)=\\ker N\\cap V^H.\n\\]\nThis is the intersection of independently defined kernel and actual invariant\nsubspaces. For normal $H$, the restricted whole-group representation and its\ninverse automorphisms are constructed. A change from $\\varphi$ to $\\varphi h$,\nwith $h\\in H$, leaves the restricted operator itself unchanged.","ms.}\\phantomsection\\label{book:x-construction-12}\nLet $K$ be a field, $G$ a group, and $\\nu:G\\to K^\\times$ a character.\nThe new datum consists of an actual representation $r$ and nilpotent $N$ with\n$r(g)N=\\nu(g)Nr(g)$ for every $g$. For a supplied subgroup $H$, define\n\\[\n E(D,H)=\\ker N\\cap V^H.\n\\]\nThis is the intersection of independently defined kernel and actual invariant\nsubspaces. For normal $H$, the restricted whole-group representation and its\ninverse automorphisms are constructed. A change from $\\varphi$ to $\\varphi h$,\nwith $h\\in H$, leaves the restricted operator itself unchanged.","character.\nThe new datum consists of an actual representation $r$ and nilpotent $N$ with\n$r(g)N=\\nu(g)Nr(g)$ for every $g$. For a supplied subgroup $H$, define\n\\[\n E(D,H)=\\ker N\\cap V^H.\n\\]\nThis is the intersection of independently defined kernel and actual invariant\nsubspaces. For normal $H$, the restricted whole-group representation and its\ninverse automorphisms are constructed. A change from $\\varphi$ to $\\varphi h$,\nwith $h\\in H$, leaves the restricted operator itself unchanged.","he new datum consists of an actual representation $r$ and nilpotent $N$ with\n$r(g)N=\\nu(g)Nr(g)$ for every $g$. For a supplied subgroup $H$, define\n\\[\n E(D,H)=\\ker N\\cap V^H.\n\\]\nThis is the intersection of independently defined kernel and actual invariant\nsubspaces. For normal $H$, the restricted whole-group representation and its\ninverse automorphisms are constructed. A change from $\\varphi$ to $\\varphi h$,\nwith $h\\in H$, leaves the restricted operator itself unchanged.","sts of an actual representation $r$ and nilpotent $N$ with\n$r(g)N=\\nu(g)Nr(g)$ for every $g$. For a supplied subgroup $H$, define\n\\[\n E(D,H)=\\ker N\\cap V^H.\n\\]\nThis is the intersection of independently defined kernel and actual invariant\nsubspaces. For normal $H$, the restricted whole-group representation and its\ninverse automorphisms are constructed. A change from $\\varphi$ to $\\varphi h$,\nwith $h\\in H$, leaves the restricted operator itself unchanged.","For a finite family of finite-dimensional constituents and natural multiplicities,\nthe actual local space of the constructed sum is linearly equivalent to the sum of the constituent local spaces, equivariantly\nfor every group element. Its characteristic polynomials multiply, its dimensions\nand traces add, and maps intertwining both the group and $N$ restrict to the\nlocal spaces. The explicit bounded-copy inclusion and retraction still split\nthere. Finite inertia image and an open inertia kernel are preserved.","Writing $Q_D(T)$ for the reversed local characteristic polynomial, Lean proves\n\\[\n Q_D(T)=\\det(1-T r(\\varphi)\\mid E(D,H)),\\qquad Q_D(0)=1,\n \\qquad\n Q_{\\bigoplus_iD_i^{\\oplus m_i}}(T)=\\prod_iQ_{D_i}(T)^{m_i}.\n\\]\nThe abstract formulas do not need characteristic zero. Normality and finite\ndimension appear explicitly where used. Arithmetic identification of $G,H,\\nu$\nand $\\varphi$ remains separate. In the geometric-Frobenius convention,\n$\\nu(\\varphi)$ is the inverse residue cardinality. It is this unit, rather than\nthe residue cardinality, that enters the earlier special-chain parameter.","l characteristic polynomial, Lean proves\n\\[\n Q_D(T)=\\det(1-T r(\\varphi)\\mid E(D,H)),\\qquad Q_D(0)=1,\n \\qquad\n Q_{\\bigoplus_iD_i^{\\oplus m_i}}(T)=\\prod_iQ_{D_i}(T)^{m_i}.\n\\]\nThe abstract formulas do not need characteristic zero. Normality and finite\ndimension appear explicitly where used. Arithmetic identification of $G,H,\\nu$\nand $\\varphi$ remains separate. In the geometric-Frobenius convention,\n$\\nu(\\varphi)$ is the inverse residue cardinality. It is this unit, rather than\nthe residue cardinality, that enters the earlier special-chain parameter.","An explicit action of $C_2\\times\\mathbb Z$ has $\\nu(\\varphi)=1/3$,\n$F=\\operatorname{diag}(2,6,18)$ and $Ne_1=e_0$, $Ne_2=e_1$, $Ne_0=0$.\nTrivial inertia gives local dimension one and denominator $1-2T$. Scalar sign\ninertia gives dimension zero and denominator one. The entire $F$ and $N$ are\nequal in these two examples, and $N^2\\ne0$. Thus inertia cannot be omitted.\nConversely, unramified chains of lengths one and three have equal Euler\ndenominators while only the first has $N=0$. Euler factors therefore do not\nidentify complete monodromy data.","name{diag}(2,6,18)$ and $Ne_1=e_0$, $Ne_2=e_1$, $Ne_0=0$.\nTrivial inertia gives local dimension one and denominator $1-2T$. Scalar sign\ninertia gives dimension zero and denominator one. The entire $F$ and $N$ are\nequal in these two examples, and $N^2\\ne0$. Thus inertia cannot be omitted.\nConversely, unramified chains of lengths one and three have equal Euler\ndenominators while only the first has $N=0$. Euler factors therefore do not\nidentify complete monodromy data.","The remaining arithmetic comparison is\n$\\operatorname{WD}(\\bigoplus_i V_i^{\\oplus m_i})\\cong\n\\bigoplus_i\\operatorname{WD}(V_i)^{\\oplus m_i}$ for the actual attached functor,\nintertwining the Weil action and monodromy, together with the intended coefficient\nand Hodge--Tate comparisons. These are not supplied by the abstract datum.\nClaim 9.2 remains partial at its full arithmetic scope.","Recovered multiplicities determine the semisimple isomorphism type relative\nto the specified constituents, not a canonical embedding or projector inside\na repeated isotypic component. For example, the two projectors\n$\\operatorname{diag}(1,0)$ and $\\operatorname{diag}(0,1)$ on a two-dimensional\ntrivial representation are distinct equivariant idempotents with trace one.","\\begin{numericalcertificate}[Multiplicity recovery]\\label{px:9.3}\nThe original companion matrix was not retrieved or rerun. A new exact\nrealization uses the trivial character and two independent sign characters\nof $(\\mathbf Z/2\\mathbf Z)^2$, evaluated at $(0,0),(1,0),(0,1)$:\n\\[\n C=\\begin{pmatrix}1&1&1","\\begin{numericalcertificate}[Multiplicity recovery]\\label{px:9.3}\nThe original companion matrix was not retrieved or rerun. A new exact\nrealization uses the trivial character and two independent sign characters\nof $(\\mathbf Z/2\\mathbf Z)^2$, evaluated at $(0,0),(1,0),(0,1)$:\n\\[\n C=\\begin{pmatrix}1&1&1\\\\1&-1&1\\\\1&1&-1\\end{pmatrix},\n \\qquad \\det C=4,\n \\qquad C\\begin{pmatrix}2\\\\0\\\\1\\end{pmatrix}\n       =\\begin{pmatrix}3\\\\3\\\\1\\end{pmatrix}.\n\\]\nLean checks that these are the traces of actual one-dimensional\nrepresentations, recovers $(2,0,1)$ uniquely, rejects a target whose unique\nsolution is $(1/2,0,0)$, and rejects $(4,0,1)$ under capacities $(3,2,2)$.\nThe separator becomes singular modulo two. These are finite-image\nrepresentation checks, not fabricated Frobenius data for an arithmetic carrier.\n\\end{numericalcertificate}","1\\end{pmatrix}.\n\\]\nLean checks that these are the traces of actual one-dimensional\nrepresentations, recovers $(2,0,1)$ uniquely, rejects a target whose unique\nsolution is $(1/2,0,0)$, and rejects $(4,0,1)$ under capacities $(3,2,2)$.\nThe separator becomes singular modulo two. These are finite-image\nrepresentation checks, not fabricated Frobenius data for an arithmetic carrier.\n\\end{numericalcertificate}","\\paragraph{Finite-exponential reconstruction and a cubic inverse.}\\phantomsection\\label{book:x-construction-13}\nLet $K$ be a characteristic-zero field and let $(r,N)$ be supplied group data\nwith nilpotent $N$ and $r(g)N=\\nu(g)Nr(g)$. For a supplied scalar map satisfying\n\\[\nt(gh)=t(g)+\\nu(g)t(h),\n\\]\nthe formalization constructs\n\\[\n\\rho(g)=\\exp(t(g)N)r(g),\\qquad\n\\exp(aN)=\\sum_{j=0}^{d-1}\\frac{a^j}{j!}N^j\\quad(N^d=0).\n\\]\nLean proves the representation laws, actual finite-copy and split-map\ncompatibility, and joint continuity from continuity of $N$, $r$ and $t$ with\nthe stated continuous vector-space operations. The monodromy fixed kernel\nintersected with subgroup invariants is unchanged by this twist. For normal\ninertia, its restricted action and finite-dimensional Euler denominator agree.\nTwisting again by $-","\\paragraph{Finite-exponential reconstruction and a cubic inverse.}\\phantomsection\\label{book:x-construction-13}\nLet $K$ be a characteristic-zero field and let $(r,N)$ be supplied group data\nwith nilpotent $N$ and $r(g)N=\\nu(g)Nr(g)$. For a supplied scalar map satisfying\n\\[\nt(gh)=t(g)+\\nu(g)t(h),\n\\]\nthe formalization constructs\n\\[\n\\rho(g)=\\exp(t(g)N)r(g),\\qquad\n\\exp(aN)=\\sum_{j=0}^{d-1}\\frac{a^j}{j!}N^j\\quad(N^d=0).\n\\]\nLean proves the representation laws, actual finite-copy and split-map\ncompatibility, and joint continuity from continuity of $N$, $r$ and $t$ with\nthe stated continuous vector-space operations. The monodromy fixed kernel\nintersected with subgroup invariants is unchanged by this twist. For normal\ninertia, its restricted action and finite-dimensional Euler denominator agree.\nTwisting again by $-t$ recovers the datum with $N$ still supplied.","ld and let $(r,N)$ be supplied group data\nwith nilpotent $N$ and $r(g)N=\\nu(g)Nr(g)$. For a supplied scalar map satisfying\n\\[\nt(gh)=t(g)+\\nu(g)t(h),\n\\]\nthe formalization constructs\n\\[\n\\rho(g)=\\exp(t(g)N)r(g),\\qquad\n\\exp(aN)=\\sum_{j=0}^{d-1}\\frac{a^j}{j!}N^j\\quad(N^d=0).\n\\]\nLean proves the representation laws, actual finite-copy and split-map\ncompatibility, and joint continuity from continuity of $N$, $r$ and $t$ with\nthe stated continuous vector-space operations. The monodromy fixed kernel\nintersected with subgroup invariants is unchanged by this twist. For normal\ninertia, its restricted action and finite-dimensional Euler denominator agree.\nTwisting again by $-t$ recovers the datum with $N$ still supplied.","d let $(r,N)$ be supplied group data\nwith nilpotent $N$ and $r(g)N=\\nu(g)Nr(g)$. For a supplied scalar map satisfying\n\\[\nt(gh)=t(g)+\\nu(g)t(h),\n\\]\nthe formalization constructs\n\\[\n\\rho(g)=\\exp(t(g)N)r(g),\\qquad\n\\exp(aN)=\\sum_{j=0}^{d-1}\\frac{a^j}{j!}N^j\\quad(N^d=0).\n\\]\nLean proves the representation laws, actual finite-copy and split-map\ncompatibility, and joint continuity from continuity of $N$, $r$ and $t$ with\nthe stated continuous vector-space operations. The monodromy fixed kernel\nintersected with subgroup invariants is unchanged by this twist. For normal\ninertia, its restricted action and finite-dimensional Euler denominator agree.\nTwisting again by $-t$ recovers the datum with $N$ still supplied.","r,N)$ be supplied group data\nwith nilpotent $N$ and $r(g)N=\\nu(g)Nr(g)$. For a supplied scalar map satisfying\n\\[\nt(gh)=t(g)+\\nu(g)t(h),\n\\]\nthe formalization constructs\n\\[\n\\rho(g)=\\exp(t(g)N)r(g),\\qquad\n\\exp(aN)=\\sum_{j=0}^{d-1}\\frac{a^j}{j!}N^j\\quad(N^d=0).\n\\]\nLean proves the representation laws, actual finite-copy and split-map\ncompatibility, and joint continuity from continuity of $N$, $r$ and $t$ with\nthe stated continuous vector-space operations. The monodromy fixed kernel\nintersected with subgroup invariants is unchanged by this twist. For normal\ninertia, its restricted action and finite-dimensional Euler denominator agree.\nTwisting again by $-t$ recovers the datum with $N$ still supplied.",".\n\\]\nLean proves the representation laws, actual finite-copy and split-map\ncompatibility, and joint continuity from continuity of $N$, $r$ and $t$ with\nthe stated continuous vector-space operations. The monodromy fixed kernel\nintersected with subgroup invariants is unchanged by this twist. For normal\ninertia, its restricted action and finite-dimensional Euler denominator agree.\nTwisting again by $-t$ recovers the datum with $N$ still supplied.","tation laws, actual finite-copy and split-map\ncompatibility, and joint continuity from continuity of $N$, $r$ and $t$ with\nthe stated continuous vector-space operations. The monodromy fixed kernel\nintersected with subgroup invariants is unchanged by this twist. For normal\ninertia, its restricted action and finite-dimensional Euler denominator agree.\nTwisting again by $-t$ recovers the datum with $N$ still supplied.","The \\hyperref[book:x-construction-13]{finite-exponential reconstruction} also established an extracted inverse on the explicit class $N^3=0$:\n\\[\n\\log_2 U=(U-I)-\\tfrac12(U-I)^2,\\qquad\n a^{-1}\\log_2(\\exp(aN))=N\\quad(a\\ne0).\n\\]\nBoth exponential/logarithmic inverse identities are checked on the cubic\nnilpotent and unipotent loci. If a supplied common sample $u$ has\n$r_D(u)=r_E(u)=I$ and $t(u)\\ne0$, an intertwiner of the reconstructed cubic\nactions recovers both original group equivariance and monodromy equivariance.\nThe general nilpotence exponential theorem is not restricted to this cubic\nclass, but the extracted inverse in the \\hyperref[book:x-construction-13]{finite-exponential reconstruction} was. The \\hyperref[book:x-construction-20]{finite logarithmic recovery argument} below removes\nthat restriction by using the full finite","The \\hyperref[book:x-construction-13]{finite-exponential reconstruction} also established an extracted inverse on the explicit class $N^3=0$:\n\\[\n\\log_2 U=(U-I)-\\tfrac12(U-I)^2,\\qquad\n a^{-1}\\log_2(\\exp(aN))=N\\quad(a\\ne0).\n\\]\nBoth exponential/logarithmic inverse identities are checked on the cubic\nnilpotent and unipotent loci. If a supplied common sample $u$ has\n$r_D(u)=r_E(u)=I$ and $t(u)\\ne0$, an intertwiner of the reconstructed cubic\nactions recovers both original group equivariance and monodromy equivariance.\nThe general nilpotence exponential theorem is not restricted to this cubic\nclass, but the extracted inverse in the \\hyperref[book:x-construction-13]{finite-exponential reconstruction} was. The \\hyperref[book:x-construction-20]{finite logarithmic recovery argument} below removes\nthat restriction by using the full finite logarithm.","The rational affine-group example has $u=(1,1)$, $d=(0,1/3)$ and\n\\[\n\\rho(u)=\\begin{pmatrix}1&1&1/2","The rational affine-group example has $u=(1,1)$, $d=(0,1/3)$ and\n\\[\n\\rho(u)=\\begin{pmatrix}1&1&1/2\\\\0&1&1\\\\0&0&1\\end{pmatrix},\n\\qquad \\rho(d)=\\operatorname{diag}(1,3,9).\n\\]\nReversing the factor order or dropping the quadratic term gives checked\nfailures. For the length-four chain, the quadratic logarithm has erroneous\n$e_3$-to-$e_0$ coefficient $-1/3$, so its cubic restriction is substantive.\nA zero-parameter sample also fails to recover $N$.","0&0&1\\end{pmatrix},\n\\qquad \\rho(d)=\\operatorname{diag}(1,3,9).\n\\]\nReversing the factor order or dropping the quadratic term gives checked\nfailures. For the length-four chain, the quadratic logarithm has erroneous\n$e_3$-to-$e_0$ coefficient $-1/3$, so its cubic restriction is substantive.\nA zero-parameter sample also fails to recover $N$.","No arithmetic group, tame cocycle or required sample is constructed by these\nstatements. The attached comparison for every intended representation,\ncoefficient comparisons and Hodge--Tate/period data remain separate. Finite\ninertia image and an open inertia kernel of $r$ are not asserted to persist\nunder this twist to $\\rho$. Claim 9.2 remains partial.","Let \\(E\\) be a field of characteristic zero and let \\(T_1,\\ldots,T_N\\) be commuting operators on a finite-dimensional \\(E\\)-vector space \\(A\\), each diagonalizable over \\(E\\).  Thus the selected spectra split over the coefficient field.  Let \\(a_i\\in E\\) be a selected target scalar for \\(T_i\\), possibly absent from its spectrum.  Define\n\\[\ne_i\n=\n\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a_i\\}}\n\\frac{T_i-\\mu I}{a_i-\\mu},\n\\qquad\ne_T=\\prod_{i=1}^{N}e_i,\n\\]\nwith \\(e_i=0\\) if \\(a_i\\) is absent.",",\\ldots,T_N\\) be commuting operators on a finite-dimensional \\(E\\)-vector space \\(A\\), each diagonalizable over \\(E\\).  Thus the selected spectra split over the coefficient field.  Let \\(a_i\\in E\\) be a selected target scalar for \\(T_i\\), possibly absent from its spectrum.  Define\n\\[\ne_i\n=\n\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a_i\\}}\n\\frac{T_i-\\mu I}{a_i-\\mu},\n\\qquad\ne_T=\\prod_{i=1}^{N}e_i,\n\\]\nwith \\(e_i=0\\) if \\(a_i\\) is absent.","be commuting operators on a finite-dimensional \\(E\\)-vector space \\(A\\), each diagonalizable over \\(E\\).  Thus the selected spectra split over the coefficient field.  Let \\(a_i\\in E\\) be a selected target scalar for \\(T_i\\), possibly absent from its spectrum.  Define\n\\[\ne_i\n=\n\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a_i\\}}\n\\frac{T_i-\\mu I}{a_i-\\mu},\n\\qquad\ne_T=\\prod_{i=1}^{N}e_i,\n\\]\nwith \\(e_i=0\\) if \\(a_i\\) is absent.","\\begin{theorem}[Exact finite occupancy trace]\\label{px:10.1}\nThe operator \\(e_T\\) is the projector onto the joint eigenspace\n\\[\nA_T\n=\n\\{v:T_iv=a_iv\\text{ for every }i\\}.\n\\]\nMoreover,\n\\[\n\\boxed{\n\\Tr(e_T)=\\rank(e_T)=\\dim A_T\\in\\Z_{\\ge0}.\n}\n\\]\nThe trace is identified with its unique nonnegative integer preimage under \\(\\Z\\hookrightarrow E\\).  Thus \\(\\Tr(e_T)>0\\) refers to that integer, not to an ordering of an arbitrary field \\(E\\).  The finite packet is occupied exactly when this integer is positive.  In positive characteristic the field-valued trace can vanish for a nonzero projector. The rank test, not a positive field trace, must then be used.\n\\end{theorem}","occupancy trace]\\label{px:10.1}\nThe operator \\(e_T\\) is the projector onto the joint eigenspace\n\\[\nA_T\n=\n\\{v:T_iv=a_iv\\text{ for every }i\\}.\n\\]\nMoreover,\n\\[\n\\boxed{\n\\Tr(e_T)=\\rank(e_T)=\\dim A_T\\in\\Z_{\\ge0}.\n}\n\\]\nThe trace is identified with its unique nonnegative integer preimage under \\(\\Z\\hookrightarrow E\\).  Thus \\(\\Tr(e_T)>0\\) refers to that integer, not to an ordering of an arbitrary field \\(E\\).  The finite packet is occupied exactly when this integer is positive.  In positive characteristic the field-valued trace can vanish for a nonzero projector. The rank test, not a positive field trace, must then be used.\n\\end{theorem}","\\begin{proof}\nFor each operator, its finite set of actual eigenvalues is used without\nmultiplicity. The polynomial\n\\[\n p_{i,a}(X)=\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a\\}}\n (a-\\mu)^{-1}(X-\\mu)\n\\]\nis used when $a$ occurs, and the zero polynomial otherwise. For an eigenvector\n$T_i v=bv$, evaluation gives\n\\[\n p_{i,a}(T_i)v=\\begin{cases}v&b=a,","\\begin{proof}\nFor each operator, its finite set of actual eigenvalues is used without\nmultiplicity. The polynomial\n\\[\n p_{i,a}(X)=\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a\\}}\n (a-\\mu)^{-1}(X-\\mu)\n\\]\nis used when $a$ occurs, and the zero polynomial otherwise. For an eigenvector\n$T_i v=bv$, evaluation gives\n\\[\n p_{i,a}(T_i)v=\\begin{cases}v&b=a,\\\\0&b\\ne a.\\end{cases}\n\\]\nBecause the eigenspaces span $A$, this proves that $e_i=p_{i,a_i}(T_i)$ maps\nall of $A$ into the selected eigenspace and fixes that eigenspace pointwise.\nIt therefore proves the individual projector and image assertions, including\nan absent target.","0&b\\ne a.\\end{cases}\n\\]\nBecause the eigenspaces span $A$, this proves that $e_i=p_{i,a_i}(T_i)$ maps\nall of $A$ into the selected eigenspace and fixes that eigenspace pointwise.\nIt therefore proves the individual projector and image assertions, including\nan absent target.","Polynomials in commuting operators commute. If projections $P,Q$ onto\n$U,W$ commute, then $PQ$ maps into both $U$ and $W$, and fixes every vector\nin $U\\cap W$. Induction proves that the actual ordered product $e_T$ is a\nprojection onto $A_T$. Commutation also makes the result independent of the\nordering. An empty list of conditions gives the identity.","For a projection onto a subspace, the direct sum of its image and kernel\nidentifies it with $I\\oplus0$. Thus its trace in $E$ is the natural dimension\nof its image cast into $E$. In characteristic zero this cast is injective,\nso there is a unique natural integer represented by the trace. That integer\nis positive exactly when the joint eigenspace contains a nonzero vector.\n\\end{proof}","\\paragraph{Naturality and finite hypotheses.}\nIf $F T_i=U_i F$ and the source operators split, then\n$F e_{T,a}=e_{U,a}F$. This follows first on source eigenvectors and then on\ntheir span. In particular, a change of basis conjugates the constructed\nprojector and preserves its trace. Adding conditions decreases the actual\njoint eigenspace and cannot increase its dimension. These statements do not\nassume an orthonormal basis or a positive definite inner product.","\\paragraph{A specified finite realization.}\nOn $\\Q^4$, take the diagonal operators with diagonal lists\n\\[\n(1,1,1,0),\\quad(2,2,0,0),\\quad(3,0,0,0),\\quad(0,4,4,4)\n\\]\nand target scalars $(1,2,3,4)$. All four targets occur individually. Imposing\nthese conditions successively gives actual projector ranks $3,2,1,0$.\nThe last packet is empty despite individual feasibility. The formal code\nchecks these operators, projector formulas, traces and dimensions. This is\na newly specified realization, not a claim to have recovered the original\ncompanion implementation.","\\paragraph{Why the hypotheses cannot be omitted.}\nFor\n\\[\n P=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\\qquad\n Q=\\tfrac12\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix},\n\\]\neach operator is idempotent, but $PQ$ is not idempotent and has trace $1/2$.\nFor the Jordan block $J=\\left(\\begin{smallmatrix}1&1","1&1\\end{pmatrix},\n\\]\neach operator is idempotent, but $PQ$ is not idempotent and has trace $1/2$.\nFor the Jordan block $J=\\left(\\begin{smallmatrix}1&1","1&1\\end{pmatrix},\n\\]\neach operator is idempotent, but $PQ$ is not idempotent and has trace $1/2$.\nFor the Jordan block $J=\\left(\\begin{smallmatrix}1&1\\\\0&1\\end{smallmatrix}\\right)$,\nthe singleton-spectrum Lagrange polynomial is $1$, so the formula produces\nthe identity although the eigenspace at $1$ is only a line. Both failures\nare exact formal counterexamples to dropping the corresponding hypotheses,\nnot counterexamples to the theorem with those hypotheses retained.","0&1\\end{smallmatrix}\\right)$,\nthe singleton-spectrum Lagrange polynomial is $1$, so the formula produces\nthe identity although the eigenspace at $1$ is only a line. Both failures\nare exact formal counterexamples to dropping the corresponding hypotheses,\nnot counterexamples to the theorem with those hypotheses retained.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.77\\linewidth]{15_finite_occupancy_ranks.pdf}\n\\caption{A consistent sequence of finite conditions narrows the carrier from ranks \\(3\\) to \\(2\\) to \\(1\\).  One inconsistent condition forces the joint projector to zero.}\n\\end{figure}","\\begin{corollary}[Exact location of the remaining existence problem]\\label{px:11.2}\nInverse-limit reconstruction alone is not a nonemptiness theorem.  The existence requirements must retain both residual companions and liftability through every finite depth:\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{companion existence: }&X_{\\lambda,1}(\\Compat)\\ne\\varnothing\n\\text{ for every }\\lambda;\\\\\n\\text{finite-depth liftability: }&X_{\\lambda,m}(\\Compat)\\ne\\varnothing\n\\text{ for every }\\lambda,m\\ge2;\\\\\n\\text{finite occupancy: }&\\Tr(e_T)>0\n\\text{ for every required global packet.}\n\\end{aligned}}\n\\]\nFormal reconstruction does not create a missing residual root, repair an empty higher level, or force global occupancy.  Nonempty finite levels with finite branching supply a branch, after which inverse-limit closure reconstructs an admissible object.\n\\end{corollary}","\\paragraph{An occupied node can have no child.}\nThe actual order-two representation with generator\n\\[\n A=\\begin{pmatrix}1&2\\\\2&3\\end{pmatrix}\\pmod4\n\\]\nsatisfies $A^2=I$ and $\\det(XI-A)=X^2-1$. It has no lift modulo $8$ with\nthe same characteristic polynomial. Indeed, every lift has entries congruent\nto $1,2,2,3$ modulo $4$. Trace zero then forces its determinant to be $3$\nmodulo $8$, whereas $X^2-1$ requires determinant $-1=7$ modulo $8$.\nBoth the occupied node and the absence of any next-level matrix or representation\nwith the prescribed polynomial are Lean-checked. Other branches of the same\nunrestricted involution forest survive, so this example does not contradict\nfinite-branch compactness.","2&3\\end{pmatrix}\\pmod4\n\\]\nsatisfies $A^2=I$ and $\\det(XI-A)=X^2-1$. It has no lift modulo $8$ with\nthe same characteristic polynomial. Indeed, every lift has entries congruent\nto $1,2,2,3$ modulo $4$. Trace zero then forces its determinant to be $3$\nmodulo $8$, whereas $X^2-1$ requires determinant $-1=7$ modulo $8$.\nBoth the occupied node and the absence of any next-level matrix or representation\nwith the prescribed polynomial are Lean-checked. Other branches of the same\nunrestricted involution forest survive, so this example does not contradict\nfinite-branch compactness.","\\paragraph{Local properties need their own limit argument.}\nOver a domain, for a nonzero $\\pi\\in I$, the representations of the additive\ninteger group given by\n\\[\n \\rho_n(g)=\\begin{pmatrix}1&\\pi^n g","\\paragraph{Local properties need their own limit argument.}\nOver a domain, for a nonzero $\\pi\\in I$, the representations of the additive\ninteger group given by\n\\[\n \\rho_n(g)=\\begin{pmatrix}1&\\pi^n g\\\\0&1\\end{pmatrix}\n\\]\nare all nontrivial and converge pointwise in the ideal-adic topology to the\ntrivial representation. The convergence and the failure of limit preservation\nfor nontriviality are formally checked. Thus not every property of approximating\nintegral representations can be asserted for a limit. This example is not\nclaimed to disprove stability for any particular intended Galois local condition.","For each supplied required packet $p$, let $A_p$ be its finite-dimensional carrier\nover a characteristic-zero field $E_p$, and let $e_p$ be the actual joint spectral\nprojector for its finite list of split commuting conditions. No order on $E_p$\nis assumed. The source notation $\\Tr(e_p)>0$ means that the unique natural number\n$n_p$ with $\\Tr(e_p)=(n_p:E_p)$ is positive. The checked projector formula identifies\n$n_p$ with the dimension of the actual joint eigenspace.","ied required packet $p$, let $A_p$ be its finite-dimensional carrier\nover a characteristic-zero field $E_p$, and let $e_p$ be the actual joint spectral\nprojector for its finite list of split commuting conditions. No order on $E_p$\nis assumed. The source notation $\\Tr(e_p)>0$ means that the unique natural number\n$n_p$ with $\\Tr(e_p)=(n_p:E_p)$ is positive. The checked projector formula identifies\n$n_p$ with the dimension of the actual joint eigenspace.","An admissible realization here includes continuous coefficient-field members and\nmembership of every finite reduction in its designated admissible class.\nUnder the completion hypotheses and these supplied carrier hypotheses, Lean proves\n\\[\n\\begin{aligned}\n&\\exists\\,\\rho,\\ (x_p)_p:\\quad\n  \\rho\\text{ is an admissible realization of the full chosen spine},\\quad\n  0\\ne x_p\\in\\operatorname{im}(e_p)\\text{ for every }p\\\\\n&\\quad\\Longleftrightarrow\\quad\n \\bigl(\\forall\\lambda,\\ X_{\\lambda,1}(\\Compat)\\ne\\varnothing\\bigr)\n \\ \\land\\ \\bigl(\\forall\\lambda,\\ \\forall m\\ge2,\n               X_{\\lambda,m}(\\Compat)\\ne\\varnothing\\bigr)\n \\ \\land\\ \\bigl(\\forall p,\\ n_p>0\\bigr).\n\\end{aligned}\n\\]\nThe packet index may be infinite. Each $x_p$ belongs to its own supplied carrier.\nNo common vector across distinct carriers or embedding of the reconstructed\nsystem into those carriers is asserted. Neither an unknown carrier nor positive\ntrace is constructed without the corresponding hypotheses.","&\\quad\\Longleftrightarrow\\quad\n \\bigl(\\forall\\lambda,\\ X_{\\lambda,1}(\\Compat)\\ne\\varnothing\\bigr)\n \\ \\land\\ \\bigl(\\forall\\lambda,\\ \\forall m\\ge2,\n               X_{\\lambda,m}(\\Compat)\\ne\\varnothing\\bigr)\n \\ \\land\\ \\bigl(\\forall p,\\ n_p>0\\bigr).\n\\end{aligned}\n\\]\nThe packet index may be infinite. Each $x_p$ belongs to its own supplied carrier.\nNo common vector across distinct carriers or embedding of the reconstructed\nsystem into those carriers is asserted. Neither an unknown carrier nor positive\ntrace is constructed without the corresponding hypotheses.","\\paragraph{Every required packet is necessary.}\nFor each omitted index $N\\in\\mathbb N$, consider one-dimensional rational carriers\nwith the identity operator, requiring eigenvalue $0$ at $N$ and eigenvalue $1$\nat every other index. The actual spectral projector has trace $0$ at $N$ and\ntrace $1$ elsewhere. Lean proves that any predetermined finite sample of packet\nindices can miss an empty required packet: choose $N$ outside that sample.\nThe family depends on the omitted index. This does not assert that one fixed\nfamily passes every finite prefix while failing global occupancy.","equired packet is necessary.}\nFor each omitted index $N\\in\\mathbb N$, consider one-dimensional rational carriers\nwith the identity operator, requiring eigenvalue $0$ at $N$ and eigenvalue $1$\nat every other index. The actual spectral projector has trace $0$ at $N$ and\ntrace $1$ elsewhere. Lean proves that any predetermined finite sample of packet\nindices can miss an empty required packet: choose $N$ outside that sample.\nThe family depends on the omitted index. This does not assert that one fixed\nfamily passes every finite prefix while failing global occupancy.","For \\(q=3\\) and formal dimension \\(d=2\\), the finite levels have cardinalities\n\\[\n1,9,81,729,6561,\\ldots.\n\\]\nThe seed zeta function is\n\\[\nZ(T)=\\frac{T}{1-9T}.\n\\]","For weights \\((1,2,4)\\), the counts are\n\\[\n1,2,8,32,256,2048,\\ldots.\n\\]\nThe first coordinate is active immediately, the second one level later, and the third at depth five.  The exact rational function generated by the code is\n\\[\nZ_{(1,2,4)}(T)\n=\n\\frac{T(1-6T-8T^2-32T^3)}{1-8T}.\n\\]\nThe negative signs in this presentation arise from moving the denominator to \\(1-8T\\). Every power-series coefficient is positive.","For \\(x^2=0\\) over a residue field of size three, the counts are\n\\[\n1,3,3,9,9,27,27,81,\\ldots,\n\\]\nand\n\\[\nZ(T)=\\frac{T+3T^2}{1-3T^2}.\n\\]\nThe exact branching dimension is \\(1/2\\).","Recovered carrier multiplicities & \\((2,0,1)\\)","Finite occupancy ranks & \\((3,2,1,0)\\)","The complete automated suite contains \\(28\\) tests.  They cover residual partition, no crossing, inverse-limit digits, weighted and singular zeta coefficients, directed source components, cohomology packets, companion forests, common-spine consistency, finite Frobenius separation, multiplicity recovery, and exact occupancy projectors.","Singular power zeta and dimension & proved & scalar equation \\(x^e=0\\)","The residual seed theory becomes precise once vertical and horizontal generation are separated.  Positive-depth perturbations can be complete within one formal component.  They cannot change the residual root.  The universal arithmetic seed is therefore a forest rather than one branch:\n\\[\n\\boxed{\n\\text{residual roots}\n\\longrightarrow\n\\text{finite deformation levels}\n\\longrightarrow\n\\text{inverse-limit branches}.\n}\n\\]","The residual module partitions coordinates modulo \\(p\\), verifies compatibility\n\\[\nx_{m+1}\\equiv x_m\\pmod{p^m},\n\\]\nand reconstructs the digit \\(a_{m-1}\\) by\n\\[\na_{m-1}=\\left\\lfloor\\frac{x_m}{p^{m-1}}\\right\\rfloor\\pmod p.\n\\]","For canonical representatives $0\\le x_m<p^m$, set $x_0=0$. Compatibility also gives\n$a_{m-1}=(x_m-x_{m-1})/p^{m-1}\\in\\{0,\\ldots,p-1\\}$.\nThe floor in the first formula is essential. For example, the compatible\nbase-five residues $4,19$ give the second digit $3$, not the rational number $19/5$.\nThe executable companion uses the same integer floor division.","For an integral polynomial \\(f\\), the finite root forest is obtained by exact enumeration of\n\\[\nf(x)\\equiv0\\pmod{p^m}\n\\]\nand by connecting a child \\(x_{m+1}\\) to its reduction modulo \\(p^m\\).  This is intended for small exact examples, not large deformation rings.","Modular Gaussian elimination computes the ranks of the differentials.  For\n\\[\nC^{i-1}\\xrightarrow{d_{i-1}}C^i\\xrightarrow{d_i}C^{i+1},\n\\]\nthe finite dimension is evaluated as\n\\[\n\\dim H^i\n=\n\\dim C^i-\\rank d_i-\\rank d_{i-1}.\n\\]\nPacket subcomplexes are computed separately and checked against the total.","\\paragraph{Finite-group comparison with precision loss.}\\phantomsection\\label{book:x-construction-17}\nLet $R$ be a commutative domain, $\\Gamma$ a finite group, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finit","\\paragraph{Finite-group comparison with precision loss.}\\phantomsection\\label{book:x-construction-17}\nLet $R$ be a commutative domain, $\\Gamma$ a finite group, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this","\\paragraph{Finite-group comparison with precision loss.}\\phantomsection\\label{book:x-construction-17}\nLet $R$ be a commutative domain, $\\Gamma$ a finite group, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","mparison with precision loss.}\\phantomsection\\label{book:x-construction-17}\nLet $R$ be a commutative domain, $\\Gamma$ a finite group, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","precision loss.}\\phantomsection\\label{book:x-construction-17}\nLet $R$ be a commutative domain, $\\Gamma$ a finite group, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","a commutative domain, $\\Gamma$ a finite group, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","\\Gamma$ a finite group, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","p, and $\\rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","rho,\\tau$\nactual integral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","tegral matrix representations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","entations. Suppose $a=|\\Gamma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","ma|\\ne0$ in $R$ and\n$a\\mid\\pi^e$. An equivariant matrix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","trix modulo $(\\pi)^{e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","e+k}$ lifts to an\nintegral equivariant matrix with the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","h the prescribed reduction modulo $(\\pi)^k$.\nIndeed, lift its entries to $A$ and form\n\\[\nS=\\sum_{g\\in\\Gamma}\\rho(g)A\\tau(g^{-1}).\n\\]\nThe finite equations imply $S-aA=\\pi^{e+k}E$ with $E$ integral. Choose $b$\nwith $\\pi^e=ab$ and put $P=A+b\\pi^kE$. Then $aP=S$, so cancellation of\n$a\\ne0$ proves equivariance. Over a local ring, if $\\pi$ is a nonunit,\n$k>0$ and the supplied comparison is invertible, $P$ is an integral unit.\nNeither completeness nor a finite residue field is needed for this step.","For a DVR, $e=v_\\pi(|\\Gamma|)$ is derived from the actual group order.\nThe single precision $e+1$, independent of rank, detects integral\nisomorphism. It makes reduction injective on integral isomorphism classes,\nand finite coefficient quotients give finitely many integral classes in\nfixed rank. This is a formalization of classical finite-group rigidity\n(Maranda, 1953, Theorem~2), with an explicit proof of the comparison's\nprecision loss.","For complete local coefficients with finite quotients, downward-compatible\nallowed representations occupied at every precision, and the required\nfinite-group type at depth $e+1$, reconstruction followed by this comparison\nproduces the required integral representation. Continuity is proved for a\nfinite discrete source and for pullback through a fixed continuous map to\nthat source. Occupancy at only one precision is insufficient. The result\nlifts comparisons between actual integral representations. It does not lift\narbitrary quotient representations or supply the missing arithmetic packet\ninterpretation.","For $C_2$ over $\\mathbb Z_2$, take\n\\[\n\\tau(g)=\\begin{pmatrix}1&0","For $C_2$ over $\\mathbb Z_2$, take\n\\[\n\\tau(g)=\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix},\\qquad\n\\rho(g)=\\begin{pmatrix}1&2\\\\0&-1\\end{pmatrix},\\qquad\nA=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}.\n\\]\nThe matrix $A$ intertwines modulo $4$ and its average is\n$P=\\left(\\begin{smallmatrix}1&-1","0&1\\end{pmatrix}.\n\\]\nThe matrix $A$ intertwines modulo $4$ and its average is\n$P=\\left(\\begin{smallmatrix}1&-1","0&1\\end{pmatrix}.\n\\]\nThe matrix $A$ intertwines modulo $4$ and its average is\n$P=\\left(\\begin{smallmatrix}1&-1\\\\0&1\\end{smallmatrix}\\right)$.\nNo exact intertwiner can retain $A$ modulo $4$, although $P$ retains it\nmodulo $2$. The actual trivial and sign representations also agree modulo\n$2$ but are not integrally isomorphic. These checked examples show why\nresidual-only detection and full preservation of the input precision cannot\nbe asserted here.","0&1\\end{smallmatrix}\\right)$.\nNo exact intertwiner can retain $A$ modulo $4$, although $P$ retains it\nmodulo $2$. The actual trivial and sign representations also agree modulo\n$2$ but are not integrally isomorphic. These checked examples show why\nresidual-only detection and full preservation of the input precision cannot\nbe asserted here.","\\paragraph{Lifting a summand inside a supplied finite-group carrier.}\\phantomsection\\label{book:x-construction-18}\nLet $R$ be a commutative integral domain complete and separated for the\n$(\\pi)$-adic filtration. Let $\\Gamma$ be a finite group, and let\n$\\rho:\\Gamma\\to\\mathrm{GL}_n(R)$ be an actual integral representation.\nAssume that $a=|\\Gamma|\\ne0$ in $R$ and $a\\mid\\pi^e$. For $k>0$, suppose\n\\[\n\\bar E\\in M_n\\bigl(R/(\\pi)^{e+k}\\bigr),\\qquad\n\\bar E^2=\\bar E,\\qquad\n\\bar\\rho(g)\\bar E=\\bar E\\bar\\rho(g)\\quad(g\\in\\Gamma).\n\\]\nThere is an actual integral matrix $E$ satisfying\n\\[\nE^2=E,\\qquad \\rho(g)E=E\\rho(g),\\qquad\nE\\bmod(\\pi)^k=\\bar E\\bmod(\\pi)^k.\n\\]\nNo integral summand or coherent family of quotient projectors is assumed.\nNo local-ring assumption or finiteness of the residue field is needed.","ete and separated for the\n$(\\pi)$-adic filtration. Let $\\Gamma$ be a finite group, and let\n$\\rho:\\Gamma\\to\\mathrm{GL}_n(R)$ be an actual integral representation.\nAssume that $a=|\\Gamma|\\ne0$ in $R$ and $a\\mid\\pi^e$. For $k>0$, suppose\n\\[\n\\bar E\\in M_n\\bigl(R/(\\pi)^{e+k}\\bigr),\\qquad\n\\bar E^2=\\bar E,\\qquad\n\\bar\\rho(g)\\bar E=\\bar E\\bar\\rho(g)\\quad(g\\in\\Gamma).\n\\]\nThere is an actual integral matrix $E$ satisfying\n\\[\nE^2=E,\\qquad \\rho(g)E=E\\rho(g),\\qquad\nE\\bmod(\\pi)^k=\\bar E\\bmod(\\pi)^k.\n\\]\nNo integral summand or coherent family of quotient projectors is assumed.\nNo local-ring assumption or finiteness of the residue field is needed.","d for the\n$(\\pi)$-adic filtration. Let $\\Gamma$ be a finite group, and let\n$\\rho:\\Gamma\\to\\mathrm{GL}_n(R)$ be an actual integral representation.\nAssume that $a=|\\Gamma|\\ne0$ in $R$ and $a\\mid\\pi^e$. For $k>0$, suppose\n\\[\n\\bar E\\in M_n\\bigl(R/(\\pi)^{e+k}\\bigr),\\qquad\n\\bar E^2=\\bar E,\\qquad\n\\bar\\rho(g)\\bar E=\\bar E\\bar\\rho(g)\\quad(g\\in\\Gamma).\n\\]\nThere is an actual integral matrix $E$ satisfying\n\\[\nE^2=E,\\qquad \\rho(g)E=E\\rho(g),\\qquad\nE\\bmod(\\pi)^k=\\bar E\\bmod(\\pi)^k.\n\\]\nNo integral summand or coherent family of quotient projectors is assumed.\nNo local-ring assumption or finiteness of the residue field is needed.","To prove this, apply the preceding averaging correction with both\nrepresentations equal to $\\rho$. This produces an exact equivariant matrix\n$A$ with the prescribed reduction at depth $k$, so that\n$A^2-A\\in(\\pi)^kM_n(R)$. Averaging alone need not preserve idempotence.\nUse the integral polynomial\n\\[\nf(A)=3A^2-2A^3.\n\\]\nThe identities\n\\[\nf(A)-A=(1-2A)(A^2-A),\\qquad\nf(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3)\n\\]\nshow that successive corrections preserve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ c","To prove this, apply the preceding averaging correction with both\nrepresentations equal to $\\rho$. This produces an exact equivariant matrix\n$A$ with the prescribed reduction at depth $k$, so that\n$A^2-A\\in(\\pi)^kM_n(R)$. Averaging alone need not preserve idempotence.\nUse the integral polynomial\n\\[\nf(A)=3A^2-2A^3.\n\\]\nThe identities\n\\[\nf(A)-A=(1-2A)(A^2-A),\\qquad\nf(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3)\n\\]\nshow that successive corrections preserve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$,","To prove this, apply the preceding averaging correction with both\nrepresentations equal to $\\rho$. This produces an exact equivariant matrix\n$A$ with the prescribed reduction at depth $k$, so that\n$A^2-A\\in(\\pi)^kM_n(R)$. Averaging alone need not preserve idempotence.\nUse the integral polynomial\n\\[\nf(A)=3A^2-2A^3.\n\\]\nThe identities\n\\[\nf(A)-A=(1-2A)(A^2-A),\\qquad\nf(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3)\n\\]\nshow that successive corrections preserve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$, so the correction also\nworks in residue cha","To prove this, apply the preceding averaging correction with both\nrepresentations equal to $\\rho$. This produces an exact equivariant matrix\n$A$ with the prescribed reduction at depth $k$, so that\n$A^2-A\\in(\\pi)^kM_n(R)$. Averaging alone need not preserve idempotence.\nUse the integral polynomial\n\\[\nf(A)=3A^2-2A^3.\n\\]\nThe identities\n\\[\nf(A)-A=(1-2A)(A^2-A),\\qquad\nf(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3)\n\\]\nshow that successive corrections preserve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$, so the correction also\nworks in residue characteristic $2$.","qquad\nf(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3)\n\\]\nshow that successive corrections preserve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$, so the correction also\nworks in residue characteristic $2$.","4A-3)\n\\]\nshow that successive corrections preserve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$, so the correction also\nworks in residue characteristic $2$.","successive corrections preserve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$, so the correction also\nworks in residue characteristic $2$.","rve the retained reduction and improve\nthe idempotence defect. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$, so the correction also\nworks in residue characteristic $2$.","t. In these formulas scalar constants denote scalar\nmatrices. The Lean proof establishes at least one additional ideal digit at\neach step, constructs the resulting compatible family of finite-level matrices, and uses\ncompleteness and separatedness to recover $E$. Every matrix commuting with\n$A$ commutes with $E$. There is no division by $2$, so the correction also\nworks in residue characteristic $2$.","The actual submodules $U=\\operatorname{im}E$ and $W=\\ker E$ are invariant,\nand addition gives an equivariant isomorphism\n\\[\nU\\oplus W\\simeq R^n.\n\\]\nThe inclusion $i:U\\hookrightarrow R^n$ and retraction $r(v)=Ev$ are\nconstructed as linear maps and satisfy $ri=1_U$ and $ir=E$. Their\nequivariance is proved. Coordinatewise reduction maps $U$ onto\n$\\operatorname{im}(\\bar E\\bmod(\\pi)^k)$ and maps $W$ onto\n$\\ker(\\bar E\\bmod(\\pi)^k)$. Both equalities are surjectivity statements\nonto the actual submodules over the finite-level quotient, not merely containment. The same\nprojector defines a summand after pullback through any group homomorphism\nto $\\Gamma$.","actual submodules $U=\\operatorname{im}E$ and $W=\\ker E$ are invariant,\nand addition gives an equivariant isomorphism\n\\[\nU\\oplus W\\simeq R^n.\n\\]\nThe inclusion $i:U\\hookrightarrow R^n$ and retraction $r(v)=Ev$ are\nconstructed as linear maps and satisfy $ri=1_U$ and $ir=E$. Their\nequivariance is proved. Coordinatewise reduction maps $U$ onto\n$\\operatorname{im}(\\bar E\\bmod(\\pi)^k)$ and maps $W$ onto\n$\\ker(\\bar E\\bmod(\\pi)^k)$. Both equalities are surjectivity statements\nonto the actual submodules over the finite-level quotient, not merely containment. The same\nprojector defines a summand after pullback through any group homomorphism\nto $\\Gamma$.","d satisfy $ri=1_U$ and $ir=E$. Their\nequivariance is proved. Coordinatewise reduction maps $U$ onto\n$\\operatorname{im}(\\bar E\\bmod(\\pi)^k)$ and maps $W$ onto\n$\\ker(\\bar E\\bmod(\\pi)^k)$. Both equalities are surjectivity statements\nonto the actual submodules over the finite-level quotient, not merely containment. The same\nprojector defines a summand after pullback through any group homomorphism\nto $\\Gamma$.","For $R=\\mathbb Z_2$ and the sign carrier\n$\\rho(s)=\\operatorname{diag}(1,-1)$, the projector\n\\[\n\\bar E=\\begin{pmatrix}1&2","For $R=\\mathbb Z_2$ and the sign carrier\n$\\rho(s)=\\operatorname{diag}(1,-1)$, the projector\n\\[\n\\bar E=\\begin{pmatrix}1&2\\\\0&0\\end{pmatrix}\\pmod4\n\\]\nis equivariant and the theorem constructs a nonzero proper equivariant\nprojector retaining its reduction modulo $2$. No exact equivariant\nendomorphism retains this prescribed matrix modulo $4$. At the insufficient\ninput depth $1$, the projector\n$\\left(\\begin{smallmatrix}1&1","0&0\\end{pmatrix}\\pmod4\n\\]\nis equivariant and the theorem constructs a nonzero proper equivariant\nprojector retaining its reduction modulo $2$. No exact equivariant\nendomorphism retains this prescribed matrix modulo $4$. At the insufficient\ninput depth $1$, the projector\n$\\left(\\begin{smallmatrix}1&1","0&0\\end{pmatrix}\\pmod4\n\\]\nis equivariant and the theorem constructs a nonzero proper equivariant\nprojector retaining its reduction modulo $2$. No exact equivariant\nendomorphism retains this prescribed matrix modulo $4$. At the insufficient\ninput depth $1$, the projector\n$\\left(\\begin{smallmatrix}1&1\\\\0&0\\end{smallmatrix}\\right)\\pmod2$\nhas no exact equivariant lift retaining its reduction. These are statements\nabout the supplied carrier and prescribed projectors, not about nonexistence\nof all integral summands.","\\paragraph{Simultaneous projector-family lifting and rigidity.}\\phantomsection\\label{book:x-construction-19}\nKeep the complete domain, supplied finite-group carrier and group-order\nhypotheses of the \\hyperref[book:x-construction-18]{supplied-carrier summand-lifting construction}. Let $J$ be any finite label set and $k>0$. Suppose\n$(\\bar E_i)_{i\\in J}$ is a complete orthogonal family of equivariant\nidempotents modulo $(\\pi)^{e+k}$. There exists an integral equivariant\nfamily $(E_i)_{i\\in J}$ satisfying\n\\[\nE_i^2=E_i,\\qquad E_iE_j=0\\ (i\\ne j),\\qquad\n\\sum_{i\\in J}E_i=1,\\qquad\nE_i\\bmod(\\pi)^k=\\bar E_i\\bmod(\\pi)^k.\n\\]\nThe loss is at most $e$, independent of $|J|$ and the ambient matrix rank.\nThere is no local-ring or finite-residue-field hypothesis for existence.\nThe formal theorem also treats empty label sets whenever their input\nequations can hold.","us projector-family lifting and rigidity.}\\phantomsection\\label{book:x-construction-19}\nKeep the complete domain, supplied finite-group carrier and group-order\nhypotheses of the \\hyperref[book:x-construction-18]{supplied-carrier summand-lifting construction}. Let $J$ be any finite label set and $k>0$. Suppose\n$(\\bar E_i)_{i\\in J}$ is a complete orthogonal family of equivariant\nidempotents modulo $(\\pi)^{e+k}$. There exists an integral equivariant\nfamily $(E_i)_{i\\in J}$ satisfying\n\\[\nE_i^2=E_i,\\qquad E_iE_j=0\\ (i\\ne j),\\qquad\n\\sum_{i\\in J}E_i=1,\\qquad\nE_i\\bmod(\\pi)^k=\\bar E_i\\bmod(\\pi)^k.\n\\]\nThe loss is at most $e$, independent of $|J|$ and the ambient matrix rank.\nThere is no local-ring or finite-residue-field hypothesis for existence.\nThe formal theorem also treats empty label sets whenever their input\nequations can hold.","abel{book:x-construction-19}\nKeep the complete domain, supplied finite-group carrier and group-order\nhypotheses of the \\hyperref[book:x-construction-18]{supplied-carrier summand-lifting construction}. Let $J$ be any finite label set and $k>0$. Suppose\n$(\\bar E_i)_{i\\in J}$ is a complete orthogonal family of equivariant\nidempotents modulo $(\\pi)^{e+k}$. There exists an integral equivariant\nfamily $(E_i)_{i\\in J}$ satisfying\n\\[\nE_i^2=E_i,\\qquad E_iE_j=0\\ (i\\ne j),\\qquad\n\\sum_{i\\in J}E_i=1,\\qquad\nE_i\\bmod(\\pi)^k=\\bar E_i\\bmod(\\pi)^k.\n\\]\nThe loss is at most $e$, independent of $|J|$ and the ambient matrix rank.\nThere is no local-ring or finite-residue-field hypothesis for existence.\nThe formal theorem also treats empty label sets whenever their input\nequations can hold.","Keep the complete domain, supplied finite-group carrier and group-order\nhypotheses of the \\hyperref[book:x-construction-18]{supplied-carrier summand-lifting construction}. Let $J$ be any finite label set and $k>0$. Suppose\n$(\\bar E_i)_{i\\in J}$ is a complete orthogonal family of equivariant\nidempotents modulo $(\\pi)^{e+k}$. There exists an integral equivariant\nfamily $(E_i)_{i\\in J}$ satisfying\n\\[\nE_i^2=E_i,\\qquad E_iE_j=0\\ (i\\ne j),\\qquad\n\\sum_{i\\in J}E_i=1,\\qquad\nE_i\\bmod(\\pi)^k=\\bar E_i\\bmod(\\pi)^k.\n\\]\nThe loss is at most $e$, independent of $|J|$ and the ambient matrix rank.\nThere is no local-ring or finite-residue-field hypothesis for existence.\nThe formal theorem also treats empty label sets whenever their input\nequations can hold.","-carrier summand-lifting construction}. Let $J$ be any finite label set and $k>0$. Suppose\n$(\\bar E_i)_{i\\in J}$ is a complete orthogonal family of equivariant\nidempotents modulo $(\\pi)^{e+k}$. There exists an integral equivariant\nfamily $(E_i)_{i\\in J}$ satisfying\n\\[\nE_i^2=E_i,\\qquad E_iE_j=0\\ (i\\ne j),\\qquad\n\\sum_{i\\in J}E_i=1,\\qquad\nE_i\\bmod(\\pi)^k=\\bar E_i\\bmod(\\pi)^k.\n\\]\nThe loss is at most $e$, independent of $|J|$ and the ambient matrix rank.\nThere is no local-ring or finite-residue-field hypothesis for existence.\nThe formal theorem also treats empty label sets whenever their input\nequations can hold.","ting construction}. Let $J$ be any finite label set and $k>0$. Suppose\n$(\\bar E_i)_{i\\in J}$ is a complete orthogonal family of equivariant\nidempotents modulo $(\\pi)^{e+k}$. There exists an integral equivariant\nfamily $(E_i)_{i\\in J}$ satisfying\n\\[\nE_i^2=E_i,\\qquad E_iE_j=0\\ (i\\ne j),\\qquad\n\\sum_{i\\in J}E_i=1,\\qquad\nE_i\\bmod(\\pi)^k=\\bar E_i\\bmod(\\pi)^k.\n\\]\nThe loss is at most $e$, independent of $|J|$ and the ambient matrix rank.\nThere is no local-ring or finite-residue-field hypothesis for existence.\nThe formal theorem also treats empty label sets whenever their input\nequations can hold.","Average every supplied matrix once. Inductively let $C$ be the complement\nof the previously lifted projectors and correct the next averaged matrix\ninside the corner defined by $C$. The integral polynomial correction\npreserves the retained reduction and commutation with $C$ and the carrier.\nMultiplication by $C$ gives the required supported projector. This consumes\nno further precision. Once every label has been lifted, the complementary\nidempotent reduces to zero at positive depth. Adic separatedness forces it\nto vanish, proving the sum-to-identity equation.","The actual invariant images $U_i=\\operatorname{im}E_i$ give an equivariant\nlinear isomorphism\n\\[\n\\bigoplus_{i\\in J}U_i\\simeq R^n,\n\\qquad (v_i)_i\\longmapsto\\sum_i v_i,\n\\qquad v\\longmapsto(E_iv)_i.\n\\]\nLean represents the finite direct sum by the finite product of the image\nsubmodules. The existing coordinatewise reduction theorem maps every\n$U_i$ onto the corresponding reduced projector image. No chosen bases or\nunproved freeness of the images are used.","For rigidity alone, let $R$ be a commutative local ring, let $\\pi$ be a\nnonunit, and let $E,F$ be complete equivariant families on the same carrier\nwith $E_i\\equiv F_i\\pmod{(\\pi)^k}$ for every label and $k>0$. Then\n\\[\nU=\\sum_{i\\in J}F_iE_i,\\qquad\nU\\equiv1\\pmod{(\\pi)^k},\\qquad UE_i=F_iU.\n\\]\nReduction reflects invertibility in this setting, so $U$ is an integral\nunit and $UE_iU^{-1}=F_i$. It commutes with the carrier and induces actual\nequivariant isomorphisms $\\operatorname{im}E_i\\simeq\\operatorname{im}F_i$.\nThis rigidity theorem requires neither a finite acting group, a domain,\nnor completeness. More generally a unit-reflecting coefficient map suffices.\nThe combined result proves existence and uniqueness up to this l","For rigidity alone, let $R$ be a commutative local ring, let $\\pi$ be a\nnonunit, and let $E,F$ be complete equivariant families on the same carrier\nwith $E_i\\equiv F_i\\pmod{(\\pi)^k}$ for every label and $k>0$. Then\n\\[\nU=\\sum_{i\\in J}F_iE_i,\\qquad\nU\\equiv1\\pmod{(\\pi)^k},\\qquad UE_i=F_iU.\n\\]\nReduction reflects invertibility in this setting, so $U$ is an integral\nunit and $UE_iU^{-1}=F_i$. It commutes with the carrier and induces actual\nequivariant isomorphisms $\\operatorname{im}E_i\\simeq\\operatorname{im}F_i$.\nThis rigidity theorem requires neither a finite acting group, a domain,\nnor completeness. More generally a unit-reflecting coefficient map suffices.\nThe combined result proves existence and uniqueness up to this labeled\nconjugacy. Exact uniqueness of t","For rigidity alone, let $R$ be a commutative local ring, let $\\pi$ be a\nnonunit, and let $E,F$ be complete equivariant families on the same carrier\nwith $E_i\\equiv F_i\\pmod{(\\pi)^k}$ for every label and $k>0$. Then\n\\[\nU=\\sum_{i\\in J}F_iE_i,\\qquad\nU\\equiv1\\pmod{(\\pi)^k},\\qquad UE_i=F_iU.\n\\]\nReduction reflects invertibility in this setting, so $U$ is an integral\nunit and $UE_iU^{-1}=F_i$. It commutes with the carrier and induces actual\nequivariant isomorphisms $\\operatorname{im}E_i\\simeq\\operatorname{im}F_i$.\nThis rigidity theorem requires neither a finite acting group, a domain,\nnor completeness. More generally a unit-reflecting coefficient map suffices.\nThe combined result proves existence and uniqueness up to this labeled\nconjugacy. Exact uniqueness of the matrices is false.","i$ be a\nnonunit, and let $E,F$ be complete equivariant families on the same carrier\nwith $E_i\\equiv F_i\\pmod{(\\pi)^k}$ for every label and $k>0$. Then\n\\[\nU=\\sum_{i\\in J}F_iE_i,\\qquad\nU\\equiv1\\pmod{(\\pi)^k},\\qquad UE_i=F_iU.\n\\]\nReduction reflects invertibility in this setting, so $U$ is an integral\nunit and $UE_iU^{-1}=F_i$. It commutes with the carrier and induces actual\nequivariant isomorphisms $\\operatorname{im}E_i\\simeq\\operatorname{im}F_i$.\nThis rigidity theorem requires neither a finite acting group, a domain,\nnor completeness. More generally a unit-reflecting coefficient map suffices.\nThe combined result proves existence and uniqueness up to this labeled\nconjugacy. Exact uniqueness of the matrices is false.","For the continuous $\\mathbb Z_2$ carrier\n$\\rho(s)=\\operatorname{diag}(1,-1,1)$, conjugate the three coordinate\nprojectors by $1+2e_{01}$ and reduce modulo $4$. They form a complete\nequivariant family. The general theorem constructs three nonzero integral\nsummands retaining this family modulo $2$. No exact equivariant family\nretains the whole prescribed family modulo $4$. Separate checked examples\nshow that independent equivariant idempotent lifts can lose orthogonality,\nand that distinct complete families with the same reduction can be related\nby the asserted identity-congruent conjugacy.","\\paragraph{General finite logarithmic recovery.}\\phantomsection\\label{book:x-construction-20}\nLet $A$ be a rational algebra, without any commutativity assumption on $A$.\nFor a unipotent element $U$ and a positive integer $d$ satisfying\n$(U-1)^d=0$, put\n\\[\nL_d(U)=\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}(U-1)^j.\n\\]\nThe value is independent of the chosen annihilating exponent. The finite\nlogarithm and the nilpotent exponential satisfy\n\\[\nL(\\exp N)=N,\\qquad \\exp(L(U))=U.\n\\]\nBoth maps preserve every annihilating exponent, and hence preserve the\nexact nilpotency class of $N$ and $\\exp(N)-1$, or of $U-1$ and $L(U)$.\nThe Lean finite sum includes its zero-index term with coefficient zero.\nThe displayed mathematical sum starts at one. The formal statements also\nhandle the zero algebra and the exponent-z","\\paragraph{General finite logarithmic recovery.}\\phantomsection\\label{book:x-construction-20}\nLet $A$ be a rational algebra, without any commutativity assumption on $A$.\nFor a unipotent element $U$ and a positive integer $d$ satisfying\n$(U-1)^d=0$, put\n\\[\nL_d(U)=\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}(U-1)^j.\n\\]\nThe value is independent of the chosen annihilating exponent. The finite\nlogarithm and the nilpotent exponential satisfy\n\\[\nL(\\exp N)=N,\\qquad \\exp(L(U))=U.\n\\]\nBoth maps preserve every annihilating exponent, and hence preserve the\nexact nilpotency class of $N$ and $\\exp(N)-1$, or of $U-1$ and $L(U)$.\nThe Lean finite sum includes its zero-index term with coefficient zero.\nThe displayed mathematical sum starts at one. The formal statements also\nhandle the zero algebra and the exponent-zero case.","mic recovery.}\\phantomsection\\label{book:x-construction-20}\nLet $A$ be a rational algebra, without any commutativity assumption on $A$.\nFor a unipotent element $U$ and a positive integer $d$ satisfying\n$(U-1)^d=0$, put\n\\[\nL_d(U)=\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}(U-1)^j.\n\\]\nThe value is independent of the chosen annihilating exponent. The finite\nlogarithm and the nilpotent exponential satisfy\n\\[\nL(\\exp N)=N,\\qquad \\exp(L(U))=U.\n\\]\nBoth maps preserve every annihilating exponent, and hence preserve the\nexact nilpotency class of $N$ and $\\exp(N)-1$, or of $U-1$ and $L(U)$.\nThe Lean finite sum includes its zero-index term with coefficient zero.\nThe displayed mathematical sum starts at one. The formal statements also\nhandle the zero algebra and the exponent-zero case.","For a unipotent element $U$ and a positive integer $d$ satisfying\n$(U-1)^d=0$, put\n\\[\nL_d(U)=\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}(U-1)^j.\n\\]\nThe value is independent of the chosen annihilating exponent. The finite\nlogarithm and the nilpotent exponential satisfy\n\\[\nL(\\exp N)=N,\\qquad \\exp(L(U))=U.\n\\]\nBoth maps preserve every annihilating exponent, and hence preserve the\nexact nilpotency class of $N$ and $\\exp(N)-1$, or of $U-1$ and $L(U)$.\nThe Lean finite sum includes its zero-index term with coefficient zero.\nThe displayed mathematical sum starts at one. The formal statements also\nhandle the zero algebra and the exponent-zero case.","nipotent element $U$ and a positive integer $d$ satisfying\n$(U-1)^d=0$, put\n\\[\nL_d(U)=\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}(U-1)^j.\n\\]\nThe value is independent of the chosen annihilating exponent. The finite\nlogarithm and the nilpotent exponential satisfy\n\\[\nL(\\exp N)=N,\\qquad \\exp(L(U))=U.\n\\]\nBoth maps preserve every annihilating exponent, and hence preserve the\nexact nilpotency class of $N$ and $\\exp(N)-1$, or of $U-1$ and $L(U)$.\nThe Lean finite sum includes its zero-index term with coefficient zero.\nThe displayed mathematical sum starts at one. The formal statements also\nhandle the zero algebra and the exponent-zero case.","$ and a positive integer $d$ satisfying\n$(U-1)^d=0$, put\n\\[\nL_d(U)=\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}(U-1)^j.\n\\]\nThe value is independent of the chosen annihilating exponent. The finite\nlogarithm and the nilpotent exponential satisfy\n\\[\nL(\\exp N)=N,\\qquad \\exp(L(U))=U.\n\\]\nBoth maps preserve every annihilating exponent, and hence preserve the\nexact nilpotency class of $N$ and $\\exp(N)-1$, or of $U-1$ and $L(U)$.\nThe Lean finite sum includes its zero-index term with coefficient zero.\nThe displayed mathematical sum starts at one. The formal statements also\nhandle the zero algebra and the exponent-zero case.","sitive integer $d$ satisfying\n$(U-1)^d=0$, put\n\\[\nL_d(U)=\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}(U-1)^j.\n\\]\nThe value is independent of the chosen annihilating exponent. The finite\nlogarithm and the nilpotent exponential satisfy\n\\[\nL(\\exp N)=N,\\qquad \\exp(L(U))=U.\n\\]\nBoth maps preserve every annihilating exponent, and hence preserve the\nexact nilpotency class of $N$ and $\\exp(N)-1$, or of $U-1$ and $L(U)$.\nThe Lean finite sum includes its zero-index term with coefficient zero.\nThe displayed mathematical sum starts at one. The formal statements also\nhandle the zero algebra and the exponent-zero case.","Let $K$ be a characteristic-zero field and $N$ an endomorphism with\n$N^d=0$. For $t\\ne0$ the actual recovery formula is\n\\[\nN=\\frac1t\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}\n       \\bigl(\\exp(tN)-1\\bigr)^j.\n\\]\nAll coefficients lie in $K$. The vector spaces need not be finite\ndimensional, but nilpotence still means that some finite power of the\nwhole operator is zero. A linear map intertwines two nilpotent operators exactly\nwhen it intertwines their exponential samples at a common nonzero\nparameter. Also\n\\[\n\\ker\\bigl(\\exp(tN)-1\\bigr)=\\ker N.\n\\]","ry formula is\n\\[\nN=\\frac1t\\sum_{j=1}^{d-1}\\frac{(-1)^{j+1}}{j}\n       \\bigl(\\exp(tN)-1\\bigr)^j.\n\\]\nAll coefficients lie in $K$. The vector spaces need not be finite\ndimensional, but nilpotence still means that some finite power of the\nwhole operator is zero. A linear map intertwines two nilpotent operators exactly\nwhen it intertwines their exponential samples at a common nonzero\nparameter. Also\n\\[\n\\ker\\bigl(\\exp(tN)-1\\bigr)=\\ker N.\n\\]","For supplied covariance data and the twisted cocycle of the \\hyperref[book:x-construction-13]{finite-exponential reconstruction}, let\n$\\rho_D(g)=\\exp(t(g)N_D)r_D(g)$. If a common element $u$ satisfies\n$r_D(u)=r_E(u)=1$ and $t(u)\\ne0$, every intertwiner of $\\rho_D$ and\n$\\rho_E$ recovers both the original $r$-intertwiner and the $N$-intertwiner.\nThere is no cubic nilpotence restriction. This recovers equivariant\nprojectors for both original components. Reconstructed-action isomorphisms\ninduce actual isomorphisms of the inertia-fixed monodromy kernels. Their local characteristic polynomials\nagree under the existing finite-dimensional and normal-subgroup hypotheses.","For actual finite original inertia images the common kernel sample is\nderived. Suppose $u\\in H$ satisfies $\\nu(u)=1$ and $t(u)\\ne0$ and both\n$r_D(H)$ and $r_E(H)$ are finite. Their sampled actions have positive\nfinite orders $a,b$. At $u^{ab}$ both original actions are trivial, and\n\\[\nt(u^{ab})=(ab)t(u)\\ne0\n\\]\nby the cocycle identity and characteristic zero. Thus the same full\nintertwiner equivalence holds using the actual finite-image hypotheses,\nwithout a separately supplied common trivial-action sample. The initial\nnonzero cocycle sample and the arithmetic interpretation of the data\nremain requirements.","Actual affine representations with monodromy chains of every finite\nlength invoke these theorems. In particular a length-six example has\n$N^5\\ne0$ and $N^6=0$ and is recovered from its actual reconstructed action.\nThe length-four example still defeats the old quadratic logarithm, while\nthe full finite logarithm recovers it.","The sample hypotheses retain substantive content. A zero parameter loses\n$N$. For unrestricted supplied algebraic data, one can also absorb\n$\\exp(tN)$ into $r$ and set the new monodromy to zero. The reconstructed\nrepresentation is unchanged. The absorbed original action fails the\nrequired trivial-action sample condition. This is a counterexample in the\nalgebraic datum class, not a failure of uniqueness of arithmetic\nWeil--Deligne attachment with finite original inertia action.","\\paragraph{Interpreted quadratic arithmetic completion.}\\phantomsection\\label{book:x-construction-21}\nFix the continuous quotient\n\\[\nq:G_{\\Q}\\longrightarrow C_2\n\\]\nobtained by the action on a chosen square root $i$ of $-1$ in the algebraic\nclosure. Its surjectivity and the equivalence\n\\[\nq(\\sigma)=1\\quad\\Longleftrightarrow\\quad\\sigma(i)=i\n\\]\nare proved using the existing actual quadratic Galois construction.\nFor a commutative ring $B$, define the arithmetic packet\n\\[\n\\mathcal P_q(B)=\n\\{\\chi:G_{\\Q}\\longrightarrow B^\\times\\mid\\ker q\\subseteq\\ker\\chi\\},\n\\]\nwhere the maps are group homomorphisms. Unique descent through the actual\nsurjective map $q$ is derived, rather than supplied. Every member has finite\nimage and is continuous for the coefficient topology, since its descent\nhas the finite discrete domain $C_2$.","thmetic completion.}\\phantomsection\\label{book:x-construction-21}\nFix the continuous quotient\n\\[\nq:G_{\\Q}\\longrightarrow C_2\n\\]\nobtained by the action on a chosen square root $i$ of $-1$ in the algebraic\nclosure. Its surjectivity and the equivalence\n\\[\nq(\\sigma)=1\\quad\\Longleftrightarrow\\quad\\sigma(i)=i\n\\]\nare proved using the existing actual quadratic Galois construction.\nFor a commutative ring $B$, define the arithmetic packet\n\\[\n\\mathcal P_q(B)=\n\\{\\chi:G_{\\Q}\\longrightarrow B^\\times\\mid\\ker q\\subseteq\\ker\\chi\\},\n\\]\nwhere the maps are group homomorphisms. Unique descent through the actual\nsurjective map $q$ is derived, rather than supplied. Every member has finite\nimage and is continuous for the coefficient topology, since its descent\nhas the finite discrete domain $C_2$.","\\label{book:x-construction-21}\nFix the continuous quotient\n\\[\nq:G_{\\Q}\\longrightarrow C_2\n\\]\nobtained by the action on a chosen square root $i$ of $-1$ in the algebraic\nclosure. Its surjectivity and the equivalence\n\\[\nq(\\sigma)=1\\quad\\Longleftrightarrow\\quad\\sigma(i)=i\n\\]\nare proved using the existing actual quadratic Galois construction.\nFor a commutative ring $B$, define the arithmetic packet\n\\[\n\\mathcal P_q(B)=\n\\{\\chi:G_{\\Q}\\longrightarrow B^\\times\\mid\\ker q\\subseteq\\ker\\chi\\},\n\\]\nwhere the maps are group homomorphisms. Unique descent through the actual\nsurjective map $q$ is derived, rather than supplied. Every member has finite\nimage and is continuous for the coefficient topology, since its descent\nhas the finite discrete domain $C_2$.","ts surjectivity and the equivalence\n\\[\nq(\\sigma)=1\\quad\\Longleftrightarrow\\quad\\sigma(i)=i\n\\]\nare proved using the existing actual quadratic Galois construction.\nFor a commutative ring $B$, define the arithmetic packet\n\\[\n\\mathcal P_q(B)=\n\\{\\chi:G_{\\Q}\\longrightarrow B^\\times\\mid\\ker q\\subseteq\\ker\\chi\\},\n\\]\nwhere the maps are group homomorphisms. Unique descent through the actual\nsurjective map $q$ is derived, rather than supplied. Every member has finite\nimage and is continuous for the coefficient topology, since its descent\nhas the finite discrete domain $C_2$.","a commutative ring $B$, define the arithmetic packet\n\\[\n\\mathcal P_q(B)=\n\\{\\chi:G_{\\Q}\\longrightarrow B^\\times\\mid\\ker q\\subseteq\\ker\\chi\\},\n\\]\nwhere the maps are group homomorphisms. Unique descent through the actual\nsurjective map $q$ is derived, rather than supplied. Every member has finite\nimage and is continuous for the coefficient topology, since its descent\nhas the finite discrete domain $C_2$.","Let $R$ be any commutative ring and $B$ any commutative $R$-algebra. The\ncoordinate ring $A_R=R[X]/(X^2-1)$ represents precisely this specified\npacket:\n\\[\n\\Hom_{R\\text{-alg}}(A_R,B)\n\\simeq\\{x\\in B:x^2=1\\}\n\\simeq\\Hom(C_2,B^\\times)\n\\simeq\\mathcal P_q(B).\n\\]\nThe first map evaluates the coordinate. Its inverse is the quotient\nuniversal property. A root is a unit with itself as inverse and determines\na character by its value on the generator of $C_2$. Finally pull back\nalong $q$. All inverse identities and naturality under coefficient\nhomomorphisms are proved. In particular, the point reductions and actual\narithmetic character reductions coincide. There is no invertibility-of-two,\nreducedness or nontri","Let $R$ be any commutative ring and $B$ any commutative $R$-algebra. The\ncoordinate ring $A_R=R[X]/(X^2-1)$ represents precisely this specified\npacket:\n\\[\n\\Hom_{R\\text{-alg}}(A_R,B)\n\\simeq\\{x\\in B:x^2=1\\}\n\\simeq\\Hom(C_2,B^\\times)\n\\simeq\\mathcal P_q(B).\n\\]\nThe first map evaluates the coordinate. Its inverse is the quotient\nuniversal property. A root is a unit with itself as inverse and determines\na character by its value on the generator of $C_2$. Finally pull back\nalong $q$. All inverse identities and naturality under coefficient\nhomomorphisms are proved. In particular, the point reductions and actual\narithmetic character reductions coincide. There is no invertibility-of-two,\nreducedness or nontriviality hypothesis on $B$. No sm","Let $R$ be any commutative ring and $B$ any commutative $R$-algebra. The\ncoordinate ring $A_R=R[X]/(X^2-1)$ represents precisely this specified\npacket:\n\\[\n\\Hom_{R\\text{-alg}}(A_R,B)\n\\simeq\\{x\\in B:x^2=1\\}\n\\simeq\\Hom(C_2,B^\\times)\n\\simeq\\mathcal P_q(B).\n\\]\nThe first map evaluates the coordinate. Its inverse is the quotient\nuniversal property. A root is a unit with itself as inverse and determines\na character by its value on the generator of $C_2$. Finally pull back\nalong $q$. All inverse identities and naturality under coefficient\nhomomorphisms are proved. In particular, the point reductions and actual\narithmetic character reductions coincide. There is no invertibility-of-two,\nreducedness or nontriviality hypothesis on $B$. No smoothness or\n\\'{e}tal","Let $R$ be any commutative ring and $B$ any commutative $R$-algebra. The\ncoordinate ring $A_R=R[X]/(X^2-1)$ represents precisely this specified\npacket:\n\\[\n\\Hom_{R\\text{-alg}}(A_R,B)\n\\simeq\\{x\\in B:x^2=1\\}\n\\simeq\\Hom(C_2,B^\\times)\n\\simeq\\mathcal P_q(B).\n\\]\nThe first map evaluates the coordinate. Its inverse is the quotient\nuniversal property. A root is a unit with itself as inverse and determines\na character by its value on the generator of $C_2$. Finally pull back\nalong $q$. All inverse identities and naturality under coefficient\nhomomorphisms are proved. In particular, the point reductions and actual\narithmetic character reductions coincide. There is no invertibility-of-two,\nreducedness or nontriviality hypothesis on $B$. No smoothness or\n\\'{e}taleness statement at two is used or asserted.","ommutative $R$-algebra. The\ncoordinate ring $A_R=R[X]/(X^2-1)$ represents precisely this specified\npacket:\n\\[\n\\Hom_{R\\text{-alg}}(A_R,B)\n\\simeq\\{x\\in B:x^2=1\\}\n\\simeq\\Hom(C_2,B^\\times)\n\\simeq\\mathcal P_q(B).\n\\]\nThe first map evaluates the coordinate. Its inverse is the quotient\nuniversal property. A root is a unit with itself as inverse and determines\na character by its value on the generator of $C_2$. Finally pull back\nalong $q$. All inverse identities and naturality under coefficient\nhomomorphisms are proved. In particular, the point reductions and actual\narithmetic character reductions coincide. There is no invertibility-of-two,\nreducedness or nontriviality hypothesis on $B$. No smoothness or\n\\'{e}taleness statement at two is used or asserted.","rdinate ring $A_R=R[X]/(X^2-1)$ represents precisely this specified\npacket:\n\\[\n\\Hom_{R\\text{-alg}}(A_R,B)\n\\simeq\\{x\\in B:x^2=1\\}\n\\simeq\\Hom(C_2,B^\\times)\n\\simeq\\mathcal P_q(B).\n\\]\nThe first map evaluates the coordinate. Its inverse is the quotient\nuniversal property. A root is a unit with itself as inverse and determines\na character by its value on the generator of $C_2$. Finally pull back\nalong $q$. All inverse identities and naturality under coefficient\nhomomorphisms are proved. In particular, the point reductions and actual\narithmetic character reductions coincide. There is no invertibility-of-two,\nreducedness or nontriviality hypothesis on $B$. No smoothness or\n\\'{e}taleness statement at two is used or asserted.","rsal property. A root is a unit with itself as inverse and determines\na character by its value on the generator of $C_2$. Finally pull back\nalong $q$. All inverse identities and naturality under coefficient\nhomomorphisms are proved. In particular, the point reductions and actual\narithmetic character reductions coincide. There is no invertibility-of-two,\nreducedness or nontriviality hypothesis on $B$. No smoothness or\n\\'{e}taleness statement at two is used or asserted.","Let $I\\subseteq R$ and suppose that $R$ is $I$-adically complete. Using\nthe checked ring-map reconstruction on the quadratic coordinate ring,\nLean proves\n\\[\n\\mathcal P_q(R)\\simeq\\varprojlim_{m\\ge0}\\mathcal P_q(R/I^m).\n\\]\nThe zero-precision quotient is included only for the formal indexing.\nThe inverse constructs an actual arithmetic character, with exactly the\nprescribed reductions. It is unique, and its continuity is derived from\nits factorization through the same finite quotient.","There is also a completion theorem without chosen compatible characters.\nSuppose that $R/I^m$ is finite for every $m\\ge1$, and that\n$C_m\\subseteq\\mathcal P_q(R/I^m)$ are allowed subsets such that every\nreduction of an allowed member is allowed. Then\n\\[\n\\begin{split}\n&\\exists\\chi\\in\\mathcal P_q(R),\\quad\n  \\chi\\text{ is continuous and }\\chi\\bmod I^m\\in C_m\n  \\text{ for all }m\\ge1","There is also a completion theorem without chosen compatible characters.\nSuppose that $R/I^m$ is finite for every $m\\ge1$, and that\n$C_m\\subseteq\\mathcal P_q(R/I^m)$ are allowed subsets such that every\nreduction of an allowed member is allowed. Then\n\\[\n\\begin{split}\n&\\exists\\chi\\in\\mathcal P_q(R),\\quad\n  \\chi\\text{ is continuous and }\\chi\\bmod I^m\\in C_m\n  \\text{ for all }m\\ge1\\\\\n&\\hspace{3em}\\Longleftrightarrow\\quad\n  C_m\\ne\\varnothing\\text{ for all }m\\ge1.\n\\end{split}\n\\]\nThe root interpretation makes the allowed finite levels genuinely finite.\nFinite-branch compactness chooses the compatible roots, and the ring\nreconstruction gives their integral root. Naturality turns this into the\nrequired arithmetic character. Surjectivity of transitions and individual\nliftability of every finite node are not assumed. The root-level theorem\nneeds only the allowed root subsets to be finite.","For $R=\\Z_2$, Lean derives finiteness of all positive maximal-ideal\nquotients and applies this completion theorem. Since $\\Z_2$ is a domain,\nits only roots of $X^2-1$ are $1$ and $-1$. Thus the integral packet has\nexactly two distinct characters: the trivial character and the sign\ncharacter describing the action on $i$. They have the same reduction\nmodulo two and distinct reductions modulo four.","The distinction between finite occupancy and continuation is witnessed\nby an actual finite arithmetic character. The roots modulo eight are\n$1,3,5,7$. The root $3$ defines a member of\n$\\mathcal P_q(\\Z/8\\Z)$, but no member of\n$\\mathcal P_q(\\Z/16\\Z)$ reduces to it. Indeed the only scalar lifts of\n$3$ are $3$ and $11$, and both square to $9$ modulo sixteen. The formal\nproof also classifies all roots modulo sixteen and the reductions with\nchildren. Consequently a nonempty finite node need not lie on an\ninfinite branch, even in this explicitly interpreted arithmetic packet.","The \\hyperref[book:x-construction-21]{fixed-quadratic-packet interpretation} completes this restricted arithmetic interpretation and its\nconditional completion. The packet is defined by one fixed quotient. It\nis not the collection of all quadratic characters of $G_{\\Q}$. No local\nramification criterion, Hodge--Tate weight, Weil--Deligne attachment, or\ncross-coefficient local invariant was established by the \\hyperref[book:x-construction-21]{fixed-quadratic-packet interpretation}. The full\nmanuscript still requires its remaining packet identifications and\narithmetic comparisons. Universal positive occupancy remains conjectural.","\\paragraph{Actual quadratic local conditions and completion.}\\phantomsection\\label{book:x-construction-22}\nContinue with the fixed packet $\\mathcal P_q$ above. For a\ncharacteristic-zero field $K$, let\n\\[\nj_K:G_K\\longrightarrow G_{\\Q}\n\\]\nbe the actual field-induced homomorphism between absolute Galois groups.\nIt includes a chosen embedding of algebraic closures. Let $i_K$ be the\nimage of the chosen square root $i$ under that embedding. Lean proves\ncompatibility with the Galois actions and hence\n\\[\n(q\\circ j_K)(\\sigma)=1\n\\quad\\Longleftrightarrow\\quad \\sigma(i_K)=i_K.\n\\]\nThe fixed-field theorem for the algebraic closure then gives\n\\[\nq\\circ j_K=1\n\\quad\\Longleftrightarrow\\quad\n\\exists x\\in K,\\ x^2=-1.\n\\]\nIf no such root exists, the local quotient is surjective onto $C_2$.\nThese are statements about the actual local Galois map, not a supplied\na","\\paragraph{Actual quadratic local conditions and completion.}\\phantomsection\\label{book:x-construction-22}\nContinue with the fixed packet $\\mathcal P_q$ above. For a\ncharacteristic-zero field $K$, let\n\\[\nj_K:G_K\\longrightarrow G_{\\Q}\n\\]\nbe the actual field-induced homomorphism between absolute Galois groups.\nIt includes a chosen embedding of algebraic closures. Let $i_K$ be the\nimage of the chosen square root $i$ under that embedding. Lean proves\ncompatibility with the Galois actions and hence\n\\[\n(q\\circ j_K)(\\sigma)=1\n\\quad\\Longleftrightarrow\\quad \\sigma(i_K)=i_K.\n\\]\nThe fixed-field theorem for the algebraic closure then gives\n\\[\nq\\circ j_K=1\n\\quad\\Longleftrightarrow\\quad\n\\exists x\\in K,\\ x^2=-1.\n\\]\nIf no such root exists, the local quotient is surjective onto $C_2$.\nThese are statements about the actual local Galois map, not a supplied\narithmetic interpretation of a scalar label.","ual quadratic local conditions and completion.}\\phantomsection\\label{book:x-construction-22}\nContinue with the fixed packet $\\mathcal P_q$ above. For a\ncharacteristic-zero field $K$, let\n\\[\nj_K:G_K\\longrightarrow G_{\\Q}\n\\]\nbe the actual field-induced homomorphism between absolute Galois groups.\nIt includes a chosen embedding of algebraic closures. Let $i_K$ be the\nimage of the chosen square root $i$ under that embedding. Lean proves\ncompatibility with the Galois actions and hence\n\\[\n(q\\circ j_K)(\\sigma)=1\n\\quad\\Longleftrightarrow\\quad \\sigma(i_K)=i_K.\n\\]\nThe fixed-field theorem for the algebraic closure then gives\n\\[\nq\\circ j_K=1\n\\quad\\Longleftrightarrow\\quad\n\\exists x\\in K,\\ x^2=-1.\n\\]\nIf no such root exists, the local quotient is surjective onto $C_2$.\nThese are statements about the actual local Galois map, not a supplied\narithmetic interpretation of a scalar label.","n-22}\nContinue with the fixed packet $\\mathcal P_q$ above. For a\ncharacteristic-zero field $K$, let\n\\[\nj_K:G_K\\longrightarrow G_{\\Q}\n\\]\nbe the actual field-induced homomorphism between absolute Galois groups.\nIt includes a chosen embedding of algebraic closures. Let $i_K$ be the\nimage of the chosen square root $i$ under that embedding. Lean proves\ncompatibility with the Galois actions and hence\n\\[\n(q\\circ j_K)(\\sigma)=1\n\\quad\\Longleftrightarrow\\quad \\sigma(i_K)=i_K.\n\\]\nThe fixed-field theorem for the algebraic closure then gives\n\\[\nq\\circ j_K=1\n\\quad\\Longleftrightarrow\\quad\n\\exists x\\in K,\\ x^2=-1.\n\\]\nIf no such root exists, the local quotient is surjective onto $C_2$.\nThese are statements about the actual local Galois map, not a supplied\narithmetic interpretation of a scalar label.","e the actual field-induced homomorphism between absolute Galois groups.\nIt includes a chosen embedding of algebraic closures. Let $i_K$ be the\nimage of the chosen square root $i$ under that embedding. Lean proves\ncompatibility with the Galois actions and hence\n\\[\n(q\\circ j_K)(\\sigma)=1\n\\quad\\Longleftrightarrow\\quad \\sigma(i_K)=i_K.\n\\]\nThe fixed-field theorem for the algebraic closure then gives\n\\[\nq\\circ j_K=1\n\\quad\\Longleftrightarrow\\quad\n\\exists x\\in K,\\ x^2=-1.\n\\]\nIf no such root exists, the local quotient is surjective onto $C_2$.\nThese are statements about the actual local Galois map, not a supplied\narithmetic interpretation of a scalar label.","be the\nimage of the chosen square root $i$ under that embedding. Lean proves\ncompatibility with the Galois actions and hence\n\\[\n(q\\circ j_K)(\\sigma)=1\n\\quad\\Longleftrightarrow\\quad \\sigma(i_K)=i_K.\n\\]\nThe fixed-field theorem for the algebraic closure then gives\n\\[\nq\\circ j_K=1\n\\quad\\Longleftrightarrow\\quad\n\\exists x\\in K,\\ x^2=-1.\n\\]\nIf no such root exists, the local quotient is surjective onto $C_2$.\nThese are statements about the actual local Galois map, not a supplied\narithmetic interpretation of a scalar label.","For every commutative coefficient ring $B$ and $\\chi\\in\\mathcal P_q(B)$,\ndefine $\\chi_K=\\chi\\circ j_K$. The local character is continuous and\ncommutes with coefficient change. The precise local admissibility\ncondition is\n\\[\n\\operatorname{LocallyTrivial}_K(\\chi)\n\\quad\\Longleftrightarrow\\quad \\chi_K=1.\n\\]\nDescent through the fixed quadratic quotient proves\n\\[\n\\operatorname{LocallyTrivial}_K(\\chi)\n\\quad\\Longleftrightarrow\\quad\n\\bigl(\\exists x\\in K,\\ x^2=-1\\bigr)\\ \\lor\\ \\chi=1.\n\\]\nNo reducedness, characteristic-zero or domain hypothesis is imposed on\n$B$. In particular, the formula applies to the finite coefficient\nquotients used in completion.","h coefficient change. The precise local admissibility\ncondition is\n\\[\n\\operatorname{LocallyTrivial}_K(\\chi)\n\\quad\\Longleftrightarrow\\quad \\chi_K=1.\n\\]\nDescent through the fixed quadratic quotient proves\n\\[\n\\operatorname{LocallyTrivial}_K(\\chi)\n\\quad\\Longleftrightarrow\\quad\n\\bigl(\\exists x\\in K,\\ x^2=-1\\bigr)\\ \\lor\\ \\chi=1.\n\\]\nNo reducedness, characteristic-zero or domain hypothesis is imposed on\n$B$. In particular, the formula applies to the finite coefficient\nquotients used in completion.","The p-adic square-root criterion is proved for every prime:\n\\[\n\\bigl(\\exists x\\in\\Q_p,\\ x^2=-1\\bigr)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4.\n\\]\nA root satisfies $\\lvert x\\rvert_p^2=1$ and is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the","The p-adic square-root criterion is proved for every prime:\n\\[\n\\bigl(\\exists x\\in\\Q_p,\\ x^2=-1\\bigr)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4.\n\\]\nA root satisfies $\\lvert x\\rvert_p^2=1$ and is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactl","The p-adic square-root criterion is proved for every prime:\n\\[\n\\bigl(\\exists x\\in\\Q_p,\\ x^2=-1\\bigr)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4.\n\\]\nA root satisfies $\\lvert x\\rvert_p^2=1$ and is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are","The p-adic square-root criterion is proved for every prime:\n\\[\n\\bigl(\\exists x\\in\\Q_p,\\ x^2=-1\\bigr)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4.\n\\]\nA root satisfies $\\lvert x\\rvert_p^2=1$ and is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis","The p-adic square-root criterion is proved for every prime:\n\\[\n\\bigl(\\exists x\\in\\Q_p,\\ x^2=-1\\bigr)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4.\n\\]\nA root satisfies $\\lvert x\\rvert_p^2=1$ and is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis nontrivial on the full local Galois groups at","atisfies $\\lvert x\\rvert_p^2=1$ and is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis nontrivial on the full local Galois groups at three and two.","lvert x\\rvert_p^2=1$ and is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis nontrivial on the full local Galois groups at three and two.","d is therefore integral.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis nontrivial on the full local Galois groups at three and two.","l.\nReduction modulo three rules out a root in $\\Q_3$, and reduction modulo\nfour rules out a root in $\\Q_2$. At an odd prime, any residue root of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis nontrivial on the full local Galois groups at three and two.","ot of\n$X^2+1$ has nonzero derivative and lifts by the pinned Mathlib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis nontrivial on the full local Galois groups at three and two.","ib Hensel\ntheorem. The finite-field square criterion then gives the stated\ncongruence. An explicit Hensel application at the residue root $2$ modulo $5$ supplies\na root in $\\Z_5$. Consequently\n\\[\n\\boxed{\n\\operatorname{LocallyTrivial}_{\\Q_p}(\\chi)\n\\quad\\Longleftrightarrow\\quad p\\equiv1\\pmod4\\ \\lor\\ \\chi=1.\n}\n\\]\nThe arithmetic prime $p$ is independent of the coefficient prime. For\nthe two integral $\\Z_2$-valued characters, the sign character is locally\ntrivial exactly at primes congruent to one modulo four. Both distinct\nglobal characters are locally trivial at five, while the sign character\nis nontrivial on the full local Galois groups at three and two.","The inertia theorem uses actual residue actions. For a valuation subring\n$A$ of an extension field $L/K$, let $D_A$ be its decomposition subgroup\nin $\\operatorname{Aut}_K(L)$ and put\n\\[\nI_A=\\ker\\bigl(D_A\\longrightarrow\\operatorname{Aut}(k_A)\\bigr),\n\\]\nwhere $k_A$ is the residue field. If $2$ is a unit in $A$, every member\nof $I_A$ fixes every $y\\in L$ with $y^2=-1$. Indeed $y^{-1}=-y$, so\n$y$ belongs to $A$ and is a unit there. An automorphism sends $y$ to\n$y$ or $-y$. Their residues are distinct because $2$ and $y$ are units,\nwhereas inertia fixes residues. The second possibility is excluded.","tia theorem uses actual residue actions. For a valuation subring\n$A$ of an extension field $L/K$, let $D_A$ be its decomposition subgroup\nin $\\operatorname{Aut}_K(L)$ and put\n\\[\nI_A=\\ker\\bigl(D_A\\longrightarrow\\operatorname{Aut}(k_A)\\bigr),\n\\]\nwhere $k_A$ is the residue field. If $2$ is a unit in $A$, every member\nof $I_A$ fixes every $y\\in L$ with $y^2=-1$. Indeed $y^{-1}=-y$, so\n$y$ belongs to $A$ and is a unit there. An automorphism sends $y$ to\n$y$ or $-y$. Their residues are distinct because $2$ and $y$ are units,\nwhereas inertia fixes residues. The second possibility is excluded.","valuation subring\n$A$ of an extension field $L/K$, let $D_A$ be its decomposition subgroup\nin $\\operatorname{Aut}_K(L)$ and put\n\\[\nI_A=\\ker\\bigl(D_A\\longrightarrow\\operatorname{Aut}(k_A)\\bigr),\n\\]\nwhere $k_A$ is the residue field. If $2$ is a unit in $A$, every member\nof $I_A$ fixes every $y\\in L$ with $y^2=-1$. Indeed $y^{-1}=-y$, so\n$y$ belongs to $A$ and is a unit there. An automorphism sends $y$ to\n$y$ or $-y$. Their residues are distinct because $2$ and $y$ are units,\nwhereas inertia fixes residues. The second possibility is excluded.","on subring\n$A$ of an extension field $L/K$, let $D_A$ be its decomposition subgroup\nin $\\operatorname{Aut}_K(L)$ and put\n\\[\nI_A=\\ker\\bigl(D_A\\longrightarrow\\operatorname{Aut}(k_A)\\bigr),\n\\]\nwhere $k_A$ is the residue field. If $2$ is a unit in $A$, every member\nof $I_A$ fixes every $y\\in L$ with $y^2=-1$. Indeed $y^{-1}=-y$, so\n$y$ belongs to $A$ and is a unit there. An automorphism sends $y$ to\n$y$ or $-y$. Their residues are distinct because $2$ and $y$ are units,\nwhereas inertia fixes residues. The second possibility is excluded.","ing\n$A$ of an extension field $L/K$, let $D_A$ be its decomposition subgroup\nin $\\operatorname{Aut}_K(L)$ and put\n\\[\nI_A=\\ker\\bigl(D_A\\longrightarrow\\operatorname{Aut}(k_A)\\bigr),\n\\]\nwhere $k_A$ is the residue field. If $2$ is a unit in $A$, every member\nof $I_A$ fixes every $y\\in L$ with $y^2=-1$. Indeed $y^{-1}=-y$, so\n$y$ belongs to $A$ and is a unit there. An automorphism sends $y$ to\n$y$ or $-y$. Their residues are distinct because $2$ and $y$ are units,\nwhereas inertia fixes residues. The second possibility is excluded.","D_A$ be its decomposition subgroup\nin $\\operatorname{Aut}_K(L)$ and put\n\\[\nI_A=\\ker\\bigl(D_A\\longrightarrow\\operatorname{Aut}(k_A)\\bigr),\n\\]\nwhere $k_A$ is the residue field. If $2$ is a unit in $A$, every member\nof $I_A$ fixes every $y\\in L$ with $y^2=-1$. Indeed $y^{-1}=-y$, so\n$y$ belongs to $A$ and is a unit there. An automorphism sends $y$ to\n$y$ or $-y$. Their residues are distinct because $2$ and $y$ are units,\nwhereas inertia fixes residues. The second possibility is excluded.","its decomposition subgroup\nin $\\operatorname{Aut}_K(L)$ and put\n\\[\nI_A=\\ker\\bigl(D_A\\longrightarrow\\operatorname{Aut}(k_A)\\bigr),\n\\]\nwhere $k_A$ is the residue field. If $2$ is a unit in $A$, every member\nof $I_A$ fixes every $y\\in L$ with $y^2=-1$. Indeed $y^{-1}=-y$, so\n$y$ belongs to $A$ and is a unit there. An automorphism sends $y$ to\n$y$ or $-y$. Their residues are distinct because $2$ and $y$ are units,\nwhereas inertia fixes residues. The second possibility is excluded.","For every prime $p$, Lean constructs a valuation subring\n$A\\subseteq\\overline{\\Q_p}$ containing the integral image of $\\Z_p$\nsuch that the induced map $\\Z_p\\to A$ is local. For odd $p$, two is\na unit. The preceding argument applies to every valuation subring\ncontaining that integral image, and hence every member of\n$\\mathcal P_q(B)$ is trivial on its inertia subgroup after actual local\nrestriction. The existence theorem supplies a locally dominating choice,\nso this assertion is not vacuous.","This inertia is the actual residue-action kernel inside $D_A$. No equality\n$D_A=G_{\\Q_p}$ is established in this block. Nor is ramification at two\nproved: the nontrivial full local character there is a separate result.\nAt three the $\\Z_2$-valued sign character is nontrivial on the full local\nGalois group but trivial on every inertia subgroup covered by the theorem.\nThus local triviality and the stated inertia triviality are distinguished\nby checked arithmetic examples.","The local conditions survive coefficient completion by a proved theorem.\nFor an $I$-adically separated commutative ring $R$,\n\\[\n\\operatorname{LocallyTrivial}_K(\\chi)\n\\quad\\Longleftrightarrow\\quad\n\\operatorname{LocallyTrivial}_K(\\chi\\bmod I^m)\n\\text{ for every }m\\ge1.\n\\]\nIf $-1$ has a square root in $K$, every packet member satisfies the\ncondition. Otherwise the condition says that the character is globally\ntrivial, and separatedness detects that equality through all reductions.\nThis proves the closure statement for the independently defined arithmetic\nproperty.","Suppose now that $R$ is $I$-adically complete and that every positive\nquotient $R/I^m$ is finite. Let $C_m\\subseteq\\mathcal P_q(R/I^m)$ be\nallowed subsets preserved by reduction, and let $(p_s)_{s\\in S}$ be any\nindexed family of arithmetic primes. The index set $S$ need not be finite.\nLean proves the simultaneous arithmetic completion criterion\n\\[\n\\begin{split}\n&\\exists\\chi\\in\\mathcal P_q(R),\\quad\n \\chi\\text{ continuous},\\quad\n \\chi\\bmod I^m\\in C_m\\ (m\\ge1),\\quad\n \\chi_{\\Q_{p_s}}=1\\ (s\\in S)","Suppose now that $R$ is $I$-adically complete and that every positive\nquotient $R/I^m$ is finite. Let $C_m\\subseteq\\mathcal P_q(R/I^m)$ be\nallowed subsets preserved by reduction, and let $(p_s)_{s\\in S}$ be any\nindexed family of arithmetic primes. The index set $S$ need not be finite.\nLean proves the simultaneous arithmetic completion criterion\n\\[\n\\begin{split}\n&\\exists\\chi\\in\\mathcal P_q(R),\\quad\n \\chi\\text{ continuous},\\quad\n \\chi\\bmod I^m\\in C_m\\ (m\\ge1),\\quad\n \\chi_{\\Q_{p_s}}=1\\ (s\\in S)\\\\\n&\\quad\\Longleftrightarrow\\quad\n \\forall m\\ge1,\\ \\exists\\psi\\in C_m,\\quad\n \\forall s\\in S,\\ p_s\\equiv1\\pmod4\\ \\lor\\ \\psi=1.\n\\end{split}\n\\]\nThe constructed character also satisfies every odd-prime valuation-inertia\ncondition stated above. Finite-branch compactness selects all finite\ncharacters together. No coherent branch or surjective transition maps\nare assumed. The new local-closure implication is proved, not supplied.\nEvery-level occupancy remains a substantive hypothesis. One nonsplit\nselected prime forces the completed global character to be trivial.","Write $\\rho_i$ for the supplied constituents, $c_i$ for the carrier multiplicities and $m_i$ for the selected multiplicities, with $m_i\\leq c_i$. At a distinguished coefficient place $\\lambda$, the input presentations have the form\n\\[\n V_{\\mathrm{car},\\lambda}\\simeq\\bigoplus_i\\rho_{i,\\lambda}^{\\oplus c_i},\n \\qquad U_{0,\\lambda}\\simeq\\bigoplus_i\\rho_{i,\\lambda}^{\\oplus m_i}.\n\\]\nThese are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family\n$\\bigoplus_i\\rho_{i,\\lambda}^{\\oplus m_i}$\nhas a split equivariant inclusion in the carrier. The finite trace sample uniquely determines the multiplicity vector in the declared constituent packet.","icities, with $m_i\\leq c_i$. At a distinguished coefficient place $\\lambda$, the input presentations have the form\n\\[\n V_{\\mathrm{car},\\lambda}\\simeq\\bigoplus_i\\rho_{i,\\lambda}^{\\oplus c_i},\n \\qquad U_{0,\\lambda}\\simeq\\bigoplus_i\\rho_{i,\\lambda}^{\\oplus m_i}.\n\\]\nThese are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family\n$\\bigoplus_i\\rho_{i,\\lambda}^{\\oplus m_i}$\nhas a split equivariant inclusion in the carrier. The finite trace sample uniquely determines the multiplicity vector in the declared constituent packet.","by the formal endpoint. A nonzero occupied conclusion additionally requires the manuscript's declared nonzero-multiplicity premise and nonzero constituents. The module-support/Fitting-ideal alternative is logically separate: its checked support formalization is inherited and is not proved again by this final module. No unrestricted companion existence, automorphy, carrier existence, or universal $p$-adic Hodge comparison follows from this conditional assembly.","\\bibitem{Chenevier2014}\nGa{\\\"e}tan Chenevier.\n\\newblock The {$p$}-adic analytic space of pseudocharacters of a profinite\n  group and {Pseudorepresentations} over arbitrary rings.\n\\newblock In Fred Diamond, Payman~L. Kassaei, and Minhyong Kim, editors, {\\em\n  Automorphic Forms and Galois Representations. Volume 1}, volume 414 of {\\em\n  London Mathematical Society Lecture Note Series}, pages 221--285. Cambridge\n  University Press, Cambridge, 2014.","\\bibitem{Denef1984}\nJan Denef.\n\\newblock The rationality of the {Poincar{\\'e}} series associated to the\n  {$p$}-adic points on a variety.\n\\newblock {\\em Inventiones Mathematicae}, 77:1--23, 1984.","\\bibitem{Kisin2010}\nMark Kisin.\n\\newblock Deformations of {$G_{\\mathbb Q_p}$} and {$\\mathrm{GL}_2(\\mathbb\n  Q_p)$} representations.\n\\newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors,\n  {\\em Repr{\\'e}sentations {$p$}-adiques de Groupes {$p$}-adiques II:\n  Repr{\\'e}sentations de {$\\mathrm{GL}_2(\\mathbb Q_p)$} et\n  {$(\\varphi,\\Gamma)$}-modules}, volume 330 of {\\em Ast{\\'e}risque}, pages\n  511--528. Soci{\\'e}t{\\'e} Math{\\'e}matique de France, Paris, 2010.","\\bibitem{LarsenPink1992}\nMichael Larsen and Richard Pink.\n\\newblock On {$\\ell$}-independence of algebraic monodromy groups in compatible\n  systems of representations.\n\\newblock {\\em Inventiones Mathematicae}, 107(3):603--636, 1992.","\\bibitem{Mazur1989}\nBarry Mazur.\n\\newblock Deforming {Galois} representations.\n\\newblock In Yasutaka Ihara, Kenneth Ribet, and Jean-Pierre Serre, editors,\n  {\\em Galois Groups over {$\\mathbb Q$}}, volume~16 of {\\em Mathematical\n  Sciences Research Institute Publications}, pages 385--437. Springer, New\n  York, 1989.","\\bibitem{Serre1998}\nJean-Pierre Serre.\n\\newblock {\\em Abelian {$\\ell$}-adic Representations and Elliptic Curves}.\n\\newblock A K Peters, Wellesley, Massachusetts, 1998.","\\paragraph{Scope and notation.}\nLet $K$ be a number field of degree $d\\geq1$, with ring of integers\n$\\OO=\\OO_K$. Write $\\Nm I=\\#(\\OO/I)$ for the absolute norm of a nonzero\nintegral ideal and $n(x)=\\Nm_{K/\\mathbb Q}(x)\\in\\ZZ$ for $x\\in\\OO$.\nThus $\\Nm(x)=\\absn{n(x)}$. Let $\\iota$ be the mixed embedding in its\nEuclidean coordinates, $L(J)=\\iota(J)$, and $D$ the actual unit-norm\nfundamental cone used in the preceding counting block. Its norm function\nscales by $t^d$ under a positive real dilation $t$.\nThe full-paper ledger remains \\textbf{17 checked claims and 3 partial claims}.","egree $d\\geq1$, with ring of integers\n$\\OO=\\OO_K$. Write $\\Nm I=\\#(\\OO/I)$ for the absolute norm of a nonzero\nintegral ideal and $n(x)=\\Nm_{K/\\mathbb Q}(x)\\in\\ZZ$ for $x\\in\\OO$.\nThus $\\Nm(x)=\\absn{n(x)}$. Let $\\iota$ be the mixed embedding in its\nEuclidean coordinates, $L(J)=\\iota(J)$, and $D$ the actual unit-norm\nfundamental cone used in the preceding counting block. Its norm function\nscales by $t^d$ under a positive real dilation $t$.\nThe full-paper ledger remains \\textbf{17 checked claims and 3 partial claims}.","$\\OO=\\OO_K$. Write $\\Nm I=\\#(\\OO/I)$ for the absolute norm of a nonzero\nintegral ideal and $n(x)=\\Nm_{K/\\mathbb Q}(x)\\in\\ZZ$ for $x\\in\\OO$.\nThus $\\Nm(x)=\\absn{n(x)}$. Let $\\iota$ be the mixed embedding in its\nEuclidean coordinates, $L(J)=\\iota(J)$, and $D$ the actual unit-norm\nfundamental cone used in the preceding counting block. Its norm function\nscales by $t^d$ under a positive real dilation $t$.\nThe full-paper ledger remains \\textbf{17 checked claims and 3 partial claims}.","\\paragraph{Actual additive residues.}\nFix nonzero integral ideals $M\\subseteq J$, and put $q=[J:M]=\\#(J/M)$.\nThe formal quotient of ideal lattices is identified with the additive\ncongruence relation $x-a\\in M$ on actual elements of $J$. In particular,\nthe residue label is not an independent parameter attached to a count.","\\begin{proposition}\\label{prop:residue-count}\nThere is a constant $B\\geq0$, depending on $K,M,J$, such that, for every\n$a\\in J/M$ and every real $t\\geq1$,\n\\[\n \\left|\\#\\{x\\in J:x\\bmod M=a,\\ \\iota(x)\\in tD\\}\n -\\frac{\\vol(D)}{q\\,\\covol(L(J))}t^d\\right|\n \\leq B(t^{d-1}+1).\n\\]\nThe same constant works for all residues $a$.\n\\end{proposition}","The estimate is proved from the actual boundary charts and the fixed-lattice\ndiscrepancy theorem. The index factor comes from the covolume identity\n$\\covol(L(M))=q\\,\\covol(L(J))$.\nFor a complex weight $w:J/M\\to\\CC$, summing the estimates gives the leading\nterm\n\\[\n \\frac{\\vol(D)t^d}{q\\,\\covol(L(J))}\\sum_{a\\in J/M}w(a),\n\\]\nwith error at most $B\\sum_a\\absn{w(a)}(t^{d-1}+1)$.\nIf the finite sum of weights is zero, only that error remains.\nWriting $Y=t^d$ gives an error $O(Y^{1-1/d}+1)$ for the actual norm threshold.\nThese assertions include $d=1$.","charts and the fixed-lattice\ndiscrepancy theorem. The index factor comes from the covolume identity\n$\\covol(L(M))=q\\,\\covol(L(J))$.\nFor a complex weight $w:J/M\\to\\CC$, summing the estimates gives the leading\nterm\n\\[\n \\frac{\\vol(D)t^d}{q\\,\\covol(L(J))}\\sum_{a\\in J/M}w(a),\n\\]\nwith error at most $B\\sum_a\\absn{w(a)}(t^{d-1}+1)$.\nIf the finite sum of weights is zero, only that error remains.\nWriting $Y=t^d$ gives an error $O(Y^{1-1/d}+1)$ for the actual norm threshold.\nThese assertions include $d=1$.","attice\ndiscrepancy theorem. The index factor comes from the covolume identity\n$\\covol(L(M))=q\\,\\covol(L(J))$.\nFor a complex weight $w:J/M\\to\\CC$, summing the estimates gives the leading\nterm\n\\[\n \\frac{\\vol(D)t^d}{q\\,\\covol(L(J))}\\sum_{a\\in J/M}w(a),\n\\]\nwith error at most $B\\sum_a\\absn{w(a)}(t^{d-1}+1)$.\nIf the finite sum of weights is zero, only that error remains.\nWriting $Y=t^d$ gives an error $O(Y^{1-1/d}+1)$ for the actual norm threshold.\nThese assertions include $d=1$.","tity\n$\\covol(L(M))=q\\,\\covol(L(J))$.\nFor a complex weight $w:J/M\\to\\CC$, summing the estimates gives the leading\nterm\n\\[\n \\frac{\\vol(D)t^d}{q\\,\\covol(L(J))}\\sum_{a\\in J/M}w(a),\n\\]\nwith error at most $B\\sum_a\\absn{w(a)}(t^{d-1}+1)$.\nIf the finite sum of weights is zero, only that error remains.\nWriting $Y=t^d$ gives an error $O(Y^{1-1/d}+1)$ for the actual norm threshold.\nThese assertions include $d=1$.","\\newpage\n\\paragraph{The residue norm and its sign.}\nFor $m\\in\\NN$, reducing the multiplication matrices of $\\OO$ modulo $m$\nand taking their determinants constructs a monoid homomorphism\n\\[\n \\overline n:\\OO/m\\OO\\longrightarrow\\ZZ/m\\ZZ,\n \\qquad \\overline n([x])=n(x)\\bmod m.\n\\]\nThe construction is proved independent of the integral basis.\nFor $m\\ne0$, its source is finite of cardinality $m^d$.\nConsequently $x-y\\in m\\OO$ implies $n(x)\\equiv n(y)\\pmod m$.\nNo surjectivity of $\\overline n$ is asserted.","\\paragraph{The residue norm and its sign.}\nFor $m\\in\\NN$, reducing the multiplication matrices of $\\OO$ modulo $m$\nand taking their determinants constructs a monoid homomorphism\n\\[\n \\overline n:\\OO/m\\OO\\longrightarrow\\ZZ/m\\ZZ,\n \\qquad \\overline n([x])=n(x)\\bmod m.\n\\]\nThe construction is proved independent of the integral basis.\nFor $m\\ne0$, its source is finite of cardinality $m^d$.\nConsequently $x-y\\in m\\OO$ implies $n(x)\\equiv n(y)\\pmod m$.\nNo surjectivity of $\\overline n$ is asserted.","norm and its sign.}\nFor $m\\in\\NN$, reducing the multiplication matrices of $\\OO$ modulo $m$\nand taking their determinants constructs a monoid homomorphism\n\\[\n \\overline n:\\OO/m\\OO\\longrightarrow\\ZZ/m\\ZZ,\n \\qquad \\overline n([x])=n(x)\\bmod m.\n\\]\nThe construction is proved independent of the integral basis.\nFor $m\\ne0$, its source is finite of cardinality $m^d$.\nConsequently $x-y\\in m\\OO$ implies $n(x)\\equiv n(y)\\pmod m$.\nNo surjectivity of $\\overline n$ is asserted.","the multiplication matrices of $\\OO$ modulo $m$\nand taking their determinants constructs a monoid homomorphism\n\\[\n \\overline n:\\OO/m\\OO\\longrightarrow\\ZZ/m\\ZZ,\n \\qquad \\overline n([x])=n(x)\\bmod m.\n\\]\nThe construction is proved independent of the integral basis.\nFor $m\\ne0$, its source is finite of cardinality $m^d$.\nConsequently $x-y\\in m\\OO$ implies $n(x)\\equiv n(y)\\pmod m$.\nNo surjectivity of $\\overline n$ is asserted.","The relation to the geometric sign is also proved. If $\\sigma$ ranges over\nreal embeddings and $\\tau$ over one embedding from each complex pair, then\n\\[\n n(x)=\\prod_{\\sigma}\\sigma(x)\\prod_{\\tau}\\absn{\\tau(x)}^2.\n\\]\nFor $x\\ne0$, the second factor is positive. Thus the sign of $n(x)$ is\nexactly the sign of the real-coordinate product. Cutting $D$ into its\npositive and negative portions $D_+,D_-$ adds no boundary outside the\nexisting frontier, and both portions retain the proved boundary atlas.\nThey are disjoint and their union is $D$.","lex pair, then\n\\[\n n(x)=\\prod_{\\sigma}\\sigma(x)\\prod_{\\tau}\\absn{\\tau(x)}^2.\n\\]\nFor $x\\ne0$, the second factor is positive. Thus the sign of $n(x)$ is\nexactly the sign of the real-coordinate product. Cutting $D$ into its\npositive and negative portions $D_+,D_-$ adds no boundary outside the\nexisting frontier, and both portions retain the proved boundary atlas.\nThey are disjoint and their union is $D$.","For $\\varepsilon\\in\\{1,-1\\}$ define\n$\\overline n_{\\varepsilon}(a)=\\varepsilon\\overline n(a)$.\nOn the corresponding sign portion,\n\\[\n \\Nm(x)\\bmod m=\\overline n_{\\varepsilon}([x]).\n\\]\nThe sign cannot be dropped: $-1\\equiv2\\pmod3$, whereas\n$\\Nm(-1)=1$ and $\\Nm(2)=2$ for principal ideals of $\\ZZ$.","\\paragraph{The exact ideal filter.}\nSuppose $IJ=(x)$, $n(x)$ has the indicated sign, and\n$\\gcd(\\Nm J,m)=1$. For $c\\in\\ZZ/m\\ZZ$,\n\\[\n \\Nm I\\equiv c\\pmod m\n \\quad\\Longleftrightarrow\\quad\n \\overline n_{\\varepsilon}([x])=c(\\Nm J\\bmod m).\n\\]\nThis gives an actual finite residue target, which is allowed to be empty.\nThe product identity before cancellation does not require coprimality.","Let $[J]=C^{-1}$ in $\\Cl(\\OO)$, and let $\\mathcal G(J)$ denote the\nactual fundamental-cone generator set. For every predicate $P:\\RR\\to\\mathrm{Prop}$\nand every real $X$, the norm-preserving correspondence gives\n\\begin{align*}\n &\\#\\{x\\in\\mathcal G(J):\\Nm(x)\\leq X\\Nm J,\\ P(\\Nm(x))\\}","Let $[J]=C^{-1}$ in $\\Cl(\\OO)$, and let $\\mathcal G(J)$ denote the\nactual fundamental-cone generator set. For every predicate $P:\\RR\\to\\mathrm{Prop}$\nand every real $X$, the norm-preserving correspondence gives\n\\begin{align*}\n &\\#\\{x\\in\\mathcal G(J):\\Nm(x)\\leq X\\Nm J,\\ P(\\Nm(x))\\}\\\\\n &\\qquad=\\#\\mu(K)\\,\n \\#\\{I\\ne0:[I]=C,\\ \\Nm I\\leq X,\\ P(\\Nm I\\,\\Nm J)\\}.\n\\end{align*}\nThe same norm-preserving equivalence carries arbitrary complex weights.\nIn particular, write $G_{J,\\chi}(Y)$ for the sum of\n$\\chi(\\Nm(x)\\bmod m)$ over these generators with $\\Nm(x)\\leq Y$, and\n$T_{C,\\chi}(X)$ for the corresponding sum over ideals in class $C$.\nFor multiplicative $\\chi$ preserving $0$ and $1$, the exact identity is\n\\begin{equation}\\label{eq:weighted-torsion}\n G_{J,\\chi}(X\\Nm J)\n =\\#\\mu(K)\\,\\chi(\\Nm J\\bmod m)\\,T_{C,\\chi}(X).\n\\end{equation}\nBoth signs are included before applying this full roots-of-unity factor.\nA root of unity can interchange the sign portions, so that factor is not\nautomatically valid for either portion separately.","&\\qquad=\\#\\mu(K)\\,\n \\#\\{I\\ne0:[I]=C,\\ \\Nm I\\leq X,\\ P(\\Nm I\\,\\Nm J)\\}.\n\\end{align*}\nThe same norm-preserving equivalence carries arbitrary complex weights.\nIn particular, write $G_{J,\\chi}(Y)$ for the sum of\n$\\chi(\\Nm(x)\\bmod m)$ over these generators with $\\Nm(x)\\leq Y$, and\n$T_{C,\\chi}(X)$ for the corresponding sum over ideals in class $C$.\nFor multiplicative $\\chi$ preserving $0$ and $1$, the exact identity is\n\\begin{equation}\\label{eq:weighted-torsion}\n G_{J,\\chi}(X\\Nm J)\n =\\#\\mu(K)\\,\\chi(\\Nm J\\bmod m)\\,T_{C,\\chi}(X).\n\\end{equation}\nBoth signs are included before applying this full roots-of-unity factor.\nA root of unity can interchange the sign portions, so that factor is not\nautomatically valid for either portion separately.","=C,\\ \\Nm I\\leq X,\\ P(\\Nm I\\,\\Nm J)\\}.\n\\end{align*}\nThe same norm-preserving equivalence carries arbitrary complex weights.\nIn particular, write $G_{J,\\chi}(Y)$ for the sum of\n$\\chi(\\Nm(x)\\bmod m)$ over these generators with $\\Nm(x)\\leq Y$, and\n$T_{C,\\chi}(X)$ for the corresponding sum over ideals in class $C$.\nFor multiplicative $\\chi$ preserving $0$ and $1$, the exact identity is\n\\begin{equation}\\label{eq:weighted-torsion}\n G_{J,\\chi}(X\\Nm J)\n =\\#\\mu(K)\\,\\chi(\\Nm J\\bmod m)\\,T_{C,\\chi}(X).\n\\end{equation}\nBoth signs are included before applying this full roots-of-unity factor.\nA root of unity can interchange the sign portions, so that factor is not\nautomatically valid for either portion separately.","\\newpage\n\\paragraph{Ordinary class characters: an unconditional power saving.}\nThe preceding block proved, for every ordinary ideal class $C$,\n\\[\n A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\\qquad\n c_K=\\frac{2^{r_1}(2\\pi)^{r_2}R_K}{\\#\\mu(K)\\sqrt{\\absn{\\Delta_K}}}.\n\\]\nHere $A_C$ counts actual nonzero integral ideals, $r_1,r_2$ count the real\nand complex places, $R_K$ is the regulator, and $\\Delta_K$ is the discriminant. For a weight\n$w:\\Cl(\\OO)\\to\\CC$, define the actual finite ideal sum\n\\[\n S_w(X)=\\sum_{\\substack{I\\ne0","ary class characters: an unconditional power saving.}\nThe preceding block proved, for every ordinary ideal class $C$,\n\\[\n A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\\qquad\n c_K=\\frac{2^{r_1}(2\\pi)^{r_2}R_K}{\\#\\mu(K)\\sqrt{\\absn{\\Delta_K}}}.\n\\]\nHere $A_C$ counts actual nonzero integral ideals, $r_1,r_2$ count the real\nand complex places, $R_K$ is the regulator, and $\\Delta_K$ is the discriminant. For a weight\n$w:\\Cl(\\OO)\\to\\CC$, define the actual finite ideal sum\n\\[\n S_w(X)=\\sum_{\\substack{I\\ne0","e preceding block proved, for every ordinary ideal class $C$,\n\\[\n A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\\qquad\n c_K=\\frac{2^{r_1}(2\\pi)^{r_2}R_K}{\\#\\mu(K)\\sqrt{\\absn{\\Delta_K}}}.\n\\]\nHere $A_C$ counts actual nonzero integral ideals, $r_1,r_2$ count the real\nand complex places, $R_K$ is the regulator, and $\\Delta_K$ is the discriminant. For a weight\n$w:\\Cl(\\OO)\\to\\CC$, define the actual finite ideal sum\n\\[\n S_w(X)=\\sum_{\\substack{I\\ne0\\\\\\Nm I\\leq X}}w([I]).\n\\]\nExact grouping by ideal class gives $S_w(X)=\\sum_C w(C)A_C(X)$.\nIf $\\sum_Cw(C)=0$, the common leading coefficient cancels, so there is\n$B_w\\geq0$ such that\n\\[\n \\absn{S_w(X)}\\leq B_wX^{1-1/d}\\qquad(X\\geq1).\n\\]\nFor every nontrivial character $\\chi:\\Cl(\\OO)\\to\\CC^{\\times}$,\nthe finite character sum is zero. Thus the displayed estimate applies\nwithout an additional arithmetic cancellation hypothesis.\nThis conclusion concerns ordinary class characters. It does not identify\nthem with the further norm-residue characters needed for Frobenius separation.","Nm I\\leq X}}w([I]).\n\\]\nExact grouping by ideal class gives $S_w(X)=\\sum_C w(C)A_C(X)$.\nIf $\\sum_Cw(C)=0$, the common leading coefficient cancels, so there is\n$B_w\\geq0$ such that\n\\[\n \\absn{S_w(X)}\\leq B_wX^{1-1/d}\\qquad(X\\geq1).\n\\]\nFor every nontrivial character $\\chi:\\Cl(\\OO)\\to\\CC^{\\times}$,\nthe finite character sum is zero. Thus the displayed estimate applies\nwithout an additional arithmetic cancellation hypothesis.\nThis conclusion concerns ordinary class characters. It does not identify\nthem with the further norm-residue characters needed for Frobenius separation.","\\paragraph{An actual Mellin endpoint.}\nThe counting functions are constant on the integer norm steps. Their\nmeasurability, and hence that of $S_w$, is proved from this fact.\nFor zero-sum $w$, define\n\\[\n F_w(h)=(1+h)\\int_1^{\\infty}x^{-2-h}S_w(x)\\,dx\n \\qquad(h\\geq0).\n\\]\nSince $1-1/d<1$, the logarithmic first moment\n\\[\n M_w=\\int_1^{\\infty}(1+\\log x)\\,\\absn{x^{-2}S_w(x)}\\,dx\n\\]\nis finite. Lean proves the explicit estimate\n$\\absn{F_w(h)-F_w(0)}\\leq hM_w$ for every $h\\geq0$, and therefore\n$F_w(h)=F_w(0)+O(h)$ as $h\\downarrow0$.\nThe function here is the displayed integral of the actual ideal sum.\nIts identification with the intended arithmetic Euler product is a separate step.","llin endpoint.}\nThe counting functions are constant on the integer norm steps. Their\nmeasurability, and hence that of $S_w$, is proved from this fact.\nFor zero-sum $w$, define\n\\[\n F_w(h)=(1+h)\\int_1^{\\infty}x^{-2-h}S_w(x)\\,dx\n \\qquad(h\\geq0).\n\\]\nSince $1-1/d<1$, the logarithmic first moment\n\\[\n M_w=\\int_1^{\\infty}(1+\\log x)\\,\\absn{x^{-2}S_w(x)}\\,dx\n\\]\nis finite. Lean proves the explicit estimate\n$\\absn{F_w(h)-F_w(0)}\\leq hM_w$ for every $h\\geq0$, and therefore\n$F_w(h)=F_w(0)+O(h)$ as $h\\downarrow0$.\nThe function here is the displayed integral of the actual ideal sum.\nIts identification with the intended arithmetic Euler product is a separate step.","constant on the integer norm steps. Their\nmeasurability, and hence that of $S_w$, is proved from this fact.\nFor zero-sum $w$, define\n\\[\n F_w(h)=(1+h)\\int_1^{\\infty}x^{-2-h}S_w(x)\\,dx\n \\qquad(h\\geq0).\n\\]\nSince $1-1/d<1$, the logarithmic first moment\n\\[\n M_w=\\int_1^{\\infty}(1+\\log x)\\,\\absn{x^{-2}S_w(x)}\\,dx\n\\]\nis finite. Lean proves the explicit estimate\n$\\absn{F_w(h)-F_w(0)}\\leq hM_w$ for every $h\\geq0$, and therefore\n$F_w(h)=F_w(0)+O(h)$ as $h\\downarrow0$.\nThe function here is the displayed integral of the actual ideal sum.\nIts identification with the intended arithmetic Euler product is a separate step.","norm steps. Their\nmeasurability, and hence that of $S_w$, is proved from this fact.\nFor zero-sum $w$, define\n\\[\n F_w(h)=(1+h)\\int_1^{\\infty}x^{-2-h}S_w(x)\\,dx\n \\qquad(h\\geq0).\n\\]\nSince $1-1/d<1$, the logarithmic first moment\n\\[\n M_w=\\int_1^{\\infty}(1+\\log x)\\,\\absn{x^{-2}S_w(x)}\\,dx\n\\]\nis finite. Lean proves the explicit estimate\n$\\absn{F_w(h)-F_w(0)}\\leq hM_w$ for every $h\\geq0$, and therefore\n$F_w(h)=F_w(0)+O(h)$ as $h\\downarrow0$.\nThe function here is the displayed integral of the actual ideal sum.\nIts identification with the intended arithmetic Euler product is a separate step.","rability, and hence that of $S_w$, is proved from this fact.\nFor zero-sum $w$, define\n\\[\n F_w(h)=(1+h)\\int_1^{\\infty}x^{-2-h}S_w(x)\\,dx\n \\qquad(h\\geq0).\n\\]\nSince $1-1/d<1$, the logarithmic first moment\n\\[\n M_w=\\int_1^{\\infty}(1+\\log x)\\,\\absn{x^{-2}S_w(x)}\\,dx\n\\]\nis finite. Lean proves the explicit estimate\n$\\absn{F_w(h)-F_w(0)}\\leq hM_w$ for every $h\\geq0$, and therefore\n$F_w(h)=F_w(0)+O(h)$ as $h\\downarrow0$.\nThe function here is the displayed integral of the actual ideal sum.\nIts identification with the intended arithmetic Euler product is a separate step.","For a finite family of nontrivial ordinary class characters, suppose functions\n$p,Z$ satisfy $h\\,p(h)\\to P\\ne0$, $h\\,Z(h)\\to Q\\ne0$, and, for all sufficiently\nsmall $h>0$,\n\\[\n Z(h)=\\left(p(h)\\prod_i F_{\\chi_i}(h)\\right)^e,\n \\qquad e\\in\\NN.\n\\]\nThe checked real-axis criterion then gives $e=1$ and\n$F_{\\chi_i}(0)\\ne0$ for every $i$. The pole limits and product identity\nremain hypotheses of this application.","\\newpage\n\\paragraph{An actual unit action on ideal residues.}\nFix $m>0$ and a nonzero integral ideal $J$.\nMultiplication by $u\\in(\\OO/m\\OO)^{\\times}$ gives a proved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,","\\newpage\n\\paragraph{An actual unit action on ideal residues.}\nFix $m>0$ and a nonzero integral ideal $J$.\nMultiplication by $u\\in(\\OO/m\\OO)^{\\times}$ gives a proved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means ne","\\newpage\n\\paragraph{An actual unit action on ideal residues.}\nFix $m>0$ and a nonzero integral ideal $J$.\nMultiplication by $u\\in(\\OO/m\\OO)^{\\times}$ gives a proved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Thei","\\newpage\n\\paragraph{An actual unit action on ideal residues.}\nFix $m>0$ and a nonzero integral ideal $J$.\nMultiplication by $u\\in(\\OO/m\\OO)^{\\times}$ gives a proved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Their vanishing follows from the actual p","\\newpage\n\\paragraph{An actual unit action on ideal residues.}\nFix $m>0$ and a nonzero integral ideal $J$.\nMultiplication by $u\\in(\\OO/m\\OO)^{\\times}$ gives a proved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Their vanishing follows from the actual permutation.","integral ideal $J$.\nMultiplication by $u\\in(\\OO/m\\OO)^{\\times}$ gives a proved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Their vanishing follows from the actual permutation.","on by $u\\in(\\OO/m\\OO)^{\\times}$ gives a proved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Their vanishing follows from the actual permutation.","ved permutation of\n$J/mJ$. Its inverse follows from $(ab-1)x\\in mJ$ when\n$ab-1\\in m\\OO$ and $x\\in J$. No surjectivity of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Their vanishing follows from the actual permutation.","ty of\n$J/mJ\\to\\OO/m\\OO$ or coprimality condition on $J$ is used.\nThe well-defined residue norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Their vanishing follows from the actual permutation.","norm on this quotient satisfies\n\\[\n \\overline n_J(u\\cdot a)=\\overline n(u)\\,\\overline n_J(a).\n\\]\nFor a multiplicative $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserving $1$, an exhibited\nunit with\n\\begin{equation}\\label{eq:unit-witness}\n u\\in(\\OO/m\\OO)^{\\times},\\qquad \\chi(\\overline n(u))\\ne1\n\\end{equation}\ntherefore forces\n\\[\n \\sum_{a\\in J/mJ}\\chi\\bigl(s\\,\\overline n_J(a)\\bigr)=0\n \\qquad(s\\in\\ZZ/m\\ZZ).\n\\]\nTaking $s=1,-1$ gives the two finite means needed by the sign-corrected\nnorm weights. Their vanishing follows from the actual permutation.","\\paragraph{The assembled ideal norm-character estimate.}\n\\begin{proposition}\\label{prop:norm-character}\nLet $C\\in\\Cl(\\OO)$, let $J\\ne0$ be an integral ideal with $[J]=C^{-1}$,\nand let $m>0$. Let $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserve multiplication,\n$0$, and $1$. Assume a residue-ring unit satisfying\n\\eqref{eq:unit-witness}, and assume\n$\\chi(\\Nm J\\bmod m)\\ne0$. Then\n\\[\n T_{C,\\chi}(X)\n =\\sum_{\\substack{I\\ne0,\\ [I]=C","\\paragraph{The assembled ideal norm-character estimate.}\n\\begin{proposition}\\label{prop:norm-character}\nLet $C\\in\\Cl(\\OO)$, let $J\\ne0$ be an integral ideal with $[J]=C^{-1}$,\nand let $m>0$. Let $\\chi:\\ZZ/m\\ZZ\\to\\CC$ preserve multiplication,\n$0$, and $1$. Assume a residue-ring unit satisfying\n\\eqref{eq:unit-witness}, and assume\n$\\chi(\\Nm J\\bmod m)\\ne0$. Then\n\\[\n T_{C,\\chi}(X)\n =\\sum_{\\substack{I\\ne0,\\ [I]=C\\\\\\Nm I\\leq X}}\n       \\chi(\\Nm I\\bmod m)\n =O_{K,J,m,\\chi}(X^{1-1/d}).\n\\]\nMore precisely, some $B\\geq0$ bounds its absolute value by\n$BX^{1-1/d}$ for every real $X\\geq1$.\n\\end{proposition}","Nm I\\leq X}}\n       \\chi(\\Nm I\\bmod m)\n =O_{K,J,m,\\chi}(X^{1-1/d}).\n\\]\nMore precisely, some $B\\geq0$ bounds its absolute value by\n$BX^{1-1/d}$ for every real $X\\geq1$.\n\\end{proposition}","The assembly identifies the residue sum with the sum over actual generators,\nrecombines the sign portions, and uses $Y=t^d$ to obtain\n$\\absn{G_{J,\\chi}(Y)}\\leq B_0(Y^{1-1/d}+1)$.\nEquation~\\eqref{eq:weighted-torsion} then permits division by the nonzero\nfactor $\\#\\mu(K)\\chi(\\Nm J\\bmod m)$.\nThe final theorem supplies its finite means and counting estimates internally.\nFor $d=1$ the exponent is zero and the conclusion is a bounded sum, so the\nargument does not require $d>1$.","tors,\nrecombines the sign portions, and uses $Y=t^d$ to obtain\n$\\absn{G_{J,\\chi}(Y)}\\leq B_0(Y^{1-1/d}+1)$.\nEquation~\\eqref{eq:weighted-torsion} then permits division by the nonzero\nfactor $\\#\\mu(K)\\chi(\\Nm J\\bmod m)$.\nThe final theorem supplies its finite means and counting estimates internally.\nFor $d=1$ the exponent is zero and the conclusion is a bounded sum, so the\nargument does not require $d>1$.","\\paragraph{What remains open.}\nNontriviality of $\\chi$ on the larger residue ring does not supply\n\\eqref{eq:unit-witness}: its restriction to the image of the norm on units\nmay be trivial. Witness existence for the intended characters, their\nremaining norm-image cases, and suitable inverse-class representatives with\n$\\chi(\\Nm J)\\ne0$ are still arithmetic obligations.\nMeasurability and the Mellin attachment for these norm-character sums remain\nto be checked. The compiled Mellin results above concern ordinary class characters.\nThe resulting estimates must also be attached to the required Euler-product\nidentities and prime-occurrence argument. The conditional pole criterion\ndoes not supply those identities. Claims 9.1, 9.2, and 11.1 remain partia","\\paragraph{What remains open.}\nNontriviality of $\\chi$ on the larger residue ring does not supply\n\\eqref{eq:unit-witness}: its restriction to the image of the norm on units\nmay be trivial. Witness existence for the intended characters, their\nremaining norm-image cases, and suitable inverse-class representatives with\n$\\chi(\\Nm J)\\ne0$ are still arithmetic obligations.\nMeasurability and the Mellin attachment for these norm-character sums remain\nto be checked. The compiled Mellin results above concern ordinary class characters.\nThe resulting estimates must also be attached to the required Euler-product\nidentities and prime-occurrence argument. The conditional pole criterion\ndoes not supply those identities. Claims 9.1, 9.2, and 11.1 remain partial,\nincluding the separate local compatibility obligations. The ledger stays\n\\textbf{17 checked / 3 partial}.","\\special{pdf:trailerid [<0e9d864536f570d7480471e0f99695cb><0e9d864536f570d7480471e0f99695cb>]}\n\\begin{abstract}\nLet \\(K\\) be a nonarchimedean local field. We resolve several finite layers that occur inside the local Langlands correspondence for general linear groups. Frobenius semisimple Weil Deligne parameters first split into irreducible Weil atoms and finite monodromy chains. In the tame case, character projectors show that every irreducible atom is one transitive Frobenius orbit, and Mackey theory gives an exact greatest common divisor and least common multiple rule for tensor packets. Regular finite field characters produce primitive cuspidal projectors and finite Gauss phases. For wild data, projective intertwiners recover an alternating commutator form. Its radical free quotient determines a finite He","The local Langlands correspondence for \\(\\GL_n(K)\\) is an established theorem. It matches irreducible smooth representations of \\(\\GL_n(K)\\) with Frobenius semisimple Weil Deligne representations of dimension \\(n\\). Its proof and normalization draw on global methods, harmonic analysis, types, local factors, and the geometry of Shimura varieties \\cite{Henniart2000,HarrisTaylor2001,BushnellHenniart2006}. This chapter asks a narrower structural question. Which parts of the local correspondence admit finite packet coordinates whose projectors, transfers, phases, and gluing laws can be proved directly?","Let \\(W_K\\) be the Weil group of \\(K\\), let \\(I_K\\) be inertia, let \\(P_K\\) be wild inertia, and choose a geometric Frobenius lift \\(\\Phi\\). A Weil Deligne representation is a pair \\((r,N)\\) with\n\\[\nr(w)Nr(w)^{-1}=\\abs{w}_K N.\n\\]","\\begin{standardfact}[Indecomposable monodromy blocks]\\label{fact:spdecomp}\nEvery Frobenius semisimple Weil Deligne representation is a direct sum of blocks\n\\[\n\\operatorname{Sp}_m(\\tau),\n\\]\nwhere \\(\\tau\\) is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length \\(m\\).\n\\end{standardfact}","\\begin{definition}[Monodromy packet chain]\\label{def:monodromy_chain}\nLet \\(V_0,\\ldots,V_{m-1}\\) be copies of the space of \\(\\tau\\). Put\n\\[\nV=\\bigoplus_{j=0}^{m-1}V_j.\n\\]\nA monodromy packet chain is a nilpotent operator \\(N\\) with\n\\[\nN(V_j)\\subseteq V_{j-1}\n\\]\nfor \\(j>0\\), with \\(N(V_0)=0\\), together with a Frobenius operator whose conjugation scales \\(N\\) by \\(q^{-1}\\).\n\\end{definition}","\\begin{proposition}[Exact block reconstruction]\\label{prop:monodromy_reconstruction}\nLet \\(P_j\\) be the projector onto \\(V_j\\). Then\n\\[\nN=\\sum_{j=1}^{m-1}P_{j-1}NP_j,\n\\qquad\nN^m=0.\n\\]\nEvery power \\(N^r\\) is the sum of the packet paths of length \\(r\\) along the chain.\n\\end{proposition}","\\begin{proof}\nThe projectors resolve the identity. Since \\(N\\) maps only from \\(V_j\\) to \\(V_{j-1}\\), every block except the displayed nearest neighbor blocks vanishes. A product of \\(r\\) nonzero blocks moves down exactly \\(r\\) levels. No path of length \\(m\\) exists, so \\(N^m=0\\).\n\\end{proof}","\\begin{proposition}[Scalar covariance model]\\label{prop:scalar_covariance}\nLet\n\\[\nF=\\operatorname{diag}(1,q,q^2,\\ldots,q^{m-1})\n\\]\nand let \\(N_{j-1,j}=1\\). Then\n\\[\nFNF^{-1}=q^{-1}N.\n\\]\n\\end{proposition}","\\begin{proof}\nThe only nonzero entry in column \\(j\\) and row \\(j-1\\) is multiplied by \\(q^{j-1}q^{-j}=q^{-1}\\) under conjugation.\n\\end{proof}","\\begin{theorem}[Formal atomic reconstruction]\\label{thm:formal_atomic}\nAssume that every irreducible Weil representation \\(\\tau\\) has been assigned a supercuspidal representation \\(\\sigma_\\tau\\), compatibly with isomorphism and unramified twist. Then the rules\n\\[\n\\operatorname{Sp}_m(\\tau)\n\\longmapsto\n\\operatorname{St}_m(\\sigma_\\tau)\n\\]\nand normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.\n\\end{theorem}","\\begin{remark}\\label{rem:atomic_boundary}\n\\Cref{thm:formal_atomic} is a reconstruction theorem. It does not produce the atomic assignment \\(\\tau\\mapsto\\sigma_\\tau\\), and it does not prove the required local factor identities.\n\\end{remark}","Assume that \\(P_K\\) acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let \\(t\\) generate a quotient of order \\(M\\), and put \\(\\zeta_M=\\e^{2\\pi i/M}\\). For \\(j\\in\\ZZ/M\\ZZ\\), define\n\\[\nP_j\n=\n\\frac1M\n\\sum_{a=0}^{M-1}\n\\zeta_M^{-ja}r(t)^a.\n\\]","\\begin{proposition}[Tame character projectors]\\label{prop:tame_projectors}\nThe operators \\(P_j\\) are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by\n\\[\n\\Phi P_j\\Phi^{-1}=P_{qj}.\n\\]\n\\end{proposition}","\\begin{proof}\nThe first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of \\cref{sec:wd}, the relation is \\(\\Phi^{-1}t\\Phi=t^q\\). Equivalently, \\(\\Phi t\\Phi^{-1}=t^{q^{-1}}\\), with inverse taken modulo \\(M\\) since \\(\\gcd(q,M)=1\\). Conjugating the Fourier sum and changing its summation index therefore gives \\(\\Phi P_j\\Phi^{-1}=P_{qj}\\). The inverse Frobenius convention reverses the label direction while preserving the same orbits.\n\\end{proof}","\\begin{theorem}[Tame orbit induction]\\label{thm:tame_induction}\nLet \\(\\tau\\) be an irreducible complex representation of \\(W_K\\) with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length \\(f\\). If \\(K_f/K\\) is the unramified extension of degree \\(f\\), there is a character \\(\\xi\\) of \\(W_{K_f}\\) such that\n\\[\n\\boxed{\n\\tau\\cong\\Ind_{W_{K_f}}^{W_K}\\xi.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe nonzero tame packets are permuted by Frobenius. A union of Frobenius orbits is stable under \\(W_K\\), so irreducibility forces one orbit. Choose one packet \\(V_\\theta\\). Its stabilizer is \\(W_{K_f}=\\langle I_K,\\Phi^f\\rangle\\). A nonzero proper stable subspace of \\(V_\\theta\\) would generate a nonzero proper \\(W_K\\) stable subspace after summing its Frobenius translates. Thus \\(V_\\theta\\) is irreducible for the stabilizer. Inertia acts on it by the scalar character \\(\\theta\\), and the remaining action is generated by \\(\\Phi^f\\). The stabilizer action on this packet is therefore one dimensional. Induction reconstructs the complete orbit.\n\\end{proof}","orbits is stable under \\(W_K\\), so irreducibility forces one orbit. Choose one packet \\(V_\\theta\\). Its stabilizer is \\(W_{K_f}=\\langle I_K,\\Phi^f\\rangle\\). A nonzero proper stable subspace of \\(V_\\theta\\) would generate a nonzero proper \\(W_K\\) stable subspace after summing its Frobenius translates. Thus \\(V_\\theta\\) is irreducible for the stabilizer. Inertia acts on it by the scalar character \\(\\theta\\), and the remaining action is generated by \\(\\Phi^f\\). The stabilizer action on this packet is therefore one dimensional. Induction reconstructs the complete orbit.\n\\end{proof}","ces one orbit. Choose one packet \\(V_\\theta\\). Its stabilizer is \\(W_{K_f}=\\langle I_K,\\Phi^f\\rangle\\). A nonzero proper stable subspace of \\(V_\\theta\\) would generate a nonzero proper \\(W_K\\) stable subspace after summing its Frobenius translates. Thus \\(V_\\theta\\) is irreducible for the stabilizer. Inertia acts on it by the scalar character \\(\\theta\\), and the remaining action is generated by \\(\\Phi^f\\). The stabilizer action on this packet is therefore one dimensional. Induction reconstructs the complete orbit.\n\\end{proof}","\\begin{corollary}[Dimension and regularity]\\label{cor:tame_dimension}\nThe dimension of \\(\\tau\\) equals the Frobenius orbit length of its tame inertia character. The inducing character is regular, meaning that no smaller positive power of Frobenius fixes it.\n\\end{corollary}","\\begin{proof}\nThe induced representation has one line for every Frobenius translate of the chosen character. A shorter orbit would produce a proper stabilizer and contradict the minimality of \\(f\\).\n\\end{proof}","\\begin{proposition}[Dual and self dual tame packets]\\label{prop:tame_dual}\nFor a tame atom\n\\[\n\\tau_\\xi=\\Ind_{W_{K_f}}^{W_K}\\xi,\n\\]\nwe have\n\\[\n\\tau_\\xi^\\vee\n\\cong\n\\Ind_{W_{K_f}}^{W_K}\\xi^{-1}.\n\\]\nThe atom is self dual exactly when\n\\[\n\\xi^{-1}=\\xi^{\\Phi^j}\n\\]\nfor some \\(j\\in\\ZZ/f\\ZZ\\).\n\\end{proposition}","\\begin{proof}\nDuality commutes with finite index induction, giving the first statement. Two regular induced tame atoms are isomorphic exactly when their inducing characters lie in the same Frobenius orbit. Applying this to \\(\\xi\\) and \\(\\xi^{-1}\\) gives the criterion.\n\\end{proof}","\\begin{proposition}[Unramified twist]\\label{prop:tame_twist}\nLet \\(\\nu\\) be an unramified character of \\(W_K\\). Then\n\\[\n\\tau_\\xi\\otimes\\nu\n\\cong\n\\Ind_{W_{K_f}}^{W_K}\n\\left(\n\\xi\\,\\nu|_{W_{K_f}}\n\\right).\n\\]\nThe tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar \\(a\\), the twisted cycle has closing scalar\n\\[\na\\nu(\\Phi)^f.\n\\]\n\\end{proposition}","\\begin{proof}\nThe tensor identity follows from the projection formula for induction. Since \\(\\nu\\) is trivial on inertia, it does not change any tame character packet. It multiplies every Frobenius step by \\(\\nu(\\Phi)\\), so a closed cycle of length \\(f\\) gains the factor \\(\\nu(\\Phi)^f\\).\n\\end{proof}","Characters of \\(\\FF_{q^n}^{\\times}\\) are indexed by exponents modulo \\(q^n-1\\), and Frobenius acts by multiplication by \\(q\\).","\\begin{proposition}[Möbius orbit count]\\label{prop:mobius_count}\nThe number of regular Frobenius orbits of length \\(n\\) is\n\\[\n\\boxed{\nc_n(q)\n=\n\\frac1n\n\\sum_{d\\mid n}\n\\mu(d)\n\\bigl(q^{n/d}-1\\bigr).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nFor every divisor \\(r\\mid n\\), the characters fixed by \\(\\Frob^r\\) are exactly those descending to \\(\\FF_{q^r}^{\\times}\\), and there are \\(q^r-1\\) of them. Möbius inversion gives the number of characters of exact orbit length \\(n\\). Division by \\(n\\) gives the number of orbits.\n\\end{proof}","\\begin{example}[Degree four over \\(\\FF_2\\)]\\label{ex:degree_four}\nThe regular exponent orbits modulo fifteen are\n\\[\n(1,2,4,8),\n\\qquad\n(3,6,12,9),\n\\qquad\n(7,14,13,11).\n\\]\nThe formula gives\n\\[\n\\frac14\\bigl((2^4-1)-(2^2-1)\\bigr)=3.\n\\]\n\\end{example}","Let\n\\[\nG_n=\\GL_n(\\FF_q),\n\\qquad\nT_n\\cong\\FF_{q^n}^{\\times}\n\\]\nwith \\(T_n\\) an elliptic maximal torus.","\\begin{standardfact}[Regular elliptic torus characters]\\label{fact:deligne_lusztig}\nA regular character \\(\\theta:T_n\\to\\CC^{\\times}\\) determines an irreducible cuspidal representation \\(\\kappa_\\theta\\) of \\(G_n\\). Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of \\(G_n\\) occurs in this way.\n\\end{standardfact}","This is the finite field input from Deligne and Lusztig theory \\cite{DeligneLusztig1976}. We use standard finite group character theory for the central idempotents \\cite{Serre1977,Isaacs1976}. Once \\(\\kappa_\\theta\\) is known, its block is selected by a central idempotent.","\\begin{proposition}[Primitive cuspidal idempotent]\\label{prop:cuspidal_idempotent}\nLet \\(\\chi_\\theta\\) be the irreducible character of \\(\\kappa_\\theta\\), and let \\(d_\\theta=\\chi_\\theta(1)\\). Then\n\\[\ne_\\theta\n=\n\\frac{d_\\theta}{\\abs{G_n}}\n\\sum_{g\\in G_n}\n\\chi_\\theta(g^{-1})g\n\\]\nis the primitive central idempotent selecting the \\(\\kappa_\\theta\\) block in \\(\\CC[G_n]\\).\n\\end{proposition}","\\begin{proof}\nOn an irreducible representation of character \\(\\psi\\), the central element acts by a scalar. Taking traces shows that the scalar is\n\\[\n\\frac{d_\\theta}{\\abs{G_n}\\psi(1)}\n\\sum_{g\\in G_n}\n\\chi_\\theta(g^{-1})\\psi(g).\n\\]\nCharacter orthogonality makes this scalar one for \\(\\psi=\\chi_\\theta\\) and zero otherwise. The element is therefore central, idempotent, and primitive.\n\\end{proof}","\\begin{proposition}[Parabolic annihilation]\\label{prop:parabolic_annihilation}\nLet \\(P=LU\\) be a proper parabolic subgroup of \\(G_n\\), and put\n\\[\ne_U=\\frac1{\\abs U}\\sum_{u\\in U}u.\n\\]\nIf \\(\\kappa_\\theta\\) is cuspidal, then\n\\[\n\\boxed{\ne_Ue_\\theta=e_\\theta e_U=0.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe operator \\(e_U\\) projects onto the subspace of \\(U\\) fixed vectors. A cuspidal representation has no nonzero fixed vector under the unipotent radical of a proper parabolic. Thus \\(e_U\\) acts by zero on the \\(\\kappa_\\theta\\) block. Since \\(e_\\theta\\) is central, both products vanish.\n\\end{proof}","The depth zero construction inflates a finite cuspidal representation to \\(\\GL_n(\\OO_K)\\), extends it to the standard compact modulo center subgroup, and then compactly induces it to \\(\\GL_n(K)\\). This is classical type theory \\cite{BushnellKutzko1993,BushnellHenniart2006}.","The induced tame Weil atom has an elementary cycle invariant. In an orbit basis \\(e_0,\\ldots,e_{n-1}\\), write\n\\[\n\\tau(\\Phi)e_j=e_{j+1}\n\\]\nfor \\(j<n-1\\), and\n\\[\n\\tau(\\Phi)e_{n-1}=ae_0.\n\\]","\\begin{proposition}[Frobenius cycle polynomial]\\label{prop:frobenius_cycle}\nThe Frobenius transfer matrix has characteristic polynomial\n\\[\nX^n-a\n\\]\nand determinant\n\\[\n\\boxed{\n\\det\\tau(\\Phi)=(-1)^{n-1}a.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe matrix is the companion matrix of \\(X^n-a\\). Its only nonzero determinant term is the closed \\(n\\) cycle. That cycle has sign \\((-1)^{n-1}\\), and the closing edge contributes \\(a\\).\n\\end{proof}","\\begin{corollary}[Unramified orientation character]\\label{cor:orientation}\nThe sign \\((-1)^{n-1}\\) is the orientation of the closed Frobenius packet cycle. It is unchanged by rescaling the individual orbit basis vectors.\n\\end{corollary}","Let\n\\[\n\\tau_\\xi=\\Ind_{W_{K_n}}^{W_K}\\xi,\n\\qquad\n\\tau_\\eta=\\Ind_{W_{K_m}}^{W_K}\\eta,\n\\]\nwhere \\(K_n/K\\) and \\(K_m/K\\) are unramified. Put\n\\[\nd=\\gcd(n,m),\n\\qquad\nL=\\lcm(n,m).\n\\]","\\begin{theorem}[Greatest common divisor and least common multiple packets]\\label{thm:gcd_lcm}\nThere is a decomposition\n\\[\n\\boxed{\n\\tau_\\xi\\otimes\\tau_\\eta\n\\cong\n\\bigoplus_{i=0}^{d-1}\n\\Ind_{W_{K_L}}^{W_K}\n\\left(\n\\xi_{L,n}\\eta_{L,m}^{\\Phi^i}\n\\right),\n}\n\\]\nwhere \\(\\xi_{L,n}\\) and \\(\\eta_{L,m}\\) are the norm lifts to \\(K_L\\).\n\\end{theorem}","\\begin{proof}\nApply Mackey decomposition \\cite{Mackey1952}. The double cosets\n\\[\nW_{K_n}\\backslash W_K/W_{K_m}\n\\]\nare the orbits of simultaneous translation on \\(\\ZZ/n\\ZZ\\times\\ZZ/m\\ZZ\\). There are \\(d\\) orbits, each of length \\(L\\). The stabilizer of a point in each orbit is \\(W_{K_L}\\). The second character is shifted by the chosen Frobenius representative. Summing the induced terms over the double cosets gives the decomposition.\n\\end{proof}","\\begin{corollary}[Packet count and dimension]\\label{cor:gcd_lcm_count}\nThe tensor product has \\(d\\) induced interaction packets, each with carrier dimension and Frobenius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","e tensor product has \\(d\\) induced interaction packets, each with carrier dimension and Frobenius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","action packets, each with carrier dimension and Frobenius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","benius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","ength \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","Represent the residue characters by exponents\n\\[\na\\pmod{q^n-1},\n\\qquad\nb\\pmod{q^m-1}.\n\\]","\\begin{proposition}[Refined exponent formula]\\label{prop:refined_exponent}\nThe \\(i\\)th interaction packet on \\(\\FF_{q^L}^{\\times}\\) has exponent\n\\[\n\\boxed{\nc_i\n=\na\\frac{q^L-1}{q^n-1}\n+bq^i\\frac{q^L-1}{q^m-1}\n\\pmod{q^L-1}.\n}\n\\]\nIt is trivial on tame inertia exactly when \\(c_i=0\\).\n\\end{proposition}","\\begin{proof}\nNorm lifting a character exponent from \\(\\FF_{q^n}^{\\times}\\) to \\(\\FF_{q^L}^{\\times}\\) multiplies the exponent by \\((q^L-1)/(q^n-1)\\). The Frobenius shift multiplies the second exponent by \\(q^i\\). Multiplication of characters adds exponents. Triviality is equivalent to the resulting exponent being zero modulo \\(q^L-1\\).\n\\end{proof}","Let \\(V\\) be one interaction packet. If its inertia character is nontrivial, then \\(V^{I_K}=0\\). If the inertia character is trivial, Frobenius acts on the invariant packet by a companion cycle.","\\begin{proposition}[Local polynomial of one tame packet]\\label{prop:tame_local_polynomial}\nSuppose an interaction packet has orbit length \\(r\\) and closing scalar \\(a\\). Its local polynomial is\n\\[\n\\det\\left(I-u\\Phi\\mid V^{I_K}\\right)\n=\n\\begin{cases}\n1-a u^r,&\\text{if the inertia character is trivial},","\\begin{proposition}[Local polynomial of one tame packet]\\label{prop:tame_local_polynomial}\nSuppose an interaction packet has orbit length \\(r\\) and closing scalar \\(a\\). Its local polynomial is\n\\[\n\\det\\left(I-u\\Phi\\mid V^{I_K}\\right)\n=\n\\begin{cases}\n1-a u^r,&\\text{if the inertia character is trivial},\\\\\n1,&\\text{otherwise}.\n\\end{cases}\n\\]\n\\end{proposition}","\\begin{proof}\nFor nontrivial inertia, the invariant space is zero. In the trivial case, Frobenius on the invariant packet is the companion matrix of \\(X^r-a\\). Substituting \\(X=u^{-1}\\) and multiplying by \\(u^r\\) gives \\(1-a u^r\\).\n\\end{proof}","\\begin{corollary}[Tame tensor local factor]\\label{cor:tame_tensor_factor}\nThe local factor of the tame tensor product is the product of\n\\[\n(1-a_i u^{r_i})^{-1}\n\\]\nover precisely those refined packets whose inertia exponent vanishes.\n\\end{corollary}","\\begin{proposition}[Trivial tensor packet criterion]\\label{prop:trivial_tensor_packet}\nFor every refined interaction packet, let \\(r_i\\) be its cycle length and \\(a_i\\) its closing scalar. The dimension of the \\(W_K\\) fixed space in \\(\\tau_\\xi\\otimes\\tau_\\eta\\) equals the number of packets satisfying both\n\\[\nc_i=0\n\\qquad\\text{and}\\qquad\na_i=1.\n\\]\n\\end{proposition}","\\begin{proof}\nA nonzero fixed vector must lie in an inertia trivial packet, so \\(c_i=0\\). On that packet, Frobenius is a companion cycle with characteristic polynomial \\(X^{r_i}-a_i\\). Its fixed space is the eigenspace for eigenvalue one. This eigenspace is one dimensional exactly when \\(a_i=1\\), and it is zero otherwise. Summing over the refined packets proves the formula.\n\\end{proof}","\\begin{corollary}[Tame Hom test]\\label{cor:tame_hom}\nFor irreducible tame atoms,\n\\[\n\\dim\\Hom_{W_K}(\\tau_\\xi^\\vee,\\tau_\\eta)\n\\]\nis zero or one, and it is one exactly when the refined tensor packet test in \\cref{prop:trivial_tensor_packet} finds one trivial packet.\n\\end{corollary}","\\begin{proof}\nThe Hom space is the fixed space of \\(\\tau_\\xi\\otimes\\tau_\\eta\\). Schur's lemma gives dimension zero or one, and \\cref{prop:trivial_tensor_packet} gives the packet criterion.\n\\end{proof}","\\begin{proposition}[Conductor of a nontrivial tame atom]\\label{prop:tame_conductor}\nLet \\(\\tau\\) be an irreducible tame Weil representation of dimension \\(n>1\\). Then\n\\[\n\\Swan(\\tau)=0,\n\\qquad\n\\dim\\tau^{I_K}=0,\n\\qquad\n\\Art(\\tau)=n.\n\\]\n\\end{proposition}","\\begin{proof}\nWild inertia acts trivially, so the Swan conductor is zero. An inertia fixed vector would generate a nonzero Frobenius stable sum of trivial character packets. Irreducibility would then make inertia act trivially on all of \\(\\tau\\). An irreducible representation of the quotient generated by Frobenius would be one dimensional, contrary to \\(n>1\\). The Artin conductor is therefore\n\\[\n\\dim\\tau-\\dim\\tau^{I_K}+\\Swan(\\tau)=n.\n\\]\n\\end{proof}","Let \\(k=\\FF_q\\), let \\(\\psi:k\\to\\CC^{\\times}\\) be a nontrivial additive character, and let \\(\\chi:k^{\\times}\\to\\CC^{\\times}\\) be a multiplicative character. Define\n\\[\nG(\\chi,\\psi)\n=\n\\sum_{x\\in k^{\\times}}\\chi(x)\\psi(x).\n\\]\nThe standard theory of Gauss and Jacobi sums will be used in the form recorded in \\cite{DavenportHasse1935,BerndtEvansWilliams1998}.","\\begin{proposition}[Gauss magnitude]\\label{prop:gauss_magnitude}\nIf \\(\\chi\\) is nontrivial, then\n\\[\n\\boxed{\n\\abs{G(\\chi,\\psi)}=q^{1/2}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nCompute\n\\[\n\\abs{G(\\chi,\\psi)}^2\n=\n\\sum_{x,y\\in k^{\\times}}\n\\chi(xy^{-1})\\psi(x-y).\n\\]\nWrite \\(x=ty\\). The inner sum over \\(y\\) equals \\(q-1\\) when \\(t=1\\) and equals \\(-1\\) otherwise. Since \\(\\sum_{t\\in k^{\\times}}\\chi(t)=0\\), the result is \\(q\\).\n\\end{proof}","\\begin{definition}[Normalized Gauss phase]\\label{def:gauss_phase}\nFor nontrivial \\(\\chi\\), define\n\\[\n\\gamma(\\chi,\\psi)\n=\nq^{-1/2}G(\\chi,\\psi).\n\\]\nThen \\(\\abs{\\gamma(\\chi,\\psi)}=1\\).\n\\end{definition}","\\begin{proposition}[Duality of Gauss phases]\\label{prop:gauss_duality}\nFor nontrivial \\(\\chi\\),\n\\[\n\\boxed{\n\\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)\n=\n\\chi(-1).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe standard Gauss identity gives\n\\[\nG(\\chi,\\psi)G(\\chi^{-1},\\psi)=\\chi(-1)q.\n\\]\nDivide by \\(q\\).\n\\end{proof}","\\begin{standardfact}[Hasse and Davenport lifting]\\label{fact:hasse_davenport}\nLet \\(k_r/k\\) be the extension of degree \\(r\\). Lift \\(\\chi\\) by the norm and \\(\\psi\\) by the trace. Then the lifted Gauss sum satisfies the Hasse and Davenport product relation, with a sign determined by the chosen additive character normalization.\n\\end{standardfact}","Let \\(p\\) be odd. Let \\(X\\) be an \\(r\\) dimensional vector space over \\(\\FF_p\\), and put\n\\[\nV=X\\oplus X^*.\n\\]\nDefine the alternating form\n\\[\nb\\bigl((x,\\xi),(y,\\eta)\\bigr)\n=\n\\eta(x)-\\xi(y).\n\\]\nLet \\(Z=\\FF_p\\), and define the Heisenberg group\n\\[\n\\frakH(V)=V\\times Z\n\\]\nwith multiplication\n\\[\n(v,z)(w,t)\n=\n\\left(v+w,z+t+\\frac12 b(v,w)\\right).\n\\]\nFix the additive character\n\\[\n\\psi(z)=\\e^{2\\pi iz/p}.\n\\]","\\begin{definition}[Schrödinger operators]\\label{def:schrodinger}\nOn \\(\\calH=\\CC^X\\), define\n\\[\n\\bigl(\\rho(x,\\xi,z)f\\bigr)(u)\n=\n\\psi\\left(z+\\xi(u)+\\frac12\\xi(x)\\right)f(u+x).\n\\]\n\\end{definition}","\\begin{theorem}[Finite Schrödinger representation]\\label{thm:schrodinger_rep}\nThe map \\(\\rho:\\frakH(V)\\to\\GL(\\calH)\\) is a representation. It is irreducible, its center acts by \\(\\psi\\), and\n\\[\n\\dim\\calH=p^r=\\sqrt{\\abs V}.\n\\]\n\\end{theorem}","\\begin{proof}\nA direct calculation gives\n\\[\n\\rho(v,z)\\rho(w,t)\n=\n\\rho\\left(v+w,z+t+\\frac12b(v,w)\\right).\n\\]\nTranslations by \\(X\\) and multiplications by characters from \\(X^*\\) generate the full matrix algebra on \\(\\CC^X\\). Indeed, the character multipliers give all diagonal projectors by finite Fourier inversion, and translations move each coordinate line to every other coordinate line. The commutant is therefore scalar, so the representation is irreducible. The dimension is \\(\\abs X=p^r\\).\n\\end{proof}","\\frac12b(v,w)\\right).\n\\]\nTranslations by \\(X\\) and multiplications by characters from \\(X^*\\) generate the full matrix algebra on \\(\\CC^X\\). Indeed, the character multipliers give all diagonal projectors by finite Fourier inversion, and translations move each coordinate line to every other coordinate line. The commutant is therefore scalar, so the representation is irreducible. The dimension is \\(\\abs X=p^r\\).\n\\end{proof}","\\begin{proposition}[Character support]\\label{prop:heisenberg_trace}\nThe character of the Schrödinger representation satisfies\n\\[\n\\boxed{\n\\Tr\\rho(v,z)\n=\n\\begin{cases}\np^r\\psi(z),&v=0,\\\\\n0,&v\\ne0.\n\\end{cases}\n}\n\\]\n\\end{proposition}","\\begin{proof}\nIf the translation component \\(x\\) is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If \\(x=0\\) but \\(\\xi\\ne0\\), the trace is\n\\[\n\\psi(z)\\sum_{u\\in X}\\psi(\\xi(u))=0.\n\\]\nWhen \\(x=\\xi=0\\), the operator is the scalar \\(\\psi(z)\\).\n\\end{proof}","Let \\(L\\subseteq V\\) be maximal isotropic. Choose a subgroup lift \\(\\widetilde L\\subseteq\\frakH(V)\\) on which the central coordinate is zero.","\\begin{proposition}[Isotropic averaging projector]\\label{prop:isotropic_selector}\nThe operator\n\\[\nP_L\n=\n\\frac1{\\abs L}\n\\sum_{\\ell\\in\\widetilde L}\n\\rho(\\ell)\n\\]\nis an orthogonal projector. In the Schrödinger model with \\(L=X\\), it has rank one and selects the constant functions.\n\\end{proposition}","\\begin{proof}\nThe lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For \\(L=X\\), the average is the translation average\n\\[\n(P_Lf)(u)\n=\n\\frac1{\\abs X}\n\\sum_{x\\in X}f(u+x),\n\\]\nwhich is the constant function with value equal to the mean of \\(f\\). Its range is one dimensional.\n\\end{proof}","\\begin{proof}\nThe projector rank equals its trace. The identity element contributes \\(p^r\\), while every nonidentity element of the lifted isotropic subgroup is noncentral and has trace zero by \\cref{prop:heisenberg_trace}. Since \\(\\abs L=p^r\\), we get\n\\[\n\\Tr P_L\n=\n\\frac{p^r}{p^r}=1.\n\\]\nThus the projector has rank one.\n\\end{proof}","Let \\(A\\) be a finite abelian group, and let\n\\[\nc:A\\times A\\to\\CC^{\\times}\n\\]\nbe a normalized two cocycle. The twisted group algebra \\(\\CC^c[A]\\) has basis \\((u_a)_{a\\in A}\\) and multiplication\n\\[\nu_au_b=c(a,b)u_{a+b}.\n\\]\nDefine the commutator bicharacter\n\\[\nb_c(a,b)=c(a,b)c(b,a)^{-1}.\n\\]","\\begin{proposition}[Scalar choice invariance]\\label{prop:cocycle_choice}\nReplacing \\(u_a\\) by \\(\\lambda_au_a\\), with \\(\\lambda_a\\in\\CC^{\\times}\\), changes \\(c\\) by a coboundary and leaves \\(b_c\\) unchanged.\n\\end{proposition}","\\begin{proof}\nThe new cocycle is\n\\[\nc'(a,b)\n=\n\\lambda_a\\lambda_b\\lambda_{a+b}^{-1}c(a,b).\n\\]\nThe scalar coboundary factor is symmetric in \\(a\\) and \\(b\\), so it cancels from \\(c'(a,b)c'(b,a)^{-1}\\).\n\\end{proof}","\\begin{proposition}[Alternating commutator form]\\label{prop:commutator_form}\nThe map \\(b_c\\) is an alternating bicharacter. Its radical\n\\[\nR=\\{a\\in A:b_c(a,x)=1\\text{ for every }x\\in A\\}\n\\]\nis a subgroup, and \\(b_c\\) descends to a nondegenerate alternating bicharacter on \\(A/R\\).\n\\end{proposition}","\\begin{proof}\nAssociativity of the twisted product gives multiplicativity of scalar commutators in each variable. We have \\(b_c(a,a)=1\\) and \\(b_c(b,a)=b_c(a,b)^{-1}\\). The radical is the kernel of the homomorphism from \\(A\\) to its character group defined by \\(a\\mapsto b_c(a,\\cdot)\\). The induced pairing on the quotient is nondegenerate by construction.\n\\end{proof}","\\begin{theorem}[Nondegenerate twisted group algebra]\\label{thm:twisted_matrix}\nAssume that \\(b_c\\) is nondegenerate. Then \\(\\abs A\\) is a square, and\n\\[\n\\boxed{\n\\CC^c[A]\n\\cong\nM_{\\sqrt{\\abs A}}(\\CC).\n}\n\\]\nConsequently, \\(\\CC^c[A]\\) has one irreducible module up to isomorphism, of dimension \\(\\sqrt{\\abs A}\\).\n\\end{theorem}","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) p","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A fi","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central sim","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebr","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\a","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size i","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","s \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","e and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","aps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","onjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","\\begin{corollary}[Radical free envelope]\\label{cor:radical_free}\nFor an arbitrary scalar projective action of \\(A\\), the quotient\n\\[\n\\boxed{\nA^{\\mathrm{symp}}=A/\\rad b_c\n}\n\\]\nis the maximal quotient detected by noncommuting projective intertwiners. If \\(A\\) is an elementary abelian \\(p\\) group, then \\(A^{\\mathrm{symp}}\\) is a symplectic \\(\\FF_p\\) space.\n\\end{corollary}","\\begin{proof}\nEvery radical element commutes with all projective generators, so it contributes only central data. Quotienting by the radical leaves the nondegenerate commutator form. In the elementary abelian case, multiplicative values in the group of \\(p\\)th roots can be written as an alternating bilinear form over \\(\\FF_p\\).\n\\end{proof}","We now isolate a finite Clifford setting in which the quotient projective action is defined on a multiplicity space. This avoids assigning a quotient action directly to arbitrary representatives of \\(G/N\\).","\\begin{definition}[One break Clifford datum]\\label{def:one_break_datum}\nA one break Clifford datum consists of finite groups and representations\n\\[\nN\\triangleleft G,\n\\qquad\nA=G/N,\n\\qquad\n\\Pi:G\\to\\GL(\\calX),\n\\qquad\n\\rho:N\\to\\GL(W),\n\\]\nwith the following properties.","\\begin{enumerate}\n\\item The quotient \\(A\\) is a finite elementary abelian \\(p\\) group.\n\\item The representations \\(\\Pi\\) and \\(\\rho\\) are irreducible.\n\\item The isomorphism class of \\(\\rho\\) is fixed by \\(G\\).\n\\item The restriction of \\(\\Pi\\) to \\(N\\) is \\(\\rho\\) isotypic.\n\\item The quotient records one declared ramification break.\n\\end{enumerate}","The multiplicity space is\n\\[\nM_\\rho=\\Hom_N(W,\\calX).\n\\]\n\\end{definition}","The isotypic condition gives an \\(N\\) equivariant evaluation isomorphism\n\\[\nW\\otimes M_\\rho\n\\longrightarrow\n\\calX.\n\\]\nChoose a section \\(s:A\\to G\\). For every \\(a\\in A\\), choose an intertwiner \\(I_a\\) from \\(\\rho\\) to its conjugate by \\(s(a)\\). Under the evaluation isomorphism, there is a unique operator \\(T_a\\) on \\(M_\\rho\\) such that\n\\[\n\\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators \\(T_a\\) form a projective representation of \\(A\\).","n \\(N\\) equivariant evaluation isomorphism\n\\[\nW\\otimes M_\\rho\n\\longrightarrow\n\\calX.\n\\]\nChoose a section \\(s:A\\to G\\). For every \\(a\\in A\\), choose an intertwiner \\(I_a\\) from \\(\\rho\\) to its conjugate by \\(s(a)\\). Under the evaluation isomorphism, there is a unique operator \\(T_a\\) on \\(M_\\rho\\) such that\n\\[\n\\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators \\(T_a\\) form a projective representation of \\(A\\).","\\begin{proposition}[Clifford multiplicity action]\\label{prop:clifford_action}\nThe projective equivalence class of the action \\((T_a)_{a\\in A}\\) is independent of the section, the intertwiners, and the chosen isotypic evaluation isomorphism. The projective action is irreducible.\n\\end{proposition}","\\begin{proof}\nChanging an intertwiner \\(I_a\\) by a scalar changes \\(T_a\\) by the inverse scalar. Changing the isotypic evaluation isomorphism conjugates all \\(T_a\\) by one element of \\(\\GL(M_\\rho)\\). Changing the section multiplies \\(s(a)\\) by an element of \\(N\\). That factor is absorbed by the \\(\\rho\\) action on \\(W\\), leaving a scalar change on the multiplicity operator by Schur's lemma. Thus all choices give projectively equivalent actions.","If a nonzero proper subspace of \\(M_\\rho\\) were stable under all projective operators \\(T_a\\), then its tensor product with \\(W\\) would give a nonzero proper \\(G\\) stable subspace of \\(\\calX\\). This contradicts irreducibility of \\(\\Pi\\).\n\\end{proof}","\\begin{definition}[Clifford cocycle class]\\label{def:clifford_class}\nChoose scalars \\(c(a,b)\\) such that\n\\[\nT_aT_b=c(a,b)T_{a+b}.\n\\]\nThe cohomology class\n\\[\n[c_\\rho]\\in H^2(A,\\CC^{\\times})\n\\]\nis the Clifford cocycle class of the datum.\n\\end{definition}","\\begin{proposition}[Intrinsic commutator pairing]\\label{prop:one_break_form}\nThe pairing\n\\[\nb_\\rho(a,b)\n=\nc(a,b)c(b,a)^{-1}\n\\]\nis independent of every choice in \\cref{def:one_break_datum}. It is an alternating bicharacter\n\\[\nb_\\rho:A\\times A\\to\\mu_p.\n\\]\n\\end{proposition}","\\begin{proof}\nThe projective equivalence class is intrinsic by \\cref{prop:clifford_action}. Conjugation does not change scalar commutators, and scalar changes alter the cocycle by a coboundary. The commutator pairing is therefore unchanged by \\cref{prop:cocycle_choice}. Its alternating and bilinear properties follow from \\cref{prop:commutator_form}. Since every element of \\(A\\) has order \\(p\\), every value has order dividing \\(p\\).\n\\end{proof}","uivalence class is intrinsic by \\cref{prop:clifford_action}. Conjugation does not change scalar commutators, and scalar changes alter the cocycle by a coboundary. The commutator pairing is therefore unchanged by \\cref{prop:cocycle_choice}. Its alternating and bilinear properties follow from \\cref{prop:commutator_form}. Since every element of \\(A\\) has order \\(p\\), every value has order dividing \\(p\\).\n\\end{proof}","\\begin{definition}[One break symplectic envelope]\\label{def:one_break_envelope}\nThe one break symplectic envelope is\n\\[\n\\boxed{\nV_\\rho=A/\\rad b_\\rho.\n}\n\\]\n\\end{definition}","\\begin{theorem}[Nondegenerate one break packet]\\label{thm:one_break_matrix}\nIf \\(b_\\rho\\) is nondegenerate, then\n\\[\n\\CC^{c_\\rho}[A]\n\\cong\n\\End(M_\\rho),\n\\qquad\n\\dim M_\\rho=\\sqrt{\\abs A}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe projective action gives a nonzero representation of the twisted group algebra on \\(M_\\rho\\). The algebra is a full matrix algebra by \\cref{thm:twisted_matrix}. Its unique irreducible module has dimension \\(\\sqrt{\\abs A}\\). The projective multiplicity action is irreducible by \\cref{prop:clifford_action}, so it is that module.\n\\end{proof}","\\begin{proposition}[Isomorphism invariance]\\label{prop:one_break_invariance}\nThe cocycle class and the equivariant isometry class of \\(V_\\rho\\) depend only on the isomorphism class of the one break Clifford datum.\n\\end{proposition}","\\begin{proposition}[Tame stabilizer action]\\label{prop:tame_stabilizer}\nSuppose a finite group \\(C\\) acts by automorphisms of the one break Clifford datum. Then \\(C\\) acts on \\(V_\\rho\\) by symplectic automorphisms.\n\\end{proposition}","\\begin{corollary}[Heisenberg representation attached to one break]\\label{cor:one_break_heisenberg}\nFix a nontrivial central character. The envelope \\(V_\\rho\\) determines one irreducible finite Heisenberg representation up to isomorphism.\n\\end{corollary}","Let \\(\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) w","Let \\(\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galo","Let \\(\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","n subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","e vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","rivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","F_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\begin{standardfact}[Type side Heisenberg quotient]\\label{fact:type_heisenberg}\nThe pairing \\(h_\\theta\\) is nondegenerate and alternating. There is one irreducible representation \\(\\eta_\\theta\\) of \\(J_\\theta^1\\) containing \\(\\theta\\), and its dimension is\n\\[\n\\sqrt{\\abs{J_\\theta^1/H_\\theta^1}}.\n\\]\n\\end{standardfact}","This is part of the simple character and Heisenberg representation theory used in the construction of types \\cite{BushnellKutzko1993}. The extension of \\(\\eta_\\theta\\) from \\(J_\\theta^1\\) to \\(J_\\theta\\) is not unique before a normalization is chosen.","\\begin{proposition}[Type side projective commutator]\\label{prop:type_commutator}\nThe projective action of \\(J_\\theta/J_\\theta^1\\) on the Heisenberg representation preserves the commutator form on \\(V_\\theta\\). Any two choices of projective lifts give the same alternating form.\n\\end{proposition}","\\begin{proof}\nConjugation by \\(J_\\theta\\) preserves \\(\\theta\\), so it preserves the commutator pairing defined from \\(\\theta([x,y])\\). Two choices of projective lifts differ by scalars. The scalar commutator is therefore unchanged by \\cref{prop:cocycle_choice}.\n\\end{proof}","The full bridge would construct an intrinsic Galois envelope for every wild atom and identify it canonically with \\(V_\\theta\\). We prove the comparison after the required symplectic identification has been supplied.","\\begin{enumerate}\n\\item a one break Galois envelope \\((V_\\rho,b_\\rho)\\)\n\\item a type side quotient \\((V_\\theta,h_\\theta)\\)\n\\item an equivariant symplectic isometry\n\\[\n\\iota:V_\\rho\\longrightarrow V_\\theta\n\\]\nfor a finite cyclic stabilizer \\(C=\\langle c\\rangle\\)\n\\item the same nontrivial central character on the two Heisenberg groups\n\\item when \\(p=2\\), compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting \\(\\iota\\).\n\\end{enumerate}\n\\end{definition}","_\\rho)\\)\n\\item a type side quotient \\((V_\\theta,h_\\theta)\\)\n\\item an equivariant symplectic isometry\n\\[\n\\iota:V_\\rho\\longrightarrow V_\\theta\n\\]\nfor a finite cyclic stabilizer \\(C=\\langle c\\rangle\\)\n\\item the same nontrivial central character on the two Heisenberg groups\n\\item when \\(p=2\\), compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting \\(\\iota\\).\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Restricted Heisenberg bridge]\\label{thm:restricted_bridge}\nA restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map\n\\[\nU:\\calH_\\rho\\longrightarrow\\calH_\\theta\n\\]\nsuch that\n\\[\nU\\omega_\\rho(h)\n=\n\\omega_\\theta(\\iota_H(h))U\n\\]\nfor every Heisenberg element \\(h\\), and every other such map is a scalar multiple of \\(U\\).\n\\end{theorem}","ridge]\\label{thm:restricted_bridge}\nA restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map\n\\[\nU:\\calH_\\rho\\longrightarrow\\calH_\\theta\n\\]\nsuch that\n\\[\nU\\omega_\\rho(h)\n=\n\\omega_\\theta(\\iota_H(h))U\n\\]\nfor every Heisenberg element \\(h\\), and every other such map is a scalar multiple of \\(U\\).\n\\end{theorem}","\\begin{proof}\nWhen \\(p\\) is odd, the symplectic isometry and the common central character induce the standard isomorphism \\(\\iota_H\\) of the two Heisenberg groups. When \\(p=2\\), the required lift of \\(\\iota\\) is part of the bridge datum because the alternating form alone does not determine the central extension. Pulling the type side representation back through \\(\\iota_H\\) gives an irreducible Heisenberg representation with the same central character as the Galois side representation. The finite Stone von Neumann theorem gives an isomorphism. Schur's lemma gives uniqueness up to scalar.\n\\end{proof}","The stabilizer actions need not agree after choosing \\(U\\). Their discrepancy is scalar because both actions implement the same automorphisms of the irreducible Heisenberg representation.","\\begin{proposition}[Cyclic discrepancy character]\\label{prop:cyclic_discrepancy}\nAssume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group \\(C\\). Then there is a character\n\\[\n\\delta:C\\to\\CC^{\\times}\n\\]\nsuch that\n\\[\nU\\Omega_\\rho(g)\n=\n\\delta(g)\\Omega_\\theta(g)U\n\\]\nfor every \\(g\\in C\\).\n\\end{proposition}","\\begin{proof}\nFor each \\(g\\), the operator\n\\[\n\\Omega_\\theta(g)^{-1}U\\Omega_\\rho(g)U^{-1}\n\\]\ncommutes with the type side Heisenberg representation. It is therefore scalar. Since both stabilizer actions are honest representations, multiplication of group elements shows that the scalars form a character.\n\\end{proof}","\\begin{lemma}[Linearization over a cyclic stabilizer]\\label{lem:cyclic_linearization}\nEvery finite dimensional projective representation of a finite cyclic group over \\(\\CC\\) is projectively equivalent to an honest representation.\n\\end{lemma}","\\begin{proof}\nLet \\(C=\\langle c\\rangle\\) have order \\(m\\), and let \\(T\\) be a lift of the projective generator. Since \\(c^m=1\\), the operator \\(T^m\\) is scalar. Choose \\(\\lambda\\in\\CC^{\\times}\\) with \\(\\lambda^mT^m=I\\). Replacing \\(T\\) by \\(\\lambda T\\) gives an honest cyclic representation.\n\\end{proof}","\\begin{theorem}[One phase closes the cyclic bridge]\\label{thm:one_phase_bridge}\nIn a restricted bridge datum, choose linearizations of the cyclic stabilizer actions. If the two actions have the same scalar phase on the generator \\(c\\), then the Heisenberg intertwiner can be chosen to intertwine the full semidirect product action.\n\\end{theorem}","\\begin{proof}\nBy \\cref{prop:cyclic_discrepancy}, the discrepancy is a character \\(\\delta\\). Equality of the generator phase gives \\(\\delta(c)=1\\). Since \\(c\\) generates \\(C\\), the character is trivial. The chosen map \\(U\\) then intertwines every stabilizer element as well as the Heisenberg group.\n\\end{proof}","\\begin{remark}\\label{rem:bridge_scope}\nThe theorem does not construct \\(\\iota\\). It proves that, once an equivariant symplectic isometry exists, the remaining finite comparison is one dimensional for a cyclic stabilizer.\n\\end{remark}","Let \\(p\\) be odd, let \\((V,b)\\) be a finite symplectic \\(\\FF_p\\) space of dimension \\(2r\\), and let\n\\[\ng\\in\\Sp(V)\n\\]\nhave no fixed vector. Fix one of the standard finite Weil normalizations used in the literature \\cite{Gerardin1977,Neuhauser2002,GurevichHadani2007}.","\\begin{standardfact}[Fixed point free Weil trace]\\label{fact:weil_trace}\nUnder the declared normalization,\n\\[\n\\boxed{\n\\Tr\\omega(g)\n=\n\\left(\\frac{-1}{p}\\right)^r\n\\left(\\frac{\\det(g-I)}{p}\\right).\n}\n\\]\nHere \\((\\frac{\\cdot}{p})\\) denotes the quadratic character of \\(\\FF_p^{\\times}\\).\n\\end{standardfact}","\\begin{definition}[Odd finite Weil phase]\\label{def:odd_phase}\nFor fixed point free \\(g\\), define\n\\[\n\\Phi_p(g,V)\n=\n\\left(\\frac{-1}{p}\\right)^r\n\\left(\\frac{\\det(g-I)}{p}\\right).\n\\]\n\\end{definition}","\\begin{proposition}[Symplectic invariance]\\label{prop:phase_invariance}\nIf\n\\[\n\\iota:(V,b)\\longrightarrow(V',b')\n\\]\nis a symplectic isometry and \\(g'=\\iota g\\iota^{-1}\\), then\n\\[\n\\Phi_p(g,V)=\\Phi_p(g',V').\n\\]\n\\end{proposition}","\\begin{proof}\nThe dimensions agree, and conjugate matrices have the same determinant of \\(g-I\\).\n\\end{proof}","\\begin{proposition}[Orthogonal multiplicativity]\\label{prop:odd_multiplicativity}\nSuppose\n\\[\nV=V_1\\perpbig\\cdots\\perpbig V_s\n\\]\nand \\(g\\) preserves every summand. If no \\(g|_{V_j}\\) has a fixed vector, then\n\\[\n\\boxed{\n\\Phi_p(g,V)\n=\n\\prod_{j=1}^{s}\n\\Phi_p(g|_{V_j},V_j).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe half dimensions add, and\n\\[\n\\det(g-I\\mid V)\n=\n\\prod_j\\det(g-I\\mid V_j).\n\\]\nThe quadratic character is multiplicative.\n\\end{proof}","\\begin{corollary}[Characteristic polynomial certificate]\\label{cor:characteristic_phase}\nLet\n\\[\nP_g(X)=\\det(XI-g\\mid V).\n\\]\nThen\n\\[\n\\boxed{\n\\Phi_p(g,V)\n=\n\\left(\\frac{-1}{p}\\right)^r\n\\left(\\frac{P_g(1)}{p}\\right).\n}\n\\]\n\\end{corollary}","\\begin{proof}\nSince \\(\\dim V\\) is even,\n\\[\n\\det(g-I)=\\det(I-g)=P_g(1).\n\\]\n\\end{proof}","\\begin{proposition}[Fixed and moving decomposition]\\label{prop:fixed_moving}\nAssume that \\(g\\in\\Sp(V)\\) has order prime to \\(p\\). Then\n\\[\nV=\\ker(g-I)\\perpbig\\im(g-I).\n\\]\nThe restriction of the symplectic form to \\(\\im(g-I)\\) is nondegenerate, and \\(g-I\\) is invertible on that moving summand.\n\\end{proposition}","\\begin{proof}\nThe order hypothesis makes \\(g\\) semisimple, so\n\\[\nV=\\ker(g-I)\\oplus\\im(g-I).\n\\]\nIf \\(x\\in\\ker(g-I)\\) and \\(y=(g-I)z\\), then\n\\[\nb(x,y)=b(x,gz)-b(x,z)=b(g^{-1}x,z)-b(x,z)=0.\n\\]\nThus the two summands are orthogonal. Nondegeneracy on each summand follows from nondegeneracy on \\(V\\). The last statement is immediate from the direct sum decomposition.\n\\end{proof}","\\begin{definition}[Moving phase]\\label{def:moving_phase}\nUnder the hypotheses of \\cref{prop:fixed_moving}, define\n\\[\n\\Phi_p^{\\mathrm{mov}}(g,V)\n=\n\\Phi_p\\left(g|_{\\im(g-I)},\\im(g-I)\\right).\n\\]\n\\end{definition}","\\begin{proposition}[Hyperbolic phase formula]\\label{prop:hyperbolic_phase}\nLet \\(U\\) be a finite dimensional \\(\\FF_p\\) space, let \\(a\\in\\GL(U)\\) have no eigenvalue one, and let \\(g\\) act on\n\\[\nH(U)=U\\oplus U^*\n\\]\nby\n\\[\ng=(a,(a^{-1})^*).\n\\]\nThen\n\\[\n\\boxed{\n\\Phi_p(g,H(U))\n=\n\\left(\\frac{\\det a}{p}\\right).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nPut \\(d=\\dim U\\). We have\n\\[\n\\det\\bigl((a^{-1})^*-I\\bigr)\n=\n(-1)^d(\\det a)^{-1}\\det(a-I).\n\\]\nHence\n\\[\n\\det(g-I)\n=\n(-1)^d(\\det a)^{-1}\\det(a-I)^2.\n\\]\nMultiplying its quadratic character by the normalization factor \\((\\frac{-1}{p})^d\\) cancels the sign and the square. Since a quadratic character equals its inverse, the result is \\((\\frac{\\det a}{p})\\).\n\\end{proof}","\\begin{corollary}[Phase only bridge test]\\label{cor:phase_only_test}\nTwo odd characteristic moving packets have the same normalized phase whenever their moving half dimensions agree and the determinants of \\(g-I\\) lie in the same square class.\n\\end{corollary}","\\begin{proof}\nApply the definition of \\(\\Phi_p\\).\n\\end{proof}","\\begin{remark}\\label{rem:fixed_space}\nWhen \\(g\\) has fixed vectors, one first separates the fixed part and applies the formula to the moving quotient. Additional normalization data may enter the full trace on the fixed part.\n\\end{remark}","Let \\((V,b)\\) be a symplectic vector space over \\(\\FF_2\\). The alternating form alone is not enough to determine the finite phase.","\\begin{definition}[Quadratic refinement]\\label{def:quadratic_refinement}\nA quadratic refinement of \\(b\\) is a map\n\\[\nq:V\\to\\FF_2\n\\]\nsatisfying\n\\[\nq(x+y)-q(x)-q(y)=b(x,y).\n\\]\n\\end{definition}","\\begin{proposition}[Affine space of refinements]\\label{prop:refinement_affine}\nIf one quadratic refinement exists, then the set of all quadratic refinements is an affine space over \\(V^*\\).\n\\end{proposition}","\\begin{proof}\nThe difference of two refinements has zero polar form, so it is additive and hence linear over \\(\\FF_2\\). Conversely, adding a linear form does not change the polar form.\n\\end{proof}","\\begin{definition}[Arf invariant]\\label{def:arf}\nChoose a symplectic basis \\((e_i,f_i)_{i=1}^{r}\\). Define\n\\[\n\\Arf(q)\n=\n\\sum_{i=1}^{r}q(e_i)q(f_i)\n\\in\\FF_2.\n\\]\n\\end{definition}","This is the classical classification of nonsingular quadratic forms over \\(\\FF_2\\) \\cite{MilnorHusemoller1973}.","\\begin{theorem}[Dyadic Gauss sign]\\label{thm:arf_gauss}\nLet \\(\\dim V=2r\\), and let \\(q\\) be nondegenerate. Then\n\\[\n\\boxed{\n\\sum_{x\\in V}(-1)^{q(x)}\n=\n(-1)^{\\Arf(q)}2^r.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nOrthogonally decompose \\((V,q)\\) into two dimensional nonsingular planes. On a plane with Arf invariant zero, the four values of \\((-1)^q\\) sum to \\(2\\). On a plane with Arf invariant one, they sum to \\(-2\\). The total sum is the product of the plane sums. The parity of the number of anisotropic planes is the Arf invariant.\n\\end{proof}","\\begin{corollary}[Orthogonal dyadic multiplicativity]\\label{cor:arf_multiplicative}\nFor an orthogonal sum\n\\[\n(V,q)=(V_1,q_1)\\perpbig(V_2,q_2),\n\\]\nwe have\n\\[\n\\Arf(q)=\\Arf(q_1)+\\Arf(q_2)\n\\]\nand the normalized Gauss sign is the product of the two normalized signs.\n\\end{corollary}","\\begin{proposition}[The form alone does not determine the phase]\\label{prop:dyadic_need_q}\nOn every nonzero symplectic \\(\\FF_2\\) space, there are quadratic refinements of both Arf invariants. Their polar forms agree, but their normalized Gauss signs are opposite.\n\\end{proposition}","\\begin{proposition}[Translation of quadratic refinements]\\label{prop:quadratic_translation}\nFor \\(v\\in V\\), put\n\\[\nq_v(x)=q(x)+b(v,x).\n\\]\nThen\n\\[\n\\boxed{\n\\Arf(q_v)=\\Arf(q)+q(v).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nUsing the quadratic identity,\n\\[\nq_v(x)=q(x+v)+q(v).\n\\]\nTherefore\n\\[\n\\sum_x(-1)^{q_v(x)}\n=\n(-1)^{q(v)}\n\\sum_x(-1)^{q(x)}.\n\\]\nCompare the signs in \\cref{thm:arf_gauss}.\n\\end{proof}","\\begin{theorem}[Number of dyadic refinements]\\label{thm:quadratic_counts}\nLet \\(\\dim V=2r\\). The number of quadratic refinements with Arf invariant zero is\n\\[\n\\boxed{\n2^{r-1}(2^r+1),\n}\n\\]\nand the number with Arf invariant one is\n\\[\n\\boxed{\n2^{r-1}(2^r-1).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nEvery refinement is \\(q_v\\) for a unique \\(v\\in V\\), because \\(b\\) identifies \\(V\\) with \\(V^*\\). By \\cref{prop:quadratic_translation}, the Arf invariant changes according to \\(q(v)\\). Let \\(Z\\) and \\(O\\) be the numbers of zeros and ones of \\(q\\). Then\n\\[\nZ+O=2^{2r}\n\\]\nand \\cref{thm:arf_gauss} gives\n\\[\nZ-O=(-1)^{\\Arf(q)}2^r.\n\\]\nSolving gives the two displayed numbers, with their roles exchanged when the starting refinement has Arf invariant one. Since translation runs through every refinement, the final counts are independent of the starting choice.\n\\end{proof}","Let \\(\\Gamma\\) be a connected finite graph. At every vertex \\(v\\), place finite dimensional complex vector spaces \\(V_v\\) and \\(W_v\\) of the same dimension. For every oriented edge \\(e:u\\to v\\), let\n\\[\nA_e:V_u\\longrightarrow V_v,\n\\qquad\nB_e:W_u\\longrightarrow W_v\n\\]\nbe invertible. The reversed edge carries the inverse map.","\\begin{definition}[Path transport]\\label{def:path_transport}\nFor an oriented path\n\\[\n\\gamma=(v_0\\to v_1\\to\\cdots\\to v_m),\n\\]\ndefine\n\\[\nA_\\gamma=A_{v_m v_{m-1}}\\cdots A_{v_1v_0},\n\\qquad\nB_\\gamma=B_{v_m v_{m-1}}\\cdots B_{v_1v_0}.\n\\]\nFor a loop based at \\(r\\), these are the based holonomies.\n\\end{definition}","\\begin{theorem}[Matrix holonomy criterion]\\label{thm:matrix_holonomy}\nFix a root \\(r\\) and an isomorphism\n\\[\nU_r:V_r\\longrightarrow W_r.\n\\]\nThe following conditions are equivalent.","\\begin{enumerate}\n\\item There are isomorphisms \\(U_v:V_v\\to W_v\\), with the declared \\(U_r\\), such that\n\\[\nU_vA_e=B_eU_u\n\\]\nfor every edge \\(e:u\\to v\\).\n\\item For one fundamental cycle basis based at \\(r\\),\n\\[\nU_rA_\\gamma=B_\\gamma U_r.\n\\]\n\\item The same identity holds for every based loop \\(\\gamma\\).\n\\end{enumerate}\n\\end{theorem}","Conversely, choose a spanning tree. Starting from \\(U_r\\), define \\(U_v\\) by transport along the unique tree path from \\(r\\) to \\(v\\). The edge equations hold on every tree edge by construction. Every non tree edge closes one fundamental cycle. The edge equation on that edge is equivalent to the root holonomy identity for its fundamental cycle. Condition two therefore gives all remaining edge equations.\n\\end{proof}","\\begin{corollary}[Unbased gauge classification]\\label{cor:unbased_holonomy}\nThere exists some family of vertex isomorphisms if and only if the two holonomy representations\n\\[\n\\pi_1(\\Gamma,r)\\longrightarrow\\GL(V_r),\n\\qquad\n\\pi_1(\\Gamma,r)\\longrightarrow\\GL(W_r)\n\\]\nare conjugate.\n\\end{corollary}","\\begin{proof}\nApply \\cref{thm:matrix_holonomy} and allow \\(U_r\\) to vary.\n\\end{proof}","\\begin{corollary}[Scalar and projective gluing]\\label{cor:scalar_holonomy}\nIf every based loop acts by a scalar, finite gluing is equivalent to equality of the scalar holonomies on a fundamental cycle basis. The number of independent tests is\n\\[\n\\boxed{\n\\abs E-\\abs V+1.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nA connected graph has a spanning tree with \\(\\abs V-1\\) edges. Every remaining edge gives one fundamental cycle. In the scalar case, conjugacy has no effect.\n\\end{proof}","\\begin{proposition}[Exact edge residual from cycle defect]\\label{prop:holonomy_residual}\nFix a spanning tree and construct the vertex maps by tree transport from \\(U_r\\). Let \\(e:u\\to v\\) be a non tree edge. Let \\(p_u\\) and \\(p_v\\) be the tree paths from the root to \\(u\\) and \\(v\\). Define the fundamental cycle defect\n\\[\nD_e\n=\nU_rA_{p_v}^{-1}A_eA_{p_u}\n-\nB_{p_v}^{-1}B_eB_{p_u}U_r.\n\\]\nThen the edge residual is exactly\n\\[\n\\boxed{\nU_vA_e-B_eU_u\n=\nB_{p_v}D_eA_{p_u}^{-1}.\n}\n\\]\n\\end{proposition}","ct edge residual from cycle defect]\\label{prop:holonomy_residual}\nFix a spanning tree and construct the vertex maps by tree transport from \\(U_r\\). Let \\(e:u\\to v\\) be a non tree edge. Let \\(p_u\\) and \\(p_v\\) be the tree paths from the root to \\(u\\) and \\(v\\). Define the fundamental cycle defect\n\\[\nD_e\n=\nU_rA_{p_v}^{-1}A_eA_{p_u}\n-\nB_{p_v}^{-1}B_eB_{p_u}U_r.\n\\]\nThen the edge residual is exactly\n\\[\n\\boxed{\nU_vA_e-B_eU_u\n=\nB_{p_v}D_eA_{p_u}^{-1}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nSubstitute\n\\[\nU_u=B_{p_u}U_rA_{p_u}^{-1},\n\\qquad\nU_v=B_{p_v}U_rA_{p_v}^{-1}\n\\]\ninto the left side and factor \\(B_{p_v}\\) on the left and \\(A_{p_u}^{-1}\\) on the right.\n\\end{proof}","\\begin{corollary}[Quantitative finite gluing]\\label{cor:quantitative_holonomy}\nWith the notation of \\cref{prop:holonomy_residual},\n\\[\n\\norm{U_vA_e-B_eU_u}\n\\leq\n\\norm{B_{p_v}}\\norm{D_e}\\norm{A_{p_u}^{-1}}.\n\\]\nThus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.\n\\end{corollary}","\\begin{definition}[Orthogonal break decomposition]\\label{def:break_decomposition}\nAn orthogonal break decomposition is a symplectic direct sum\n\\[\nV=\\perpbig_{s\\in S}V_s\n\\]\nthat is preserved by the declared tame stabilizer. Each \\(V_s\\) is the finite symplectic packet assigned to one break.\n\\end{definition}","\\begin{proposition}[Heisenberg tensor factorization]\\label{prop:heisenberg_tensor}\nFix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands:\n\\[\n\\boxed{\n\\omega_V\n\\cong\n\\bigotimes_{s\\in S}\\omega_{V_s}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe Heisenberg group of an orthogonal sum is the central product of the Heisenberg groups of the summands. The tensor product representation has the declared central character and has dimension\n\\[\n\\prod_s\\sqrt{\\abs{V_s}}\n=\n\\sqrt{\\abs V}.\n\\]\nIt is irreducible by the finite Stone von Neumann theorem, so it is the required representation.\n\\end{proof}","\\begin{corollary}[Breakwise phase product]\\label{cor:break_phase}\nIn odd residue characteristic, let \\(g\\in\\Sp(V)\\) preserve every break packet and suppose its restriction to \\(M=\\im(g-I)\\) is fixed point free. Put \\(g_s=g|_{V_s}\\) and \\(M_s=\\im(g_s-I)\\). The moving restrictions are symplectic, and\n\\[\n\\boxed{\n\\Phi_p(g|_M,M)\n=\n\\prod_{s\\in S}\\Phi_p(g_s|_{M_s},M_s).\n}\n\\]\nIn residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.\n\\end{corollary}","\\begin{proof}\nThe invariant orthogonal decomposition gives \\(M=\\mathop{\\perp}\\limits_{s\\in S}M_s\\). Symplectic invariance gives \\(M^\\perp=\\ker(g-I)\\). Fixed point freeness on \\(M\\) therefore makes the restricted form nondegenerate on \\(M\\) and on each \\(M_s\\). Each restriction of \\(g_s-I\\) to \\(M_s\\) is invertible, so \\cref{prop:odd_multiplicativity} applies to the moving spaces. A zero moving space contributes one. In characteristic two, apply \\cref{cor:arf_multiplicative}.\n\\end{proof}","Local class field theory identifies one dimensional Weil characters with characters of \\(K^{\\times}\\) \\cite{Serre1967,Tate1979}. The tame induction theorem extends this explicit picture to higher dimensional tame atoms. The local polynomial and conductor calculations in \\cref{prop:tame_local_polynomial,prop:tame_conductor} therefore give explicit finite invariants for the declared tame packets. Finite Gauss sums give the residue field phase. The full local factor, however, also includes ramification depth, Haar measure normalization, the additive character conductor, and wild interaction data.","\\begin{proof}\nThe dimension and orbit follow from \\cref{thm:tame_induction}. The cycle polynomial is \\cref{prop:frobenius_cycle}. For rank greater than one, the conductor is \\cref{prop:tame_conductor}. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula \\(\\operatorname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum. For a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","ivial inertia and one otherwise, by the formula \\(\\operatorname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum. For a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","otherwise, by the formula \\(\\operatorname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum. For a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","rname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum. For a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","Fix this chapter's reciprocity, additive-character, local-factor, and phase\nnormalizations. Let \\(\\mathcal A\\) be the declared class of irreducible wild\nWeil atoms under consideration. For \\(\\rho\\in\\mathcal A\\), the supplied\nassembled finite datum \\(d_\\rho\\) consists of the following inputs,\nhistorically denoted H1--H4:\n\\begin{enumerate}\n\\item an intrinsic finite break decomposition on the Galois side;\n\\item the declared equivariant symplectic bridges to the supplied breakwise\ntype-side quotients;\n\\item compatible odd or dyadic central-extension and phase data on every\nbreak, including every prescribed finite marking;\n\\item a compatible transport system with vanishing matrix holonomy\nobstruction.\n\\end{enumerate}\nThese are finite comparison inputs. Supplying separate breakwise target\ntypes does not","Fix this chapter's reciprocity, additive-character, local-factor, and phase\nnormalizations. Let \\(\\mathcal A\\) be the declared class of irreducible wild\nWeil atoms under consideration. For \\(\\rho\\in\\mathcal A\\), the supplied\nassembled finite datum \\(d_\\rho\\) consists of the following inputs,\nhistorically denoted H1--H4:\n\\begin{enumerate}\n\\item an intrinsic finite break decomposition on the Galois side;\n\\item the declared equivariant symplectic bridges to the supplied breakwise\ntype-side quotients;\n\\item compatible odd or dyadic central-extension and phase data on every\nbreak, including every prescribed finite marking;\n\\item a compatible transport system with vanishing matrix holonomy\nobstruction.\n\\end{enumerate}\nThese are finite comparison inputs. Supplying separate breakwise target\ntypes does not itself supply one common arithmetic candidate. If a common\ncandidate is already supplied, that is a","Fix this chapter's reciprocity, additive-character, local-factor, and phase\nnormalizations. Let \\(\\mathcal A\\) be the declared class of irreducible wild\nWeil atoms under consideration. For \\(\\rho\\in\\mathcal A\\), the supplied\nassembled finite datum \\(d_\\rho\\) consists of the following inputs,\nhistorically denoted H1--H4:\n\\begin{enumerate}\n\\item an intrinsic finite break decomposition on the Galois side;\n\\item the declared equivariant symplectic bridges to the supplied breakwise\ntype-side quotients;\n\\item compatible odd or dyadic central-extension and phase data on every\nbreak, including every prescribed finite marking;\n\\item a compatible transport system with vanishing matrix holonomy\nobstruction.\n\\end{enumerate}\nThese are finite comparison inputs. Supplying separate breakwise target\ntypes does not itself supply one common arithmetic candidate. If a common\ncandidate is already supplied, that is a stronger restricted hypothesis.","An \\emph{arithmetic realization witness}\n\\(\\mathcal R(d_\\rho,\\pi)\\) consists of actual arithmetic inducing data\nfor an admissible supercuspidal \\(\\pi\\), together with explicit\ncomparison and recovery maps carrying the prescribed assembled finite\npacket. The maps preserve its declared stabilizer action, all character values\nexplicitly included in the supplied datum, central-extension and phase\ndata, and compatible transports.\nRecovery returns that same \\(\\pi\\), with its fixed central character and\nnoncompact completion. An enlarged actual inducing carrier is permitted.\nThe whole packet is not required to be the original maximal-simple-type\nquotient. Any stronger native-marking or numerical-layer requirement is\npart of the declared datum and must be verifie","An \\emph{arithmetic realization witness}\n\\(\\mathcal R(d_\\rho,\\pi)\\) consists of actual arithmetic inducing data\nfor an admissible supercuspidal \\(\\pi\\), together with explicit\ncomparison and recovery maps carrying the prescribed assembled finite\npacket. The maps preserve its declared stabilizer action, all character values\nexplicitly included in the supplied datum, central-extension and phase\ndata, and compatible transports.\nRecovery returns that same \\(\\pi\\), with its fixed central character and\nnoncompact completion. An enlarged actual inducing carrier is permitted.\nThe whole packet is not required to be the original maximal-simple-type\nquotient. Any stronger native-marking or numerical-layer requirement is\npart of the declared datum and must be verified separately.","ho,\\pi)\\) consists of actual arithmetic inducing data\nfor an admissible supercuspidal \\(\\pi\\), together with explicit\ncomparison and recovery maps carrying the prescribed assembled finite\npacket. The maps preserve its declared stabilizer action, all character values\nexplicitly included in the supplied datum, central-extension and phase\ndata, and compatible transports.\nRecovery returns that same \\(\\pi\\), with its fixed central character and\nnoncompact completion. An enlarged actual inducing carrier is permitted.\nThe whole packet is not required to be the original maximal-simple-type\nquotient. Any stronger native-marking or numerical-layer requirement is\npart of the declared datum and must be verified separately.","\\begingroup\n\\renewcommand{\\theconjecture}{66}\n\\begin{conjecture}[Arithmetic packet realization]\n\\label{conj:arithmetic_realization}\\label{thm:conditional_atomic}\nFor every \\(\\rho\\in\\mathcal A\\) with its supplied H1--H4 assembled\nfinite datum \\(d_\\rho\\), there exists an actual admissible supercuspidal\nrepresentation \\(\\pi\\) of \\(\\GL_{\\dim\\rho}(K)\\) and an arithmetic\nrealization witness \\(\\mathcal R(d_\\rho,\\pi)\\).\n\\end{conjecture}\n\\endgroup","\\begin{proposition}[Comparison of one realized candidate]\n\\label{prop:realized_candidate_comparison}\nLet \\(\\rho\\in\\mathcal A\\), and let \\(\\pi\\) be one actual admissible\nsupercuspidal representation of \\(\\GL_{\\dim\\rho}(K)\\) with\n\\(\\mathcal R(d_\\rho,\\pi)\\). Suppose H5 holds for this same \\(\\pi\\):\nits required pair local factors equal those prescribed for \\(\\rho\\)\non the stipulated characterizing family of twists, with the fixed\nnormalizations. Assume the stated local-factor characterization applies\nto that family and class. Then \\(\\pi\\) is isomorphic to the representation\nassigned to \\(\\rho\\) by the established local Langlands correspondence.\n\\end{proposition}","\\(\\pi\\) be one actual admissible\nsupercuspidal representation of \\(\\GL_{\\dim\\rho}(K)\\) with\n\\(\\mathcal R(d_\\rho,\\pi)\\). Suppose H5 holds for this same \\(\\pi\\):\nits required pair local factors equal those prescribed for \\(\\rho\\)\non the stipulated characterizing family of twists, with the fixed\nnormalizations. Assume the stated local-factor characterization applies\nto that family and class. Then \\(\\pi\\) is isomorphic to the representation\nassigned to \\(\\rho\\) by the established local Langlands correspondence.\n\\end{proposition}","tation of \\(\\GL_{\\dim\\rho}(K)\\) with\n\\(\\mathcal R(d_\\rho,\\pi)\\). Suppose H5 holds for this same \\(\\pi\\):\nits required pair local factors equal those prescribed for \\(\\rho\\)\non the stipulated characterizing family of twists, with the fixed\nnormalizations. Assume the stated local-factor characterization applies\nto that family and class. Then \\(\\pi\\) is isomorphic to the representation\nassigned to \\(\\rho\\) by the established local Langlands correspondence.\n\\end{proposition}","\\begin{enumerate}\n\\item Construct a choice free finite symplectic envelope from an arbitrary irreducible wild inertia atom, with its ramification filtration and tame stabilizer action.\n\\item Prove a canonical equivariant symplectic isometry from that envelope to the type side quotient \\(J_\\theta^1/H_\\theta^1\\).\n\\item Recover the canonical beta extension phase in every break, including the required dyadic quadratic refinement.\n\\item Prove matrix holonomy compatibility for the resulting packet transport system.\n\\item Compare the full family of pair \\(L\\), epsilon, and gamma factors without invoking the established correspondence.\n\\end{enumerate}","n and tame stabilizer action.\n\\item Prove a canonical equivariant symplectic isometry from that envelope to the type side quotient \\(J_\\theta^1/H_\\theta^1\\).\n\\item Recover the canonical beta extension phase in every break, including the required dyadic quadratic refinement.\n\\item Prove matrix holonomy compatibility for the resulting packet transport system.\n\\item Compare the full family of pair \\(L\\), epsilon, and gamma factors without invoking the established correspondence.\n\\end{enumerate}","\\begin{example}[A degree four tame orbit]\\label{ex:tame_four_again}\nOver \\(\\FF_2\\), the exponent orbit\n\\[\n1\\longmapsto2\\longmapsto4\\longmapsto8\\longmapsto1\n\\]\nhas length four. It therefore gives a tame atom induced from the unramified extension of degree four. The Frobenius transfer has one four cycle, and the number of regular degree four orbits is three by \\cref{prop:mobius_count}.\n\\end{example}","\\begin{example}[Tensor packets of lengths four and six]\\label{ex:tensor_46}\nFor \\(n=4\\) and \\(m=6\\),\n\\[\n\\gcd(4,6)=2,\n\\qquad\n\\lcm(4,6)=12.\n\\]\nThe tensor product splits into two interaction packets of length twelve. Their total dimension is \\(2\\cdot12=24\\), equal to the product \\(4\\cdot6\\).\n\\end{example}","\\begin{example}[Rank one Heisenberg group]\\label{ex:heisenberg_rank_one}\nTake \\(V=\\FF_3^2\\). The Heisenberg group has order \\(3^3=27\\), and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.\n\\end{example}","\\begin{example}[Nondegenerate twisted algebra]\\label{ex:twisted_plane}\nLet \\(A=\\FF_p^2\\) with its standard alternating form. A cocycle with that commutator form gives\n\\[\n\\CC^c[A]\\cong M_p(\\CC).\n\\]\nThe unique irreducible projective representation has dimension \\(p\\).\n\\end{example}","\\begin{example}[Dyadic phase ambiguity]\\label{ex:dyadic_plane}\nOn a symplectic plane over \\(\\FF_2\\), choose refinements \\(q_0\\) and \\(q_1\\) with Arf invariants zero and one. Their polar forms are the same, while\n\\[\n\\sum_x(-1)^{q_0(x)}=2,\n\\qquad\n\\sum_x(-1)^{q_1(x)}=-2.\n\\]\nThus the alternating packet alone cannot select the dyadic sign.\n\\end{example}","The later constructions provide actual arithmetic candidates and exact\nfinite packet recovery in substantial restricted families. They do not\nestablish \\cref{conj:arithmetic_realization} for every admitted source.\nTheir detailed ordinary proofs are typeset in the appendices, and the\noriginal reviewed Markdown files accompany this edition. In these\nappendices \\(F\\) denotes the base local field, while \\(K=F[\\beta]\\)\ndenotes a parameter field. These are local notation conventions.\nAll references to ``claim 66'' in the retained notes mean the general\nendpoint now isolated as Conjecture 66.","ater constructions provide actual arithmetic candidates and exact\nfinite packet recovery in substantial restricted families. They do not\nestablish \\cref{conj:arithmetic_realization} for every admitted source.\nTheir detailed ordinary proofs are typeset in the appendices, and the\noriginal reviewed Markdown files accompany this edition. In these\nappendices \\(F\\) denotes the base local field, while \\(K=F[\\beta]\\)\ndenotes a parameter field. These are local notation conventions.\nAll references to ``claim 66'' in the retained notes mean the general\nendpoint now isolated as Conjecture 66.","\\begin{theorem}[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its comp","\\begin{theorem}[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already","\\begin{theorem}[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","amified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","egrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","Assume actually supplied labeled full-character extensions with whole\ntwo-dimensional symplectic quotients whose balanced complex-scalar\ncentral product has \\(d\\) planes, one vertex, and trivial stabilizer.\nThen the same \\(\\pi\\) admits a full scalar-pushout realization of that\npacket and exact recovery of its unchanged inducing representation.\nThe prescribed central character must lie in the covered tame class.\nNo literal embedding of the old matrix groups, canonical native marking,\nor prescribed numerical host-depth placement is inferred.\n\\end{theorem}","\\begin{proof}\nThe explicit Artin--Schreier recursion and its normalized filtered lift\nconstruct the maximal parameter and literal defining sequence. See\n\\cref{app:odd_tower}. Its trace transport identity makes every native\ntrace zero, rather than relying on amplification by \\(p\\). Native-rank\nfull-character construction, full beta-determinant transfer, trace-zero\ncorrection, and maximal-field character extension give the scalar\ntriviality and central completion. See \\cref{app:odd_character}.\nThe actual leading symplectic layer has dimension\n\\((p-1)p^{d-1}\\ge2d\\) and survives every later approximation\n(\\cref{app:odd_capacity}). A labeled symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retaine","See\n\\cref{app:odd_tower}. Its trace transport identity makes every native\ntrace zero, rather than relying on amplification by \\(p\\). Native-rank\nfull-character construction, full beta-determinant transfer, trace-zero\ncorrection, and maximal-field character extension give the scalar\ntriviality and central completion. See \\cref{app:odd_character}.\nThe actual leading symplectic layer has dimension\n\\((p-1)p^{d-1}\\ge2d\\) and survives every later approximation\n(\\cref{app:odd_capacity}). A labeled symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retained induction/idempotent recovery returns the same\n\\(\\Lambda\\) and \\(\\pi\\), as proved in \\cref{app:odd_synthesis}.\n\\end{proof}","ty}). A labeled symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retained induction/idempotent recovery returns the same\n\\(\\Lambda\\) and \\(\\pi\\), as proved in \\cref{app:odd_synthesis}.\n\\end{proof}","symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retained induction/idempotent recovery returns the same\n\\(\\Lambda\\) and \\(\\pi\\), as proved in \\cref{app:odd_synthesis}.\n\\end{proof}","For \\(p>3\\), a native minimal rank-\\(p\\) type quotient has dimension\n\\(p-1\\), so it is not automatically a plane. The theorem requires\ngenuinely supplied whole-plane data. Independence of source tensor\nfactors is also an input. An arbitrary single-factor wild determinant\nneed not be tame.","\\begin{theorem}[Restricted dyadic marked assembly realization]\n\\label{thm:dyadic_arithmetic}\nLet \\(F=\\FF_2((t))\\), fix an additive character of conductor\n\\(t\\OO_F\\), fix \\(d\\ge1\\), and let \\(h_1<\\cdots<h_d\\) be positive odd integers.\nPut \\(q_i=3h_i\\), \\(N=2^d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","c}\nLet \\(F=\\FF_2((t))\\), fix an additive character of conductor\n\\(t\\OO_F\\), fix \\(d\\ge1\\), and let \\(h_1<\\cdots<h_d\\) be positive odd integers.\nPut \\(q_i=3h_i\\), \\(N=2^d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","(t\\OO_F\\), fix \\(d\\ge1\\), and let \\(h_1<\\cdots<h_d\\) be positive odd integers.\nPut \\(q_i=3h_i\\), \\(N=2^d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","For actually supplied full marked whole-plane scalar-pushout data with\none vertex, trivial stabilizer, and prescribed source central character\ntrivial on \\(U_F^1\\), an arithmetic assembly carrier exists\nat every length and recovers the same candidate: inside its original\ntype for \\(d\\ge3\\), and in explicit enlarged inducing carriers for\n\\(d=1,2\\). Every supplied full character value and transported whole\nfinite subgroup, lift, square, and phase marking is retained at the\nrepresented marking level. Any central-character kernel already factored\nout remains recorded separately. Scalar pushout does not recover its\ninvisible elements. This does\nnot identify the transported marking with an independently prescribed\ncanonical native marking.\n\\end{theorem}","\\begin{proof}\nThe Artin--Schreier filtered lift has native trace one. Consequently\neach native correction \\(\\delta\\) satisfies\n\\[\nv_F(\\operatorname{tr}\\delta)+s\\ge1 \\quad (s=1),\n\\]\nwhich makes its additive scalar correction trivial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\). See \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","ently\neach native correction \\(\\delta\\) satisfies\n\\[\nv_F(\\operatorname{tr}\\delta)+s\\ge1 \\quad (s=1),\n\\]\nwhich makes its additive scalar correction trivial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\). See \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","h makes its additive scalar correction trivial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\). See \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","ial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\). See \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\). See \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","At rank two, with \\(q=2m+1\\), enlarge\n\\(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\). No literal extraspecial quotient is claimed.","two, with \\(q=2m+1\\), enlarge\n\\(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\). No literal extraspecial quotient is claimed.","2m+1\\), enlarge\n\\(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\). No literal extraspecial quotient is claimed.","(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\). No literal extraspecial quotient is claimed.","\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\). No literal extraspecial quotient is claimed.","-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\). No literal extraspecial quotient is claimed.","At both ranks \\(K^\\times\\) normalizes \\(P\\). Thus\n\\(\\mathcal N=K^\\times J\\) is normal of index two in\n\\(\\mathcal M=K^\\times P\\), and\n\\[\n\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda,\n\\qquad\n\\pi\\simeq\\cInd_{\\mathcal M}^{\\GL_N(F)}\\Sigma.\n\\]\nInduction in stages preserves the full original \\(\\Lambda\\), its\ncentral character, and its noncompact root. See\n\\cref{app:rank4_stages,app:lowrank_contract}. The full marked\nscalar-pushout comparison applies to these actual enlarged pairs as\nwell as to the original higher-rank types (\\cref{app:dyadic_markings}).\nThe full synthesis and scope are in \\cref{app:dyadic_synthesis}.\n\\end{proof}","The original low-rank maximal-type capacity obstruction remains true.\nThe new pair is a common arithmetic assembly carrier. It is not renamed\nas the original maximal-simple-character pair. For an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","mon arithmetic assembly carrier. It is not renamed\nas the original maximal-simple-character pair. For an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","ter pair. For an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","r an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","For the same candidates the constructions give\n\\[\n\\begin{array}{c|c|c}\n & p\\text{ odd} & p=2\\\\ \\hline\n\\dim_{\\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\\\\n\\Swan(\\pi)&p^{d-1}(p+1)h_d&3\\cdot2^{d-1}h_d\n\\end{array}\n\\]\nand torsion number one. Here \\(\\Swan\\) denotes the Swan conductor,\nand \\(a\\) below denotes the ordinary pair conductor. An epsilon exponent\nfor a specified additive character is a different quantity. The separate\nprogress paper, Sections 22--24, reconciles these conventions and the\nactual-order hypotheses. Their self-pair conductor formulas are\n\\[\n\\begin{aligned}\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i\n &&(p\\text{ odd}),","\\Swan(\\pi)&p^{d-1}(p+1)h_d&3\\cdot2^{d-1}h_d\n\\end{array}\n\\]\nand torsion number one. Here \\(\\Swan\\) denotes the Swan conductor,\nand \\(a\\) below denotes the ordinary pair conductor. An epsilon exponent\nfor a specified additive character is a different quantity. The separate\nprogress paper, Sections 22--24, reconciles these conventions and the\nactual-order hypotheses. Their self-pair conductor formulas are\n\\[\n\\begin{aligned}\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i\n &&(p\\text{ odd}),","i)&p^{d-1}(p+1)h_d&3\\cdot2^{d-1}h_d\n\\end{array}\n\\]\nand torsion number one. Here \\(\\Swan\\) denotes the Swan conductor,\nand \\(a\\) below denotes the ordinary pair conductor. An epsilon exponent\nfor a specified additive character is a different quantity. The separate\nprogress paper, Sections 22--24, reconciles these conventions and the\nactual-order hypotheses. Their self-pair conductor formulas are\n\\[\n\\begin{aligned}\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i\n &&(p\\text{ odd}),\\\\\na(\\pi^\\vee\\times\\pi)\n &=N^2-1+\\sum_{i=1}^d3\\cdot4^{i-1}h_i\n &&(p=2).\n\\end{aligned}\n\\]\nThese identities do not prove H5, identify an endo-class, or construct\na canonical source-to-host marking. At rank eight, the retained\nskipped-degree host has the same displayed numerical profiles as the\nfull tower and a smaller Heisenberg quotient. The explicit construction,\nrather than numerical matching alone, supplies the arithmetic object.","Nontrivial prescribed symplectic stabilizer actions, independently fixed\ncanonical native markings, arbitrary source/transport diagrams, and any\nrequired numerical layer placement remain outside the restricted results.\nThe current totally ramified host cannot implement nontrivial\nprime-to-\\(p\\) symplectic actions by the stronger character-preserving\ninner-normalizer mechanism. This obstruction is stated precisely in\n\\cref{app:stabilizer_contract}. That stronger mechanism is not silently\nimposed on the original conjecture. H5 remains conditional for the same\nfixed candidate.","general & 2\\^{}d & 2\\^{}d−2 & \\(2^{2^{d-1}-1}\\) & 2\\^{}d &\nSufficient iff d\\textgreater=3","The H staircase uses the corresponding t\\_j. Hence the exact\ncontribution of transition j to \\(J^1/H^1\\) is","One can also obtain this by subtracting the exact logarithmic indices of\nH and J in 1+P. It is an equality of the group index\n\\textbar \\(J^1/H^1\\)\\textbar, not a guess from the visible\nleading matrix support. BK 3.4.1 makes the resulting commutator pairing\nnondegenerate.","The exponents coincide because N is odd. Hence \\(J^1=H^1\\),\neta is one-dimensional, and no two-dimensional symplectic source plane\ncan embed into this quotient.","The construction retains the actual canonical beta extension and can be\ncompleted with any prescribed central quasicharacter omega\\_F trivial on\nU\\_F\\^{}1. Because \\(k_F^\\times=\\FF_2^\\times\\) is trivial,\nsuch an omega\\_F is unramified. Thus this covers every tamely ramified\ncentral quasicharacter for this particular residue field.","They are the compatible native truncations of (5), because the outer\nextension, uniformizer, and lift depend on the same retained largest h\nat each stage. For \\(1\\le j<d\\), in rank 2\\^{}(j+1), put","It is well defined by the intersection condition. Choose M\\textgreater=a\nwith chi\\textbar U\\_K\\^{}M=1. Then chi\\_0 is a character of a subgroup\nof the finite abelian group \\(U_K^1/U_K^M\\). Extend it by\nadjoining generators and choosing the required complex roots, and pull\nit back. The resulting chi\\_K extends chi and is trivial on U\\_F\\^{}1.","Section~4 constructs chi\\_E on U\\_E\\^{}1 extending phi and trivial on\nU\\_F\\^{}1. Define theta\\_1 on \\(H^1=U_E^1\\) U\\_(A\\_1)\\^{}a\nby chi\\_E on the first factor and psi\\_alpha on the second. The\nrestrictions agree on the intersection. BK 3.2.1 gives the actual full\nsimple character, with theta\\_1\\textbar U\\_F\\^{}1=1.","Here \\(\\OO_K^\\times=U_K^1\\) because the residue field is\nF\\_2. Since U\\_K\\^{}1 is contained in J\\^{}1, we have","Let omega\\_F be any smooth quasicharacter trivial on U\\_F\\^{}1. Since\n\\(\\OO_F^\\times=U_F^1\\), it is determined by its value\nomega\\_F(t). The actual maximal type extends to K\\^{}times J.\nConcretely, (K\\^{}times J)/J is infinite cyclic, generated by a\nK-uniformizer. An intertwiner for its conjugation action on eta gives an\nextension Lambda\\_0. Denote its central character by omega\\_0.","This note does not compute \\(J^1/H^1\\) or prove that it\ncontains a particular dyadic source packet. It does not establish a\nquadratic refinement, canonical phase matching, nontrivial stabilizer\naction, general transport graph, source admission, or H5. In particular,\nan actual supercuspidal host with the required central character is not\nasserted to be a completed realization of the original dyadic packet\nwithout those separate proofs.","Let \\(F=\\mathbb F_2((t))\\), fix an additive character \\(\\psi_F\\) of\nconductor \\(t\\mathcal O_F\\), and take increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","((t))\\), fix an additive character \\(\\psi_F\\) of\nconductor \\(t\\mathcal O_F\\), and take increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","r \\(\\psi_F\\) of\nconductor \\(t\\mathcal O_F\\), and take increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","The actual stratum admits a full simple character \\(\\theta\\) trivial on\n\\(U_F^1\\). Its canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) of \\(\\mathrm{GL}_N(F)\\)\nwith any prescribed smooth central quasicharacter \\(\\omega\\) trivial on\n\\(U_F^1\\), provided the noncompact completion is free or already\ncompatible. Over this residue field, such \\(\\omega\\) is unramified.","For this same constructed candidate, \\[\n\\begin{aligned}\nt(\\pi)&=1,& \\operatorname{Sw}(\\pi)&=3\\,2^{d-1}h_d,\\\\\n\\dim_{\\mathbb F_2}W&=N-2,&\n\\dim\\eta&=2^{N/2-1},\\qquad W=J^1/H^1,\\\\\na(\\pi^\\vee\\times\\pi)\n &=N^2-1+\\sum_{i=1}^d3\\,4^{i-1}h_i.\n\\end{aligned}\n\\] Matching these profiles does not identify the source, its endo-class,\nor the pair factors in original H5.","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base","ve coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","\\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","ramified of degree two, and \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","erse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","ey satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","At an inductive field step \\(d\\ge2\\), set \\(n=2^{d-1}q_d\\),\n\\(R=2^{d-2}q_{d-1}\\). Then \\(n>2R\\) and \\(-2R>-n-q_1\\). BK Remark 2.2.7\ngives the maximal field generated by the actual displayed matrix.\nLiteral truncation and the filtered lift give the defining sequence.\nNonzero native trace also proves separability: the irreducible\ncharacteristic polynomial has a nonzero coefficient of \\(X^{N-1}\\), so\nits derivative is nonzero. See \\path{tower/DYADIC_AS_TOWER167.md}\nfor all field and filtration details.","The replacement for trace zero is the depth inequality \\[\n\\boxed{v_F(\\operatorname{tr}\\delta)+s\\ge1.}\n\\] It makes the additive correction trivial on scalar \\(U_F^s\\), because\n\\((u-1)\\operatorname{tr}\\delta\\in t\\mathcal O_F\\). Our native\ncorrections have trace one, so the inequality holds at \\(s=1\\). This is\nthe exact source-to-target transfer of the weighted-visibility principle. The Euler coefficient theorem is motivation, not an input to\nthe arithmetic proof. In characteristic two, the weights are uniformizer\ndepths, rather than integer powers of two in the coefficient field.","At the minimal rank-two base, extend the forced character on\n\\(U_E^{(q+1)/2}\\) to \\(U_E^1\\), imposing triviality on \\(U_F^1\\). The\nprescriptions agree on their intersection because the parameter trace is\none and \\(\\psi_F(u-1)=1\\). This is an actual finite abelian character\nextension and gives the full minimal simple character.","For a later native step, first transfer the full beta-determinant unit\ncharacter to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar\nunits, \\[\n\\Theta_j(uI)=\\theta_j(u^2)=1.\n\\] For odd critical level \\(r\\), use \\(m=\\lfloor r/2\\rfloor\\) and\n\\(D=H^{m+1}(\\beta)=H^{m+1}(\\gamma)\\). The endpoint is valid because\n\\(2(m+1)-r=1\\): the cross term for the additive character lies in the radical on\nwhich the matrix additive character is trivial. BK 3.3.18 supplies\n\\(\\vartheta=\\Theta_j|_D\\,\\psi_{\\beta-\\gamma}\\). Integral trace makes it\ntrivial on \\(D\\cap U_F^1\\).","nit\ncharacter to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar\nunits, \\[\n\\Theta_j(uI)=\\theta_j(u^2)=1.\n\\] For odd critical level \\(r\\), use \\(m=\\lfloor r/2\\rfloor\\) and\n\\(D=H^{m+1}(\\beta)=H^{m+1}(\\gamma)\\). The endpoint is valid because\n\\(2(m+1)-r=1\\): the cross term for the additive character lies in the radical on\nwhich the matrix additive character is trivial. BK 3.3.18 supplies\n\\(\\vartheta=\\Theta_j|_D\\,\\psi_{\\beta-\\gamma}\\). Integral trace makes it\ntrivial on \\(D\\cap U_F^1\\).","\\[\n\\Theta_j(uI)=\\theta_j(u^2)=1.\n\\] For odd critical level \\(r\\), use \\(m=\\lfloor r/2\\rfloor\\) and\n\\(D=H^{m+1}(\\beta)=H^{m+1}(\\gamma)\\). The endpoint is valid because\n\\(2(m+1)-r=1\\): the cross term for the additive character lies in the radical on\nwhich the matrix additive character is trivial. BK 3.3.18 supplies\n\\(\\vartheta=\\Theta_j|_D\\,\\psi_{\\beta-\\gamma}\\). Integral trace makes it\ntrivial on \\(D\\cap U_F^1\\).","The next parameter field is maximal. Extend the forced character on its\ndeep units together with the trivial character of \\(U_F^1\\), then apply\nBK 3.2.5 to obtain the next full character. All new extensions occur at\nmaximal native rank. No unverified determinant-image extension in a\nnonmaximal centralizer is used.","For the final totally ramified field, \\(J/J^1=\\mathbb F_2^\\times=1\\), so\n\\(J=J^1\\) and the canonical beta extension is \\(\\eta\\). Extend to\n\\(K^\\times J\\), and choose the noncompact scalar to match \\(\\omega(t)\\).\nThe needed \\(N\\)-th root exists in \\(\\mathbb C^\\times\\). The compact\ntype stays fixed. All choices precede H5. See\n\\path{central/DYADIC_FULL_CHARACTER167.md} for the complete proof.","The dyadic floors cannot be replaced by the odd-prime dimension formula.\nAt stage \\(j\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even, whereas \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","be replaced by the odd-prime dimension formula.\nAt stage \\(j\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even, whereas \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","nsion formula.\nAt stage \\(j\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even, whereas \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even, whereas \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","Now assume \\(d\\ge3\\), the one-vertex contract, declared \\(C=\\{1\\}\\), and\nactually supplied full character extensions with labeled whole-plane\nquotients. Retain their complete dyadic finite central-extension\nmarkings, including their specified central embeddings, lifts, square\nvalues, refinements and phases. Any central-character kernel already\nfactored out must be recorded. A scalar pushout cannot recover invisible\nkernel elements.","Under the scalar-pushout realization contract, with no independently\nfixed conflicting host marking, choose a labeled symplectic embedding\ninto \\(W\\). Matching commutators gives a center-fixing isomorphism\nbetween the source \\(\\mathbb C^\\times\\) extension and the corresponding\nhost restriction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\). A complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","the scalar-pushout realization contract, with no independently\nfixed conflicting host marking, choose a labeled symplectic embedding\ninto \\(W\\). Matching commutators gives a center-fixing isomorphism\nbetween the source \\(\\mathbb C^\\times\\) extension and the corresponding\nhost restriction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\). A complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","C^\\times\\) extension and the corresponding\nhost restriction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\). A complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","triction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\). A complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\). A complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","\\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\). A complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","If a genuine host-native quadratic refinement and its finite\n\\(\\mu_2\\)-central extension are independently supplied, quadratic\nembedding has a sharper condition. With otherwise free embeddings, at\nrank eight the equal-dimensional source and host must have equal Arf\ninvariants. For \\(d\\ge4\\), the host has at least one spare symplectic\nplane, so either source Arf type embeds. If the original embedding is\npreassigned, it must instead satisfy the pointwise equation\n\\(q_W(f(v))=q_{\\rm src}(v)\\). These conclusions require the actual host\nmarking as input. Dimension does not manufacture it.","\\)-central extension are independently supplied, quadratic\nembedding has a sharper condition. With otherwise free embeddings, at\nrank eight the equal-dimensional source and host must have equal Arf\ninvariants. For \\(d\\ge4\\), the host has at least one spare symplectic\nplane, so either source Arf type embeds. If the original embedding is\npreassigned, it must instead satisfy the pointwise equation\n\\(q_W(f(v))=q_{\\rm src}(v)\\). These conclusions require the actual host\nmarking as input. Dimension does not manufacture it.","The full finite proofs are in \\path{markings/DYADIC_MARKINGS167.md}.\nIn particular, the D8 and Q8 unmarked complex pushouts coincide while\ntheir whole marked finite \\(\\mu_2\\)-central extensions differ. Merely\nrecording the central \\(\\{\\pm1\\}\\) subgroup would lose that distinction.","The rank-two and rank-four carrier failures cannot be repaired by\nchanging the maximal simple parameter while keeping the specified Swan\nexponent and torsion number. At rank two, odd Swan forces a minimal\nodd-level stratum and \\(J^1=H^1\\). At rank four, Swan \\(6h_2\\) with\n\\(h_2\\) odd forces the degree pattern \\(4\\to2\\). The maximal quadratic derived field forces an odd lower critical level. Every such\nmaximal-type carrier has dimension two, below the required four. Scalar\ntwists preserve this capacity.","At rank eight, numerical profiles alone also fail to determine capacity.\n\\path{capacity/DYADIC_CAPACITY167.md}, Section 6, constructs an\nactual skipped-degree \\(8\\to2\\) host with the same Swan, torsion and\nself-pair conductor but quotient dimension four rather than six. Our\nsuccessful result uses the explicit full tower, rather than inferring\nits profile from invariants.","\\begin{itemize}\n\\tightlist\n\\item\n  Actual fields, full characters and central completion in every rank\n  \\(2^d\\).\n\\item\n  Full marked scalar-pushout packet realization for the stated family\n  when \\(d\\ge3\\), with \\(C=1\\), one vertex, and transported markings\n  retained.\n\\item\n  A conditional quadratic embedding into an independently supplied\n  native marked host, with the exact Arf or pointwise condition above.\n\\end{itemize}","The arbitrary-length field and full-character arguments are ordinary\nproofs. The exact rank-eight run for \\(h=(1,3,5)\\) checks actual matrix\nidentities, normalized lifts, leading and adjacent-correction grades\n\\(-60,-18,-3\\), native traces one and ambient traces \\(0,0,1\\). It\noutputs a matrix with 25 nonzero Laurent polynomial entries. The\ncorresponding critical exponents are proved symbolically. The run does\nnot prove simplicity, full-character existence, or universal marked\nrealization. Those conclusions use the separate proofs and their stated\nscope. No Lean build was run.","The maximal-type carrier is not the only arithmetic carrier available in\nthe constructed representation. The following constructions keep the\noriginal \\(\\beta,\\theta,\\Lambda\\) and \\(\\pi\\) fixed.","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnon","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full characte","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmeti","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\nactio","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","e \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","\\subset U_A^{m+1}\\) lies in the forced domain of the additive character, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","The identity compact-induction double coset realizes \\(E\\) inside the\nsame \\(\\pi\\), as functions supported on \\((K^\\times J)P\\). The complete\ngroup and recovery proof is\n\\path{lowrank/RANK2_ENLARGED_CARRIER167.md}.","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has expo","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exac","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. Th","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg","actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\","imation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\…","atorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_E…","e{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED…","l level is \\(r=3h_1=2m+1\\). The half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.",") is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","ible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\). A literal extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","At both low ranks, \\(K^\\times\\) normalizes the new \\(P\\). In rank two\nthis follows directly from normalization of \\(A\\) and its radical. In\nrank four, a \\(K\\)-uniformizer is approximated by the derived chain\nshift \\(R\\) times a unit of \\(B\\), with relative error in \\(P_A^{n-r}\\).\nIts conjugation error on \\(U_B^m\\) lies in \\(U_A^{n-r+m}\\subset J\\),\nwhich proves normalization. The exact argument is\n\\path{lowrank/RANK4_INDUCTION_BY_STAGES167.md}.","Consequently \\(\\mathcal M=K^\\times P\\) is an actual open group compact\nmodulo the center, \\(\\mathcal N=K^\\times J\\) is normal of index two in\n\\(\\mathcal M\\), and \\[\n\\Sigma=\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}\\Lambda,\n\\qquad \\Sigma|_P\\simeq E,\n\\qquad\n\\boxed{\\pi\\simeq\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G\\Sigma.}\n\\] Restriction to \\(P\\) is irreducible, so \\(\\Sigma\\) is irreducible.\nInduction in stages proves the last equality for the original\n\\(\\Lambda\\), including its fixed central character and noncompact root.\nThis is an actual enlarged inducing representation. It changes neither\nthe candidate nor its original canonical beta-extension data.","group compact\nmodulo the center, \\(\\mathcal N=K^\\times J\\) is normal of index two in\n\\(\\mathcal M\\), and \\[\n\\Sigma=\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}\\Lambda,\n\\qquad \\Sigma|_P\\simeq E,\n\\qquad\n\\boxed{\\pi\\simeq\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G\\Sigma.}\n\\] Restriction to \\(P\\) is irreducible, so \\(\\Sigma\\) is irreducible.\nInduction in stages proves the last equality for the original\n\\(\\Lambda\\), including its fixed central character and noncompact root.\nThis is an actual enlarged inducing representation. It changes neither\nthe candidate nor its original canonical beta-extension data.","There is also a recovery from compact data explicitly recording that\nroot. With \\(A_0=\\Lambda(\\varpi_K)\\) and\n\\(X=\\operatorname{c\\!-\\!Ind}_J^G\\eta\\), define\n\\((Tf)(g)=A_0^{-1}f(\\varpi_Kg)\\). Then \\(X/(T-1)X\\simeq\\pi\\). After\nimposing the fixed central character, \\(\\mathcal N/(F^\\times J)\\) is\ncyclic of order \\(N\\), and the finite average\n\\(N^{-1}\\sum_{i=0}^{N-1}T^i\\) selects this same extension. These\noperators act on complex representation spaces, so their scalar division\nby \\(N\\) is valid. This is a different recovery from the Block159\nprojector onto a sector of \\(\\eta\\). The exact kernel and finite\nprojector are proved in\n\\path{lowrank/ENLARGED_CARRIER_CONTRACT167.md}.","_0=\\Lambda(\\varpi_K)\\) and\n\\(X=\\operatorname{c\\!-\\!Ind}_J^G\\eta\\), define\n\\((Tf)(g)=A_0^{-1}f(\\varpi_Kg)\\). Then \\(X/(T-1)X\\simeq\\pi\\). After\nimposing the fixed central character, \\(\\mathcal N/(F^\\times J)\\) is\ncyclic of order \\(N\\), and the finite average\n\\(N^{-1}\\sum_{i=0}^{N-1}T^i\\) selects this same extension. These\noperators act on complex representation spaces, so their scalar division\nby \\(N\\) is valid. This is a different recovery from the Block159\nprojector onto a sector of \\(\\eta\\). The exact kernel and finite\nprojector are proved in\n\\path{lowrank/ENLARGED_CARRIER_CONTRACT167.md}.","1}f(\\varpi_Kg)\\). Then \\(X/(T-1)X\\simeq\\pi\\). After\nimposing the fixed central character, \\(\\mathcal N/(F^\\times J)\\) is\ncyclic of order \\(N\\), and the finite average\n\\(N^{-1}\\sum_{i=0}^{N-1}T^i\\) selects this same extension. These\noperators act on complex representation spaces, so their scalar division\nby \\(N\\) is valid. This is a different recovery from the Block159\nprojector onto a sector of \\(\\eta\\). The exact kernel and finite\nprojector are proved in\n\\path{lowrank/ENLARGED_CARRIER_CONTRACT167.md}.","Apply the marked scalar-pushout theorem of Section 4 to the\n\\textbf{actual new} pair \\((H_0,P)\\) and its carrier \\(E\\). It has\nexactly the capacity of the whole source at each low rank. All full\nsource character values and the transported whole finite subgroup,\nlifts, squares and phase markings can be attached and retained. The\ndisplayed \\(\\Sigma\\) and induction in stages recover the same candidate.","The old H2 break targets remain their supplied actual simple-character\nquotients. The enlarged pair is their common arithmetic assembly carrier. It is not renamed as the original maximal-simple \\(H^1,J^1\\)\npair. If a stronger reading requires the whole packet itself to be the\nquotient of a maximal simple character in the same candidate, the\nearlier dimension obstruction still applies. Likewise, agreement with an\nindependently fixed native quadratic marking must be checked separately.\nThe contract note proves this distinction explicitly.","The weighted dyadic mechanism now gives actual arithmetic fields, full\ncharacters and tame central completion in every rank \\(2^d\\) of the\nstated increasing odd-break family. Under the one-vertex, \\(C=1\\), full\nmarked scalar-pushout \\textbf{assembly} contract, the packet carrier can\nbe realized at every length: inside the original type for \\(d\\ge3\\), and\nin the explicit enlarged inducing carrier at \\(d=1,2\\). The source's\nprescribed central character must belong to the covered tame class. In\nparticular no arbitrary single-factor wild determinant is assumed to be\ntrivial.","weighted dyadic mechanism now gives actual arithmetic fields, full\ncharacters and tame central completion in every rank \\(2^d\\) of the\nstated increasing odd-break family. Under the one-vertex, \\(C=1\\), full\nmarked scalar-pushout \\textbf{assembly} contract, the packet carrier can\nbe realized at every length: inside the original type for \\(d\\ge3\\), and\nin the explicit enlarged inducing carrier at \\(d=1,2\\). The source's\nprescribed central character must belong to the covered tame class. In\nparticular no arbitrary single-factor wild determinant is assumed to be\ntrivial.","B-bimodularity gives s(Lift\\_E(C))=C. The lift preserves exact\nvaluation: u\\_0\\^{}(-1) is an E-unit, and expansion in 1,z gives\ncoefficient valuations distinct modulo 2. Each coefficient entry has the\nsame grade in the interleaved L-chain and the E-chain. In particular\n\\(s(P^r)=Q^r\\).","Then an actual maximal simple type in pi, with Heisenberg quotient\n\\(W=J^1/H^1\\), contains a nondegenerate symplectic\nF\\_p-subspace S of dimension","For k=F\\_p this is exactly (p−1)p\\^{}(r−1). The subspace is the image of\nthe concrete congruence group U\\^{}(N/2)(A) in \\(J^1/H^1\\).\nIt is present regardless of the later critical exponents, their\nparities, or the number of intermediate approximation fields.","Put P=rad(A), \\(U^a(A)=1+P^a\\) for a\\textgreater=1, and\nm=N/2.","For H, BK 3.1.9(ii) states that\n\\(H^a(\\beta_i)=H^a(\\beta_{i+1})\\) as soon as","Here H\\^{}m means \\(H^1\\cap U^m(A)\\). Therefore the image S of\nU\\^{}m(A) in W is canonically","For u,v in U\\^{}m(A), their group commutator is in\n\\(U^{2m}(A)=U^N(A)\\), which lies in U\\^{}(m+1)(A). The\nterminal approximation satisfies","The right side is the Heisenberg commutator pairing for the minimal\nsimple stratum of gamma. Its quotient\n\\(J^1(\\gamma)/H^1(\\gamma)\\) is precisely\n\\(U^m(A)/H^m(\\gamma)\\), because\n\\(J^1(\\gamma)=U^1(B_{\\mathrm{order}})U^m(A)\\), with\n\\(U^1(B_{\\mathrm{order}})\\subset H^1(\\gamma)\\). BK's nondegeneracy theorem for\nsimple-character commutators therefore says that the form on S is\nnondegenerate.","For general k, this is a nondegenerate alternating F\\_p-form of\ndimension \\([k:\\FF_p](n-n/p)\\). For \\(k=\\FF_p\\), it has dimension \\(n-n/p\\). This\nproves the theorem.","\\begin{enumerate}\n\\def\\labelenumi{\\arabic{enumi}.}\n\\item\n  K\\_j=F{[}b\\_j{]} is a totally ramified maximal field of degree p\\^{}j,\n  normalizing the native complete period-p\\^{}j chain A\\_j.\n\\item\n  For \\(1\\le j<d\\), in rank p\\^{}(j+1), gamma\\_j=b\\_j tensor I\\_p is a simple\n  approximation to b\\_(j+1), at the critical level\n  r\\_j=q\\_(d-j)\\textgreater0 of b\\_(j+1). They are equivalent as strata\n  at that level.\n\\item\n  For \\(1\\le j<d\\), delta\\_j=b\\_(j+1)-gamma\\_j has grade -r\\_j and F-matrix trace zero.\n\\item\n  At native rank p, b\\_1 is multiplication by alpha in the explicit\n  Artin--Schreier field","enumi{\\arabic{enumi}.}\n\\item\n  K\\_j=F{[}b\\_j{]} is a totally ramified maximal field of degree p\\^{}j,\n  normalizing the native complete period-p\\^{}j chain A\\_j.\n\\item\n  For \\(1\\le j<d\\), in rank p\\^{}(j+1), gamma\\_j=b\\_j tensor I\\_p is a simple\n  approximation to b\\_(j+1), at the critical level\n  r\\_j=q\\_(d-j)\\textgreater0 of b\\_(j+1). They are equivalent as strata\n  at that level.\n\\item\n  For \\(1\\le j<d\\), delta\\_j=b\\_(j+1)-gamma\\_j has grade -r\\_j and F-matrix trace zero.\n\\item\n  At native rank p, b\\_1 is multiplication by alpha in the explicit\n  Artin--Schreier field","Indeed, on U\\_K\\^{}a U\\_F\\^{}1 define chi\\_0(ku)=chi(k). Agreement on\nthe intersection makes this a character. Choose M\\textgreater=a such\nthat chi is trivial on U\\_K\\^{}M. The character chi\\_0 descends to a\nsubgroup of the finite abelian group \\(U_K^1/U_K^M\\). It\nextends to that whole group by successively adjoining generators and\nselecting complex roots. Pulling it back proves the assertion. This also\ndescribes an actual finite procedure for making the extension choices.","For the native minimal stratum, \\(H^1=U_E^1\\)\nU\\_(A\\_1)\\^{}(a). Define theta\\_1 by chi\\_E on U\\_E\\^{}1 and by\npsi\\_alpha on U\\_(A\\_1)\\^{}a. They agree on the intersection, so BK 3.2.1\nsupplies the actual full simple character. It satisfies\ntheta\\_1\\textbar U\\_F\\^{}1=1.","Let \\(p\\) be odd, \\(F=\\mathbb F_p((t))\\), and \\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\\qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","\\(F=\\mathbb F_p((t))\\), and \\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\\qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","dots<h_d,\\qquad p\\nmid h_i,\\qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","For this actual stratum there exists a full simple character \\(\\theta\\)\nwith \\(\\theta|_{U_F^1}=1\\). Retaining its canonical beta extension, the\nresidue character and noncompact extension can be selected so that the\nresulting irreducible admissible supercuspidal \\(\\pi\\) has any\nprescribed smooth central quasicharacter \\(\\omega\\) trivial on\n\\(U_F^1\\). These completion choices must be free or already compatible. Incompatible immutably supplied choices are not replaced.","The same candidate has \\[\n\\begin{aligned}\n\\operatorname{rank}(\\pi)&=N, &t(\\pi)&=1,\\\\\n\\operatorname{Sw}(\\pi)&=p^{d-1}(p+1)h_d, &\n\\dim_{\\mathbb F_p}(J^1/H^1)&=N-1,\\\\\n\\dim\\eta&=p^{(N-1)/2},\\\\\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.\n\\end{aligned}\n\\] These numerical identities do not identify a Weil representation, an\nendo-class, or the pair epsilon factors required by H5.","At a recursive base \\((L,t_L)\\), introduce \\[\nE=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q_d}.\n\\] This is a separable totally ramified extension of degree \\(p\\), with\n\\(v_E(\\alpha)=-q_d\\) and \\(\\operatorname{Tr}_{E/L}\\alpha=0\\). Choose\n\\(1\\le b<p\\) with \\(q_db\\equiv-1\\pmod p\\), set \\(a=(q_db+1)/p\\), and put\n\\(z=t_L^a\\alpha^b\\). Then \\(z\\) is an explicit uniformizer. Write its\nEisenstein polynomial as \\(f(X)=X^p+\\cdots+a_0\\).","recursive base \\((L,t_L)\\), introduce \\[\nE=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q_d}.\n\\] This is a separable totally ramified extension of degree \\(p\\), with\n\\(v_E(\\alpha)=-q_d\\) and \\(\\operatorname{Tr}_{E/L}\\alpha=0\\). Choose\n\\(1\\le b<p\\) with \\(q_db\\equiv-1\\pmod p\\), set \\(a=(q_db+1)/p\\), and put\n\\(z=t_L^a\\alpha^b\\). Then \\(z\\) is an explicit uniformizer. Write its\nEisenstein polynomial as \\(f(X)=X^p+\\cdots+a_0\\).","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stron","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying an","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).",", where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","P_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","me{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","appa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","For the inductive field step \\(d\\ge2\\), put \\(n=p^{d-1}q_d\\) and\n\\(R=p^{d-2}q_{d-1}\\). The derived matrix is the recursively constructed\nmaximal-field parameter, with \\(k_0=-q_1\\). The inequalities \\(n>pR>2R\\)\nimply \\(-2R>-n-q_1\\). Bushnell--Kutzko Remark 2.2.7 therefore applies to\nthe actual displayed matrix. It supplies its maximal field,\nnormalization, and critical exponent. Valuation preservation and literal\ntruncation of the recursion supply the whole defining sequence. The full\ndetails, quotient count, and conductor dependencies are in\n\\path{tower/TRACE_ZERO_AS_TOWER166.md} and\n\\path{tower/RECURSIVE_TOWER166.md}.","The full-character argument uses a specific order of operations. First\nconstruct a full character at native rank \\(p^j\\), trivial on \\(U_F^1\\).\nTransfer its full beta-determinant unit character to multiplicity \\(p\\),\nusing BK 3.6.13(ii) and 3.6.14. On a scalar \\(u\\in U_F^1\\), this gives\n\\(\\theta_j(u^p)=1\\). Next add the actual trace-zero correction, using BK\n3.3.18. It remains trivial on the scalar intersection of its domain.\nFinally extend at the next \\textbf{maximal native parameter field}: the\nforced deep character and the trivial character of \\(U_F^1\\) agree on\ntheir intersection, so they extend through a finite abelian unit\nquotient. BK 3.2.5 constructs the full simple character.","s full beta-determinant unit character to multiplicity \\(p\\),\nusing BK 3.6.13(ii) and 3.6.14. On a scalar \\(u\\in U_F^1\\), this gives\n\\(\\theta_j(u^p)=1\\). Next add the actual trace-zero correction, using BK\n3.3.18. It remains trivial on the scalar intersection of its domain.\nFinally extend at the next \\textbf{maximal native parameter field}: the\nforced deep character and the trivial character of \\(U_F^1\\) agree on\ntheir intersection, so they extend through a finite abelian unit\nquotient. BK 3.2.5 constructs the full simple character.","This procedure never substitutes an unjustified extension through a\nnonmaximal centralizer. It proves full-character compatibility,\nincluding shallow units, rather than merely cancellation of a leading\ntrace. The residue factor fixes \\(\\omega\\) on \\(\\mathbb F_p^\\times\\), and an\n\\(N\\)-th-root choice for the noncompact generator fixes \\(\\omega(t)\\).\nEvery choice is recorded before H5. See\n\\path{central/TRACE_ZERO_CHARACTER_INDUCTION166.md} for the\ncomplete induction.","Assume the one-vertex scalar-pushout contract: there are \\(d\\) labeled,\nactually supplied full character extensions with whole symplectic\nquotients \\(V_i\\) of dimension two over \\(\\mathbb F_p\\). Their balanced\n\\(\\mathbb C^\\times\\)-central product has quotient\n\\(V_{\\mathrm{src}}=\\bigoplus_iV_i\\) and Heisenberg carrier dimension\n\\(p^d=N\\). Initially take the declared stabilizer \\(C=\\{1\\}\\).","For \\(p=3\\), this applies directly to the retained cubic family with\nwhole rank-three break-type quotients. For \\(p>3\\), it applies only when\nthe supplied plane data genuinely satisfy the stated original bridge\ncontract. A native minimal rank-\\(p\\) type quotient has dimension\n\\(p-1\\), so it must not automatically be called a plane.","The actual candidate from Section 1 has a nondegenerate leading subspace\n\\[\nS\\subset J^1/H^1,\\qquad\n\\dim_{\\mathbb F_p}S=(p-1)p^{d-1}\\ge2d.\n\\] Here \\(S\\) is the image of the concrete congruence subgroup at half\nthe leading even level. BK's high-depth equality preserves this subgroup\nand its commutator pairing through every later approximation. See\n\\path{capacity/LEADING_LAYER_CAPACITY166.md}, Section 1. This part\nof that note applies to the direct constructor and does not require its\nseparate classical application of arithmetic induction.","Choose a labeled symplectic embedding \\(V_{\\mathrm{src}}\\to S\\). For\neach original full character form \\[\n\\mathcal P_i=(\\mathbb C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\] The balanced product and the corresponding host restriction have the\nsame alternating commutator. The center-fixing extension isomorphism,\nobtained with matched Darboux lifts and complex roots, therefore\npreserves every supplied value \\(\\theta_i(h)\\), including the shallow\nvalues.","mbedding \\(V_{\\mathrm{src}}\\to S\\). For\neach original full character form \\[\n\\mathcal P_i=(\\mathbb C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\] The balanced product and the corresponding host restriction have the\nsame alternating commutator. The center-fixing extension isomorphism,\nobtained with matched Darboux lifts and complex roots, therefore\npreserves every supplied value \\(\\theta_i(h)\\), including the shallow\nvalues.","Write \\(W=V_{\\mathrm{src}}\\perp V_{\\mathrm{src}}^\\perp\\). Finite\nStone--von Neumann gives\n\\(\\eta_W\\simeq\\eta_{\\mathrm{src}}\\otimes\\eta_\\perp\\). Choose an\nisotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra \\(e\\) has rank \\(N\\). The\nretained induction/idempotent theorem recovers the \\textbf{same}\ninducing representation \\(\\Lambda\\) from \\(e\\Lambda\\), and then the same\n\\(\\pi\\) by compact induction. No candidate is changed at recovery.","\\(W=V_{\\mathrm{src}}\\perp V_{\\mathrm{src}}^\\perp\\). Finite\nStone--von Neumann gives\n\\(\\eta_W\\simeq\\eta_{\\mathrm{src}}\\otimes\\eta_\\perp\\). Choose an\nisotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra \\(e\\) has rank \\(N\\). The\nretained induction/idempotent theorem recovers the \\textbf{same}\ninducing representation \\(\\Lambda\\) from \\(e\\Lambda\\), and then the same\n\\(\\pi\\) by compact induction. No candidate is changed at recovery.","athrm{src}}^\\perp\\). Finite\nStone--von Neumann gives\n\\(\\eta_W\\simeq\\eta_{\\mathrm{src}}\\otimes\\eta_\\perp\\). Choose an\nisotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra \\(e\\) has rank \\(N\\). The\nretained induction/idempotent theorem recovers the \\textbf{same}\ninducing representation \\(\\Lambda\\) from \\(e\\Lambda\\), and then the same\n\\(\\pi\\) by compact induction. No candidate is changed at recovery.","When the retained independent extraspecial tensor source has \\(d\\ge2\\),\nits full determinant is trivial on wild inertia: each factor determinant\nhas order dividing \\(p\\), and tensor determinants raise it to\n\\(p^{d-1}\\). Thus its central character belongs to the class in Section\n1. For \\(d=1\\), wild triviality of the source determinant must be\nchecked separately. Independence of the tensor factors is an input, not\na consequence of distinct \\(h_i\\).","nsor source has \\(d\\ge2\\),\nits full determinant is trivial on wild inertia: each factor determinant\nhas order dividing \\(p\\), and tensor determinants raise it to\n\\(p^{d-1}\\). Thus its central character belongs to the class in Section\n1. For \\(d=1\\), wild triviality of the source determinant must be\nchecked separately. Independence of the tensor factors is an input, not\na consequence of distinct \\(h_i\\).","There is also a positive result beyond \\(C=\\{1\\}\\). Suppose the finite\nprime-to-\\(p\\) stabilizer acts trivially on the source scalar extension,\nand its prescribed honest operators on the \\textbf{entire} rank-\\(N\\)\nWeil carrier are \\(\\chi(c)I_N\\), each lying in the actual inertia image.\nAssume the candidate's central character is the determinant of that same\nsource under the fixed rank-one reciprocity convention.","Then a finite cover \\(\\widetilde C\\twoheadrightarrow C\\) acts by actual\ncentral matrices in the unchanged candidate. Its represented operators\ndescend to exactly \\(\\chi\\) on \\(C\\). Hence the attachment and recovery\nabove are equivariant for that action. A direct matrix homomorphism from\n\\(C\\) itself is not asserted and can fail.","Indeed, for residue cardinality \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert tha","Indeed, for residue cardinality \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar ch","Indeed, for residue cardinality \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","y \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","hi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","The old fixed-field central argument genuinely failed. At \\(p=3\\),\n\\((h_1,h_2)=(1,5)\\), its naturally transferred full character takes the\nvalue \\(\\zeta_3\\ne1\\) on \\(1+t^2\\), although the desired wild center is\ntrivial. A shallow character choice cannot repair a forced value. The\ntwo-stage changing-field repair is retained in\n\\path{central/CENTRAL_REPAIR166.md}. The trace-zero tower above\nresolves this central issue uniformly in every length.","The construction does not use an inverse established correspondence or\nthe classical arithmetic-induction/descent candidate to select \\(\\pi\\).\nIt does use the named BK construction and type theorems, the retained\ntype/formal-degree conductor formulas, and exact sector recovery. The\ncapacity note separately retains an alternative consequence for a\ncandidate supplied by the substantial classical arithmetic constructor. That is a different route and is not an input to Section 1.","\\begin{enumerate}\n\\def\\labelenumi{\\arabic{enumi}.}\n\\tightlist\n\\item\n  \\textbf{Nontrivial prescribed symplectic stabilizer actions.} For this\n  totally ramified full-degree host, the stronger inner-normalizer\n  interpretation forces prime-to-\\(p\\) actions on the quotient to be\n  trivial. It therefore cannot implement a genuinely nontrivial supplied\n  symplectic action. The raw original contract does not automatically\n  require that stronger interpretation. It still needs an actual valid\n  arithmetic comparison, not relabeling of external operators.\n\\item\n  \\textbf{Arbitrary admissible source packets and transport diagrams.}\n  The proved attachment has the whole-plane, one-vertex, finite\n  scalar-pushout hypotheses above. It does not establish all source\n  classes, arbitrary graphs, or prescribed numerical layer placement.\n\\item\n  \\textbf{The dyadic construction with all markings.} A separate\n  field-level recursion and some capacit","f{Arbitrary admissible source packets and transport diagrams.}\n  The proved attachment has the whole-plane, one-vertex, finite\n  scalar-pushout hypotheses above. It does not establish all source\n  classes, arbitrary graphs, or prescribed numerical layer placement.\n\\item\n  \\textbf{The dyadic construction with all markings.} A separate\n  field-level recursion and some capacity statements exist at \\(p=2\\).\n  The odd-prime trace-zero/full-character theorem here does not settle\n  that case.\n\\end{enumerate}","The arbitrary-length theorem is proved symbolically and independently\nreviewed. The exact \\(p=3,d=2,(h_1,h_2)=(1,2)\\) computation in\n\\texttt{tower/as\\_rank9check/} verifies the displayed matrix, trace\nidentities, corestriction, and levels. It is a finite check, not the\nuniversal proof. The separate rank-27 run in \\texttt{tower/run\\_rank27/}\nchecks the earlier non-trace-zero recursion and must not be described as\nan AS central completion example. No Lean build was run.","Then v\\_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic\nminimal polynomial \\(f(X)=X^p+a_{p-1}X^{p-1}+\\cdots+a_0\\)\nis Eisenstein. It is an explicit polynomial: compute the product over\nthe p conjugates","and reduce coefficients using \\(\\alpha^p-\\alpha=t_L^{-q}\\).\nThe invariant coefficients belong to L. This is a finite algebraic\ninstruction, with no choice of an unknown extension or matrix.","Indeed v\\_E(a\\_k)\\textgreater=p for 1\\textless=k\\textless p, so the\ncontribution indexed by a\\_k has valuation at least k greater than the\nfirst contribution. Applying s entrywise on E\\^{}m gives a B-bimodule\ncorestriction from End\\_L(E\\^{}m) to B=End\\_E(E\\^{}m). With the complete\ninterleaved period-pm L-chain and period-m E-chain, it preserves grades\nand satisfies \\(s(P^r)=Q^r\\).","equal to its E-grade, and distinct i have valuations distinct modulo p.\nThus the minimum cannot cancel. This also proves the stated surjectivity\n\\(s(P^r)=Q^r\\) without a compactness assumption or an assumption that a lift exists.","The field degrees and critical levels of \\(B^{[j]}\\) are exactly","Write R=rad(A), \\(U_A^a=1+R^a\\) for a\\textgreater=1, and\n\\(U_K^a=1+\\mathfrak p_K^a\\). Let theta be a full simple character.\nSince q is odd and the residue field is F2, the actual maximal simple\ntype group and character groups coincide:","The new Heisenberg pair is an actual pair of compact subgroups of\nGL2(F), but is not asserted to be the BK \\(H^1/J^1\\) pair of\na maximal simple type. Whether the original H2 realization relation\nallows this enlargement is a separate obligation.","K× normalizes A and every power of its radical, so U\\_K\\^{}1 normalizes\nU\\_A\\^{}m. Therefore P is a subgroup and is compact open. It is\ncontained in U\\_A\\^{}1 because m\\textgreater=1 and\n\\(U_K^1\\subset U_A^1\\).","For if ku∈U\\_A\\^{}m with k∈U\\_K\\^{}1 and u∈U\\_A\\^{}(m+1), then\n\\(k\\in K^\\times\\cap U_A^m=U_K^m\\). The reverse inclusion is immediate.\nThe complete period-two A-grade \\(R^m/R^{m+1}\\) has\nF2-dimension 2, and its field grade \\(\\mathfrak p_K^m/\\mathfrak p_K^{m+1}\\) has\ndimension 1. Thus","Finally, if kj∈P with k∈K× and j∈J, then \\(k\\in Pj^{-1}\\subset U_A^1\\).\nHence k∈U\\_K\\^{}1 and kj∈J. Therefore","As g²∈J, conjugation by g² fixes the character theta. Thus\n\\(\\lambda^g=\\lambda^{-1}=\\lambda\\). It follows that","The previously reviewed negative statement concerned eta of the original\nBK \\(J^1/H^1\\). That quotient remains zero. The new positive\nstatement concerns a different, explicitly constructed arithmetic pair\ninside the same ambient representation, with exact recovery that preserves the noncompact normalization.","Use \\(U_A^j=1+P_A^j\\) and \\(U_B^j=1+Q^j\\).\nBK 3.1.7--9 give the actual groups","The quotient \\(U_B^m/U_B^{m+1}\\) has dimension two. By\nSection~2 the image of J cap U\\_B\\^{}m on this grade is exactly its\ncentralizing line, of dimension one. For the converse inclusion needed\nhere, the B-approximation of a K-element of grade m differs from it by\ngrade m+n-r\\textgreater a. That error lies in U\\_A\\^{}a subset J. Thus\nthe whole centralizing line really occurs in the intersection, not only\nin an abstract associated graded space. Consequently","Its leading grade is outside the centralizing line, so u represents the\nnontrivial coset of P/J. Since \\(u^2=1+x^2\\) in\ncharacteristic two, we have u\\^{}2 in U\\_B\\^{}(2m) subset U\\_B\\^{}(m+1)\nsubset H.","First, lambda is invariant under u. Indeed u\\^{}2 belongs to J and theta\nis J-invariant, so \\(\\lambda^u=\\lambda^{-1}=\\lambda\\) by (3).\nConsequently H0 is normal in P, and theta0 is P-invariant.","Write J=H U\\_A\\^{}a. Since lambda is a character and J-invariant, it\nkills {[}H,H{]} and {[}H,U\\_A\\^{}a{]}. It also kills\n\\([U_A^a,U_A^a]\\), which lies in U\\_A\\^{}(2a). Hence\nlambda({[}J,J{]})=1. It kills every square in J as well: squares of H\nhave lambda-value one, squares of U\\_A\\^{}a lie in U\\_A\\^{}(2a), and the\nmixed commutator contributions have just been handled.","The quotient \\(K^\\times/F^\\times\\) is compact because K/F is a\nfinite totally ramified extension, and P is compact. Hence M is compact\nmodulo F\\^{}times. It is open in G because it contains the open subgroup\nJ.","\\begin{abstract}\nLet \\(\\bar\\rho\\) be a residual Galois representation and let \\(R\\) be a global deformation ring with declared local conditions. We separate the global automorphy problem into residual occupancy, deformation support, Hecke occupancy, and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \\(R\\cong\\TT\\) transports support, Fitting ideals, and point fibers, but a quotient m","\\begin{abstract}\nLet \\(\\bar\\rho\\) be a residual Galois representation and let \\(R\\) be a global deformation ring with declared local conditions. We separate the global automorphy problem into residual occupancy, deformation support, Hecke occupancy, and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \\(R\\cong\\TT\\) transports support, Fitting ideals, and point fibers, but a quotient map can erase an entire component. Finite Hecke alg","and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \\(R\\cong\\TT\\) transports support, Fitting ideals, and point fibers, but a quotient map can erase an entire component. Finite Hecke algebras are resolved into canonical Artin packets, so simultaneous generalized eigenspaces are available without assuming semisimplicity. Nested packets in one finite carrier stabilize. For changing finite carriers, nonzero stable cores give compatible inverse limit vectors. These results give a pointwise criterion: a characteristic zero deformation point in automorphic support produces a Hecke eigensystem. With attached Galois representations, local global compatibility, and classicality, the point is automorphic. A fixed compatible system then propagates the conc…","Automorphy lifting joins several kinds of mathematics that are easy to conflate. A residual representation must first occur in an automorphic carrier. The chosen characteristic zero representation must lie on a deformation problem controlled by local conditions. A patching argument must show that the automorphic module reaches the relevant part of the deformation space. A comparison such as \\(R\\cong\\TT\\) must identify the arithmetic and Hecke coordinates. Finally, the resulting Hecke point must be classical and must carry the expected Galois representation with the required local compatibility.","The last part assembles these statements into a conditional theorem. A characteristic zero point of \\(R\\) is automorphic when it lies in the support of the automorphic carrier, the comparison with the Hecke algebra is available, and the corresponding Hecke fiber satisfies the declared realization theorem. If the point belongs to a fixed strictly compatible system, equality of Frobenius polynomials propagates the automorphic identification to every coefficient place by Chebotarev density and Brauer Nesbitt \\cite{BrauerNesbitt1941,Neukirch1999,Gee2022}.","Local quotient conditions are developed in \\cref{sec:local_conditions}. Adequacy and residual occupancy appear in \\cref{sec:adequacy}. Auxiliary group algebras are treated in \\cref{sec:auxiliary}. Ordinary and derived patching support appear in \\cref{sec:patching_support}. The effect of \\(R\\cong\\TT\\) is isolated in \\cref{sec:rt}. Finite Hecke packets and infinite occupancy are developed in \\cref{sec:hecke_packets}. Pointwise automorphy and compatible systems appear in \\cref{sec:pointwise,sec:compatible}. The coefficient prime two is discussed in \\cref{sec:two}. The final conditional theorem is stated in \\cref{sec:completion}.","Throughout, \\(S\\) is a finite set of places. Frobenius elements are geometric Frobenius elements. All Hecke polynomials, attached representations, and common algebraic spines use that same convention and the normalization specified by the declared realization theorem.","Let \\(F\\) be a number field. Fix a prime \\(p\\), a finite extension \\(E/\\QQ_p\\), its valuation ring \\(\\OO\\), uniformizer \\(\\varpi\\), and residue field \\(k\\). Let\n\\[\n\\bar\\rho:G_{F,S}\\longrightarrow\\GL_n(k)\n\\]\nbe a continuous residual representation, where \\(S\\) is a finite set of places containing every place at which the deformation problem is allowed to ramify.","\\begin{definition}[Global deformation datum]\\label{def:global_datum}\nA global deformation datum consists of the following objects.\n\\begin{enumerate}\n\\item A residual representation \\(\\bar\\rho\\).\n\\item For every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","l deformation datum]\\label{def:global_datum}\nA global deformation datum consists of the following objects.\n\\begin{enumerate}\n\\item A residual representation \\(\\bar\\rho\\).\n\\item For every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","m}\nA global deformation datum consists of the following objects.\n\\begin{enumerate}\n\\item A residual representation \\(\\bar\\rho\\).\n\\item For every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","\\begin{standardfact}[Represented deformation problem]\\label{fact:represented_deformation}\nUnder the usual finiteness and Schur type hypotheses, framed lifts of \\(\\bar\\rho\\) are represented by complete Noetherian local \\(\\OO\\) algebras. A declared local deformation condition is represented by a quotient when the corresponding subfunctor is representable. Tangent and obstruction spaces are described by the appropriate first and second cohomology groups.\n\\end{standardfact}","\\begin{definition}[Residual automorphic occupancy]\\label{def:residual_occupancy}\nThe residual datum \\((\\bar\\rho,\\tau)\\) is automorphically occupied when a nonzero localized automorphic or cohomological Hecke carrier realizes the residual representation and the declared local conditions.\n\\end{definition}","\\begin{theorem}[Completed tensor quotient]\\label{thm:tensor_quotient}\nLet \\(R_1,\\ldots,R_s\\) be complete Noetherian local \\(\\OO\\) algebras with common residue field \\(k\\). Let \\(I_i\\subseteq R_i\\) be closed ideals. Put\n\\[\nR^{\\mathrm{loc}}=R_1\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}R_s\n\\]\nand let \\(J\\) be the closed ideal generated by the images of the \\(I_i\\). Then there is a canonical isomorphism\n\\[\n\\boxed{\n(R_1/I_1)\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}(R_s/I_s)\n\\cong\nR^{\\mathrm{loc}}/J.\n}\n\\]\n\\end{theorem}","gin{theorem}[Completed tensor quotient]\\label{thm:tensor_quotient}\nLet \\(R_1,\\ldots,R_s\\) be complete Noetherian local \\(\\OO\\) algebras with common residue field \\(k\\). Let \\(I_i\\subseteq R_i\\) be closed ideals. Put\n\\[\nR^{\\mathrm{loc}}=R_1\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}R_s\n\\]\nand let \\(J\\) be the closed ideal generated by the images of the \\(I_i\\). Then there is a canonical isomorphism\n\\[\n\\boxed{\n(R_1/I_1)\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}(R_s/I_s)\n\\cong\nR^{\\mathrm{loc}}/J.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nWrite \\(S=\\bigotimes_{i=1}^{s}R_i\\) for the ordinary tensor product. Its ideal of definition is\n\\[\n\\fraka=\\frakm_{\\OO}S+\\sum_i\\frakm_{R_i}S,\n\\qquad L=\\sum_i I_iS.\n\\]\nThe scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism\n\\[\nS/L\\cong\\bigotimes_{i=1}^{s}(R_i/I_i),\n\\]\ncarrying the image of \\(\\fraka\\) to the defining ideal on the right. The quotient topology on \\(S/L\\) is precisely this image ideal topology.","nary tensor product. Its ideal of definition is\n\\[\n\\fraka=\\frakm_{\\OO}S+\\sum_i\\frakm_{R_i}S,\n\\qquad L=\\sum_i I_iS.\n\\]\nThe scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism\n\\[\nS/L\\cong\\bigotimes_{i=1}^{s}(R_i/I_i),\n\\]\ncarrying the image of \\(\\fraka\\) to the defining ideal on the right. The quotient topology on \\(S/L\\) is precisely this image ideal topology.","Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring \\(S\\) and ideal \\(L\\), there is a canonical bicontinuous ring isomorphism\n\\[\n\\widehat S\\big/\\overline{L\\widehat S}\\cong\\widehat{S/L}.\n\\]\nIndeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from \\(S/L\\) to \\(\\widehat S/\\overline{L\\widehat S}\\) extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the quotient of a complete countably based additive group is complete. Its closed defining subgroup makes it Hausdorff. The two composites agree with the identity on dense quotient representatives, so continuity proves both inverse laws.","Applying this comparison to the displayed ordinary tensor isomorphism proves the claim. The closed extended ideal is exactly \\(J\\), since extension commutes with the finite sum of the factor ideals. The maps preserve coefficients and each canonical factor map. The common residue field identifies \\(S/\\fraka\\) with \\(k\\). Finite generation of \\(\\fraka\\), supplied by the Noetherian factor hypotheses, identifies the completed topology with its maximal ideal topology. Proper quotients retain locality. If a condition is the unit ideal, the ring comparison remains valid with the zero ring, without treating it as a local ring with residue field \\(k\\).\n\\end{proof}","l factor map. The common residue field identifies \\(S/\\fraka\\) with \\(k\\). Finite generation of \\(\\fraka\\), supplied by the Noetherian factor hypotheses, identifies the completed topology with its maximal ideal topology. Proper quotients retain locality. If a condition is the unit ideal, the ring comparison remains valid with the zero ring, without treating it as a local ring with residue field \\(k\\).\n\\end{proof}","\\begin{corollary}[Simultaneous local conditions]\\label{cor:simultaneous_local}\nA continuous point of \\(R^{\\mathrm{loc}}\\) with Hausdorff target factors continuously through \\(R^{\\mathrm{loc}}/J\\) if and only if its restriction to every \\(R_i\\) factors continuously through \\(R_i/I_i\\). Each factor is unique. If the original point is local, its quotient factor is local.\n\\end{corollary}","\\begin{proof}\nLet \\(f:R^{\\mathrm{loc}}\\to A\\) be the point. Since \\(A\\) is Hausdorff and \\(f\\) is continuous, \\(\\ker f\\) is closed. Thus it contains the closed ideal \\(J\\) if and only if it contains the algebraic ideal generated by the images of the \\(I_i\\). This is equivalent to every local restriction killing \\(I_i\\). The universal property of the actual ideal quotient gives the factors, and the quotient topology gives their continuity. Surjectivity of the quotient map gives uniqueness. If \\(f\\) reflects units, lift an element of the quotient to the original ring to see that its factor also reflects units.\n\\end{proof}","ontinuous, \\(\\ker f\\) is closed. Thus it contains the closed ideal \\(J\\) if and only if it contains the algebraic ideal generated by the images of the \\(I_i\\). This is equivalent to every local restriction killing \\(I_i\\). The universal property of the actual ideal quotient gives the factors, and the quotient topology gives their continuity. Surjectivity of the quotient map gives uniqueness. If \\(f\\) reflects units, lift an element of the quotient to the original ring to see that its factor also reflects units.\n\\end{proof}","\\begin{proposition}[Global base change for local conditions]\\label{prop:global_local_basechange}\nIn the global deformation datum of \\cref{def:global_datum}, let \\(R^\\square\\) be a complete Noetherian local \\(\\OO\\) algebra receiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","he global deformation datum of \\cref{def:global_datum}, let \\(R^\\square\\) be a complete Noetherian local \\(\\OO\\) algebra receiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","um of \\cref{def:global_datum}, let \\(R^\\square\\) be a complete Noetherian local \\(\\OO\\) algebra receiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","eiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective.","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchan","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the te","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defi","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","teq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","ies the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","l by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","For a continuous algebra point \\(x:B\\to C\\) with complete Hausdorff target, the kernel of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change. It does not supply representability of","For a continuous algebra point \\(x:B\\to C\\) with complete Hausdorff target, the kernel of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change. It does not supply representability of a new arithmetic deformation problem.\n\\end{proof}","he kernel of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change. It does not supply representability of a new arithmetic deformation problem.\n\\end{proof}","of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change. It does not supply representability of a new arithmetic deformation problem.\n\\end{proof}","The local coefficient points in \\cref{prop:global_local_basechange} are test rings in the represented deformation category. Later field valued points \\(x:R\\to E'\\) are algebraic or generic specializations, and continuity of \\(\\rho_x\\) is imposed separately. Such a field specialization need not be a local ring map: under \\(\\OO\\to E\\), a uniformizer becomes a unit.","\\begin{proposition}[Idempotents in a local ring]\\label{prop:local_idempotent}\nIf \\(R\\) is a commutative local ring, then its only idempotents are zero and one.\n\\end{proposition}","\\begin{proof}\nLet \\(e^2=e\\). Since \\(e+(1-e)=1\\), at least one of \\(e\\) and \\(1-e\\) is a unit. If \\(e\\) is a unit, then \\(e=1\\). If \\(1-e\\) is a unit, then \\(e=0\\).\n\\end{proof}","\\begin{remark}\\label{rem:components_not_idempotents}\nAn irreducible component of \\(\\Spec R\\) for a local deformation ring need not be open and closed. It therefore need not be selected by an idempotent. Components and local conditions should be described by quotient ideals and module support. Idempotent packet formulas are reserved for finite Artin algebras or other settings with an actual product decomposition.\n\\end{remark}","\\begin{standardfact}[Adequate image]\\label{fact:adequacy}\nLet \\(G\\subseteq\\GL(V)\\) act absolutely irreducibly over a finite field \\(k\\) of characteristic \\(p\\). Under the standard definition, adequacy imposes vanishing conditions on \\(H^1(G,k)\\) and \\(H^1(G,\\ad^0V)\\), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of \\(\\ad^0V\\).\n\\end{standardfact}","Let \\(G=S_3\\), let \\(k=\\FF_7\\), and let \\(V\\) be the standard two dimensional representation. In the basis\n\\[\nv_1=(1,-1,0),\n\\qquad\nv_2=(0,1,-1),\n\\]\na transposition and a three cycle act by\n\\[\ns=\n\\begin{pmatrix}\n6&1","Let \\(G=S_3\\), let \\(k=\\FF_7\\), and let \\(V\\) be the standard two dimensional representation. In the basis\n\\[\nv_1=(1,-1,0),\n\\qquad\nv_2=(0,1,-1),\n\\]\na transposition and a three cycle act by\n\\[\ns=\n\\begin{pmatrix}\n6&1\\\\\n0&1\n\\end{pmatrix},\n\\qquad\nt=\n\\begin{pmatrix}\n0&6\\\\\n1&6\n\\end{pmatrix}.\n\\]","\\begin{theorem}[Finite adequacy certificate]\\label{thm:s3_adequacy}\nThe standard representation of \\(S_3\\) over \\(\\FF_7\\) is adequate. More precisely,\n\\[\nH^0(S_3,\\ad^0V)=0,\n\\qquad\nH^1(S_3,\\FF_7)=0,\n\\qquad\nH^1(S_3,\\ad^0V)=0,\n\\]\nand\n\\[\n\\ad^0V\\cong\\operatorname{sgn}\\oplus V.\n\\]\nThe sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.\n\\end{theorem}","\\begin{proof}\nThe coordinate map\n\\[\n(a,b)\\longmapsto(a,-a+b,-b)\n\\]\nidentifies the displayed matrices with the permutation action on the sum zero subspace of \\(\\FF_7^3\\). This action is faithful. The four matrices \\(1,s,t,st\\) span \\(M_2\\) over every field of characteristic seven: the determinant of their coordinate matrix is \\(4\\ne0\\). A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles are respectively\n\\[\n(X-1)^2,\\qquad (X-1)(X+1),\\qquad (X-2)(X-4).\n\\]\nThus all eigenvalues already lie in \\(\\FF_7\\).","). This action is faithful. The four matrices \\(1,s,t,st\\) span \\(M_2\\) over every field of characteristic seven: the determinant of their coordinate matrix is \\(4\\ne0\\). A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles are respectively\n\\[\n(X-1)^2,\\qquad (X-1)(X+1),\\qquad (X-2)(X-4).\n\\]\nThus all eigenvalues already lie in \\(\\FF_7\\).","d of characteristic seven: the determinant of their coordinate matrix is \\(4\\ne0\\). A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles are respectively\n\\[\n(X-1)^2,\\qquad (X-1)(X+1),\\qquad (X-2)(X-4).\n\\]\nThus all eigenvalues already lie in \\(\\FF_7\\).","A trace zero matrix fixed by conjugation by both \\(s\\) and \\(t\\) is zero, as follows by comparing its four entries. This proves \\(H^0(S_3,\\ad^0V)=0\\). For any \\(\\FF_7[S_3]\\) module \\(M\\) and one cocycle \\(f\\), put\n\\[\nu=-6^{-1}\\sum_{h\\in S_3}f(h).\n\\]\nSumming \\(f(gh)=g f(h)+f(g)\\) over \\(h\\) gives \\(gu-u=f(g)\\). Hence every one cocycle is a coboundary and \\(H^1(S_3,M)=0\\), giving both asserted first cohomology vanishings.","Write\n\\[\nW(a,b,c)=\\begin{pmatrix}a&b\\\\c&-a\\end{pmatrix},\n\\qquad\nZ=W(2,3,4),\\quad A=W(1,1,2),\\quad B=W(1,5,6).\n\\]\nThe three coordinate columns have determinant \\(3\\ne0\\). Under conjugation,\n\\[\nsZ=-Z,\\quad tZ=Z,\\qquad\nsA=-A,\\quad sB=A+B,\\quad tA=B,\\quad tB=-A-B.\n\\]\nConsequently \\((z,(a,b))\\mapsto zZ+aA+bB\\) is an equivariant isomorphism from \\(\\operatorname{sgn}\\oplus V\\) to \\(\\ad^0V\\).","c&-a\\end{pmatrix},\n\\qquad\nZ=W(2,3,4),\\quad A=W(1,1,2),\\quad B=W(1,5,6).\n\\]\nThe three coordinate columns have determinant \\(3\\ne0\\). Under conjugation,\n\\[\nsZ=-Z,\\quad tZ=Z,\\qquad\nsA=-A,\\quad sB=A+B,\\quad tA=B,\\quad tB=-A-B.\n\\]\nConsequently \\((z,(a,b))\\mapsto zZ+aA+bB\\) is an equivariant isomorphism from \\(\\operatorname{sgn}\\oplus V\\) to \\(\\ad^0V\\).","The sign witness is \\(W_{\\mathrm{sgn}}=Z\\), while\n\\(W_{\\mathrm{std}}=W(6,4,3)=3A+3B\\) belongs to the standard summand. The corresponding projectors and trace pairings are\n\\[\nP_{t,2}=\\begin{pmatrix}2&4","The sign witness is \\(W_{\\mathrm{sgn}}=Z\\), while\n\\(W_{\\mathrm{std}}=W(6,4,3)=3A+3B\\) belongs to the standard summand. The corresponding projectors and trace pairings are\n\\[\nP_{t,2}=\\begin{pmatrix}2&4\\\\3&6\\end{pmatrix},\n\\qquad \\Tr(P_{t,2}W_{\\mathrm{sgn}})=3\\ne0,\n\\]\n\\[\nP_{s,1}=\\begin{pmatrix}0&4","3&6\\end{pmatrix},\n\\qquad \\Tr(P_{t,2}W_{\\mathrm{sgn}})=3\\ne0,\n\\]\n\\[\nP_{s,1}=\\begin{pmatrix}0&4\\\\0&1\\end{pmatrix},\n\\qquad \\Tr(P_{s,1}W_{\\mathrm{std}})=6\\ne0.\n\\]\nFor an operator \\(D\\) with \\((D-\\alpha I)(D-\\beta I)=0\\) and \\(\\alpha\\ne\\beta\\), the canonical projector is\n\\[\nP_{D,\\alpha}=(\\alpha-\\beta)^{-1}(D-\\beta I).\n\\]\nIts image equals \\(\\ker((D-\\alpha I)^n)\\) for every \\(n\\ge1\\), and it kills \\(\\ker((D-\\beta I)^n)\\). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","0&1\\end{pmatrix},\n\\qquad \\Tr(P_{s,1}W_{\\mathrm{std}})=6\\ne0.\n\\]\nFor an operator \\(D\\) with \\((D-\\alpha I)(D-\\beta I)=0\\) and \\(\\alpha\\ne\\beta\\), the canonical projector is\n\\[\nP_{D,\\alpha}=(\\alpha-\\beta)^{-1}(D-\\beta I).\n\\]\nIts image equals \\(\\ker((D-\\alpha I)^n)\\) for every \\(n\\ge1\\), and it kills \\(\\ker((D-\\beta I)^n)\\). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use\n\\[\nP_{s,-1}=\\begin{pmatrix}1&3\\\\0&0\\end{pmatrix},\n\\qquad\nP_{tst^{-1},1}=\\begin{pmatrix}1&0\\\\4&0\\end{pmatrix}.\n\\]\nThe four trace pairings with these matrices have trivial common kernel in \\(M_2(\\FF_7)\\), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy clause and completes the certificate.\n\\end{proof}","4&0\\end{pmatrix}.\n\\]\nThe four trace pairings with these matrices have trivial common kernel in \\(M_2(\\FF_7)\\), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy clause and completes the certificate.\n\\end{proof}","Let \\(\\Delta\\) be a finite abelian \\(p\\) group and let\n\\[\n\\Lambda=\\OO[\\Delta].\n\\]\nThe augmentation map is\n\\[\n\\epsilon:\\Lambda\\longrightarrow\\OO,\n\\qquad\n\\sum_{g\\in\\Delta}a_gg\\longmapsto\\sum_{g\\in\\Delta}a_g.\n\\]\nIts kernel is the augmentation ideal \\(\\fraka\\).","\\begin{proposition}[Functorial group algebra transfer]\\label{prop:group_algebra_transfer}\nA group homomorphism \\(q:\\Delta'\\to\\Delta\\) induces an \\(\\OO\\) algebra homomorphism\n\\[\nq_*:\\OO[\\Delta']\\longrightarrow\\OO[\\Delta].\n\\]\nIf \\(q\\) is surjective, then \\(q_*\\) is surjective. Augmentation commutes with the transfer:\n\\[\n\\epsilon_\\Delta q_*=\\epsilon_{\\Delta'}.\n\\]\n\\end{proposition}","\\begin{proof}\nExtend \\(q\\) linearly from the group basis. Multiplicativity follows from \\(q(gh)=q(g)q(h)\\). Surjectivity follows because every basis element of \\(\\OO[\\Delta]\\) has a preimage. Both augmentation maps send every group element to one, which gives the final identity.\n\\end{proof}","\\begin{theorem}[Residual augmentation algebra]\\label{thm:residual_augmentation_algebra}\nSuppose\n\\[\n\\Delta\\cong\n\\prod_{i=1}^{r}C_{p^{n_i}}.\n\\]\nChoose generators \\(\\gamma_i\\), and let \\(X_i=\\gamma_i-1\\). Then\n\\[\n\\boxed{\nk[\\Delta]\n\\cong\nk[X_1,\\ldots,X_r]\n\\big/\n(X_1^{p^{n_1}},\\ldots,X_r^{p^{n_r}}).\n}\n\\]\nThe augmentation ideal is generated by the \\(X_i\\), and \\(k[\\Delta]\\) is local.\n\\end{theorem}","\\begin{proof}\nIn characteristic \\(p\\),\n\\[\n(1+X_i)^{p^{n_i}}-1=X_i^{p^{n_i}}.\n\\]\nThus the assignment \\(X_i\\mapsto\\gamma_i-1\\) gives a surjective homomorphism from the displayed truncated polynomial ring to \\(k[\\Delta]\\). Both sides have dimension \\(\\prod_ip^{n_i}=\\abs{\\Delta}\\) over \\(k\\), so the map is an isomorphism. The quotient by the ideal generated by the \\(X_i\\) is \\(k\\). Since each \\(X_i\\) is nilpotent, that ideal is the unique maximal ideal.\n\\end{proof}","\\begin{corollary}[No residual character splitting]\\label{cor:no_residual_character_splitting}\nThe residual group algebra \\(k[\\Delta]\\) has no nontrivial idempotents. In particular, character projectors do not split a finite \\(p\\) group algebra in residue characteristic \\(p\\).\n\\end{corollary}","\\begin{proof}\nThe first assertion follows from locality in \\cref{thm:residual_augmentation_algebra}. For the general finite \\(p\\) group assertion, let \\(e\\in k[G]\\) be idempotent and write \\(\\epsilon:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","llows from locality in \\cref{thm:residual_augmentation_algebra}. For the general finite \\(p\\) group assertion, let \\(e\\in k[G]\\) be idempotent and write \\(\\epsilon:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","ssertion, let \\(e\\in k[G]\\) be idempotent and write \\(\\epsilon:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","on:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","A vector in the regular representation fixed by all left translations has constant coefficients. Consequently \\(ve=\\epsilon(e)v=0\\). On the other hand, \\(e^2=e\\) implies \\(we=w\\) for every \\(w\\in W\\), hence \\(ve=v\\), a contradiction. Thus every idempotent of augmentation zero is zero. Since \\(\\epsilon(e)^2=\\epsilon(e)\\) in the field \\(k\\), either \\(\\epsilon(e)=0\\) or \\(\\epsilon(e)=1\\). In the latter case, apply the preceding argument to \\(1-e\\).\n\\end{proof}","d by all left translations has constant coefficients. Consequently \\(ve=\\epsilon(e)v=0\\). On the other hand, \\(e^2=e\\) implies \\(we=w\\) for every \\(w\\in W\\), hence \\(ve=v\\), a contradiction. Thus every idempotent of augmentation zero is zero. Since \\(\\epsilon(e)^2=\\epsilon(e)\\) in the field \\(k\\), either \\(\\epsilon(e)=0\\) or \\(\\epsilon(e)=1\\). In the latter case, apply the preceding argument to \\(1-e\\).\n\\end{proof}","For \\(M\\otimes_{\\Lambda,\\epsilon}\\OO\\), we use a right \\(\\Lambda\\) module \\(M\\). The left module convention is \\(\\OO\\otimes_{\\Lambda,\\epsilon}M\\). Throughout, \\(\\fraka M\\) denotes the augmentation relation submodule: it is generated by \\(ma\\) for right modules and by \\(am\\) for left modules, with \\(a\\in\\fraka\\). This convention also applies to nonabelian groups in \\cref{cor:dyadic_coinvariant_recovery}. For abelian groups, both tensor orders agree over the commutative group algebra.","\\begin{theorem}[Coinvariant recovery]\\label{thm:coinvariant_recovery}\nLet \\(M\\) be a free \\(\\Lambda\\) module of rank \\(d\\). Then\n\\[\n\\boxed{\nM/\\fraka M\n\\cong\nM\\otimes_{\\Lambda,\\epsilon}\\OO\n\\cong\n\\OO^d.\n}\n\\]\nThe statement is valid for every coefficient prime, including two.\n\\end{theorem}","\\begin{proof}\nChoose a \\(\\Lambda\\) basis of \\(M\\). Tensoring the resulting isomorphism \\(M\\cong\\Lambda^d\\) along augmentation gives \\(\\OO^d\\). Tensoring by \\(\\Lambda/\\fraka\\) is the same as quotienting by \\(\\fraka M\\).\n\\end{proof}","\\begin{standardfact}[Iwasawa limit]\\label{fact:iwasawa_limit}\nFor \\(\\Delta_m=(\\ZZ/p^m\\ZZ)^r\\), the inverse limit of \\(\\OO[\\Delta_m]\\) is the Iwasawa algebra\n\\[\n\\OO[[\\ZZ_p^r]]\\cong\\OO[[X_1,\\ldots,X_r]],\n\\]\nwhere \\(X_i\\) corresponds to a chosen topological generator minus one \\cite{RibesZalesskii2010}.\n\\end{standardfact}","The augmentation formalism is uniform in \\(p\\). The difficult coefficient two questions lie elsewhere. Residual image hypotheses, local deformation geometry, and classicality may change at two even though coinvariant recovery remains exact.","\\begin{definition}[Support and near faithfulness]\\label{def:support_near_faithful}\nLet \\(R\\) be a commutative ring and let \\(M\\) be an \\(R\\) module. Its support is\n\\[\n\\Supp_R(M)\n=\n\\{\\frakp\\in\\Spec R:M_{\\frakp}\\neq0\\}.\n\\]\nIf \\(R\\) is Noetherian and \\(M\\) is finite, then \\(M\\) is \\emph{nearly faithful} when \\(\\Ann_R(M)\\) is a nilpotent ideal.\n\\end{definition}","\\begin{theorem}[Finite module support]\\label{thm:finite_module_support}\nLet \\(R\\) be a commutative ring and let \\(M\\) be a finite \\(R\\) module. Then\n\\[\n\\boxed{\n\\Supp_R(M)\n=\nV\\bigl(\\Ann_R(M)\\bigr).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLet \\(\\frakp\\in\\Spec R\\). If \\(\\Ann_R(M)\\not\\subseteq\\frakp\\), choose\n\\(a\\in\\Ann_R(M)\\setminus\\frakp\\). The element \\(a\\) becomes a unit in\n\\(R_{\\frakp}\\), so \\(M_{\\frakp}=0\\).","Conversely, suppose \\(M_{\\frakp}=0\\). Choose generators\n\\(m_1,\\ldots,m_d\\) of \\(M\\). For each \\(i\\), there is an element\n\\(s_i\\notin\\frakp\\) such that \\(s_im_i=0\\). The product\n\\(s=s_1\\cdots s_d\\) lies outside \\(\\frakp\\) and annihilates every generator.\nThus \\(s\\in\\Ann_R(M)\\setminus\\frakp\\). This proves the equality.\n\\end{proof}","\\begin{theorem}[Fitting support criterion]\\label{thm:fitting_support}\nLet \\(R\\) be Noetherian and let \\(M\\) be a finite \\(R\\) module. Then\n\\[\n\\boxed{\n\\Supp_R(M)\n=\nV\\bigl(\\Fitt_0^R(M)\\bigr).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nIf \\(M=0\\), the statement follows from the usual convention \\(\\Fitt_0(0)=R\\). Assume \\(M\\neq0\\), and choose a finite presentation\n\\[\nR^m\\xrightarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","from the usual convention \\(\\Fitt_0(0)=R\\). Assume \\(M\\neq0\\), and choose a finite presentation\n\\[\nR^m\\xrightarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","nite presentation\n\\[\nR^m\\xrightarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","tarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","\\begin{theorem}[Higher Fitting strata]\\label{thm:higher_fitting_strata}\nLet \\(R\\) be Noetherian and let \\(M\\) be finite. For \\(r\\geq1\\), put\n\\[\n\\mu_{\\frakp}(M)\n=\n\\dim_{\\kappa(\\frakp)}\n\\left(M\\otimes_R\\kappa(\\frakp)\\right).\n\\]\nThen\n\\[\n\\boxed{\n\\left\\{\n\\frakp\\in\\Spec R:\n\\mu_{\\frakp}(M)\\geq r\n\\right\\}\n=\nV\\bigl(\\Fitt_{r-1}^R(M)\\bigr).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nUse a finite presentation with target rank \\(n\\). If \\(r>n\\), every fiber has dimension at most \\(n\\), so the left-hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right-hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","proof}\nUse a finite presentation with target rank \\(n\\). If \\(r>n\\), every fiber has dimension at most \\(n\\), so the left-hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right-hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","every fiber has dimension at most \\(n\\), so the left-hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right-hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","\\(n\\), so the left-hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right-hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","\\begin{proposition}[Fitting ideals and base change]\\label{prop:fitting_base_change}\nLet \\(R\\to S\\) be a ring map and let \\(M\\) be a finitely presented \\(R\\) module. Then for every \\(j\\geq0\\),\n\\[\n\\boxed{\n\\Fitt_j^S(M\\otimes_RS)\n=\n\\Fitt_j^R(M)S.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nBase change carries a finite presentation matrix for \\(M\\) to a presentation matrix for \\(M\\otimes_RS\\). Every defining minor is carried to the corresponding minor over \\(S\\).\n\\end{proof}","\\begin{theorem}[Near faithfulness is full support]\\label{thm:near_faithful_full_support}\nLet \\(R\\) be Noetherian and let \\(M\\) be a finite \\(R\\) module. The following conditions are equivalent.\n\\begin{enumerate}\n\\item \\(M\\) is nearly faithful.\n\\item \\(\\Supp_R(M)=\\Spec R\\).\n\\item \\(\\Ann_R(M)\\subseteq\\nil(R)\\).\n\\end{enumerate}\nIf \\(R\\) is reduced, these conditions are equivalent to faithfulness.\n\\end{theorem}","\\begin{proof}\nBy \\cref{thm:finite_module_support}, full support is equivalent to\n\\[\nV\\bigl(\\Ann_R(M)\\bigr)=\\Spec R.\n\\]\nThis holds exactly when \\(\\Ann_R(M)\\) is contained in every prime ideal, hence in the nilradical. Because \\(R\\) is Noetherian, the ideal \\(\\Ann_R(M)\\) is finitely generated. An ideal generated by finitely many nilpotent elements is nilpotent. This proves the equivalence of the first three conditions. If \\(R\\) is reduced, its nilradical is zero.\n\\end{proof}",":finite_module_support}, full support is equivalent to\n\\[\nV\\bigl(\\Ann_R(M)\\bigr)=\\Spec R.\n\\]\nThis holds exactly when \\(\\Ann_R(M)\\) is contained in every prime ideal, hence in the nilradical. Because \\(R\\) is Noetherian, the ideal \\(\\Ann_R(M)\\) is finitely generated. An ideal generated by finitely many nilpotent elements is nilpotent. This proves the equivalence of the first three conditions. If \\(R\\) is reduced, its nilradical is zero.\n\\end{proof}","\\begin{corollary}[Support on every irreducible component]\\label{cor:every_component_support}\nUnder the hypotheses of \\cref{thm:near_faithful_full_support}, a nearly faithful module is nonzero at the generic point of every irreducible component of \\(\\Spec R\\).\n\\end{corollary}","\\begin{proof}\nEvery generic point is a prime of \\(R\\), and full support contains every prime.\n\\end{proof}","\\begin{theorem}[Support after a quotient]\\label{thm:quotient_support}\nLet \\(R\\) be Noetherian, let \\(I\\subseteq R\\) be an ideal, and let \\(M\\) be a finite \\(R\\) module. Under the canonical identification\n\\[\n\\Spec(R/I)\\cong V(I)\\subseteq\\Spec R,\n\\]\none has\n\\[\n\\boxed{\n\\Supp_{R/I}(M/IM)\n=\n\\Supp_R(M)\\cap V(I).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLet \\(\\frakp\\supseteq I\\). Localizing gives\n\\[\n(M/IM)_{\\frakp}\n\\cong\nM_{\\frakp}/I_{\\frakp}M_{\\frakp}.\n\\]\nIf \\(M_{\\frakp}=0\\), the quotient is zero. If \\(M_{\\frakp}\\neq0\\), then\n\\(I_{\\frakp}\\) lies in the maximal ideal of the local ring \\(R_{\\frakp}\\). Nakayama's lemma shows that\n\\(M_{\\frakp}/I_{\\frakp}M_{\\frakp}\\neq0\\). The stated equality follows.\n\\end{proof}","\\begin{definition}[Ordinary patching datum]\\label{def:ordinary_patching_datum}\nAn ordinary patching datum consists of a Noetherian ring \\(R_\\infty\\), an ideal\n\\(\\fraka\\subseteq R_\\infty\\), a finite \\(R_\\infty\\) module \\(M_\\infty\\), and identifications\n\\[\nR\\cong R_\\infty/\\fraka,\n\\qquad\nM\\cong M_\\infty/\\fraka M_\\infty.\n\\]\nIn automorphy lifting applications, \\(R_\\infty\\) is a patched deformation ring and \\(M_\\infty\\) is a patched automorphic module. The definition itself contains no automorphy assertion.\n\\end{definition}","definition}[Ordinary patching datum]\\label{def:ordinary_patching_datum}\nAn ordinary patching datum consists of a Noetherian ring \\(R_\\infty\\), an ideal\n\\(\\fraka\\subseteq R_\\infty\\), a finite \\(R_\\infty\\) module \\(M_\\infty\\), and identifications\n\\[\nR\\cong R_\\infty/\\fraka,\n\\qquad\nM\\cong M_\\infty/\\fraka M_\\infty.\n\\]\nIn automorphy lifting applications, \\(R_\\infty\\) is a patched deformation ring and \\(M_\\infty\\) is a patched automorphic module. The definition itself contains no automorphy assertion.\n\\end{definition}","\\begin{theorem}[Ordinary patching support descent]\\label{thm:ordinary_support_descent}\nLet \\((R_\\infty,\\fraka,M_\\infty,R,M)\\) be an ordinary patching datum. If\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty,\n\\]\nthen\n\\[\n\\boxed{\n\\Supp_R(M)=\\Spec R.\n}\n\\]\nEquivalently, if \\(M_\\infty\\) is nearly faithful, then \\(M\\) is nearly faithful. If \\(R\\) is reduced, then \\(M\\) is faithful.\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:quotient_support} with \\(I=\\fraka\\). The support of the quotient is\n\\[\n\\Spec R_\\infty\\cap V(\\fraka)=V(\\fraka),\n\\]\nwhich identifies with \\(\\Spec R\\). The remaining statements follow from\n\\cref{thm:near_faithful_full_support}.\n\\end{proof}","\\begin{corollary}[Componentwise finite level occupancy]\\label{cor:componentwise_occupancy}\nIn the setting of \\cref{thm:ordinary_support_descent}, every irreducible component of\n\\(\\Spec R\\) is contained in the support of \\(M\\). In particular, no component of the finite level deformation ring is erased by the quotient from the patched level.\n\\end{corollary}","\\begin{proof}\nThe module \\(M\\) has full support by \\cref{thm:ordinary_support_descent}.\n\\end{proof}","\\begin{example}[Freeness over auxiliary variables is not enough]\\label{ex:freeness_not_support}\nLet \\(k\\) be a field, let \\(S=k[[t]]\\), and put\n\\[\nR=S[[x]]/(x(x-t)).\n\\]\nThe ring \\(R\\) is reduced and has two irreducible components. Let\n\\[\nM=R/(x)\\cong S.\n\\]\nThen \\(M\\) is free of rank one over \\(S\\), but\n\\[\n\\Supp_R(M)=V(x)\n\\]\nis only one component of \\(\\Spec R\\). Thus freeness over the auxiliary power series ring does not imply full support over the deformation ring.\n\\end{example}","\\begin{proof}\nThe two nonassociate prime factors \\(x\\) and \\(x-t\\) occur with multiplicity one, so the principal quotient is reduced. The displayed isomorphism makes \\(M\\) free over \\(S\\). Its annihilator as an \\(R\\) module is \\((x)\\), hence\n\\[\n\\Supp_R(M)=V(x)\n\\]\nby \\cref{thm:finite_module_support}. The generic point \\((x-t)\\) of the other component does not contain \\(x\\), so it is not in the support.\n\\end{proof}","\\begin{definition}[Derived support]\\label{def:derived_support}\nLet \\(R\\) be a commutative ring and let \\(C\\) be a perfect complex of \\(R\\) modules. Its derived support is\n\\[\n\\Supp_R(C)\n=\n\\left\\{\n\\frakp\\in\\Spec R:\nC\\mathbin{\\mathop{\\otimes}\\limits_R^{\\mathbf L}}\\kappa(\\frakp)\n\\not\\simeq0\n\\right\\}.\n\\]\n\\end{definition}","\\begin{standardfact}[Cohomological description of derived support]\\label{fact:derived_support_cohomology}\nIf \\(R\\) is Noetherian and \\(C\\) is perfect, then\n\\[\n\\Supp_R(C)\n=\n\\bigcup_i\\Supp_R H^i(C).\n\\]\nIn particular, derived support is closed and is empty exactly when \\(C\\simeq0\\) \\cite{Weibel1994,Eisenbud1995}.\n\\end{standardfact}","\\begin{theorem}[Derived quotient support]\\label{thm:derived_quotient_support}\nLet \\(R\\) be a commutative ring, let \\(I\\subseteq R\\) be an ideal, and let \\(C\\) be a perfect complex over \\(R\\). Under\n\\(\\Spec(R/I)\\cong V(I)\\), one has\n\\[\n\\boxed{\n\\Supp_{R/I}\n\\left(\nC\\mathbin{\\mathop{\\otimes}\\limits_R^{\\mathbf L}}R/I\n\\right)\n=\n\\Supp_R(C)\\cap V(I).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nChoose a bounded complex \\(P\\) of finite projective \\(R\\) modules representing \\(C\\). Scalar extension to a commutative \\(R\\) algebra \\(S\\) is computed degree by degree as \\(S\\otimes_RP\\). The canonical symmetry of tensor products gives the displayed order \\(C\\otimes_R^{\\mathbf L}S\\). Scalar extension preserves the bounds, finite generation, and projectivity of the terms. A derived isomorphism between two such projective models lifts to a homotopy equivalence, which scalar extension preserves. The resulting derived object and its residue field fibers are therefore independent of the chosen model.","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the su","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on co","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated a","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","omorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","ong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","y flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","\\begin{definition}[Derived patching datum]\\label{def:derived_patching_datum}\nA derived patching datum consists of a Noetherian ring \\(R_\\infty\\), an ideal\n\\(\\fraka\\), a perfect complex \\(C_\\infty\\) over \\(R_\\infty\\), and identifications\n\\[\nR\\cong R_\\infty/\\fraka,\n\\qquad\nC\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R.\n\\]\n\\end{definition}","\\begin{theorem}[Derived patching support descent]\\label{thm:derived_support_descent}\nLet \\((R_\\infty,\\fraka,C_\\infty,R,C)\\) be a derived patching datum. If\n\\[\n\\Supp_{R_\\infty}(C_\\infty)=\\Spec R_\\infty,\n\\]\nthen\n\\[\n\\boxed{\n\\Supp_R(C)=\\Spec R.\n}\n\\]\nFor every point \\(x:R\\to E'\\) to a field, the derived fiber\n\\[\nC\\mathbin{\\mathop{\\otimes}\\limits_{R,x}^{\\mathbf L}}E'\n\\]\nis nonzero.\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:derived_quotient_support} to the patched complex and use the actual ring identification \\(R\\cong R_\\infty/\\fraka\\). Full patched support restricts to \\(V(\\fraka)\\), which corresponds to all of \\(\\Spec R\\). The supplied derived isomorphism with \\(C\\) preserves this support by the projective model argument in that theorem.","For a field valued point \\(x:R\\to E'\\), put \\(\\frakp=\\ker x\\). The point factors as \\(R\\to\\kappa(\\frakp)\\to E'\\), where the second map is a field embedding. Tensor association on a projective model identifies the fiber at \\(x\\) with the extension to \\(E'\\) of the fiber at \\(\\frakp\\). The latter is nonzero by full support. Faithful flatness of the field extension reflects exactness in every degree, so the fiber at \\(x\\) is nonzero as well. This applies to every field valued point, with no closed point restriction.\n\\end{proof}","point \\(x:R\\to E'\\), put \\(\\frakp=\\ker x\\). The point factors as \\(R\\to\\kappa(\\frakp)\\to E'\\), where the second map is a field embedding. Tensor association on a projective model identifies the fiber at \\(x\\) with the extension to \\(E'\\) of the fiber at \\(\\frakp\\). The latter is nonzero by full support. Faithful flatness of the field extension reflects exactness in every degree, so the fiber at \\(x\\) is nonzero as well. This applies to every field valued point, with no closed point restriction.\n\\end{proof}","\\begin{corollary}[A nonzero cohomological degree survives]\\label{cor:derived_degree_survives}\nUnder the hypotheses of \\cref{thm:derived_support_descent}, for every field valued point \\(x:R\\to E'\\), there is an integer \\(i\\) such that\n\\[\nH^i\n\\left(\nC\\mathbin{\\mathop{\\otimes}\\limits_{R,x}^{\\mathbf L}}E'\n\\right)\n\\neq0.\n\\]\n\\end{corollary}","\\section{Transport through an isomorphism \\texorpdfstring{\\(R\\cong\\TT\\)}{R equals T}}\\label{sec:rt}","\\begin{theorem}[Support and Fitting transport]\\label{thm:rt_support_transport}\nLet \\(\\varphi:R\\to\\TT\\) be an isomorphism of Noetherian rings. Let \\(M\\) be a finite \\(\\TT\\) module, viewed as an \\(R\\) module through \\(\\varphi\\). Then the homeomorphism\n\\[\n\\varphi^*:\\Spec\\TT\\longrightarrow\\Spec R\n\\]\nsatisfies\n\\[\n\\boxed{\n\\varphi^*\\bigl(\\Supp_{\\TT}(M)\\bigr)\n=\n\\Supp_R(M).\n}\n\\]\nFor every \\(j\\geq0\\),\n\\[\n\\boxed{\n\\varphi\n\\left(\n\\Fitt_j^R(M)\n\\right)\n=\n\\Fitt_j^{\\TT}(M).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLocalization commutes with transport along an isomorphism, so a localization of \\(M\\) vanishes on one side exactly when the corresponding localization vanishes on the other. This proves the support identity. A finite presentation matrix over \\(R\\) is carried entry by entry to a presentation matrix over \\(\\TT\\). The minors generating each Fitting ideal are therefore carried to the corresponding minors.\n\\end{proof}","\\begin{corollary}[Full support is invariant under \\texorpdfstring{\\(R\\cong\\TT\\)}{R equals T}]\\label{cor:rt_full_support}\nIn the setting of \\cref{thm:rt_support_transport}, \\(M\\) has full \\(R\\) support if and only if it has full \\(\\TT\\) support.\n\\end{corollary}","\\begin{proposition}[A quotient can erase a component]\\label{prop:quotient_erases_component}\nLet\n\\[\nR=k[[x,y]]/(xy)\n\\]\nand let\n\\[\n\\TT=R/(x)\\cong k[[y]].\n\\]\nViewed as an \\(R\\) module, \\(\\TT\\) has support \\(V(x)\\), which misses the generic point \\((y)\\) of the other irreducible component. Thus a surjection \\(R\\twoheadrightarrow\\TT\\) does not by itself imply full component occupancy.\n\\end{proposition}","\\begin{proof}\nThe annihilator of the \\(R\\) module \\(\\TT\\) is \\((x)\\). Hence its support is \\(V(x)\\) by \\cref{thm:finite_module_support}. The prime \\((y)\\) does not contain \\(x\\), so it is not in this support.\n\\end{proof}","\\begin{theorem}[Field fiber criterion]\\label{thm:field_fiber_criterion}\nLet \\(R\\) be a commutative ring, let \\(M\\) be a finite \\(R\\) module, and let\n\\(x:R\\to E'\\) be a homomorphism to a field. Put \\(\\frakp=\\ker x\\). Then\n\\[\n\\boxed{\nM\\otimes_{R,x}E'\\neq0\n\\quad\\Longleftrightarrow\\quad\n\\frakp\\in\\Supp_R(M).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe map \\(x\\) factors through the local ring \\(R_{\\frakp}\\) and then through an embedding of the residue field \\(\\kappa(\\frakp)\\) into \\(E'\\). Since \\(M\\) is finite, Nakayama's lemma gives\n\\[\nM_{\\frakp}\\neq0\n\\quad\\Longleftrightarrow\\quad\nM\\otimes_R\\kappa(\\frakp)\\neq0.\n\\]\nExtension of scalars from \\(\\kappa(\\frakp)\\) to \\(E'\\) preserves nonzero vector spaces.\n\\end{proof}","\\begin{corollary}[Supported deformation points give nonzero Hecke fibers]\\label{cor:supported_hecke_fiber}\nLet \\(R\\cong\\TT\\), and let \\(M\\) be a finite Hecke module. If a point\n\\(x:R\\to E'\\) lies in \\(\\Supp_R(M)\\), then the corresponding Hecke fiber\n\\[\nM\\otimes_{\\TT,x}E'\n\\]\nis nonzero.\n\\end{corollary}","\\begin{standardfact}[Artin decomposition]\\label{fact:artin_decomposition}\nLet \\(E'\\) be a field and let \\(\\TT\\) be a finite dimensional commutative\n\\(E'\\) algebra. Then\n\\[\n\\boxed{\n\\TT\n\\cong\n\\prod_{\\frakm\\in\\Max\\TT}\\TT_{\\frakm}.\n}\n\\]\nEach factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents \\((e_{\\frakm})\\) such that\n\\[\ne_{\\frakm}e_{\\mathfrak n}=0\n\\quad(\\frakm\\neq\\mathfrak n),\n\\qquad\n\\sum_{\\frakm}e_{\\frakm}=1,\n\\]\nand \\(e_{\\frakm}\\TT\\cong\\TT_{\\frakm}\\) \\cite{AtiyahMacdonald1969,Matsumura1989}.\n\\end{standardfact}","Let \\(E'\\) be a field and let \\(\\TT\\) be a finite dimensional commutative\n\\(E'\\) algebra. Then\n\\[\n\\boxed{\n\\TT\n\\cong\n\\prod_{\\frakm\\in\\Max\\TT}\\TT_{\\frakm}.\n}\n\\]\nEach factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents \\((e_{\\frakm})\\) such that\n\\[\ne_{\\frakm}e_{\\mathfrak n}=0\n\\quad(\\frakm\\neq\\mathfrak n),\n\\qquad\n\\sum_{\\frakm}e_{\\frakm}=1,\n\\]\nand \\(e_{\\frakm}\\TT\\cong\\TT_{\\frakm}\\) \\cite{AtiyahMacdonald1969,Matsumura1989}.\n\\end{standardfact}","\\begin{theorem}[Canonical Hecke packet decomposition]\\label{thm:canonical_hecke_packets}\nLet \\(\\TT\\) be a finite dimensional commutative algebra over a field \\(E'\\), and let \\(V\\) be a finite \\(\\TT\\) module. Then\n\\[\n\\boxed{\nV\n=\n\\bigoplus_{\\frakm\\in\\Max\\TT}e_{\\frakm}V.\n}\n\\]\nFor every maximal ideal \\(\\frakm\\), the canonical map gives\n\\[\n\\boxed{\ne_{\\frakm}V\\cong V_{\\frakm}.}\n\\]\nThe packet \\(e_{\\frakm}V\\) is nonzero exactly when \\(\\frakm\\in\\Supp_{\\TT}(V)\\).\n\\end{theorem}","packet decomposition]\\label{thm:canonical_hecke_packets}\nLet \\(\\TT\\) be a finite dimensional commutative algebra over a field \\(E'\\), and let \\(V\\) be a finite \\(\\TT\\) module. Then\n\\[\n\\boxed{\nV\n=\n\\bigoplus_{\\frakm\\in\\Max\\TT}e_{\\frakm}V.\n}\n\\]\nFor every maximal ideal \\(\\frakm\\), the canonical map gives\n\\[\n\\boxed{\ne_{\\frakm}V\\cong V_{\\frakm}.}\n\\]\nThe packet \\(e_{\\frakm}V\\) is nonzero exactly when \\(\\frakm\\in\\Supp_{\\TT}(V)\\).\n\\end{theorem}","\\begin{proof}\nThe idempotent relations in \\cref{fact:artin_decomposition} give the direct sum decomposition. Localization at \\(\\frakm\\) kills every factor except the \\(\\frakm\\) factor, where \\(e_{\\frakm}\\) becomes one. Thus the localization map identifies \\(e_{\\frakm}V\\) with \\(V_{\\frakm}\\). The final statement is the definition of support.\n\\end{proof}","\\begin{corollary}[Simultaneous generalized eigenspaces]\\label{cor:simultaneous_generalized_eigenspaces}\nAssume that \\(E'\\) is algebraically closed and that \\(\\TT\\) is generated by commuting operators \\(T_1,\\ldots,T_d\\) on \\(V\\). For a maximal ideal\n\\[\n\\frakm\n=\n(T_1-\\lambda_1,\\ldots,T_d-\\lambda_d),\n\\]\nthere is an integer \\(N\\geq1\\) such that\n\\[\n\\boxed{\ne_{\\frakm}V\n=\n\\left\\{\nv\\in V:\n(T_i-\\lambda_i)^Nv=0\n\\text{ for every }i\n\\right\\}.\n}\n\\]\nThus the canonical Artin packet is the simultaneous generalized eigenspace.\n\\end{corollary}","\\begin{proof}\nThe maximal ideal of the Artin local algebra \\(\\TT_{\\frakm}\\) is nilpotent. Hence a common power of every \\(T_i-\\lambda_i\\) kills the local packet. On every other local factor, at least one \\(T_i-\\lambda_i\\) is a unit, so no nonzero vector can satisfy all the displayed nilpotence conditions.","Conversely, suppose every \\((T_i-\\lambda_i)^N\\) kills \\(v\\). Every monomial of total degree greater than \\(d(N-1)\\) contains one factor to exponent at least \\(N\\). Therefore a power of \\(\\frakm\\) kills \\(v\\), which places \\(v\\) in the \\(\\frakm\\) primary packet.\n\\end{proof}","\\begin{corollary}[Rank and trace detection in characteristic zero]\\label{cor:rank_trace_detection}\nAssume that \\(E'\\) has characteristic zero. For every maximal Hecke packet,\n\\[\n\\boxed{\n\\Tr_{E'}(e_{\\frakm}|V)\n=\n\\rank_{E'}(e_{\\frakm}|V)\n=\n\\dim_{E'}e_{\\frakm}V.\n}\n\\]\nIn particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.\n\\end{corollary}","\\begin{proof}\nAn idempotent linear map has minimal polynomial dividing \\(X(X-1)\\), which has distinct roots. It is diagonalizable with eigenvalues zero and one. Its trace and rank both equal the multiplicity of the eigenvalue one. A positive integer is nonzero in characteristic zero.\n\\end{proof}","\\begin{example}[A nonsemisimple Hecke packet]\\label{ex:nonsemisimple_hecke_packet}\nLet\n\\[\n\\TT\n=\nE'[u]/(u^2)\n\\times\nE'[v]/(v^3)\n\\]\nand let \\(V=\\TT\\) be the regular module. The two canonical central idempotents split\n\\[\nV=V_u\\oplus V_v\n\\]\nwith dimensions two and three. Multiplication by \\(u\\) is nilpotent on \\(V_u\\), and multiplication by \\(v\\) is nilpotent on \\(V_v\\). Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.\n\\end{example}","\\begin{proof}\nThe product decomposition gives the two central idempotents \\((1,0)\\) and \\((0,1)\\). Their ranges are the two local factors. The classes of \\(u\\) and \\(v\\) are nilpotent of orders two and three, respectively.\n\\end{proof}","\\begin{theorem}[Nested packets in one finite carrier]\\label{thm:nested_finite_carrier}\nLet \\(V\\) be a finite dimensional vector space. Let \\(e_1,e_2,\\ldots\\) be idempotents such that\n\\[\ne_{m+1}e_m=e_me_{m+1}=e_{m+1}\n\\]\nfor every \\(m\\). If \\(e_mV\\neq0\\) for every \\(m\\), then the descending chain stabilizes and\n\\[\n\\boxed{\n\\bigcap_{m\\geq1}e_mV\n\\neq0.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe displayed relations give\n\\[\ne_{m+1}V\\subseteq e_mV.\n\\]\nThe nonnegative dimensions form a nonincreasing sequence of positive integers, so they eventually become constant. Once two consecutive dimensions agree, the corresponding inclusion is equality. Thus the ranges stabilize at a nonzero subspace, which is their intersection.\n\\end{proof}","\\begin{corollary}[Finite witness of incompatibility]\\label{cor:finite_witness_incompatibility}\nUnder the hypotheses of \\cref{thm:nested_finite_carrier}, if\n\\[\n\\bigcap_{m\\geq1}e_mV=0,\n\\]\nthen \\(e_mV=0\\) for some finite \\(m\\).\n\\end{corollary}","\\begin{definition}[Stable core]\\label{def:stable_core}\nLet\n\\[\nV_1\\longleftarrow V_2\\longleftarrow V_3\\longleftarrow\\cdots\n\\]\nbe an inverse system of finite dimensional vector spaces. For \\(n\\geq m\\), write\n\\(f_{n,m}:V_n\\to V_m\\). The stable core at level \\(m\\) is\n\\[\nY_m\n=\n\\bigcap_{n\\geq m}\\im(f_{n,m}).\n\\]\n\\end{definition}","\\begin{lemma}[Surjectivity on stable cores]\\label{lem:stable_core_surjectivity}\nFor every \\(m\\), the transition map \\(V_{m+1}\\to V_m\\) induces a surjection\n\\[\n\\boxed{\nY_{m+1}\\twoheadrightarrow Y_m.\n}\n\\]\n\\end{lemma}","\\begin{proof}\nFix \\(y\\in Y_m\\). For \\(n\\geq m+1\\), let\n\\[\nA_n\n=\n\\left\\{\nz\\in\\im(f_{n,m+1}):\nf_{m+1,m}(z)=y\n\\right\\}.\n\\]\nEach \\(A_n\\) is a nonempty affine subspace of \\(V_{m+1}\\), and the sequence\n\\((A_n)\\) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in \\(Y_{m+1}\\) and maps to \\(y\\).\n\\end{proof}","y\\in Y_m\\). For \\(n\\geq m+1\\), let\n\\[\nA_n\n=\n\\left\\{\nz\\in\\im(f_{n,m+1}):\nf_{m+1,m}(z)=y\n\\right\\}.\n\\]\nEach \\(A_n\\) is a nonempty affine subspace of \\(V_{m+1}\\), and the sequence\n\\((A_n)\\) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in \\(Y_{m+1}\\) and maps to \\(y\\).\n\\end{proof}","\\begin{theorem}[Nonzero stable core criterion]\\label{thm:nonzero_stable_core}\nLet \\((V_m)\\) be an inverse system of finite dimensional vector spaces. If\n\\[\nY_m\\neq0\n\\]\nfor every \\(m\\), then\n\\[\n\\boxed{\n\\varprojlim_mV_m\n\\neq0.\n}\n\\]\nMore precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of \\(Y_1\\).\n\\end{theorem}","\\begin{proof}\nChoose \\(0\\neq y_1\\in Y_1\\). By \\cref{lem:stable_core_surjectivity}, choose recursively\n\\(y_{m+1}\\in Y_{m+1}\\) mapping to \\(y_m\\). Every \\(y_m\\) is nonzero because it maps to the nonzero vector \\(y_1\\). The sequence \\((y_m)\\) is a nonzero element of the inverse limit.\n\\end{proof}","\\begin{corollary}[Surjective systems]\\label{cor:surjective_inverse_system}\nIf every transition map \\(V_{m+1}\\to V_m\\) is surjective and every \\(V_m\\) is nonzero, then the inverse limit is nonzero.\n\\end{corollary}","\\begin{proof}\nSurjectivity gives \\(Y_m=V_m\\). Apply \\cref{thm:nonzero_stable_core}.\n\\end{proof}","\\begin{example}[Tail escape in an infinite carrier]\\label{ex:tail_escape}\nLet \\(V=\\ell^2(\\NN)\\), and let \\(P_m\\) be the orthogonal projection onto the closed span of\n\\[\ne_m,e_{m+1},e_{m+2},\\ldots.\n\\]\nEvery range \\(P_mV\\) is nonzero and the ranges are nested, but\n\\[\n\\boxed{\n\\bigcap_{m\\geq1}P_mV=0.\n}\n\\]\nThus the finite carrier hypothesis in \\cref{thm:nested_finite_carrier} cannot be removed.\n\\end{example}","\\begin{proof}\nA vector in every tail range has its first \\(m-1\\) coordinates equal to zero for every \\(m\\). All of its coordinates vanish.\n\\end{proof}","\\begin{definition}[Automorphic Hecke datum]\\label{def:automorphic_hecke_datum}\nAn automorphic Hecke datum consists of the following objects.\n\\begin{enumerate}\n\\item A commutative \\(\\OO\\) algebra \\(\\TT\\).\n\\item A finite \\(\\TT\\) module \\(H\\), or a perfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","efinition}[Automorphic Hecke datum]\\label{def:automorphic_hecke_datum}\nAn automorphic Hecke datum consists of the following objects.\n\\begin{enumerate}\n\\item A commutative \\(\\OO\\) algebra \\(\\TT\\).\n\\item A finite \\(\\TT\\) module \\(H\\), or a perfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","te \\(\\TT\\) module \\(H\\), or a perfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","erfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","\\begin{standardfact}[Chebotarev and Brauer Nesbitt identification]\\label{fact:chebotarev_brauer_nesbitt}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let\n\\[\n\\rho_1,\\rho_2:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nbe continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then\n\\[\n\\boxed{\\rho_1\\cong\\rho_2.}\n\\]\nThis follows from Chebotarev density and the Brauer Nesbitt theorem\n\\cite{BrauerNesbitt1941,Neukirch1999,Gee2022}.\n\\end{standardfact}","[Chebotarev and Brauer Nesbitt identification]\\label{fact:chebotarev_brauer_nesbitt}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let\n\\[\n\\rho_1,\\rho_2:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nbe continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then\n\\[\n\\boxed{\\rho_1\\cong\\rho_2.}\n\\]\nThis follows from Chebotarev density and the Brauer Nesbitt theorem\n\\cite{BrauerNesbitt1941,Neukirch1999,Gee2022}.\n\\end{standardfact}","\\begin{theorem}[Support gives a Hecke eigensystem]\\label{thm:support_gives_hecke_system}\nLet \\(R\\cong\\TT\\), let \\(H\\) be a finite \\(\\TT\\) module, and let\n\\(x:R\\to E'\\) be a field valued point. If\n\\[\n\\ker x\\in\\Supp_R(H),\n\\]\nthen\n\\[\n\\boxed{\nH_x:=H\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nEvery element of \\(\\TT\\) acts on \\(H_x\\) through the scalar prescribed by \\(x\\).\n\\end{theorem}","\\begin{proof}\nNonvanishing follows from \\cref{cor:supported_hecke_fiber}. The tensor product relation\n\\[\nth\\otimes1=h\\otimes x(t)\n\\]\nshows that \\(t\\in\\TT\\) acts through \\(x(t)\\).\n\\end{proof}","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nw","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nf","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irre…","galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","pological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","us ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","uire both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","tion below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","me the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","g\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}",")\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","tomorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","e attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","\\begin{proof}\nThe coefficients of the characteristic polynomials of \\(\\rho_x\\) and \\(\\rho_{\\pi_x}\\) are continuous and constant on conjugacy classes. Their equality on the declared dense union of Frobenius conjugacy classes therefore extends to every element of \\(G_{F,S}\\), by Hausdorffness. A semisimplification has the same characteristic polynomials as the original representation. The algebraic Brauer Nesbitt input in \\cref{fact:chebotarev_brauer_nesbitt} thus gives \\(\\rho_x^{\\mathrm{ss}}\\cong\\rho_{\\pi_x}\\). If \\(\\rho_x\\) is irreducible, it is already semisimple.\n\\end{proof}","x\\) and \\(\\rho_{\\pi_x}\\) are continuous and constant on conjugacy classes. Their equality on the declared dense union of Frobenius conjugacy classes therefore extends to every element of \\(G_{F,S}\\), by Hausdorffness. A semisimplification has the same characteristic polynomials as the original representation. The algebraic Brauer Nesbitt input in \\cref{fact:chebotarev_brauer_nesbitt} thus gives \\(\\rho_x^{\\mathrm{ss}}\\cong\\rho_{\\pi_x}\\). If \\(\\rho_x\\) is irreducible, it is already semisimple.\n\\end{proof}","nstant on conjugacy classes. Their equality on the declared dense union of Frobenius conjugacy classes therefore extends to every element of \\(G_{F,S}\\), by Hausdorffness. A semisimplification has the same characteristic polynomials as the original representation. The algebraic Brauer Nesbitt input in \\cref{fact:chebotarev_brauer_nesbitt} thus gives \\(\\rho_x^{\\mathrm{ss}}\\cong\\rho_{\\pi_x}\\). If \\(\\rho_x\\) is irreducible, it is already semisimple.\n\\end{proof}","\\begin{corollary}[Full support gives pointwise automorphy on the realized locus]\\label{cor:full_support_pointwise_automorphy}\nIn the setting of \\cref{thm:pointwise_galois_realization}, suppose\n\\[\n\\Supp_R(H)=\\Spec R.\n\\]\nFor every characteristic zero point \\(x:R\\to E'\\) with \\(\\ker x\\in U\\subseteq\\Spec R\\), assume the coefficient topology and continuity hypotheses of \\cref{thm:pointwise_galois_realization} and the declared classicality and Galois realization input. Then every such point is automorphic in the precise sense of \\cref{thm:pointwise_galois_realization}.\n\\end{corollary}","\\begin{proof}\nEvery point of \\(U\\) lies in the support, so \\cref{thm:support_gives_hecke_system} gives a nonzero Hecke fiber. Apply the declared realization input and then \\cref{thm:pointwise_galois_realization}.\n\\end{proof}","\\begin{theorem}[Derived pointwise realization]\\label{thm:derived_pointwise_realization}\nLet \\(e:R\\cong\\TT\\), and let \\(C\\) be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for \\(C\\), supported in degrees \\([a,b]\\). For every characteristic zero point \\(x:R\\to E'\\), write \\(y=x\\circ e^{-1}\\). Then some integer \\(i\\in[a,b]\\) has\n\\[\nH^i\\left(C\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\right)\\neq0.\n\\]\nEvery \\(t\\in\\TT\\) acts on this cohomology through \\(y(t)\\).","zation}\nLet \\(e:R\\cong\\TT\\), and let \\(C\\) be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for \\(C\\), supported in degrees \\([a,b]\\). For every characteristic zero point \\(x:R\\to E'\\), write \\(y=x\\circ e^{-1}\\). Then some integer \\(i\\in[a,b]\\) has\n\\[\nH^i\\left(C\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\right)\\neq0.\n\\]\nEvery \\(t\\in\\TT\\) acts on this cohomology through \\(y(t)\\).","e a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for \\(C\\), supported in degrees \\([a,b]\\). For every characteristic zero point \\(x:R\\to E'\\), write \\(y=x\\circ e^{-1}\\). Then some integer \\(i\\in[a,b]\\) has\n\\[\nH^i\\left(C\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\right)\\neq0.\n\\]\nEvery \\(t\\in\\TT\\) acts on this cohomology through \\(y(t)\\).","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the spec","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irredu","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached repres","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","[a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","\\begin{proof}\nFull derived support gives a nonzero fiber at \\(y\\). Scalar extension of the chosen projective model preserves its degree bounds. Its cohomology therefore vanishes below \\(a\\) and above \\(b\\). Since vanishing in every degree is equivalent to being zero in the derived category, some degree \\(i\\in[a,b]\\) has nonzero cohomology. Choose a nonzero vector in that cohomology module.","Multiplication by \\(t\\in\\TT\\) is a cochain endomorphism of the original model. In the scalar extension the tensor relation identifies this endomorphism with \\(y(t)\\) times the identity. Passing to the derived category and then to cohomology preserves this equality. Thus the chosen vector is a common Hecke eigenvector with character \\(y\\).","The interval inclusion puts this eigensystem in the declared classicality range. Apply the realization input to obtain an occurring automorphic representation and its continuous semisimple attached representation. Its Frobenius characteristic polynomials agree with those of \\(\\rho_x\\) on the prescribed dense set. Continuity extends the equality to every group element. The supplied semisimplification preserves these polynomials, so the general Brauer Nesbitt input gives an invertible linear intertwiner. If \\(\\rho_x\\) is irreducible, it is already semisimple and the same comparison applies directly.\n\\end{proof}","btain an occurring automorphic representation and its continuous semisimple attached representation. Its Frobenius characteristic polynomials agree with those of \\(\\rho_x\\) on the prescribed dense set. Continuity extends the equality to every group element. The supplied semisimplification preserves these polynomials, so the general Brauer Nesbitt input gives an invertible linear intertwiner. If \\(\\rho_x\\) is irreducible, it is already semisimple and the same comparison applies directly.\n\\end{proof}","\\begin{definition}[Algebraic Frobenius spine]\\label{def:frobenius_spine}\nAn algebraic Frobenius spine consists of a number field \\(L\\), a finite set of places \\(S\\) of \\(F\\), and polynomials\n\\[\nP_v(X)\\in L[X]\n\\]\nfor every finite place \\(v\\notin S\\), together with any additional local labels required by the declared compatibility problem.\n\\end{definition}","\\begin{definition}[Finite companion forest]\\label{def:companion_forest}\nFix a coefficient place \\(\\lambda\\) of \\(L\\). A finite companion forest for the spine is an inverse system of finite sets\n\\[\nX_{\\lambda,1}\n\\longleftarrow\nX_{\\lambda,2}\n\\longleftarrow\nX_{\\lambda,3}\n\\longleftarrow\\cdots\n\\]\nwhose points are admissible realizations modulo \\(\\lambda^m\\), with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.\n\\end{definition}","\\begin{theorem}[Companion branch criterion]\\label{thm:companion_branch_criterion}\nSuppose every level \\(X_{\\lambda,m}\\) is finite and nonempty, and every point reduces along the displayed forest to the fixed residual root. Then\n\\[\n\\boxed{\n\\varprojlim_mX_{\\lambda,m}\n\\neq\\varnothing.\n}\n\\]\n\\end{theorem}","\\begin{corollary}[Empty finite companion obstruction]\\label{cor:empty_companion_obstruction}\nIf \\(X_{\\lambda,m}=\\varnothing\\) for some \\(m\\), then no complete \\(\\lambda\\) adic realization admissible for this forest exists.\n\\end{corollary}","Fix a topological group \\(\\Gamma\\), a finite matrix index set \\(D\\), and a set \\(\\Lambda\\) of coefficient indices. For each \\(\\lambda\\), let \\(\\OO_\\lambda\\) be a commutative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","bda\\), let \\(\\OO_\\lambda\\) be a commutative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","\\(\\OO_\\lambda\\) be a commutative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","utative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","For every positive integer \\(n\\), let \\(X_{\\lambda,n}\\) be a finite nonempty set of conjugacy classes of actual continuous representations\n\\[\n\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda/I_\\lambda^n).\n\\]\nConjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that \\(X_{\\lambda,1}\\) consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.","Choose elements \\(\\gamma_v\\in\\Gamma\\) on a sample set \\(\\Omega\\), with declared good subsets \\(\\Omega_\\lambda\\subseteq\\Omega\\). In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivia","Choose elements \\(\\gamma_v\\in\\Gamma\\) on a sample set \\(\\Omega\\), with declared good subsets \\(\\Omega_\\lambda\\subseteq\\Omega\\). In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condi","Choose elements \\(\\gamma_v\\in\\Gamma\\) on a sample set \\(\\Omega\\), with declared good subsets \\(\\Omega_\\lambda\\subseteq\\Omega\\). In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","thmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","us elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","or each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","y an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","Finally, put\n\\[\nB_\\lambda=\\OO_\\lambda[x_{g,a,b}:g\\in\\Gamma,\\ a,b\\in D].\n\\]\nFor each declared local condition index \\(u\\), supply an ideal \\(J_{\\lambda,u}\\subseteq B_\\lambda\\). Evaluation at a representation sends \\(x_{g,a,b}\\) to its \\((a,b)\\) matrix entry at \\(g\\). Assume that evaluation at every admissible finite representative kills \\(J_{\\lambda,u}\\), with polynomial coefficients reduced modulo \\(I_\\lambda^n\\). For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitrary arithmetic condition is identified with such an ideal without a separate argument.","ally, put\n\\[\nB_\\lambda=\\OO_\\lambda[x_{g,a,b}:g\\in\\Gamma,\\ a,b\\in D].\n\\]\nFor each declared local condition index \\(u\\), supply an ideal \\(J_{\\lambda,u}\\subseteq B_\\lambda\\). Evaluation at a representation sends \\(x_{g,a,b}\\) to its \\((a,b)\\) matrix entry at \\(g\\). Assume that evaluation at every admissible finite representative kills \\(J_{\\lambda,u}\\), with polynomial coefficients reduced modulo \\(I_\\lambda^n\\). For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitrary arithmetic condition is identified with such an ideal without a separate argument.","\\begin{theorem}[Conditional simultaneous companion completion]\\label{thm:simultaneous_companion_completion}\nUnder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","neous companion completion]\\label{thm:simultaneous_companion_completion}\nUnder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","ous_companion_completion}\nUnder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","nder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","a\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","Completeness and separatedness identify \\(\\GL_D(\\OO_\\lambda)\\) with the compatible inverse limit of the quotient matrix groups. Lift entries and inverse entries to construct this identification. The two reduction and reconstruction identities hold entrywise. Finite quotient continuity implies continuity for the ideal adic topology, including the matrix inverse, so the reconstructed representation is continuous.","Each coefficient of its characteristic polynomial agrees with the prescribed integral coefficient modulo every positive power of \\(I_\\lambda\\). Separatedness gives equality of the full polynomials. The same argument applied to the matrix entries of \\(\\rho_\\lambda(g)-1\\) gives triviality on \\(H_\\lambda\\). Continuous coefficient change and the matching equations for \\(P_{\\lambda,v}\\) give the asserted common spine over the local fields.","For \\(p\\in J_{\\lambda,u}\\), evaluation commutes with reduction. Its value is zero modulo every positive power of \\(I_\\lambda\\), hence is zero integrally by separatedness. Therefore the evaluation homomorphism kills the whole ideal and factors uniquely through \\(B_\\lambda/J_{\\lambda,u}\\). Choose a realization for each \\(\\lambda\\).\n\\end{proof}","\\begin{theorem}[One member propagation in a fixed system]\\label{thm:one_member_propagation}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","stem]\\label{thm:one_member_propagation}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","ation}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","a\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFor every \\(\\lambda\\), the two semisimple representations have the same unramified characteristic polynomials at all places outside one finite set. Apply \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{proof}","\\begin{proposition}[Finite Frobenius separator]\\label{prop:finite_frobenius_separator}\nLet \\(t_1,\\ldots,t_r\\) be linearly independent functions on a set \\(\\Omega\\) with values in a field \\(E'\\). Then there are points\n\\(\\omega_1,\\ldots,\\omega_r\\in\\Omega\\) such that the matrix\n\\[\n\\left(t_i(\\omega_j)\\right)_{j,i}\n\\]\nis invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.\n\\end{proposition}","\\begin{proof}\nLet\n\\[\nW\n=\n\\operatorname{span}\n\\left\\{\n(t_1(\\omega),\\ldots,t_r(\\omega)):\\omega\\in\\Omega\n\\right\\}\n\\subseteq E'^r.\n\\]\nIf \\(W\\neq E'^r\\), a nonzero linear functional on \\(E'^r\\) would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus \\(W=E'^r\\). Choose \\(r\\) evaluation vectors forming a basis.\n\\end{proof}","\\begin{proposition}[Failure of integral sign projectors]\\label{prop:dyadic_sign_projectors}\nLet \\(C_2=\\langle g\\rangle\\). Over \\(\\QQ_2\\), the two sign projectors are\n\\[\n\\boxed{\ne_+=\\frac{1+g}{2},\n\\qquad\ne_-=\\frac{1-g}{2}.\n}\n\\]\nNeither belongs to \\(\\ZZ_2[C_2]\\). After reduction modulo two,\n\\[\n\\boxed{\n\\FF_2[C_2]\n\\cong\n\\FF_2[X]/(X^2),\n\\qquad\nX=g-1.\n}\n\\]\nThis ring is local and has no nontrivial idempotents.\n\\end{proposition}","\\begin{proof}\nThe displayed rational elements are the unique primitive idempotents because they project to the two characters of \\(C_2\\). Their coefficients contain \\(1/2\\), so they are not integral. In characteristic two,\n\\[\n(g-1)^2=g^2-2g+1=0.\n\\]\nThe map \\(X\\mapsto g-1\\) gives the displayed isomorphism. The quotient by \\((X)\\) is \\(\\FF_2\\), and \\((X)\\) is nilpotent. Thus the ring is local.\n\\end{proof}","\\begin{corollary}[Dyadic coinvariant recovery]\\label{cor:dyadic_coinvariant_recovery}\nLet \\(\\Delta\\) be a finite two group, let \\(\\Lambda=\\OO[\\Delta]\\) with \\(\\OO\\) a finite extension of \\(\\ZZ_2\\), and let \\(M\\) be finite free over \\(\\Lambda\\). Then\n\\[\n\\boxed{\nM\\otimes_{\\Lambda,\\epsilon}\\OO\n\\cong\nM/\\fraka M\n}\n\\]\nis free over \\(\\OO\\) of the same \\(\\Lambda\\) rank. No semisimple character decomposition is required.\n\\end{corollary}","\\begin{proof}\nUse the right module convention fixed in \\cref{sec:auxiliary}. Quotienting by all balancing relations identifies \\(M\\otimes_{\\Lambda,\\epsilon}\\OO\\) with \\(M/\\fraka M\\), without any commutativity assumption on \\(\\Delta\\). Applying \\(\\epsilon\\) to the coordinates of a finite \\(\\Lambda\\) basis gives an \\(\\OO\\) basis with the same index set.\n\\end{proof}","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation w","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation;…","point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of bot…","datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\…","llowing objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy …","ented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, den…","l deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and g…","ng \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Ne…","\\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retaine…","]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition…","g\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition}","phism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition}","ecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Ordinary conditional automorphic completion]\\label{thm:ordinary_conditional_completion}\nLet \\(x:R\\to E'\\) admit an ordinary completion datum. Assume \\(E'\\) is a Hausdorff topological field of characteristic zero with continuous ring operations, and that both \\(\\rho_x=x\\circ\\rho_R\\) and its attached comparison representation are continuous. Then\n\\[\n\\boxed{\nH\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nThe specialization\n\\[\n\\rho_x=x\\circ\\rho_R\n\\]\nis identified after semisimplification with the Galois representation attached to a classical automorphic representation in that fiber. If \\(\\rho_x\\) is irreducible, it is itself automorphic.\n\\end{theorem}","conditional automorphic completion]\\label{thm:ordinary_conditional_completion}\nLet \\(x:R\\to E'\\) admit an ordinary completion datum. Assume \\(E'\\) is a Hausdorff topological field of characteristic zero with continuous ring operations, and that both \\(\\rho_x=x\\circ\\rho_R\\) and its attached comparison representation are continuous. Then\n\\[\n\\boxed{\nH\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nThe specialization\n\\[\n\\rho_x=x\\circ\\rho_R\n\\]\nis identified after semisimplification with the Galois representation attached to a classical automorphic representation in that fiber. If \\(\\rho_x\\) is irreducible, it is itself automorphic.\n\\end{theorem}","ological field of characteristic zero with continuous ring operations, and that both \\(\\rho_x=x\\circ\\rho_R\\) and its attached comparison representation are continuous. Then\n\\[\n\\boxed{\nH\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nThe specialization\n\\[\n\\rho_x=x\\circ\\rho_R\n\\]\nis identified after semisimplification with the Galois representation attached to a classical automorphic representation in that fiber. If \\(\\rho_x\\) is irreducible, it is itself automorphic.\n\\end{theorem}","\\begin{proof}\nFull patched support descends to full support of \\(H\\) over \\(R\\) by\n\\cref{thm:ordinary_support_descent}. The isomorphism \\(R\\cong\\TT\\) transports that support by \\cref{thm:rt_support_transport}. The field fiber is nonzero by\n\\cref{thm:support_gives_hecke_system}. The declared classicality and realization theorem supplies an automorphic representation in the fiber and an attached Galois representation. The comparison with \\(\\rho_x\\) follows from \\cref{thm:pointwise_galois_realization}.\n\\end{proof}","s to full support of \\(H\\) over \\(R\\) by\n\\cref{thm:ordinary_support_descent}. The isomorphism \\(R\\cong\\TT\\) transports that support by \\cref{thm:rt_support_transport}. The field fiber is nonzero by\n\\cref{thm:support_gives_hecke_system}. The declared classicality and realization theorem supplies an automorphic representation in the fiber and an attached Galois representation. The comparison with \\(\\rho_x\\) follows from \\cref{thm:pointwise_galois_realization}.\n\\end{proof}","\\begin{definition}[Derived completion datum at a point]\\label{def:derived_completion_datum}\nA derived completion datum is defined in the same way, with a perfect patched complex\n\\(C_\\infty\\) of full derived support and a finite level automorphic complex\n\\[\nC\n\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","n{definition}[Derived completion datum at a point]\\label{def:derived_completion_datum}\nA derived completion datum is defined in the same way, with a perfect patched complex\n\\(C_\\infty\\) of full derived support and a finite level automorphic complex\n\\[\nC\n\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","ll derived support and a finite level automorphic complex\n\\[\nC\n\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","meq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","\\begin{theorem}[Derived conditional automorphic completion]\\label{thm:derived_conditional_completion}\nLet \\(x:R\\to E'\\) admit a derived completion datum, with Hecke identification \\(e:R\\cong\\TT\\), and put \\(y=x\\circ e^{-1}\\). Then\n\\[\n\\boxed{\nC\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\not\\simeq0.\n}\n\\]\nSome integer cohomological degree contains a nonzero Hecke eigensystem with character \\(y\\). Its degree lies between the bounds of any chosen finite projective model for \\(C\\).","in{theorem}[Derived conditional automorphic completion]\\label{thm:derived_conditional_completion}\nLet \\(x:R\\to E'\\) admit a derived completion datum, with Hecke identification \\(e:R\\cong\\TT\\), and put \\(y=x\\circ e^{-1}\\). Then\n\\[\n\\boxed{\nC\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\not\\simeq0.\n}\n\\]\nSome integer cohomological degree contains a nonzero Hecke eigensystem with character \\(y\\). Its degree lies between the bounds of any chosen finite projective model for \\(C\\).","l_completion}\nLet \\(x:R\\to E'\\) admit a derived completion datum, with Hecke identification \\(e:R\\cong\\TT\\), and put \\(y=x\\circ e^{-1}\\). Then\n\\[\n\\boxed{\nC\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\not\\simeq0.\n}\n\\]\nSome integer cohomological degree contains a nonzero Hecke eigensystem with character \\(y\\). Its degree lies between the bounds of any chosen finite projective model for \\(C\\).","t topology, continuous specialization, supplied semisimplification, and common Frobenius polynomial hypotheses in \\cref{thm:derived_pointwise_realization}. If the declared classicality and Galois realization theorem applies to this eigensystem and supplies an occurring classical automorphic representation with a continuous semisimple attached representation and those Frobenius polynomials, then \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation. This pointwise applicability condition does not require a uniform classicality assertion in other degrees.\n\\end{theorem}","plification, and common Frobenius polynomial hypotheses in \\cref{thm:derived_pointwise_realization}. If the declared classicality and Galois realization theorem applies to this eigensystem and supplies an occurring classical automorphic representation with a continuous semisimple attached representation and those Frobenius polynomials, then \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation. This pointwise applicability condition does not require a uniform classicality assertion in other degrees.\n\\end{theorem}","\\begin{proof}\nFull support descends from the patched perfect complex by \\cref{thm:derived_support_descent}. Scalar transport through \\(e\\), followed by the actual derived isomorphism with the automorphic Hecke complex, gives full support for \\(C\\). Its fiber at \\(y\\) is therefore nonzero. The bounded model and induced Hecke action argument in \\cref{thm:derived_pointwise_realization} supplies a nonzero cohomology eigenvector in an integer degree within the model's bounds.","If realization applies to this eigensystem, it supplies the occurring automorphic object and a continuous semisimple attached representation with the declared Frobenius polynomials. The representation comparison in \\cref{thm:derived_pointwise_realization} then identifies the supplied semisimplification of \\(\\rho_x\\) with that attached representation. Only applicability to this eigensystem is used. A classicality theorem for every degree in a prescribed interval is unnecessary.\n\\end{proof}","\\begin{theorem}[Compatible system completion from one automorphic member]\\label{thm:system_completion}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with common Frobenius spine. Suppose the member at \\(\\lambda_0\\) admits an ordinary or derived completion datum and is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same Frobenius spine. Then\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\quad\n\\text{for every }\\lambda\\in\\Lambda.\n}\n\\]\n\\end{theorem}","automorphic member]\\label{thm:system_completion}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with common Frobenius spine. Suppose the member at \\(\\lambda_0\\) admits an ordinary or derived completion datum and is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same Frobenius spine. Then\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\quad\n\\text{for every }\\lambda\\in\\Lambda.\n}\n\\]\n\\end{theorem}","etion}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with common Frobenius spine. Suppose the member at \\(\\lambda_0\\) admits an ordinary or derived completion datum and is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same Frobenius spine. Then\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\quad\n\\text{for every }\\lambda\\in\\Lambda.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe completion theorem at \\(\\lambda_0\\) gives the automorphic representation \\(\\pi\\). Apply \\cref{thm:one_member_propagation} to the two compatible systems.\n\\end{proof}","\\begin{theorem}[Five gate formulation]\\label{thm:five_gate_completion}\nFix a characteristic zero deformation point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\it","\\begin{theorem}[Five gate formulation]\\label{thm:five_gate_completion}\nFix a characteristic zero deformation point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Co","on}\nFix a characteristic zero deformation point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does trans…","on point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with th…","e chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived","ic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realiza…","ing \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theor…","em \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classi…","An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representat…","eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a conti…","le or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard…","complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general B…","ng map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fac…","f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebota…","he carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rh…","gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","t \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","ation gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","tached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","nuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","otarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","For propagation, suppose both a semisimple compatible family \\(\\{\\rho_\\lambda\\}\\) containing \\(\\rho_x\\) and a family \\(\\{\\rho_{\\pi,\\lambda}\\}\\) attached to this same \\(\\pi\\) already exist. Require the distinguished members to be \\(\\rho_x\\) and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of \\cref{thm:one_member_propagation}, with that spine agreeing with the specialized Hecke polynomials at the distinguished place. Then\n\\[\n\\rho_\\lambda\\cong\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\qquad\\text{for every }\\lambda.\n\\]\n\\end{theorem}","da}\\}\\) attached to this same \\(\\pi\\) already exist. Require the distinguished members to be \\(\\rho_x\\) and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of \\cref{thm:one_member_propagation}, with that spine agreeing with the specialized Hecke polynomials at the distinguished place. Then\n\\[\n\\rho_\\lambda\\cong\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\qquad\\text{for every }\\lambda.\n\\]\n\\end{theorem}","The comparison inclusion transports this full support to \\(H\\). More specifically, \\(y\\circ f=x\\) identifies the contraction of \\(\\ker y\\) with \\(\\ker x\\), so support at the source point gives support at the Hecke point. In the ordinary case, \\cref{thm:field_fiber_criterion} gives a nonzero actual tensor fiber. In the derived case, the support criterion gives a nonzero actual derived fiber and hence a nonzero cohomology vector. Its integer degree lies within the bounds of the chosen finite projective model. In both cases, multiplication on the original carrier induces the scalar \\(y(t)\\) on the specialized vector for every \\(t\\in\\TT\\).","ives support at the Hecke point. In the ordinary case, \\cref{thm:field_fiber_criterion} gives a nonzero actual tensor fiber. In the derived case, the support criterion gives a nonzero actual derived fiber and hence a nonzero cohomology vector. Its integer degree lies within the bounds of the chosen finite projective model. In both cases, multiplication on the original carrier induces the scalar \\(y(t)\\) on the specialized vector for every \\(t\\in\\TT\\).","ary case, \\cref{thm:field_fiber_criterion} gives a nonzero actual tensor fiber. In the derived case, the support criterion gives a nonzero actual derived fiber and hence a nonzero cohomology vector. Its integer degree lies within the bounds of the chosen finite projective model. In both cases, multiplication on the original carrier induces the scalar \\(y(t)\\) on the specialized vector for every \\(t\\in\\TT\\).","Finally, retain the same \\(\\pi\\) in its attached family. The distinguished member identities connect both families to the pointwise construction, while the common algebraic spine gives the required Frobenius polynomials at every coefficient place. Apply \\cref{thm:one_member_propagation} to these two existing families. The construction does not produce a missing companion or an attached compatible family.\n\\end{proof}","An isomorphism \\(R\\cong\\TT\\) transports support and Fitting data & An \\(R\\cong\\TT\\) theorem outside published hypotheses","The preserved release records complete mapped Lean statements for all 49 internal numbered claims of its revised manuscript. The present traditional edition retains those proof bodies, exposes the hypotheses on the local map and coefficient target in \\cref{prop:global_local_basechange}, makes the \\(r>n\\) convention explicit in the proof of \\cref{thm:higher_fitting_strata}, states all the pointwise comparison inputs in \\cref{def:ordinary_completion_datum}, and declares a finite ramification set and coherent geometric Frobenius normalization. These editorial clarifications are recorded separately from the inherited eight amendments. They are not counted as newly compiled Lean certificates. No new statement amendment was needed for the completion and tensor results. The eight inherited amendments remain explicitly separate from the original wording, and the cited arithmetic inputs remain outside the internal claim census. Formal proof coverage is recorded claim by claim in the roadmap. These","The new checks for \\cref{thm:residual_augmentation_algebra,cor:no_residual_character_splitting,cor:rank_trace_detection,prop:dyadic_sign_projectors} construct the full multivariable truncated polynomial algebra isomorphism and identify its augmentation generators. The residual group algebra is local for abelian \\(p\\) groups in characteristic \\(p\\). For each supplied Hecke idempotent, the trace is the image dimension cast into the coefficient field. For that idempotent, occupancy is equivalent to positive rank and, in characteristic zero, nonzero trace. The dyadic calculation gives the two rational sign projectors, proves that neither lies in the integral group algebra, and identifies the residual algebra with \\(\\FF_2[X]/(X^2)\\).","oups in characteristic \\(p\\). For each supplied Hecke idempotent, the trace is the image dimension cast into the coefficient field. For that idempotent, occupancy is equivalent to positive rank and, in characteristic zero, nonzero trace. The dyadic calculation gives the two rational sign projectors, proves that neither lies in the integral group algebra, and identifies the residual algebra with \\(\\FF_2[X]/(X^2)\\).","Corollary~\\ref{cor:no_residual_character_splitting} is now checked for every finite \\(p\\) group, including nonabelian groups, over every field of characteristic \\(p\\). The proof uses a finite prime field orbit span and a nonzero fixed vector. The original unrestricted statement is restored, and the earlier explicit abelian restriction is no longer needed.","The checks for \\cref{prop:quotient_erases_component,thm:canonical_hecke_packets,cor:simultaneous_generalized_eigenspaces} now include the actual formal power series quotient and the canonical Hecke packet maps. In the node \\(k[[x,y]]/(xy)\\), quotienting by \\((x)\\) gives \\(k[[y]]\\), with support \\(V(x)\\). Lean verifies both irreducible components and the exclusion of the generic point \\((y)\\). The components can still meet, so this is not a claim that their supports are disjoint.","For the Hecke results, the canonical ring decomposition in \\cref{fact:artin_decomposition} remains a declared external input. Its coordinates are the actual localization maps. The module direct sum, the packet identification with the localized module, and the formula \\(v\\mapsto v/1\\) are constructed in Lean. Nilpotence in the selected Artin local factor, together with invertibility of a generator on each other factor, gives a single positive exponent for the simultaneous generalized eigenspace. The result includes the explicit generators \\(T_i-\\lambda_i\\), agrees with the packet used in the trace theorem, and does not assume semisimplicity.","the actual localization maps. The module direct sum, the packet identification with the localized module, and the formula \\(v\\mapsto v/1\\) are constructed in Lean. Nilpotence in the selected Artin local factor, together with invertibility of a generator on each other factor, gives a single positive exponent for the simultaneous generalized eigenspace. The result includes the explicit generators \\(T_i-\\lambda_i\\), agrees with the packet used in the trace theorem, and does not assume semisimplicity.","The original comparison wording did not explicitly supply a compatible Hecke point over the fixed coefficient field. Support preservation alone is insufficient: extension from \\(\\mathbb Q\\) to \\(\\mathbb Q(i)\\) preserves full support, but there is no ring map \\(\\mathbb Q(i)\\to\\mathbb Q\\) extending the identity. The revision therefore states \\(y\\circ f=x\\) and exposes the arithmetic comparison and existing-family inputs. Residual occupancy, representability and the applicable patching theorem remain supplied arithmetic provenance. The original full statement remains partial. The completed tensor quotients and intrinsic Fitting ideals are now checked as described above, including the full Fitting transport clause.","ted. Perfect complexes are actual derived category objects equipped with bounded finite projective models and derived isomorphisms to those models. Scalar extension is constructed on cochain complexes. Its associativity, preservation of homotopy equivalences, and independence under actual derived isomorphisms are checked. The quotient support formula uses the actual map of spectra and its image \\(V(I)\\). Every field valued point factors through the residue field at its kernel, and faithful flatness of field extensions preserves and reflects acyclicity. The surviving integer degree is expressed using the actual derived cohomology functor, with a corresponding witness in the specialized cochain model. These proofs do not require the external cohomological support description in \\cref{fact:derived_support_cohomology}. The external classicality theorems remain supplied arithmetic inputs. The conditional derived realization argument is checked as described above.","The adequacy checks prove \\cref{thm:s3_adequacy} as originally stated. They construct the faithful standard representation from the actual permutation module, prove irreducibility after extension to the algebraic closure, and split every group element's characteristic polynomial over \\(\\FF_7\\). The trace zero adjoint is the kernel of matrix trace with its conjugation action, and its sign plus standard decomposition is an actual equivariant linear isomorphism. The averaging proof is connected to Mathlib's actual first group cohomology, while the invariant calculation gives actual degree zero vanishing. Canonical projectors are identified with generalized eigenspace projectors for every positive exponent. Their trace pairings detect every nonzero endomorphism and hence every irreducible adjoint submodule. The two printed constituent witnesses and their traces are checked exactly. The adequacy definition remains a cited convention. No external adequacy theorem is assumed in this f","\\begin{abstract}\nThis chapter develops the signal realization of the projector-resolved seed calculus. The impulse $\\delta_0$ is taken as a primitive signal seed and its shift orbit supplies the address system. Residue projectors split a sequence into exact polyphase channels, nested divisibility gives exact refinement, and every finite impulse-response operator becomes a matrix of packet transfers. The standard perfect-reconstruction condition for a filter bank is therefore an analysis--synthesis identity in the common operator language of the series. Haar analysis and every finite-depth wavelet-packet transform are treated on complex $\\ell^2$, together with concrete finite-signal models and a predict--update lifting step. The edge problem is separated into two statements. When a discontinuity packet is known an","e into exact polyphase channels, nested divisibility gives exact refinement, and every finite impulse-response operator becomes a matrix of packet transfers. The standard perfect-reconstruction condition for a filter bank is therefore an analysis--synthesis identity in the common operator language of the series. Haar analysis and every finite-depth wavelet-packet transform are treated on complex $\\ell^2$, together with concrete finite-signal models and a predict--update lifting step. The edge problem is separated into two statements. When a discontinuity packet is known and retained exactly, approximation error is exactly the approximation error of the remaining smooth packet. No spectral approximation of the jump is performed. When the packet is estimated, a symmetric-difference bound isolates mask error, amplitude error, and smooth-background error. Numerical studies compare global Fourier truncation, exact known-packet reconstruction, and a deliberately narrow positive-contrast estimator in one and two dimensions, including curved boundaries, corners, and multiple amplitudes. The …","Let $\\ell^2(\\bbZ;\\bbC)$, abbreviated $\\ell^2(\\bbZ)$, be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript $*$ on a bounded operator denotes its Hilbert adjoint. Write $\\delta_n$ for the unit impulse at address $n$ and let the bilateral shift be\n\\[\n(Sx)[n]=x[n-1].\n\\]\nThen\n\\[\nS^n\\delta_0=\\delta_n.\n\\]\nThe primitive signal seed is therefore $s=\\delta_0$, the address operators are the shift words $S^n$, and the completion is the $\\ell^2$ norm.","reviated $\\ell^2(\\bbZ)$, be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript $*$ on a bounded operator denotes its Hilbert adjoint. Write $\\delta_n$ for the unit impulse at address $n$ and let the bilateral shift be\n\\[\n(Sx)[n]=x[n-1].\n\\]\nThen\n\\[\nS^n\\delta_0=\\delta_n.\n\\]\nThe primitive signal seed is therefore $s=\\delta_0$, the address operators are the shift words $S^n$, and the completion is the $\\ell^2$ norm.","summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript $*$ on a bounded operator denotes its Hilbert adjoint. Write $\\delta_n$ for the unit impulse at address $n$ and let the bilateral shift be\n\\[\n(Sx)[n]=x[n-1].\n\\]\nThen\n\\[\nS^n\\delta_0=\\delta_n.\n\\]\nThe primitive signal seed is therefore $s=\\delta_0$, the address operators are the shift words $S^n$, and the completion is the $\\ell^2$ norm.",". The inner product is conjugate-linear in its first argument. A superscript $*$ on a bounded operator denotes its Hilbert adjoint. Write $\\delta_n$ for the unit impulse at address $n$ and let the bilateral shift be\n\\[\n(Sx)[n]=x[n-1].\n\\]\nThen\n\\[\nS^n\\delta_0=\\delta_n.\n\\]\nThe primitive signal seed is therefore $s=\\delta_0$, the address operators are the shift words $S^n$, and the completion is the $\\ell^2$ norm.","\\begin{theorem}[Impulse seed theorem]\\label{result:01}\nThe closed shift-generated hull of $\\delta_0$ is all of $\\ell^2(\\bbZ)$:\n\\[\n\\boxed{\n\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in\\bbZ\\}}^{\\,\\ell^2}\n=\\ell^2(\\bbZ).\n}\n\\]\nEvery finitely supported signal has the exact finite expression\n\\[\nx=\\sum_{n\\in\\bbZ}x[n]S^n\\delta_0.\n\\]\n\\end{theorem}","\\begin{proof}\nThe shift orbit is the standard orthonormal basis $\\{\\delta_n\\}_{n\\in\\bbZ}$. For $x\\in\\ell^2$, the finite truncations\n\\[\nx_F=\\sum_{n\\in F}x[n]\\delta_n,\\qquad F\\subset\\bbZ\\text{ finite},\n\\]\nsatisfy $\\|x-x_F\\|_2^2=\\sum_{n\\notin F}|x[n]|^2\\to0$ as the finite sets exhaust $\\bbZ$. Thus the unordered Hilbert-space sum $\\sum_nx[n]\\delta_n$ converges to $x$, and the finite-support statement is exact. The shift is a surjective isometry because $n\\mapsto n-1$ is a bijection of $\\bbZ$, with inverse shift $n\\mapsto n+1$. This proves the closed-span assertion.\n\\end{proof}","mal basis $\\{\\delta_n\\}_{n\\in\\bbZ}$. For $x\\in\\ell^2$, the finite truncations\n\\[\nx_F=\\sum_{n\\in F}x[n]\\delta_n,\\qquad F\\subset\\bbZ\\text{ finite},\n\\]\nsatisfy $\\|x-x_F\\|_2^2=\\sum_{n\\notin F}|x[n]|^2\\to0$ as the finite sets exhaust $\\bbZ$. Thus the unordered Hilbert-space sum $\\sum_nx[n]\\delta_n$ converges to $x$, and the finite-support statement is exact. The shift is a surjective isometry because $n\\mapsto n-1$ is a bijection of $\\bbZ$, with inverse shift $n\\mapsto n+1$. This proves the closed-span assertion.\n\\end{proof}","}$. For $x\\in\\ell^2$, the finite truncations\n\\[\nx_F=\\sum_{n\\in F}x[n]\\delta_n,\\qquad F\\subset\\bbZ\\text{ finite},\n\\]\nsatisfy $\\|x-x_F\\|_2^2=\\sum_{n\\notin F}|x[n]|^2\\to0$ as the finite sets exhaust $\\bbZ$. Thus the unordered Hilbert-space sum $\\sum_nx[n]\\delta_n$ converges to $x$, and the finite-support statement is exact. The shift is a surjective isometry because $n\\mapsto n-1$ is a bijection of $\\bbZ$, with inverse shift $n\\mapsto n+1$. This proves the closed-span assertion.\n\\end{proof}","inite truncations\n\\[\nx_F=\\sum_{n\\in F}x[n]\\delta_n,\\qquad F\\subset\\bbZ\\text{ finite},\n\\]\nsatisfy $\\|x-x_F\\|_2^2=\\sum_{n\\notin F}|x[n]|^2\\to0$ as the finite sets exhaust $\\bbZ$. Thus the unordered Hilbert-space sum $\\sum_nx[n]\\delta_n$ converges to $x$, and the finite-support statement is exact. The shift is a surjective isometry because $n\\mapsto n-1$ is a bijection of $\\bbZ$, with inverse shift $n\\mapsto n+1$. This proves the closed-span assertion.\n\\end{proof}","This realization fixes the language used below:\n\\[\n\\begin{gathered}\n\\text{impulse seed}\\longrightarrow\\text{shift address}\\longrightarrow\\text{packet projection}\\longrightarrow\\text{filter transfer}\\\\\n\\longrightarrow\\text{refinement}\\longrightarrow\\text{reconstruction}.\n\\end{gathered}\n\\]","\\begin{remark}[Several completions]\\label{core:record:02}\nThe same algebraic seed description applies to finite sequences, $\\ell^1$, $\\ell^2$, periodic sequences, tempered distributions, and multidimensional lattices. The topology decides which infinite reconstructions are legitimate. Finite-dimensional experiments in this chapter use ordinary Euclidean norms. Theorem statements specify the ambient norm when it matters.\n\\end{remark}","Fix an integer $N\\ge1$. For $0\\le r<N$, define\n\\[\n(P_{N,r}x)[n]\n=\n\\mathbf 1_{n\\equiv r\\, (\\mathrm{mod}\\,N)}x[n].\n\\]\nThe corresponding decimated channel is\n\\[\nx_r[q]=x[Nq+r].\n\\]","\\begin{theorem}[Polyphase projector theorem]\\label{result:02}\nThe maps $P_{N,r}$ are bounded orthogonal projectors on $\\ell^2$, with operator norm at most one, and satisfy\n\\[\nP_{N,r}P_{N,s}=\\delta_{rs}P_{N,r},\n\\qquad\n\\sum_{r=0}^{N-1}P_{N,r}=I.\n\\]\nConsequently,\n\\[\n\\boxed{x=\\sum_{r=0}^{N-1}P_{N,r}x}\n\\]\nwith no approximation.\n\\end{theorem}","\\begin{proof}\nEach integer belongs to exactly one residue class modulo $N$. Multiplying the two indicator functions gives zero for distinct classes and the same indicator for equal classes. Summing all residue indicators gives one at every address. Moreover,\n\\[\n\\|P_{N,r}x\\|_2^2=\\sum_{n\\equiv r\\ (\\mathrm{mod}\\,N)}|x[n]|^2\\le\\|x\\|_2^2,\n\\qquad\n\\langle P_{N,r}x,y\\rangle=\\langle x,P_{N,r}y\\rangle.\n\\]\nThe mask is real-valued, so the inner-product identity follows term by term. Hence each projector is self-adjoint, and the ranges of different residue projectors are orthogonal.\n\\end{proof}","Define the analysis and synthesis maps\n\\[\n\\cA_Nx=(x_0,\\ldots,x_{N-1}),\n\\qquad\n\\cS_N(x_0,\\ldots,x_{N-1})[Nq+r]=x_r[q].\n\\]","\\begin{corollary}[Analysis--synthesis identity]\\label{result:03}\n\\[\n\\boxed{\\cS_N\\cA_N=I.}\n\\]\nOn the product of the channel spaces one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summabil","\\begin{corollary}[Analysis--synthesis identity]\\label{result:03}\n\\[\n\\boxed{\\cS_N\\cA_N=I.}\n\\]\nOn the product of the channel spaces one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, w","\\begin{corollary}[Analysis--synthesis identity]\\label{result:03}\n\\[\n\\boxed{\\cS_N\\cA_N=I.}\n\\]\nOn the product of the channel spaces one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","bel{result:03}\n\\[\n\\boxed{\\cS_N\\cA_N=I.}\n\\]\nOn the product of the channel spaces one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","}\n\\[\n\\boxed{\\cS_N\\cA_N=I.}\n\\]\nOn the product of the channel spaces one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","_N\\cA_N=I.}\n\\]\nOn the product of the channel spaces one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","of the channel spaces one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","one also has $\\cA_N\\cS_N=I$. Equip this finite product with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","oduct with the Hilbert norm\n\\[\n\\|(x_0,\\ldots,x_{N-1})\\|_{\\mathrm{ch}}^2\n=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\nThen $\\cA_N$ is a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","a linear isometric equivalence and\n\\[\n\\|x\\|_2^2=\\sum_{r=0}^{N-1}\\|x_r\\|_2^2.\n\\]\n\\end{corollary}\n\\begin{proof}\nEuclidean division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","n division gives a unique pair $(r,q)$ with $0\\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\\mapsto(n\\bmod N,n\\mathbin{\\mathrm{div}}N)$ and $(r,q)\\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\\ell^2(\\{0,\\ldots,N-1\\}\\times\\bbZ)$, which realizes exactly this Hilbert norm.\n\\end{proof}","Suppose $N,M\\ge1$ and $N\\mid M$. Every coarse phase has a disjoint family of fine descendants:\n\\[\n\\operatorname{Child}_{M}(N,r)\n=\n\\{s:0\\le s<M,\\ s\\equiv r\\pmod N\\}.\n\\]","\\begin{theorem}[Exact nested refinement]\\label{result:04}\nFor $N,M\\ge1$ and $N\\mid M$,\n\\[\n\\boxed{\nP_{N,r}\n=\n\\sum_{\\substack{0\\le s<M","\\begin{theorem}[Exact nested refinement]\\label{result:04}\nFor $N,M\\ge1$ and $N\\mid M$,\n\\[\n\\boxed{\nP_{N,r}\n=\n\\sum_{\\substack{0\\le s<M\\\\s\\equiv r\\, (\\mathrm{mod}\\,N)}}P_{M,s}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nAn integer $n$ satisfies $n\\equiv r\\pmod N$ exactly when its unique residue $s=n\\bmod M$ lies among the listed descendants. The fine indicators are disjoint, so their sum is the coarse indicator.\n\\end{proof}","Let $h$ be a finite impulse response and\n\\[\ny=h*x,\n\\qquad\ny[n]=\\sum_{q\\in\\bbZ}h[q]x[n-q].\n\\]\nResolve both input and output into $N$ phases. For output phase $k$ and input phase $s$, define the transfer kernel\n\\[\n\\boxed{\nh_{ks}[m]=h[Nm+k-s].\n}\n\\]\nOut-of-range taps are understood to be zero.","\\begin{theorem}[Exact FIR polyphase transfer law]\\label{result:05}\nThe output channels satisfy\n\\[\n\\boxed{\ny_k[n]\n=\n\\sum_{s=0}^{N-1}\\sum_{m\\in\\bbZ}\nh_{ks}[m]x_s[n-m].\n}\n\\]\nFor the numerical channel transform, take $z\\in\\bbC\\setminus\\{0\\}$ and assume\n\\[\n\\sum_{n\\in\\bbZ}|x_s[n]z^{-n}|<\\infty\\quad(0\\le s<N).\n\\]\nDefine $X_s(z)=\\sum_nx_s[n]z^{-n}$. The output series then also converge absolutely, and\n\\[\n\\bm Y(z)=\\bm H^{(N)}(z)\\bm X(z),\n\\qquad\nH^{(N)}_{ks}(z)=\\sum_m h[Nm+k-s]z^{-m}.\n\\]\n\\end{theorem}","\\begin{proof}\nEvaluate $y$ at address $Nn+k$ and partition every input index by its residue $s$:\n\\[\n\\begin{aligned}\ny[Nn+k]\n&=\\sum_qh[q]x[Nn+k-q]","\\begin{proof}\nEvaluate $y$ at address $Nn+k$ and partition every input index by its residue $s$:\n\\[\n\\begin{aligned}\ny[Nn+k]\n&=\\sum_qh[q]x[Nn+k-q]\\\\\n&=\\sum_{s=0}^{N-1}\\sum_m\nh[Nm+k-s]x[N(n-m)+s].\n\\end{aligned}\n\\]\nFor a fixed output phase $k$, the change of tap address $q=Nm+k-s$ is a bijection between $\\bbZ$ and the pairs $(s,m)$ with $0\\le s<N$: choose $s\\equiv k-q\\pmod N$ and then $m=(q-k+s)/N$. Thus this is a reindexing of the finite filter support, valid for every bilateral input sequence, without a convergence assumption on $x$.","&=\\sum_{s=0}^{N-1}\\sum_m\nh[Nm+k-s]x[N(n-m)+s].\n\\end{aligned}\n\\]\nFor a fixed output phase $k$, the change of tap address $q=Nm+k-s$ is a bijection between $\\bbZ$ and the pairs $(s,m)$ with $0\\le s<N$: choose $s\\equiv k-q\\pmod N$ and then $m=(q-k+s)/N$. Thus this is a reindexing of the finite filter support, valid for every bilateral input sequence, without a convergence assumption on $x$.","The last factor is $x_s[n-m]$, giving the time-domain formula. Under the additional transform hypothesis, each delayed channel has transform\n\\[\n\\sum_nx_s[n-m]z^{-n}\n=z^{-m}\\sum_jx_s[j]z^{-j}.\n\\]\nAbsolute summability permits this bijective reindexing, and only finitely many delayed channels are combined. Taking their finite linear combination gives the matrix formula. The time-domain theorem therefore remains valid even when no numerical bilateral transform exists.\n\\end{proof}","\\begin{remark}[Operator-block form]\\label{core:record:07}\nIf $H$ denotes convolution by $h$, the same identity is\n\\[\nH=\\sum_{k,s}P_{N,k}HP_{N,s}.\n\\]\nThus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.\n\\end{remark}","Work with finite impulse-response banks. Let $\\mathcal L=\\bbC[z,z^{-1}]$ be the Laurent-polynomial algebra, with the monomial $z^{-m}$ acting on a bilateral sequence by $x[n]\\mapsto x[n-m]$. Let $\\bm E\\in\\operatorname{Mat}_{N\\times N}(\\mathcal L)$ and $\\bm R\\in\\operatorname{Mat}_{N\\times N}(\\mathcal L)$ be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero $z$ where the input transforms are absolutely convergent, this gives\n\\[\n\\widehat{\\bm X}(z)=\\bm R(z)\\bm E(z)\\bm X(z).\n\\]","ks. Let $\\mathcal L=\\bbC[z,z^{-1}]$ be the Laurent-polynomial algebra, with the monomial $z^{-m}$ acting on a bilateral sequence by $x[n]\\mapsto x[n-m]$. Let $\\bm E\\in\\operatorname{Mat}_{N\\times N}(\\mathcal L)$ and $\\bm R\\in\\operatorname{Mat}_{N\\times N}(\\mathcal L)$ be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero $z$ where the input transforms are absolutely convergent, this gives\n\\[\n\\widehat{\\bm X}(z)=\\bm R(z)\\bm E(z)\\bm X(z).\n\\]","with the monomial $z^{-m}$ acting on a bilateral sequence by $x[n]\\mapsto x[n-m]$. Let $\\bm E\\in\\operatorname{Mat}_{N\\times N}(\\mathcal L)$ and $\\bm R\\in\\operatorname{Mat}_{N\\times N}(\\mathcal L)$ be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero $z$ where the input transforms are absolutely convergent, this gives\n\\[\n\\widehat{\\bm X}(z)=\\bm R(z)\\bm E(z)\\bm X(z).\n\\]","\\begin{theorem}[Perfect-reconstruction criterion]\\label{result:06}\nFor $d\\in\\bbZ$, such a bank reconstructs every input up to a delay of $d$ \\emph{channel indices} exactly when the following Laurent-polynomial identity holds:\n\\[\n\\boxed{\n\\bm R(z)\\bm E(z)=z^{-d}I_N.\n}\n\\]\nIf the right side is $c z^{-d}I_N$ with $c\\ne0$, reconstruction holds up to gain $c$ and channel delay $d$. After interleaving, this is a delay of $Nd$ original samples. A negative delay denotes an advance.\n\\end{theorem}","rfect-reconstruction criterion]\\label{result:06}\nFor $d\\in\\bbZ$, such a bank reconstructs every input up to a delay of $d$ \\emph{channel indices} exactly when the following Laurent-polynomial identity holds:\n\\[\n\\boxed{\n\\bm R(z)\\bm E(z)=z^{-d}I_N.\n}\n\\]\nIf the right side is $c z^{-d}I_N$ with $c\\ne0$, reconstruction holds up to gain $c$ and channel delay $d$. After interleaving, this is a delay of $Nd$ original samples. A negative delay denotes an advance.\n\\end{theorem}","\\begin{proof}\nFinite convolution represents Laurent-polynomial multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let $\\bm B=\\bm R\\bm E-cz^{-d}I_N$ and suppose its action vanishes on every channel sequence. Insert an impulse at channel $s$, address zero, with every other channel zero. Output channel $k$ reads the coefficients of the Laurent polynomial $B_{ks}$ at all addresses. All of these coefficients vanish, so $B_{ks}=0$ for every $k,s$. The representation is faithful, proving necessity as a polynomial identity rather than merely equality at one evaluation point. The displayed unit-gain case is $c=1$.\n\\end{proof}","oof}\nFinite convolution represents Laurent-polynomial multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let $\\bm B=\\bm R\\bm E-cz^{-d}I_N$ and suppose its action vanishes on every channel sequence. Insert an impulse at channel $s$, address zero, with every other channel zero. Output channel $k$ reads the coefficients of the Laurent polynomial $B_{ks}$ at all addresses. All of these coefficients vanish, so $B_{ks}=0$ for every $k,s$. The representation is faithful, proving necessity as a polynomial identity rather than merely equality at one evaluation point. The displayed unit-gain case is $c=1$.\n\\end{proof}","multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let $\\bm B=\\bm R\\bm E-cz^{-d}I_N$ and suppose its action vanishes on every channel sequence. Insert an impulse at channel $s$, address zero, with every other channel zero. Output channel $k$ reads the coefficients of the Laurent polynomial $B_{ks}$ at all addresses. All of these coefficients vanish, so $B_{ks}=0$ for every $k,s$. The representation is faithful, proving necessity as a polynomial identity rather than merely equality at one evaluation point. The displayed unit-gain case is $c=1$.\n\\end{proof}","utivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let $\\bm B=\\bm R\\bm E-cz^{-d}I_N$ and suppose its action vanishes on every channel sequence. Insert an impulse at channel $s$, address zero, with every other channel zero. Output channel $k$ reads the coefficients of the Laurent polynomial $B_{ks}$ at all addresses. All of these coefficients vanish, so $B_{ks}=0$ for every $k,s$. The representation is faithful, proving necessity as a polynomial identity rather than merely equality at one evaluation point. The displayed unit-gain case is $c=1$.\n\\end{proof}","e sums. Thus the matrix identity implies the asserted action. Conversely, let $\\bm B=\\bm R\\bm E-cz^{-d}I_N$ and suppose its action vanishes on every channel sequence. Insert an impulse at channel $s$, address zero, with every other channel zero. Output channel $k$ reads the coefficients of the Laurent polynomial $B_{ks}$ at all addresses. All of these coefficients vanish, so $B_{ks}=0$ for every $k,s$. The representation is faithful, proving necessity as a polynomial identity rather than merely equality at one evaluation point. The displayed unit-gain case is $c=1$.\n\\end{proof}","Insert an impulse at channel $s$, address zero, with every other channel zero. Output channel $k$ reads the coefficients of the Laurent polynomial $B_{ks}$ at all addresses. All of these coefficients vanish, so $B_{ks}=0$ for every $k,s$. The representation is faithful, proving necessity as a polynomial identity rather than merely equality at one evaluation point. The displayed unit-gain case is $c=1$.\n\\end{proof}","\\paragraph{Original-sample delay.}\nFor an arbitrary original-sample delay $D\\in\\bbZ$, the output phase $k$ has\n\\[\ny_k[n]=x_{r_k}[n+q_k],\\qquad\nr_k=(k-D)\\bmod N,\\qquad q_k=(k-D)\\mathbin{\\mathrm{div}}N.\n\\]\nIts channel matrix has the sole nonzero entry $z^{q_k}$ in row $k$, column $r_k$. In particular, a one-sample delay at $N=2$ has matrix\n\\[\n\\begin{pmatrix}0&z^{-1}","\\paragraph{Original-sample delay.}\nFor an arbitrary original-sample delay $D\\in\\bbZ$, the output phase $k$ has\n\\[\ny_k[n]=x_{r_k}[n+q_k],\\qquad\nr_k=(k-D)\\bmod N,\\qquad q_k=(k-D)\\mathbin{\\mathrm{div}}N.\n\\]\nIts channel matrix has the sole nonzero entry $z^{q_k}$ in row $k$, column $r_k$. In particular, a one-sample delay at $N=2$ has matrix\n\\[\n\\begin{pmatrix}0&z^{-1}\\\\1&0\\end{pmatrix},\n\\]\nwhereas $z^{-1}I_2$ delays the original signal by two samples. The general address identity is valid for negative sample addresses as well.","1&0\\end{pmatrix},\n\\]\nwhereas $z^{-1}I_2$ delays the original signal by two samples. The general address identity is valid for negative sample addresses as well.","The Haar analysis matrix is\n\\[\n\\bm E_{\\mathrm{H}}\n=\n\\frac1{\\sqrt2}\n\\begin{pmatrix}\n1&1\\\\\n1&-1\n\\end{pmatrix}.\n\\]\nIt is orthogonal, so $\\bm R_{\\mathrm{H}}=\\bm E_{\\mathrm{H}}^{\\mathsf T}$ and\n\\[\n\\bm R_{\\mathrm{H}}\\bm E_{\\mathrm{H}}=I_2.\n\\]\nThe channels are\n\\[\na[n]=\\frac{x[2n]+x[2n+1]}{\\sqrt2},\n\\qquad\nd[n]=\\frac{x[2n]-x[2n+1]}{\\sqrt2}.\n\\]","1&-1\n\\end{pmatrix}.\n\\]\nIt is orthogonal, so $\\bm R_{\\mathrm{H}}=\\bm E_{\\mathrm{H}}^{\\mathsf T}$ and\n\\[\n\\bm R_{\\mathrm{H}}\\bm E_{\\mathrm{H}}=I_2.\n\\]\nThe channels are\n\\[\na[n]=\\frac{x[2n]+x[2n+1]}{\\sqrt2},\n\\qquad\nd[n]=\\frac{x[2n]-x[2n+1]}{\\sqrt2}.\n\\]","\\begin{corollary}[Haar energy identity]\\label{result:07}\n\\[\n\\boxed{\n\\|x\\|_2^2=\\|a\\|_2^2+\\|d\\|_2^2.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nFor any $u,v\\in\\bbC$, the parallelogram identity gives\n\\[\n\\left|\\frac{u+v}{\\sqrt2}\\right|^2+\n\\left|\\frac{u-v}{\\sqrt2}\\right|^2=|u|^2+|v|^2.\n\\]\nApply this to $u=x[2n]$, $v=x[2n+1]$ and sum over $n\\in\\bbZ$. Each summand is nonnegative. The right side is $\\|x\\|_2^2$ by the two-channel address equivalence. This also proves that the literal low and detail channels belong to $\\ell^2$. The real Haar matrix is unitary over $\\bbC$, so the finite-signal statement is the same calculation with a finite sum.\n\\end{proof}",", the parallelogram identity gives\n\\[\n\\left|\\frac{u+v}{\\sqrt2}\\right|^2+\n\\left|\\frac{u-v}{\\sqrt2}\\right|^2=|u|^2+|v|^2.\n\\]\nApply this to $u=x[2n]$, $v=x[2n+1]$ and sum over $n\\in\\bbZ$. Each summand is nonnegative. The right side is $\\|x\\|_2^2$ by the two-channel address equivalence. This also proves that the literal low and detail channels belong to $\\ell^2$. The real Haar matrix is unitary over $\\bbC$, so the finite-signal statement is the same calculation with a finite sum.\n\\end{proof}","Split $x$ into even and odd channels $e,o$. Define\n\\[\nd=o-e,\n\\qquad\ns=e+\\frac12d.\n\\]\nThe inverse is\n\\[\ne=s-\\frac12d,\n\\qquad\no=d+e.\n\\]","\\begin{theorem}[Exact lifting reconstruction]\\label{result:08}\nThe predict--update map $(e,o)\\mapsto(s,d)$ is invertible with determinant one and the displayed inverse. Consequently the associated finite transform reconstructs every even-length signal exactly.\n\\end{theorem}","\\begin{proof}\nThe forward matrix is\n\\[\n\\begin{pmatrix}s\\\\d\\end{pmatrix}\n=\n\\begin{pmatrix}1/2&1/2\\\\-1&1\\end{pmatrix}\n\\begin{pmatrix}e\\\\o\\end{pmatrix},\n\\]\nwhose determinant is one. Direct substitution gives the inverse formulas.\n\\end{proof}","On $\\ell^2(\\bbZ;\\bbC)$, apply the Haar analysis to both children at each stage. For a finite input, assume its length is divisible by $2^\\ell$ when a depth-$\\ell$ transform is used. A binary word $\\alpha\\in\\{0,1\\}^{\\ell}$ addresses a depth-$\\ell$ packet, denoted $W_{\\ell,\\alpha}x$. Appending $0$ or $1$ selects the low or high child.","\\begin{theorem}[Wavelet-packet reconstruction and energy]\\label{result:09}\nAt every finite depth $\\ell$,\n\\[\n\\boxed{\nx=\\sum_{\\alpha\\in\\{0,1\\}^{\\ell}}W_{\\ell,\\alpha}^{*}W_{\\ell,\\alpha}x,\n}\n\\]\nand\n\\[\n\\boxed{\n\\|x\\|_2^2\n=\n\\sum_{\\alpha\\in\\{0,1\\}^{\\ell}}\n\\|W_{\\ell,\\alpha}x\\|_2^2.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nWrite $H_0,H_1$ for the literal low and detail operators. The preceding energy identity implies $\\|H_bx\\|_2\\le\\|x\\|_2$, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth $\\ell$, apply the one-level identity to each $W_{\\ell,\\alpha}x$ and sum. This proves it at depth $\\ell+1$.","Set $A=\\sum_\\alpha W_{\\ell,\\alpha}^*W_{\\ell,\\alpha}$. This is a finite sum of bounded operators. The energy result says $\\langle x,Ax\\rangle=\\langle x,x\\rangle$ for every $x$. Complex polarization, applied also to $x+y$ and $x+\\ii y$, gives $\\langle y,(A-I)x\\rangle=0$ for all $x,y$, hence $A=I$. Applying the operator identity to $x$ proves synthesis. This argument uses the actual Hilbert adjoints of the decimation operators, not a separately postulated inverse symbol.\n\\end{proof}","Let\n\\[\nf=s+j\n\\]\nwhere $s$ belongs to a smooth approximation class and $j$ is a known discontinuity packet. Let $Q_m$ be any approximation operator applied only to $s$, and define\n\\[\n\\widetilde f_m=Q_ms+j.\n\\]","\\begin{theorem}[Exact known-edge theorem]\\label{result:10}\nFor every norm for which the expressions are defined,\n\\[\n\\boxed{\nf-\\widetilde f_m=s-Q_ms,\n\\qquad\n\\|f-\\widetilde f_m\\|=\\|s-Q_ms\\|.\n}\n\\]\nNo approximation of $j$ occurs.\n\\end{theorem}","\\begin{proof}\nSubtract:\n\\[\nf-\\widetilde f_m=(s+j)-(Q_ms+j)=s-Q_ms.\n\\]\nTaking the norm gives the second identity.\n\\end{proof}","\\begin{corollary}[Several exact edge packets]\\label{result:11}\nIf\n\\[\nf=s+\\sum_{q=1}^{M}j_q\n\\]\nand every $j_q$ is retained exactly, then\n\\[\n\\left\\|f-\\left(Q_ms+\\sum_qj_q\\right)\\right\\|\n=\\|s-Q_ms\\|.\n\\]\nCurved edges, corners, and different jump amplitudes do not alter the identity.\n\\end{corollary}\n\\begin{proof}\nThe finite sum $j=\\sum_{q=1}^M j_q$ is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family $M=0$ is included with sum zero.\n\\end{proof}","lt:11}\nIf\n\\[\nf=s+\\sum_{q=1}^{M}j_q\n\\]\nand every $j_q$ is retained exactly, then\n\\[\n\\left\\|f-\\left(Q_ms+\\sum_qj_q\\right)\\right\\|\n=\\|s-Q_ms\\|.\n\\]\nCurved edges, corners, and different jump amplitudes do not alter the identity.\n\\end{corollary}\n\\begin{proof}\nThe finite sum $j=\\sum_{q=1}^M j_q$ is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family $M=0$ is included with sum zero.\n\\end{proof}","Let $(X,\\Sigma,\\mu)$ be a measure space, with measurable masks $\\Omega,\\widehat\\Omega\\subseteq X$ and scalar amplitudes $a,\\widehat a\\in\\bbC$ (real amplitudes are a special case). Suppose a scalar jump packet is\n\\[\nj=a\\mathbf 1_{\\Omega},\n\\]\nand an estimated packet is\n\\[\n\\widehat j=\\widehat a\\mathbf 1_{\\widehat\\Omega}.\n\\]","\\begin{theorem}[Packet perturbation bound]\\label{result:12}\nFor $1\\le p<\\infty$, interpreted first as an inequality of extended $L^p$ seminorms,\n\\[\n\\boxed{\n\\|j-\\widehat j\\|_{L^p}\n\\le\n|a-\\widehat a|\\,\\mu(\\Omega)^{1/p}\n+\n|\\widehat a|\\,\\mu(\\Omega\\triangle\\widehat\\Omega)^{1/p}.\n}\n\\]\nIf $\\mu(\\Omega)$ and $\\mu(\\Omega\\triangle\\widehat\\Omega)$ are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued $L^p$ inequality. For the pointwise supremum, and consequently also the essential $L^\\infty$ seminorm,\n\\[\n\\|j-\\widehat j\\|_{\\infty}\n\\le |a-\\widehat a|+|\\widehat a|\\,\n\\mathbf 1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}.\n\\]\n\\end{theorem}","bound]\\label{result:12}\nFor $1\\le p<\\infty$, interpreted first as an inequality of extended $L^p$ seminorms,\n\\[\n\\boxed{\n\\|j-\\widehat j\\|_{L^p}\n\\le\n|a-\\widehat a|\\,\\mu(\\Omega)^{1/p}\n+\n|\\widehat a|\\,\\mu(\\Omega\\triangle\\widehat\\Omega)^{1/p}.\n}\n\\]\nIf $\\mu(\\Omega)$ and $\\mu(\\Omega\\triangle\\widehat\\Omega)$ are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued $L^p$ inequality. For the pointwise supremum, and consequently also the essential $L^\\infty$ seminorm,\n\\[\n\\|j-\\widehat j\\|_{\\infty}\n\\le |a-\\widehat a|+|\\widehat a|\\,\n\\mathbf 1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}.\n\\]\n\\end{theorem}","f extended $L^p$ seminorms,\n\\[\n\\boxed{\n\\|j-\\widehat j\\|_{L^p}\n\\le\n|a-\\widehat a|\\,\\mu(\\Omega)^{1/p}\n+\n|\\widehat a|\\,\\mu(\\Omega\\triangle\\widehat\\Omega)^{1/p}.\n}\n\\]\nIf $\\mu(\\Omega)$ and $\\mu(\\Omega\\triangle\\widehat\\Omega)$ are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued $L^p$ inequality. For the pointwise supremum, and consequently also the essential $L^\\infty$ seminorm,\n\\[\n\\|j-\\widehat j\\|_{\\infty}\n\\le |a-\\widehat a|+|\\widehat a|\\,\n\\mathbf 1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}.\n\\]\n\\end{theorem}","orms,\n\\[\n\\boxed{\n\\|j-\\widehat j\\|_{L^p}\n\\le\n|a-\\widehat a|\\,\\mu(\\Omega)^{1/p}\n+\n|\\widehat a|\\,\\mu(\\Omega\\triangle\\widehat\\Omega)^{1/p}.\n}\n\\]\nIf $\\mu(\\Omega)$ and $\\mu(\\Omega\\triangle\\widehat\\Omega)$ are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued $L^p$ inequality. For the pointwise supremum, and consequently also the essential $L^\\infty$ seminorm,\n\\[\n\\|j-\\widehat j\\|_{\\infty}\n\\le |a-\\widehat a|+|\\widehat a|\\,\n\\mathbf 1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}.\n\\]\n\\end{theorem}","\\begin{proof}\nWrite\n\\[\na\\mathbf 1_{\\Omega}-\\widehat a\\mathbf 1_{\\widehat\\Omega}\n=(a-\\widehat a)\\mathbf 1_{\\Omega}\n+\\widehat a(\\mathbf 1_{\\Omega}-\\mathbf 1_{\\widehat\\Omega}).\n\\]\nApply the triangle inequality and use\n\\[\n\\|\\mathbf 1_{\\Omega}-\\mathbf 1_{\\widehat\\Omega}\\|_{L^p}\n=\\mu(\\Omega\\triangle\\widehat\\Omega)^{1/p}.\n\\]\nFor the supremum estimate, $|\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}|\\le1$ pointwise, and this function vanishes everywhere when the symmetric difference is empty. The same pointwise bound also bounds the essential supremum. Nonempty null symmetric differences need not contribute to the essential norm. The stated bound remains valid in that case and need not be sharp. Measurability justifies the $L^p$ triangle inequality and the indicator norm formula.\n\\end{proof}","r the supremum estimate, $|\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}|\\le1$ pointwise, and this function vanishes everywhere when the symmetric difference is empty. The same pointwise bound also bounds the essential supremum. Nonempty null symmetric differences need not contribute to the essential norm. The stated bound remains valid in that case and need not be sharp. Measurability justifies the $L^p$ triangle inequality and the indicator norm formula.\n\\end{proof}","If $s,\\widehat s$ are measurable smooth-background models, the same triangle inequality gives, for finite $p$,\n\\[\n\\boxed{\n\\|f-(\\widehat s+\\widehat j)\\|_p\n\\le\n\\|s-\\widehat s\\|_p\n+|a-\\widehat a|\\mu(\\Omega)^{1/p}\n+|\\widehat a|\\mu(\\Omega\\triangle\\widehat\\Omega)^{1/p}.\n}\n\\]\nThis is the correct separation of errors. Detection error is not hidden inside a generic reconstruction score.","\\begin{certificate}[One-dimensional errors]\\label{core:record:18}\nFor $1024$ samples on $[-1,1)$ and Fourier cutoff $m=18$, the retained frequency indices are $-18,\\ldots,18$: 37 bins including the zero-frequency bin. The measured values were\n\\[\n\\begin{array}{lrr}\n\\toprule\n\\text{method}&\\RMSE&\\text{maximum absolute error}","\\begin{certificate}[One-dimensional errors]\\label{core:record:18}\nFor $1024$ samples on $[-1,1)$ and Fourier cutoff $m=18$, the retained frequency indices are $-18,\\ldots,18$: 37 bins including the zero-frequency bin. The measured values were\n\\[\n\\begin{array}{lrr}\n\\toprule\n\\text{method}&\\RMSE&\\text{maximum absolute error}\\\\\n\\midrule\n\\text{global Fourier truncation}&5.4023\\times10^{-2}&3.4968\\times10^{-1}\\\\\n\\text{known edge packet}&1.3692\\times10^{-4}&1.4503\\times10^{-3}\\\\\n\\text{estimated edge packet}&2.4029\\times10^{-2}&5.9530\\times10^{-2}\\\\\n\\bottomrule\n\\end{array}\n\\]\nThe simple estimator recovered the binary mask exactly in this deliberately separated two-class example. That result is a property of this test, not a universal guarantee.\n\\end{certificate}","\\begin{certificate}[Two-dimensional errors]\\label{core:record:19}\nFor the $192\\times192$ disk experiment, a square Fourier mask retains indices $|k_x|,|k_y|\\le16$, or $33^2=1089$ bins. The reported errors are\n\\[\n\\begin{array}{lrr}\n\\toprule\n\\text{method}&\\text{full-image RMSE}&\\text{boundary-band RMSE}","\\begin{certificate}[Two-dimensional errors]\\label{core:record:19}\nFor the $192\\times192$ disk experiment, a square Fourier mask retains indices $|k_x|,|k_y|\\le16$, or $33^2=1089$ bins. The reported errors are\n\\[\n\\begin{array}{lrr}\n\\toprule\n\\text{method}&\\text{full-image RMSE}&\\text{boundary-band RMSE}\\\\\n\\midrule\n\\text{global Fourier truncation}&3.7643\\times10^{-2}&1.8541\\times10^{-1}\\\\\n\\text{known edge packet}&6.4752\\times10^{-5}&1.2709\\times10^{-5}\\\\\n\\text{estimated edge packet}&3.4043\\times10^{-3}&8.8834\\times10^{-3}\\\\\n\\bottomrule\n\\end{array}\n\\]\nThe mask intersection-over-union was one in this positive-contrast synthetic case. The amplitude was estimated rather than inserted from the generating model.\n\\end{certificate}","\\begin{certificate}[Multiedge errors]\\label{core:record:20}\nThe global reconstruction had RMSE $3.3740\\times10^{-2}$ and maximum error $3.2807\\times10^{-1}$. The known-packet reconstruction had only the remaining smooth-background error. The experiment checks the finite implementation of the theorem for a composite packet. It is not a data-driven multiregion segmentation result.\n\\end{certificate}","\\midrule\n16&$1.7999\\times10^{-1}$&$6.6522\\times10^{-2}$","32&$3.8414\\times10^{-2}$&$4.9153\\times10^{-2}$","64&$1.6267\\times10^{-2}$&$3.8056\\times10^{-2}$","128&$1.7387\\times10^{-3}$&$2.5631\\times10^{-2}$","256&$1.5870\\times10^{-4}$&$1.7424\\times10^{-2}$","\\begin{definition}[Complete packet pruning]\\label{core:record:22}\nLet $\\mathcal T$ be a finite rooted refinement tree.  Every node $v$ carries a projector $P_v$, the root satisfies $P_{\\varnothing}=I$, and every internal node satisfies\n\\[\nP_v=\\sum_{u\\in\\operatorname{ch}(v)}P_u,\n\\qquad\nP_uP_w=0\\quad(u\\ne w\\text{ in }\\operatorname{ch}(v)).\n\\]\nA set $\\Lambda$ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets $\\Lambda$ exactly once.\n\\end{definition}","ore:record:22}\nLet $\\mathcal T$ be a finite rooted refinement tree.  Every node $v$ carries a projector $P_v$, the root satisfies $P_{\\varnothing}=I$, and every internal node satisfies\n\\[\nP_v=\\sum_{u\\in\\operatorname{ch}(v)}P_u,\n\\qquad\nP_uP_w=0\\quad(u\\ne w\\text{ in }\\operatorname{ch}(v)).\n\\]\nA set $\\Lambda$ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets $\\Lambda$ exactly once.\n\\end{definition}","\\begin{theorem}[Exact synthesis from every complete pruning]\\label{result:13}\nFor every complete pruning $\\Lambda$,\n\\[\n\\boxed{\\sum_{v\\in\\Lambda}P_v=I,}\n\\qquad\n\\boxed{x=\\sum_{v\\in\\Lambda}P_vx.}\n\\]\nThus changing the selected tree changes the coordinates but does not introduce reconstruction error.\n\\end{theorem}","\\begin{proof}\nStarting from the root identity $P_{\\varnothing}=I$, replace every unselected internal node by the exact sum of its children.  If a complete pruning contains the root it contains no other node. Otherwise its restriction to each child subtree is itself a complete pruning: every terminal path through that child must be met exactly once. Induction on the finite tree therefore justifies the substitution and its termination precisely at $\\Lambda$. Child subtrees have distinct node addresses, so their selected-node sets form a disjoint union. The sum identity uses only root identity and parent-to-child conservation. Sibling orthogonality gives the additional interpretation as nonoverlapping projector packets.  Applying the resulting identity to $","varnothing}=I$, replace every unselected internal node by the exact sum of its children.  If a complete pruning contains the root it contains no other node. Otherwise its restriction to each child subtree is itself a complete pruning: every terminal path through that child must be met exactly once. Induction on the finite tree therefore justifies the substitution and its termination precisely at $\\Lambda$. Child subtrees have distinct node addresses, so their selected-node sets form a disjoint union. The sum identity uses only root identity and parent-to-child conservation. Sibling orthogonality gives the additional interpretation as nonoverlapping projector packets.  Applying the resulting identity to $x$ gives the synthesis formula.\n\\end{proof}","the finite tree therefore justifies the substitution and its termination precisely at $\\Lambda$. Child subtrees have distinct node addresses, so their selected-node sets form a disjoint union. The sum identity uses only root identity and parent-to-child conservation. Sibling orthogonality gives the additional interpretation as nonoverlapping projector packets.  Applying the resulting identity to $x$ gives the synthesis formula.\n\\end{proof}","Let $y$ denote the observed data.  For a complete pruning $\\Lambda$, consider any finite criterion\n\\[\nJ_y(\\Lambda)\n=\nR_y(\\Lambda)\n+\\eta L_y(\\Lambda)\n+\\tau K(\\Lambda),\n\\]\nwhere $R_y$ is a representation cost, $L_y$ is a cross-packet leakage cost, and $K$ is a tree-complexity penalty.  Here $R_y,L_y,K$ take finite real values, and $\\eta,\\tau$ are fixed real weights (usually nonnegative in applications). No additivity is needed for the first theorem.","\\begin{theorem}[Finite globally optimal stable selector]\\label{result:14}\nLet $\\mathfrak P(\\mathcal T)$ be the finite set of complete prunings of $\\mathcal T$.","\\begin{enumerate}[label=(\\roman*)]\n\\item A global minimizer\n\\[\n\\Lambda_y^*\\in\\operatorname*{argmin}_{\\Lambda\\in\\mathfrak P(\\mathcal T)}J_y(\\Lambda)\n\\]\nalways exists.\n\\item If the minimizer is unique and at least one competing pruning exists, its gap\n\\[\n\\Gamma_y\n=\n\\min_{\\Lambda\\ne\\Lambda_y^*}\n\\bigl(J_y(\\Lambda)-J_y(\\Lambda_y^*)\\bigr)\n\\]\nis attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced.\n\\item In the case with competitors, if new data $y'$ satisfy\n\\[\n\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}\n|J_{y'}(\\Lambda)-J_y(\\Lambda)|\n<\\frac{\\Gamma_y}{2},\n\\]\nthen $\\Lambda_{y'}^*=\\Lambda_y^*$.\n\\item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.\n\\end{enumerate}\n\\end{theorem}","tem If the minimizer is unique and at least one competing pruning exists, its gap\n\\[\n\\Gamma_y\n=\n\\min_{\\Lambda\\ne\\Lambda_y^*}\n\\bigl(J_y(\\Lambda)-J_y(\\Lambda_y^*)\\bigr)\n\\]\nis attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced.\n\\item In the case with competitors, if new data $y'$ satisfy\n\\[\n\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}\n|J_{y'}(\\Lambda)-J_y(\\Lambda)|\n<\\frac{\\Gamma_y}{2},\n\\]\nthen $\\Lambda_{y'}^*=\\Lambda_y^*$.\n\\item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.\n\\end{enumerate}\n\\end{theorem}","imizer is unique and at least one competing pruning exists, its gap\n\\[\n\\Gamma_y\n=\n\\min_{\\Lambda\\ne\\Lambda_y^*}\n\\bigl(J_y(\\Lambda)-J_y(\\Lambda_y^*)\\bigr)\n\\]\nis attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced.\n\\item In the case with competitors, if new data $y'$ satisfy\n\\[\n\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}\n|J_{y'}(\\Lambda)-J_y(\\Lambda)|\n<\\frac{\\Gamma_y}{2},\n\\]\nthen $\\Lambda_{y'}^*=\\Lambda_y^*$.\n\\item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.\n\\end{enumerate}\n\\end{theorem}","da_y^*}\n\\bigl(J_y(\\Lambda)-J_y(\\Lambda_y^*)\\bigr)\n\\]\nis attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced.\n\\item In the case with competitors, if new data $y'$ satisfy\n\\[\n\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}\n|J_{y'}(\\Lambda)-J_y(\\Lambda)|\n<\\frac{\\Gamma_y}{2},\n\\]\nthen $\\Lambda_{y'}^*=\\Lambda_y^*$.\n\\item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.\n\\end{enumerate}\n\\end{theorem}","subset of the finite node set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for any competing pruning $\\Lambda\\ne\\Lambda_y^*$,\n\\[\n\\begin{aligned}\nJ_{y'}(\\Lambda)-J_{y'}(\\Lambda_y^*)\n&\\ge\nJ_y(\\Lambda)-J_y(\\Lambda_y^*)\n-2\\sup_{\\Xi}|J_{y'}(\\Xi)-J_y(\\Xi)|","set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for any competing pruning $\\Lambda\\ne\\Lambda_y^*$,\n\\[\n\\begin{aligned}\nJ_{y'}(\\Lambda)-J_{y'}(\\Lambda_y^*)\n&\\ge\nJ_y(\\Lambda)-J_y(\\Lambda_y^*)\n-2\\sup_{\\Xi}|J_{y'}(\\Xi)-J_y(\\Xi)|\\\\\n&>\\Gamma_y-\\Gamma_y=0.\n\\end{aligned}\n\\]\nHence the same pruning remains uniquely optimal.  Exact synthesis depends only on completeness of the pruning, not on the cost used to select it.\n\\end{proof}","\\begin{theorem}[Dynamic-programming best packet basis]\\label{result:15}\nSuppose\n\\[\nJ_y(\\Lambda)=\\sum_{v\\in\\Lambda}c_v(y).\n\\]\nFor a node $v$, define recursively\n\\[\nV_y(v)\n=\n\\begin{cases}\nc_v(y),&v\\text{ terminal},","\\begin{theorem}[Dynamic-programming best packet basis]\\label{result:15}\nSuppose\n\\[\nJ_y(\\Lambda)=\\sum_{v\\in\\Lambda}c_v(y).\n\\]\nFor a node $v$, define recursively\n\\[\nV_y(v)\n=\n\\begin{cases}\nc_v(y),&v\\text{ terminal},\\\\[2mm]\n\\displaystyle\n\\min\\left\\{c_v(y),\\sum_{u\\in\\operatorname{ch}(v)}V_y(u)\\right\\},\n&v\\text{ internal}.\n\\end{cases}\n\\]\nStop at $v$ when $c_v(y)\\le\\sum_uV_y(u)$, including equality, and refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node costs supplied and child values cached, the evaluator uses one comparison per internal node, one visit per node, and at most $|\\mathcal T|-1$ additions. Its arithmetic-and-visit count is therefore at most $3|\\mathcal T|$. This is a unit-cost statement about the specified evaluator. It excludes evaluation of the local costs, bit complexity of scalar arithmetic, and materialization or sorting of packet-address strings.\n\\end{theorem}","[2mm]\n\\displaystyle\n\\min\\left\\{c_v(y),\\sum_{u\\in\\operatorname{ch}(v)}V_y(u)\\right\\},\n&v\\text{ internal}.\n\\end{cases}\n\\]\nStop at $v$ when $c_v(y)\\le\\sum_uV_y(u)$, including equality, and refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node costs supplied and child values cached, the evaluator uses one comparison per internal node, one visit per node, and at most $|\\mathcal T|-1$ additions. Its arithmetic-and-visit count is therefore at most $3|\\mathcal T|$. This is a unit-cost statement about the specified evaluator. It excludes evaluation of the local costs, bit complexity of scalar arithmetic, and materialization or sorting of packet-address strings.\n\\end{theorem}","style\n\\min\\left\\{c_v(y),\\sum_{u\\in\\operatorname{ch}(v)}V_y(u)\\right\\},\n&v\\text{ internal}.\n\\end{cases}\n\\]\nStop at $v$ when $c_v(y)\\le\\sum_uV_y(u)$, including equality, and refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node costs supplied and child values cached, the evaluator uses one comparison per internal node, one visit per node, and at most $|\\mathcal T|-1$ additions. Its arithmetic-and-visit count is therefore at most $3|\\mathcal T|$. This is a unit-cost statement about the specified evaluator. It excludes evaluation of the local costs, bit complexity of scalar arithmetic, and materialization or sorting of packet-address strings.\n\\end{theorem}","internal}.\n\\end{cases}\n\\]\nStop at $v$ when $c_v(y)\\le\\sum_uV_y(u)$, including equality, and refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node costs supplied and child values cached, the evaluator uses one comparison per internal node, one visit per node, and at most $|\\mathcal T|-1$ additions. Its arithmetic-and-visit count is therefore at most $3|\\mathcal T|$. This is a unit-cost statement about the specified evaluator. It excludes evaluation of the local costs, bit complexity of scalar arithmetic, and materialization or sorting of packet-address strings.\n\\end{theorem}","\\begin{proof}\nProceed upward from the terminal nodes.  Every complete pruning below an internal node $v$ has exactly two possible forms: it either stops at $v$, with cost $c_v(y)$, or it refines and is the union of complete child prunings, whose minimum total cost is the sum of the already computed child minima.  The recursion therefore gives the exact minimum on every subtree, in particular at the root. This characterization also proves both directions of the correspondence between recursive cuts and complete covering antichains, for arbitrary positive finite branching, including unary branches. Summing cached child values charges at most one addition per child edge. A finite rooted tree has $|\\mathcal T|-1$ such edges. The number of internal nodes is at most $|\\mathcal T|$, which gives the","\\begin{proof}\nProceed upward from the terminal nodes.  Every complete pruning below an internal node $v$ has exactly two possible forms: it either stops at $v$, with cost $c_v(y)$, or it refines and is the union of complete child prunings, whose minimum total cost is the sum of the already computed child minima.  The recursion therefore gives the exact minimum on every subtree, in particular at the root. This characterization also proves both directions of the correspondence between recursive cuts and complete covering antichains, for arbitrary positive finite branching, including unary branches. Summing cached child values charges at most one addition per child edge. A finite rooted tree has $|\\mathcal T|-1$ such edges. The number of internal nodes is at most $|\\mathcal T|$, which gives the stated work bound. Ties do not affect optimality, but t","\\begin{proof}\nProceed upward from the terminal nodes.  Every complete pruning below an internal node $v$ has exactly two possible forms: it either stops at $v$, with cost $c_v(y)$, or it refines and is the union of complete child prunings, whose minimum total cost is the sum of the already computed child minima.  The recursion therefore gives the exact minimum on every subtree, in particular at the root. This characterization also proves both directions of the correspondence between recursive cuts and complete covering antichains, for arbitrary positive finite branching, including unary branches. Summing cached child values charges at most one addition per child edge. A finite rooted tree has $|\\mathcal T|-1$ such edges. The number of internal nodes is at most $|\\mathcal T|$, which gives the stated work bound. Ties do not affect optimality, but the stop-on-ties rule","ld minima.  The recursion therefore gives the exact minimum on every subtree, in particular at the root. This characterization also proves both directions of the correspondence between recursive cuts and complete covering antichains, for arbitrary positive finite branching, including unary branches. Summing cached child values charges at most one addition per child edge. A finite rooted tree has $|\\mathcal T|-1$ such edges. The number of internal nodes is at most $|\\mathcal T|$, which gives the stated work bound. Ties do not affect optimality, but the stop-on-ties rule makes the selected pruning unambiguous.\n\\end{proof}","subtree, in particular at the root. This characterization also proves both directions of the correspondence between recursive cuts and complete covering antichains, for arbitrary positive finite branching, including unary branches. Summing cached child values charges at most one addition per child edge. A finite rooted tree has $|\\mathcal T|-1$ such edges. The number of internal nodes is at most $|\\mathcal T|$, which gives the stated work bound. Ties do not affect optimality, but the stop-on-ties rule makes the selected pruning unambiguous.\n\\end{proof}","\\begin{corollary}[Local perturbation certificate]\\label{result:16}\nAssume\n\\[\n|c_v(y')-c_v(y)|\\le\\varepsilon_v.\n\\]\nDefine\n\\[\nE(v)\n=\n\\begin{cases}\n\\varepsilon_v,&v\\text{ terminal},","\\begin{corollary}[Local perturbation certificate]\\label{result:16}\nAssume\n\\[\n|c_v(y')-c_v(y)|\\le\\varepsilon_v.\n\\]\nDefine\n\\[\nE(v)\n=\n\\begin{cases}\n\\varepsilon_v,&v\\text{ terminal},\\\\[1mm]\n\\displaystyle\n\\max\\left\\{\\varepsilon_v,\n\\sum_{u\\in\\operatorname{ch}(v)}E(u)\\right\\},&v\\text{ internal}.\n\\end{cases}\n\\]\nFor the complete prunings $\\mathfrak P_v$ below $v$, the envelope is exactly\n\\[\nE(v)=\\max_{\\Lambda\\in\\mathfrak P_v}\\sum_{a\\in\\Lambda}\\varepsilon_a.\n\\]\nThe maximum is attained. Every complete-pruning cost changes by at most $E(\\varnothing)$, and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors,\n\\[\n\\boxed{\\Gamma_y>2E(\\varnothing)}\n\\]\nis a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event o","[1mm]\n\\displaystyle\n\\max\\left\\{\\varepsilon_v,\n\\sum_{u\\in\\operatorname{ch}(v)}E(u)\\right\\},&v\\text{ internal}.\n\\end{cases}\n\\]\nFor the complete prunings $\\mathfrak P_v$ below $v$, the envelope is exactly\n\\[\nE(v)=\\max_{\\Lambda\\in\\mathfrak P_v}\\sum_{a\\in\\Lambda}\\varepsilon_a.\n\\]\nThe maximum is attained. Every complete-pruning cost changes by at most $E(\\varnothing)$, and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors,\n\\[\n\\boxed{\\Gamma_y>2E(\\varnothing)}\n\\]\nis a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event of probability at least $1-\\delta$, and the","[1mm]\n\\displaystyle\n\\max\\left\\{\\varepsilon_v,\n\\sum_{u\\in\\operatorname{ch}(v)}E(u)\\right\\},&v\\text{ internal}.\n\\end{cases}\n\\]\nFor the complete prunings $\\mathfrak P_v$ below $v$, the envelope is exactly\n\\[\nE(v)=\\max_{\\Lambda\\in\\mathfrak P_v}\\sum_{a\\in\\Lambda}\\varepsilon_a.\n\\]\nThe maximum is attained. Every complete-pruning cost changes by at most $E(\\varnothing)$, and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors,\n\\[\n\\boxed{\\Gamma_y>2E(\\varnothing)}\n\\]\nis a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event of probability at least $1-\\delta$, and the displayed","[1mm]\n\\displaystyle\n\\max\\left\\{\\varepsilon_v,\n\\sum_{u\\in\\operatorname{ch}(v)}E(u)\\right\\},&v\\text{ internal}.\n\\end{cases}\n\\]\nFor the complete prunings $\\mathfrak P_v$ below $v$, the envelope is exactly\n\\[\nE(v)=\\max_{\\Lambda\\in\\mathfrak P_v}\\sum_{a\\in\\Lambda}\\varepsilon_a.\n\\]\nThe maximum is attained. Every complete-pruning cost changes by at most $E(\\varnothing)$, and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors,\n\\[\n\\boxed{\\Gamma_y>2E(\\varnothing)}\n\\]\nis a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event of probability at least $1-\\delta$, and the displayed strict-gap condition holds, the same selector is recovered with probability at least $1-\\delta$. I","[1mm]\n\\displaystyle\n\\max\\left\\{\\varepsilon_v,\n\\sum_{u\\in\\operatorname{ch}(v)}E(u)\\right\\},&v\\text{ internal}.\n\\end{cases}\n\\]\nFor the complete prunings $\\mathfrak P_v$ below $v$, the envelope is exactly\n\\[\nE(v)=\\max_{\\Lambda\\in\\mathfrak P_v}\\sum_{a\\in\\Lambda}\\varepsilon_a.\n\\]\nThe maximum is attained. Every complete-pruning cost changes by at most $E(\\varnothing)$, and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors,\n\\[\n\\boxed{\\Gamma_y>2E(\\varnothing)}\n\\]\nis a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event of probability at least $1-\\delta$, and the displayed strict-gap condition holds, the same selector is recovered with probability at least $1-\\delta$. Individual nodewise probabilities alone do not give this joint confidence without an additi","&v\\text{ internal}.\n\\end{cases}\n\\]\nFor the complete prunings $\\mathfrak P_v$ below $v$, the envelope is exactly\n\\[\nE(v)=\\max_{\\Lambda\\in\\mathfrak P_v}\\sum_{a\\in\\Lambda}\\varepsilon_a.\n\\]\nThe maximum is attained. Every complete-pruning cost changes by at most $E(\\varnothing)$, and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors,\n\\[\n\\boxed{\\Gamma_y>2E(\\varnothing)}\n\\]\nis a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event of probability at least $1-\\delta$, and the displayed strict-gap condition holds, the same selector is recovered with probability at least $1-\\delta$. Individual nodewise probabilities alone do not give this joint confidence without an additional argument, such as a union bound.\n\\end{corollary}","y, for a unique optimum with competitors,\n\\[\n\\boxed{\\Gamma_y>2E(\\varnothing)}\n\\]\nis a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event of probability at least $1-\\delta$, and the displayed strict-gap condition holds, the same selector is recovered with probability at least $1-\\delta$. Individual nodewise probabilities alone do not give this joint confidence without an additional argument, such as a union bound.\n\\end{corollary}","t the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \\emph{all} local bounds simultaneously on one event of probability at least $1-\\delta$, and the displayed strict-gap condition holds, the same selector is recovered with probability at least $1-\\delta$. Individual nodewise probabilities alone do not give this joint confidence without an additional argument, such as a union bound.\n\\end{corollary}","\\begin{proof}\nFor any pruning of the subtree rooted at $v$, either $v$ itself is selected, producing perturbation at most $\\varepsilon_v$, or the pruning is the disjoint union of child prunings, producing perturbation at most the sum of the child envelopes.  The same decomposition applied to the nonnegative weights $\\varepsilon_a$ proves equality with the maximum weight: at each node choose either the stop option or a maximizing pruning in each child, whichever gives the larger total. Finiteness ensures these choices attain their maxima. Evaluating each cost at a minimizer of the other cost bounds the difference of the two optimal values by the uniform envelope. The selector statement follows from the strict-gap theorem. The finite deterministic c","\\begin{proof}\nFor any pruning of the subtree rooted at $v$, either $v$ itself is selected, producing perturbation at most $\\varepsilon_v$, or the pruning is the disjoint union of child prunings, producing perturbation at most the sum of the child envelopes.  The same decomposition applied to the nonnegative weights $\\varepsilon_a$ proves equality with the maximum weight: at each node choose either the stop option or a maximizing pruning in each child, whichever gives the larger total. Finiteness ensures these choices attain their maxima. Evaluating each cost at a minimizer of the other cost bounds the difference of the two optimal values by the uniform envelope. The selector statement follows from the strict-gap theorem. The finite deterministic comparisons d","\\begin{proof}\nFor any pruning of the subtree rooted at $v$, either $v$ itself is selected, producing perturbation at most $\\varepsilon_v$, or the pruning is the disjoint union of child prunings, producing perturbation at most the sum of the child envelopes.  The same decomposition applied to the nonnegative weights $\\varepsilon_a$ proves equality with the maximum weight: at each node choose either the stop option or a maximizing pruning in each child, whichever gives the larger total. Finiteness ensures these choices attain their maxima. Evaluating each cost at a minimizer of the other cost bounds the difference of the two optimal values by the uniform envelope. The selector statement follows from the strict-gap theorem. The finite deterministic comparisons defining the selector make its preservation event measurable when th","\\begin{proof}\nFor any pruning of the subtree rooted at $v$, either $v$ itself is selected, producing perturbation at most $\\varepsilon_v$, or the pruning is the disjoint union of child prunings, producing perturbation at most the sum of the child envelopes.  The same decomposition applied to the nonnegative weights $\\varepsilon_a$ proves equality with the maximum weight: at each node choose either the stop option or a maximizing pruning in each child, whichever gives the larger total. Finiteness ensures these choices attain their maxima. Evaluating each cost at a minimizer of the other cost bounds the difference of the two optimal values by the uniform envelope. The selector statement follows from the strict-gap theorem. The finite deterministic comparisons defining the selector make its preservation event measurable when the random node costs are measurable. The joint local-error event is contained in the selector-preservation event, so monotonicity of probability proves the probabilistic assertion.\n\\end{proof}","\\begin{certificate}[Adaptive-tree optimum]\\label{core:record:27}\nFor the depth-five example, the complete binary tree has $458{,}330$ admissible prunings.  Dynamic programming and exhaustive enumeration returned the same minimum with residual $0$.  The selected pruning was\n\\[\n\\{000,001,0100,0101,011,1\\},\n\\]\nand mixed-depth synthesis reconstructed the signal with maximum residual $8.88\\times10^{-16}$.  The optimum gap was $3.1909\\times10^{-4}$, while twice the declared perturbation envelope was $6.4\\times10^{-5}$, so the finite selector is certified stable.\n\\end{certificate}","ve-tree optimum]\\label{core:record:27}\nFor the depth-five example, the complete binary tree has $458{,}330$ admissible prunings.  Dynamic programming and exhaustive enumeration returned the same minimum with residual $0$.  The selected pruning was\n\\[\n\\{000,001,0100,0101,011,1\\},\n\\]\nand mixed-depth synthesis reconstructed the signal with maximum residual $8.88\\times10^{-16}$.  The optimum gap was $3.1909\\times10^{-4}$, while twice the declared perturbation envelope was $6.4\\times10^{-5}$, so the finite selector is certified stable.\n\\end{certificate}","plete binary tree has $458{,}330$ admissible prunings.  Dynamic programming and exhaustive enumeration returned the same minimum with residual $0$.  The selected pruning was\n\\[\n\\{000,001,0100,0101,011,1\\},\n\\]\nand mixed-depth synthesis reconstructed the signal with maximum residual $8.88\\times10^{-16}$.  The optimum gap was $3.1909\\times10^{-4}$, while twice the declared perturbation envelope was $6.4\\times10^{-5}$, so the finite selector is certified stable.\n\\end{certificate}","Linear time invariance is not required for exact packet coordinates. Let $A$ and $B$ be formal power series over a commutative coefficient ring (in particular $\\bbC$). Set\n\\[\nA(z)=\\sum_{n\\ge0}a_nz^n,\n\\qquad\nA_i=P_{N,i}A.\n\\]\nAll phase indices range from $0$ to $N-1$. For another representation $B(z)$, multiplication gives a nonlinear or bilinear interaction.","\\begin{theorem}[Cyclic product law]\\label{result:17}\n\\[\n\\boxed{\nP_{N,k}(AB)\n=\n\\sum_{i+j\\equiv k\\, (\\mathrm{mod}\\,N)}A_iB_j.\n}\n\\]\nFor $q\\in\\bbN$, the $q$th-power identity is\n\\[\nP_{N,k}(A^q)\n=\n\\sum_{i_1+\\cdots+i_q\\equiv k}\nA_{i_1}\\cdots A_{i_q}.\n\\]\n\\end{theorem}","\\begin{proof}\nEvery coefficient in $A_iB_j$ has degree congruent to $i+j$ modulo $N$. At any fixed degree $n$, the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on $q$ and the finite product law. At $q=0$, the index set contains the single empty tuple, its product is $1$, and its residue sum is $0$. Hence the right side is $1$ for $k=0$ and $0$ otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.\n\\end{proof}","n{proof}\nEvery coefficient in $A_iB_j$ has degree congruent to $i+j$ modulo $N$. At any fixed degree $n$, the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on $q$ and the finite product law. At $q=0$, the index set contains the single empty tuple, its product is $1$, and its residue sum is $0$. Hence the right side is $1$ for $k=0$ and $0$ otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.\n\\end{proof}","n $A_iB_j$ has degree congruent to $i+j$ modulo $N$. At any fixed degree $n$, the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on $q$ and the finite product law. At $q=0$, the index set contains the single empty tuple, its product is $1$, and its residue sum is $0$. Hence the right side is $1$ for $k=0$ and $0$ otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.\n\\end{proof}","nt to $i+j$ modulo $N$. At any fixed degree $n$, the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on $q$ and the finite product law. At $q=0$, the index set contains the single empty tuple, its product is $1$, and its residue sum is $0$. Hence the right side is $1$ for $k=0$ and $0$ otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.\n\\end{proof}","+j$ modulo $N$. At any fixed degree $n$, the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on $q$ and the finite product law. At $q=0$, the index set contains the single empty tuple, its product is $1$, and its residue sum is $0$. Hence the right side is $1$ for $k=0$ and $0$ otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.\n\\end{proof}","$N$. At any fixed degree $n$, the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on $q$ and the finite product law. At $q=0$, the index set contains the single empty tuple, its product is $1$, and its residue sum is $0$. Hence the right side is $1$ for $k=0$ and $0$ otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.76\\textwidth]{14_nonlinear_phase_coupling.pdf}\n\\caption{Norms of pairwise product interactions for five residue phases. The label in each cell gives the output phase $(i+j)\\bmod5$.}\n\\label{fig:nonlinear-coupling}\n\\end{figure}","For a finite nonlinear shift operator\n\\[\nF(x)[n]=x[n]+\\alpha x[n]x[n-1],\n\\]\nthe exact packet coordinates are simply\n\\[\nF_r((x_s)_s)=P_{N,r}F\\!\\left(\\sum_sP_{N,s}x\\right).\n\\]\nAnalysis followed by this packet system and synthesis reproduces the direct nonlinear output exactly. The numerical residual in the companion implementation was zero to machine representation.","Let $Z_0>0$ be a reference impedance and\n\\[\nz=\\frac{Z}{Z_0}=r+\\ii x.\n\\]\nThe reflection coefficient is\n\\[\n\\boxed{\n\\Gam=\\frac{z-1}{z+1},\n\\qquad\nz=\\frac{1+\\Gam}{1-\\Gam}.\n}\n\\]","\\begin{classical}[Cayley geometry]\\label{core:record:30}\nThe map sends the finite passive half-plane $r\\ge0$ onto the closed unit disk with $\\Gam=1$ omitted. Adjoining the point at infinity in impedance adds that missing boundary point. The open right half-plane maps conformally onto the open disk. Constant resistance and reactance become the familiar Smith circles. This conformal geometry is classical and is not replaced by polyphase refinement.\n\\end{classical}","Writing $\\Gam=u+\\ii v$, the grid equations are\n\\[\n\\left(u-\\frac{r}{1+r}\\right)^2+v^2=\\frac1{(1+r)^2},\n\\]\nand, for $x\\ne0$,\n\\[\n(u-1)^2+\\left(v-\\frac1x\\right)^2=\\frac1{x^2}.\n\\]","Fix\n\\[\n\\omega_N=e^{2\\pi\\ii/N}\n\\]\nand rotate the reflection coefficient by\n\\[\nR_{N,k}(\\Gam)=\\omega_N^k\\Gam.\n\\]","\\begin{theorem}[Polyphase M\\\"obius action]\\label{result:18}\nRegard normalized impedance as the projective line $\\mathbb P^1(\\bbC)=\\bbC\\cup\\{\\infty\\}$. For $N\\ge1$, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is\n\\[\n\\boxed{\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n}\n\\]\nAs maps on $\\mathbb P^1(\\bbC)$ it satisfies\n\\[\nT_{N,k}\\circ T_{N,\\ell}=T_{N,k+\\ell\\, (\\mathrm{mod}\\,N)}.\n\\]\n\\end{theorem}","obius action]\\label{result:18}\nRegard normalized impedance as the projective line $\\mathbb P^1(\\bbC)=\\bbC\\cup\\{\\infty\\}$. For $N\\ge1$, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is\n\\[\n\\boxed{\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n}\n\\]\nAs maps on $\\mathbb P^1(\\bbC)$ it satisfies\n\\[\nT_{N,k}\\circ T_{N,\\ell}=T_{N,k+\\ell\\, (\\mathrm{mod}\\,N)}.\n\\]\n\\end{theorem}","\\begin{proof}\nLet $C(z)=(z-1)/(z+1)$. Then\n\\[\nT_{N,k}=C^{-1}\\circ R_{N,k}\\circ C.\n\\]\nFor a division-free construction, represent a point by a nonzero pair $[u:v]$, identifying $[cu:cv]=[u:v]$ for $c\\ne0$. With $w=\\omega_N^k$, use the matrix\n\\[\nA(w)=\\frac12\\begin{pmatrix}1+w&1-w","\\begin{proof}\nLet $C(z)=(z-1)/(z+1)$. Then\n\\[\nT_{N,k}=C^{-1}\\circ R_{N,k}\\circ C.\n\\]\nFor a division-free construction, represent a point by a nonzero pair $[u:v]$, identifying $[cu:cv]=[u:v]$ for $c\\ne0$. With $w=\\omega_N^k$, use the matrix\n\\[\nA(w)=\\frac12\\begin{pmatrix}1+w&1-w\\\\1-w&1+w\\end{pmatrix}.\n\\]\nIt has determinant $w\\ne0$, preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives $A(w)A(w')=A(ww')$ and $A(1)=I$. Since $\\omega_N^N=1$, this proves the cyclic composition law modulo $N$. The Cayley coordinate of $[u:v]$ is $[u-v:u+v]$, and applying $A(w)$ changes this pair to $[w(u-v):u+v]$. This proves the conjugacy also at poles. For $[z:1]$, taking the coordinate ratio gives the displayed finite-chart formula.\n\\end{proof}","1-w&1+w\\end{pmatrix}.\n\\]\nIt has determinant $w\\ne0$, preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives $A(w)A(w')=A(ww')$ and $A(1)=I$. Since $\\omega_N^N=1$, this proves the cyclic composition law modulo $N$. The Cayley coordinate of $[u:v]$ is $[u-v:u+v]$, and applying $A(w)$ changes this pair to $[w(u-v):u+v]$. This proves the conjugacy also at poles. For $[z:1]$, taking the coordinate ratio gives the displayed finite-chart formula.\n\\end{proof}","It has determinant $w\\ne0$, preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives $A(w)A(w')=A(ww')$ and $A(1)=I$. Since $\\omega_N^N=1$, this proves the cyclic composition law modulo $N$. The Cayley coordinate of $[u:v]$ is $[u-v:u+v]$, and applying $A(w)$ changes this pair to $[w(u-v):u+v]$. This proves the conjugacy also at poles. For $[z:1]$, taking the coordinate ratio gives the displayed finite-chart formula.\n\\end{proof}","\\begin{corollary}[Impedance--admittance is the two-phase split]\\label{result:19}\nFor $N=2$ and finite $z\\ne0$,\n\\[\n\\boxed{T_{2,1}(z)=z^{-1}.}\n\\]\nOn the full projective line it exchanges $0$ and $\\infty$. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance.\n\\end{corollary}\n\\begin{proof}\n$\\omega_2=\\exp(\\pi\\ii)=-1$, so $A(\\omega_2)[u:v]=[v:u]$. The finite ratio is $1/z$ when $z\\ne0$. The coordinate swap also gives $[0:1]\\leftrightarrow[1:0]$. No ordinary division by zero is used to represent infinity.\n\\end{proof}","rollary}[Impedance--admittance is the two-phase split]\\label{result:19}\nFor $N=2$ and finite $z\\ne0$,\n\\[\n\\boxed{T_{2,1}(z)=z^{-1}.}\n\\]\nOn the full projective line it exchanges $0$ and $\\infty$. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance.\n\\end{corollary}\n\\begin{proof}\n$\\omega_2=\\exp(\\pi\\ii)=-1$, so $A(\\omega_2)[u:v]=[v:u]$. The finite ratio is $1/z$ when $z\\ne0$. The coordinate swap also gives $[0:1]\\leftrightarrow[1:0]$. No ordinary division by zero is used to represent infinity.\n\\end{proof}","o-phase split]\\label{result:19}\nFor $N=2$ and finite $z\\ne0$,\n\\[\n\\boxed{T_{2,1}(z)=z^{-1}.}\n\\]\nOn the full projective line it exchanges $0$ and $\\infty$. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance.\n\\end{corollary}\n\\begin{proof}\n$\\omega_2=\\exp(\\pi\\ii)=-1$, so $A(\\omega_2)[u:v]=[v:u]$. The finite ratio is $1/z$ when $z\\ne0$. The coordinate swap also gives $[0:1]\\leftrightarrow[1:0]$. No ordinary division by zero is used to represent infinity.\n\\end{proof}","For $0\\le\\rho<1$ and angle $\\theta$,\n\\[\n\\Gam=\\rho e^{\\ii\\theta}\n\\]\ngives\n\\[\nz(\\rho,\\theta)\n=\\frac{1+\\rho e^{\\ii\\theta}}{1-\\rho e^{\\ii\\theta}},\n\\]\nwith\n\\[\nr(\\rho,\\theta)=\\frac{1-\\rho^2}{1-2\\rho\\cos\\theta+\\rho^2},\n\\qquad\nx(\\rho,\\theta)=\\frac{2\\rho\\sin\\theta}{1-2\\rho\\cos\\theta+\\rho^2}.\n\\]\nA fixed $\\rho$ is a constant-mismatch circle. Refining $N$ increases angular resolution without changing $\\rho$.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.78\\textwidth]{12_smith_nested_refinement.pdf}\n\\caption{The same reflection circle at phase orders $2,4,8,16,32$. Each coarse phase ray appears in every finer grid whose order is a multiple of its order. The exact residue-projector refinement law concerns coefficient packets, not sums of impedance points.}\n\\label{fig:smith-refinement}\n\\end{figure}","For a lossless transmission line,\n\\[\n\\Gam_{\\rm in}=\\Gam_L e^{-2\\ii\\beta\\ell}.\n\\]\nOne $N$-phase step corresponds to a line-length increment $\\lambda/(2N)$, up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","Write\n\\[\nN=2^sM,\n\\qquad M\\text{ odd}.\n\\]\nThe cyclic index splits by the Chinese remainder theorem:\n\\[\n\\bbZ/N\\bbZ\n\\cong\n\\bbZ/2^s\\bbZ\\times\\bbZ/M\\bbZ.\n\\]","\\begin{proposition}[Dyadic--odd packet factorization]\\label{result:20}\nAfter identifying the residue labels by the Chinese remainder map, the phase projector factors as\n\\[\n\\boxed{\nP_{N,j}=P_{2^s,j_2}P_{M,j_o}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe two congruence conditions are independent because $2^s$ and $M$ are coprime. Their indicator functions multiply to the indicator of the unique residue class modulo $N$ satisfying both.\n\\end{proof}","The first dyadic step is exceptional in Smith coordinates because it is the inversion $z\\mapsto z^{-1}$. Higher dyadic levels refine the open/short duality into quarter-, eighth-, and finer phase sectors.","Away from its pole $\\Gam=1$, the inverse Cayley map has derivative\n\\[\n\\frac{dz}{d\\Gam}=\\frac{2}{(1-\\Gam)^2},\n\\]\nwhile normalized admittance has derivative proportional to $(1+\\Gam)^{-2}$. Away from $\\Gam=\\pm1$, this motivates the symmetric sensitivity\n\\[\n\\boxed{\nw_{\\rm Smith}(\\Gam)\n=\n\\frac1{|1-\\Gam|^2}\n+\n\\frac1{|1+\\Gam|^2}.\n}\n\\]","\\begin{conjecture}[Edge-packet approximation class]\\label{core:record:35}\nThere should exist natural function spaces in which a signal is represented as\n\\[\nf=s+\\sum_q a_q\\mathbf1_{\\Omega_q},\n\\]\nwith separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation error with discontinuity-location error.\n\\end{conjecture}","\\begin{certificate}[Finite numerical residuals]\\label{core:record:36}\nThe largest recorded residuals were\n\\[\n\\begin{array}{lr}\n\\toprule\n\\text{identity}&\\text{residual}\\\\\n\\midrule\n\\text{polyphase reconstruction}&0\\\\\n\\text{nested refinement}&0\\\\\n\\text{polyphase convolution}&4.44\\times10^{-16}\\\\\n\\text{Haar reconstruction}&8.88\\times10^{-16}\\\\\n\\text{wavelet-packet reconstruction}&2.22\\times10^{-15}\\\\\n\\text{adaptive-tree optimum versus exhaustive search}&0\\\\\n\\text{adaptive mixed-depth synthesis}&8.88\\times10^{-16}\\\\\n\\text{lifting reconstruction}&4.44\\times10^{-16}\\\\\n\\text{Smith rotation conjugacy}&3.34\\times10^{-16}\\\\\n\\text{Smith impedance--admittance duality}&0\\\\\n\\text{nonlinear product-packet reconstruction}&1.33\\times10^{-15}\\\\\n\\text{nonlinear shifted-filter reconstruction}&0\\\\\n\\bottomrule\n\\end{array}\n\\]\nThe Haar and wavelet energy residuals were $5.68\\times10^{-14}$ and $2.27\\times10^{-13}$, respectively.\n\\end{certificate}","\\bottomrule\n\\end{array}\n\\]\nThe Haar and wavelet energy residuals were $5.68\\times10^{-14}$ and $2.27\\times10^{-13}$, respectively.\n\\end{certificate}","The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly:\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{seed}&=\\delta_0,\\\\\n\\text{address}&=S^n,\\\\\n\\text{packet}&=P_{N,r}x,\\\\\n\\text{transfer}&=P_{N,k}HP_{N,s},\\\\\n\\text{refinement}&=P_{N,r}=\\sum_{s\\succ r}P_{M,s},\\\\\n\\text{synthesis}&=\\sum_rP_{N,r}x,\\\\\n\\text{nonlinearity}&=P_{N,r}F\\left(\\sum_sP_{N,s}x\\right).\n\\end{aligned}\n}\n\\]\nFilter banks, wavelet packets, adaptive complete prunings, edge packets, and Smith coordinates inhabit the same operator grammar, but their claims differ. Polyphase and filter-bank identities are exact. Wavelet perfect reconstruction is exact. The known-edge theorem is exact conditional on the decomposition. Estimated-edge performance depends on a detection model. The Smith chart remains the classical Cayley geometry equipped with an exact cyclic refinement algebra. Keeping those distinctions explicit is part of the calculus, not an editorial afterthought.","Given $x$, $h$, and $N$:","\\begin{enumerate}\n\\item Form $x_s[n]=x[Nn+s]$ for $0\\le s<N$.\n\\item Form every transfer kernel $h_{ks}[m]=h[Nm+k-s]$.\n\\item Compute\n\\[\ny_k[n]=\\sum_s\\sum_mh_{ks}[m]x_s[n-m].\n\\]\n\\item Interleave the $y_k$ channels.\n\\end{enumerate}","Given a decomposition $f=s+j$:","\\begin{enumerate}\n\\item Apply the selected smooth approximation $Q_m$ to $s$ only.\n\\item Retain $j$ exactly.\n\\item Return $Q_ms+j$.\n\\end{enumerate}","The error certificate is $\\|s-Q_ms\\|$.","\\midrule\n1D rectangular packet&$5.4023\\times10^{-2}$&$1.3692\\times10^{-4}$&$2.4029\\times10^{-2}$","2D disk packet&$3.7643\\times10^{-2}$&$6.4752\\times10^{-5}$&$3.4043\\times10^{-3}$","2D multiedge packet&$3.3740\\times10^{-2}$&\\multicolumn{2}{c}{known composite packet only}","\\midrule\n\\endhead\n$s$&$\\delta_0$&primitive signal seed","$A_n$&$S^n$&address operator","$P_{N,r}$&residue mask&phase packet projector","$T_{ks}$&$P_{N,k}HP_{N,s}$&input-to-output phase transfer","$E,R$&analysis and synthesis matrices&coordinate conversion and reconstruction","$W(\\gamma)$&filter or path weight&refinement or implementation cost","$\\Delta(F)$&coarse/fine signal defect&aliasing, leakage, or estimator inconsistency","$\\operatorname{Hol}(\\gamma)$&closed phase transfer&cycle consistency in a network","$\\Gam$&reflection coefficient&Smith-disk coordinate","\\midrule\\endhead\n\\ref{result:01}&HilbertPackets&Complex $\\ell^2$ impulse expansion, finite-support identity, unitary shifts, and closed shift-generated span.","\\ref{result:06}&FilterBank; FiniteTransforms&Finite Laurent/FIR bank composition, faithful action on sequences, gain-delay reconstruction criterion, and the $Nd$ original-sample delay.","\\ref{result:07}&HaarChannels; HaarHilbert&Literal even/odd input samples, normalized complex low/detail channels, and $\\ell^2$ energy equality.","\\ref{result:12}&MaskPerturbation&Measurable-mask symmetric-difference norm, extended finite-$p$ error bound, pointwise and essential supremum bounds, and smooth-background error.","\\ref{result:19}&SmithCyclic&Two-phase coordinate swap, $0\\leftrightarrow\\infty$, and finite impedance inversion when $z\\ne0$.","\\paragraph{Finite filters and numerical transforms.}\nThe filter-bank proof is an algebraic theorem about finite Laurent-polynomial kernels acting faithfully on bilateral sequences. Equality at a single complex value of $z$ would be insufficient to prove a polynomial identity. The impulse argument avoids that error. The separate transform theorem proves the finite-filter action agrees with evaluation wherever the weighted input series converge absolutely. An arbitrary bilateral $\\ell^2$ sequence need not have an absolutely convergent numerical $z$-transform. The all-sequence time-domain theorem makes no such assumption.","gebraic theorem about finite Laurent-polynomial kernels acting faithfully on bilateral sequences. Equality at a single complex value of $z$ would be insufficient to prove a polynomial identity. The impulse argument avoids that error. The separate transform theorem proves the finite-filter action agrees with evaluation wherever the weighted input series converge absolutely. An arbitrary bilateral $\\ell^2$ sequence need not have an absolutely convergent numerical $z$-transform. The all-sequence time-domain theorem makes no such assumption.","hfully on bilateral sequences. Equality at a single complex value of $z$ would be insufficient to prove a polynomial identity. The impulse argument avoids that error. The separate transform theorem proves the finite-filter action agrees with evaluation wherever the weighted input series converge absolutely. An arbitrary bilateral $\\ell^2$ sequence need not have an absolutely convergent numerical $z$-transform. The all-sequence time-domain theorem makes no such assumption.","\\paragraph{Hilbert operators and finite experiments.}\nThe coordinate maps, literal Haar channels, and packet operators are constructed on complex $\\ell^2$. Square summability and boundedness are established before adjoints are used. The finite Python experiments separately check indexing, reconstruction, and energy at finite lengths. They illustrate these operators but are not the objects executed by the Lean kernel. Likewise, the Python Smith routine evaluates only finite charts. The Lean projective construction supplies the pole-safe universal statement.","For the one-dimensional low-pass experiment, ``cutoff 18'' means 37 signed frequency bins, not 18 retained coefficients. The disk experiment uses a square $33\\times33$ frequency mask, not a radial cutoff. The sparse comparison uses largest-coefficient budgets instead of either low-pass convention. The full-image and boundary-band RMSEs are normalized by the number of samples in the respective comparison set. The supplied inpainting routines perform a fixed finite number of periodic neighbor-averaging iterations. The term ``harmonic filling'' describes that iteration, not a separately proved exact continuum Dirichlet solver.","The second aim is to separate representation from recovery. If a signal has already been written as\n\\[\nf=s+j\n\\]\nwith a smooth part \\(s\\) and a singular packet \\(j\\), then\n\\[\n\\widetilde f=Qs+j\n\\]\nsatisfies\n\\[\nf-\\widetilde f=s-Qs.\n\\]\nThis identity is exact, but it does not discover \\(j\\). A useful signal theory must therefore distinguish three errors. The first is the approximation error of the smooth component. The second is the error in the estimated singular packet. The third is the amplification caused by applying a reconstruction operator to that packet error. The middle sections of this part isolate all three terms.","The singular packets considered here range from indicator functions of regions to truncated powers\n\\[\n\\frac{(t-\\tau)_+^m}{m!}.\n\\]\nThe latter encode jumps in derivatives of order \\(m\\). They permit a piecewise smooth function to be reduced to a globally Sobolev remainder under explicit trace assumptions. This gives a precise version of the informal statement that a discontinuity should be represented directly rather than approximated by a global smooth basis.","The first result is an exact coordinate theorem. Let \\(L\\) be a finite index sublattice of \\(\\ZZ^d\\), and let \\(R\\) contain one representative from every coset. The analysis map\n\\[\nx\\longmapsto (x_r)_{r\\in R},\n\\qquad\nx_r(\\ell)=x(r+\\ell),\n\\]\nis an isometric isomorphism on every \\(\\ell^p\\) space. This statement includes the usual one-dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If \\(L'\\leq L\\), the coarse packets split exactly into their fine descendants. A convolution operator becomes a matrix of convolution operators on \\(L\\), with entries\n\\[\nh_{rs}(m)=h(r-s+m).\n\\]\nThe matrix is therefore a coordinate representation of the original operator, not a reduced model.","a finite index sublattice of \\(\\ZZ^d\\), and let \\(R\\) contain one representative from every coset. The analysis map\n\\[\nx\\longmapsto (x_r)_{r\\in R},\n\\qquad\nx_r(\\ell)=x(r+\\ell),\n\\]\nis an isometric isomorphism on every \\(\\ell^p\\) space. This statement includes the usual one-dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If \\(L'\\leq L\\), the coarse packets split exactly into their fine descendants. A convolution operator becomes a matrix of convolution operators on \\(L\\), with entries\n\\[\nh_{rs}(m)=h(r-s+m).\n\\]\nThe matrix is therefore a coordinate representation of the original operator, not a reduced model.","R},\n\\qquad\nx_r(\\ell)=x(r+\\ell),\n\\]\nis an isometric isomorphism on every \\(\\ell^p\\) space. This statement includes the usual one-dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If \\(L'\\leq L\\), the coarse packets split exactly into their fine descendants. A convolution operator becomes a matrix of convolution operators on \\(L\\), with entries\n\\[\nh_{rs}(m)=h(r-s+m).\n\\]\nThe matrix is therefore a coordinate representation of the original operator, not a reduced model.",")=x(r+\\ell),\n\\]\nis an isometric isomorphism on every \\(\\ell^p\\) space. This statement includes the usual one-dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If \\(L'\\leq L\\), the coarse packets split exactly into their fine descendants. A convolution operator becomes a matrix of convolution operators on \\(L\\), with entries\n\\[\nh_{rs}(m)=h(r-s+m).\n\\]\nThe matrix is therefore a coordinate representation of the original operator, not a reduced model.","The filter bank results distinguish exact reconstruction from quantitative near reconstruction. Exact recovery with delay \\(d\\) is the pointwise matrix identity\n\\[\nR(z)E(z)=z^{-d}I.\n\\]\nWhen this identity fails, the reconstruction error is controlled by the essential supremum of the delay compensated multiplier defect. If the filters themselves are perturbed, \\cref{thm:filter_perturbation} separates the analysis error, the synthesis error, and their product. The frame bound in \\cref{thm:frame_perturbation} shows that a lower singular value gap survives every sufficiently small perturbation. This makes the distinction between algebraic invertibility and stable numerical reconstruction explicit.","The packet tree results treat every admissible pruning, not only complete trees of a fixed depth. Unitary local splits give a unitary global transform. Local frame bounds propagate recursively in the nonunitary case. If the cost of a pruning is additive over retained nodes, then\n\\[\nF(v)\n=\n\\min\\left\\{\nc(v),\n\\sum_{w\\in\\operatorname{child}(v)}F(w)\n\\right\\}\n\\]\ngives the exact optimal tree. The cost and the selected basis are stable when the node costs change by less than the gap separating the optimal pruning from its competitors.","The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order \\(m\\) at \\(\\tau_q\\), then the packet\n\\[\n\\jump{f^{(m)}}_{\\tau_q}\n\\frac{(t-\\tau_q)_+^m}{m!}\n\\]\nremoves exactly that trace jump. Subtracting all packets through order \\(r-1\\) leaves a global \\(W^{r,p}\\) remainder. The packet coefficients are uniquely determined by the one-sided trace jumps. This gives a finite and explicit singular part for the class covered by \\cref{thm:higher_order_packet_decomposition}.","The reconstruction formula with an estimated packet is\n\\[\nf-\\widehat f_m\n=\n(I-Q_m)s+(I-Q_m)(j-\\widehat j).\n\\]\nThis identity identifies the two terms that must be controlled. The exact packet formulas in \\cref{sec:packet_error} then evaluate or bound \\(\\|j-\\widehat j\\|\\) from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus \\(|\\tau-\\widehat\\tau|^{1/p}\\) in finite \\(L^p\\), while every positive order truncated power packet is Lipschitz in its location.","reconstruction formula with an estimated packet is\n\\[\nf-\\widehat f_m\n=\n(I-Q_m)s+(I-Q_m)(j-\\widehat j).\n\\]\nThis identity identifies the two terms that must be controlled. The exact packet formulas in \\cref{sec:packet_error} then evaluate or bound \\(\\|j-\\widehat j\\|\\) from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus \\(|\\tau-\\widehat\\tau|^{1/p}\\) in finite \\(L^p\\), while every positive order truncated power packet is Lipschitz in its location.","packet is\n\\[\nf-\\widehat f_m\n=\n(I-Q_m)s+(I-Q_m)(j-\\widehat j).\n\\]\nThis identity identifies the two terms that must be controlled. The exact packet formulas in \\cref{sec:packet_error} then evaluate or bound \\(\\|j-\\widehat j\\|\\) from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus \\(|\\tau-\\widehat\\tau|^{1/p}\\) in finite \\(L^p\\), while every positive order truncated power packet is Lipschitz in its location.","The principal recovery theorem uses adjacent sample contrasts. Let\n\\[\nb_h=Mh+2\\varepsilon\n\\]\nbe the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If\n\\[\na_{\\min}>2b_h,\n\\]\nthen every threshold between \\(b_h\\) and \\(a_{\\min}-b_h\\) detects exactly the cells containing jumps. The jump locations are recovered within \\(h/2\\), and each amplitude error is at most \\(b_h\\). The probability theorem replaces the deterministic noise term by\n\\[\n2\\sigma\\sqrt{\\log(2n/\\delta)}.\n\\]\nTwo half shifted Haar grids provide a second recovery mechanism. Every point is at least one quarter of a cell away from the boundary of one of the grids, so the Haar coefficient gap can be made independent of dyadic alignment.","recovery theorem uses adjacent sample contrasts. Let\n\\[\nb_h=Mh+2\\varepsilon\n\\]\nbe the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If\n\\[\na_{\\min}>2b_h,\n\\]\nthen every threshold between \\(b_h\\) and \\(a_{\\min}-b_h\\) detects exactly the cells containing jumps. The jump locations are recovered within \\(h/2\\), and each amplitude error is at most \\(b_h\\). The probability theorem replaces the deterministic noise term by\n\\[\n2\\sigma\\sqrt{\\log(2n/\\delta)}.\n\\]\nTwo half shifted Haar grids provide a second recovery mechanism. Every point is at least one quarter of a cell away from the boundary of one of the grids, so the Haar coefficient gap can be made independent of dyadic alignment.","on\n\\]\nbe the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If\n\\[\na_{\\min}>2b_h,\n\\]\nthen every threshold between \\(b_h\\) and \\(a_{\\min}-b_h\\) detects exactly the cells containing jumps. The jump locations are recovered within \\(h/2\\), and each amplitude error is at most \\(b_h\\). The probability theorem replaces the deterministic noise term by\n\\[\n2\\sigma\\sqrt{\\log(2n/\\delta)}.\n\\]\nTwo half shifted Haar grids provide a second recovery mechanism. Every point is at least one quarter of a cell away from the boundary of one of the grids, so the Haar coefficient gap can be made independent of dyadic alignment.","\\midrule\n\\(L\\leq\\ZZ^d\\) & a finite index sublattice","\\(R\\) & a set of representatives for \\(\\ZZ^d/L\\)","\\(P_r\\) & the coordinate projector onto the coset \\(r+L\\)","\\(\\Acal_L,\\Scal_L\\) & lattice analysis and interleaving synthesis","\\(E,R\\) & filter bank analysis and synthesis symbols","\\(Q_m\\) & a bounded linear approximation operator acting on the smooth part","\\(j,\\widehat j\\) & the true and estimated singular packets","\\(H_\\tau\\) & the step packet \\(\\one_{[\\tau,1]}\\)","\\(\\phi_{m,\\tau}\\) & the truncated power packet \\((t-\\tau)_+^m/m!\\)","\\(h\\) & the grid spacing in the recovery theorems","Let \\(1\\leq p\\leq\\infty\\). For a countable set \\(G\\), write \\(\\ell^p(G)\\) for the usual complex sequence space. For an integer \\(d\\geq1\\), when \\(G=\\ZZ^d\\), translation by \\(a\\in\\ZZ^d\\) is\n\\[\n(T_ax)(n)=x(n-a).\n\\]\nLet \\(\\delta_0\\) be the impulse at the origin.","\\begin{theorem}[Impulse generation]\\label{thm:impulse_generation}\nFor \\(1\\leq p<\\infty\\),\n\\[\n\\overline{\\Span\\{T_a\\delta_0:a\\in\\ZZ^d\\}}^{\\ell^p}\n=\n\\ell^p(\\ZZ^d).\n\\]\nThe same statement holds in \\(c_0(\\ZZ^d)\\) with the supremum norm. In \\(\\ell^\\infty(\\ZZ^d)\\), the closed span of the translation orbit is exactly \\(c_0(\\ZZ^d)\\).\n\\end{theorem}","\\begin{proof}\nThe translation orbit is the standard coordinate family \\((\\delta_a)_{a\\in\\ZZ^d}\\). Its finite span is the space of finitely supported sequences. Finitely supported sequences are dense in \\(\\ell^p\\) for finite \\(p\\) and in \\(c_0\\). Their closure in the supremum norm is \\(c_0\\), which is a proper closed subspace of \\(\\ell^\\infty\\).\n\\end{proof}","\\begin{remark}\\label{rem:infinity_boundary}\nThe distinction at \\(p=\\infty\\) is structural. The impulse generates every finitely supported signal algebraically, but it is not a topological cyclic seed for all bounded signals.\n\\end{remark}","\\begin{proposition}[Finite periodic signals]\\label{prop:finite_periodic}\nLet \\(G=\\ZZ^d/L\\), where \\(L\\leq\\ZZ^d\\) has finite index. The translation orbit of the impulse at the identity is a basis of \\(\\CC^G\\). In particular, one impulse generates every signal on the finite periodic lattice.\n\\end{proposition}","\\begin{proof}\nTranslation sends the impulse at the identity to the impulse at each element of \\(G\\). These impulses form the standard basis of \\(\\CC^G\\).\n\\end{proof}","Let \\(L\\leq\\ZZ^d\\) be a sublattice of finite index \\(q\\). Choose a set \\(R\\subset\\ZZ^d\\) containing one representative from every coset in \\(\\ZZ^d/L\\). Thus\n\\[\n\\ZZ^d=\\bigsqcup_{r\\in R}(r+L).\n\\]\nFor \\(x:\\ZZ^d\\to\\CC\\), its channel at \\(r\\) is the sequence on \\(L\\)\n\\[\nx_r(\\ell)=x(r+\\ell).\n\\]","\\begin{definition}[Lattice packet]\\label{def:lattice_packet}\nFor \\(r\\in R\\), define the coordinate projector\n\\[\n(P_rx)(n)=\\one_{r+L}(n)x(n).\n\\]\nThe family \\((P_r)_{r\\in R}\\) is called the packet system associated with \\(L\\) and \\(R\\).\n\\end{definition}","For finite \\(p\\), equip \\(\\bigoplus_{r\\in R}\\ell^p(L)\\) with\n\\[\n\\|(u_r)\\|^p=\\sum_{r\\in R}\\|u_r\\|_p^p.\n\\]\nFor \\(p=\\infty\\), use the maximum norm.","\\begin{theorem}[Exact lattice polyphase decomposition]\\label{thm:lattice_polyphase}\nFor every \\(1\\leq p\\leq\\infty\\), the analysis map\n\\[\n\\Acal_L:\\ell^p(\\ZZ^d)\\longrightarrow\\bigoplus_{r\\in R}\\ell^p(L),\n\\qquad\n\\Acal_Lx=(x_r)_{r\\in R},\n\\]\nis an isometric isomorphism. Its inverse is the interleaving map\n\\[\n\\Scal_L(u)(r+\\ell)=u_r(\\ell).\n\\]\nThe projectors satisfy\n\\[\nP_rP_s=\\delta_{rs}P_r,\n\\qquad\n\\sum_{r\\in R}P_r=I.\n\\]\n\\end{theorem}","\\begin{proof}\nEvery address has a unique representation \\(r+\\ell\\). For finite \\(p\\),\n\\[\n\\|x\\|_p^p\n=\n\\sum_{r\\in R}\\sum_{\\ell\\in L}|x(r+\\ell)|^p\n=\n\\sum_{r\\in R}\\|x_r\\|_p^p.\n\\]\nThe corresponding supremum identity is immediate when \\(p=\\infty\\). Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.\n\\end{proof}","\\begin{corollary}[One dimensional residue coordinates]\\label{cor:one_dimensional_polyphase}\nFor \\(L=N\\ZZ\\) and \\(R=\\{0,\\ldots,N-1\\}\\),\n\\[\nx_r(k)=x(Nk+r)\n\\]\nand\n\\[\n\\ell^p(\\ZZ)\\cong\\bigoplus_{r=0}^{N-1}\\ell^p(\\ZZ)\n\\]\nisometrically.\n\\end{corollary}","\\begin{proposition}[Translation covariance]\\label{prop:translation_covariance}\nFix \\(a\\in\\ZZ^d\\). For every \\(r\\in R\\), write uniquely\n\\[\nr-a=r_a+\\lambda_{a,r},\n\\qquad\nr_a\\in R,\n\\quad\n\\lambda_{a,r}\\in L.\n\\]\nThen\n\\[\n(\\Acal_LT_ax)_r(\\ell)\n=\nx_{r_a}(\\ell+\\lambda_{a,r}).\n\\]\nThus a fine translation permutes the lattice packets and applies an internal translation on \\(L\\).\n\\end{proposition}","\\begin{proof}\nEvaluate at \\(r+\\ell\\):\n\\[\n(T_ax)(r+\\ell)\n=x(r+\\ell-a)\n=x(r_a+\\ell+\\lambda_{a,r}).\n\\]\nThis is the stated channel formula.\n\\end{proof}","Let \\(L'\\leq L\\leq\\ZZ^d\\), with both indices finite. Choose representative systems \\(R\\) for \\(\\ZZ^d/L\\) and \\(R'\\) for \\(\\ZZ^d/L'\\). For \\(r\\in R\\), define\n\\[\n\\operatorname{Child}(r)\n=\n\\{s\\in R':s+L'\\subset r+L\\}.\n\\]","\\begin{theorem}[Exact lattice refinement]\\label{thm:lattice_refinement}\nFor every coarse packet,\n\\[\nP_r^{L}\n=\n\\sum_{s\\in\\operatorname{Child}(r)}P_s^{L'}.\n\\]\nThe fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.\n\\end{theorem}","\\begin{proof}\nThe coset \\(r+L\\) is the disjoint union of the \\(L'\\) cosets contained in it. Multiplying by the corresponding indicator functions gives the projector identity. A nested partition of a coset is independent of whether the intermediate level is inserted, which proves transitivity.\n\\end{proof}","\\begin{example}[Rectangular image phases]\\label{ex:image_phases}\nFor\n\\[\nL=M\\ZZ\\times N\\ZZ\\subset\\ZZ^2,\n\\]\none may take\n\\[\nR=\\{0,\\ldots,M-1\\}\\times\\{0,\\ldots,N-1\\}.\n\\]\nThe analysis map separates an image into \\(MN\\) exact phases. Refining \\((M,N)\\) to \\((M',N')\\), where \\(M\\mid M'\\) and \\(N\\mid N'\\), splits every coarse phase into\n\\[\n\\frac{M'}M\\frac{N'}N\n\\]\nfine phases.\n\\end{example}","Let \\(h\\in\\ell^1(\\ZZ^d)\\) and \\(x\\in\\ell^p(\\ZZ^d)\\), with \\(1\\leq p\\leq\\infty\\), and define\n\\[\n(Hx)(n)\n=(h*x)(n)\n=\\sum_{q\\in\\ZZ^d}h(q)x(n-q).\n\\]\nYoung's inequality gives\n\\[\n\\|Hx\\|_p\\leq\\|h\\|_1\\|x\\|_p\n\\]\nfor every \\(1\\leq p\\leq\\infty\\).","\\begin{definition}[Packet transfer kernel]\\label{def:packet_transfer_kernel}\nFor output residue \\(r\\in R\\) and input residue \\(s\\in R\\), define a kernel on \\(L\\) by\n\\[\nh_{rs}(m)=h(r-s+m),\n\\qquad\nm\\in L.\n\\]\n\\end{definition}","\\begin{theorem}[Exact convolution transfer law]\\label{thm:convolution_transfer}\nLet \\(y=h*x\\). Then\n\\[\ny_r(\\ell)\n=\n\\sum_{s\\in R}\\sum_{m\\in L}h_{rs}(m)x_s(\\ell-m).\n\\]\nEquivalently,\n\\[\n\\Acal_LH\\Acal_L^{-1}\n=\n(H_{rs})_{r,s\\in R},\n\\]\nwhere \\(H_{rs}\\) is convolution on \\(L\\) by \\(h_{rs}\\).\n\\end{theorem}","\\begin{proof}\nEvaluate the output at \\(r+\\ell\\). Write every input address as \\(s+\\ell'\\). The convolution offset is\n\\[\n(r+\\ell)-(s+\\ell')\n=(r-s)+(\\ell-\\ell').\n\\]\nSetting \\(m=\\ell-\\ell'\\) gives the formula.\n\\end{proof}","\\begin{proposition}[Transfer norm control]\\label{prop:transfer_norm_control}\nFor every output residue \\(r\\),\n\\[\n\\sum_{s\\in R}\\|h_{rs}\\|_{\\ell^1(L)}\n=\n\\|h\\|_{\\ell^1(\\ZZ^d)}.\n\\]\nThe packet transfer matrix acts on the polyphase direct sum with norm at most \\(\\|h\\|_1\\) on every \\(\\ell^p\\) space.\n\\end{proposition}","\\begin{proof}\nFor fixed \\(r\\), every \\(q\\in\\ZZ^d\\) has a unique representation \\(q=r-s+m\\) with \\(s\\in R\\) and \\(m\\in L\\). This proves the equality of the absolute sums. The operator norm bound also follows by conjugating Young's inequality through the isometric analysis map.\n\\end{proof}","\\begin{proposition}[Composition of transfer matrices]\\label{prop:transfer_composition}\nIf \\(G\\) and \\(H\\) are convolution operators with packet matrices \\((G_{rs})\\) and \\((H_{rs})\\), then the packet matrix of \\(GH\\) is\n\\[\n(GH)_{rt}\n=\n\\sum_{s\\in R}G_{rs}H_{st}.\n\\]\nIn one-dimensional transform notation, the corresponding matrix symbols multiply pointwise.\n\\end{proposition}","\\begin{proof}\nConjugation by \\(\\Acal_L\\) preserves composition:\n\\[\n\\Acal_LGH\\Acal_L^{-1}\n=\n(\\Acal_LG\\Acal_L^{-1})(\\Acal_LH\\Acal_L^{-1}).\n\\]\nReading the \\((r,t)\\) block gives the formula.\n\\end{proof}","For \\(L=N\\ZZ\\) with \\(N\\geq1\\), first take \\(|z|=1\\). The assumption \\(h\\in\\ell^1(\\ZZ)\\) makes the following symbol series absolutely convergent. Define\n\\[\nH_{ks}(z)=\\sum_{m\\in\\ZZ}h(Nm+k-s)z^{-m}.\n\\]\nFor \\(x\\in\\ell^1(\\ZZ)\\), the channel transforms converge absolutely on the unit circle and satisfy\n\\[\n\\bm Y(z)=\\bm H_N(z)\\bm X(z).\n\\]\nFor \\(x\\in\\ell^2(\\ZZ)\\), the same identity holds almost everywhere in the \\(L^2\\) Fourier sense. For finite impulse-response kernels it also holds at nonzero \\(z\\) wherever the input channel transforms are weighted absolutely summable. This is the classical polyphase matrix. It is the exact block matrix of the original convolution operator.","ZZ\\) with \\(N\\geq1\\), first take \\(|z|=1\\). The assumption \\(h\\in\\ell^1(\\ZZ)\\) makes the following symbol series absolutely convergent. Define\n\\[\nH_{ks}(z)=\\sum_{m\\in\\ZZ}h(Nm+k-s)z^{-m}.\n\\]\nFor \\(x\\in\\ell^1(\\ZZ)\\), the channel transforms converge absolutely on the unit circle and satisfy\n\\[\n\\bm Y(z)=\\bm H_N(z)\\bm X(z).\n\\]\nFor \\(x\\in\\ell^2(\\ZZ)\\), the same identity holds almost everywhere in the \\(L^2\\) Fourier sense. For finite impulse-response kernels it also holds at nonzero \\(z\\) wherever the input channel transforms are weighted absolutely summable. This is the classical polyphase matrix. It is the exact block matrix of the original convolution operator.","the following symbol series absolutely convergent. Define\n\\[\nH_{ks}(z)=\\sum_{m\\in\\ZZ}h(Nm+k-s)z^{-m}.\n\\]\nFor \\(x\\in\\ell^1(\\ZZ)\\), the channel transforms converge absolutely on the unit circle and satisfy\n\\[\n\\bm Y(z)=\\bm H_N(z)\\bm X(z).\n\\]\nFor \\(x\\in\\ell^2(\\ZZ)\\), the same identity holds almost everywhere in the \\(L^2\\) Fourier sense. For finite impulse-response kernels it also holds at nonzero \\(z\\) wherever the input channel transforms are weighted absolutely summable. This is the classical polyphase matrix. It is the exact block matrix of the original convolution operator.","\\begin{corollary}[Finite impulse response closure]\\label{cor:fir_closure}\nIf \\(h\\) has finite support, every packet kernel \\(h_{ks}\\) has finite support. Matrix multiplication of polyphase symbols agrees with composition of the original finite impulse response operators.\n\\end{corollary}","\\begin{example}[A four phase finite impulse response operator]\\label{ex:four_phase_fir}\nLet\n\\[\nh(0)=1,\n\\quad h(1)=2,\n\\quad h(2)=3,\n\\quad h(3)=4,\n\\quad h(4)=5,\n\\]\nwith all other taps zero. For four residue channels, the transfer symbol is\n\\[\n\\bm H_4(z)\n=\n\\begin{pmatrix}\n1+5z^{-1}&4z^{-1}&3z^{-1}&2z^{-1}","\\begin{example}[A four phase finite impulse response operator]\\label{ex:four_phase_fir}\nLet\n\\[\nh(0)=1,\n\\quad h(1)=2,\n\\quad h(2)=3,\n\\quad h(3)=4,\n\\quad h(4)=5,\n\\]\nwith all other taps zero. For four residue channels, the transfer symbol is\n\\[\n\\bm H_4(z)\n=\n\\begin{pmatrix}\n1+5z^{-1}&4z^{-1}&3z^{-1}&2z^{-1}\\\\\n2&1+5z^{-1}&4z^{-1}&3z^{-1}\\\\\n3&2&1+5z^{-1}&4z^{-1}\\\\\n4&3&2&1+5z^{-1}\n\\end{pmatrix}.\n\\]\nThe delay term records the tap that crosses a coarse cell boundary. No approximation is introduced by the matrix representation.\n\\end{example}","Throughout this section the banks act boundedly on the relevant polyphase \\(\\ell^2\\) spaces. A finite channel analysis bank acts in Fourier coordinates by a matrix \\(E(z)\\). A synthesis bank acts by a matrix \\(R(z)\\). The matrices may be square or rectangular. Their entries are Laurent polynomials for finite filters and bounded measurable functions for general stable filters.","\\begin{theorem}[Perfect reconstruction criterion]\\label{thm:perfect_reconstruction}\nLet \\(E(z)\\) be an \\(M\\) by \\(N\\) analysis matrix and let \\(R(z)\\) be an \\(N\\) by \\(M\\) synthesis matrix. The bank reconstructs every input with a delay of \\(d\\) coarse channel samples, equivalent to \\(Nd\\) original samples in the one-dimensional \\(N\\)-phase convention, exactly if and only if\n\\[\nR(z)E(z)=z^{-d}I_N\n\\]\nfor almost every point on the unit circle. A nonzero scalar factor gives exact reconstruction up to gain and delay.\n\\end{theorem}","\\begin{proof}\nThe channel Fourier transform turns analysis followed by synthesis into multiplication by \\(R(z)E(z)\\). If the matrix identity holds, inverse transformation gives the delayed input. Conversely, if every input is reconstructed with that delay, the difference matrix annihilates every vector valued Fourier datum and must vanish almost everywhere.\n\\end{proof}","\\begin{theorem}[Quantitative imperfect reconstruction]\\label{thm:imperfect_reconstruction}\nDefine the delay compensated defect\n\\[\nD(\\e^{\\ii\\omega})\n=\n\\e^{\\ii d\\omega}R(\\e^{\\ii\\omega})E(\\e^{\\ii\\omega})-I_N.\n\\]\nIf\n\\[\n\\varepsilon_{\\mathrm{rec}}\n=\n\\esssup_{\\omega}\\|D(\\e^{\\ii\\omega})\\|_{\\mathrm{op}}<\\infty,\n\\]\nthen the delay compensated reconstruction \\(\\widetilde x\\) satisfies\n\\[\n\\|\\widetilde x-x\\|_2\n\\leq\n\\varepsilon_{\\mathrm{rec}}\\|x\\|_2.\n\\]\nThe constant is sharp as an operator norm.\n\\end{theorem}","\\begin{proof}\nIn polyphase Fourier coordinates, the error is \\(D(\\e^{\\ii\\omega})\\bm X(\\e^{\\ii\\omega})\\). Pointwise multiplication and Plancherel's theorem give the upper bound. The operator norm of a measurable matrix multiplier is its essential supremum, which proves sharpness.\n\\end{proof}","\\begin{example}[A reconstruction certificate]\\label{ex:reconstruction_certificate}\nSuppose a two channel bank has delay compensated defect\n\\[\nD(\\e^{\\ii\\omega})\n=\n\\begin{pmatrix}\n0.01\\cos\\omega&0\\\\\n0&0.02\\sin\\omega\n\\end{pmatrix}.\n\\]\nThen\n\\[\n\\esssup_\\omega\\|D(\\e^{\\ii\\omega})\\|_{\\mathrm{op}}\n=0.02.\n\\]\nEvery input therefore has relative reconstruction error at most two percent in \\(\\ell^2\\). The estimate is independent of the input length.\n\\end{example}","0&0.02\\sin\\omega\n\\end{pmatrix}.\n\\]\nThen\n\\[\n\\esssup_\\omega\\|D(\\e^{\\ii\\omega})\\|_{\\mathrm{op}}\n=0.02.\n\\]\nEvery input therefore has relative reconstruction error at most two percent in \\(\\ell^2\\). The estimate is independent of the input length.\n\\end{example}","\\begin{theorem}[Stability under filter perturbation]\\label{thm:filter_perturbation}\nAssume \\(RE=z^{-d}I_N\\). Let\n\\[\n\\widetilde E=E+\\Delta E,\n\\qquad\n\\widetilde R=R+\\Delta R.\n\\]\nSet\n\\[\n\\begin{aligned}\ne_0&=\\esssup\\|E\\|_{\\mathrm{op}},\n& r_0&=\\esssup\\|R\\|_{\\mathrm{op}},","\\begin{theorem}[Stability under filter perturbation]\\label{thm:filter_perturbation}\nAssume \\(RE=z^{-d}I_N\\). Let\n\\[\n\\widetilde E=E+\\Delta E,\n\\qquad\n\\widetilde R=R+\\Delta R.\n\\]\nSet\n\\[\n\\begin{aligned}\ne_0&=\\esssup\\|E\\|_{\\mathrm{op}},\n& r_0&=\\esssup\\|R\\|_{\\mathrm{op}},\\\\\n\\delta_E&=\\esssup\\|\\Delta E\\|_{\\mathrm{op}},\n& \\delta_R&=\\esssup\\|\\Delta R\\|_{\\mathrm{op}}.\n\\end{aligned}\n\\]\nThen the delay compensated reconstruction defect obeys\n\\[\n\\varepsilon_{\\mathrm{rec}}\n\\leq\nr_0\\delta_E+e_0\\delta_R+\\delta_R\\delta_E.\n\\]\n\\end{theorem}","\\delta_E&=\\esssup\\|\\Delta E\\|_{\\mathrm{op}},\n& \\delta_R&=\\esssup\\|\\Delta R\\|_{\\mathrm{op}}.\n\\end{aligned}\n\\]\nThen the delay compensated reconstruction defect obeys\n\\[\n\\varepsilon_{\\mathrm{rec}}\n\\leq\nr_0\\delta_E+e_0\\delta_R+\\delta_R\\delta_E.\n\\]\n\\end{theorem}","\\begin{proof}\nExpand\n\\[\n\\widetilde R\\widetilde E-RE\n=\nR\\Delta E+\\Delta R E+\\Delta R\\Delta E.\n\\]\nTake pointwise operator norms, then the essential supremum. Multiplication by the delay factor has unit modulus.\n\\end{proof}","\\begin{theorem}[Frame bounds from the polyphase symbol]\\label{thm:frame_bounds}\nAssume that\n\\[\nA I_N\n\\leq\nE(\\e^{\\ii\\omega})^*E(\\e^{\\ii\\omega})\n\\leq\nB I_N\n\\]\nfor almost every \\(\\omega\\), where \\(0<A\\leq B<\\infty\\). Then the analysis bank satisfies\n\\[\nA\\|x\\|_2^2\n\\leq\n\\|\\Acal_E x\\|_2^2\n\\leq\nB\\|x\\|_2^2.\n\\]\nThe pointwise matrix\n\\[\nR_{\\rm can}(\\e^{\\ii\\omega})\n=\n(E^*E)^{-1}E^*\n\\]\nis a bounded left inverse and gives the canonical dual synthesis bank.\n\\end{theorem}","\\begin{proof}\nApply the matrix inequality to the polyphase Fourier vector at every frequency and integrate. Plancherel's theorem gives the frame inequalities. The lower bound makes \\(E^*E\\) invertible almost everywhere, and direct multiplication gives \\(R_{\\rm can}E=I_N\\). The same spectral bounds show that the canonical dual is bounded.\n\\end{proof}","\\begin{corollary}[Coefficient noise under canonical synthesis]\\label{cor:canonical_noise}\nUnder the hypotheses of \\cref{thm:frame_bounds}, suppose the observed coefficient vector is\n\\[\nc=\\Acal_Ex+\\nu.\n\\]\nCanonical synthesis gives\n\\[\n\\widehat x=R_{\\mathrm{can}}c\n\\]\nand\n\\[\n\\|\\widehat x-x\\|_2\n\\leq\n\\frac1{\\sqrt A}\\|\\nu\\|_2.\n\\]\n\\end{corollary}","\\begin{proof}\nThe smallest singular value of \\(E\\) is at least \\(\\sqrt A\\), so the pointwise norm of its pseudoinverse is at most \\(1/\\sqrt A\\). Apply Plancherel's theorem to \\(R_{\\mathrm{can}}\\nu\\).\n\\end{proof}","\\begin{theorem}[Perturbation of frame bounds]\\label{thm:frame_perturbation}\nUnder the hypotheses of \\cref{thm:frame_bounds}, let \\(\\widetilde E=E+\\Delta E\\) and suppose\n\\[\n\\delta=\\esssup_\\omega\\|\\Delta E(\\e^{\\ii\\omega})\\|_{\\mathrm{op}}<\\sqrt A.\n\\]\nThen \\(\\widetilde E\\) is a frame analysis bank with bounds\n\\[\n(\\sqrt A-\\delta)^2\n\\quad\\text{and}\\quad\n(\\sqrt B+\\delta)^2.\n\\]\n\\end{theorem}","\\begin{proof}\nFor every vector \\(v\\),\n\\[\n\\|\\widetilde Ev\\|\n\\geq\n\\|Ev\\|-\\|\\Delta Ev\\|\n\\geq\n(\\sqrt A-\\delta)\\|v\\|.\n\\]\nThe upper bound follows from\n\\[\n\\|\\widetilde Ev\\|\n\\leq\n(\\sqrt B+\\delta)\\|v\\|.\n\\]\nSquare the inequalities and integrate in frequency.\n\\end{proof}","\\begin{remark}\\label{rem:canonical_dual_support}\nEven when \\(E\\) is a Laurent polynomial matrix, the canonical dual need not have finite impulse response. Finite support of both analysis and synthesis is an additional algebraic condition.\n\\end{remark}","\\begin{corollary}[Paraunitary bank]\\label{cor:paraunitary}\nIf \\(E(z)^*E(z)=I_N\\) on the unit circle, then analysis is an isometry and synthesis may be chosen as \\(R(z)=E(z)^*\\).\n\\end{corollary}","\\begin{proof}\nUse \\cref{thm:frame_bounds} with \\(A=B=1\\), then multiply the matrices.\n\\end{proof}","The Haar matrix is\n\\[\nE_H\n=\n\\frac1{\\sqrt2}\n\\begin{pmatrix}\n1&1\\\\\n1&-1\n\\end{pmatrix}.\n\\]\nFor even and odd samples \\(e(n)=x(2n)\\) and \\(o(n)=x(2n+1)\\), the low and high coordinates are\n\\[\na(n)=\\frac{e(n)+o(n)}{\\sqrt2},\n\\qquad\nd(n)=\\frac{e(n)-o(n)}{\\sqrt2}.\n\\]","1&-1\n\\end{pmatrix}.\n\\]\nFor even and odd samples \\(e(n)=x(2n)\\) and \\(o(n)=x(2n+1)\\), the low and high coordinates are\n\\[\na(n)=\\frac{e(n)+o(n)}{\\sqrt2},\n\\qquad\nd(n)=\\frac{e(n)-o(n)}{\\sqrt2}.\n\\]","\\begin{corollary}[Haar reconstruction and energy]\\label{cor:haar_reconstruction}\nThe Haar bank reconstructs exactly and\n\\[\n\\|x\\|_2^2=\\|a\\|_2^2+\\|d\\|_2^2.\n\\]\n\\end{corollary}","\\begin{proof}\nThe matrix \\(E_H\\) is unitary.\n\\end{proof}","Let \\(P(z)\\) and \\(Q(z)\\) be Laurent polynomials. Define\n\\[\nL_P(z)\n=\n\\begin{pmatrix}\n1&0","Let \\(P(z)\\) and \\(Q(z)\\) be Laurent polynomials. Define\n\\[\nL_P(z)\n=\n\\begin{pmatrix}\n1&0\\\\\nP(z)&1\n\\end{pmatrix},\n\\qquad\nU_Q(z)\n=\n\\begin{pmatrix}\n1&Q(z)\\\\\n0&1\n\\end{pmatrix}.\n\\]","\\begin{theorem}[Exact lifting update]\\label{thm:lifting_update}\nThe lifting matrices are invertible with\n\\[\nL_P^{-1}=L_{-P},\n\\qquad\nU_Q^{-1}=U_{-Q}.\n\\]\nIf \\(R_0E_0=I\\), then every finite product \\(V\\) of lifting matrices gives another exact bank by\n\\[\nE=VE_0,\n\\qquad\nR=R_0V^{-1}.\n\\]\nEvery lifting step has determinant one.\n\\end{theorem}","\\begin{proof}\nDirect multiplication proves the inverse formulas and the determinant statement. The reconstructed product is\n\\[\nRE\n=R_0V^{-1}VE_0\n=R_0E_0\n=I.\n\\]\n\\end{proof}","\\begin{example}[A two neighbor predict and update transform]\\label{ex:predict_update}\nSplit a sequence into even and odd samples. Define\n\\[\nd(n)\n=o(n)-\\frac{e(n)+e(n+1)}2\n\\]\nand\n\\[\ns(n)\n=e(n)+\\frac{d(n-1)+d(n)}4.\n\\]\nThe inverse is obtained in reverse order:\n\\[\ne(n)\n=s(n)-\\frac{d(n-1)+d(n)}4,\n\\]\nthen\n\\[\no(n)\n=d(n)+\\frac{e(n)+e(n+1)}2.\n\\]\nThus reconstruction is exact over every coefficient field in which two is invertible. This is a standard compactly supported biorthogonal lifting pattern \\cite{CohenDaubechiesFeauveau1992,Sweldens1996,DaubechiesSweldens1998}.\n\\end{example}","\\begin{definition}[Unitary packet tree]\\label{def:unitary_packet_tree}\nA finite unitary packet tree consists of Hilbert spaces \\(\\Hcal_v\\) at its vertices and, at every internal vertex \\(v\\), a unitary map\n\\[\nU_v:\\Hcal_v\\longrightarrow\\bigoplus_{w\\in\\operatorname{child}(v)}\\Hcal_w.\n\\]\nAn admissible pruning is a set of vertices that meets every root to terminal path exactly once.\n\\end{definition}","\\begin{theorem}[Reconstruction on a pruned packet tree]\\label{thm:pruned_packet_tree}\nEvery admissible pruning \\(\\Lcal\\) defines a unitary analysis map\n\\[\n\\Acal_{\\Lcal}:\\Hcal_{\\rm root}\n\\longrightarrow\n\\bigoplus_{v\\in\\Lcal}\\Hcal_v.\n\\]\nIts inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular,\n\\[\n\\|x\\|^2\n=\n\\sum_{v\\in\\Lcal}\\|x_v\\|^2.\n\\]\n\\end{theorem}","\\begin{proof}\nProceed by induction on the depth of the retained subtree. If the root is retained, the statement is immediate. If the root is split, apply the unitary map \\(U_{\\rm root}\\), then apply the induction hypothesis independently in each child subtree. A direct sum of unitary maps is unitary. Reversing the maps with adjoints gives synthesis.\n\\end{proof}","\\begin{corollary}[Coefficient perturbations in a unitary tree]\\label{cor:unitary_tree_noise}\nLet \\(c=\\Acal_{\\Lcal}x\\) be the leaf coefficients of an admissible unitary pruning, and let \\(\\widehat c=c+e\\). If \\(\\widehat x\\) is synthesized from \\(\\widehat c\\), then\n\\[\n\\|\\widehat x-x\\|=\\|e\\|.\n\\]\nThus coefficient quantization and coefficient noise are neither amplified nor attenuated by exact unitary synthesis.\n\\end{corollary}","\\begin{proof}\nThe synthesis map is unitary by \\cref{thm:pruned_packet_tree}. Hence\n\\[\n\\widehat x-x\n=\\Acal_{\\Lcal}^{-1}e\n\\]\nhas the same norm as \\(e\\).\n\\end{proof}","\\begin{proposition}[Bounds for nonunitary local splits]\\label{prop:nonunitary_tree_bounds}\nSuppose every internal node has an analysis map \\(U_v\\) and constants \\(0\\leq A_v\\leq B_v<\\infty\\) satisfying\n\\[\nA_v\\|x\\|^2\n\\leq\n\\sum_{w\\in\\operatorname{child}(v)}\\|(U_vx)_w\\|^2\n\\leq\nB_v\\|x\\|^2.\n\\]\nDefine recursively\n\\[\na(v)=\n\\begin{cases}\n1,&v\\text{ retained},","\\begin{proposition}[Bounds for nonunitary local splits]\\label{prop:nonunitary_tree_bounds}\nSuppose every internal node has an analysis map \\(U_v\\) and constants \\(0\\leq A_v\\leq B_v<\\infty\\) satisfying\n\\[\nA_v\\|x\\|^2\n\\leq\n\\sum_{w\\in\\operatorname{child}(v)}\\|(U_vx)_w\\|^2\n\\leq\nB_v\\|x\\|^2.\n\\]\nDefine recursively\n\\[\na(v)=\n\\begin{cases}\n1,&v\\text{ retained},\\\\\nA_v\\min_w a(w),&v\\text{ split},\n\\end{cases}\n\\]\nand\n\\[\nb(v)=\n\\begin{cases}\n1,&v\\text{ retained},","A_v\\min_w a(w),&v\\text{ split},\n\\end{cases}\n\\]\nand\n\\[\nb(v)=\n\\begin{cases}\n1,&v\\text{ retained},\\\\\nB_v\\max_w b(w),&v\\text{ split}.\n\\end{cases}\n\\]\nThen the leaf analysis of the pruned tree has bounds \\(a({\\rm root})\\) and \\(b({\\rm root})\\).\n\\end{proposition}","B_v\\max_w b(w),&v\\text{ split}.\n\\end{cases}\n\\]\nThen the leaf analysis of the pruned tree has bounds \\(a({\\rm root})\\) and \\(b({\\rm root})\\).\n\\end{proposition}","\\begin{proof}\nAssume the child subtree bounds have been proved. Then\n\\[\n\\sum_{w}\\sum_{\\ell\\in\\Lcal_w}\\|x_\\ell\\|^2\n\\geq\n\\min_w a(w)\n\\sum_w\\|(U_vx)_w\\|^2\n\\geq\nA_v\\min_w a(w)\\|x\\|^2.\n\\]\nThe upper bound follows in the same way with the maximum and \\(B_v\\). Induction completes the proof.\n\\end{proof}","Assign a real cost \\(c(v)\\) to retaining each node. For an admissible pruning \\(\\Lcal\\), let\n\\[\nC(\\Lcal)=\\sum_{v\\in\\Lcal}c(v).\n\\]\nThe cost may measure entropy, coefficient count, an approximation penalty, or any other additive local objective.","\\begin{theorem}[Exact additive best basis recursion]\\label{thm:best_basis}\nFor each vertex define\n\\[\nF(v)\n=\n\\min\\left\\{\n c(v),\n \\sum_{w\\in\\operatorname{child}(v)}F(w)\n\\right\\},\n\\]\nwith \\(F(v)=c(v)\\) at terminal vertices. Then \\(F(v)\\) is the minimum cost among all admissible prunings of the subtree rooted at \\(v\\). A minimizing pruning is recovered by retaining \\(v\\) when the first term is no larger than the second and splitting it otherwise.\n\\end{theorem}","\\begin{proof}\nEvery admissible pruning of the subtree has exactly two forms. It either retains \\(v\\), with cost \\(c(v)\\), or splits \\(v\\) and independently chooses an admissible pruning in every child subtree. The minimum in the second case is the sum of the child minima. This gives the recursion and the reconstruction rule. In an equality case both choices minimize, and the stated rule retains the node.\n\\end{proof}","\\begin{theorem}[Stability of the optimal cost]\\label{thm:best_basis_cost_stability}\nLet \\(c\\) and \\(\\widetilde c\\) be two node cost functions. For an admissible pruning \\(\\Lcal\\), set\n\\[\n\\varepsilon(\\Lcal)\n=\n\\sum_{v\\in\\Lcal}|c(v)-\\widetilde c(v)|\n\\]\nand let\n\\[\n\\varepsilon_*\n=\n\\max_{\\Lcal}\\varepsilon(\\Lcal).\n\\]\nIf \\(F_*\\) and \\(\\widetilde F_*\\) are the two optimal root costs, then\n\\[\n|F_*-\\widetilde F_*|\n\\leq\n\\varepsilon_*.\n\\]\n\\end{theorem}","\\begin{proof}\nFor every admissible pruning,\n\\[\n|C(\\Lcal)-\\widetilde C(\\Lcal)|\n\\leq\n\\varepsilon_*.\n\\]\nEvaluate \\(\\widetilde C\\) on a minimizer for \\(C\\) to obtain \\(\\widetilde F_*\\leq F_*+\\varepsilon_*\\). Reverse the roles to obtain \\(F_*\\leq\\widetilde F_*+\\varepsilon_*\\).\n\\end{proof}","\\begin{corollary}[Stability of the selected basis]\\label{cor:best_basis_selection_stability}\nAssume the optimal pruning \\(\\Lcal_*\\) for \\(c\\) is unique and at least one competing pruning exists. Define \\(g=\\min_{\\Lcal\\ne\\Lcal_*}(C(\\Lcal)-C(\\Lcal_*))>0\\). If\n\\[\n2\\varepsilon_*<g,\n\\]\nthen the optimal pruning for \\(\\widetilde c\\) is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.\n\\end{corollary}","\\begin{proof}\nThe cost of the original minimizer can increase by at most \\(\\varepsilon_*\\), while the cost of every competitor can decrease by at most \\(\\varepsilon_*\\). The original gap therefore remains positive.\n\\end{proof}","For \\(m\\geq0\\) and \\(0<\\tau<1\\), define\n\\[\n\\phi_{m,\\tau}(t)\n=\n\\frac{(t-\\tau)_+^m}{m!}.\n\\]\nThus \\(\\phi_{0,\\tau}=\\one_{[\\tau,1]}\\). On each side of \\(\\tau\\), the packet is a polynomial. Its derivative of order \\(m\\) has a unit jump at \\(\\tau\\).","\\begin{definition}[Sobolev step packet class]\\label{def:step_packet_class}\nFix \\(r\\geq1\\), \\(1\\leq p\\leq\\infty\\), and an integer \\(Q\\geq0\\). The class \\(\\Fcal_{r,p}(Q)\\) consists of functions\n\\[\nf=s+\\sum_{q=1}^{Q}a_q\\phi_{0,\\tau_q}\n\\]\nwith \\(s\\in W^{r,p}(0,1)\\) and\n\\[\n0<\\tau_1<\\cdots<\\tau_Q<1.\n\\]\nNo lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.\n\\end{definition}","\\begin{theorem}[Higher order singular packet decomposition]\\label{thm:higher_order_packet_decomposition}\nLet\n\\[\n0=\\tau_0<\\tau_1<\\cdots<\\tau_Q<\\tau_{Q+1}=1.\n\\]\nSuppose the restriction of \\(f\\) to every interval \\((\\tau_q,\\tau_{q+1})\\) belongs to \\(W^{r,p}\\), and choose the one-dimensional Sobolev representatives whose derivatives through order \\(r-1\\) have one-sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","egin{theorem}[Higher order singular packet decomposition]\\label{thm:higher_order_packet_decomposition}\nLet\n\\[\n0=\\tau_0<\\tau_1<\\cdots<\\tau_Q<\\tau_{Q+1}=1.\n\\]\nSuppose the restriction of \\(f\\) to every interval \\((\\tau_q,\\tau_{q+1})\\) belongs to \\(W^{r,p}\\), and choose the one-dimensional Sobolev representatives whose derivatives through order \\(r-1\\) have one-sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","_packet_decomposition}\nLet\n\\[\n0=\\tau_0<\\tau_1<\\cdots<\\tau_Q<\\tau_{Q+1}=1.\n\\]\nSuppose the restriction of \\(f\\) to every interval \\((\\tau_q,\\tau_{q+1})\\) belongs to \\(W^{r,p}\\), and choose the one-dimensional Sobolev representatives whose derivatives through order \\(r-1\\) have one-sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","Suppose the restriction of \\(f\\) to every interval \\((\\tau_q,\\tau_{q+1})\\) belongs to \\(W^{r,p}\\), and choose the one-dimensional Sobolev representatives whose derivatives through order \\(r-1\\) have one-sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","_{q+1})\\) belongs to \\(W^{r,p}\\), and choose the one-dimensional Sobolev representatives whose derivatives through order \\(r-1\\) have one-sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","e the one-dimensional Sobolev representatives whose derivatives through order \\(r-1\\) have one-sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","hose derivatives through order \\(r-1\\) have one-sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","sided traces at the endpoints. Define\n\\[\na_{q,m}\n=\n\\jump{f^{(m)}}_{\\tau_q}\n=\nf^{(m)}(\\tau_q+)-f^{(m)}(\\tau_q-)\n\\]\nand\n\\[\nj_f(t)\n=\n\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}a_{q,m}\\phi_{m,\\tau_q}(t).\n\\]\nThen\n\\[\ns_f=f-j_f\n\\]\nhas matching one-sided derivatives through order \\(r-1\\) at every breakpoint and belongs to \\(W^{r,p}(0,1)\\). Its weak derivative of order \\(r\\) equals the piecewise classical derivative \\(f^{(r)}\\) almost everywhere.\n\\end{theorem}","\\begin{proof}\nFor \\(0\\leq \\ell\\leq m\\), the classical derivative of \\(\\phi_{m,\\tau}\\) on the two open sides is\n\\[\n\\phi_{m,\\tau}^{(\\ell)}(t)\n=\n\\frac{(t-\\tau)_+^{m-\\ell}}{(m-\\ell)!}.\n\\]\nThis derivative has zero jump when \\(\\ell<m\\) and a unit jump when \\(\\ell=m\\). Derivatives of order larger than \\(m\\) vanish on both open sides. It follows that subtracting the term \\(a_{q,m}\\phi_{m,\\tau_q}\\) removes exactly the jump in the derivative of order \\(m\\) without changing the other classical trace jumps. Hence \\(s_f\\) has matching traces through order \\(r-1\\).","ssical derivative of \\(\\phi_{m,\\tau}\\) on the two open sides is\n\\[\n\\phi_{m,\\tau}^{(\\ell)}(t)\n=\n\\frac{(t-\\tau)_+^{m-\\ell}}{(m-\\ell)!}.\n\\]\nThis derivative has zero jump when \\(\\ell<m\\) and a unit jump when \\(\\ell=m\\). Derivatives of order larger than \\(m\\) vanish on both open sides. It follows that subtracting the term \\(a_{q,m}\\phi_{m,\\tau_q}\\) removes exactly the jump in the derivative of order \\(m\\) without changing the other classical trace jumps. Hence \\(s_f\\) has matching traces through order \\(r-1\\).","s\n\\[\n\\phi_{m,\\tau}^{(\\ell)}(t)\n=\n\\frac{(t-\\tau)_+^{m-\\ell}}{(m-\\ell)!}.\n\\]\nThis derivative has zero jump when \\(\\ell<m\\) and a unit jump when \\(\\ell=m\\). Derivatives of order larger than \\(m\\) vanish on both open sides. It follows that subtracting the term \\(a_{q,m}\\phi_{m,\\tau_q}\\) removes exactly the jump in the derivative of order \\(m\\) without changing the other classical trace jumps. Hence \\(s_f\\) has matching traces through order \\(r-1\\).","\\tau)_+^{m-\\ell}}{(m-\\ell)!}.\n\\]\nThis derivative has zero jump when \\(\\ell<m\\) and a unit jump when \\(\\ell=m\\). Derivatives of order larger than \\(m\\) vanish on both open sides. It follows that subtracting the term \\(a_{q,m}\\phi_{m,\\tau_q}\\) removes exactly the jump in the derivative of order \\(m\\) without changing the other classical trace jumps. Hence \\(s_f\\) has matching traces through order \\(r-1\\).","On each partition interval, \\(j_f\\) is a polynomial of degree at most \\(r-1\\), so its weak derivative of order \\(r\\) is zero there. A piecewise \\(W^{r,p}\\) function whose traces through order \\(r-1\\) match across every breakpoint belongs to \\(W^{r,p}(0,1)\\). Repeated integration by parts on the partition intervals shows that all boundary terms cancel. Thus the weak derivative of order \\(r\\) is the piecewise weak derivative of \\(f\\).\n\\end{proof}","f\\) is a polynomial of degree at most \\(r-1\\), so its weak derivative of order \\(r\\) is zero there. A piecewise \\(W^{r,p}\\) function whose traces through order \\(r-1\\) match across every breakpoint belongs to \\(W^{r,p}(0,1)\\). Repeated integration by parts on the partition intervals shows that all boundary terms cancel. Thus the weak derivative of order \\(r\\) is the piecewise weak derivative of \\(f\\).\n\\end{proof}","\\begin{corollary}[Uniqueness of the trace packet]\\label{cor:trace_packet_uniqueness}\nFix the breakpoints \\(\\tau_q\\). If\n\\[\nf=s+\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}b_{q,m}\\phi_{m,\\tau_q}\n\\]\nwith \\(s\\in W^{r,p}(0,1)\\), then\n\\[\nb_{q,m}=\\jump{f^{(m)}}_{\\tau_q}.\n\\]\nThe finite truncated power packet is therefore the unique packet supported at the declared breakpoints that removes all trace jumps through order \\(r-1\\).\n\\end{corollary}","gin{corollary}[Uniqueness of the trace packet]\\label{cor:trace_packet_uniqueness}\nFix the breakpoints \\(\\tau_q\\). If\n\\[\nf=s+\\sum_{q=1}^{Q}\\sum_{m=0}^{r-1}b_{q,m}\\phi_{m,\\tau_q}\n\\]\nwith \\(s\\in W^{r,p}(0,1)\\), then\n\\[\nb_{q,m}=\\jump{f^{(m)}}_{\\tau_q}.\n\\]\nThe finite truncated power packet is therefore the unique packet supported at the declared breakpoints that removes all trace jumps through order \\(r-1\\).\n\\end{corollary}","\\begin{proof}\nEvery representative in \\(W^{r,p}(0,1)\\) has derivatives through order \\(r-1\\) that admit absolutely continuous representatives on compact subintervals, with matching one-sided traces at each interior point. Taking the jump of the derivative of order \\(m\\) therefore removes the contribution of \\(s\\). The packet \\(\\phi_{m,\\tau_q}\\) has unit jump in its derivative of order \\(m\\), while all other declared packets have zero jump at that derivative order and breakpoint. Hence the remaining jump is exactly \\(b_{q,m}\\).\n\\end{proof}","ous representatives on compact subintervals, with matching one-sided traces at each interior point. Taking the jump of the derivative of order \\(m\\) therefore removes the contribution of \\(s\\). The packet \\(\\phi_{m,\\tau_q}\\) has unit jump in its derivative of order \\(m\\), while all other declared packets have zero jump at that derivative order and breakpoint. Hence the remaining jump is exactly \\(b_{q,m}\\).\n\\end{proof}","Let \\(X\\) be a normed linear space and let \\(Q:X\\to X\\) be bounded and linear. The next identity is elementary, but it is the central separation principle.","\\begin{theorem}[Exact retained packet identity]\\label{thm:retained_packet_identity}\nLet \\(f=s+j\\) in \\(X\\). Define\n\\[\n\\widetilde f=Qs+j.\n\\]\nThen\n\\[\n\\boxed{\nf-\\widetilde f=(I-Q)s.\n}\n\\]\nConsequently,\n\\[\n\\|f-\\widetilde f\\|=\\|s-Qs\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nSubtract:\n\\[\nf-\\widetilde f\n=s+j-Qs-j\n=(I-Q)s.\n\\]\n\\end{proof}","\\begin{remark}\\label{rem:retained_packet_boundary}\nThe theorem assumes that the packet \\(j\\) is known. It does not supply a detector. It also does not say that every representation of \\(f\\) as \\(s+j\\) is useful. Once a meaningful packet has been identified, its approximation error can be removed from the smooth approximation problem exactly.\n\\end{remark}","\\begin{corollary}[Sobolev rate after packet separation]\\label{cor:sobolev_rate}\nSuppose \\(Q_m\\) satisfies\n\\[\n\\|s-Q_ms\\|_{L^p}\n\\leq\nC2^{-mr}\\|s\\|_{W^{r,p}}\n\\]\nfor every \\(s\\in W^{r,p}(0,1)\\). If \\(f=s+j\\) and \\(j\\) is retained, then\n\\[\n\\|f-(Q_ms+j)\\|_{L^p}\n\\leq\nC2^{-mr}\\|s\\|_{W^{r,p}}.\n\\]\nThe same conclusion applies to the higher order decomposition in \\cref{thm:higher_order_packet_decomposition}.\n\\end{corollary}","\\begin{theorem}[Reconstruction with an estimated packet]\\label{thm:estimated_packet_reconstruction}\nLet \\(Q_m:X\\to X\\) be linear, let \\(f=s+j\\), let \\(\\widehat j\\) be an estimated packet, and define\n\\[\n\\widehat f_m\n=\nQ_m(f-\\widehat j)+\\widehat j.\n\\]\nThen\n\\[\n\\boxed{\nf-\\widehat f_m\n=\n(I-Q_m)s+(I-Q_m)(j-\\widehat j).\n}\n\\]\nIf \\(Q_m\\) is bounded, then\n\\[\n\\|f-\\widehat f_m\\|\n\\leq\n\\|s-Q_ms\\|\n+\n(1+\\|Q_m\\|)\\|j-\\widehat j\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nSince \\(f-\\widehat j=s+(j-\\widehat j)\\),\n\\[\n\\widehat f_m\n=\nQ_ms+Q_m(j-\\widehat j)+\\widehat j.\n\\]\nSubtract this expression from \\(s+j\\). The norm bound follows from\n\\[\n\\|(I-Q_m)u\\|\n\\leq\n(1+\\|Q_m\\|)\\|u\\|.\n\\]\n\\end{proof}","\\begin{theorem}[Exact scalar region formula]\\label{thm:scalar_region_formula}\nLet \\(E,F\\) be measurable subsets of a measure space, let \\(a,b\\in\\CC\\), and let \\(1\\leq p<\\infty\\). Then\n\\[\n\\begin{aligned}\n\\|a\\one_E-b\\one_F\\|_p^p\n={}&\n|a-b|^p\\mu(E\\cap F)\n+|a|^p\\mu(E\\setminus F)","\\begin{theorem}[Exact scalar region formula]\\label{thm:scalar_region_formula}\nLet \\(E,F\\) be measurable subsets of a measure space, let \\(a,b\\in\\CC\\), and let \\(1\\leq p<\\infty\\). Then\n\\[\n\\begin{aligned}\n\\|a\\one_E-b\\one_F\\|_p^p\n={}&\n|a-b|^p\\mu(E\\cap F)\n+|a|^p\\mu(E\\setminus F)\\\\\n&+|b|^p\\mu(F\\setminus E).\n\\end{aligned}\n\\]\nFor equal amplitudes,\n\\[\n\\|a\\one_E-a\\one_F\\|_p\n=\n|a|\\mu(E\\mathbin\\triangle F)^{1/p}.\n\\]\n\\end{theorem}","&+|b|^p\\mu(F\\setminus E).\n\\end{aligned}\n\\]\nFor equal amplitudes,\n\\[\n\\|a\\one_E-a\\one_F\\|_p\n=\n|a|\\mu(E\\mathbin\\triangle F)^{1/p}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe sets \\(E\\cap F\\), \\(E\\setminus F\\), and \\(F\\setminus E\\) are disjoint. The difference takes the values \\(a-b\\), \\(a\\), and \\(-b\\) on these three sets and is zero elsewhere. Integrating its \\(p\\)th power gives the formula.\n\\end{proof}","\\begin{corollary}[Location error for one step]\\label{cor:step_location_error}\nLet \\(0\\leq\\tau,\\widehat\\tau\\leq1\\) and \\(H_\\tau=\\one_{[\\tau,1]}\\). For \\(1\\leq p<\\infty\\),\n\\[\n\\|aH_\\tau-aH_{\\widehat\\tau}\\|_p\n=\n|a||\\tau-\\widehat\\tau|^{1/p}.\n\\]\nIn the supremum norm, the error equals \\(|a|\\) whenever \\(\\tau\\neq\\widehat\\tau\\).\n\\end{corollary}","\\begin{proof}\nThe symmetric difference of the two intervals has length \\(|\\tau-\\widehat\\tau|\\). The supremum statement follows because the two steps differ by \\(a\\) on a nonempty interval.\n\\end{proof}","\\begin{theorem}[Location stability for higher order packets]\\label{thm:higher_order_location}\nLet \\(m\\geq1\\), \\(1\\leq p<\\infty\\), and \\(0\\leq\\tau,\\widehat\\tau\\leq1\\). Then\n\\[\n\\|\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\\|_{L^p(0,1)}\n\\leq\n\\frac{|\\tau-\\widehat\\tau|}\n{(m-1)!((m-1)p+1)^{1/p}}.\n\\]\nFor \\(p=\\infty\\),\n\\[\n\\|\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\\|_\\infty\n\\leq\n\\frac{|\\tau-\\widehat\\tau|}{(m-1)!}.\n\\]\nThus location error is Lipschitz for every derivative jump packet of positive order, while a value jump has only the finite \\(L^p\\) modulus in \\cref{cor:step_location_error}.\n\\end{theorem}","y\\), and \\(0\\leq\\tau,\\widehat\\tau\\leq1\\). Then\n\\[\n\\|\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\\|_{L^p(0,1)}\n\\leq\n\\frac{|\\tau-\\widehat\\tau|}\n{(m-1)!((m-1)p+1)^{1/p}}.\n\\]\nFor \\(p=\\infty\\),\n\\[\n\\|\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\\|_\\infty\n\\leq\n\\frac{|\\tau-\\widehat\\tau|}{(m-1)!}.\n\\]\nThus location error is Lipschitz for every derivative jump packet of positive order, while a value jump has only the finite \\(L^p\\) modulus in \\cref{cor:step_location_error}.\n\\end{theorem}","\\begin{proof}\nAssume \\(\\tau<\\widehat\\tau\\). Differentiation with respect to the location parameter gives\n\\[\n\\frac{\\partial}{\\partial u}\\phi_{m,u}(t)\n=-\\phi_{m-1,u}(t)\n\\]\nfor almost every \\(t\\). Hence\n\\[\n\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\n=\n\\int_\\tau^{\\widehat\\tau}\\phi_{m-1,u}\\dd u.\n\\]\nMinkowski's inequality gives\n\\[\n\\|\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\\|_p\n\\leq\n|\\tau-\\widehat\\tau|\n\\sup_u\\|\\phi_{m-1,u}\\|_p.\n\\]\nThe supremum is bounded by the norm of \\(t^{m-1}/(m-1)!\\) on \\([0,1]\\), which is the stated constant. The supremum norm argument is identical.\n\\end{proof}","ion with respect to the location parameter gives\n\\[\n\\frac{\\partial}{\\partial u}\\phi_{m,u}(t)\n=-\\phi_{m-1,u}(t)\n\\]\nfor almost every \\(t\\). Hence\n\\[\n\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\n=\n\\int_\\tau^{\\widehat\\tau}\\phi_{m-1,u}\\dd u.\n\\]\nMinkowski's inequality gives\n\\[\n\\|\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\\|_p\n\\leq\n|\\tau-\\widehat\\tau|\n\\sup_u\\|\\phi_{m-1,u}\\|_p.\n\\]\nThe supremum is bounded by the norm of \\(t^{m-1}/(m-1)!\\) on \\([0,1]\\), which is the stated constant. The supremum norm argument is identical.\n\\end{proof}","e location parameter gives\n\\[\n\\frac{\\partial}{\\partial u}\\phi_{m,u}(t)\n=-\\phi_{m-1,u}(t)\n\\]\nfor almost every \\(t\\). Hence\n\\[\n\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\n=\n\\int_\\tau^{\\widehat\\tau}\\phi_{m-1,u}\\dd u.\n\\]\nMinkowski's inequality gives\n\\[\n\\|\\phi_{m,\\tau}-\\phi_{m,\\widehat\\tau}\\|_p\n\\leq\n|\\tau-\\widehat\\tau|\n\\sup_u\\|\\phi_{m-1,u}\\|_p.\n\\]\nThe supremum is bounded by the norm of \\(t^{m-1}/(m-1)!\\) on \\([0,1]\\), which is the stated constant. The supremum norm argument is identical.\n\\end{proof}","Let \\((E_i)_{i=0}^{I}\\) and \\((F_j)_{j=0}^{J}\\) be measurable partitions of the same domain, up to null sets. Let \\(a_i\\) and \\(b_j\\) be the corresponding values. The index zero may be used for a background region.","\\begin{theorem}[Overlap matrix formula]\\label{thm:overlap_matrix}\nDefine\n\\[\nM_{ij}=\\mu(E_i\\cap F_j).\n\\]\nFor \\(1\\leq p<\\infty\\),\n\\[\n\\left\\|\n\\sum_{i=0}^{I}a_i\\one_{E_i}\n-\n\\sum_{j=0}^{J}b_j\\one_{F_j}\n\\right\\|_p^p\n=\n\\sum_{i=0}^{I}\\sum_{j=0}^{J}|a_i-b_j|^pM_{ij}.\n\\]\nThe row sums of \\(M\\) are \\(\\mu(E_i)\\), and the column sums are \\(\\mu(F_j)\\).\n\\end{theorem}","\\begin{proof}\nThe intersections \\(E_i\\cap F_j\\) form a measurable partition. On such an intersection, the two piecewise constant functions have values \\(a_i\\) and \\(b_j\\). Integrate the constant error over every intersection. The marginal identities follow by summing the partition pieces in a fixed row or column.\n\\end{proof}","\\begin{corollary}[Background and packet error]\\label{cor:background_packet_error}\nSuppose the true and estimated packets use the same amplitude list \\((a_i)_{i=0}^{I}\\), but possibly different regions. Then\n\\[\n\\|j-\\widehat j\\|_p^p\n=\n\\sum_{i,k}|a_i-a_k|^p\\mu(E_i\\cap F_k).\n\\]\nOnly off-diagonal overlaps contribute.\n\\end{corollary}","\\begin{proof}\nApply \\cref{thm:overlap_matrix} with \\(b_k=a_k\\). The diagonal terms vanish.\n\\end{proof}","For a closed set \\(\\Gamma\\), write\n\\[\nN_\\delta(\\Gamma)\n=\n\\{x:\\dist(x,\\Gamma)<\\delta\\}.\n\\]","\\begin{definition}[Linear tube bound]\\label{def:linear_tube_bound}\nA boundary \\(\\Gamma\\) has a linear tube bound if there are constants \\(C_\\Gamma\\) and \\(\\delta_0>0\\) such that\n\\[\n\\mu(N_\\delta(\\Gamma))\n\\leq\nC_\\Gamma\\delta\n\\]\nfor \\(0<\\delta<\\delta_0\\).\n\\end{definition}","\\begin{theorem}[Geometric packet bound]\\label{thm:geometric_packet_bound}\nLet \\(E\\) be a region with boundary \\(\\Gamma\\), and suppose\n\\[\nE\\mathbin\\triangle F\n\\subseteq\nN_\\delta(\\Gamma).\n\\]\nIf \\(\\Gamma\\) has a linear tube bound and \\(0<\\delta<\\delta_0\\) lies in its stated range, then for \\(1\\leq p<\\infty\\),\n\\[\n\\|a\\one_E-a\\one_F\\|_p\n\\leq\n|a|(C_\\Gamma\\delta)^{1/p}.\n\\]\nIf amplitudes \\(a\\) and \\(b\\) differ, then\n\\[\n\\|a\\one_E-b\\one_F\\|_p\n\\leq\n|a-b|\\mu(E\\cap F)^{1/p}\n+\n(|a|+|b|)(C_\\Gamma\\delta)^{1/p}.\n\\]\n\\end{theorem}","m}[Geometric packet bound]\\label{thm:geometric_packet_bound}\nLet \\(E\\) be a region with boundary \\(\\Gamma\\), and suppose\n\\[\nE\\mathbin\\triangle F\n\\subseteq\nN_\\delta(\\Gamma).\n\\]\nIf \\(\\Gamma\\) has a linear tube bound and \\(0<\\delta<\\delta_0\\) lies in its stated range, then for \\(1\\leq p<\\infty\\),\n\\[\n\\|a\\one_E-a\\one_F\\|_p\n\\leq\n|a|(C_\\Gamma\\delta)^{1/p}.\n\\]\nIf amplitudes \\(a\\) and \\(b\\) differ, then\n\\[\n\\|a\\one_E-b\\one_F\\|_p\n\\leq\n|a-b|\\mu(E\\cap F)^{1/p}\n+\n(|a|+|b|)(C_\\Gamma\\delta)^{1/p}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe equal amplitude case follows from \\cref{thm:scalar_region_formula} and the tube inclusion. For unequal amplitudes, split the error into its value mismatch on \\(E\\cap F\\) and the two disagreement pieces. Apply the triangle inequality and the same tube estimate.\n\\end{proof}","\\begin{example}[A perturbed disk]\\label{ex:perturbed_disk}\nLet \\(E\\) and \\(F\\) be concentric disks of radii \\(r\\) and \\(r+\\delta\\) in \\(\\RR^2\\), with \\(\\delta>0\\). Then\n\\[\n\\mu(E\\mathbin\\triangle F)\n=\n\\pi\\bigl((r+\\delta)^2-r^2\\bigr)\n=\n2\\pi r\\delta+\\pi\\delta^2.\n\\]\nFor equal amplitude \\(a\\), the exact \\(L^2\\) packet error is\n\\[\n|a|\\sqrt{2\\pi r\\delta+\\pi\\delta^2}.\n\\]\nThis gives the expected square root dependence on boundary displacement.\n\\end{example}","ooth bound. The gap is a condition on the observation model, rather than an automatic consequence of packet coordinates.\n\\end{keyidea}\n\\begin{figure}[htbp]\\centering\n\\includegraphics[width=.93\\linewidth]{monograph_figures/jump_contrast_gap.pdf}\n\\caption{A sufficient threshold interval from exact rational contrast bounds. With the displayed model parameters, smooth and noise contrasts are at most $1/10$ and jump contrasts are at least $9/10$. Every threshold strictly between those values separates the cells under the sampling assumptions stated below. The figure does not assert an optimal threshold for other observation models.}\\label{monograph:fig:contrast}\n\\end{figure}","he observation model, rather than an automatic consequence of packet coordinates.\n\\end{keyidea}\n\\begin{figure}[htbp]\\centering\n\\includegraphics[width=.93\\linewidth]{monograph_figures/jump_contrast_gap.pdf}\n\\caption{A sufficient threshold interval from exact rational contrast bounds. With the displayed model parameters, smooth and noise contrasts are at most $1/10$ and jump contrasts are at least $9/10$. Every threshold strictly between those values separates the cells under the sampling assumptions stated below. The figure does not assert an optimal threshold for other observation models.}\\label{monograph:fig:contrast}\n\\end{figure}","Let\n\\[\nt_k=kh,\n\\qquad\nk=0,\\ldots,n,\n\\qquad\nh=\\frac1n.\n\\]\nAssume\n\\[\nf(t)\n=\ns(t)+\\sum_{q=1}^{Q}a_qH_{\\tau_q}(t),\n\\qquad\nH_\\tau(t)=\\one_{[\\tau,1]}(t),\n\\]\nwhere \\(s\\) is Lipschitz with constant \\(M\\), and \\(Q\\geq1\\) is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe\n\\[\ny_k=f(t_k)+\\eta_k.\n\\]\nDefine the adjacent contrast\n\\[\nD_k=y_{k+1}-y_k.\n\\]","t_k=kh,\n\\qquad\nk=0,\\ldots,n,\n\\qquad\nh=\\frac1n.\n\\]\nAssume\n\\[\nf(t)\n=\ns(t)+\\sum_{q=1}^{Q}a_qH_{\\tau_q}(t),\n\\qquad\nH_\\tau(t)=\\one_{[\\tau,1]}(t),\n\\]\nwhere \\(s\\) is Lipschitz with constant \\(M\\), and \\(Q\\geq1\\) is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe\n\\[\ny_k=f(t_k)+\\eta_k.\n\\]\nDefine the adjacent contrast\n\\[\nD_k=y_{k+1}-y_k.\n\\]","me\n\\[\nf(t)\n=\ns(t)+\\sum_{q=1}^{Q}a_qH_{\\tau_q}(t),\n\\qquad\nH_\\tau(t)=\\one_{[\\tau,1]}(t),\n\\]\nwhere \\(s\\) is Lipschitz with constant \\(M\\), and \\(Q\\geq1\\) is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe\n\\[\ny_k=f(t_k)+\\eta_k.\n\\]\nDefine the adjacent contrast\n\\[\nD_k=y_{k+1}-y_k.\n\\]","\\begin{lemma}[Contrast gap]\\label{lem:contrast_gap}\nAssume \\(|\\eta_k|\\leq\\varepsilon\\) and set\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nIf the cell \\((t_k,t_{k+1})\\) contains no jump, then\n\\[\n|D_k|\\leq b_h.\n\\]\nIf it contains one jump of amplitude \\(a_q\\), then\n\\[\n|D_k-a_q|\\leq b_h\n\\]\nand therefore\n\\[\n|D_k|\\geq |a_q|-b_h.\n\\]\n\\end{lemma}","\\begin{proof}\nIn a cell without a jump,\n\\[\nD_k=s(t_{k+1})-s(t_k)+\\eta_{k+1}-\\eta_k.\n\\]\nThe Lipschitz and noise bounds give \\(|D_k|\\leq Mh+2\\varepsilon\\). In a cell with one jump, the same formula has the additional term \\(a_q\\), which proves the second estimate.\n\\end{proof}","\\begin{theorem}[Deterministic jump cell recovery]\\label{thm:deterministic_jump_recovery}\nLet\n\\[\na_{\\min}=\\min_q|a_q|\n\\]\nand assume\n\\[\na_{\\min}>2b_h.\n\\]\nChoose a threshold \\(\\theta\\) satisfying\n\\[\nb_h<\\theta<a_{\\min}-b_h.\n\\]\nThen\n\\[\n\\widehat\\Kcal\n=\n\\{k:|D_k|>\\theta\\}\n\\]\nis exactly the set of grid cells containing jumps. If \\(\\widehat\\tau_q\\) is the midpoint of the detected cell and \\(\\widehat a_q=D_k\\), then after matching by order,\n\\[\n|\\widehat\\tau_q-\\tau_q|\n\\leq\n\\frac h2\n\\]\nand\n\\[\n|\\widehat a_q-a_q|\n\\leq\nb_h.\n\\]\n\\end{theorem}","l recovery]\\label{thm:deterministic_jump_recovery}\nLet\n\\[\na_{\\min}=\\min_q|a_q|\n\\]\nand assume\n\\[\na_{\\min}>2b_h.\n\\]\nChoose a threshold \\(\\theta\\) satisfying\n\\[\nb_h<\\theta<a_{\\min}-b_h.\n\\]\nThen\n\\[\n\\widehat\\Kcal\n=\n\\{k:|D_k|>\\theta\\}\n\\]\nis exactly the set of grid cells containing jumps. If \\(\\widehat\\tau_q\\) is the midpoint of the detected cell and \\(\\widehat a_q=D_k\\), then after matching by order,\n\\[\n|\\widehat\\tau_q-\\tau_q|\n\\leq\n\\frac h2\n\\]\nand\n\\[\n|\\widehat a_q-a_q|\n\\leq\nb_h.\n\\]\n\\end{theorem}","y}\nLet\n\\[\na_{\\min}=\\min_q|a_q|\n\\]\nand assume\n\\[\na_{\\min}>2b_h.\n\\]\nChoose a threshold \\(\\theta\\) satisfying\n\\[\nb_h<\\theta<a_{\\min}-b_h.\n\\]\nThen\n\\[\n\\widehat\\Kcal\n=\n\\{k:|D_k|>\\theta\\}\n\\]\nis exactly the set of grid cells containing jumps. If \\(\\widehat\\tau_q\\) is the midpoint of the detected cell and \\(\\widehat a_q=D_k\\), then after matching by order,\n\\[\n|\\widehat\\tau_q-\\tau_q|\n\\leq\n\\frac h2\n\\]\nand\n\\[\n|\\widehat a_q-a_q|\n\\leq\nb_h.\n\\]\n\\end{theorem}","\\begin{proof}\nBy \\cref{lem:contrast_gap}, every cell without a jump has contrast at most \\(b_h<\\theta\\). Every cell with a jump has contrast at least \\(a_{\\min}-b_h>\\theta\\). Hence the detected set is exact. The midpoint of a cell is within \\(h/2\\) of every point in that cell. The amplitude estimate is the second inequality in \\cref{lem:contrast_gap}.\n\\end{proof}","\\begin{corollary}[Jump count and sign recovery]\\label{cor:jump_count_sign}\nUnder the hypotheses of \\cref{thm:deterministic_jump_recovery}, the number of detected cells equals \\(Q\\), and\n\\[\n\\sgn(\\widehat a_q)=\\sgn(a_q)\n\\]\nfor every real valued jump.\n\\end{corollary}","\\begin{proof}\nExact cell recovery gives the jump count. Since\n\\[\n|\\widehat a_q-a_q|\\leq b_h<|a_q|,\n\\]\nthe amplitude estimate cannot cross zero.\n\\end{proof}","\\begin{corollary}[Packet reconstruction after detection]\\label{cor:reconstruction_after_detection}\nLet\n\\[\nj=\\sum_{q=1}^{Q}a_qH_{\\tau_q},\n\\qquad\n\\widehat j=\\sum_{q=1}^{Q}\\widehat a_qH_{\\widehat\\tau_q}.\n\\]\nUnder the hypotheses of \\cref{thm:deterministic_jump_recovery}, for \\(1\\leq p<\\infty\\),\n\\[\n\\|j-\\widehat j\\|_p\n\\leq\nQb_h\n+\n\\left(\n\\frac h2\\sum_{q=1}^{Q}|a_q|^p\n\\right)^{1/p}.\n\\]\nIf \\(Q_m\\) is bounded and linear and\n\\[\n\\widehat f_m=Q_m(f-\\widehat j)+\\widehat j,\n\\]\nthen\n\\[\n\\begin{aligned}\n\\|f-\\widehat f_m\\|_p\n\\leq{}&\n\\|s-Q_ms\\|_p","cket reconstruction after detection]\\label{cor:reconstruction_after_detection}\nLet\n\\[\nj=\\sum_{q=1}^{Q}a_qH_{\\tau_q},\n\\qquad\n\\widehat j=\\sum_{q=1}^{Q}\\widehat a_qH_{\\widehat\\tau_q}.\n\\]\nUnder the hypotheses of \\cref{thm:deterministic_jump_recovery}, for \\(1\\leq p<\\infty\\),\n\\[\n\\|j-\\widehat j\\|_p\n\\leq\nQb_h\n+\n\\left(\n\\frac h2\\sum_{q=1}^{Q}|a_q|^p\n\\right)^{1/p}.\n\\]\nIf \\(Q_m\\) is bounded and linear and\n\\[\n\\widehat f_m=Q_m(f-\\widehat j)+\\widehat j,\n\\]\nthen\n\\[\n\\begin{aligned}\n\\|f-\\widehat f_m\\|_p\n\\leq{}&\n\\|s-Q_ms\\|_p","ruction_after_detection}\nLet\n\\[\nj=\\sum_{q=1}^{Q}a_qH_{\\tau_q},\n\\qquad\n\\widehat j=\\sum_{q=1}^{Q}\\widehat a_qH_{\\widehat\\tau_q}.\n\\]\nUnder the hypotheses of \\cref{thm:deterministic_jump_recovery}, for \\(1\\leq p<\\infty\\),\n\\[\n\\|j-\\widehat j\\|_p\n\\leq\nQb_h\n+\n\\left(\n\\frac h2\\sum_{q=1}^{Q}|a_q|^p\n\\right)^{1/p}.\n\\]\nIf \\(Q_m\\) is bounded and linear and\n\\[\n\\widehat f_m=Q_m(f-\\widehat j)+\\widehat j,\n\\]\nthen\n\\[\n\\begin{aligned}\n\\|f-\\widehat f_m\\|_p\n\\leq{}&\n\\|s-Q_ms\\|_p\\\\\n&+(1+\\|Q_m\\|)\n\\left[\nQb_h+\n\\left(\n\\frac h2\\sum_q|a_q|^p\n\\right)^{1/p}\n\\right].\n\\end{aligned}\n\\]\n\\end{corollary}","\\begin{proof}\nWrite\n\\[\nj-\\widehat j\n=\n\\sum_q(a_q-\\widehat a_q)H_{\\widehat\\tau_q}\n+\n\\sum_q a_q(H_{\\tau_q}-H_{\\widehat\\tau_q}).\n\\]\nThe first sum has norm at most \\(\\sum_q|a_q-\\widehat a_q|\\leq Qb_h\\). The location error interval for the \\(q\\)th jump is contained in its detected grid cell and has length at most \\(h/2\\). Distinct detected cells are disjoint, so the location errors have disjoint supports. Their \\(p\\)th powers therefore add exactly, which gives the displayed location term. The reconstruction estimate follows from \\cref{thm:estimated_packet_reconstruction}.\n\\end{proof}","}\nWrite\n\\[\nj-\\widehat j\n=\n\\sum_q(a_q-\\widehat a_q)H_{\\widehat\\tau_q}\n+\n\\sum_q a_q(H_{\\tau_q}-H_{\\widehat\\tau_q}).\n\\]\nThe first sum has norm at most \\(\\sum_q|a_q-\\widehat a_q|\\leq Qb_h\\). The location error interval for the \\(q\\)th jump is contained in its detected grid cell and has length at most \\(h/2\\). Distinct detected cells are disjoint, so the location errors have disjoint supports. Their \\(p\\)th powers therefore add exactly, which gives the displayed location term. The reconstruction estimate follows from \\cref{thm:estimated_packet_reconstruction}.\n\\end{proof}","\\begin{remark}[Why finite \\(L^p\\) matters]\\label{rem:finite_p_recovery}\nA shifted step does not converge in the supremum norm until the location is exact. The finite \\(L^p\\) error does converge because the disagreement is confined to an interval whose length tends to zero.\n\\end{remark}","\\begin{corollary}[Deterministic consistency]\\label{cor:deterministic_consistency}\nAssume \\(h\\to0\\) along grids satisfying the preceding sampling assumptions, \\(\\varepsilon_h\\to0\\), the number of jumps is fixed, and the nonzero jump amplitudes and positive minimum separation remain fixed. Then the detected packet converges to the true packet in every finite \\(L^p\\) norm. If \\(Q_m\\) has a vanishing smooth approximation error and uniformly bounded norm, the complete reconstruction also converges in \\(L^p\\).\n\\end{corollary}","g grids satisfying the preceding sampling assumptions, \\(\\varepsilon_h\\to0\\), the number of jumps is fixed, and the nonzero jump amplitudes and positive minimum separation remain fixed. Then the detected packet converges to the true packet in every finite \\(L^p\\) norm. If \\(Q_m\\) has a vanishing smooth approximation error and uniformly bounded norm, the complete reconstruction also converges in \\(L^p\\).\n\\end{corollary}","\\begin{proof}\nThe gap condition holds for sufficiently small \\(h\\). Then \\(b_h\\to0\\), and the bound in \\cref{cor:reconstruction_after_detection} tends to zero.\n\\end{proof}","\\begin{example}[A numerical threshold certificate]\\label{ex:jump_threshold_certificate}\nTake \\(h=1/256\\), \\(M=0.5\\), and \\(\\varepsilon=0.002\\). Then\n\\[\nb_h\n=\\frac{0.5}{256}+0.004\n<0.00596.\n\\]\nIf every jump has magnitude at least \\(0.08\\), then \\(0.08>2b_h\\). Any threshold between \\(0.00596\\) and \\(0.07404\\) recovers the exact jump cells. The location error is at most\n\\[\n\\frac1{512}\n\\]\nand every amplitude error is less than \\(0.00596\\).\n\\end{example}","hreshold certificate]\\label{ex:jump_threshold_certificate}\nTake \\(h=1/256\\), \\(M=0.5\\), and \\(\\varepsilon=0.002\\). Then\n\\[\nb_h\n=\\frac{0.5}{256}+0.004\n<0.00596.\n\\]\nIf every jump has magnitude at least \\(0.08\\), then \\(0.08>2b_h\\). Any threshold between \\(0.00596\\) and \\(0.07404\\) recovers the exact jump cells. The location error is at most\n\\[\n\\frac1{512}\n\\]\nand every amplitude error is less than \\(0.00596\\).\n\\end{example}","Assume the noise variables are independent, centered, and \\(\\sigma\\) sub-Gaussian:\n\\[\n\\mathbb E\\e^{\\lambda\\eta_k}\n\\leq\n\\e^{\\sigma^2\\lambda^2/2}.\n\\]","\\begin{theorem}[Probability recovery bound]\\label{thm:probability_recovery}\nFix \\(0<\\delta<1\\) and define\n\\[\nt_{n,\\delta}\n=\n2\\sigma\n\\sqrt{\\log\\left(\\frac{2n}{\\delta}\\right)}\n\\]\nand\n\\[\nb_{h,\\delta}=Mh+t_{n,\\delta}.\n\\]\nWith probability at least \\(1-\\delta\\), every contrast satisfies the conclusions of \\cref{lem:contrast_gap} with \\(b_h\\) replaced by \\(b_{h,\\delta}\\).","If\n\\[\na_{\\min}>2b_{h,\\delta},\n\\]\nthen thresholding between \\(b_{h,\\delta}\\) and \\(a_{\\min}-b_{h,\\delta}\\) recovers every jump cell with probability at least \\(1-\\delta\\). The same location and amplitude bounds hold with \\(b_{h,\\delta}\\).\n\\end{theorem}","\\begin{proof}\nIf \\(\\sigma=0\\), every centered zero-sub-Gaussian noise variable vanishes almost surely, and the noise-free deterministic conclusion applies. For \\(\\sigma>0\\), the difference \\(\\eta_{k+1}-\\eta_k\\) is sub-Gaussian with variance proxy \\(2\\sigma^2\\), so\n\\[\n\\mathbb P\\bigl(|\\eta_{k+1}-\\eta_k|>t\\bigr)\n\\leq\n2\\exp\\left(-\\frac{t^2}{4\\sigma^2}\\right).\n\\]\nA union bound over the \\(n\\) cells gives\n\\[\n\\mathbb P\\left(\n\\max_k|\\eta_{k+1}-\\eta_k|>t_{n,\\delta}\n\\right)\n\\leq\n\\delta.\n\\]\nOn the complementary event, repeat the deterministic proof.\n\\end{proof}","in{proof}\nIf \\(\\sigma=0\\), every centered zero-sub-Gaussian noise variable vanishes almost surely, and the noise-free deterministic conclusion applies. For \\(\\sigma>0\\), the difference \\(\\eta_{k+1}-\\eta_k\\) is sub-Gaussian with variance proxy \\(2\\sigma^2\\), so\n\\[\n\\mathbb P\\bigl(|\\eta_{k+1}-\\eta_k|>t\\bigr)\n\\leq\n2\\exp\\left(-\\frac{t^2}{4\\sigma^2}\\right).\n\\]\nA union bound over the \\(n\\) cells gives\n\\[\n\\mathbb P\\left(\n\\max_k|\\eta_{k+1}-\\eta_k|>t_{n,\\delta}\n\\right)\n\\leq\n\\delta.\n\\]\nOn the complementary event, repeat the deterministic proof.\n\\end{proof}","\\begin{corollary}[Repeated measurements]\\label{cor:repeated_measurements}\nSuppose \\(R\\) observations are averaged at every sample location, with all noise variables independent across sites and repetitions and each centered and \\(\\sigma\\) sub-Gaussian. Then \\cref{thm:probability_recovery} holds with \\(\\sigma\\) replaced by \\(\\sigma/\\sqrt R\\). In particular, a sufficient recovery condition is\n\\[\na_{\\min}\n>\n2Mh+\n\\frac{4\\sigma}{\\sqrt R}\n\\sqrt{\\log\\left(\\frac{2n}{\\delta}\\right)}.\n\\]\n\\end{corollary}","\\begin{proof}\nThe average of \\(R\\) independent \\(\\sigma\\) sub-Gaussian variables is \\(\\sigma/\\sqrt R\\) sub-Gaussian. Substitute this parameter in \\cref{thm:probability_recovery}.\n\\end{proof}","\\begin{remark}\\label{rem:noise_scaling}\nThe displayed uniform noise certificate scales as \\(\\sigma\\sqrt{\\log n}\\). At fixed positive \\(\\sigma\\), it does not imply consistency of this detector. Decreasing the noise scale or averaging repeated measurements gives sufficient regimes. Other averaging schemes require their own analysis.\n\\end{remark}","\\begin{corollary}[Probability consistency]\\label{cor:probability_consistency}\nLet \\(\\sigma_n\\sqrt{\\log n}\\to0\\) and \\(h=1/n\\). For every fixed \\(0<\\delta<1\\), along grids satisfying the preceding sampling assumptions, all jump cells are recovered with probability at least \\(1-\\delta\\) for sufficiently large \\(n\\). For the fixed finite jump model above, along grids satisfying the preceding sampling assumptions, the packet error converges to zero in probability in every finite \\(L^p\\) norm.\n\\end{corollary}","\\(\\sigma_n\\sqrt{\\log n}\\to0\\) and \\(h=1/n\\). For every fixed \\(0<\\delta<1\\), along grids satisfying the preceding sampling assumptions, all jump cells are recovered with probability at least \\(1-\\delta\\) for sufficiently large \\(n\\). For the fixed finite jump model above, along grids satisfying the preceding sampling assumptions, the packet error converges to zero in probability in every finite \\(L^p\\) norm.\n\\end{corollary}","\\begin{proof}\nFor each fixed \\(\\delta>0\\), \\(b_{h,\\delta}\\to0\\). Apply \\cref{thm:probability_recovery,cor:reconstruction_after_detection}. For any error tolerance \\(\\epsilon>0\\), the high-probability bound is eventually smaller than \\(\\epsilon\\), so the limit superior of the probability that the packet error exceeds \\(\\epsilon\\) is at most \\(\\delta\\). Since \\(\\delta\\) is arbitrary, convergence is in probability.\n\\end{proof}","\\begin{proposition}[Vector jump recovery]\\label{prop:vector_jump_recovery}\nLet the values lie in a normed space \\(V\\). Assume\n\\[\n\\|s(t)-s(u)\\|_V\\leq M|t-u|,\n\\qquad\n\\|\\eta_k\\|_V\\leq\\varepsilon,\n\\]\nand let the jump vectors satisfy \\(\\|a_q\\|_V\\geq a_{\\min}\\). Thresholding \\(\\|D_k\\|_V\\) gives the same recovery theorem under the condition\n\\[\na_{\\min}>2(Mh+2\\varepsilon).\n\\]\n\\end{proposition}","The adjacent difference detector is local in the sample domain. Wavelet coefficients and higher differences give related multiscale packets. In the Haar subsections the signal is defined on \\(\\RR\\). Here the step convention is \\(H_\\tau(t)=\\one_{[\\tau,\\infty)}(t)\\), whose restriction to \\((0,1)\\) agrees with the preceding interval step. A chosen extension beyond the observation interval must use this half-line step convention, and the smooth extension must satisfy the stated Lipschitz hypothesis on every displayed wavelet support.","Let\n\\[\nI_{j,k}\n=\n[k2^{-j},(k+1)2^{-j})\n\\]\nand let \\(\\psi_{j,k}\\) be the normalized Haar wavelet supported on \\(I_{j,k}\\), positive on the left half and negative on the right half.","\\begin{theorem}[Smooth Haar coefficient bound]\\label{thm:smooth_haar_bound}\nIf \\(s\\) is Lipschitz with constant \\(M\\), then\n\\[\n|\\langle s,\\psi_{j,k}\\rangle|\n\\leq\n\\frac M4\\,2^{-3j/2}.\n\\]\n\\end{theorem}","\\begin{proof}\nLet \\(L=2^{-j}\\) and let \\(a\\) be the left endpoint of the support. Pair every point \\(t\\in[a,a+L/2]\\) with \\(t+L/2\\). Then\n\\[\n\\langle s,\\psi_{j,k}\\rangle\n=\nL^{-1/2}\n\\int_a^{a+L/2}\n\\overline{s(t)-s(t+L/2)}\\,\\dd t.\n\\]\nThe integrand has magnitude at most \\(ML/2\\), and the integration interval has length \\(L/2\\). The result is \\(ML^{3/2}/4\\).\n\\end{proof}","\\begin{theorem}[Exact Haar coefficient of a step]\\label{thm:haar_step_coefficient}\nLet \\(\\tau\\in I_{j,k}\\), let \\(L=2^{-j}\\), and write\n\\[\nu=\\frac{\\tau-kL}{L}\\in[0,1].\n\\]\nThen\n\\[\n\\left|\n\\left\\langle\n aH_\\tau,\\psi_{j,k}\n\\right\\rangle\n\\right|\n=\n|a|L^{1/2}\\min\\{u,1-u\\}.\n\\]\nEvery Haar coefficient whose support does not cross \\(\\tau\\) is zero.\n\\end{theorem}","\\begin{proof}\nAway from the jump, the step is constant on the support and the Haar wavelet has zero mean. When the jump lies in the support, integrate the positive and negative halves directly. If \\(u\\leq1/2\\), the magnitude is \\(|a|uL^{1/2}\\). If \\(u\\geq1/2\\), it is \\(|a|(1-u)L^{1/2}\\).\n\\end{proof}","\\begin{corollary}[Scale separation]\\label{cor:haar_scale_separation}\nSuppose \\(f=s+aH_\\tau+\\eta\\), where \\(s\\) is \\(M\\) Lipschitz and \\(\\|\\eta\\|_\\infty\\leq\\varepsilon\\). If the jump lies at relative distance at least \\(\\rho\\) from both ends of \\(I_{j,k}\\), then\n\\[\n|\\langle f,\\psi_{j,k}\\rangle|\n\\geq\n\\left(\n|a|\\rho\n-\\varepsilon\n-\\frac M4L\n\\right)L^{1/2}.\n\\]\nA wavelet support without a jump satisfies\n\\[\n|\\langle f,\\psi_{j,k}\\rangle|\n\\leq\n\\left(\n\\varepsilon+\\frac M4L\n\\right)L^{1/2}.\n\\]\nThus a threshold gap exists whenever\n\\[\n|a|\\rho\n>\n2\\varepsilon+\\frac M2L.\n\\]\n\\end{corollary}","gin{corollary}[Scale separation]\\label{cor:haar_scale_separation}\nSuppose \\(f=s+aH_\\tau+\\eta\\), where \\(s\\) is \\(M\\) Lipschitz and \\(\\|\\eta\\|_\\infty\\leq\\varepsilon\\). If the jump lies at relative distance at least \\(\\rho\\) from both ends of \\(I_{j,k}\\), then\n\\[\n|\\langle f,\\psi_{j,k}\\rangle|\n\\geq\n\\left(\n|a|\\rho\n-\\varepsilon\n-\\frac M4L\n\\right)L^{1/2}.\n\\]\nA wavelet support without a jump satisfies\n\\[\n|\\langle f,\\psi_{j,k}\\rangle|\n\\leq\n\\left(\n\\varepsilon+\\frac M4L\n\\right)L^{1/2}.\n\\]\nThus a threshold gap exists whenever\n\\[\n|a|\\rho\n>\n2\\varepsilon+\\frac M2L.\n\\]\n\\end{corollary}","itz and \\(\\|\\eta\\|_\\infty\\leq\\varepsilon\\). If the jump lies at relative distance at least \\(\\rho\\) from both ends of \\(I_{j,k}\\), then\n\\[\n|\\langle f,\\psi_{j,k}\\rangle|\n\\geq\n\\left(\n|a|\\rho\n-\\varepsilon\n-\\frac M4L\n\\right)L^{1/2}.\n\\]\nA wavelet support without a jump satisfies\n\\[\n|\\langle f,\\psi_{j,k}\\rangle|\n\\leq\n\\left(\n\\varepsilon+\\frac M4L\n\\right)L^{1/2}.\n\\]\nThus a threshold gap exists whenever\n\\[\n|a|\\rho\n>\n2\\varepsilon+\\frac M2L.\n\\]\n\\end{corollary}","\\begin{proof}\nCombine \\cref{thm:smooth_haar_bound,thm:haar_step_coefficient}. The noise coefficient is at most\n\\[\n\\|\\eta\\|_\\infty\\|\\psi_{j,k}\\|_1\n=\n\\varepsilon L^{1/2}.\n\\]\nThe final inequality makes the lower bound on a jump support larger than the upper bound on a smooth support.\n\\end{proof}","\\begin{figure}\n\\centering\n\\includegraphics[width=0.68\\textwidth]{10_haar_scale_separation.pdf}\n\\caption{The smooth contribution decays like \\(L^{3/2}\\), while a separated jump contribution decays like \\(L^{1/2}\\).}\n\\label{fig:haar_scale_separation}\n\\end{figure}","\\begin{theorem}[Half cell shift guarantee]\\label{thm:half_shift_guarantee}\nFix a cell length \\(L>0\\). Consider the grid with boundaries \\(kL\\) and the grid with boundaries \\((k+1/2)L\\). Every point \\(\\tau\\in\\RR\\) lies in a cell of at least one grid at distance at least \\(L/4\\) from both cell boundaries.\n\\end{theorem}","\\begin{proof}\nReduce \\(\\tau\\) modulo \\(L\\) and write its representative as \\(x\\in[0,L)\\). If\n\\[\nL/4\\leq x\\leq 3L/4,\n\\]\nthen the distance from \\(x\\) to both boundaries of its cell in the original grid is at least \\(L/4\\). If \\(x<L/4\\) or \\(x>3L/4\\), then \\(x\\) lies in the cell of the shifted grid bounded by \\(-L/2\\) and \\(L/2\\), or in its translate, at distance at least \\(L/4\\) from both shifted boundaries. One of the two grids therefore has the required property.\n\\end{proof}","\\begin{corollary}[Alignment free two grid Haar detection]\\label{cor:two_grid_haar_detection}\nAssume the support length \\(L\\) is smaller than the minimum jump separation. Analyze the signal with Haar wavelets on the original grid and on the half shifted grid. Every jump appears in a support with \\(\\rho\\geq1/4\\) in one of the two systems. Therefore a threshold gap exists independently of alignment whenever\n\\[\n\\frac{|a|}{4}\n>\n2\\varepsilon+\\frac M2L.\n\\]\n\\end{corollary}","ollary}[Alignment free two grid Haar detection]\\label{cor:two_grid_haar_detection}\nAssume the support length \\(L\\) is smaller than the minimum jump separation. Analyze the signal with Haar wavelets on the original grid and on the half shifted grid. Every jump appears in a support with \\(\\rho\\geq1/4\\) in one of the two systems. Therefore a threshold gap exists independently of alignment whenever\n\\[\n\\frac{|a|}{4}\n>\n2\\varepsilon+\\frac M2L.\n\\]\n\\end{corollary}","\\begin{proof}\nApply \\cref{thm:half_shift_guarantee} and then \\cref{cor:haar_scale_separation} with \\(\\rho=1/4\\). The separation assumption prevents a selected support from containing two jumps.\n\\end{proof}","Let\n\\[\n(\\Delta x)(n)=x(n+1)-x(n).\n\\]\nThen\n\\[\n\\Delta^m x(n)\n=\n\\sum_{r=0}^{m}\n(-1)^{m-r}\\binom mr x(n+r).\n\\]","\\begin{theorem}[Polynomial annihilation and boundary support]\\label{thm:polynomial_annihilation}\nLet \\(x\\) be a sequence that agrees with a polynomial of degree at most \\(d\\) on each interval of a finite partition of \\(\\ZZ\\). Then\n\\[\n\\Delta^{d+1}x(n)=0\n\\]\nwhenever the complete stencil\n\\[\n\\{n,n+1,\\ldots,n+d+1\\}\n\\]\nlies inside one polynomial piece. Hence \\(\\Delta^{d+1}x\\) is supported in a band of at most \\(d+1\\) addresses before each breakpoint.\n\\end{theorem}","eorem}[Polynomial annihilation and boundary support]\\label{thm:polynomial_annihilation}\nLet \\(x\\) be a sequence that agrees with a polynomial of degree at most \\(d\\) on each interval of a finite partition of \\(\\ZZ\\). Then\n\\[\n\\Delta^{d+1}x(n)=0\n\\]\nwhenever the complete stencil\n\\[\n\\{n,n+1,\\ldots,n+d+1\\}\n\\]\nlies inside one polynomial piece. Hence \\(\\Delta^{d+1}x\\) is supported in a band of at most \\(d+1\\) addresses before each breakpoint.\n\\end{theorem}","\\begin{proof}\nThe \\((d+1)\\)st forward difference of a polynomial of degree at most \\(d\\) is zero. The only stencils for which this argument does not apply are those crossing a breakpoint.\n\\end{proof}","\\begin{proposition}[Noise amplification by differences]\\label{prop:difference_noise}\nIf \\(\\|\\eta\\|_\\infty\\leq\\varepsilon\\), then\n\\[\n\\|\\Delta^m\\eta\\|_\\infty\n\\leq\n2^m\\varepsilon.\n\\]\nTherefore no false response above \\(2^{d+1}\\varepsilon\\) can occur on a stencil lying inside one polynomial piece.\n\\end{proposition}","\\begin{proof}\nThe absolute values of the binomial coefficients in \\(\\Delta^m\\) sum to \\(2^m\\).\n\\end{proof}","\\begin{corollary}[Robust breakpoint bands]\\label{cor:robust_breakpoint_bands}\nAssume every true breakpoint produces at least one response of magnitude \\(\\rho\\) in its \\((d+1)\\)st difference band. If\n\\[\n\\rho>2^{d+2}\\varepsilon,\n\\]\nthen thresholding at any level between \\(2^{d+1}\\varepsilon\\) and \\(\\rho-2^{d+1}\\varepsilon\\) detects a nonempty cluster near every breakpoint and creates no cluster away from the breakpoint bands.\n\\end{corollary}","\\begin{proof}\nUse \\cref{thm:polynomial_annihilation,prop:difference_noise}. Noise cannot cross the threshold away from a breakpoint. At a response of magnitude \\(\\rho\\), the perturbed magnitude remains at least \\(\\rho-2^{d+1}\\varepsilon\\), which is above the threshold.\n\\end{proof}","\\begin{standardfact}[Classical square-wave Gibbs behavior]\\label{sf:gibbs_behavior}\nLet \\(q(x)=\\tfrac12\\operatorname{sign}(\\sin x)\\), whose jump at zero has size one. For \\(n\\geq1\\), its partial sum through the first \\(n\\) positive odd frequencies is\n\\[\nQ_n(x)=\\frac{2}{\\pi}\\sum_{k=0}^{n-1}\\frac{\\sin((2k+1)x)}{2k+1}.\n\\]\nAt \\(x_n=\\pi/(2n)\\),\n\\[\nx_n\\longrightarrow0,\\qquad Q_n(x_n)\\longrightarrow\\frac{\\operatorname{Si}(\\pi)}{\\pi}\n=\\frac12+G,\\qquad G\\approx0.089489872236,\n\\]\nwhere \\(\\operatorname{Si}(z)=\\int_0^z\\sin(t)/t\\,dt\\) and \\(G=\\operatorname{Si}(\\pi)/\\pi-1/2>0\\).\nThus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude \\(a\\) gives an overshoot or undershoot of magnitude \\(|a|G\\).","Let \\(q(x)=\\tfrac12\\operatorname{sign}(\\sin x)\\), whose jump at zero has size one. For \\(n\\geq1\\), its partial sum through the first \\(n\\) positive odd frequencies is\n\\[\nQ_n(x)=\\frac{2}{\\pi}\\sum_{k=0}^{n-1}\\frac{\\sin((2k+1)x)}{2k+1}.\n\\]\nAt \\(x_n=\\pi/(2n)\\),\n\\[\nx_n\\longrightarrow0,\\qquad Q_n(x_n)\\longrightarrow\\frac{\\operatorname{Si}(\\pi)}{\\pi}\n=\\frac12+G,\\qquad G\\approx0.089489872236,\n\\]\nwhere \\(\\operatorname{Si}(z)=\\int_0^z\\sin(t)/t\\,dt\\) and \\(G=\\operatorname{Si}(\\pi)/\\pi-1/2>0\\).\nThus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude \\(a\\) gives an overshoot or undershoot of magnitude \\(|a|G\\).","whose jump at zero has size one. For \\(n\\geq1\\), its partial sum through the first \\(n\\) positive odd frequencies is\n\\[\nQ_n(x)=\\frac{2}{\\pi}\\sum_{k=0}^{n-1}\\frac{\\sin((2k+1)x)}{2k+1}.\n\\]\nAt \\(x_n=\\pi/(2n)\\),\n\\[\nx_n\\longrightarrow0,\\qquad Q_n(x_n)\\longrightarrow\\frac{\\operatorname{Si}(\\pi)}{\\pi}\n=\\frac12+G,\\qquad G\\approx0.089489872236,\n\\]\nwhere \\(\\operatorname{Si}(z)=\\int_0^z\\sin(t)/t\\,dt\\) and \\(G=\\operatorname{Si}(\\pi)/\\pi-1/2>0\\).\nThus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude \\(a\\) gives an overshoot or undershoot of magnitude \\(|a|G\\).","pi}\\sum_{k=0}^{n-1}\\frac{\\sin((2k+1)x)}{2k+1}.\n\\]\nAt \\(x_n=\\pi/(2n)\\),\n\\[\nx_n\\longrightarrow0,\\qquad Q_n(x_n)\\longrightarrow\\frac{\\operatorname{Si}(\\pi)}{\\pi}\n=\\frac12+G,\\qquad G\\approx0.089489872236,\n\\]\nwhere \\(\\operatorname{Si}(z)=\\int_0^z\\sin(t)/t\\,dt\\) and \\(G=\\operatorname{Si}(\\pi)/\\pi-1/2>0\\).\nThus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude \\(a\\) gives an overshoot or undershoot of magnitude \\(|a|G\\).","\\(x_n=\\pi/(2n)\\),\n\\[\nx_n\\longrightarrow0,\\qquad Q_n(x_n)\\longrightarrow\\frac{\\operatorname{Si}(\\pi)}{\\pi}\n=\\frac12+G,\\qquad G\\approx0.089489872236,\n\\]\nwhere \\(\\operatorname{Si}(z)=\\int_0^z\\sin(t)/t\\,dt\\) and \\(G=\\operatorname{Si}(\\pi)/\\pi-1/2>0\\).\nThus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude \\(a\\) gives an overshoot or undershoot of magnitude \\(|a|G\\).","For the limit, put \\(u_k=(k+\\tfrac12)/n\\). The identity\n\\[\nQ_n(x_n)=\\frac{1}{\\pi n}\\sum_{k=0}^{n-1}\\frac{\\sin(\\pi u_k)}{u_k}\n\\longrightarrow\\frac1\\pi\\int_0^1\\frac{\\sin(\\pi u)}u\\,du\n=\\frac{\\operatorname{Si}(\\pi)}{\\pi}\n\\]\nis a midpoint Riemann sum, since the integrand has continuous value \\(\\pi\\) at zero. This is the normalized classical example in \\cite[Section~6.16(i), equations~6.16.1--6.16.4]{DLMFGibbs}.\n\\end{standardfact}","\\begin{theorem}[Uniform convergence with exact jump packets]\\label{thm:uniform_packet_convergence}\nLet \\(f=s+j\\) on a compact domain, where \\(j\\) is any bounded singular packet. Suppose \\(Q_ms\\to s\\) uniformly. Then\n\\[\n\\widetilde f_m=Q_ms+j\n\\]\nconverges uniformly to \\(f\\), even when \\(f\\) is discontinuous:\n\\[\n\\|f-\\widetilde f_m\\|_\\infty\n=\n\\|s-Q_ms\\|_\\infty\n\\longrightarrow0.\n\\]\nThere is no error proportional to the jump size because the jump is shared exactly by \\(f\\) and \\(\\widetilde f_m\\).\n\\end{theorem}","hm:uniform_packet_convergence}\nLet \\(f=s+j\\) on a compact domain, where \\(j\\) is any bounded singular packet. Suppose \\(Q_ms\\to s\\) uniformly. Then\n\\[\n\\widetilde f_m=Q_ms+j\n\\]\nconverges uniformly to \\(f\\), even when \\(f\\) is discontinuous:\n\\[\n\\|f-\\widetilde f_m\\|_\\infty\n=\n\\|s-Q_ms\\|_\\infty\n\\longrightarrow0.\n\\]\nThere is no error proportional to the jump size because the jump is shared exactly by \\(f\\) and \\(\\widetilde f_m\\).\n\\end{theorem}","_packet_convergence}\nLet \\(f=s+j\\) on a compact domain, where \\(j\\) is any bounded singular packet. Suppose \\(Q_ms\\to s\\) uniformly. Then\n\\[\n\\widetilde f_m=Q_ms+j\n\\]\nconverges uniformly to \\(f\\), even when \\(f\\) is discontinuous:\n\\[\n\\|f-\\widetilde f_m\\|_\\infty\n=\n\\|s-Q_ms\\|_\\infty\n\\longrightarrow0.\n\\]\nThere is no error proportional to the jump size because the jump is shared exactly by \\(f\\) and \\(\\widetilde f_m\\).\n\\end{theorem}","\\Needspace{20\\baselineskip}\n\\begin{theorem}[Estimated packet Gibbs bound]\\label{thm:estimated_packet_gibbs}\nLet \\(Q_m\\) be linear and uniformly bounded on \\(L^p\\), with\n\\[\n\\sup_m\\|Q_m\\|\\leq C_Q.\n\\]\nFor \\(f=s+j\\) and an estimated packet \\(\\widehat j_m\\), define\n\\[\n\\widehat f_m\n=Q_m(f-\\widehat j_m)+\\widehat j_m.\n\\]\nThen\n\\[\n\\|f-\\widehat f_m\\|_p\n\\leq\n\\|s-Q_ms\\|_p\n+(1+C_Q)\\|j-\\widehat j_m\\|_p.\n\\]\nConsequently, smooth approximation and packet recovery may be improved independently. No claim of Gibbs removal is justified unless both terms are controlled in the norm under consideration.\n\\end{theorem}","\\begin{corollary}[Boundary band error]\\label{cor:boundary_band_error}\nLet \\(\\Gamma\\) be the singular boundary and assume\n\\[\n\\mu(N_\\delta(\\Gamma))\\leq C_\\Gamma\\delta\n\\]\nfor small \\(\\delta\\). For finite \\(p\\),\n\\[\n\\|f-\\widetilde f_m\\|_{L^p(N_\\delta(\\Gamma))}\n\\leq\n(C_\\Gamma\\delta)^{1/p}\n\\|s-Q_ms\\|_\\infty.\n\\]\n\\end{corollary}","\\begin{proof}\nThe error equals \\(s-Q_ms\\). Bound it by its supremum and multiply by the measure of the boundary tube.\n\\end{proof}","\\begin{example}[Square wave Fourier tail]\\label{ex:square_wave_tail}\nLet\n\\[\nj(t)=\\one_{[-a,a]}(t)\n\\]\non the circle and let \\(s\\) be smooth. If \\(Q_m\\) is a Fourier partial sum, then \\(Q_m(s+j)\\) has the ordinary oscillatory boundary layer. In contrast,\n\\[\nQ_ms+j\n\\]\nhas error exactly \\(s-Q_ms\\). The visual comparison in \\cref{fig:gibbs_comparison} illustrates this change in approximation target. It is not a new Fourier summability theorem.\n\\end{example}","\\begin{theorem}[Residue law for Cauchy products]\\label{thm:cauchy_residue_law}\nLet\n\\[\nF(z)=\\sum_{n\\geq0}f_nz^n,\n\\qquad\nG(z)=\\sum_{n\\geq0}g_nz^n\n\\]\nbe formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo \\(N\\):\n\\[\nF=\\sum_{r=0}^{N-1}F_r,\n\\qquad\nG=\\sum_{s=0}^{N-1}G_s.\n\\]\nIf \\(H=FG\\), then its residue packet \\(H_t\\) is\n\\[\n\\boxed{\nH_t\n=\n\\sum_{r+s\\equiv t\\!\\!\\pmod N}F_rG_s.\n}\n\\]\n\\end{theorem}","rem}[Residue law for Cauchy products]\\label{thm:cauchy_residue_law}\nLet\n\\[\nF(z)=\\sum_{n\\geq0}f_nz^n,\n\\qquad\nG(z)=\\sum_{n\\geq0}g_nz^n\n\\]\nbe formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo \\(N\\):\n\\[\nF=\\sum_{r=0}^{N-1}F_r,\n\\qquad\nG=\\sum_{s=0}^{N-1}G_s.\n\\]\nIf \\(H=FG\\), then its residue packet \\(H_t\\) is\n\\[\n\\boxed{\nH_t\n=\n\\sum_{r+s\\equiv t\\!\\!\\pmod N}F_rG_s.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nEvery monomial in \\(F_rG_s\\) has degree congruent to \\(r+s\\) modulo \\(N\\). Conversely, every coefficient of degree congruent to \\(t\\) is formed from pairs whose residues sum to \\(t\\). Grouping the Cauchy product by this residue gives the formula.\n\\end{proof}","\\begin{theorem}[Shifted pointwise monomial law]\\label{thm:shifted_monomial_law}\nLet\n\\[\ny(n)=\\prod_{j=1}^{m}x(n-a_j)\n\\]\nand write\n\\[\na_j=Nc_j+b_j,\n\\qquad\n0\\leq b_j<N.\n\\]\nFor output residue \\(r\\), define\n\\[\nr_j\\equiv r-b_j\\pmod N,\n\\qquad\n0\\leq r_j<N,\n\\]\nand let\n\\[\n\\kappa_{r,b_j}\n=\n\\begin{cases}\n1,&r<b_j,","\\begin{theorem}[Shifted pointwise monomial law]\\label{thm:shifted_monomial_law}\nLet\n\\[\ny(n)=\\prod_{j=1}^{m}x(n-a_j)\n\\]\nand write\n\\[\na_j=Nc_j+b_j,\n\\qquad\n0\\leq b_j<N.\n\\]\nFor output residue \\(r\\), define\n\\[\nr_j\\equiv r-b_j\\pmod N,\n\\qquad\n0\\leq r_j<N,\n\\]\nand let\n\\[\n\\kappa_{r,b_j}\n=\n\\begin{cases}\n1,&r<b_j,\\\\\n0,&r\\geq b_j.\n\\end{cases}\n\\]\nThen the output channel is\n\\[\n\\boxed{\ny_r(q)\n=\n\\prod_{j=1}^{m}\nx_{r_j}\n\\bigl(q-c_j-\\kappa_{r,b_j}\\bigr).\n}\n\\]\n\\end{theorem}","0,&r\\geq b_j.\n\\end{cases}\n\\]\nThen the output channel is\n\\[\n\\boxed{\ny_r(q)\n=\n\\prod_{j=1}^{m}\nx_{r_j}\n\\bigl(q-c_j-\\kappa_{r,b_j}\\bigr).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFor \\(n=Nq+r\\),\n\\[\nn-a_j\n=N(q-c_j)+r-b_j.\n\\]\nIf \\(r\\geq b_j\\), this address has residue \\(r-b_j\\) and coarse index \\(q-c_j\\). If \\(r<b_j\\), add \\(N\\) to the residue and subtract one from the coarse index. Multiplying the resulting channel values gives the formula.\n\\end{proof}","\\begin{example}[Quadratic nearest neighbor interaction]\\label{ex:quadratic_neighbor}\nFor\n\\[\ny(n)=x(n)x(n-1)\n\\]\nand modulus \\(N\\),\n\\[\ny_r(q)\n=\n\\begin{cases}\nx_r(q)x_{r-1}(q),&r\\geq1,","\\begin{example}[Quadratic nearest neighbor interaction]\\label{ex:quadratic_neighbor}\nFor\n\\[\ny(n)=x(n)x(n-1)\n\\]\nand modulus \\(N\\),\n\\[\ny_r(q)\n=\n\\begin{cases}\nx_r(q)x_{r-1}(q),&r\\geq1,\\\\\nx_0(q)x_{N-1}(q-1),&r=0.\n\\end{cases}\n\\]\nThe carry at residue zero is the nonlinear analogue of a transfer crossing a coarse cell boundary.\n\\end{example}","\\midrule\n16 & \\(1.7999\\times10^{-1}\\) & \\(6.6522\\times10^{-2}\\)","32 & \\(3.8414\\times10^{-2}\\) & \\(4.9153\\times10^{-2}\\)","64 & \\(1.6267\\times10^{-2}\\) & \\(3.8056\\times10^{-2}\\)","128 & \\(1.7387\\times10^{-3}\\) & \\(2.5631\\times10^{-2}\\)","256 & \\(1.5870\\times10^{-4}\\) & \\(1.7424\\times10^{-2}\\)","\\midrule\none-dimensional step & \\(5.4023\\times10^{-2}\\) & \\(1.3692\\times10^{-4}\\) & \\(2.4029\\times10^{-2}\\)","two-dimensional disk & \\(3.7643\\times10^{-2}\\) & \\(6.4752\\times10^{-5}\\) & \\(3.4043\\times10^{-3}\\)","multiple regions & \\(3.3740\\times10^{-2}\\) & \\(5.2118\\times10^{-7}\\) & not evaluated","\\begin{interpretation}[title={Keep the rotation and its chart separate}]\nFor the phase factorization below, $M$ and $N$ are positive phase orders. The reflection coordinate rotates in the unit disk. The Cayley map expresses that same motion as a fractional linear impedance action. A pole of the chart does not mean that the projective action fails. This realization is independent of the separated jump detector.\n\\end{interpretation}","Let\n\\[\n\\mathbb H\n=\n\\{z\\in\\CC:\\Re z>0\\}\n\\]\nand define\n\\[\nC(z)=\\frac{z-1}{z+1}.\n\\]\nIt maps \\(\\mathbb H\\) conformally onto the unit disk \\(\\mathbb D\\), with inverse\n\\[\nC^{-1}(w)=\\frac{1+w}{1-w}.\n\\]\nFor \\(\\theta\\in\\RR\\), let\n\\[\nR_\\theta(w)=\\e^{\\ii\\theta}w.\n\\]","\\begin{theorem}[Cyclic M\\\"obius action]\\label{thm:cyclic_mobius}\nThe pullback\n\\[\n\\Phi_\\theta\n=C^{-1}R_\\theta C\n\\]\nis the fractional linear map\n\\[\n\\boxed{\n\\Phi_\\theta(z)\n=\n\\frac{(1+\\e^{\\ii\\theta})z+(1-\\e^{\\ii\\theta})}\n{(1-\\e^{\\ii\\theta})z+(1+\\e^{\\ii\\theta})}.\n}\n\\]\nIt preserves \\(\\mathbb H\\) and satisfies\n\\[\n\\Phi_\\theta\\Phi_\\varphi\n=\n\\Phi_{\\theta+\\varphi}.\n\\]\nFor an integer \\(N\\geq1\\) and \\(\\theta=2\\pi/N\\), the map has order \\(N\\).\n\\end{theorem}","us action]\\label{thm:cyclic_mobius}\nThe pullback\n\\[\n\\Phi_\\theta\n=C^{-1}R_\\theta C\n\\]\nis the fractional linear map\n\\[\n\\boxed{\n\\Phi_\\theta(z)\n=\n\\frac{(1+\\e^{\\ii\\theta})z+(1-\\e^{\\ii\\theta})}\n{(1-\\e^{\\ii\\theta})z+(1+\\e^{\\ii\\theta})}.\n}\n\\]\nIt preserves \\(\\mathbb H\\) and satisfies\n\\[\n\\Phi_\\theta\\Phi_\\varphi\n=\n\\Phi_{\\theta+\\varphi}.\n\\]\nFor an integer \\(N\\geq1\\) and \\(\\theta=2\\pi/N\\), the map has order \\(N\\).\n\\end{theorem}","\\begin{proof}\nSubstitute \\(\\e^{\\ii\\theta}(z-1)/(z+1)\\) into \\(C^{-1}\\) and simplify. Preservation of \\(\\mathbb H\\) follows because \\(C\\) maps it to \\(\\mathbb D\\), rotation preserves \\(\\mathbb D\\), and \\(C^{-1}\\) maps back. The group law follows by conjugating the rotation group law. When \\(\\theta=2\\pi/N\\), the \\(N\\)th power is the identity. Conversely, if its \\(k\\)th power is the identity, conjugation by \\(C\\) and evaluation at \\(w=1/2\\) give \\(\\e^{2\\pi\\ii k/N}=1\\), so \\(N\\) divides \\(k\\). Hence the order is exactly \\(N\\).\n\\end{proof}","of}\nSubstitute \\(\\e^{\\ii\\theta}(z-1)/(z+1)\\) into \\(C^{-1}\\) and simplify. Preservation of \\(\\mathbb H\\) follows because \\(C\\) maps it to \\(\\mathbb D\\), rotation preserves \\(\\mathbb D\\), and \\(C^{-1}\\) maps back. The group law follows by conjugating the rotation group law. When \\(\\theta=2\\pi/N\\), the \\(N\\)th power is the identity. Conversely, if its \\(k\\)th power is the identity, conjugation by \\(C\\) and evaluation at \\(w=1/2\\) give \\(\\e^{2\\pi\\ii k/N}=1\\), so \\(N\\) divides \\(k\\). Hence the order is exactly \\(N\\).\n\\end{proof}","\\theta}(z-1)/(z+1)\\) into \\(C^{-1}\\) and simplify. Preservation of \\(\\mathbb H\\) follows because \\(C\\) maps it to \\(\\mathbb D\\), rotation preserves \\(\\mathbb D\\), and \\(C^{-1}\\) maps back. The group law follows by conjugating the rotation group law. When \\(\\theta=2\\pi/N\\), the \\(N\\)th power is the identity. Conversely, if its \\(k\\)th power is the identity, conjugation by \\(C\\) and evaluation at \\(w=1/2\\) give \\(\\e^{2\\pi\\ii k/N}=1\\), so \\(N\\) divides \\(k\\). Hence the order is exactly \\(N\\).\n\\end{proof}","+1)\\) into \\(C^{-1}\\) and simplify. Preservation of \\(\\mathbb H\\) follows because \\(C\\) maps it to \\(\\mathbb D\\), rotation preserves \\(\\mathbb D\\), and \\(C^{-1}\\) maps back. The group law follows by conjugating the rotation group law. When \\(\\theta=2\\pi/N\\), the \\(N\\)th power is the identity. Conversely, if its \\(k\\)th power is the identity, conjugation by \\(C\\) and evaluation at \\(w=1/2\\) give \\(\\e^{2\\pi\\ii k/N}=1\\), so \\(N\\) divides \\(k\\). Hence the order is exactly \\(N\\).\n\\end{proof}","implify. Preservation of \\(\\mathbb H\\) follows because \\(C\\) maps it to \\(\\mathbb D\\), rotation preserves \\(\\mathbb D\\), and \\(C^{-1}\\) maps back. The group law follows by conjugating the rotation group law. When \\(\\theta=2\\pi/N\\), the \\(N\\)th power is the identity. Conversely, if its \\(k\\)th power is the identity, conjugation by \\(C\\) and evaluation at \\(w=1/2\\) give \\(\\e^{2\\pi\\ii k/N}=1\\), so \\(N\\) divides \\(k\\). Hence the order is exactly \\(N\\).\n\\end{proof}","rvation of \\(\\mathbb H\\) follows because \\(C\\) maps it to \\(\\mathbb D\\), rotation preserves \\(\\mathbb D\\), and \\(C^{-1}\\) maps back. The group law follows by conjugating the rotation group law. When \\(\\theta=2\\pi/N\\), the \\(N\\)th power is the identity. Conversely, if its \\(k\\)th power is the identity, conjugation by \\(C\\) and evaluation at \\(w=1/2\\) give \\(\\e^{2\\pi\\ii k/N}=1\\), so \\(N\\) divides \\(k\\). Hence the order is exactly \\(N\\).\n\\end{proof}","ecause \\(C\\) maps it to \\(\\mathbb D\\), rotation preserves \\(\\mathbb D\\), and \\(C^{-1}\\) maps back. The group law follows by conjugating the rotation group law. When \\(\\theta=2\\pi/N\\), the \\(N\\)th power is the identity. Conversely, if its \\(k\\)th power is the identity, conjugation by \\(C\\) and evaluation at \\(w=1/2\\) give \\(\\e^{2\\pi\\ii k/N}=1\\), so \\(N\\) divides \\(k\\). Hence the order is exactly \\(N\\).\n\\end{proof}","\\begin{corollary}[Impedance and admittance]\\label{cor:impedance_admittance}\nFor \\(N=2\\), \\(\\theta=\\pi\\) and\n\\[\n\\Phi_\\pi(z)=\\frac1z.\n\\]\nThus the two phase action is normalized impedance and admittance inversion.\n\\end{corollary}","\\begin{proof}\nInsert \\(\\e^{\\ii\\pi}=-1\\) in the formula of \\cref{thm:cyclic_mobius}.\n\\end{proof}","\\begin{proposition}[Radial and phase coordinates]\\label{prop:radial_phase}\nWrite\n\\[\nC(z)=\\rho\\e^{\\ii\\varphi},\n\\qquad\n0\\leq\\rho<1.\n\\]\nThen\n\\[\nC(\\Phi_\\theta(z))\n=\n\\rho\\e^{\\ii(\\varphi+\\theta)}.\n\\]\nThe radial mismatch \\(\\rho\\) is invariant and the phase coordinate advances by \\(\\theta\\).\n\\end{proposition}","\\begin{proof}\nThis is the defining conjugacy \\(C\\Phi_\\theta=R_\\theta C\\).\n\\end{proof}","\\begin{proposition}[Coprime phase factorization]\\label{prop:phase_factorization}\nLet \\(M\\) and \\(N\\) be coprime. The cyclic phase labels modulo \\(MN\\) are equivalent to pairs of labels modulo \\(M\\) and modulo \\(N\\):\n\\[\n\\ZZ/(MN)\\ZZ\n\\cong\n\\ZZ/M\\ZZ\\times\\ZZ/N\\ZZ.\n\\]\nUnder this identification, the rotation by \\(2\\pi/(MN)\\) is determined by its two projected cyclic actions.\n\\end{proposition}","\\begin{proof}\nThis is the Chinese remainder theorem applied to the additive phase label. A label modulo \\(MN\\) is uniquely determined by its two residues, and addition of labels is preserved.\n\\end{proof}","The higher order singular packet theorem removes finitely many trace jumps. A more complete approximation theory should connect packet norms to geometric measure quantities in \\(\\BV\\), piecewise Sobolev spaces, and free discontinuity models. It should also determine when a redundant packet dictionary can identify the derivative order of a singularity rather than merely its location.","The resulting chain is\n\\[\n\\begin{gathered}\n\\text{lattice coordinates}\n\\longrightarrow\n\\text{transfer blocks}\n\\longrightarrow\n\\text{stable filter banks}\n\\longrightarrow\n\\text{packet trees}\n\\\\\n\\longrightarrow\n\\text{singular packet model}\n\\longrightarrow\n\\text{local recovery}\n\\longrightarrow\n\\text{smooth approximation rate}.\n\\end{gathered}\n\\]\nEvery arrow is exact or quantitatively controlled under the hypotheses stated in the corresponding result.","tion 11, Section 13.}\nA packet calculus can reorganize a signal exactly, choose among finitely many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows.\n\\vfig{three_worlds}\n\\head{A signal that can be checked by hand}\nOur main finite signal is $f=s+j\\in\\QQ^{16}$. The background repeats\n\\[\n\\tfrac14(0,1,2,1,0,-1,-2,-1),\n\\]\nand $j[n]=3$ at $6\\le n<12$, with $j[n]=0$ elsewhere. This is a discrete background model, not an assertion of continuum smoothness. Its energy is exactly $57$. We use it for residue coordinates, Haar transforms, coefficient deletion, and known-packet cancellation.\n\\head{A tree whose alternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate or…","reorganize a signal exactly, choose among finitely many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows.\n\\vfig{three_worlds}\n\\head{A signal that can be checked by hand}\nOur main finite signal is $f=s+j\\in\\QQ^{16}$. The background repeats\n\\[\n\\tfrac14(0,1,2,1,0,-1,-2,-1),\n\\]\nand $j[n]=3$ at $6\\le n<12$, with $j[n]=0$ elsewhere. This is a discrete background model, not an assertion of continuum smoothness. Its energy is exactly $57$. We use it for residue coordinates, Haar transforms, coefficient deletion, and known-packet cancellation.\n\\head{A tree whose alternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate orbit with no hidden numerical poles}\nThe four…","ng finitely many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows.\n\\vfig{three_worlds}\n\\head{A signal that can be checked by hand}\nOur main finite signal is $f=s+j\\in\\QQ^{16}$. The background repeats\n\\[\n\\tfrac14(0,1,2,1,0,-1,-2,-1),\n\\]\nand $j[n]=3$ at $6\\le n<12$, with $j[n]=0$ elsewhere. This is a discrete background model, not an assertion of continuum smoothness. Its energy is exactly $57$. We use it for residue coordinates, Haar transforms, coefficient deletion, and known-packet cancellation.\n\\head{A tree whose alternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate orbit with no hidden numerical poles}\nThe four reflection points $\\Gamma=i^k/2$ map t…","many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows.\n\\vfig{three_worlds}\n\\head{A signal that can be checked by hand}\nOur main finite signal is $f=s+j\\in\\QQ^{16}$. The background repeats\n\\[\n\\tfrac14(0,1,2,1,0,-1,-2,-1),\n\\]\nand $j[n]=3$ at $6\\le n<12$, with $j[n]=0$ elsewhere. This is a discrete background model, not an assertion of continuum smoothness. Its energy is exactly $57$. We use it for residue coordinates, Haar transforms, coefficient deletion, and known-packet cancellation.\n\\head{A tree whose alternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate orbit with no hidden numerical poles}\nThe four reflection points $\\Gamma=i^k/2$ map to four exact…","ns, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows.\n\\vfig{three_worlds}\n\\head{A signal that can be checked by hand}\nOur main finite signal is $f=s+j\\in\\QQ^{16}$. The background repeats\n\\[\n\\tfrac14(0,1,2,1,0,-1,-2,-1),\n\\]\nand $j[n]=3$ at $6\\le n<12$, with $j[n]=0$ elsewhere. This is a discrete background model, not an assertion of continuum smoothness. Its energy is exactly $57$. We use it for residue coordinates, Haar transforms, coefficient deletion, and known-packet cancellation.\n\\head{A tree whose alternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate orbit with no hidden numerical poles}\nThe four reflection points $\\Gamma=i^k/2$ map to four exact impedances. A sec…","w for each, so that the reader can compute rather than only follow arrows.\n\\vfig{three_worlds}\n\\head{A signal that can be checked by hand}\nOur main finite signal is $f=s+j\\in\\QQ^{16}$. The background repeats\n\\[\n\\tfrac14(0,1,2,1,0,-1,-2,-1),\n\\]\nand $j[n]=3$ at $6\\le n<12$, with $j[n]=0$ elsewhere. This is a discrete background model, not an assertion of continuum smoothness. Its energy is exactly $57$. We use it for residue coordinates, Haar transforms, coefficient deletion, and known-packet cancellation.\n\\head{A tree whose alternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate orbit with no hidden numerical poles}\nThe four reflection points $\\Gamma=i^k/2$ map to four exact impedances. A second orbit passes through infinity. Homogeneous coordinates make that point an ordinary part of the construction.\n\\takeawa…","ts\n\\[\n\\tfrac14(0,1,2,1,0,-1,-2,-1),\n\\]\nand $j[n]=3$ at $6\\le n<12$, with $j[n]=0$ elsewhere. This is a discrete background model, not an assertion of continuum smoothness. Its energy is exactly $57$. We use it for residue coordinates, Haar transforms, coefficient deletion, and known-packet cancellation.\n\\head{A tree whose alternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate orbit with no hidden numerical poles}\nThe four reflection points $\\Gamma=i^k/2$ map to four exact impedances. A second orbit passes through infinity. Homogeneous coordinates make that point an ordinary part of the construction.\n\\takeaway{Exact coordinates preserve information. An optimizer chooses coordinates. An estimator must still identify information not already supplied.}","lternatives can all be seen}\nThe full binary tree of depth three has only $26$ complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal score for signal quality.\n\\head{A coordinate orbit with no hidden numerical poles}\nThe four reflection points $\\Gamma=i^k/2$ map to four exact impedances. A second orbit passes through infinity. Homogeneous coordinates make that point an ordinary part of the construction.\n\\takeaway{Exact coordinates preserve information. An optimizer chooses coordinates. An estimator must still identify information not already supplied.}","midrule\n$S,\\delta_0$ & Sample shift and unit impulse & A finite truncation of every $\\ell^2$ signal","$P_{N,r}f$ & A masked signal on the original lattice & The shorter decimated channel","$f_r[q]$ & The value $f[Nq+r]$ & An average of neighboring samples","$D$ & One channel-index delay & Necessarily one original-sample delay","$E(D),R(D)$ & Finite Laurent analysis and synthesis matrices & Values at one test frequency","$W_\\alpha$ & The addressed Haar analysis operator & An edge detector by itself","$W_\\alpha^*W_\\alpha$ & A projector on the original signal space & A scalar packet energy","$\\Lambda$ & A complete covering antichain of a finite tree & An arbitrary collection of nodes","$j,\\widehat j$ & Known and estimated discontinuity packets & Interchangeable hypotheses","$\\Gamma,z$ & Reflection coefficient and normalized impedance & Coefficients that can be linearly interchanged","bottomrule\n\\end{tabularx}\n\\head{Three arithmetic conventions}\nFIR examples use bilateral sequences with finite support and \\emph{zero extension}. Haar examples use vectors of length $16$, with lengths divisible by the required powers of two. The three-tap smoother and nonlinear neighbor example use \\emph{periodic} indexing. These boundary conventions are stated where they enter. None is silently substituted for another.\nThe polynomial variable $t$ on \\pagerefV{20} indexes coefficients. The delay symbol $D$ acts on channels. The Smith variable $z$ is impedance. Keeping them separate prevents a visual resemblance from becoming an invalid identity.\n\\takeaway{A correct formula needs the correct object, product, norm, and boundary convention. A familiar matrix does not supply them automatically.}","conventions}\nFIR examples use bilateral sequences with finite support and \\emph{zero extension}. Haar examples use vectors of length $16$, with lengths divisible by the required powers of two. The three-tap smoother and nonlinear neighbor example use \\emph{periodic} indexing. These boundary conventions are stated where they enter. None is silently substituted for another.\nThe polynomial variable $t$ on \\pagerefV{20} indexes coefficients. The delay symbol $D$ acts on channels. The Smith variable $z$ is impedance. Keeping them separate prevents a visual resemblance from becoming an invalid identity.\n\\takeaway{A correct formula needs the correct object, product, norm, and boundary convention. A familiar matrix does not supply them automatically.}","e support and \\emph{zero extension}. Haar examples use vectors of length $16$, with lengths divisible by the required powers of two. The three-tap smoother and nonlinear neighbor example use \\emph{periodic} indexing. These boundary conventions are stated where they enter. None is silently substituted for another.\nThe polynomial variable $t$ on \\pagerefV{20} indexes coefficients. The delay symbol $D$ acts on channels. The Smith variable $z$ is impedance. Keeping them separate prevents a visual resemblance from becoming an invalid identity.\n\\takeaway{A correct formula needs the correct object, product, norm, and boundary convention. A familiar matrix does not supply them automatically.}","examples use vectors of length $16$, with lengths divisible by the required powers of two. The three-tap smoother and nonlinear neighbor example use \\emph{periodic} indexing. These boundary conventions are stated where they enter. None is silently substituted for another.\nThe polynomial variable $t$ on \\pagerefV{20} indexes coefficients. The delay symbol $D$ acts on channels. The Smith variable $z$ is impedance. Keeping them separate prevents a visual resemblance from becoming an invalid identity.\n\\takeaway{A correct formula needs the correct object, product, norm, and boundary convention. A familiar matrix does not supply them automatically.}","\\vpage{3}{Decimation is a permutation, not a loss}{EXACT ADDRESSES / THE RUNNING SIGNAL}{Chapter V: Theorem 2.1, Corollary 2.2.}\nFor four phases, write every sample address uniquely as $n=4q+r$ with $0\\le r<4$. Analysis stores $f[n]$ at channel address $(r,q)$. Synthesis puts that exact value back at $4q+r$.\n\\vfig{addresses}\n\\platecaption{Every cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=","\\vpage{3}{Decimation is a permutation, not a loss}{EXACT ADDRESSES / THE RUNNING SIGNAL}{Chapter V: Theorem 2.1, Corollary 2.2.}\nFor four phases, write every sample address uniquely as $n=4q+r$ with $0\\le r<4$. Analysis stores $f[n]$ at channel address $(r,q)$. Synthesis puts that exact value back at $4q+r$.\n\\vfig{addresses}\n\\platecaption{Every cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=","\\vpage{3}{Decimation is a permutation, not a loss}{EXACT ADDRESSES / THE RUNNING SIGNAL}{Chapter V: Theorem 2.1, Corollary 2.2.}\nFor four phases, write every sample address uniquely as $n=4q+r$ with $0\\le r<4$. Analysis stores $f[n]$ at channel address $(r,q)$. Synthesis puts that exact value back at $4q+r$.\n\\vfig{addresses}\n\\platecaption{Every cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the c","\\vpage{3}{Decimation is a permutation, not a loss}{EXACT ADDRESSES / THE RUNNING SIGNAL}{Chapter V: Theorem 2.1, Corollary 2.2.}\nFor four phases, write every sample address uniquely as $n=4q+r$ with $0\\le r<4$. Analysis stores $f[n]$ at channel address $(r,q)$. Synthesis puts that exact value back at $4q+r$.\n\\vfig{addresses}\n\\platecaption{Every cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the channel tuple is the sum-of-","\\vpage{3}{Decimation is a permutation, not a loss}{EXACT ADDRESSES / THE RUNNING SIGNAL}{Chapter V: Theorem 2.1, Corollary 2.2.}\nFor four phases, write every sample address uniquely as $n=4q+r$ with $0\\le r<4$. Analysis stores $f[n]$ at channel address $(r,q)$. Synthesis puts that exact value back at $4q+r$.\n\\vfig{addresses}\n\\platecaption{Every cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the channel tuple is the sum-of-squares Hilbert norm, not the maximum of the cha","y cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the channel tuple is the sum-of-squares Hilbert norm, not the maximum of the channel norms. This distinction matters in the infinite-dimensional statement.\n\\head{Negative addresses are ordinary addresses}\nEuclidean division, rather than truncation toward zero, gives\n$-1=4(-1)+3$ and $-5=4(-2)+3$. The finite tests exercise both positive and negative sample addresses. The algebraic identity applies before any plotting window or finite buffer is chosen.\n\\takeaway{Downsampling one channel discards information. Splitt…","nown jump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the channel tuple is the sum-of-squares Hilbert norm, not the maximum of the channel norms. This distinction matters in the infinite-dimensional statement.\n\\head{Negative addresses are ordinary addresses}\nEuclidean division, rather than truncation toward zero, gives\n$-1=4(-1)+3$ and $-5=4(-2)+3$. The finite tests exercise both positive and negative sample addresses. The algebraic identity applies before any plotting window or finite buffer is chosen.\n\\takeaway{Downsampling one channel discards information. Splitting into all residue channels is an invertible address change.}","ump packet, not an estimated feature or a frequency band.}\n\\head{What has changed? Only where the values are stored}\nThe complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector $P_{4,r}f$ is different: it keeps the original lattice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the channel tuple is the sum-of-squares Hilbert norm, not the maximum of the channel norms. This distinction matters in the infinite-dimensional statement.\n\\head{Negative addresses are ordinary addresses}\nEuclidean division, rather than truncation toward zero, gives\n$-1=4(-1)+3$ and $-5=4(-2)+3$. The finite tests exercise both positive and negative sample addresses. The algebraic identity applies before any plotting window or finite buffer is chosen.\n\\takeaway{Downsampling one channel discards information. Splitting into all residue channels is an invertible address change.}","ice and writes zeros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the channel tuple is the sum-of-squares Hilbert norm, not the maximum of the channel norms. This distinction matters in the infinite-dimensional statement.\n\\head{Negative addresses are ordinary addresses}\nEuclidean division, rather than truncation toward zero, gives\n$-1=4(-1)+3$ and $-5=4(-2)+3$. The finite tests exercise both positive and negative sample addresses. The algebraic identity applies before any plotting window or finite buffer is chosen.\n\\takeaway{Downsampling one channel discards information. Splitting into all residue channels is an invertible address change.}","ros outside phase $r$.\n\\[\n\\mathcal S_4\\mathcal A_4f=f,\n\\qquad\n\\|f\\|_2^2=\\sum_{r=0}^3\\|f_r\\|_2^2=57.\n\\]\nThe norm on the channel tuple is the sum-of-squares Hilbert norm, not the maximum of the channel norms. This distinction matters in the infinite-dimensional statement.\n\\head{Negative addresses are ordinary addresses}\nEuclidean division, rather than truncation toward zero, gives\n$-1=4(-1)+3$ and $-5=4(-2)+3$. The finite tests exercise both positive and negative sample addresses. The algebraic identity applies before any plotting window or finite buffer is chosen.\n\\takeaway{Downsampling one channel discards information. Splitting into all residue channels is an invertible address change.}","\\vpage{4}{Refine packets on a common lattice}{EXACT REFINEMENT / WHAT CAN BE ADDED}{Chapter V: Theorem 2.3.}\nAn even sample belongs to exactly one of residues $0$ and $2$ modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2}.\n\\]\n\\vfig{refinement}\n\\platecaption{The common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f","\\vpage{4}{Refine packets on a common lattice}{EXACT REFINEMENT / WHAT CAN BE ADDED}{Chapter V: Theorem 2.3.}\nAn even sample belongs to exactly one of residues $0$ and $2$ modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2}.\n\\]\n\\vfig{refinement}\n\\platecaption{The common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$","\\vpage{4}{Refine packets on a common lattice}{EXACT REFINEMENT / WHAT CAN BE ADDED}{Chapter V: Theorem 2.3.}\nAn even sample belongs to exactly one of residues $0$ and $2$ modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2}.\n\\]\n\\vfig{refinement}\n\\platecaption{The common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition,","of residues $0$ and $2$ modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2}.\n\\]\n\\vfig{refinement}\n\\platecaption{The common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","ur. Therefore the coarse even mask is the sum of two disjoint fine masks:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2}.\n\\]\n\\vfig{refinement}\n\\platecaption{The common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","sum of two disjoint fine masks:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2}.\n\\]\n\\vfig{refinement}\n\\platecaption{The common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","disjoint fine masks:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2}.\n\\]\n\\vfig{refinement}\n\\platecaption{The common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","s between the displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","displayed cells.}\n\\head{Why the coordinate convention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","nvention matters}\nThe decimated channel $f_0[q]=f[2q]$ has its own index $q$. At modulus four the descendants are $f[4q]$ and $f[4q+2]$. These shorter arrays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","rays must be interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","interleaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M","leaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples.\nFor example, at the first descendant address the values $f[0]=0$ and $f[2]=1/2$ are two separate values. Their sum $1/2$ cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement.\n\\head{The general formula}\nWhen $N$ divides $M$,\n\\[\nP_{N,r}=\\sum_{\\substack{0\\le a<M\\\\a\\equiv r\\pmod N}}P_{M,a}.\n\\]\nThe divisibility hypothesis ensures that every fine residue class lies entirely inside one coarse class. Without it, this particular descendant relation is not the right refinement rule.\n\\takeaway{Finer coordinates can preserve the object exactly. Compressing their values to averages is an additional operation, with an additional information-loss question.}","\\vpage{5}{A filter becomes a matrix of actual delays}{FIR TRANSFER / A FOUR-TAP EXAMPLE}{Chapter V: Theorem 3.1.}\nTake the causal filter $h=(1,2,-1,1)$, with taps at $0,1,2,3$, acting by $y[n]=\\sum_qh[q]f[n-q]$. Resolve the input into even and odd channels. Let $D$ denote one delay in the \\emph{channel} index.\n\\vfig{fir_transfer}\n\\head{Reading one matrix entry}\nThe odd-input contribution to the even output is $(2D+D^2)f_1$. The tap at original delay one reaches the preceding odd channel sample. The tap at delay three reaches the odd channel two positions back. These are different delays despite coming from the same input phase.\nThe two channel equations are\n\\[\n\\begin{aligned}\ny_0[n]&=f_0[n]-f_0[n-1]+2f_1[n-1]+f_1[n-2],","lter becomes a matrix of actual delays}{FIR TRANSFER / A FOUR-TAP EXAMPLE}{Chapter V: Theorem 3.1.}\nTake the causal filter $h=(1,2,-1,1)$, with taps at $0,1,2,3$, acting by $y[n]=\\sum_qh[q]f[n-q]$. Resolve the input into even and odd channels. Let $D$ denote one delay in the \\emph{channel} index.\n\\vfig{fir_transfer}\n\\head{Reading one matrix entry}\nThe odd-input contribution to the even output is $(2D+D^2)f_1$. The tap at original delay one reaches the preceding odd channel sample. The tap at delay three reaches the odd channel two positions back. These are different delays despite coming from the same input phase.\nThe two channel equations are\n\\[\n\\begin{aligned}\ny_0[n]&=f_0[n]-f_0[n-1]+2f_1[n-1]+f_1[n-2],","note one delay in the \\emph{channel} index.\n\\vfig{fir_transfer}\n\\head{Reading one matrix entry}\nThe odd-input contribution to the even output is $(2D+D^2)f_1$. The tap at original delay one reaches the preceding odd channel sample. The tap at delay three reaches the odd channel two positions back. These are different delays despite coming from the same input phase.\nThe two channel equations are\n\\[\n\\begin{aligned}\ny_0[n]&=f_0[n]-f_0[n-1]+2f_1[n-1]+f_1[n-2],\\\\\ny_1[n]&=2f_0[n]+f_0[n-1]+f_1[n]-f_1[n-1].\n\\end{aligned}\n\\]\nInterleaving gives exactly the direct convolution, including its boundary samples under zero extension.\n\\head{The general address calculation}\nFor $N$ phases, the block from input $s$ to output $k$ has kernel\n$h_{ks}[m]=h[Nm+k-s]$. This comes from the exact identity\n$Nn+k-q=N(n-m)+s$. It is a finite reindexing of taps, so no numerical transform of the input is needed. Evaluation of a bilateral transform is a separate step with the summability hypotheses stated in the manuscript.\n\\takeaway{The matrix does not approximate the filter. Its entries record which input phase is …","y_1[n]&=2f_0[n]+f_0[n-1]+f_1[n]-f_1[n-1].\n\\end{aligned}\n\\]\nInterleaving gives exactly the direct convolution, including its boundary samples under zero extension.\n\\head{The general address calculation}\nFor $N$ phases, the block from input $s$ to output $k$ has kernel\n$h_{ks}[m]=h[Nm+k-s]$. This comes from the exact identity\n$Nn+k-q=N(n-m)+s$. It is a finite reindexing of taps, so no numerical transform of the input is needed. Evaluation of a bilateral transform is a separate step with the summability hypotheses stated in the manuscript.\n\\takeaway{The matrix does not approximate the filter. Its entries record which input phase is used, how far back to read, and what coefficient to multiply.}","\\vpage{6}{One sample is not one channel step}{DELAY UNITS / AN IMPULSE MAKES THE DIFFERENCE VISIBLE}{Chapter V: Theorem 4.1.}\nA one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis inc","\\vpage{6}{One sample is not one channel step}{DELAY UNITS / AN IMPULSE MAKES THE DIFFERENCE VISIBLE}{Chapter V: Theorem 4.1.}\nA one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advanc","\\vpage{6}{One sample is not one channel step}{DELAY UNITS / AN IMPULSE MAKES THE DIFFERENCE VISIBLE}{Chapter V: Theorem 4.1.}\nA one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed","\\vpage{6}{One sample is not one channel step}{DELAY UNITS / AN IMPULSE MAKES THE DIFFERENCE VISIBLE}{Chapter V: Theorem 4.1.}\nA one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original si","N IMPULSE MAKES THE DIFFERENCE VISIBLE}{Chapter V: Theorem 4.1.}\nA one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit …","E DIFFERENCE VISIBLE}{Chapter V: Theorem 4.1.}\nA one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phas…","ay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one or…","hases and wraps one of them into the preceding channel index. Its matrix is $J(D)$, not $DI_2$.\n\\vfig{sample_delay}\n\\head{Trace one impulse}\nThe input $\\delta_0$ has a single nonzero value in even channel address zero. Under $J(D)$ it appears in odd channel address zero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","ero: this is the original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","e original sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","inal sample $\\delta_1$. Under $DI_2$ it stays even but moves to channel address one: this is $\\delta_2$.\nThus\n\\[\nJ(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","(D)^2=DI_2,\n\\qquad\nH(D)=I_2+2J(D)-J(D)^2+J(D)^3\n\\]\nrecovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","both the off-diagonal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","onal phase swap and the channel powers.\n\\head{Signed delays need the same care}\nFor an original-sample delay $d$ at modulus $N$,\n\\[\ny_k[n]=f_{(k-d)\\bmod N}\n\\!\\left[n+\\left\\lfloor\\frac{k-d}{N}\\right\\rfloor\\right].\n\\]\nThis includes advances when $d<0$. The finite verification checks the formula on signed sample addresses. A bank with product $D^qI_N$ delays the original signal by $Nq$ samples, not $q$ samples.\n\\takeaway{Always state the unit of a delay. A phase permutation and a channel delay together can represent one original sample.}","\\vpage{7}{Perfect reconstruction is an operator identity}{FILTER BANKS / WHY A FREQUENCY SPOT-CHECK IS INSUFFICIENT}{Chapter V: Theorem 4.1.}\nConsider the finite Laurent bank\n\\[\nE(D)=\\mat{1&D\\\\0&1},\\qquad\nR(D)=\\mat{1&-D\\\\0&1}.\n\\]\nIts synthesis cancels the exact delayed cross-channel contribution introduced by analysis.\n\\vfig{perfect_reconstruction}\n\\head{Two inverses, all inputs}\nDirect polynomial multiplication gives $RE=ER=I_2$. Since $D$ acts as a shift, these identities imply perfect reconstruction on every bilateral channel sequence. With finite kernels, composition requires only finitely many operations at each address.\nThe converse uses impulses. Insert an impulse in one input channel. Each output channel lists every coefficient of the corresponding transfer polynomial. If all such responses vanish for $RE-I$, every coefficient is zero. Hence the polynomial matrix itself is zero.\n\\head{A deliberately mislea","0&1}.\n\\]\nIts synthesis cancels the exact delayed cross-channel contribution introduced by analysis.\n\\vfig{perfect_reconstruction}\n\\head{Two inverses, all inputs}\nDirect polynomial multiplication gives $RE=ER=I_2$. Since $D$ acts as a shift, these identities imply perfect reconstruction on every bilateral channel sequence. With finite kernels, composition requires only finitely many operations at each address.\nThe converse uses impulses. Insert an impulse in one input channel. Each output channel lists every coefficient of the corresponding transfer polynomial. If all such responses vanish for $RE-I$, every coefficient is zero. Hence the polynomial matrix itself is zero.\n\\head{A deliberately misleading test}\nThe matrix $B(D)=\\operatorname{diag}(D,1)$ satisfies $B(1)=I_2$. At that one transform value it appears perfect. In the time domain, however, it delays only the first channel. An impulse in that","0&1}.\n\\]\nIts synthesis cancels the exact delayed cross-channel contribution introduced by analysis.\n\\vfig{perfect_reconstruction}\n\\head{Two inverses, all inputs}\nDirect polynomial multiplication gives $RE=ER=I_2$. Since $D$ acts as a shift, these identities imply perfect reconstruction on every bilateral channel sequence. With finite kernels, composition requires only finitely many operations at each address.\nThe converse uses impulses. Insert an impulse in one input channel. Each output channel lists every coefficient of the corresponding transfer polynomial. If all such responses vanish for $RE-I$, every coefficient is zero. Hence the polynomial matrix itself is zero.\n\\head{A deliberately misleading test}\nThe matrix $B(D)=\\operatorname{diag}(D,1)$ satisfies $B(1)=I_2$. At that one transform value it appears perfect. In the time domain, however, it delays only the first channel. An impulse in that channel im","$RE=ER=I_2$. Since $D$ acts as a shift, these identities imply perfect reconstruction on every bilateral channel sequence. With finite kernels, composition requires only finitely many operations at each address.\nThe converse uses impulses. Insert an impulse in one input channel. Each output channel lists every coefficient of the corresponding transfer polynomial. If all such responses vanish for $RE-I$, every coefficient is zero. Hence the polynomial matrix itself is zero.\n\\head{A deliberately misleading test}\nThe matrix $B(D)=\\operatorname{diag}(D,1)$ satisfies $B(1)=I_2$. At that one transform value it appears perfect. In the time domain, however, it delays only the first channel. An impulse in that channel immediately distinguishes it from the identity.\nThis is why a small set of successful frequency evaluations is not automatically a proof of the full Laurent identity. A separate degree bound and a sufficient interpolation argument would be needed to turn finitely many evaluations into such a certificate.\n\\takeaway{Exact reconstruction concerns every coefficient of the transfer o…","finite kernels, composition requires only finitely many operations at each address.\nThe converse uses impulses. Insert an impulse in one input channel. Each output channel lists every coefficient of the corresponding transfer polynomial. If all such responses vanish for $RE-I$, every coefficient is zero. Hence the polynomial matrix itself is zero.\n\\head{A deliberately misleading test}\nThe matrix $B(D)=\\operatorname{diag}(D,1)$ satisfies $B(1)=I_2$. At that one transform value it appears perfect. In the time domain, however, it delays only the first channel. An impulse in that channel immediately distinguishes it from the identity.\nThis is why a small set of successful frequency evaluations is not automatically a proof of the full Laurent identity. A separate degree bound and a sufficient interpolation argument would be needed to turn finitely many evaluations into such a certificate.\n\\takeaway{Exact reconstruction concerns every coefficient of the transfer operator, not just an attractive response curve or one evaluation point.}","finitely many operations at each address.\nThe converse uses impulses. Insert an impulse in one input channel. Each output channel lists every coefficient of the corresponding transfer polynomial. If all such responses vanish for $RE-I$, every coefficient is zero. Hence the polynomial matrix itself is zero.\n\\head{A deliberately misleading test}\nThe matrix $B(D)=\\operatorname{diag}(D,1)$ satisfies $B(1)=I_2$. At that one transform value it appears perfect. In the time domain, however, it delays only the first channel. An impulse in that channel immediately distinguishes it from the identity.\nThis is why a small set of successful frequency evaluations is not automatically a proof of the full Laurent identity. A separate degree bound and a sufficient interpolation argument would be needed to turn finitely many evaluations into such a certificate.\n\\takeaway{Exact reconstruction concerns every coefficient of the transfer operator, not just an attractive response curve or one evaluation point.}","\\vpage{8}{A jump can hide from the first Haar split}{HAAR ENERGY / EDGE ALIGNMENT}{Chapter V: Corollary 4.2.}\nFor the running signal, Haar analysis gives\n\\[\na[n]=\\frac{f[2n]+f[2n+1]}{\\sqrt2},\\qquad\nd[n]=\\frac{f[2n]-f[2n+1]}{\\sqrt2}.\n\\]\nThe plot scales both channels by $\\sqrt2$ so that all displayed ordinates are rational.\n\\vfig{haar_alignment}\n\\head{The exact energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely fro","\\vpage{8}{A jump can hide from the first Haar split}{HAAR ENERGY / EDGE ALIGNMENT}{Chapter V: Corollary 4.2.}\nFor the running signal, Haar analysis gives\n\\[\na[n]=\\frac{f[2n]+f[2n+1]}{\\sqrt2},\\qquad\nd[n]=\\frac{f[2n]-f[2n+1]}{\\sqrt2}.\n\\]\nThe plot scales both channels by $\\sqrt2$ so that all displayed ordinates are rational.\n\\vfig{haar_alignment}\n\\head{The exact energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not me","A jump can hide from the first Haar split}{HAAR ENERGY / EDGE ALIGNMENT}{Chapter V: Corollary 4.2.}\nFor the running signal, Haar analysis gives\n\\[\na[n]=\\frac{f[2n]+f[2n+1]}{\\sqrt2},\\qquad\nd[n]=\\frac{f[2n]-f[2n+1]}{\\sqrt2}.\n\\]\nThe plot scales both channels by $\\sqrt2$ so that all displayed ordinates are rational.\n\\vfig{haar_alignment}\n\\head{The exact energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross…","HAAR ENERGY / EDGE ALIGNMENT}{Chapter V: Corollary 4.2.}\nFor the running signal, Haar analysis gives\n\\[\na[n]=\\frac{f[2n]+f[2n+1]}{\\sqrt2},\\qquad\nd[n]=\\frac{f[2n]-f[2n+1]}{\\sqrt2}.\n\\]\nThe plot scales both channels by $\\sqrt2$ so that all displayed ordinates are rational.\n\\vfig{haar_alignment}\n\\head{The exact energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross the jump. The next pages expose that contr…","LIGNMENT}{Chapter V: Corollary 4.2.}\nFor the running signal, Haar analysis gives\n\\[\na[n]=\\frac{f[2n]+f[2n+1]}{\\sqrt2},\\qquad\nd[n]=\\frac{f[2n]-f[2n+1]}{\\sqrt2}.\n\\]\nThe plot scales both channels by $\\sqrt2$ so that all displayed ordinates are rational.\n\\vfig{haar_alignment}\n\\head{The exact energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross the jump. The next pages expose that contribution explicitly.\n…","all displayed ordinates are rational.\n\\vfig{haar_alignment}\n\\head{The exact energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross the jump. The next pages expose that contribution explicitly.\n\\takeaway{Haar energy is exact, but a particular detail level is not a universal edge detector. Edge visibility depends on alignment, scale, and the chosen packets.}","onal.\n\\vfig{haar_alignment}\n\\head{The exact energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross the jump. The next pages expose that contribution explicitly.\n\\takeaway{Haar energy is exact, but a particular detail level is not a universal edge detector. Edge visibility depends on alignment, scale, and the chosen packets.}","act energy calculation}\nThe low channel has energy $227/4$, and the detail channel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross the jump. The next pages expose that contribution explicitly.\n\\takeaway{Haar energy is exact, but a particular detail level is not a universal edge detector. Edge visibility depends on alignment, scale, and the chosen packets.}","hannel has energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross the jump. The next pages expose that contribution explicitly.\n\\takeaway{Haar energy is exact, but a particular detail level is not a universal edge detector. Edge visibility depends on alignment, scale, and the chosen packets.}","energy $1/4$. Their sum is $57$, exactly the input energy. The normalization is essential: unnormalized sums and differences would multiply the total energy by two.\n\\head{Why is the first detail channel so small?}\nThe jump starts at sample $6$ and ends just before sample $12$. Both lie \\emph{between} the pairs $(0,1),(2,3),\\ldots,(14,15)$. Inside every pair the jump contributes either $(0,0)$ or $(3,3)$, so its first-level difference is zero. The small detail values come entirely from the discrete background.\nThis does not mean the edge is absent from the transform. The next split of the low channel compares neighboring pairs, and those comparisons do cross the jump. The next pages expose that contribution explicitly.\n\\takeaway{Haar energy is exact, but a particular detail level is not a universal edge detector. Edge visibility depends on alignment, scale, and the chosen packets.}","\\vpage{9}{Lifting preserves information, not energy}{PREDICT--UPDATE / INVERTIBILITY VERSUS ISOMETRY}{Chapter V: Theorem 4.3.}\nFor an even/odd pair $(e,o)$, form the prediction residual $d=o-e$, then update $s=e+d/2$. The inverse reads $e=s-d/2$, followed by $o=d+e$.\n\\vfig{lifting_geometry}\n\\platecaption{The square $[0,4]^2$ and its exact image under the lifting matrix have the same area. The separate pair $(2,6)$ illustrates that Euclidean energy need not be preserved.}\n\\[\n\\mat{s","ifting preserves information, not energy}{PREDICT--UPDATE / INVERTIBILITY VERSUS ISOMETRY}{Chapter V: Theorem 4.3.}\nFor an even/odd pair $(e,o)$, form the prediction residual $d=o-e$, then update $s=e+d/2$. The inverse reads $e=s-d/2$, followed by $o=d+e$.\n\\vfig{lifting_geometry}\n\\platecaption{The square $[0,4]^2$ and its exact image under the lifting matrix have the same area. The separate pair $(2,6)$ illustrates that Euclidean energy need not be preserved.}\n\\[\n\\mat{s","BILITY VERSUS ISOMETRY}{Chapter V: Theorem 4.3.}\nFor an even/odd pair $(e,o)$, form the prediction residual $d=o-e$, then update $s=e+d/2$. The inverse reads $e=s-d/2$, followed by $o=d+e$.\n\\vfig{lifting_geometry}\n\\platecaption{The square $[0,4]^2$ and its exact image under the lifting matrix have the same area. The separate pair $(2,6)$ illustrates that Euclidean energy need not be preserved.}\n\\[\n\\mat{s\\\\d}=\\mat{1/2&1/2\\\\-1&1}\\mat{e\\\\o},\n\\qquad\n\\mat{e\\\\o}=\\mat{1&-1/2\\\\1&1/2}\\mat{s\\\\d}.\n\\]\n\\head{What determinant one does establish}\nThe matrix is invertible, and in these real coordinates it preserves oriented area. The predict and update operations can be undone in reverse order. Applied to each pair, the construction gives exact finite-signal reconstruction.\n\\head{What determinant one does not establish}\nThe pair $(2,6)$ becomes $(4,4)$. Its squared Euclidean norm changes from $40$ to $32$. The lifting matrix is not unitary. Energy accounting must therefore use its actual coordinate metric or a separately normalized transform. Invertibility alone cannot replace the Haar energy identi…","d}.\n\\]\n\\head{What determinant one does establish}\nThe matrix is invertible, and in these real coordinates it preserves oriented area. The predict and update operations can be undone in reverse order. Applied to each pair, the construction gives exact finite-signal reconstruction.\n\\head{What determinant one does not establish}\nThe pair $(2,6)$ becomes $(4,4)$. Its squared Euclidean norm changes from $40$ to $32$. The lifting matrix is not unitary. Energy accounting must therefore use its actual coordinate metric or a separately normalized transform. Invertibility alone cannot replace the Haar energy identity.\nAll values in this example are rational. Integer-to-integer lifting with a rounding rule is a different algorithm. Its inverse and error conventions require their own specification and are not silently inferred from the displayed rational matrix.\n\\takeaway{Perfect reconstruction, determinant one, and norm preservation are different properties. This transform has the first two, not the third.}","\\]\n\\head{What determinant one does establish}\nThe matrix is invertible, and in these real coordinates it preserves oriented area. The predict and update operations can be undone in reverse order. Applied to each pair, the construction gives exact finite-signal reconstruction.\n\\head{What determinant one does not establish}\nThe pair $(2,6)$ becomes $(4,4)$. Its squared Euclidean norm changes from $40$ to $32$. The lifting matrix is not unitary. Energy accounting must therefore use its actual coordinate metric or a separately normalized transform. Invertibility alone cannot replace the Haar energy identity.\nAll values in this example are rational. Integer-to-integer lifting with a rounding rule is a different algorithm. Its inverse and error conventions require their own specification and are not silently inferred from the displayed rational matrix.\n\\takeaway{Perfect reconstruction, determinant one, and norm preservation are different properties. This transform has the first two, not the third.}","d{What determinant one does establish}\nThe matrix is invertible, and in these real coordinates it preserves oriented area. The predict and update operations can be undone in reverse order. Applied to each pair, the construction gives exact finite-signal reconstruction.\n\\head{What determinant one does not establish}\nThe pair $(2,6)$ becomes $(4,4)$. Its squared Euclidean norm changes from $40$ to $32$. The lifting matrix is not unitary. Energy accounting must therefore use its actual coordinate metric or a separately normalized transform. Invertibility alone cannot replace the Haar energy identity.\nAll values in this example are rational. Integer-to-integer lifting with a rounding rule is a different algorithm. Its inverse and error conventions require their own specification and are not silently inferred from the displayed rational matrix.\n\\takeaway{Perfect reconstruction, determinant one, and norm preservation are different properties. This transform has the first two, not the third.}","\\vpage{10}{The second level resolves the hidden change}{ACTUAL WAVELET PACKETS / FOUR RATIONAL CHANNELS}{Chapter V: Theorem 5.1.}\nApply Haar analysis to \\emph{both} children. A binary address records the chronological low/detail choices, with $0$ for low and $1$ for detail. Each depth-two channel contains four coefficients.\n\\vfig{depth_two}\n\\head{The edge that first-level detail missed}\nPacket $01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alp","\\vpage{10}{The second level resolves the hidden change}{ACTUAL WAVELET PACKETS / FOUR RATIONAL CHANNELS}{Chapter V: Theorem 5.1.}\nApply Haar analysis to \\emph{both} children. A binary address records the chronological low/detail choices, with $0$ for low and $1$ for detail. Each depth-two channel contains four coefficients.\n\\vfig{depth_two}\n\\head{The edge that first-level detail missed}\nPacket $01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the sho","\\vpage{10}{The second level resolves the hidden change}{ACTUAL WAVELET PACKETS / FOUR RATIONAL CHANNELS}{Chapter V: Theorem 5.1.}\nApply Haar analysis to \\emph{both} children. A binary address records the chronological low/detail choices, with $0$ for low and $1$ for detail. Each depth-two channel contains four coefficients.\n\\vfig{depth_two}\n\\head{The edge that first-level detail missed}\nPacket $01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays e","L WAVELET PACKETS / FOUR RATIONAL CHANNELS}{Chapter V: Theorem 5.1.}\nApply Haar analysis to \\emph{both} children. A binary address records the chronological low/detail choices, with $0$ for low and $1$ for detail. Each depth-two channel contains four coefficients.\n\\vfig{depth_two}\n\\head{The edge that first-level detail missed}\nPacket $01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W…","Apply Haar analysis to \\emph{both} children. A binary address records the chronological low/detail choices, with $0$ for low and $1$ for detail. Each depth-two channel contains four coefficients.\n\\vfig{depth_two}\n\\head{The edge that first-level detail missed}\nPacket $01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis …","icients.\n\\vfig{depth_two}\n\\head{The edge that first-level detail missed}\nPacket $01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A…","tail missed}\nPacket $01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A packet address specifies an operator word. To add different…","01$ is the detail child of the first low channel. It contains $-11/4$, the large change between neighboring pair averages. Packet $10$ vanishes for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A packet address specifies an operator word. To add different packets as signals, …","es for this particular signal because the first-level differences alternate in canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A packet address specifies an operator word. To add different packets as signals, synthesize them back into the common signal space first.}","canceling pairs. Neither feature is a universal identity for all inputs.\nThe packet energies are\n\\[\n49,\\qquad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A packet address specifies an operator word. To add different packets as signals, synthesize them back into the common signal space first.}","uad 31/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A packet address specifies an operator word. To add different packets as signals, synthesize them back into the common signal space first.}","/4,\\qquad 0,\\qquad 1/4,\n\\]\nwhich again sum to $57$. Two factors of $1/\\sqrt2$ explain why the depth-two coefficients are rational halves and quarters.\n\\head{Coefficients and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A packet address specifies an operator word. To add different packets as signals, synthesize them back into the common signal space first.}","and projectors are different objects}\n$W_\\alpha f$ is the short coefficient vector. The original-space component is $W_\\alpha^*W_\\alpha f$. Summing these synthesized components, not the short arrays entrywise, reconstructs $f$:\n\\[\nf=\\sum_{|\\alpha|=2}W_\\alpha^*W_\\alpha f.\n\\]\nThe finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's $\\ell^2$ statement additionally establishes boundedness and the existence of the literal Hilbert adjoints.\n\\takeaway{A packet address specifies an operator word. To add different packets as signals, synthesize them back into the common signal space first.}","discard some of its sixteen coefficients. Order them by squared magnitude, with a deterministic address-and-position rule for ties.\n\\vfig{sparse_tail}\n\\head{Every discarded square has a meaning}\nIn this fixed orthonormal basis, the squared reconstruction error equals the sum of the squared discarded coefficients. For our running signal:\n\\begin{center}\n\\begin{tabular}{r|rrrrrr}\nkept coefficients $k$ & 1&2&4&8&12&16","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versu","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar ex","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment is a","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment is a fini","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment is a finite c","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment is a finite comparison, not a proof that one basis dominates every signal class. Here we","hline\nsquared error & $59/4$&$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment is a finite comparison, not a proof that one basis dominates every signal class. Here we isolate the part that can be checked exactly: reconstruction error within one fixed finite basis.\n\\takeaway{Changing coordinates creates no error. Throwing away coefficients does, and orthogonality makes that error exactly accountable.}","$115/16$&$11/16$&$1/4$&$0$&$0$\n\\end{tabular}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment is a finite comparison, not a proof that one basis dominates every signal class. Here we isolate the part that can be checked exactly: reconstruction error within one fixed finite basis.\n\\takeaway{Changing coordinates creates no error. Throwing away coefficients does, and orthogonality makes that error exactly accountable.}","ar}\n\\end{center}\nThe four zero coefficients belong to packet $10$, so twelve coefficients suffice for exact reconstruction here. This is a property of the selected signal and basis, not a guarantee that all edge signals need only twelve coefficients.\n\\head{Do not compare incompatible budgets}\nA cutoff $m$ in a one-dimensional signed Fourier spectrum means $2m+1$ bins, including zero. A budget of $k$ largest coefficients is a different rule. Real Fourier synthesis may also require conjugate-pair closure. Neither count alone equals a bit-storage budget once amplitudes, phases, indices, or a selected tree must be encoded.\nThe manuscript's Fourier-versus-Haar experiment is a finite comparison, not a proof that one basis dominates every signal class. Here we isolate the part that can be checked exactly: reconstruction error within one fixed finite basis.\n\\takeaway{Changing coordinates creates no error. Throwing away coefficients does, and orthogonality makes that error exactly accountable.}","S / NO GAPS AND NO DOUBLE COUNTING}{Chapter V: Definition 11.1, Theorem 11.2.}\nA complete pruning chooses nodes of a finite refinement tree so that every root-to-terminal path meets exactly one selected node. The horizontal positions below are \\emph{terminal packet addresses}, not time intervals in the original signal.\n\\vfig{complete_cuts}\n\\head{Three candidate selections}\nThe complete selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual…","Theorem 11.2.}\nA complete pruning chooses nodes of a finite refinement tree so that every root-to-terminal path meets exactly one selected node. The horizontal positions below are \\emph{terminal packet addresses}, not time intervals in the original signal.\n\\vfig{complete_cuts}\n\\head{Three candidate selections}\nThe complete selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illu…","te pruning chooses nodes of a finite refinement tree so that every root-to-terminal path meets exactly one selected node. The horizontal positions below are \\emph{terminal packet addresses}, not time intervals in the original signal.\n\\vfig{complete_cuts}\n\\head{Three candidate selections}\nThe complete selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not repl…","hat every root-to-terminal path meets exactly one selected node. The horizontal positions below are \\emph{terminal packet addresses}, not time intervals in the original signal.\n\\vfig{complete_cuts}\n\\head{Three candidate selections}\nThe complete selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is…","path meets exactly one selected node. The horizontal positions below are \\emph{terminal packet addresses}, not time intervals in the original signal.\n\\vfig{complete_cuts}\n\\head{Three candidate selections}\nThe complete selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a termina…","al positions below are \\emph{terminal packet addresses}, not time intervals in the original signal.\n\\vfig{complete_cuts}\n\\head{Three candidate selections}\nThe complete selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant…","es}, not time intervals in the original signal.\n\\vfig{complete_cuts}\n\\head{Three candidate selections}\nThe complete selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}","selection $\\{0,10,11\\}$ covers every terminal branch once. The selection $\\{0,00,1\\}$ covers the $00$ branch twice: it is not an antichain. The selection $\\{00,01,10\\}$ misses the $11$ branch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}","nch entirely.\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}",".\nFor actual Haar packets, set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}","set $P_v=W_v^*W_v$. Parent-to-child conservation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}","rvation gives\n\\[\nP_v=P_{v0}+P_{v1}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}","}.\n\\]\nReplacing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}","ing unselected parents by their children until the complete pruning is reached proves $\\sum_{v\\in\\Lambda}P_v=I$. This is why the coefficients at different depths still form an exact coordinate system.\n\\head{An exhaustive finite check}\nAt depth three the pruning count obeys $a_0=1$ and $a_{d+1}=1+a_d^2$, giving $1,2,5,26$. The supplied suite checks all $26$ covering conditions and all $26$ actual $16\\times16$ projector-sum identities. These finite checks illustrate, but do not replace, the universal tree theorem.\n\\takeaway{Mixed depth is allowed. Missing a terminal path or selecting an ancestor and its descendant is not.}","\\vpage{13}{Solve every alternative without searching every alternative}{ADDITIVE COSTS / EXACT DYNAMIC PROGRAMMING}{Chapter V: Theorem 11.3, Theorem 11.4.}\nGive every node the rational stop cost shown below. A node may either stop and pay its own cost, or refine and pay the already optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. …","m 11.3, Theorem 11.4.}\nGive every node the rational stop cost shown below. A node may either stop and pay its own cost, or refine and pay the already optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled co…","ive every node the rational stop cost shown below. A node may either stop and pay its own cost, or refine and pay the already optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be writt…","ost shown below. A node may either stop and pay its own cost, or refine and pay the already optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs","n below. A node may either stop and pay its own cost, or refine and pay the already optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a …","ither stop and pay its own cost, or refine and pay the already optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n…","y its own cost, or refine and pay the already optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global …","dy optimized costs of its children. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-r…","ren. The shaded nodes form the selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-refine decomposition of a comple…","he selected complete pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-refine decomposition of a complete pruning, not from a greed…","pruning.\n\\vfig[.94]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-refine decomposition of a complete pruning, not from a greedy assumption.}","]{dynamic_program}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-refine decomposition of a complete pruning, not from a greedy assumption.}","ram}\n\\[\nV(v)=\\min\\{c_v,\\ V(v0)+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-refine decomposition of a complete pruning, not from a greedy assumption.}",")+V(v1)\\},\n\\qquad V(v)=c_v\\text{ at a terminal node}.\n\\]\n\\head{Read the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-refine decomposition of a complete pruning, not from a greedy assumption.}","ad the calculation from the bottom upward}\nAt $10$, the children cost $1+1=2$, better than stopping for $7$. At $11$, stopping costs $4$, better than $2+3=5$. The right subtree therefore costs $6$, better than its stop cost $8$. The left subtree costs $1+2=3$, better than $5$. At the root, $3+6=9$ beats $20$.\nThe unique optimum is\n\\[\n\\Lambda_* = \\{00,01,100,101,11\\},\\qquad J(\\Lambda_*)=9.\n\\]\nAll $26$ complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, making the selector deterministic. A singleton tree needs no comparison.\nThe costs here are supplied numbers. The theorem optimizes that stated additive objective. It does not prove those numbers encode the best denoising criterion. Coupled costs that cannot be written as a sum of selected-node costs need a different optimizer.\n\\takeaway{Global optimality follows from the exact stop-or-refine decomposition of a complete pruning, not from a greedy assumption.}","\\vpage{14}{How far can the costs move before the tree changes?}{A GLOBAL MARGIN / NOT A LOCAL HEURISTIC}{Chapter V: Theorem 11.3, Corollary 11.5.}\nThe best pruning on the previous page costs $9$. The cheapest competitor costs $10$. Its exact global gap is therefore $\\Gamma=1$. The plate lists every complete-pruning cost in sorted order, giving each pruning its own horizontal rank.\n\\vfig{selector_margin}\n\\head{Propagate an error budget as carefully as a cost}\nSuppose every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_","\\vpage{14}{How far can the costs move before the tree changes?}{A GLOBAL MARGIN / NOT A LOCAL HEURISTIC}{Chapter V: Theorem 11.3, Corollary 11.5.}\nThe best pruning on the previous page costs $9$. The cheapest competitor costs $10$. Its exact global gap is therefore $\\Gamma=1$. The plate lists every complete-pruning cost in sorted order, giving each pruning its own horizontal rank.\n\\vfig{selector_margin}\n\\head{Propagate an error budget as carefully as a cost}\nSuppose every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac1","\\vpage{14}{How far can the costs move before the tree changes?}{A GLOBAL MARGIN / NOT A LOCAL HEURISTIC}{Chapter V: Theorem 11.3, Corollary 11.5.}\nThe best pruning on the previous page costs $9$. The cheapest competitor costs $10$. Its exact global gap is therefore $\\Gamma=1$. The plate lists every complete-pruning cost in sorted order, giving each pruning its own horizontal rank.\n\\vfig{selector_margin}\n\\head{Propagate an error budget as carefully as a cost}\nSuppose every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emp","apter V: Theorem 11.3, Corollary 11.5.}\nThe best pruning on the previous page costs $9$. The cheapest competitor costs $10$. Its exact global gap is therefore $\\Gamma=1$. The plate lists every complete-pruning cost in sorted order, giving each pruning its own horizontal rank.\n\\vfig{selector_margin}\n\\head{Propagate an error budget as carefully as a cost}\nSuppose every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing co…","itor costs $10$. Its exact global gap is therefore $\\Gamma=1$. The plate lists every complete-pruning cost in sorted order, giving each pruning its own horizontal rank.\n\\vfig{selector_margin}\n\\head{Propagate an error budget as carefully as a cost}\nSuppose every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefor…","ning cost in sorted order, giving each pruning its own horizontal rank.\n\\vfig{selector_margin}\n\\head{Propagate an error budget as carefully as a cost}\nSuppose every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the join…","ead{Propagate an error budget as carefully as a cost}\nSuppose every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the joint good event must be measurable. Separate nodewise probability bounds do not automatically give t…","e every node cost changes by at most $\\varepsilon_v=1/32$. A pruning may select as many as eight leaves, so its total cost can change by at most $8/32=1/4$. The recursive envelope proves this without guessing which pruning will be selected:\n\\[\nE(v)=\\max\\{\\varepsilon_v,\\ E(v0)+E(v1)\\},\\qquad E(\\varnothing)=\\tfrac14.\n\\]\nFor any competitor $\\Lambda$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the joint good event must be measurable. Separate nodewise probability bounds do not automatically give the probability of that joint event.\n\\takeaway{The exact sele…","$, the changed cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the joint good event must be measurable. Separate nodewise probability bounds do not automatically give the probability of that joint event.\n\\takeaway{The exact selector is stable because its global gap exceeds the total worst-case movement of both competing costs.}","hanged cost difference is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the joint good event must be measurable. Separate nodewise probability bounds do not automatically give the probability of that joint event.\n\\takeaway{The exact selector is stable because its global gap exceeds the total worst-case movement of both competing costs.}","e is bounded below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the joint good event must be measurable. Separate nodewise probability bounds do not automatically give the probability of that joint event.\n\\takeaway{The exact selector is stable because its global gap exceeds the total worst-case movement of both competing costs.}","ed below by\n\\[\nJ'(\\Lambda)-J'(\\Lambda_*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the joint good event must be measurable. Separate nodewise probability bounds do not automatically give the probability of that joint event.\n\\takeaway{The exact selector is stable because its global gap exceeds the total worst-case movement of both competing costs.}","*)\\ge \\Gamma-2E(\\varnothing)=\\tfrac12>0.\n\\]\nThus the optimum is unchanged for \\emph{every} permitted simultaneous perturbation, not just a sampled noise realization. The envelope is over all feasible prunings, not just the five nodes in the current optimum.\n\\head{Strict inequality has content}\nIf two competing costs are $9$ and $10$, moving them to $9.5$ and $9.5$ removes uniqueness. Equality in $2E=\\Gamma$ is therefore not a unique-selector certificate. For ordinary probability, the random node costs and the joint good event must be measurable. Separate nodewise probability bounds do not automatically give the probability of that joint event.\n\\takeaway{The exact selector is stable because its global gap exceeds the total worst-case movement of both competing costs.}","\\vpage{15}{Retain the jump and approximate the other packet}{AN EXACT IDENTITY / A CONDITIONAL INPUT}{Chapter V: Theorem 6.1, Corollary 6.2.}\nReturn to the running signal $f=s+j$. For a concrete finite calculation, use the periodic three-tap smoother\n\\[\n(Qx)[n]=\\tfrac14x[n-1]+\\tfrac12x[n]+\\tfrac14x[n+1],\\qquad n\\in\\mathbb Z/16\\mathbb Z.\n\\]\nThe plots compare smoothing the full signal with retaining its supplied jump packet. Every plotted sample is a rational number computed from the displayed formula.\n\\vfig{known_edge}\n\\head{The cancellation is the theorem}\nDefine $\\widetilde f=Qs+j$. Then, sample by sample,\n\\[\n\\boxed{f-\\widetilde f=s-Qs.}\n\\]\nBy contrast, linearity gives\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itse","\\vpage{15}{Retain the jump and approximate the other packet}{AN EXACT IDENTITY / A CONDITIONAL INPUT}{Chapter V: Theorem 6.1, Corollary 6.2.}\nReturn to the running signal $f=s+j$. For a concrete finite calculation, use the periodic three-tap smoother\n\\[\n(Qx)[n]=\\tfrac14x[n-1]+\\tfrac12x[n]+\\tfrac14x[n+1],\\qquad n\\in\\mathbb Z/16\\mathbb Z.\n\\]\nThe plots compare smoothing the full signal with retaining its supplied jump packet. Every plotted sample is a rational number computed from the displayed formula.\n\\vfig{known_edge}\n\\head{The cancellation is the theorem}\nDefine $\\widetilde f=Qs+j$. Then, sample by sample,\n\\[\n\\boxed{f-\\widetilde f=s-Qs.}\n\\]\nBy contrast, linearity gives\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction","$f=s+j$. For a concrete finite calculation, use the periodic three-tap smoother\n\\[\n(Qx)[n]=\\tfrac14x[n-1]+\\tfrac12x[n]+\\tfrac14x[n+1],\\qquad n\\in\\mathbb Z/16\\mathbb Z.\n\\]\nThe plots compare smoothing the full signal with retaining its supplied jump packet. Every plotted sample is a rational number computed from the displayed formula.\n\\vfig{known_edge}\n\\head{The cancellation is the theorem}\nDefine $\\widetilde f=Qs+j$. Then, sample by sample,\n\\[\n\\boxed{f-\\widetilde f=s-Qs.}\n\\]\nBy contrast, linearity gives\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is s…","e the periodic three-tap smoother\n\\[\n(Qx)[n]=\\tfrac14x[n-1]+\\tfrac12x[n]+\\tfrac14x[n+1],\\qquad n\\in\\mathbb Z/16\\mathbb Z.\n\\]\nThe plots compare smoothing the full signal with retaining its supplied jump packet. Every plotted sample is a rational number computed from the displayed formula.\n\\vfig{known_edge}\n\\head{The cancellation is the theorem}\nDefine $\\widetilde f=Qs+j$. Then, sample by sample,\n\\[\n\\boxed{f-\\widetilde f=s-Qs.}\n\\]\nBy contrast, linearity gives\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate th…","ac12x[n]+\\tfrac14x[n+1],\\qquad n\\in\\mathbb Z/16\\mathbb Z.\n\\]\nThe plots compare smoothing the full signal with retaining its supplied jump packet. Every plotted sample is a rational number computed from the displayed formula.\n\\vfig{known_edge}\n\\head{The cancellation is the theorem}\nDefine $\\widetilde f=Qs+j$. Then, sample by sample,\n\\[\n\\boxed{f-\\widetilde f=s-Qs.}\n\\]\nBy contrast, linearity gives\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate that sum from observations.\nOur finite periodic smoother is an ill…","ancellation is the theorem}\nDefine $\\widetilde f=Qs+j$. Then, sample by sample,\n\\[\n\\boxed{f-\\widetilde f=s-Qs.}\n\\]\nBy contrast, linearity gives\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate that sum from observations.\nOur finite periodic smoother is an illustration, not a new continuum approximation theorem. Its chosen boundary convention is part of the experiment.\n\\takeaway{Known-edge reconstruction removes the jump from the approximation problem. It does not remove the need to identify the jump.}","+j$. Then, sample by sample,\n\\[\n\\boxed{f-\\widetilde f=s-Qs.}\n\\]\nBy contrast, linearity gives\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate that sum from observations.\nOur finite periodic smoother is an illustration, not a new continuum approximation theorem. Its chosen boundary convention is part of the experiment.\n\\takeaway{Known-edge reconstruction removes the jump from the approximation problem. It does not remove the need to identify the jump.}","es\n\\[\nf-Qf=(s-Qs)+(j-Qj).\n\\]\nThe second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate that sum from observations.\nOur finite periodic smoother is an illustration, not a new continuum approximation theorem. Its chosen boundary convention is part of the experiment.\n\\takeaway{Known-edge reconstruction removes the jump from the approximation problem. It does not remove the need to identify the jump.}","e second error includes a new term caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate that sum from observations.\nOur finite periodic smoother is an illustration, not a new continuum approximation theorem. Its chosen boundary convention is part of the experiment.\n\\takeaway{Known-edge reconstruction removes the jump from the approximation problem. It does not remove the need to identify the jump.}","erm caused by smoothing the discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate that sum from observations.\nOur finite periodic smoother is an illustration, not a new continuum approximation theorem. Its chosen boundary convention is part of the experiment.\n\\takeaway{Known-edge reconstruction removes the jump from the approximation problem. It does not remove the need to identify the jump.}","e discontinuity. In this finite example that term is concentrated near the two jump boundaries. The identity itself requires no linearity of the approximation applied to $s$. Linearity is used only in the comparison with $Qf$.\n\\head{The input is not discovered by subtraction}\nThe data here include the correct $j$. Replacing it with a wrong $\\widehat j$ retains the error $j-\\widehat j$, even when $s$ is approximated perfectly. Multiple amplitudes and overlapping shapes are allowed if their full sum is supplied, but the identity does not estimate that sum from observations.\nOur finite periodic smoother is an illustration, not a new continuum approximation theorem. Its chosen boundary convention is part of the experiment.\n\\takeaway{Known-edge reconstruction removes the jump from the approximation problem. It does not remove the need to identify the jump.}","\\vpage{16}{Why a Fourier cutoff cannot reproduce this jump pointwise}{A CONTINUOUS PULSE / EXACT ENDPOINT VALUES}{Chapter V: Section 6, Section 8.}\nUse a separate period-one example on $[-1/2,1/2)$:\n\\[\nj(t)=\\mathbf 1_{\\{|t|<1/4\\}},\\qquad\nS_mj(t)=\\tfrac12+\\frac2\\pi\\sum_{n=1}^{m}\\frac{\\sin(\\pi n/2)}{n}\\cos(2\\pi nt).\n\\]\nThis is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coeffi","\\vpage{16}{Why a Fourier cutoff cannot reproduce this jump pointwise}{A CONTINUOUS PULSE / EXACT ENDPOINT VALUES}{Chapter V: Section 6, Section 8.}\nUse a separate period-one example on $[-1/2,1/2)$:\n\\[\nj(t)=\\mathbf 1_{\\{|t|<1/4\\}},\\qquad\nS_mj(t)=\\tfrac12+\\frac2\\pi\\sum_{n=1}^{m}\\frac{\\sin(\\pi n/2)}{n}\\cos(2\\pi nt).\n\\]\nThis is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this","S}{Chapter V: Section 6, Section 8.}\nUse a separate period-one example on $[-1/2,1/2)$:\n\\[\nj(t)=\\mathbf 1_{\\{|t|<1/4\\}},\\qquad\nS_mj(t)=\\tfrac12+\\frac2\\pi\\sum_{n=1}^{m}\\frac{\\sin(\\pi n/2)}{n}\\cos(2\\pi nt).\n\\]\nThis is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endp…","separate period-one example on $[-1/2,1/2)$:\n\\[\nj(t)=\\mathbf 1_{\\{|t|<1/4\\}},\\qquad\nS_mj(t)=\\tfrac12+\\frac2\\pi\\sum_{n=1}^{m}\\frac{\\sin(\\pi n/2)}{n}\\cos(2\\pi nt).\n\\]\nThis is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do no…",":\n\\[\nj(t)=\\mathbf 1_{\\{|t|<1/4\\}},\\qquad\nS_mj(t)=\\tfrac12+\\frac2\\pi\\sum_{n=1}^{m}\\frac{\\sin(\\pi n/2)}{n}\\cos(2\\pi nt).\n\\]\nThis is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The …","athbf 1_{\\{|t|<1/4\\}},\\qquad\nS_mj(t)=\\tfrac12+\\frac2\\pi\\sum_{n=1}^{m}\\frac{\\sin(\\pi n/2)}{n}\\cos(2\\pi nt).\n\\]\nThis is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated …","lation below is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","low is symbolic.\n\\vfig{fourier_jump}\n\\head{One value is fixed at every cutoff}\nAt $t=\\pm1/4$, every summand vanishes because\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","se\n$\\sin(\\pi n/2)\\cos(\\pi n/2)=0$ for integer $n$. Hence\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","e\n\\[\nS_mj(\\pm1/4)=\\tfrac12 \\qquad\\text{for every }m\\ge0.\n\\]\nA larger cutoff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","ff resolves a narrower transition, but it does not make the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","ake the finite trigonometric polynomial discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","discontinuous. The two displayed cutoffs retain $11$ and $31$ signed frequency bins, respectively, not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}",", not five and fifteen coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","coefficients.\n\\head{An exact known-packet comparison}\nTake $s(t)=\\frac14\\cos(2\\pi t)+\\frac1{10}\\cos(6\\pi t)$. For $m\\ge3$ this smooth packet is already in the retained Fourier span. Thus $S_ms+j=s+j$ exactly in this special model, whereas $S_m(s+j)$ still replaces the pulse by $S_mj$.\nThe values assigned at the two endpoints matter for pointwise error, but do not affect an $L^p$ norm for finite $p$. The illustrated ringing is not a root enclosure or an asymptotic error certificate.\n\\takeaway{Increasing spectral resolution and retaining a known discontinuity are different operations, with different pointwise consequences.}","\\vpage{17}{Small location errors, different norms}{MASK GEOMETRY / TRANSLATION BY ONE SIXTEENTH}{Chapter V: Theorem 7.1.}\nIn Lebesgue measure on the line, let\n\\[\n\\Omega=[\\tfrac14,\\tfrac34],\\qquad\n\\widehat\\Omega=[\\tfrac5{16},\\tfrac{13}{16}],\\qquad a=\\widehat a=3.\n\\]\nThe two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width $1/16$.\n\\vfig{mask_translation}\n\\[\n|\\Omega\\mathbin{\\triangle}\\widehat\\Omega|=\\tfrac18,\n\\qquad j-\\widehat j=3(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\n\\begin{center}\n\\begin{tabular}{c|ccc}\nerror measure & $L^1$ norm & squared $L^2$ norm & $L^\\infty$ norm","on errors, different norms}{MASK GEOMETRY / TRANSLATION BY ONE SIXTEENTH}{Chapter V: Theorem 7.1.}\nIn Lebesgue measure on the line, let\n\\[\n\\Omega=[\\tfrac14,\\tfrac34],\\qquad\n\\widehat\\Omega=[\\tfrac5{16},\\tfrac{13}{16}],\\qquad a=\\widehat a=3.\n\\]\nThe two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width $1/16$.\n\\vfig{mask_translation}\n\\[\n|\\Omega\\mathbin{\\triangle}\\widehat\\Omega|=\\tfrac18,\n\\qquad j-\\widehat j=3(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\n\\begin{center}\n\\begin{tabular}{c|ccc}\nerror measure & $L^1$ norm & squared $L^2$ norm & $L^\\infty$ norm","ehat\\Omega=[\\tfrac5{16},\\tfrac{13}{16}],\\qquad a=\\widehat a=3.\n\\]\nThe two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width $1/16$.\n\\vfig{mask_translation}\n\\[\n|\\Omega\\mathbin{\\triangle}\\widehat\\Omega|=\\tfrac18,\n\\qquad j-\\widehat j=3(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\n\\begin{center}\n\\begin{tabular}{c|ccc}\nerror measure & $L^1$ norm & squared $L^2$ norm & $L^\\infty$ norm","6},\\tfrac{13}{16}],\\qquad a=\\widehat a=3.\n\\]\nThe two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width $1/16$.\n\\vfig{mask_translation}\n\\[\n|\\Omega\\mathbin{\\triangle}\\widehat\\Omega|=\\tfrac18,\n\\qquad j-\\widehat j=3(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\n\\begin{center}\n\\begin{tabular}{c|ccc}\nerror measure & $L^1$ norm & squared $L^2$ norm & $L^\\infty$ norm","{16}],\\qquad a=\\widehat a=3.\n\\]\nThe two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width $1/16$.\n\\vfig{mask_translation}\n\\[\n|\\Omega\\mathbin{\\triangle}\\widehat\\Omega|=\\tfrac18,\n\\qquad j-\\widehat j=3(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\n\\begin{center}\n\\begin{tabular}{c|ccc}\nerror measure & $L^1$ norm & squared $L^2$ norm & $L^\\infty$ norm","hline\nexact value & $3/8$ & $9/8$ & $3$\n\\end{tabular}\n\\end{center}\n\\head{Area shrinks while the missed jump keeps its height}\nFor a translation by $\\delta$ with $0<|\\delta|<1/2$, the symmetric difference has measure $2|\\delta|$. At finite $p$,\n\\[\n\\|j-\\widehat j\\|_p=3(2|\\delta|)^{1/p}.\n\\]\nIt tends to zero as the displacement tends to zero. The essential supremum stays $3$ for every nonzero displacement, because the error reaches height $3$ on a set of positive measure. At exactly zero displacement all norms vanish.\nA nonempty disagreement set of measure zero would be different: it can affect the pointwise supremum while leaving the essential norm unchanged. This is why the chapter distinguishes the two statements.\n\\","hline\nexact value & $3/8$ & $9/8$ & $3$\n\\end{tabular}\n\\end{center}\n\\head{Area shrinks while the missed jump keeps its height}\nFor a translation by $\\delta$ with $0<|\\delta|<1/2$, the symmetric difference has measure $2|\\delta|$. At finite $p$,\n\\[\n\\|j-\\widehat j\\|_p=3(2|\\delta|)^{1/p}.\n\\]\nIt tends to zero as the displacement tends to zero. The essential supremum stays $3$ for every nonzero displacement, because the error reaches height $3$ on a set of positive measure. At exactly zero displacement all norms vanish.\nA nonempty disagreement set of measure zero would be different: it can affect the pointwise supremum while leaving the essential norm unchanged. This is why the chapter distinguishes the two statements.\n\\takeaway","hline\nexact value & $3/8$ & $9/8$ & $3$\n\\end{tabular}\n\\end{center}\n\\head{Area shrinks while the missed jump keeps its height}\nFor a translation by $\\delta$ with $0<|\\delta|<1/2$, the symmetric difference has measure $2|\\delta|$. At finite $p$,\n\\[\n\\|j-\\widehat j\\|_p=3(2|\\delta|)^{1/p}.\n\\]\nIt tends to zero as the displacement tends to zero. The essential supremum stays $3$ for every nonzero displacement, because the error reaches height $3$ on a set of positive measure. At exactly zero displacement all norms vanish.\nA nonempty disagreement set of measure zero would be different: it can affect the pointwise supremum while leaving the essential norm unchanged. This is why the chapter distinguishes the two statements.\n\\takeaway{A shr","hline\nexact value & $3/8$ & $9/8$ & $3$\n\\end{tabular}\n\\end{center}\n\\head{Area shrinks while the missed jump keeps its height}\nFor a translation by $\\delta$ with $0<|\\delta|<1/2$, the symmetric difference has measure $2|\\delta|$. At finite $p$,\n\\[\n\\|j-\\widehat j\\|_p=3(2|\\delta|)^{1/p}.\n\\]\nIt tends to zero as the displacement tends to zero. The essential supremum stays $3$ for every nonzero displacement, because the error reaches height $3$ on a set of positive measure. At exactly zero displacement all norms vanish.\nA nonempty disagreement set of measure zero would be different: it can affect the pointwise supremum while leaving the essential norm unchanged. This is why the chapter distinguishes the two statements.\n\\takeaway{A shrinking boundary strip can give small integrated error and still contain a full-height jump error.}","\\end{tabular}\n\\end{center}\n\\head{Area shrinks while the missed jump keeps its height}\nFor a translation by $\\delta$ with $0<|\\delta|<1/2$, the symmetric difference has measure $2|\\delta|$. At finite $p$,\n\\[\n\\|j-\\widehat j\\|_p=3(2|\\delta|)^{1/p}.\n\\]\nIt tends to zero as the displacement tends to zero. The essential supremum stays $3$ for every nonzero displacement, because the error reaches height $3$ on a set of positive measure. At exactly zero displacement all norms vanish.\nA nonempty disagreement set of measure zero would be different: it can affect the pointwise supremum while leaving the essential norm unchanged. This is why the chapter distinguishes the two statements.\n\\takeaway{A shrinking boundary strip can give small integrated error and still contain a full-height jump error.}","\\vpage{18}{Separate amplitude error from mask error}{COUNTING MEASURE / AN EXACT FINITE BUDGET}{Chapter V: Theorem 7.1.}\nOn the $16$ sample addresses, let $\\Omega=\\{6,\\ldots,11\\}$ and $\\widehat\\Omega=\\{7,\\ldots,12\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, no","\\vpage{18}{Separate amplitude error from mask error}{COUNTING MEASURE / AN EXACT FINITE BUDGET}{Chapter V: Theorem 7.1.}\nOn the $16$ sample addresses, let $\\Omega=\\{6,\\ldots,11\\}$ and $\\widehat\\Omega=\\{7,\\ldots,12\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is at","\\vpage{18}{Separate amplitude error from mask error}{COUNTING MEASURE / AN EXACT FINITE BUDGET}{Chapter V: Theorem 7.1.}\nOn the $16$ sample addresses, let $\\Omega=\\{6,\\ldots,11\\}$ and $\\widehat\\Omega=\\{7,\\ldots,12\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tf","\\vpage{18}{Separate amplitude error from mask error}{COUNTING MEASURE / AN EXACT FINITE BUDGET}{Chapter V: Theorem 7.1.}\nOn the $16$ sample addresses, let $\\Omega=\\{6,\\ldots,11\\}$ and $\\widehat\\Omega=\\{7,\\ldots,12\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\wide","\\vpage{18}{Separate amplitude error from mask error}{COUNTING MEASURE / AN EXACT FINITE BUDGET}{Chapter V: Theorem 7.1.}\nOn the $16$ sample addresses, let $\\Omega=\\{6,\\ldots,11\\}$ and $\\widehat\\Omega=\\{7,\\ldots,12\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tf","Separate amplitude error from mask error}{COUNTING MEASURE / AN EXACT FINITE BUDGET}{Chapter V: Theorem 7.1.}\nOn the $16$ sample addresses, let $\\Omega=\\{6,\\ldots,11\\}$ and $\\widehat\\Omega=\\{7,\\ldots,12\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term …",",11\\}$ and $\\widehat\\Omega=\\{7,\\ldots,12\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band","\\}$. Use true amplitude $a=3$ and estimate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changi…","mate $\\widehat a=5/2$. There are six true samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the report…","ue samples, five overlapping samples, and two disagreement samples.\n\\vfig{error_budget}\nThe residual splits exactly as\n\\[\n3\\mathbf1_\\Omega-\\tfrac52\\mathbf1_{\\widehat\\Omega}\n=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, ampl…","=\\tfrac12\\mathbf1_\\Omega+\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, amplitude error, and background error are separate mathematical inputs. A single reconstruction score conceals that separation.}","\\tfrac52(\\mathbf1_\\Omega-\\mathbf1_{\\widehat\\Omega}).\n\\]\nThe missed sample has error $3$, the five shared samples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, amplitude error, and background error are separate mathematical inputs. A single reconstruction score conceals that separation.}","amples have error $1/2$, and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, amplitude error, and background error are separate mathematical inputs. A single reconstruction score conceals that separation.}","and the extra sample has error $-5/2$. With \\emph{counting measure},\n\\[\n\\|j-\\widehat j\\|_1=8,\n\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, amplitude error, and background error are separate mathematical inputs. A single reconstruction score conceals that separation.}","\\qquad \\|j-\\widehat j\\|_2^2=\\tfrac{33}{2},\n\\qquad \\|j-\\widehat j\\|_\\infty=3.\n\\]\n\\head{Read the theorem as a budget, not an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, amplitude error, and background error are separate mathematical inputs. A single reconstruction score conceals that separation.}","an equality claim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, amplitude error, and background error are separate mathematical inputs. A single reconstruction score conceals that separation.}","laim}\nHere the $L^1$ bound is attained:\n\\[\n|3-\\tfrac52|\\,|\\Omega|+\\tfrac52|\\Omega\\triangle\\widehat\\Omega|\n=\\tfrac12\\cdot6+\\tfrac52\\cdot2=8.\n\\]\nThe $L^2$ triangle bound is $\\frac12\\sqrt6+\\frac52\\sqrt2$, and need not be sharp. A background estimate adds another term $\\|s-\\widehat s\\|_p$. Cancellation between terms can make the total smaller than the bound.\nNormalized RMSE divides the $L^2$ norm by $\\sqrt{16}$. A boundary-band RMSE uses the band size instead. Changing that denominator changes the reported quantity.\n\\takeaway{Mask error, amplitude error, and background error are separate mathematical inputs. A single reconstruction score conceals that separation.}","\\vpage{19}{Geometry localizes error while data may leave it ambiguous}{A FINITE IMAGE / AN AMBIGUOUS OBSERVATION}{Chapter V: Section 7, Section 9.}\nOn the integer $8\\times8$ grid, choose the disk-shaped mask\n$\\Omega=\\{(i,j):(i-3)^2+(j-3)^2\\le5\\}$ and translate it one pixel to the right. Each mask contains $21$ pixels. Their symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $","\\vpage{19}{Geometry localizes error while data may leave it ambiguous}{A FINITE IMAGE / AN AMBIGUOUS OBSERVATION}{Chapter V: Section 7, Section 9.}\nOn the integer $8\\times8$ grid, choose the disk-shaped mask\n$\\Omega=\\{(i,j):(i-3)^2+(j-3)^2\\le5\\}$ and translate it one pixel to the right. Each mask contains $21$ pixels. Their symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiv","\\vpage{19}{Geometry localizes error while data may leave it ambiguous}{A FINITE IMAGE / AN AMBIGUOUS OBSERVATION}{Chapter V: Section 7, Section 9.}\nOn the integer $8\\times8$ grid, choose the disk-shaped mask\n$\\Omega=\\{(i,j):(i-3)^2+(j-3)^2\\le5\\}$ and translate it one pixel to the right. Each mask contains $21$ pixels. Their symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select","\\vpage{19}{Geometry localizes error while data may leave it ambiguous}{A FINITE IMAGE / AN AMBIGUOUS OBSERVATION}{Chapter V: Section 7, Section 9.}\nOn the integer $8\\times8$ grid, choose the disk-shaped mask\n$\\Omega=\\{(i,j):(i-3)^2+(j-3)^2\\le5\\}$ and translate it one pixel to the right. Each mask contains $21$ pixels. Their symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional","E / AN AMBIGUOUS OBSERVATION}{Chapter V: Section 7, Section 9.}\nOn the integer $8\\times8$ grid, choose the disk-shaped mask\n$\\Omega=\\{(i,j):(i-3)^2+(j-3)^2\\le5\\}$ and translate it one pixel to the right. Each mask contains $21$ pixels. Their symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the","ction 7, Section 9.}\nOn the integer $8\\times8$ grid, choose the disk-shaped mask\n$\\Omega=\\{(i,j):(i-3)^2+(j-3)^2\\le5\\}$ and translate it one pixel to the right. Each mask contains $21$ pixels. Their symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules …","Omega=\\{(i,j):(i-3)^2+(j-3)^2\\le5\\}$ and translate it one pixel to the right. Each mask contains $21$ pixels. Their symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules out a guarantee at the \\emph{stated} noise radius and signal pair, not all recovery…","symmetric difference contains exactly $10$.\n\\vfig{mask_geometry}\n\\head{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules out a guarantee at the \\emph{stated} noise radius and signal pair, not all recovery results. The manuscript's positive-contrast estimator is deliberately narrower than a universal segmentation method…","ead{A geometric error calculation is not a segmentation algorithm}\nFor a common amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules out a guarantee at the \\emph{stated} noise radius and signal pair, not all recovery results. The manuscript's positive-contrast estimator is deliberately narrower than a universal segmentation method.\n\\takeaway{Exact synthesis can preserve an identified boundary. It…","amplitude $a$, the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules out a guarantee at the \\emph{stated} noise radius and signal pair, not all recovery results. The manuscript's positive-contrast estimator is deliberately narrower than a universal segmentation method.\n\\takeaway{Exact synthesis can preserve an identified boundary. It cannot decide between two boundaries that the observation model makes indisting…",", the squared counting-measure error is $10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules out a guarantee at the \\emph{stated} noise radius and signal pair, not all recovery results. The manuscript's positive-contrast estimator is deliberately narrower than a universal segmentation method.\n\\takeaway{Exact synthesis can preserve an identified boundary. It cannot decide between two boundaries that the observation model makes indistinguishable.}","$10a^2$, entirely on the disagreement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules out a guarantee at the \\emph{stated} noise radius and signal pair, not all recovery results. The manuscript's positive-contrast estimator is deliberately narrower than a universal segmentation method.\n\\takeaway{Exact synthesis can preserve an identified boundary. It cannot decide between two boundaries that the observation model makes indistinguishable.}","greement pixels. The full-image mean squared error is $10a^2/64$. Both are exact, but answer different normalization questions.\nNow consider the four-sample observation\n\\[\ny=(0,\\tfrac32,3,3).\n\\]\nIt is consistent with either true signal\n\\[\nu=(0,0,3,3),\\qquad v=(0,3,3,3)\n\\]\nunder deterministic sup-norm noise at most $3/2$. Indeed $y=(u+v)/2$, and both error norms equal $3/2$. A rule receiving only $y$ and that noise bound cannot be guaranteed to select both distinct true masks correctly.\n\\head{What additional structure would have to do}\nA lower noise level, separation, contrast, geometry, or a specified prior can change the identification problem. The example rules out a guarantee at the \\emph{stated} noise radius and signal pair, not all recovery results. The manuscript's positive-contrast estimator is deliberately narrower than a universal segmentation method.\n\\takeaway{Exact synthesis can preserve an identified boundary. It cannot decide between two boundaries that the observation model makes indistinguishable.}","packet laws}{POINTWISE SIGNALS VERSUS CAUCHY COEFFICIENTS}{Chapter V: Theorem 12.1.}\nThe word ``product'' must be typed. Multiplying two signal values at the same time address is not the same operation as multiplying generating series. The distinction changes which packets can interact.\n\\vfig{product_conventions}\n\\head{Pointwise multiplication preserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identit…","US CAUCHY COEFFICIENTS}{Chapter V: Theorem 12.1.}\nThe word ``product'' must be typed. Multiplying two signal values at the same time address is not the same operation as multiplying generating series. The distinction changes which packets can interact.\n\\vfig{product_conventions}\n\\head{Pointwise multiplication preserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assu…","S}{Chapter V: Theorem 12.1.}\nThe word ``product'' must be typed. Multiplying two signal values at the same time address is not the same operation as multiplying generating series. The distinction changes which packets can interact.\n\\vfig{product_conventions}\n\\head{Pointwise multiplication preserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Bef…","he distinction changes which packets can interact.\n\\vfig{product_conventions}\n\\head{Pointwise multiplication preserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","kets can interact.\n\\vfig{product_conventions}\n\\head{Pointwise multiplication preserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","vfig{product_conventions}\n\\head{Pointwise multiplication preserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","ltiplication preserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","eserves the same address}\nFor time-residue masks, $(P_{N,i}x)[n](P_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","_{N,j}y)[n]=0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","0$ when $i\\ne j$. Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","Therefore\n\\[\nP_{N,k}(x\\cdot y)=(P_{N,k}x)\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}",")\\cdot(P_{N,k}y).\n\\]\nHere the dot means multiplication at the same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","e same sample, not convolution.\n\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","\\head{Cauchy multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","y multiplication adds degrees}\nFor formal series $A(t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","t),B(t)$, the coefficient of $t^n$ is a sum over $a+b=n$. Consequently the output residue is $i+j\\pmod N$:\n\\[\nP_{N,k}(AB)=\\sum_{i+j\\equiv k\\ (N)}(P_{N,i}A)(P_{N,j}B).\n\\]\nFor $N=3$, take $A=1+2t+3t^2$ and $B=2+t^2$. Direct multiplication gives\n\\[\nAB=2+4t+7t^2+2t^3+3t^4,\n\\qquad P_{3,0}(AB)=2+2t^3.\n\\]\nThe $2t^3$ term comes from residues $1+2$, not $0+0$. At power zero, the empty product contributes $1$ only in residue zero. These are formal coefficient identities and require no convergence assumption.\n\\takeaway{Before applying a packet multiplication rule, specify whether multiplication acts on sample values or on series coefficients.}","\\vpage{21}{Nonlinear neighbors cross channel boundaries}{FULL INPUT TUPLES / EXACT CYCLIC INDEXING}{Chapter V: Section 12.}\nFor the periodic running signal define\n\\[\nF(x)[n]=x[n]+\\tfrac12x[n]x[n-1].\n\\]\nThis is nonlinear and not a convolution operator. Nevertheless its two-channel representation is exact, provided the neighbor comes from the correct channel and index.\n\\vfig{nonlinear_shift}\nPut $e[q]=x[2q]$ and $o[q]=x[2q+1]$, with channel indices taken modulo eight. Then\n\\[\n\\begin{aligned}\nF(x)[2q]&=e[q]+\\tfrac12 e[q]o[q-1],","inear neighbors cross channel boundaries}{FULL INPUT TUPLES / EXACT CYCLIC INDEXING}{Chapter V: Section 12.}\nFor the periodic running signal define\n\\[\nF(x)[n]=x[n]+\\tfrac12x[n]x[n-1].\n\\]\nThis is nonlinear and not a convolution operator. Nevertheless its two-channel representation is exact, provided the neighbor comes from the correct channel and index.\n\\vfig{nonlinear_shift}\nPut $e[q]=x[2q]$ and $o[q]=x[2q+1]$, with channel indices taken modulo eight. Then\n\\[\n\\begin{aligned}\nF(x)[2q]&=e[q]+\\tfrac12 e[q]o[q-1],","XACT CYCLIC INDEXING}{Chapter V: Section 12.}\nFor the periodic running signal define\n\\[\nF(x)[n]=x[n]+\\tfrac12x[n]x[n-1].\n\\]\nThis is nonlinear and not a convolution operator. Nevertheless its two-channel representation is exact, provided the neighbor comes from the correct channel and index.\n\\vfig{nonlinear_shift}\nPut $e[q]=x[2q]$ and $o[q]=x[2q+1]$, with channel indices taken modulo eight. Then\n\\[\n\\begin{aligned}\nF(x)[2q]&=e[q]+\\tfrac12 e[q]o[q-1],\\\\\nF(x)[2q+1]&=o[q]+\\tfrac12 o[q]e[q].\n\\end{aligned}\n\\]\nThe even output needs the \\emph{previous} odd sample. The odd output needs the current even sample. This is the same carry in the address system that appeared in the one-sample delay matrix.\n\\head{Check two actual outputs}\nThe running values are $x[1]=1/4$, $x[2]=1/2$, and $x[3]=1/4$, so\n\\[\nF(x)[2]=\\tfrac12+\\tfrac12\\tfrac12\\tfrac14=\\tfrac9{16},\n\\qquad F(x)[3]=\\tfrac14+\\tfrac12\\tfrac14\\tfrac12=\\tfrac5{16}.\n\\]\nApplying the same formula independently within each decimated array would use the wrong neighbors. Keeping the full channel tuple avoids that loss of information. The supplied suit…","F(x)[2q+1]&=o[q]+\\tfrac12 o[q]e[q].\n\\end{aligned}\n\\]\nThe even output needs the \\emph{previous} odd sample. The odd output needs the current even sample. This is the same carry in the address system that appeared in the one-sample delay matrix.\n\\head{Check two actual outputs}\nThe running values are $x[1]=1/4$, $x[2]=1/2$, and $x[3]=1/4$, so\n\\[\nF(x)[2]=\\tfrac12+\\tfrac12\\tfrac12\\tfrac14=\\tfrac9{16},\n\\qquad F(x)[3]=\\tfrac14+\\tfrac12\\tfrac14\\tfrac12=\\tfrac5{16}.\n\\]\nApplying the same formula independently within each decimated array would use the wrong neighbors. Keeping the full channel tuple avoids that loss of information. The supplied suite compares every output sample in the two implementations using exact rational arithmetic.\nNo convergence, contraction, or approximation property follows merely from writing these exact coordinates. Those require hypotheses on the nonlinear map and the problem being solved.\n\\takeaway{Exact packet coordinates remain available for nonlinear maps. They do not turn","F(x)[2q+1]&=o[q]+\\tfrac12 o[q]e[q].\n\\end{aligned}\n\\]\nThe even output needs the \\emph{previous} odd sample. The odd output needs the current even sample. This is the same carry in the address system that appeared in the one-sample delay matrix.\n\\head{Check two actual outputs}\nThe running values are $x[1]=1/4$, $x[2]=1/2$, and $x[3]=1/4$, so\n\\[\nF(x)[2]=\\tfrac12+\\tfrac12\\tfrac12\\tfrac14=\\tfrac9{16},\n\\qquad F(x)[3]=\\tfrac14+\\tfrac12\\tfrac14\\tfrac12=\\tfrac5{16}.\n\\]\nApplying the same formula independently within each decimated array would use the wrong neighbors. Keeping the full channel tuple avoids that loss of information. The supplied suite compares every output sample in the two implementations using exact rational arithmetic.\nNo convergence, contraction, or approximation property follows merely from writing these exact coordinates. Those require hypotheses on the nonlinear map and the problem being solved.\n\\takeaway{Exact packet coordinates remain available for nonlinear maps. They do not turn cross-chann","F(x)[2q+1]&=o[q]+\\tfrac12 o[q]e[q].\n\\end{aligned}\n\\]\nThe even output needs the \\emph{previous} odd sample. The odd output needs the current even sample. This is the same carry in the address system that appeared in the one-sample delay matrix.\n\\head{Check two actual outputs}\nThe running values are $x[1]=1/4$, $x[2]=1/2$, and $x[3]=1/4$, so\n\\[\nF(x)[2]=\\tfrac12+\\tfrac12\\tfrac12\\tfrac14=\\tfrac9{16},\n\\qquad F(x)[3]=\\tfrac14+\\tfrac12\\tfrac14\\tfrac12=\\tfrac5{16}.\n\\]\nApplying the same formula independently within each decimated array would use the wrong neighbors. Keeping the full channel tuple avoids that loss of information. The supplied suite compares every output sample in the two implementations using exact rational arithmetic.\nNo convergence, contraction, or approximation property follows merely from writing these exact coordinates. Those require hypotheses on the nonlinear map and the problem being solved.\n\\takeaway{Exact packet coordinates remain available for nonlinear maps. They do not turn cross-channel interactions","F(x)[2q+1]&=o[q]+\\tfrac12 o[q]e[q].\n\\end{aligned}\n\\]\nThe even output needs the \\emph{previous} odd sample. The odd output needs the current even sample. This is the same carry in the address system that appeared in the one-sample delay matrix.\n\\head{Check two actual outputs}\nThe running values are $x[1]=1/4$, $x[2]=1/2$, and $x[3]=1/4$, so\n\\[\nF(x)[2]=\\tfrac12+\\tfrac12\\tfrac12\\tfrac14=\\tfrac9{16},\n\\qquad F(x)[3]=\\tfrac14+\\tfrac12\\tfrac14\\tfrac12=\\tfrac5{16}.\n\\]\nApplying the same formula independently within each decimated array would use the wrong neighbors. Keeping the full channel tuple avoids that loss of information. The supplied suite compares every output sample in the two implementations using exact rational arithmetic.\nNo convergence, contraction, or approximation property follows merely from writing these exact coordinates. Those require hypotheses on the nonlinear map and the problem being solved.\n\\takeaway{Exact packet coordinates remain available for nonlinear maps. They do not turn cross-channel interactions into independent channel problems.}","\\vpage{22}{One reflection circle, four very different impedances}{THE CAYLEY CHART / EXACT QUARTER TURNS}{Chapter V: Theorem 13.2.}\nUse normalized impedance $z$ and reflection coefficient $\\Gamma$:\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThe circle $|\\Gamma|=1/2$ is a constant-mismatch orbit. Its four quarter-turn points map to the exact impedances shown below.\n\\vfig{cayley_orbit}\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$\\Gamma$ & $1/2$ & $i/2$ & $-1/2$ & $-i/2$","rent impedances}{THE CAYLEY CHART / EXACT QUARTER TURNS}{Chapter V: Theorem 13.2.}\nUse normalized impedance $z$ and reflection coefficient $\\Gamma$:\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThe circle $|\\Gamma|=1/2$ is a constant-mismatch orbit. Its four quarter-turn points map to the exact impedances shown below.\n\\vfig{cayley_orbit}\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$\\Gamma$ & $1/2$ & $i/2$ & $-1/2$ & $-i/2$","nces}{THE CAYLEY CHART / EXACT QUARTER TURNS}{Chapter V: Theorem 13.2.}\nUse normalized impedance $z$ and reflection coefficient $\\Gamma$:\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThe circle $|\\Gamma|=1/2$ is a constant-mismatch orbit. Its four quarter-turn points map to the exact impedances shown below.\n\\vfig{cayley_orbit}\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$\\Gamma$ & $1/2$ & $i/2$ & $-1/2$ & $-i/2$","E CAYLEY CHART / EXACT QUARTER TURNS}{Chapter V: Theorem 13.2.}\nUse normalized impedance $z$ and reflection coefficient $\\Gamma$:\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThe circle $|\\Gamma|=1/2$ is a constant-mismatch orbit. Its four quarter-turn points map to the exact impedances shown below.\n\\vfig{cayley_orbit}\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$\\Gamma$ & $1/2$ & $i/2$ & $-1/2$ & $-i/2$","CHART / EXACT QUARTER TURNS}{Chapter V: Theorem 13.2.}\nUse normalized impedance $z$ and reflection coefficient $\\Gamma$:\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThe circle $|\\Gamma|=1/2$ is a constant-mismatch orbit. Its four quarter-turn points map to the exact impedances shown below.\n\\vfig{cayley_orbit}\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$\\Gamma$ & $1/2$ & $i/2$ & $-1/2$ & $-i/2$","EXACT QUARTER TURNS}{Chapter V: Theorem 13.2.}\nUse normalized impedance $z$ and reflection coefficient $\\Gamma$:\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThe circle $|\\Gamma|=1/2$ is a constant-mismatch orbit. Its four quarter-turn points map to the exact impedances shown below.\n\\vfig{cayley_orbit}\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$\\Gamma$ & $1/2$ & $i/2$ & $-1/2$ & $-i/2$","hline\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infini","hline\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. Th","hline\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior ma","hline\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to th","hline\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right","hline\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right half-plane.\nThe classical Smith grid supplies the geometry. The cyclic labels specify exact rotations on it. Neither a new physical law nor a new optimal sampling theorem is implied by this coordinate description.\n\\takeaway{A simple action in one chart can be nonlinear in another. Conjugacy preserves the action law, not the apparent","hline\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right half-plane.\nThe classical Smith grid supplies the geometry. The cyclic labels specify exact rotations on it. Neither a new physical law nor a new optimal sampling theorem is implied by this coordinate description.\n\\takeaway{A simple action in one chart can be nonlinear in another. Conjugacy preserves the action law, not the apparent shape of a","ne\n$z$ & $3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right half-plane.\nThe classical Smith grid supplies the geometry. The cyclic labels specify exact rotations on it. Neither a new physical law nor a new optimal sampling theorem is implied by this coordinate description.\n\\takeaway{A simple action in one chart can be nonlinear in another. Conjugacy preserves the action law, not the apparent shape of a displacem…","$3$ & $3/5+4i/5$ & $1/3$ & $3/5-4i/5$\n\\end{tabular}\n\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right half-plane.\nThe classical Smith grid supplies the geometry. The cyclic labels specify exact rotations on it. Neither a new physical law nor a new optimal sampling theorem is implied by this coordinate description.\n\\takeaway{A simple action in one chart can be nonlinear in another. Conjugacy preserves the action law, not the apparent shape of a displacement.}","\\end{center}\n\\head{The coordinates preserve structure, not Euclidean distances}\nA quarter turn is a rotation in the reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right half-plane.\nThe classical Smith grid supplies the geometry. The cyclic labels specify exact rotations on it. Neither a new physical law nor a new optimal sampling theorem is implied by this coordinate description.\n\\takeaway{A simple action in one chart can be nonlinear in another. Conjugacy preserves the action law, not the apparent shape of a displacement.}","he reflection disk. After pulling back, it is a M\\\"obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right half-plane.\nThe classical Smith grid supplies the geometry. The cyclic labels specify exact rotations on it. Neither a new physical law nor a new optimal sampling theorem is implied by this coordinate description.\n\\takeaway{A simple action in one chart can be nonlinear in another. Conjugacy preserves the action law, not the apparent shape of a displacement.}","s map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points $1/2$ and $-1/2$ become $3$ and $1/3$. The continuous impedance locus is the circle $(\\operatorname{Re}z-5/3)^2+(\\operatorname{Im}z)^2=16/9$, also drawn in the plate.\nFinite passive impedances with $\\operatorname{Re}z\\ge0$ map to the closed disk except $\\Gamma=1$. That missing point is obtained only by adjoining impedance infinity. The interior maps to the open right half-plane.\nThe classical Smith grid supplies the geometry. The cyclic labels specify exact rotations on it. Neither a new physical law nor a new optimal sampling theorem is implied by this coordinate description.\n\\takeaway{A simple action in one chart can be nonlinear in another. Conjugacy preserves the action law, not the apparent shape of a displacement.}","\\vpage{23}{A pole is a chart boundary, not a broken group action}{HOMOGENEOUS COORDINATES / NO DIVISION BY ZERO}{Chapter V: Theorem 13.2, Corollary 13.3.}\nRepresent impedance by a nonzero pair $[u:v]$, where $[cu:cv]=[u:v]$ for $c\\ne0$. A finite impedance is $[z:1]$. Infinity is $[1:0]$. For a nonzero reflection multiplier $w$, use\n\\[\nA(w)=\\frac12\\begin{pmatrix}1+w&1-w","\\vpage{23}{A pole is a chart boundary, not a broken group action}{HOMOGENEOUS COORDINATES / NO DIVISION BY ZERO}{Chapter V: Theorem 13.2, Corollary 13.3.}\nRepresent impedance by a nonzero pair $[u:v]$, where $[cu:cv]=[u:v]$ for $c\\ne0$. A finite impedance is $[z:1]$. Infinity is $[1:0]$. For a nonzero reflection multiplier $w$, use\n\\[\nA(w)=\\frac12\\begin{pmatrix}1+w&1-w\\\\1-w&1+w\\end{pmatrix}.\n\\]\n\\vfig{projective_poles}\n\\head{The action is defined before a ratio is taken}\nThe determinant is $w$, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase mul","1-w&1+w\\end{pmatrix}.\n\\]\n\\vfig{projective_poles}\n\\head{The action is defined before a ratio is taken}\nThe determinant is $w$, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and thei","1-w&1+w\\end{pmatrix}.\n\\]\n\\vfig{projective_poles}\n\\head{The action is defined before a ratio is taken}\nThe determinant is $w$, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homog","1-w&1+w\\end{pmatrix}.\n\\]\n\\vfig{projective_poles}\n\\head{The action is defined before a ratio is taken}\nThe determinant is $w$, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pair","1-w&1+w\\end{pmatrix}.\n\\]\n\\vfig{projective_poles}\n\\head{The action is defined before a ratio is taken}\nThe determinant is $w$, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","t is $w$, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","ltiplication gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","tion gives\n\\[\nA(w)A(w')=A(ww'),\\qquad A(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","(1)=I.\n\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","\\]\nFor $w=i$, the orbit through zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","hrough zero is\n\\[\n[0:1]\\ \\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","\\longmapsto\\ [-i:1]\\ \\longmapsto\\ [1:0]\n\\ \\longmapsto\\ [i:1]\\ \\longmapsto\\ [0:1].\n\\]\nThe third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","terate is $A(-1)$, which swaps the two coordinates. Thus a half turn exchanges $0$ and $\\infty$ and sends each finite nonzero $z$ to $1/z$.\nThe matched point $\\Gamma=0$, equivalently $z=1$, is fixed by every phase multiplier. It is not part of the boundary orbit above. The exact checks test homogeneous pairs and their rescaling equivalence, rather than introducing an artificial complex value for $1/0$.\n\\takeaway{Compute with homogeneous pairs first. A denominator becomes a chart-selection issue rather than an exception to the cyclic action.}","\\vpage{24}{Residue labels are not impedance averages}{DYADIC--ODD FACTORIZATION / TWELVE ADDRESSES}{Chapter V: Proposition 14.1.}\nAt phase order $12=4\\cdot3$, the map\n\\[\nk\\longmapsto(k\\bmod4,\\ k\\bmod3)\n\\]\nis a bijection from twelve residue addresses to the $4\\times3$ grid. Every cell below contains its unique original address.\n\\vfig{crt_refinement}\n\\head{The factorization is an intersection of conditions}\nFor integer sample addresses, including negative ones,\n\\[\nP_{12,k}=P_{4,k\\bmod4}\\,P_{3,k\\bmod3}.\n\\]\nThe product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their intersection a single class modulo twelve. It is not a formula for multiplying or adding impedances.\n\\head{Nested samples and nested projectors are different statements}\nThe four coarse reflection rays occur at indices $0,3,6,9$ of t","\\vpage{24}{Residue labels are not impedance averages}{DYADIC--ODD FACTORIZATION / TWELVE ADDRESSES}{Chapter V: Proposition 14.1.}\nAt phase order $12=4\\cdot3$, the map\n\\[\nk\\longmapsto(k\\bmod4,\\ k\\bmod3)\n\\]\nis a bijection from twelve residue addresses to the $4\\times3$ grid. Every cell below contains its unique original address.\n\\vfig{crt_refinement}\n\\head{The factorization is an intersection of conditions}\nFor integer sample addresses, including negative ones,\n\\[\nP_{12,k}=P_{4,k\\bmod4}\\,P_{3,k\\bmod3}.\n\\]\nThe product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their intersection a single class modulo twelve. It is not a formula for multiplying or adding impedances.\n\\head{Nested samples and nested projectors are different statements}\nThe four coarse reflection rays occur at indices $0,3,6,9$ of the twelve-ray grid. Thus the geometric sample","\\vpage{24}{Residue labels are not impedance averages}{DYADIC--ODD FACTORIZATION / TWELVE ADDRESSES}{Chapter V: Proposition 14.1.}\nAt phase order $12=4\\cdot3$, the map\n\\[\nk\\longmapsto(k\\bmod4,\\ k\\bmod3)\n\\]\nis a bijection from twelve residue addresses to the $4\\times3$ grid. Every cell below contains its unique original address.\n\\vfig{crt_refinement}\n\\head{The factorization is an intersection of conditions}\nFor integer sample addresses, including negative ones,\n\\[\nP_{12,k}=P_{4,k\\bmod4}\\,P_{3,k\\bmod3}.\n\\]\nThe product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their intersection a single class modulo twelve. It is not a formula for multiplying or adding impedances.\n\\head{Nested samples and nested projectors are different statements}\nThe four coarse reflection rays occur at indices $0,3,6,9$ of the twelve-ray grid. Thus the geometric sample set is nested. Separately, a coarse coefficient projector is the","FACTORIZATION / TWELVE ADDRESSES}{Chapter V: Proposition 14.1.}\nAt phase order $12=4\\cdot3$, the map\n\\[\nk\\longmapsto(k\\bmod4,\\ k\\bmod3)\n\\]\nis a bijection from twelve residue addresses to the $4\\times3$ grid. Every cell below contains its unique original address.\n\\vfig{crt_refinement}\n\\head{The factorization is an intersection of conditions}\nFor integer sample addresses, including negative ones,\n\\[\nP_{12,k}=P_{4,k\\bmod4}\\,P_{3,k\\bmod3}.\n\\]\nThe product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their intersection a single class modulo twelve. It is not a formula for multiplying or adding impedances.\n\\head{Nested samples and nested projectors are different statements}\nThe four coarse reflection rays occur at indices $0,3,6,9$ of the twelve-ray grid. Thus the geometric sample set is nested. Separately, a coarse coefficient projector is the sum of its fine descendants in a common coefficient space.\nThere is no corresponding rule saying that a coarse impedance is the average of its descendants. The inverse Cayley map is nonlinear. Even o…","uding negative ones,\n\\[\nP_{12,k}=P_{4,k\\bmod4}\\,P_{3,k\\bmod3}.\n\\]\nThe product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their intersection a single class modulo twelve. It is not a formula for multiplying or adding impedances.\n\\head{Nested samples and nested projectors are different statements}\nThe four coarse reflection rays occur at indices $0,3,6,9$ of the twelve-ray grid. Thus the geometric sample set is nested. Separately, a coarse coefficient projector is the sum of its fine descendants in a common coefficient space.\nThere is no corresponding rule saying that a coarse impedance is the average of its descendants. The inverse Cayley map is nonlinear. Even opposite reflection points have impedances $3$ and $1/3$ in the preceding example. Their average $5/3$ is not the impedance $1$ of their averaged reflection coefficient zero.\n\\takeaway{CRT factors addresses. Refinement conserves coefficient packets. Neither operation authorizes linear averaging through a nonlinear chart.}","reflection rays occur at indices $0,3,6,9$ of the twelve-ray grid. Thus the geometric sample set is nested. Separately, a coarse coefficient projector is the sum of its fine descendants in a common coefficient space.\nThere is no corresponding rule saying that a coarse impedance is the average of its descendants. The inverse Cayley map is nonlinear. Even opposite reflection points have impedances $3$ and $1/3$ in the preceding example. Their average $5/3$ is not the impedance $1$ of their averaged reflection coefficient zero.\n\\takeaway{CRT factors addresses. Refinement conserves coefficient packets. Neither operation authorizes linear averaging through a nonlinear chart.}","on rays occur at indices $0,3,6,9$ of the twelve-ray grid. Thus the geometric sample set is nested. Separately, a coarse coefficient projector is the sum of its fine descendants in a common coefficient space.\nThere is no corresponding rule saying that a coarse impedance is the average of its descendants. The inverse Cayley map is nonlinear. Even opposite reflection points have impedances $3$ and $1/3$ in the preceding example. Their average $5/3$ is not the impedance $1$ of their averaged reflection coefficient zero.\n\\takeaway{CRT factors addresses. Refinement conserves coefficient packets. Neither operation authorizes linear averaging through a nonlinear chart.}","ve-ray grid. Thus the geometric sample set is nested. Separately, a coarse coefficient projector is the sum of its fine descendants in a common coefficient space.\nThere is no corresponding rule saying that a coarse impedance is the average of its descendants. The inverse Cayley map is nonlinear. Even opposite reflection points have impedances $3$ and $1/3$ in the preceding example. Their average $5/3$ is not the impedance $1$ of their averaged reflection coefficient zero.\n\\takeaway{CRT factors addresses. Refinement conserves coefficient packets. Neither operation authorizes linear averaging through a nonlinear chart.}","tric sample set is nested. Separately, a coarse coefficient projector is the sum of its fine descendants in a common coefficient space.\nThere is no corresponding rule saying that a coarse impedance is the average of its descendants. The inverse Cayley map is nonlinear. Even opposite reflection points have impedances $3$ and $1/3$ in the preceding example. Their average $5/3$ is not the impedance $1$ of their averaged reflection coefficient zero.\n\\takeaway{CRT factors addresses. Refinement conserves coefficient packets. Neither operation authorizes linear averaging through a nonlinear chart.}","\\vpage{25}{Sensitivity grows before the pole is reached}{LOCAL DERIVATIVES / FINITE INCREMENTS}{Chapter V: Section 14.}\nAway from $\\Gamma=1$, the inverse Cayley map satisfies\n\\[\n\\frac{dz}{d\\Gamma}=\\frac2{(1-\\Gamma)^2}.\n\\]\nAlong the real axis, three exact derivatives already show the scale change:\n\\[\n\\Gamma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place","\\vpage{25}{Sensitivity grows before the pole is reached}{LOCAL DERIVATIVES / FINITE INCREMENTS}{Chapter V: Section 14.}\nAway from $\\Gamma=1$, the inverse Cayley map satisfies\n\\[\n\\frac{dz}{d\\Gamma}=\\frac2{(1-\\Gamma)^2}.\n\\]\nAlong the real axis, three exact derivatives already show the scale change:\n\\[\n\\Gamma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{W","\\vpage{25}{Sensitivity grows before the pole is reached}{LOCAL DERIVATIVES / FINITE INCREMENTS}{Chapter V: Section 14.}\nAway from $\\Gamma=1$, the inverse Cayley map satisfies\n\\[\n\\frac{dz}{d\\Gamma}=\\frac2{(1-\\Gamma)^2}.\n\\]\nAlong the real axis, three exact derivatives already show the scale change:\n\\[\n\\Gamma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\","hapter V: Section 14.}\nAway from $\\Gamma=1$, the inverse Cayley map satisfies\n\\[\n\\frac{dz}{d\\Gamma}=\\frac2{(1-\\Gamma)^2}.\n\\]\nAlong the real axis, three exact derivatives already show the scale change:\n\\[\n\\Gamma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints…","inverse Cayley map satisfies\n\\[\n\\frac{dz}{d\\Gamma}=\\frac2{(1-\\Gamma)^2}.\n\\]\nAlong the real axis, three exact derivatives already show the scale change:\n\\[\n\\Gamma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpo…",", three exact derivatives already show the scale change:\n\\[\n\\Gamma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpolation rule, or measurement allocation. A finite-error estimate must also control how close the…","vatives already show the scale change:\n\\[\n\\Gamma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpolation rule, or measurement allocation. A finite-error estimate must also control how close the perturbed point c…","mma=\\tfrac12:\\ 8,\\qquad\n\\Gamma=\\tfrac9{10}:\\ 200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpolation rule, or measurement allocation. A finite-error estimate must also control how close the perturbed point comes to either pole.\n\\takeaway{Use the exact …","200,\\qquad\n\\Gamma=\\tfrac{99}{100}:\\ 20000.\n\\]\n\\vfig{sensitivity}\n\\head{A differential is not the full finite-error formula}\nFor a finite increment $\\varepsilon$, with both denominators nonzero,\n\\[\nz(\\Gamma+\\varepsilon)-z(\\Gamma)\n=\\frac{2\\varepsilon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpolation rule, or measurement allocation. A finite-error estimate must also control how close the perturbed point comes to either pole.\n\\takeaway{Use the exact finite difference for an error budget. A larg","lon}{(1-\\Gamma-\\varepsilon)(1-\\Gamma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpolation rule, or measurement allocation. A finite-error estimate must also control how close the perturbed point comes to either pole.\n\\takeaway{Use the exact finite difference for an error budget. A large derivative diagnoses conditioning. Choosing an optimal algorithm requires further input.}","amma)}.\n\\]\nAt $\\Gamma=9/10$ and $\\varepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpolation rule, or measurement allocation. A finite-error estimate must also control how close the perturbed point comes to either pole.\n\\takeaway{Use the exact finite difference for an error budget. A large derivative diagnoses conditioning. Choosing an optimal algorithm requires further input.}","arepsilon=1/100$, the actual increment is $20/9$, while derivative times increment is $2$. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation.\n\\head{What a proposed weight can and cannot certify}\nImpedance is sensitive near $\\Gamma=1$, and admittance near $\\Gamma=-1$. The symmetric weight\n\\[\nw(\\Gamma)=|1-\\Gamma|^{-2}+|1+\\Gamma|^{-2}\n\\]\nreflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does not establish an optimal grid, interpolation rule, or measurement allocation. A finite-error estimate must also control how close the perturbed point comes to either pole.\n\\takeaway{Use the exact finite difference for an error budget. A large derivative diagnoses conditioning. Choosing an optimal algorithm requires further input.}","Finite packet tree & All $26$ complete prunings, actual projector sums, the unique cost-$9$ optimum, and a strict perturbation-margin certificate.","bottomrule\n\\end{tabularx}\n\\head{What each evidence layer means}\n\\textbf{Manuscript theorems.} The Part I formal correspondence maps twenty theorem-like statements to $225$ supporting Lean targets in $23$ proof modules. Its compiler and axiom records provide the retained execution evidence. The formal companion gives their scope and reproduction instructions. The additional Part II developments remain outside this twenty-result map.","\\textbf{Recorded exact examples.} The symbolic visual suite was rerun for the recorded monograph review. All $181$ rational, algebraic, symbolic, and exhaustive finite checks passed. Seven deliberately false assertions were rejected. Numerical samples draw the continuous Fourier and sensitivity curves. They are not promoted to interval enclosures or universal error bounds.","rrect packet. A finite cost optimum depends on the supplied objective. An infinite packet tree, a universal segmentation procedure, minimax recovery rates, and optimal Smith sampling require additional theorems. The core's broad edge-recovery problems are not resolved by these finite examples. Plate V.27 introduces the additional separated-jump recovery results and their explicit hypotheses.\nThe $16$-sample running signal is not the $1024$-sample manuscript experiment. The $26$-pruning example is not its $458{,}330$-pruning benchmark. Their smaller sizes allow every relevant number to be inspected exactly.\n\\takeaway{Coordinates, approximation, identification, optimization, and physical interpretation are separate layers. The examples make their interfaces visible without erasing their hypotheses.}","pends on the supplied objective. An infinite packet tree, a universal segmentation procedure, minimax recovery rates, and optimal Smith sampling require additional theorems. The core's broad edge-recovery problems are not resolved by these finite examples. Plate V.27 introduces the additional separated-jump recovery results and their explicit hypotheses.\nThe $16$-sample running signal is not the $1024$-sample manuscript experiment. The $26$-pruning example is not its $458{,}330$-pruning benchmark. Their smaller sizes allow every relevant number to be inspected exactly.\n\\takeaway{Coordinates, approximation, identification, optimization, and physical interpretation are separate layers. The examples make their interfaces visible without erasing their hypotheses.}","ite packet tree, a universal segmentation procedure, minimax recovery rates, and optimal Smith sampling require additional theorems. The core's broad edge-recovery problems are not resolved by these finite examples. Plate V.27 introduces the additional separated-jump recovery results and their explicit hypotheses.\nThe $16$-sample running signal is not the $1024$-sample manuscript experiment. The $26$-pruning example is not its $458{,}330$-pruning benchmark. Their smaller sizes allow every relevant number to be inspected exactly.\n\\takeaway{Coordinates, approximation, identification, optimization, and physical interpretation are separate layers. The examples make their interfaces visible without erasing their hypotheses.}","mentation procedure, minimax recovery rates, and optimal Smith sampling require additional theorems. The core's broad edge-recovery problems are not resolved by these finite examples. Plate V.27 introduces the additional separated-jump recovery results and their explicit hypotheses.\nThe $16$-sample running signal is not the $1024$-sample manuscript experiment. The $26$-pruning example is not its $458{,}330$-pruning benchmark. Their smaller sizes allow every relevant number to be inspected exactly.\n\\takeaway{Coordinates, approximation, identification, optimization, and physical interpretation are separate layers. The examples make their interfaces visible without erasing their hypotheses.}","\\vpage{28}{Six image phases are six exact address classes}{LATTICE COORDINATES / A RECTANGULAR EXAMPLE}{Chapter V: Theorem E3.2, Theorem E3.5, Remark E2.2.}\nTake $L=2\\ZZ\\times3\\ZZ$ inside $\\ZZ^2$, with representatives\n\\[\nR=\\{0,1\\}\\times\\{0,1,2\\}.\n\\]\nEvery address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on","\\vpage{28}{Six image phases are six exact address classes}{LATTICE COORDINATES / A RECTANGULAR EXAMPLE}{Chapter V: Theorem E3.2, Theorem E3.5, Remark E2.2.}\nTake $L=2\\ZZ\\times3\\ZZ$ inside $\\ZZ^2$, with representatives\n\\[\nR=\\{0,1\\}\\times\\{0,1,2\\}.\n\\]\nEvery address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original la","\\vpage{28}{Six image phases are six exact address classes}{LATTICE COORDINATES / A RECTANGULAR EXAMPLE}{Chapter V: Theorem E3.2, Theorem E3.5, Remark E2.2.}\nTake $L=2\\ZZ\\times3\\ZZ$ inside $\\ZZ^2$, with representatives\n\\[\nR=\\{0,1\\}\\times\\{0,1,2\\}.\n\\]\nEvery address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In","\\vpage{28}{Six image phases are six exact address classes}{LATTICE COORDINATES / A RECTANGULAR EXAMPLE}{Chapter V: Theorem E3.2, Theorem E3.5, Remark E2.2.}\nTake $L=2\\ZZ\\times3\\ZZ$ inside $\\ZZ^2$, with representatives\n\\[\nR=\\{0,1\\}\\times\\{0,1,2\\}.\n\\]\nEvery address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translate","\\vpage{28}{Six image phases are six exact address classes}{LATTICE COORDINATES / A RECTANGULAR EXAMPLE}{Chapter V: Theorem E3.2, Theorem E3.5, Remark E2.2.}\nTake $L=2\\ZZ\\times3\\ZZ$ inside $\\ZZ^2$, with representatives\n\\[\nR=\\{0,1\\}\\times\\{0,1,2\\}.\n\\]\nEvery address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\hea","ZZ^2$, with representatives\n\\[\nR=\\{0,1\\}\\times\\{0,1,2\\}.\n\\]\nEvery address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coars…","\\[\nR=\\{0,1\\}\\times\\{0,1,2\\}.\n\\]\nEvery address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, dec…","very address has a unique expression $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, no…","ession $(i,j)=(r_1,r_2)+(2q_1,3q_2)$. The six channels are $x_r(2q_1,3q_2)=x(r_1+2q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{D…","q_1,r_2+3q_2)$.\n\\vfig[.80]{ext_lattice}\n\\platecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exac…","latecaption{Each cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exact synthesis and the correct norm identity.…","cell shows its residue label. Colors identify six disjoint classes. They do not average neighboring image pixels.}\n\\head{A negative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exact synthesis and the correct norm identity.}","ative address is still unique}\nThe point $(-1,7)$ has representative $(1,1)$ and lattice displacement $(-2,6)$, hence channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exact synthesis and the correct norm identity.}","channel index $(-1,2)$. Floor division gives the correct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exact synthesis and the correct norm identity.}","ct answer in both coordinates.\nFor finite $p$, the channel norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exact synthesis and the correct norm identity.}","el norm uses the sum of the $p$th powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exact synthesis and the correct norm identity.}","powers. At $p=\\infty$, it uses their maximum. Both equal the norm on the original lattice because the cosets partition all addresses. In $\\ell^\\infty$, this coordinate identity does not imply that translates of an impulse are dense: their closure is only $c_0$.\n\\head{Refinement partitions a class again}\nFor $L'=4\\ZZ\\times6\\ZZ\\subset L$, the coarse representative $(1,2)$ has fine representatives\n\\[\n(1,2),\\ (3,2),\\ (1,5),\\ (3,5).\n\\]\nTheir original-lattice masks sum to the coarse mask. As on Plate V.4, decimated arrays are interleaved, not added entrywise.\n\\takeaway{Dimension changes the address tuple. Unique coset decomposition still supplies exact synthesis and the correct norm identity.}","\\vpage{29}{A positive singular-value gap survives perturbation}{FRAME STABILITY / AN EXACT TWO-CHANNEL BANK}{Chapter V: Theorem E5.2, Theorem E5.7, Corollary E5.6.}\nConsider the constant analysis bank\n\\[\nE=\\begin{pmatrix}1&0\\\\0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\he","0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approx","0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate i","0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=","0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturb","0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I","0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error","0&2\\end{pmatrix},\\qquad\nE^*E=\\operatorname{diag}(1,4).\n\\]\nIt has frame bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first co","me bounds $A=1$, $B=4$. For every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential s…","every perturbation with operator norm at most $\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, …","$\\delta=1/4$, the squared-norm bounds remain at least $(1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, not a few sampled frequencies.\n\\takeaway{A low…","1-1/4)^2=9/16$ and at most $(2+1/4)^2=81/16$.\n\\vfig[.87]{ext_frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, not a few sampled frequencies.\n\\takeaway{A lower singular-value gap quantifies stable recovery. A reco…","frame}\n\\head{Both bounds can be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, not a few sampled frequencies.\n\\takeaway{A lower singular-value gap quantifies stable recovery. A reconstruction defect quantifies the error of the actual synthesi…","be attained}\nChoose $\\Delta E=\\operatorname{diag}(-1/4,1/4)$. Then $\\widetilde E=\\operatorname{diag}(3/4,9/4)$ has precisely those two frame bounds. Its inverse has operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, not a few sampled frequencies.\n\\takeaway{A lower singular-value gap quantifies stable recovery. A reconstruction defect quantifies the error of the actual synthesis bank being used.}","s operator norm $4/3$. A coefficient error $\\nu=(3/4,0)$ therefore produces signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, not a few sampled frequencies.\n\\takeaway{A lower singular-value gap quantifies stable recovery. A reconstruction defect quantifies the error of the actual synthesis bank being used.}","s signal error $(1,0)$, attaining the noise bound.\nThe strict condition $\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, not a few sampled frequencies.\n\\takeaway{A lower singular-value gap quantifies stable recovery. A reconstruction defect quantifies the error of the actual synthesis bank being used.}","$\\delta<\\sqrt A$ prevents the lower bound from vanishing. At $\\delta=1$, the perturbation $\\operatorname{diag}(-1,0)$ destroys the first channel entirely.\n\\head{An inverse and an approximate inverse answer different questions}\nIf the original synthesis $R=\\operatorname{diag}(1,1/2)$ is retained after this perturbation, then\n\\[\nR\\widetilde E-I=\\operatorname{diag}(-1/4,1/8).\n\\]\nIts defect norm is $1/4$. Every input has relative reconstruction error at most that value, with equality on the first coordinate. For frequency-dependent banks the theorem uses an essential supremum on the unit circle, not a few sampled frequencies.\n\\takeaway{A lower singular-value gap quantifies stable recovery. A reconstruction defect quantifies the error of the actual synthesis bank being used.}","\\vpage{30}{Nonunitary tree bounds propagate along branches}{LOCAL FRAME CONSTANTS / A MIXED-DEPTH CERTIFICATE}{Chapter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its e","\\vpage{30}{Nonunitary tree bounds propagate along branches}{LOCAL FRAME CONSTANTS / A MIXED-DEPTH CERTIFICATE}{Chapter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may","\\vpage{30}{Nonunitary tree bounds propagate along branches}{LOCAL FRAME CONSTANTS / A MIXED-DEPTH CERTIFICATE}{Chapter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate","\\vpage{30}{Nonunitary tree bounds propagate along branches}{LOCAL FRAME CONSTANTS / A MIXED-DEPTH CERTIFICATE}{Chapter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate must surv","y tree bounds propagate along branches}{LOCAL FRAME CONSTANTS / A MIXED-DEPTH CERTIFICATE}{Chapter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate must survive the least favorable child lower bound. The upper estimate must survive the largest child upper bound. Replacing these by an un…","hes}{LOCAL FRAME CONSTANTS / A MIXED-DEPTH CERTIFICATE}{Chapter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate must survive the least favorable child lower bound. The upper estimate must survive the largest child upper bound. Replacing these by an unweighted average would presume an e…","ME CONSTANTS / A MIXED-DEPTH CERTIFICATE}{Chapter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate must survive the least favorable child lower bound. The upper estimate must survive the largest child upper bound. Replacing these by an unweighted average would presume an energy distribu…","apter V: Proposition E6.4.}\nSuppose the root split has nonnegative squared-norm bounds $A=1/2$ and $B=2$. Retain the left child. Split the right child with bounds $3/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate must survive the least favorable child lower bound. The upper estimate must survive the largest child upper bound. Replacing these by an unweighted average would presume an energy distribution that the hypotheses do not supply.\nWhen…","/4$ and $5/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate must survive the least favorable child lower bound. The upper estimate must survive the largest child upper bound. Replacing these by an unweighted average would presume an energy distribution that the hypotheses do not supply.\nWhen all local maps are unitary, every factor is one and the recursion recovers exact energy conservation. For general local maps, a positive lower bound alone establish…","/4$, then retain both grandchildren.\n\\vfig[.85]{ext_tree}\n\\head{Start with one at every retained node}\nThe left subtree has bounds $a_0=b_0=1$. The right subtree has $a_1=3/4$ and $b_1=5/4$. The recursive certificate gives\n\\[\na_{\\rm root}=\\tfrac12\\min\\{1,\\tfrac34\\}=\\tfrac38,\n\\qquad\nb_{\\rm root}=2\\max\\{1,\\tfrac54\\}=\\tfrac52.\n\\]\nConsequently the total leaf coefficient energy lies between $3/8$ and $5/2$ times the root energy. These are guaranteed bounds. No assertion that every admissible collection of local maps attains both constants is needed.\n\\head{Why a minimum and maximum appear}\nAfter the root split, its energy may distribute unevenly between the two children. The lower estimate must survive the least favorable child lower bound. The upper estimate must survive the largest child upper bound. Replacing these by an unweighted average would presume an energy distribution that the hypotheses do not supply.\nWhen all local maps are unitary, every factor is one and the recursion recovers exact energy conservation. For general local maps, a positive lower bound alone establishes injecti…","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is ex","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has ma","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative t","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for ev","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for every $1\\le","\\vpage{31}{Derivative jumps have their own packets}{TRACE DATA / A STEP AND A RAMP}{Chapter V: Theorem E7.2, Corollary E7.3.}\nOn $[0,1]$, let $H_\\tau=\\mathbf1_{[\\tau,1]}$ and take\n\\[\nf(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.\n\\]\nThe function has a value jump $2$ at $1/3$ and a first-derivative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for every $1\\le p\\le\\infty$. Its second weak derivative is $2$. The singular packet accounts for the distribut","vative jump $-3$ at $2/3$. Its packet and remainder are\n\\[\nj_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for every $1\\le p\\le\\infty$. Its second weak derivative is $2$. The singular packet accounts for the distributional boundary terms that would otherwise arise at the breakpoints.\n\\head{What the general theorem requires}\nEach piece must lie in the stated Sobolev class and possess the specified one-sided traces. Subtracting the finite trace packet then joins those pieces into a global Sobole…","_f(t)=2\\phi_{0,1/3}(t)-3\\phi_{1,2/3}(t),\\qquad s_f(t)=t^2.\n\\]\n\\vfig[.92]{ext_trace}\n\\platecaption{The displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the global smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for every $1\\le p\\le\\infty$. Its second weak derivative is $2$. The singular packet accounts for the distributional boundary terms that would otherwise arise at the breakpoints.\n\\head{What the general theorem requires}\nEach piece must lie in the stated Sobolev class and possess the specified one-sided traces. Subtracting the finite trace packet then joins those pieces into a global Sobolev remainder. This is a theorem about finitely many declared …","l smooth remainder.}\n\\head{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for every $1\\le p\\le\\infty$. Its second weak derivative is $2$. The singular packet accounts for the distributional boundary terms that would otherwise arise at the breakpoints.\n\\head{What the general theorem requires}\nEach piece must lie in the stated Sobolev class and possess the specified one-sided traces. Subtracting the finite trace packet then joins those pieces into a global Sobolev remainder. This is a theorem about finitely many declared breakpoints, not a representation of every possible singular measure or arbitrary image boundary.\n\\takeaway{A value jump uses a step. A slope jump uses a ramp. The derivative order determines the packet and its normalization.}","d{The normalization makes the trace coefficient literal}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for every $1\\le p\\le\\infty$. Its second weak derivative is $2$. The singular packet accounts for the distributional boundary terms that would otherwise arise at the breakpoints.\n\\head{What the general theorem requires}\nEach piece must lie in the stated Sobolev class and possess the specified one-sided traces. Subtracting the finite trace packet then joins those pieces into a global Sobolev remainder. This is a theorem about finitely many declared breakpoints, not a representation of every possible singular measure or arbitrary image boundary.\n\\takeaway{A value jump uses a step. A slope jump uses a ramp. The derivative order determines the packet and its normalization.}","iteral}\nThe general packet is $\\phi_{m,\\tau}(t)=(t-\\tau)_+^m/m!$. Its derivative of order $m$ has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial.\nHere the remainder has matching value and first-derivative traces and belongs to $W^{2,p}(0,1)$ for every $1\\le p\\le\\infty$. Its second weak derivative is $2$. The singular packet accounts for the distributional boundary terms that would otherwise arise at the breakpoints.\n\\head{What the general theorem requires}\nEach piece must lie in the stated Sobolev class and possess the specified one-sided traces. Subtracting the finite trace packet then joins those pieces into a global Sobolev remainder. This is a theorem about finitely many declared breakpoints, not a representation of every possible singular measure or arbitrary image boundary.\n\\takeaway{A value jump uses a step. A slope jump uses a ramp. The derivative order determines the packet and its normalization.}","\\vpage{32}{The approximation operator also acts on packet error}{ESTIMATED PACKETS / AN EXACT AMPLIFICATION EXAMPLE}{Chapter V: Theorem E7.8.}\nLet $Q$ be a bounded \\emph{linear} operator and set $e=j-\\widehat j$. Substituting $f=s+j$ into $\\widehat f=Q(f-\\widehat j)+\\widehat j$ gives\n\\[\nf-\\widehat f=(I-Q)s+(I-Q)e.\n\\]\nThe second term includes the action of $Q$ on the packet error. Linearity is required for this split.\n\\vfig[.80]{ext_amplification}\n\\head{An oblique projection can increase the error}\nIn $\\RR^2$, choose\n\\[\nQ=\\begin{pmatrix}1&2","EXAMPLE}{Chapter V: Theorem E7.8.}\nLet $Q$ be a bounded \\emph{linear} operator and set $e=j-\\widehat j$. Substituting $f=s+j$ into $\\widehat f=Q(f-\\widehat j)+\\widehat j$ gives\n\\[\nf-\\widehat f=(I-Q)s+(I-Q)e.\n\\]\nThe second term includes the action of $Q$ on the packet error. Linearity is required for this split.\n\\vfig[.80]{ext_amplification}\n\\head{An oblique projection can increase the error}\nIn $\\RR^2$, choose\n\\[\nQ=\\begin{pmatrix}1&2","V: Theorem E7.8.}\nLet $Q$ be a bounded \\emph{linear} operator and set $e=j-\\widehat j$. Substituting $f=s+j$ into $\\widehat f=Q(f-\\widehat j)+\\widehat j$ gives\n\\[\nf-\\widehat f=(I-Q)s+(I-Q)e.\n\\]\nThe second term includes the action of $Q$ on the packet error. Linearity is required for this split.\n\\vfig[.80]{ext_amplification}\n\\head{An oblique projection can increase the error}\nIn $\\RR^2$, choose\n\\[\nQ=\\begin{pmatrix}1&2\\\\0&0\\end{pmatrix},\\qquad\ns=(1,0),\\qquad e=(0,1).\n\\]\nThen $Q^2=Q$ and $Qs=s$, but $(I-Q)e=(-2,1)$. The smooth error is zero while the unit packet error becomes a reconstruction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Rec…","0&0\\end{pmatrix},\\qquad\ns=(1,0),\\qquad e=(0,1).\n\\]\nThen $Q^2=Q$ and $Qs=s$, but $(I-Q)e=(-2,1)$. The smooth error is zero while the unit packet error becomes a reconstruction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Recovering the packet and approximating the background are separate tasks, but the chos","0&0\\end{pmatrix},\\qquad\ns=(1,0),\\qquad e=(0,1).\n\\]\nThen $Q^2=Q$ and $Qs=s$, but $(I-Q)e=(-2,1)$. The smooth error is zero while the unit packet error becomes a reconstruction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Recovering the packet and approximating the background are separate tasks, but the chosen approxim","0&0\\end{pmatrix},\\qquad\ns=(1,0),\\qquad e=(0,1).\n\\]\nThen $Q^2=Q$ and $Qs=s$, but $(I-Q)e=(-2,1)$. The smooth error is zero while the unit packet error becomes a reconstruction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Recovering the packet and approximating the background are separate tasks, but the chosen approximation operator couple","0&0\\end{pmatrix},\\qquad\ns=(1,0),\\qquad e=(0,1).\n\\]\nThen $Q^2=Q$ and $Qs=s$, but $(I-Q)e=(-2,1)$. The smooth error is zero while the unit packet error becomes a reconstruction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Recovering the packet and approximating the background are separate tasks, but the chosen approximation operator couples their final errors.}","uad e=(0,1).\n\\]\nThen $Q^2=Q$ and $Qs=s$, but $(I-Q)e=(-2,1)$. The smooth error is zero while the unit packet error becomes a reconstruction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Recovering the packet and approximating the background are separate tasks, but the chosen approximation operator couples their final errors.}","2,1)$. The smooth error is zero while the unit packet error becomes a reconstruction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Recovering the packet and approximating the background are separate tasks, but the chosen approximation operator couples their final errors.}","truction error of norm $\\sqrt5$.\nFor a fully specified realization, take $j=e$ and $\\widehat j=0$. Then $f=(1,1)$, $\\widehat f=Qf=(3,0)$, and their difference is exactly $(-2,1)$.\n\\head{Use the actual operator bound}\nThe estimate that always holds in this setting is\n\\[\n\\|f-\\widehat f\\|\\leq\\|s-Qs\\|+(1+\\|Q\\|)\\|e\\|.\n\\]\nHere $\\|Q\\|=\\sqrt5$, so the bound is conservative. For an orthogonal projection, $\\|I-Q\\|\\le1$ gives a sharper estimate. The operator's geometry determines which bound can be used.\n\\takeaway{Recovering the packet and approximating the background are separate tasks, but the chosen approximation operator couples their final errors.}","\\vpage{33}{An overlap matrix is an exact error ledger}{REGION VALUES / EVERY MISCLASSIFICATION HAS A WEIGHT}{Chapter V: Theorem E8.1, Theorem E8.4.}\nLet two piecewise-constant signals have region values $a=(0,2,5)$ and $b=(0,2,5)$ on a counting-measure domain of twelve samples. Suppose their overlap counts are\n\\[\nM=\\begin{pmatrix}3&1&0","\\vpage{33}{An overlap matrix is an exact error ledger}{REGION VALUES / EVERY MISCLASSIFICATION HAS A WEIGHT}{Chapter V: Theorem E8.1, Theorem E8.4.}\nLet two piecewise-constant signals have region values $a=(0,2,5)$ and $b=(0,2,5)$ on a counting-measure domain of twelve samples. Suppose their overlap counts are\n\\[\nM=\\begin{pmatrix}3&1&0\\\\0&2&1\\\\1&0&4\\end{pmatrix}.\n\\]\nThe row totals $(4,3,5)$ are the true region sizes. The column totals are also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i","1&0&4\\end{pmatrix}.\n\\]\nThe row totals $(4,3,5)$ are the true region sizes. The column totals are also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{i","1&0&4\\end{pmatrix}.\n\\]\nThe row totals $(4,3,5)$ are the true region sizes. The column totals are also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j","1&0&4\\end{pmatrix}.\n\\]\nThe row totals $(4,3,5)$ are the true region sizes. The column totals are also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is app","1&0&4\\end{pmatrix}.\n\\]\nThe row totals $(4,3,5)$ are the true region sizes. The column totals are also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.","1&0&4\\end{pmatrix}.\n\\]\nThe row totals $(4,3,5)$ are the true region sizes. The column totals are also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeawa","1&0&4\\end{pmatrix}.\n\\]\nThe row totals $(4,3,5)$ are the true region sizes. The column totals are also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the …","e also $(4,3,5)$, although three samples are assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the overlap counts. Signal values supply the weights. Their combination gives the norm relevant to …","e assigned to the wrong values.\n\\vfig[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the overlap counts. Signal values supply the weights. Their combination gives the norm relevant to reconstruction.}","[.80]{ext_overlap}\n\\head{Equal error counts need not mean equal reconstruction error}\nThe three off-diagonal entries contribute one sample each. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the overlap counts. Signal values supply the weights. Their combination gives the norm relevant to reconstruction.}","ch. Their value differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the overlap counts. Signal values supply the weights. Their combination gives the norm relevant to reconstruction.}","lue differences have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the overlap counts. Signal values supply the weights. Their combination gives the norm relevant to reconstruction.}","s have magnitudes $2$, $3$, and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the overlap counts. Signal values supply the weights. Their combination gives the norm relevant to reconstruction.}","and $5$, so\n\\[\n\\|j-\\widehat j\\|_1=2+3+5=10,\\qquad\n\\|j-\\widehat j\\|_2^2=4+9+25=38.\n\\]\nThe counting-measure supremum error is $5$. The full-domain mean squared error is $38/12=19/6$. A mismatch fraction of $3/12$ reports a different statistic and omits all three amplitude weights.\n\\head{The formula requires partitions}\nFor measurable partitions $(E_i)$ and $(F_j)$, with $M_{ij}=\\mu(E_i\\cap F_j)$,\n\\[\n\\|j-\\widehat j\\|_p^p=\\sum_{i,j}|a_i-b_j|^pM_{ij},\\qquad 1\\le p<\\infty.\n\\]\nEach location belongs to exactly one intersection. Overlapping indicator packets that are summed at the same location must first be converted to their actual combined value regions before this partition formula is applied.\n\\takeaway{Geometry supplies the overlap counts. Signal values supply the weights. Their combination gives the norm relevant to reconstruction.}","\\vpage{34}{A shifted step and a shifted ramp have different moduli}{LOCATION STABILITY / A ONE-SIXTEENTH DISPLACEMENT}{Chapter V: Corollary E8.2, Theorem E8.3, Theorem E8.7.}\nSet $\\tau=1/4$, $\\widehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncate","\\vpage{34}{A shifted step and a shifted ramp have different moduli}{LOCATION STABILITY / A ONE-SIXTEENTH DISPLACEMENT}{Chapter V: Corollary E8.2, Theorem E8.3, Theorem E8.7.}\nSet $\\tau=1/4$, $\\widehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschit","\\vpage{34}{A shifted step and a shifted ramp have different moduli}{LOCATION STABILITY / A ONE-SIXTEENTH DISPLACEMENT}{Chapter V: Corollary E8.2, Theorem E8.3, Theorem E8.7.}\nSet $\\tau=1/4$, $\\widehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with","\\vpage{34}{A shifted step and a shifted ramp have different moduli}{LOCATION STABILITY / A ONE-SIXTEENTH DISPLACEMENT}{Chapter V: Corollary E8.2, Theorem E8.3, Theorem E8.7.}\nSet $\\tau=1/4$, $\\widehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube","LITY / A ONE-SIXTEENTH DISPLACEMENT}{Chapter V: Corollary E8.2, Theorem E8.3, Theorem E8.7.}\nSet $\\tau=1/4$, $\\widehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an e…","orem E8.3, Theorem E8.7.}\nSet $\\tau=1/4$, $\\widehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1…","heorem E8.7.}\nSet $\\tau=1/4$, $\\widehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere…","ehat\\tau=5/16$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundar…","6$, and $\\delta=1/16$. A unit step differs only on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are clos…","y on the interval between its two locations:\n\\[\n\\|H_\\tau-H_{\\widehat\\tau}\\|_2=\\sqrt\\delta=\\tfrac14,\n\\qquad \\|H_\\tau-H_{\\widehat\\tau}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube…","u}\\|_\\infty=1.\n\\]\n\\vfig[.89]{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decrea…","{ext_location}\n\\head{For a ramp, the error height shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decreases in height.}","eight shrinks too}\nWrite $r_\\tau(t)=(t-\\tau)_+$. The difference grows linearly from zero to $\\delta$ on $[\\tau,\\widehat\\tau]$ and stays at $\\delta$ on $[\\widehat\\tau,1]$. Direct integration gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decreases in height.}","ion gives\n\\[\n\\|r_\\tau-r_{\\widehat\\tau}\\|_2^2\n=\\frac{\\delta^3}{3}+(1-\\widehat\\tau)\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decreases in height.}",")\\delta^2\n=\\frac{17}{6144}<\\frac1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decreases in height.}","c1{256}=\\delta^2.\n\\]\nThus the $L^2$ ramp error is at most $\\delta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decreases in height.}","ta$, and its supremum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decreases in height.}","mum error is exactly $\\delta$. Positive-order truncated powers are Lipschitz in location, with the order-dependent constants supplied by the theorem.\n\\head{The geometric analogue needs a tube hypothesis}\nIf $E\\triangle F\\subseteq N_\\delta(\\partial E)$ and the tube measure is bounded by $C\\delta$ for the stated range $0<\\delta<\\delta_0$, then an equal-amplitude packet has finite-$p$ error at most $|a|(C\\delta)^{1/p}$. A mere visual impression that two boundaries are close does not establish the inclusion or the tube estimate.\n\\takeaway{A displaced value jump keeps its full height on a smaller set. A displaced derivative-jump packet also decreases in height.}","\\vpage{35}{A strict contrast gap locates every separated jump cell}{DETERMINISTIC RECOVERY / A NUMERICAL CERTIFICATE}{Chapter V: Lemma E9.1, Theorem E9.2, Example E9.7.}\nAssume a real signal is a Lipschitz background plus jumps. No jump occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\hea","\\vpage{35}{A strict contrast gap locates every separated jump cell}{DETERMINISTIC RECOVERY / A NUMERICAL CERTIFICATE}{Chapter V: Lemma E9.1, Theorem E9.2, Example E9.7.}\nAssume a real signal is a Lipschitz background plus jumps. No jump occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is","\\vpage{35}{A strict contrast gap locates every separated jump cell}{DETERMINISTIC RECOVERY / A NUMERICAL CERTIFICATE}{Chapter V: Lemma E9.1, Theorem E9.2, Example E9.7.}\nAssume a real signal is a Lipschitz background plus jumps. No jump occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does","age{35}{A strict contrast gap locates every separated jump cell}{DETERMINISTIC RECOVERY / A NUMERICAL CERTIFICATE}{Chapter V: Lemma E9.1, Theorem E9.2, Example E9.7.}\nAssume a real signal is a Lipschitz background plus jumps. No jump occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim …","TERMINISTIC RECOVERY / A NUMERICAL CERTIFICATE}{Chapter V: Lemma E9.1, Theorem E9.2, Example E9.7.}\nAssume a real signal is a Lipschitz background plus jumps. No jump occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies th…","9.1, Theorem E9.2, Example E9.7.}\nAssume a real signal is a Lipschitz background plus jumps. No jump occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For r…","mps. No jump occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in o…","p occurs at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell cou…","at a sample point, and each grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel…","ch grid cell contains at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which i…","ins at most one jump. With spacing $h$, background constant $M$, and sample noise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separati…","oise bounded by $\\varepsilon$, write\n\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound co…","\\[\nb_h=Mh+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location t…","+2\\varepsilon.\n\\]\nA no-jump contrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tf…","ontrast has magnitude at most $b_h$. A jump contrast differs from its true amplitude by at most $b_h$.\n\\vfig[.89]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tfrac h2\\sum_q|a_q|^p)^{1/p}$. …","]{ext_threshold}\n\\head{Use the manuscript's exact numerical inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tfrac h2\\sum_q|a_q|^p)^{1/p}$. A growing number of jumps or shrinking separation requires its own analysis.\n\\takeaway{Exact detection means exa…","al inputs}\nFor $h=1/256$, $M=1/2$, $\\varepsilon=1/500$, and $a_{\\min}=2/25$,\n\\[\nb_h=\\frac{381}{64000}<\\frac3{500},\\qquad\n2b_h<\\frac2{25}.\n\\]\nThe threshold $\\theta=1/25$ lies strictly between $b_h$ and $a_{\\min}-b_h$. Thresholding $|y_{k+1}-y_k|$ therefore identifies exactly the jump-containing cells. Their midpoints locate each jump within $1/512$, and each estimated amplitude differs by at most $381/64000$.\n\\head{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tfrac h2\\sum_q|a_q|^p)^{1/p}$. A growing number of jumps or shrinking separation requires its own analysis.\n\\takeaway{Exact detection means exact cell identification under a strict margin. Location an…","{The recovered object is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tfrac h2\\sum_q|a_q|^p)^{1/p}$. A growing number of jumps or shrinking separation requires its own analysis.\n\\takeaway{Exact detection means exact cell identification under a strict margin. Location and amplitude remain quantitative estimates with explicit errors.}","is a cell, then an estimate}\nThe theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tfrac h2\\sum_q|a_q|^p)^{1/p}$. A growing number of jumps or shrinking separation requires its own analysis.\n\\takeaway{Exact detection means exact cell identification under a strict margin. Location and amplitude remain quantitative estimates with explicit errors.}","he theorem does not claim the exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tfrac h2\\sum_q|a_q|^p)^{1/p}$. A growing number of jumps or shrinking separation requires its own analysis.\n\\takeaway{Exact detection means exact cell identification under a strict margin. Location and amplitude remain quantitative estimates with explicit errors.}","exact real-valued breakpoint from a finite grid. It supplies the correct cell and a controlled representative of that cell. For real jumps, the amplitude error is too small to reverse the sign. Two opposite jumps in one cell could cancel in the contrast, which is why the separation assumption cannot be dropped.\nFor $Q$ jumps the finite-$p$ packet bound contains both $Qb_h$ and the location term $(\\tfrac h2\\sum_q|a_q|^p)^{1/p}$. A growing number of jumps or shrinking separation requires its own analysis.\n\\takeaway{Exact detection means exact cell identification under a strict margin. Location and amplitude remain quantitative estimates with explicit errors.}","\\vpage{36}{Repeated observations shrink the noise side of the gap}{SUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGET}{Chapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11.}\nSuppose all noise observations are independent and centered, with moment-generating bound $\\mathbb E e^{\\lambda\\eta}\\le e^{\\sigma^2\\lambda^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this","\\vpage{36}{Repeated observations shrink the noise side of the gap}{SUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGET}{Chapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11.}\nSuppose all noise observations are independent and centered, with moment-generating bound $\\mathbb E e^{\\lambda\\eta}\\le e^{\\sigma^2\\lambda^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a","\\vpage{36}{Repeated observations shrink the noise side of the gap}{SUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGET}{Chapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11.}\nSuppose all noise observations are independent and centered, with moment-generating bound $\\mathbb E e^{\\lambda\\eta}\\le e^{\\sigma^2\\lambda^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim","\\vpage{36}{Repeated observations shrink the noise side of the gap}{SUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGET}{Chapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11.}\nSuppose all noise observations are independent and centered, with moment-generating bound $\\mathbb E e^{\\lambda\\eta}\\le e^{\\sigma^2\\lambda^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity.","ge{36}{Repeated observations shrink the noise side of the gap}{SUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGET}{Chapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11.}\nSuppose all noise observations are independent and centered, with moment-generating bound $\\mathbb E e^{\\lambda\\eta}\\le e^{\\sigma^2\\lambda^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The ex…","the gap}{SUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGET}{Chapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11.}\nSuppose all noise observations are independent and centered, with moment-generating bound $\\mathbb E e^{\\lambda\\eta}\\le e^{\\sigma^2\\lambda^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational …","ppose all noise observations are independent and centered, with moment-generating bound $\\mathbb E e^{\\lambda\\eta}\\le e^{\\sigma^2\\lambda^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between…","a^2/2}$. Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independen…","Averaging $R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bou…","$R$ observations at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each t…","ns at each of $n+1$ sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each tail and applie…","sample sites replaces $\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each tail and applies a union bound acro…","$\\sigma$ by $\\sigma/\\sqrt R$. A sufficient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each tail and applies a union bound across all cells. Independ…","icient recovery condition is\n\\[\na_{\\min}>2Mh+\\frac{4\\sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each tail and applies a union bound across all cells. Independence of the underlying sample noises…","sigma}{\\sqrt R}\\sqrt{\\log(2n/\\delta)}.\n\\]\nThe probability of simultaneous recovery is then at least $1-\\delta$, subject to the same separated-jump sampling model.\n\\vfig[.86]{ext_repetition}\n\\head{One explicit conservative choice}\nTake $n=256$, $M=1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each tail and applies a union bound across all cells. Independence of the underlying sample noises supplies the stated sub-Gaussian difference proxy.\nF…","1/2$, $\\sigma=1/50$, $\\delta=1/20$, and $a_{\\min}=2/25$. Since $\\log(10240)<10<(16/5)^2$, choosing $R=16$ makes the right-hand side strictly less than\n\\[\n\\frac1{256}+\\frac1{50}\\frac{16}{5}\n=\\frac{2173}{32000}<\\frac2{25}.\n\\]\nThus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies $e^{10}>10240$ using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each tail and applies a union bound across all cells. Independence of the underlying sample noises supplies the stated sub-Gaussian difference proxy.\nFor a fixed separated signal model, a noise scale satisfying $\\sigma_n\\sqrt{\\log n}\\to0$ gives the stated recovery consistency in probability. A fixed positive noise scale with one observation per site does not satisfy that condition.\n\\takeaway{Mor…","cal logarithms.\n\\head{Dependence between adjacent contrasts is allowed}\nNeighboring differences share one sample. The proof does not require those differences to be independent: it bounds each tail and applies a union bound across all cells. Independence of the underlying sample noises supplies the stated sub-Gaussian difference proxy.\nFor a fixed separated signal model, a noise scale satisfying $\\sigma_n\\sqrt{\\log n}\\to0$ gives the stated recovery consistency in probability. A fixed positive noise scale with one observation per site does not satisfy that condition.\n\\takeaway{More grid points and more repeated observations affect different terms. Finer spacing reduces background variation. Averaging reduces noise.}","\\vpage{37}{Haar sees a jump only through its position in the support}{SCALE SEPARATION / A QUANTIFIED THRESHOLD}{Chapter V: Theorem E10.1, Theorem E10.2, Corollary E10.3.}\nUse a normalized Haar wavelet on an available interval of length $L$ on the real line, positive on its left half and negative on its right half. Here $H_\\tau$ is the half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\fra","\\vpage{37}{Haar sees a jump only through its position in the support}{SCALE SEPARATION / A QUANTIFIED THRESHOLD}{Chapter V: Theorem E10.1, Theorem E10.2, Corollary E10.3.}\nUse a normalized Haar wavelet on an available interval of length $L$ on the real line, positive on its left half and negative on its right half. Here $H_\\tau$ is the half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observe","\\vpage{37}{Haar sees a jump only through its position in the support}{SCALE SEPARATION / A QUANTIFIED THRESHOLD}{Chapter V: Theorem E10.1, Theorem E10.2, Corollary E10.3.}\nUse a normalized Haar wavelet on an available interval of length $L$ on the real line, positive on its left half and negative on its right half. Here $H_\\tau$ is the half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $7","through its position in the support}{SCALE SEPARATION / A QUANTIFIED THRESHOLD}{Chapter V: Theorem E10.1, Theorem E10.2, Corollary E10.3.}\nUse a normalized Haar wavelet on an available interval of length $L$ on the real line, positive on its left half and negative on its right half. Here $H_\\tau$ is the half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every suc…","A QUANTIFIED THRESHOLD}{Chapter V: Theorem E10.1, Theorem E10.2, Corollary E10.3.}\nUse a normalized Haar wavelet on an available interval of length $L$ on the real line, positive on its left half and negative on its right half. Here $H_\\tau$ is the half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\h…","ght half. Here $H_\\tau$ is the half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmo…","Here $H_\\tau$ is the half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled seco…","he half-line step $\\mathbf1_{[\\tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled second jump can reinfo…","tau,\\infty)}$, so there is no artificial second endpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled second jump can reinforce or cancel the coefficient.\n…","ndpoint jump.\nIf $u$ is the relative jump position in the support, then\n\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled second jump can reinforce or cancel the coefficient.\n\\takeaway{Scale determines the smooth contributio…","\\[\n|\\langle aH_\\tau,\\psi\\rangle|\n=|a|\\sqrt L\\min\\{u,1-u\\}.\n\\]\nThe coefficient vanishes when the jump lies exactly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled second jump can reinforce or cancel the coefficient.\n\\takeaway{Scale determines the smooth contribution, but alignment determines the jump contribution. A detection guarantee…","tly at a support boundary.\n\\vfig[.86]{ext_haar_margin}\n\\head{A worked gap with all three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled second jump can reinforce or cancel the coefficient.\n\\takeaway{Scale determines the smooth contribution, but alignment determines the jump contribution. A detection guarantee needs both.}","ll three contributions}\nTake $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled second jump can reinforce or cancel the coefficient.\n\\takeaway{Scale determines the smooth contribution, but alignment determines the jump contribution. A detection guarantee needs both.}","Take $L=1/4$, $M=1$, noise bound $\\varepsilon=1/100$, jump size $|a|=1$, and relative boundary distance at least $\\rho=1/4$. The smooth and noise contributions together are bounded by\n\\[\n\\frac M4L^{3/2}+\\varepsilon\\sqrt L\n=\\frac1{32}+\\frac1{200}=\\frac{29}{800}.\n\\]\nThe jump contributes at least $|a|\\rho\\sqrt L=1/8$. The observed coefficient therefore has magnitude at least $71/800$. The threshold $1/16=50/800$ separates every such jump support from every support containing no jump.\n\\head{The available support is part of the hypothesis}\nA fixed extension is needed if the observation domain ends inside a desired wavelet support. Also, the selected support must contain only the specified jump. An unmodeled second jump can reinforce or cancel the coefficient.\n\\takeaway{Scale determines the smooth contribution, but alignment determines the jump contribution. A detection guarantee needs both.}","\\vpage{38}{Two half-shifted grids remove exact alignment failure}{HAAR SUPPORT GEOMETRY / A QUARTER-CELL GUARANTEE}{Chapter V: Theorem E10.4, Corollary E10.5.}\nNormalize the support length to $L=1$. The first grid has boundaries at integers. The second has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump sep","\\vpage{38}{Two half-shifted grids remove exact alignment failure}{HAAR SUPPORT GEOMETRY / A QUARTER-CELL GUARANTEE}{Chapter V: Theorem E10.4, Corollary E10.5.}\nNormalize the support length to $L=1$. The first grid has boundaries at integers. The second has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe a","SUPPORT GEOMETRY / A QUARTER-CELL GUARANTEE}{Chapter V: Theorem E10.4, Corollary E10.5.}\nNormalize the support length to $L=1$. The first grid has boundaries at integers. The second has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly …","}\nNormalize the support length to $L=1$. The first grid has boundaries at integers. The second has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, r…","o $L=1$. The first grid has boundaries at integers. The second has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relati…","boundaries at integers. The second has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThi…","he second has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful supp…","has boundaries at half-integers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per ju…","ers. For a point at position $u$ modulo one, consider its distance to the nearest boundary of each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per jump. It does not by itself sp…","f each grid.\n\\vfig[.89]{ext_two_grids}\n\\head{At least one distance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per jump. It does not by itself specify a unique rule for merging duplicate detections from two grids into exact breakpoint es…","ance is one quarter or larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per jump. It does not by itself specify a unique rule for merging duplicate detections from two grids into exact breakpoint estimates.\n\\takeaway{A second grid repairs a geometric blind spo…","larger}\nIf $1/4\\le u\\le3/4$, the original cell works. Outside that range, a shifted cell works. At $u=1/8$, the original margin is $1/8$ but the shifted cell $[-1/2,1/2)$ gives margin $3/8$. At $u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per jump. It does not by itself specify a unique rule for merging duplicate detections from two grids into exact breakpoint estimates.\n\\takeaway{A second grid repairs a geometric blind spot. The contrast inequal","$u=1/4$, both margins equal $1/4$, showing that the uniform constant cannot be increased for these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per jump. It does not by itself specify a unique rule for merging duplicate detections from two grids into exact breakpoint estimates.\n\\takeaway{A second grid repairs a geometric blind spot. The contrast inequality and separation hypotheses still supply the recovery guarantee.}","these two grids.\nRestoring $L$, every jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per jump. It does not by itself specify a unique rule for merging duplicate detections from two grids into exact breakpoint estimates.\n\\takeaway{A second grid repairs a geometric blind spot. The contrast inequality and separation hypotheses still supply the recovery guarantee.}","very jump has a support with boundary distance at least $L/4$ in one grid. If $L$ is smaller than the minimum jump separation, that support contains no other jump.\n\\head{Insert the geometric guarantee into the analytic estimate}\nThe alignment-independent gap condition is\n\\[\n\\frac{|a|}{4}>2\\varepsilon+\\frac M2L.\n\\]\nFor the numbers on Plate V.37, the right side is $1/50+1/8=29/200$, strictly below $1/4$. Hence one of the two grids supplies a detectable support for every jump, regardless of its position relative to the original grid.\nThis certifies a useful support per jump. It does not by itself specify a unique rule for merging duplicate detections from two grids into exact breakpoint estimates.\n\\takeaway{A second grid repairs a geometric blind spot. The contrast inequality and separation hypotheses still supply the recovery guarantee.}","\\vpage{39}{Higher differences annihilate polynomial interiors}{BOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAIN}{Chapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8.}\nLet $x(n)=n+2(n-3)_+$ on the integer line. Its slope changes from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Dif","\\vpage{39}{Higher differences annihilate polynomial interiors}{BOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAIN}{Chapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8.}\nLet $x(n)=n+2(n-3)_+$ on the integer line. Its slope changes from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$","\\vpage{39}{Higher differences annihilate polynomial interiors}{BOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAIN}{Chapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8.}\nLet $x(n)=n+2(n-3)_+$ on the integer line. Its slope changes from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is b","te polynomial interiors}{BOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAIN}{Chapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8.}\nLet $x(n)=n+2(n-3)_+$ on the integer line. Its slope changes from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses …","iors}{BOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAIN}{Chapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8.}\nLet $x(n)=n+2(n-3)_+$ on the integer line. Its slope changes from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the break…","NOISE GAIN}{Chapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8.}\nLet $x(n)=n+2(n-3)_+$ on the integer line. Its slope changes from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one…","V: Theorem E10.6, Proposition E10.7, Corollary E10.8.}\nLet $x(n)=n+2(n-3)_+$ on the integer line. Its slope changes from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe…","ges from one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A","rom one to three at the breakpoint $3$. The second forward difference is\n\\[\n\\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).\n\\]\n\\vfig[.87]{ext_differences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A brea…","erences}\n\\head{An exact small stencil calculation}\nThe values at $n=0,\\ldots,6$ are $(0,1,2,3,6,9,12)$. The second differences at $n=0,\\ldots,4$ are $(0,0,2,0,0)$. The only nonzero stencil straddles the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a low…","les the slope change.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a lower bound at every boundary.\n\\takeaway{A vanishing interior response is structural. Reliable detection also requires the boundary response to exceed the amplified noise.}","ge.\nFor a piecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a lower bound at every boundary.\n\\takeaway{A vanishing interior response is structural. Reliable detection also requires the boundary response to exceed the amplified noise.}","ecewise polynomial of degree at most $d$, $\\Delta^{d+1}$ vanishes whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a lower bound at every boundary.\n\\takeaway{A vanishing interior response is structural. Reliable detection also requires the boundary response to exceed the amplified noise.}","s whenever its complete stencil lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a lower bound at every boundary.\n\\takeaway{A vanishing interior response is structural. Reliable detection also requires the boundary response to exceed the amplified noise.}","l lies inside one polynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a lower bound at every boundary.\n\\takeaway{A vanishing interior response is structural. Reliable detection also requires the boundary response to exceed the amplified noise.}","ynomial piece. Candidate responses are confined to at most $d+1$ starting addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a lower bound at every boundary.\n\\takeaway{A vanishing interior response is structural. Reliable detection also requires the boundary response to exceed the amplified noise.}","g addresses before each breakpoint. They are boundary bands, not necessarily exact breakpoint locations.\n\\head{Differencing also acts on noise}\nThe absolute binomial coefficients of $\\Delta^m$ sum to $2^m$. With $\\|\\eta\\|_\\infty\\le1/16$, second-difference noise is bounded by $4/16=1/4$. The true response $2$ remains at least $7/4$, while all responses away from the breakpoint band remain at most $1/4$. Threshold one separates them.\nThe general robust-band statement assumes that each breakpoint has at least one sufficiently large true response. A breakpoint can fail that assumption through its amplitude or local configuration. Annihilation away from boundaries does not prove a lower bound at every boundary.\n\\takeaway{A vanishing interior response is structural. Reliable detection also requires the boundary response to exceed the amplified noise.}","\\vpage{40}{Conditional Gibbs control closes the reconstruction estimate}{THE TWO ERRORS / A COMPLETE READING CONTRACT}{Chapter V: Theorem E11.2, Theorem E11.3.}\nThe exact retained-packet approximation shares the same discontinuity as its target. If $f=s+j$ and $Q_ms\\to s$ uniformly, then\n\\[\n\\|f-(Q_ms+j)\\|_\\infty=\\|s-Q_ms\\|_\\infty\\longrightarrow0.\n\\]\nThe approximants themselves may be discontinuous. There is no conflict with the failure of continuous Fourier polynomials to converge uniformly to a jump.\n\\begin{center}\\includegraphics[width=.78\\linewidth]{visual_graphics/annals_gibbs_comparison.pdf}\\end{center}\n\\platecaption{Preserved vector figure from the expanded source. It illustrates the choice of approximation target. It is not an independent","\\vpage{40}{Conditional Gibbs control closes the reconstruction estimate}{THE TWO ERRORS / A COMPLETE READING CONTRACT}{Chapter V: Theorem E11.2, Theorem E11.3.}\nThe exact retained-packet approximation shares the same discontinuity as its target. If $f=s+j$ and $Q_ms\\to s$ uniformly, then\n\\[\n\\|f-(Q_ms+j)\\|_\\infty=\\|s-Q_ms\\|_\\infty\\longrightarrow0.\n\\]\nThe approximants themselves may be discontinuous. There is no conflict with the failure of continuous Fourier polynomials to converge uniformly to a jump.\n\\begin{center}\\includegraphics[width=.78\\linewidth]{visual_graphics/annals_gibbs_comparison.pdf}\\end{center}\n\\platecaption{Preserved vector figure from the expanded source. It illustrates the choice of approximation target. It is not an independent universal recov","\\vpage{40}{Conditional Gibbs control closes the reconstruction estimate}{THE TWO ERRORS / A COMPLETE READING CONTRACT}{Chapter V: Theorem E11.2, Theorem E11.3.}\nThe exact retained-packet approximation shares the same discontinuity as its target. If $f=s+j$ and $Q_ms\\to s$ uniformly, then\n\\[\n\\|f-(Q_ms+j)\\|_\\infty=\\|s-Q_ms\\|_\\infty\\longrightarrow0.\n\\]\nThe approximants themselves may be discontinuous. There is no conflict with the failure of continuous Fourier polynomials to converge uniformly to a jump.\n\\begin{center}\\includegraphics[width=.78\\linewidth]{visual_graphics/annals_gibbs_comparison.pdf}\\end{center}\n\\platecaption{Preserved vector figure from the expanded source. It illustrates the choice of approximation target. It is not an independent universal recovery certificate.}\n\\head{Estimated packets need the operator norm too}\nIf the b","rge uniformly to a jump.\n\\begin{center}\\includegraphics[width=.78\\linewidth]{visual_graphics/annals_gibbs_comparison.pdf}\\end{center}\n\\platecaption{Preserved vector figure from the expanded source. It illustrates the choice of approximation target. It is not an independent universal recovery certificate.}\n\\head{Estimated packets need the operator norm too}\nIf the bounded linear operators satisfy $\\sup_m\\|Q_m\\|_{L^p\\to L^p}\\le C_Q$, then\n\\[\n\\|f-\\widehat f_m\\|_p\n\\le\\|s-Q_ms\\|_p+(1+C_Q)\\|j-\\widehat j_m\\|_p.\n\\]\nFor an orthogonal $L^2$ projection one may use $\\|I-Q_m\\|\\le1$, sharpening the packet multiplier to one. Such a bound must be justified in the norm actually being used. In particular, uniform boundedness of Fourier partial sums in the supremum norm is not supplied by this theorem.\n\\head{What the complete guide establishes}\nPlates V.1--V.26 preserve the original worked layer. Plates V.27--V.40 explain the additional t","ludegraphics[width=.78\\linewidth]{visual_graphics/annals_gibbs_comparison.pdf}\\end{center}\n\\platecaption{Preserved vector figure from the expanded source. It illustrates the choice of approximation target. It is not an independent universal recovery certificate.}\n\\head{Estimated packets need the operator norm too}\nIf the bounded linear operators satisfy $\\sup_m\\|Q_m\\|_{L^p\\to L^p}\\le C_Q$, then\n\\[\n\\|f-\\widehat f_m\\|_p\n\\le\\|s-Q_ms\\|_p+(1+C_Q)\\|j-\\widehat j_m\\|_p.\n\\]\nFor an orthogonal $L^2$ projection one may use $\\|I-Q_m\\|\\le1$, sharpening the packet multiplier to one. Such a bound must be justified in the norm actually being used. In particular, uniform boundedness of Fourier partial sums in the supremum norm is not supplied by this theorem.\n\\head{What the complete guide establishes}\nPlates V.1--V.26 preserve the original worked layer. Plates V.27--V.40 explain the additional traditional developments, outside the inherited twenty-result Lean map. The ne","}\n\\platecaption{Preserved vector figure from the expanded source. It illustrates the choice of approximation target. It is not an independent universal recovery certificate.}\n\\head{Estimated packets need the operator norm too}\nIf the bounded linear operators satisfy $\\sup_m\\|Q_m\\|_{L^p\\to L^p}\\le C_Q$, then\n\\[\n\\|f-\\widehat f_m\\|_p\n\\le\\|s-Q_ms\\|_p+(1+C_Q)\\|j-\\widehat j_m\\|_p.\n\\]\nFor an orthogonal $L^2$ projection one may use $\\|I-Q_m\\|\\le1$, sharpening the packet multiplier to one. Such a bound must be justified in the norm actually being used. In particular, uniform boundedness of Fourier partial sums in the supremum norm is not supplied by this theorem.\n\\head{What the complete guide establishes}\nPlates V.1--V.26 preserve the original worked layer. Plates V.27--V.40 explain the additional traditional developments, outside the inherited twenty-result Lean map. The new exact examples have the","tor figure from the expanded source. It illustrates the choice of approximation target. It is not an independent universal recovery certificate.}\n\\head{Estimated packets need the operator norm too}\nIf the bounded linear operators satisfy $\\sup_m\\|Q_m\\|_{L^p\\to L^p}\\le C_Q$, then\n\\[\n\\|f-\\widehat f_m\\|_p\n\\le\\|s-Q_ms\\|_p+(1+C_Q)\\|j-\\widehat j_m\\|_p.\n\\]\nFor an orthogonal $L^2$ projection one may use $\\|I-Q_m\\|\\le1$, sharpening the packet multiplier to one. Such a bound must be justified in the norm actually being used. In particular, uniform boundedness of Fourier partial sums in the supremum norm is not supplied by this theorem.\n\\head{What the complete guide establishes}\nPlates V.1--V.26 preserve the original worked layer. Plates V.27--V.40 explain the additional traditional developments, outside the inherited twenty-result Lean map. The new exact examples have their own finite arithmetic record. They d","f{Scope and claim boundary.} The chapter proves the stated finite-dimensional semigroup, packet, Markov, Bayesian, conditional-expectation, fractional spectral, Green-kernel, Feynman--Kac, Schur--Feshbach, and survival identities. It also proves quantitative refinement-defect formulas under explicit norm hypotheses, including exact transfer of a bounded generator defect to every fractional power $0<\\alpha<1$ for bounded nonnegative self-adjoint generators. It does not claim that an arbitrary infinite-dimensional operator generates a semigroup, that every stochastic process admits a finite packet representation, that every posterior family is conjugate, that finite Bayesian sufficiency automatically extends to infinite disintegrations, or that every PDE continuum limit follows from a finite test. Domain, closability, sectoriality, measurability, tightness, and compactness hypotheses remain necessary in their respective infinite-dimensional settings. Classical sources are retained in the separate end-of-series reference ledger rather than reproduced in this chapter.\n\\end{minipage}}\n\\en…","Let $X$ be a finite-dimensional real or complex vector space, let $A\\in\\operatorname{End}(X)$, and let $s\\in X$ be a primitive seed. The basic continuous transfer is\n\\[\nT(t)=e^{tA},\\qquad t\\ge 0,\n\\]\nwhile a positive operator $L$ is more naturally written\n\\[\nH(t)=e^{-tL}.\n\\]\nBoth satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with $e^{tA}$. Positive diffusion and elliptic operators are usually written with $e^{-tL}$.","dimensional real or complex vector space, let $A\\in\\operatorname{End}(X)$, and let $s\\in X$ be a primitive seed. The basic continuous transfer is\n\\[\nT(t)=e^{tA},\\qquad t\\ge 0,\n\\]\nwhile a positive operator $L$ is more naturally written\n\\[\nH(t)=e^{-tL}.\n\\]\nBoth satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with $e^{tA}$. Positive diffusion and elliptic operators are usually written with $e^{-tL}$.","d let $s\\in X$ be a primitive seed. The basic continuous transfer is\n\\[\nT(t)=e^{tA},\\qquad t\\ge 0,\n\\]\nwhile a positive operator $L$ is more naturally written\n\\[\nH(t)=e^{-tL}.\n\\]\nBoth satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with $e^{tA}$. Positive diffusion and elliptic operators are usually written with $e^{-tL}$.","The common operator chain is\n\\[\n\\boxed{\n\\text{seed}\\longrightarrow\\text{transfer}\\longrightarrow\\text{project}\n\\longrightarrow\\text{condition}\\longrightarrow\\text{refine}\n\\longrightarrow\\text{complete}\\longrightarrow\\text{reconstruct}.\n}\n\\]","\\begin{definition}[Transfer-semigroup seed system]\\label{core:record:001}\nA finite transfer-semigroup seed system is a tuple\n\\[\n\\mathfrak S=(X,s,A,\\{P_\\alpha\\}_{\\alpha\\in I},\\tau),\n\\]\nwhere $X$ is the state space, $s$ is the seed, $A$ is a transfer generator, the $P_\\alpha$ are pairwise orthogonal idempotents summing to the identity, and $\\tau$ specifies the topology in which a completion or limit is interpreted.\n\\end{definition}","\\begin{theorem}[Finite semigroup law]\\label{core:record:002}\nFor every matrix $A$ and all $s,t\\in\\bbR$,\n\\[\n\\boxed{e^{(s+t)A}=e^{sA}e^{tA}.}\n\\]\nMoreover, $T(0)=I$ and $T'(t)=AT(t)=T(t)A$.\n\\end{theorem}","\\begin{proof}\nBoth exponentials are absolutely convergent matrix power series. Since every power of $A$ commutes with every other power, the Cauchy product gives\n\\[\ne^{sA}e^{tA}\n=\\sum_{m,n\\ge0}\\frac{s^mt^n}{m!n!}A^{m+n}\n=\\sum_{r\\ge0}\\frac{(s+t)^r}{r!}A^r.\n\\]\nTermwise differentiation is valid in finite dimension and gives the derivative identities.\n\\end{proof}","\\begin{boundary}[Infinite-dimensional generators]\\label{core:record:003}\nFor an unbounded operator, writing $e^{tA}$ is not a definition unless a semigroup-generation theorem applies. The finite matrix identities in this chapter remain algebraically correct models, but a continuum realization must state the operator domain, closure, growth bound, and topology of convergence.\n\\end{boundary}","Let\n\\[\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I.\n\\]\nFor any operator $B$, define\n\\[\nB_{\\beta\\alpha}=P_\\beta BP_\\alpha.\n\\]\nThen\n\\[\nB=\\sum_{\\alpha,\\beta}B_{\\beta\\alpha}.\n\\]\nFor the semigroup, write\n\\[\nT_{\\beta\\alpha}(t)=P_\\beta e^{tA}P_\\alpha.\n\\]","\\begin{theorem}[Packet semigroup convolution]\\label{core:record:004}\nFor every $s,t$ and every pair of packets $\\alpha,\\beta$,\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\\sum_{\\gamma\\in I}T_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThus the semigroup law becomes ordinary matrix multiplication in the packet indices.\n\\end{theorem}","\\begin{proof}\nInsert the resolution of the identity between the two semigroup factors:\n\\[\n\\begin{aligned}\nP_\\beta e^{(s+t)A}P_\\alpha\n&=P_\\beta e^{sA}\\left(\\sum_\\gamma P_\\gamma\\right)e^{tA}P_\\alpha\\\\\n&=\\sum_\\gamma\n(P_\\beta e^{sA}P_\\gamma)(P_\\gamma e^{tA}P_\\alpha).\n\\end{aligned}\n\\]\n\\end{proof}","\\begin{corollary}[Reducing packets]\\label{core:record:005}\nIf $P_\\beta AP_\\alpha=0$ for every $\\beta\\ne\\alpha$, then\n\\[\nP_\\beta e^{tA}P_\\alpha=0\n\\quad(\\beta\\ne\\alpha)\n\\]\nfor every $t$. Each packet evolves independently under $P_\\alpha AP_\\alpha$.\n\\end{corollary}","\\begin{proof}\nThe hypothesis makes $A$ block diagonal. Every power $A^m$ is block diagonal with respect to the same projectors, and therefore so is the exponential series.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.66\\textwidth]{02_packet_semigroup_blocks.pdf}\n\\caption{Norms of the packet blocks $P_\\beta e^{tA}P_\\alpha$. Off-diagonal entries quantify transfer between packets.}\n\\label{core:fig:packet-blocks}\n\\end{figure}","\\begin{remark}[Path expansion]\\label{core:record:006}\nExpanding the exponential and inserting $I=\\sum_\\alpha P_\\alpha$ between every pair of generators gives\n\\[\nP_{\\alpha_m}A^mP_{\\alpha_0}\n=\n\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}\nP_{\\alpha_m}AP_{\\alpha_{m-1}}\\cdots P_{\\alpha_1}AP_{\\alpha_0}.\n\\]\nEvery coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.\n\\end{remark}","Let $X_f$ and $X_c$ be fine and coarse spaces, with generators $A_f$ and $A_c$, and let\n\\[\nC:X_f\\to X_c\n\\]\nbe a coarsening map. Define the generator-level refinement defect\n\\[\n\\boxed{\\Delta=CA_f-A_cC.}\n\\]\nThe corresponding semigroup defect is\n\\[\n\\mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.\n\\]","\\begin{theorem}[Exact Duhamel refinement identity]\\label{core:record:007}\nFor every $t\\ge0$,\n\\[\n\\boxed{\nCe^{tA_f}-e^{tA_c}C\n=\n\\int_0^t e^{(t-r)A_c}\\Delta e^{rA_f}\\,dr.\n}\n\\]\nIn particular, $\\Delta=0$ implies exact semigroup intertwining at every time.\n\\end{theorem}","\\begin{proof}\nSet\n\\[\nF(r)=e^{(t-r)A_c}Ce^{rA_f}.\n\\]\nThen\n\\[\nF'(r)=e^{(t-r)A_c}(CA_f-A_cC)e^{rA_f}.\n\\]\nIntegrating from $0$ to $t$ gives $F(t)-F(0)=Ce^{tA_f}-e^{tA_c}C$ and proves the formula.\n\\end{proof}","\\begin{corollary}[Norm bound]\\label{core:record:008}\nSuppose\n\\[\n\\norm{e^{tA_f}}\\le M_f e^{\\omega_f t},\n\\qquad\n\\norm{e^{tA_c}}\\le M_c e^{\\omega_c t}.\n\\]\nThen\n\\[\n\\norm{\\mathfrak D(t)}\n\\le\nM_fM_c\\norm{\\Delta}\n\\int_0^t e^{\\omega_c(t-r)}e^{\\omega_fr}\\,dr.\n\\]\nFor contraction semigroups,\n\\[\n\\boxed{\\norm{\\mathfrak D(t)}\\le t\\norm{\\Delta}.}\n\\]\n\\end{corollary}","For $z$ in the common resolvent set, let\n\\[\nR_f(z)=(zI-A_f)^{-1},\n\\qquad\nR_c(z)=(zI-A_c)^{-1}.\n\\]","\\begin{theorem}[Resolvent refinement identity]\\label{core:record:009}\n\\[\n\\boxed{\nCR_f(z)-R_c(z)C\n=R_c(z)\\Delta R_f(z).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nMultiply the left side by $zI-A_c$ on the left and $zI-A_f$ on the right. The result is\n\\[\n(zI-A_c)C-C(zI-A_f)=CA_f-A_cC=\\Delta.\n\\]\nMultiplication by the two resolvents recovers the stated identity.\n\\end{proof}","Use column probability vectors. A finite continuous-time Markov generator $Q$ has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is\n\\[\nK(t)=e^{tQ}.\n\\]","\\begin{theorem}[Finite Markov semigroup]\\label{core:record:010}\nIf $Q$ has nonnegative off-diagonal entries and column sums zero, then $e^{tQ}$ has nonnegative entries and column sums one for every $t\\ge0$.\n\\end{theorem}","\\begin{proof}\nChoose $\\lambda\\ge\\max_i(-Q_{ii})$. Then $B=Q+\\lambda I$ has nonnegative entries, so $e^{tB}$ is entrywise nonnegative by its power series. Hence\n\\[\ne^{tQ}=e^{-\\lambda t}e^{tB}\n\\]\nis nonnegative. Since $\\bm 1^TQ=0$, differentiation gives\n\\[\n\\frac{d}{dt}\\bm 1^Te^{tQ}=\\bm 1^TQe^{tQ}=0,\n\\]\nso $\\bm 1^Te^{tQ}=\\bm 1^T$.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.78\\textwidth]{03_random_walk_transfer.pdf}\n\\caption{The point mass $\\delta_0$ propagated by repeated symmetric random-walk transfer.}\n\\label{core:fig:random-walk}\n\\end{figure}","\\begin{example}[Symmetric random walk]\\label{core:record:011}\nStarting from $\\delta_0$ and applying the transfer\n\\[\nT=\\frac12(S+S^{-1})\n\\]\nfor $n$ steps gives\n\\[\n\\bbP(X_n=k)=\n2^{-n}\\binom{n}{(n+k)/2}\n\\]\nwhen $k\\equiv n\\pmod2$, and zero otherwise. Hence\n\\[\n\\Exp[X_n]=0,\n\\qquad\n\\Var(X_n)=n.\n\\]\nThe parity packet is exact: all mass remains in the one residue class permitted by $n$.\n\\end{example}","Let $K$ be a column-stochastic fine transition matrix, let $C$ aggregate fine states into coarse packets, and let $\\bar K$ be a coarse transition matrix. Define\n\\[\n\\Delta=CK-\\bar KC.\n\\]","\\begin{theorem}[Exact lumpability]\\label{core:record:012}\nIf $\\Delta=0$, then\n\\[\n\\boxed{CK^n=\\bar K^nC}\n\\]\nfor every $n\\ge0$. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.\n\\end{theorem}","\\begin{proof}\nThe result is immediate for $n=0$. If $CK^n=\\bar K^nC$, then\n\\[\nCK^{n+1}=\\bar K^nCK=\\bar K^{n+1}C.\n\\]\n\\end{proof}","\\begin{theorem}[Finite-horizon defect formula]\\label{core:record:013}\nFor arbitrary $\\Delta=CK-\\bar KC$,\n\\[\n\\boxed{\nCK^n-\\bar K^nC\n=\n\\sum_{j=0}^{n-1}\\bar K^{n-1-j}\\Delta K^j.\n}\n\\]\nIf $K$ and $\\bar K$ are contractions in the chosen norm, then\n\\[\n\\norm{CK^n-\\bar K^nC}\\le n\\norm{\\Delta}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe identity follows by telescoping:\n\\[\n\\bar K^{n-1-j}CK^{j+1}-\\bar K^{n-j}CK^j\n=\\bar K^{n-1-j}(CK-\\bar KC)K^j.\n\\]\nSumming from $j=0$ to $n-1$ leaves $CK^n-\\bar K^nC$. The norm bound follows termwise.\n\\end{proof}","Let $\\Omega=\\{1,\\ldots,n\\}$ carry strictly positive probabilities $\\pi_i$, and let $\\mathcal Q=\\{B_1,\\ldots,B_m\\}$ be a partition. Define\n\\[\n(P_{\\mathcal Q}f)_i\n=\n\\frac{\\sum_{j\\in B(i)}\\pi_jf_j}{\\sum_{j\\in B(i)}\\pi_j},\n\\]\nwhere $B(i)$ is the block containing $i$.","\\begin{theorem}[Finite conditional-expectation projector]\\label{core:record:014}\nWith the weighted inner product\n\\[\n\\langle f,g\\rangle_\\pi=\\sum_i\\pi_if_i\\overline{g_i},\n\\]\nthe operator $P_{\\mathcal Q}$ satisfies\n\\[\nP_{\\mathcal Q}^2=P_{\\mathcal Q},\n\\qquad\n\\langle P_{\\mathcal Q}f,g\\rangle_\\pi\n=\n\\langle f,P_{\\mathcal Q}g\\rangle_\\pi.\n\\]\nIts range is the space of functions constant on each partition block.\n\\end{theorem}","\\begin{proof}\nAfter one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal\n\\[\n\\sum_{B\\in\\mathcal Q}\n\\frac{1}{\\pi(B)}\n\\left(\\sum_{i\\in B}\\pi_i f_i\\right)\n\\overline{\\left(\\sum_{j\\in B}\\pi_j g_j\\right)}.\n\\]\n\\end{proof}","\\begin{theorem}[Tower law and martingale packets]\\label{core:record:015}\nIf $\\mathcal Q_c$ is coarser than $\\mathcal Q_f$, then\n\\[\n\\boxed{\nP_cP_f=P_fP_c=P_c.\n}\n\\]\nFor a nested tower $P_0\\le P_1\\le\\cdots\\le P_r$, define\n\\[\nD_0=P_0,\n\\qquad\nD_j=P_j-P_{j-1}.\n\\]\nThen\n\\[\nD_iD_j=\\delta_{ij}D_j,\n\\qquad\nP_r=\\sum_{j=0}^rD_j.\n\\]\n\\end{theorem}","\\begin{proof}\nA coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from $P_iP_j=P_{\\min(i,j)}$.\n\\end{proof}","\\begin{classical}[Gaussian chaos realization]\\label{core:record:016}\nFor the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally:\n\\[\nT_tH_n=e^{-nt}H_n.\n\\]\nThus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.\n\\end{classical}","Let $p=(p_i)$ be a probability vector and let $\\lambda=(\\lambda_i)$ be a nonnegative likelihood with positive evidence\n\\[\nZ_\\lambda(p)=\\sum_i\\lambda_ip_i.\n\\]\nDefine\n\\[\n\\boxed{\n\\cB_\\lambda(p)\n=\n\\frac{\\lambda\\odot p}{Z_\\lambda(p)}.\n}\n\\]\nThis map is nonlinear solely because of its scalar reconstruction factor.","\\begin{theorem}[Independent-likelihood composition]\\label{core:record:017}\nFor nonnegative likelihoods $\\lambda$ and $\\mu$, assume $Z_\\lambda(p)>0$ and $Z_{\\lambda\\odot\\mu}(p)>0$. Equivalently, both successive updates have positive evidence. Then\n\\[\n\\boxed{\n\\cB_\\mu(\\cB_\\lambda(p))\n=\n\\cB_{\\lambda\\odot\\mu}(p).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe first normalization appears in both numerator and denominator of the second update and cancels:\n\\[\n\\frac{\\mu_i(\\lambda_ip_i/Z_\\lambda)}{\\sum_j\\mu_j\\lambda_jp_j/Z_\\lambda}\n=\n\\frac{\\mu_i\\lambda_ip_i}{\\sum_j\\mu_j\\lambda_jp_j}.\n\\]\n\\end{proof}","Let the state indices be partitioned into packets $I_\\alpha$. Define\n\\[\nZ_\\alpha=\\sum_{i\\in I_\\alpha}\\lambda_ip_i,\n\\qquad\nm_\\alpha=\\frac{Z_\\alpha}{\\sum_\\beta Z_\\beta},\n\\qquad\nq_{\\alpha,i}=\\begin{cases}\n\\lambda_ip_i/Z_\\alpha,&Z_\\alpha>0,\\ i\\in I_\\alpha,","Let the state indices be partitioned into packets $I_\\alpha$. Define\n\\[\nZ_\\alpha=\\sum_{i\\in I_\\alpha}\\lambda_ip_i,\n\\qquad\nm_\\alpha=\\frac{Z_\\alpha}{\\sum_\\beta Z_\\beta},\n\\qquad\nq_{\\alpha,i}=\\begin{cases}\n\\lambda_ip_i/Z_\\alpha,&Z_\\alpha>0,\\ i\\in I_\\alpha,\\\\\n0,&\\text{otherwise}.\n\\end{cases}\n\\]","\\begin{theorem}[Bayesian packet reconstruction]\\label{core:record:018}\n\\[\n\\boxed{\n\\cB_\\lambda(p)\n=\n\\sum_\\alpha m_\\alpha q_\\alpha.\n}\n\\]\nOnly packets with positive evidence carry a conditional probability vector. A zero-evidence packet has zero posterior mass and contributes the zero vector. The nonlinear coupling between packets occurs only through the total evidence used to normalize the packet evidences.\n\\end{theorem}","\\begin{proof}\nIf $Z_\\alpha=0$, nonnegativity implies $\\lambda_i p_i=0$ for every $i\\in I_\\alpha$, so the reconstruction identity holds with both sides zero. For $Z_\\alpha>0$ and $i\\in I_\\alpha$,\n\\[\nm_\\alpha q_{\\alpha,i}\n=\n\\frac{Z_\\alpha}{Z}\\frac{\\lambda_ip_i}{Z_\\alpha}\n=\n\\frac{\\lambda_ip_i}{Z}.\n\\]\nThe packet supports are disjoint, so summing reconstructs the full posterior.\n\\end{proof}","\\begin{theorem}[Likelihood perturbation bound]\\label{core:record:019}\nLet $p$ be fixed and let $\\lambda,\\mu\\ge0$ have positive evidences $Z_\\lambda,Z_\\mu$. Then\n\\[\n\\boxed{\n\\norm{\\cB_\\lambda(p)-\\cB_\\mu(p)}_1\n\\le\n\\frac{2\\sum_i|\\lambda_i-\\mu_i|p_i}{\\min(Z_\\lambda,Z_\\mu)}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nAdd and subtract $\\mu\\odot p/Z_\\lambda$ and use the triangle inequality:\n\\[\n\\norm{\\frac{\\lambda\\odot p}{Z_\\lambda}-\\frac{\\mu\\odot p}{Z_\\mu}}_1\n\\le\n\\frac{\\norm{(\\lambda-\\mu)\\odot p}_1}{Z_\\lambda}\n+\n\\norm{\\mu\\odot p}_1\\left|\\frac1{Z_\\lambda}-\\frac1{Z_\\mu}\\right|.\n\\]\nThe second term equals $|Z_\\lambda-Z_\\mu|/Z_\\lambda$ and is bounded by the same weighted $L^1$ difference divided by $Z_\\lambda$. Interchanging $\\lambda$ and $\\mu$ gives the minimum-evidence form.\n\\end{proof}","btract $\\mu\\odot p/Z_\\lambda$ and use the triangle inequality:\n\\[\n\\norm{\\frac{\\lambda\\odot p}{Z_\\lambda}-\\frac{\\mu\\odot p}{Z_\\mu}}_1\n\\le\n\\frac{\\norm{(\\lambda-\\mu)\\odot p}_1}{Z_\\lambda}\n+\n\\norm{\\mu\\odot p}_1\\left|\\frac1{Z_\\lambda}-\\frac1{Z_\\mu}\\right|.\n\\]\nThe second term equals $|Z_\\lambda-Z_\\mu|/Z_\\lambda$ and is bounded by the same weighted $L^1$ difference divided by $Z_\\lambda$. Interchanging $\\lambda$ and $\\mu$ gives the minimum-evidence form.\n\\end{proof}","ot p/Z_\\lambda$ and use the triangle inequality:\n\\[\n\\norm{\\frac{\\lambda\\odot p}{Z_\\lambda}-\\frac{\\mu\\odot p}{Z_\\mu}}_1\n\\le\n\\frac{\\norm{(\\lambda-\\mu)\\odot p}_1}{Z_\\lambda}\n+\n\\norm{\\mu\\odot p}_1\\left|\\frac1{Z_\\lambda}-\\frac1{Z_\\mu}\\right|.\n\\]\nThe second term equals $|Z_\\lambda-Z_\\mu|/Z_\\lambda$ and is bounded by the same weighted $L^1$ difference divided by $Z_\\lambda$. Interchanging $\\lambda$ and $\\mu$ gives the minimum-evidence form.\n\\end{proof}","Let the fine state set $I$ be partitioned into nonempty coarse packets $I_a$, $a\\in A$.  The coarsening map on probability vectors is\n\\[\n(C_\\pi p)_a=P_a=\\sum_{i\\in I_a}p_i.\n\\]\nFor a fine likelihood $\\lambda=(\\lambda_i)_{i\\in I}$ and a prior $p$, define the effective coarse likelihood\n\\[\nr_a(p)\n=\n\\frac{\\sum_{i\\in I_a}\\lambda_i p_i}{P_a}\n\\qquad(P_a>0).\n\\]\nIts value on a zero-mass packet is irrelevant and may be set to zero.","\\begin{theorem}[Exact effective-likelihood factorization]\\label{core:record:020}\nWhenever the fine evidence is positive,\n\\[\n\\boxed{\nC_\\pi\\cB_\\lambda(p)\n=\n\\cB_{r(p)}(C_\\pi p).}\n\\]\nThus the only fine information needed for the coarse posterior, beyond the coarse prior masses, is the vector of conditional packet evidences $r(p)$.\n\\end{theorem}","\\begin{proof}\nFor $P=C_\\pi p$ and $Z=\\sum_i\\lambda_ip_i=\\sum_aP_ar_a(p)$,\n\\[\n\\bigl(C_\\pi\\cB_\\lambda(p)\\bigr)_a\n=\n\\frac{\\sum_{i\\in I_a}\\lambda_ip_i}{Z}\n=\n\\frac{P_ar_a(p)}{\\sum_bP_br_b(p)}\n=\n\\bigl(\\cB_{r(p)}(P)\\bigr)_a.\n\\]\n\\end{proof}","\\begin{theorem}[Bayesian refinement criterion]\\label{core:record:021}\nAssume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive.  The identity\n\\[\n\\boxed{\nC_\\pi\\cB_\\lambda(p)\n=\n\\cB_{\\bar\\lambda}(C_\\pi p)}\n\\]\nholds for every fine prior $p$ if and only if there exists one scalar $c>0$ such that\n\\[\n\\boxed{\n\\lambda_i=c\\,\\bar\\lambda_a\n\\qquad(i\\in I_a).}\n\\]\nEquivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling $c$.\n\\end{theorem}","lihoods are strictly positive.  The identity\n\\[\n\\boxed{\nC_\\pi\\cB_\\lambda(p)\n=\n\\cB_{\\bar\\lambda}(C_\\pi p)}\n\\]\nholds for every fine prior $p$ if and only if there exists one scalar $c>0$ such that\n\\[\n\\boxed{\n\\lambda_i=c\\,\\bar\\lambda_a\n\\qquad(i\\in I_a).}\n\\]\nEquivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling $c$.\n\\end{theorem}","\\begin{proof}\nIf $\\lambda_i=c\\bar\\lambda_a$ on $I_a$, then $P_a r_a(p)=c\\bar\\lambda_aP_a$ for every prior, including zero-mass packets, and the preceding theorem gives the desired identity because Bayesian normalization removes the common scalar $c$.","Conversely, fix two fine states $i,i'\\in I_a$ and one state $j\\in I_b$ in another coarse packet.  Vary the distribution of a fixed coarse mass inside $I_a$ between $i$ and $i'$ while leaving every coarse packet mass unchanged.  The right side depends only on those coarse masses and is therefore unchanged.  The left-side posterior mass of packet $a$ depends on the conditional average of $\\lambda_i$ and $\\lambda_{i'}$.  It can be independent of that variation only if $\\lambda_i=\\lambda_{i'}$.  Hence $\\lambda$ is constant on every packet. Write the value on $I_a$ as $\\lambda_a$.","sely, fix two fine states $i,i'\\in I_a$ and one state $j\\in I_b$ in another coarse packet.  Vary the distribution of a fixed coarse mass inside $I_a$ between $i$ and $i'$ while leaving every coarse packet mass unchanged.  The right side depends only on those coarse masses and is therefore unchanged.  The left-side posterior mass of packet $a$ depends on the conditional average of $\\lambda_i$ and $\\lambda_{i'}$.  It can be independent of that variation only if $\\lambda_i=\\lambda_{i'}$.  Hence $\\lambda$ is constant on every packet. Write the value on $I_a$ as $\\lambda_a$.","in another coarse packet.  Vary the distribution of a fixed coarse mass inside $I_a$ between $i$ and $i'$ while leaving every coarse packet mass unchanged.  The right side depends only on those coarse masses and is therefore unchanged.  The left-side posterior mass of packet $a$ depends on the conditional average of $\\lambda_i$ and $\\lambda_{i'}$.  It can be independent of that variation only if $\\lambda_i=\\lambda_{i'}$.  Hence $\\lambda$ is constant on every packet. Write the value on $I_a$ as $\\lambda_a$.","he distribution of a fixed coarse mass inside $I_a$ between $i$ and $i'$ while leaving every coarse packet mass unchanged.  The right side depends only on those coarse masses and is therefore unchanged.  The left-side posterior mass of packet $a$ depends on the conditional average of $\\lambda_i$ and $\\lambda_{i'}$.  It can be independent of that variation only if $\\lambda_i=\\lambda_{i'}$.  Hence $\\lambda$ is constant on every packet. Write the value on $I_a$ as $\\lambda_a$.","n $i$ and $i'$ while leaving every coarse packet mass unchanged.  The right side depends only on those coarse masses and is therefore unchanged.  The left-side posterior mass of packet $a$ depends on the conditional average of $\\lambda_i$ and $\\lambda_{i'}$.  It can be independent of that variation only if $\\lambda_i=\\lambda_{i'}$.  Hence $\\lambda$ is constant on every packet. Write the value on $I_a$ as $\\lambda_a$.","$i'$ while leaving every coarse packet mass unchanged.  The right side depends only on those coarse masses and is therefore unchanged.  The left-side posterior mass of packet $a$ depends on the conditional average of $\\lambda_i$ and $\\lambda_{i'}$.  It can be independent of that variation only if $\\lambda_i=\\lambda_{i'}$.  Hence $\\lambda$ is constant on every packet. Write the value on $I_a$ as $\\lambda_a$.","Now use priors supported on one state of packet $a$ and one state of packet $b$.  Equality of the two coarse posterior odds gives\n\\[\n\\frac{\\lambda_a}{\\lambda_b}\n=\n\\frac{\\bar\\lambda_a}{\\bar\\lambda_b}.\n\\]\nThis holds for every pair of coarse packets, so $\\lambda_a=c\\bar\\lambda_a$ for one common $c>0$.\n\\end{proof}","\\begin{remark}[The one-packet boundary]\\label{core:record:022}\nIf the coarse partition has only one packet, every posterior coarsens to the scalar $1$, so commutation is automatic and no within-packet likelihood information can be identified.  The nontrivial criterion begins with at least two coarse packets.\n\\end{remark}","When sufficiency fails, the defect still has an exact finite factorization.  Fix a declared coarse likelihood $\\bar\\lambda$, put\n\\[\nP=C_\\pi p,\n\\qquad\n\\delta(p)=r(p)-\\bar\\lambda,\n\\qquad\n\\bar b=\\cB_{\\bar\\lambda}(P),\n\\qquad\nZ_r=\\langle r(p),P\\rangle.\n\\]","\\begin{theorem}[Bayesian information-loss packet]\\label{core:record:023}\nAssume nonnegative fine and declared coarse likelihoods, with both $Z_r>0$ and $Z_{\\bar\\lambda}>0$. The coarsening defect is\n\\[\n\\boxed{\nC_\\pi\\cB_\\lambda(p)\n-\n\\cB_{\\bar\\lambda}(P)\n=\n\\frac{1}{Z_r}\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\n\\delta(p).}\n\\]\nThe factor matrix annihilates $\\bar\\lambda$.  This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together.  Moreover,\n\\[\n\\boxed{\n\\left\\|C_\\pi\\cB_\\lambda(p)-\\cB_{\\bar\\lambda}(P)\\right\\|_1\n\\le\n\\frac{2\\sum_aP_a|\\delta_a(p)|}\n{\\min\\{Z_r,Z_{\\bar\\lambda}\\}}}\n\\]\nand\n\\[\n\\sum_aP_a|\\delta_a(p)|\n\\le\n\\sum_i p_i\\left|\\lambda_i-\\bar\\lambda_{\\pi(i)}\\right|.\n\\]\n\\end{theo","\\begin{theorem}[Bayesian information-loss packet]\\label{core:record:023}\nAssume nonnegative fine and declared coarse likelihoods, with both $Z_r>0$ and $Z_{\\bar\\lambda}>0$. The coarsening defect is\n\\[\n\\boxed{\nC_\\pi\\cB_\\lambda(p)\n-\n\\cB_{\\bar\\lambda}(P)\n=\n\\frac{1}{Z_r}\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\n\\delta(p).}\n\\]\nThe factor matrix annihilates $\\bar\\lambda$.  This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together.  Moreover,\n\\[\n\\boxed{\n\\left\\|C_\\pi\\cB_\\lambda(p)-\\cB_{\\bar\\lambda}(P)\\right\\|_1\n\\le\n\\frac{2\\sum_aP_a|\\delta_a(p)|}\n{\\min\\{Z_r,Z_{\\bar\\lambda}\\}}}\n\\]\nand\n\\[\n\\sum_aP_a|\\delta_a(p)|\n\\le\n\\sum_i p_i\\left|\\lambda_i-\\bar\\lambda_{\\pi(i)}\\right|.\n\\]\n\\end{theorem}","C_\\pi\\cB_\\lambda(p)\n-\n\\cB_{\\bar\\lambda}(P)\n=\n\\frac{1}{Z_r}\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\n\\delta(p).}\n\\]\nThe factor matrix annihilates $\\bar\\lambda$.  This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together.  Moreover,\n\\[\n\\boxed{\n\\left\\|C_\\pi\\cB_\\lambda(p)-\\cB_{\\bar\\lambda}(P)\\right\\|_1\n\\le\n\\frac{2\\sum_aP_a|\\delta_a(p)|}\n{\\min\\{Z_r,Z_{\\bar\\lambda}\\}}}\n\\]\nand\n\\[\n\\sum_aP_a|\\delta_a(p)|\n\\le\n\\sum_i p_i\\left|\\lambda_i-\\bar\\lambda_{\\pi(i)}\\right|.\n\\]\n\\end{theorem}","tes $\\bar\\lambda$.  This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together.  Moreover,\n\\[\n\\boxed{\n\\left\\|C_\\pi\\cB_\\lambda(p)-\\cB_{\\bar\\lambda}(P)\\right\\|_1\n\\le\n\\frac{2\\sum_aP_a|\\delta_a(p)|}\n{\\min\\{Z_r,Z_{\\bar\\lambda}\\}}}\n\\]\nand\n\\[\n\\sum_aP_a|\\delta_a(p)|\n\\le\n\\sum_i p_i\\left|\\lambda_i-\\bar\\lambda_{\\pi(i)}\\right|.\n\\]\n\\end{theorem}","\\begin{proof}\nThe effective-likelihood theorem reduces the defect to\n\\[\n\\cB_{\\bar\\lambda+\\delta}(P)-\\cB_{\\bar\\lambda}(P).\n\\]\nSince $Z_r=Z_{\\bar\\lambda}+P^{\\mathsf T}\\delta$, direct subtraction gives\n\\[\n\\frac{1}{Z_r}\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta.\n\\]\nAlso,\n\\[\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\bar\\lambda\n=P\\odot\\bar\\lambda-\\bar bZ_{\\bar\\lambda}=0.\n\\]\nThe first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the following estimate vanish. For $P_a>0$,\n\\[\nP_a|\\delta_a|\n=\nP_a\\left|\\sum_{i\\in I_a}\\frac{p_i}{P_a}\n(\\lambda_i-\\bar\\lambda_a)\\right|\n\\le\n\\sum_{i\\in I_a}p_i|\\lambda_i-\\bar\\lambda_a|,\n\\]\nand summing over $a$ proves the second inequality.\n\\end{proof}","r\\lambda}+P^{\\mathsf T}\\delta$, direct subtraction gives\n\\[\n\\frac{1}{Z_r}\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta.\n\\]\nAlso,\n\\[\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\bar\\lambda\n=P\\odot\\bar\\lambda-\\bar bZ_{\\bar\\lambda}=0.\n\\]\nThe first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the following estimate vanish. For $P_a>0$,\n\\[\nP_a|\\delta_a|\n=\nP_a\\left|\\sum_{i\\in I_a}\\frac{p_i}{P_a}\n(\\lambda_i-\\bar\\lambda_a)\\right|\n\\le\n\\sum_{i\\in I_a}p_i|\\lambda_i-\\bar\\lambda_a|,\n\\]\nand summing over $a$ proves the second inequality.\n\\end{proof}","P^{\\mathsf T}\\delta$, direct subtraction gives\n\\[\n\\frac{1}{Z_r}\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta.\n\\]\nAlso,\n\\[\n\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\bar\\lambda\n=P\\odot\\bar\\lambda-\\bar bZ_{\\bar\\lambda}=0.\n\\]\nThe first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the following estimate vanish. For $P_a>0$,\n\\[\nP_a|\\delta_a|\n=\nP_a\\left|\\sum_{i\\in I_a}\\frac{p_i}{P_a}\n(\\lambda_i-\\bar\\lambda_a)\\right|\n\\le\n\\sum_{i\\in I_a}p_i|\\lambda_i-\\bar\\lambda_a|,\n\\]\nand summing over $a$ proves the second inequality.\n\\end{proof}","right)\\bar\\lambda\n=P\\odot\\bar\\lambda-\\bar bZ_{\\bar\\lambda}=0.\n\\]\nThe first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the following estimate vanish. For $P_a>0$,\n\\[\nP_a|\\delta_a|\n=\nP_a\\left|\\sum_{i\\in I_a}\\frac{p_i}{P_a}\n(\\lambda_i-\\bar\\lambda_a)\\right|\n\\le\n\\sum_{i\\in I_a}p_i|\\lambda_i-\\bar\\lambda_a|,\n\\]\nand summing over $a$ proves the second inequality.\n\\end{proof}","\\begin{corollary}[Canonical coarsest likelihood-resolving partition]\\label{core:record:025}\nFor a finite family of likelihoods $\\{\\lambda^{(y)}\\}_{y\\in Y}$, define\n\\[\ni\\sim j\n\\quad\\Longleftrightarrow\\quad\n\\lambda_i^{(y)}=\\lambda_j^{(y)}\n\\text{ for every }y\\in Y.\n\\]\nThe equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivial partition on which every update closes for every prior. The one-packet partition always closes, independently of the profiles, and is excluded from this last assertion.\n\\end{corollary}","\\begin{certificate}[Bayesian refinement]\\label{core:record:026}\nFor the finite test partition $\\{0,1\\}\\sqcup\\{2,3\\}$, a blockwise-constant likelihood gave commutation residual $1.11\\times10^{-16}$.  A nonconstant likelihood produced an exact defect-factorization residual $7.29\\times10^{-17}$.  Its measured $L^1$ defect was $4.8787\\times10^{-2}$, below the certified bound $1.1016\\times10^{-1}$.  Two priors with identical coarse masses but different hidden within-packet shapes produced coarse posteriors differing by $1.2400\\times10^{-1}$, which is a direct necessity witness.\n\\end{certificate}","lockwise-constant likelihood gave commutation residual $1.11\\times10^{-16}$.  A nonconstant likelihood produced an exact defect-factorization residual $7.29\\times10^{-17}$.  Its measured $L^1$ defect was $4.8787\\times10^{-2}$, below the certified bound $1.1016\\times10^{-1}$.  Two priors with identical coarse masses but different hidden within-packet shapes produced coarse posteriors differing by $1.2400\\times10^{-1}$, which is a direct necessity witness.\n\\end{certificate}","Fix a score vector $g$. The likelihood flow is\n\\[\n\\lambda_i(t)=e^{tg_i},\n\\]\nand the Bayesian flow is\n\\[\n\\Phi_t(p)_i\n=\n\\frac{e^{tg_i}p_i}{\\sum_je^{tg_j}p_j}.\n\\]","\\begin{theorem}[Bayesian nonlinear semigroup]\\label{core:record:028}\n\\[\n\\boxed{\n\\Phi_{s+t}=\\Phi_s\\circ\\Phi_t.\n}\n\\]\nIts generator is the centered multiplicative field\n\\[\n\\boxed{\n\\frac{d}{dt}p_i(t)\n=\n\\bigl(g_i-\\langle g,p(t)\\rangle\\bigr)p_i(t).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe semigroup law is the likelihood-composition theorem with $e^{sg}e^{tg}=e^{(s+t)g}$. Differentiating the normalized exponential formula gives\n\\[\n\\dot p_i=g_ip_i-p_i\\frac{\\sum_jg_je^{tg_j}p_j(0)}{\\sum_je^{tg_j}p_j(0)}.\n\\]\nThe quotient is $\\langle g,p(t)\\rangle$.\n\\end{proof}","\\begin{example}[Beta--Binomial conjugate packet]\\label{core:record:029}\nA $\\operatorname{Beta}(a,b)$ prior and $s$ successes in $n$ trials produce\n\\[\n\\operatorname{Beta}(a+s,b+n-s).\n\\]\nThe prior exponents are additive packets: the likelihood increments them by $(s,n-s)$. In the numerical example,\n\\[\n(a,b)=(2,3),\\quad (s,n)=(7,10),\n\\]\nso the posterior is $\\operatorname{Beta}(9,6)$ with mean $0.6$.\n\\end{example}","Let $K$ be column stochastic and let $\\lambda^{(r)}$ be the likelihood associated with the $r$th observation. The filter recursion is\n\\[\n\\widehat p_r=Kp_{r-1},\n\\qquad\np_r=\\cB_{\\lambda^{(r)}}(\\widehat p_r).\n\\]","\\begin{proposition}[Prediction-update reconstruction]\\label{core:record:030}\nEvery hidden Markov filtering step is the composition of a linear Markov transfer and a normalized nonlinear likelihood transfer. Assume a probability initial vector, a column-stochastic transition matrix, and nonnegative likelihoods with positive evidence at every step. Then every $p_r$ is a probability vector.\n\\end{proposition}","\\begin{proof}\nColumn stochasticity gives $\\widehat p_r\\ge0$ and $\\bm1^T\\widehat p_r=1$. Bayesian normalization then gives $p_r\\ge0$ and $\\bm1^Tp_r=1$.\n\\end{proof}","Let $L=L^*\\ge0$ on a finite-dimensional Hilbert space. Its spectral resolution is\n\\[\nL=\\sum_j\\lambda_jP_j.\n\\]\nFor $\\alpha>0$, define\n\\[\n\\boxed{L^\\alpha=\\sum_j\\lambda_j^\\alpha P_j.}\n\\]\nThe associated fractional semigroup is\n\\[\nT_\\alpha(t)=e^{-tL^\\alpha}\n=\\sum_je^{-t\\lambda_j^\\alpha}P_j.\n\\]","\\begin{theorem}[Fractional packet calculus]\\label{core:record:031}\nFor $\\alpha,\\beta>0$,\n\\[\nL^\\alpha L^\\beta=L^{\\alpha+\\beta},\n\\]\nand for $s,t\\ge0$,\n\\[\n\\boxed{T_\\alpha(s)T_\\alpha(t)=T_\\alpha(s+t).}\n\\]\nEvery spectral packet $P_j$ reduces $L^\\alpha$ and $T_\\alpha(t)$.\n\\end{theorem}","\\begin{proof}\nAll statements hold coefficientwise on the mutually orthogonal spectral projectors. For example,\n\\[\nL^\\alpha L^\\beta\n=\\sum_{j,k}\\lambda_j^\\alpha\\lambda_k^\\beta P_jP_k\n=\\sum_j\\lambda_j^{\\alpha+\\beta}P_j.\n\\]\n\\end{proof}","\\begin{theorem}[Poisson subordination]\\label{core:record:032}\nFor $t>0$ and $L\\ge0$,\n\\[\n\\boxed{\ne^{-t\\sqrt L}\n=\n\\frac{t}{2\\sqrt\\pi}\n\\int_0^\\infty\ns^{-3/2}e^{-t^2/(4s)}e^{-sL}\\,ds.\n}\n\\]\nThe integral converges in operator norm in finite dimension.\n\\end{theorem}","\\begin{proof}\nBy the spectral theorem it suffices to prove the scalar identity\n\\[\ne^{-t\\sqrt\\lambda}\n=\n\\frac{t}{2\\sqrt\\pi}\n\\int_0^\\infty\ns^{-3/2}e^{-t^2/(4s)-\\lambda s}\\,ds\n\\]\nfor $\\lambda\\ge0$. This is the classical Gaussian integral after the substitution $u=t/(2\\sqrt s)$. Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.\n\\end{proof}","Let $A\\ge0$ act on a fine Hilbert space, let $B\\ge0$ act on a coarse Hilbert space, and let $C$ be a bounded coarsening map.  Set\n\\[\nD=CA-BC,\n\\qquad\nD_\\alpha=CA^\\alpha-B^\\alpha C,\n\\qquad 0<\\alpha<1.\n\\]","\\begin{theorem}[Fractional intertwining theorem]\\label{core:record:033}\\label{core:thm:fractional-naturality}\nFor bounded nonnegative self-adjoint $A$ and $B$,\n\\[\n\\boxed{\nD_\\alpha\n=\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty\n t^\\alpha(tI+B)^{-1}D(tI+A)^{-1}\\,dt.\n}\n\\]\nThe improper integral converges in operator norm.  Moreover,\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\nK_\\alpha\\norm{C}^{1-\\alpha}\\norm{D}^{\\alpha},\n\\qquad\nK_\\alpha=\n\\frac{2^{1-\\alpha}\\sin(\\pi\\alpha)}\n{\\pi\\alpha(1-\\alpha)}.\n}\n\\]\nIf $A,B\\ge mI$ for some $m>0$, then the stronger linear estimate holds:\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\n\\alpha m^{\\alpha-1}\\norm{D}.\n}\n\\]\nIn particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\\alpha=B^\\alpha C$.\n\\end{theorem}","actional-naturality}\nFor bounded nonnegative self-adjoint $A$ and $B$,\n\\[\n\\boxed{\nD_\\alpha\n=\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty\n t^\\alpha(tI+B)^{-1}D(tI+A)^{-1}\\,dt.\n}\n\\]\nThe improper integral converges in operator norm.  Moreover,\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\nK_\\alpha\\norm{C}^{1-\\alpha}\\norm{D}^{\\alpha},\n\\qquad\nK_\\alpha=\n\\frac{2^{1-\\alpha}\\sin(\\pi\\alpha)}\n{\\pi\\alpha(1-\\alpha)}.\n}\n\\]\nIf $A,B\\ge mI$ for some $m>0$, then the stronger linear estimate holds:\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\n\\alpha m^{\\alpha-1}\\norm{D}.\n}\n\\]\nIn particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\\alpha=B^\\alpha C$.\n\\end{theorem}","For bounded nonnegative self-adjoint $A$ and $B$,\n\\[\n\\boxed{\nD_\\alpha\n=\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty\n t^\\alpha(tI+B)^{-1}D(tI+A)^{-1}\\,dt.\n}\n\\]\nThe improper integral converges in operator norm.  Moreover,\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\nK_\\alpha\\norm{C}^{1-\\alpha}\\norm{D}^{\\alpha},\n\\qquad\nK_\\alpha=\n\\frac{2^{1-\\alpha}\\sin(\\pi\\alpha)}\n{\\pi\\alpha(1-\\alpha)}.\n}\n\\]\nIf $A,B\\ge mI$ for some $m>0$, then the stronger linear estimate holds:\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\n\\alpha m^{\\alpha-1}\\norm{D}.\n}\n\\]\nIn particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\\alpha=B^\\alpha C$.\n\\end{theorem}","\\[\n\\boxed{\nD_\\alpha\n=\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty\n t^\\alpha(tI+B)^{-1}D(tI+A)^{-1}\\,dt.\n}\n\\]\nThe improper integral converges in operator norm.  Moreover,\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\nK_\\alpha\\norm{C}^{1-\\alpha}\\norm{D}^{\\alpha},\n\\qquad\nK_\\alpha=\n\\frac{2^{1-\\alpha}\\sin(\\pi\\alpha)}\n{\\pi\\alpha(1-\\alpha)}.\n}\n\\]\nIf $A,B\\ge mI$ for some $m>0$, then the stronger linear estimate holds:\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\n\\alpha m^{\\alpha-1}\\norm{D}.\n}\n\\]\nIn particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\\alpha=B^\\alpha C$.\n\\end{theorem}","The improper integral converges in operator norm.  Moreover,\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\nK_\\alpha\\norm{C}^{1-\\alpha}\\norm{D}^{\\alpha},\n\\qquad\nK_\\alpha=\n\\frac{2^{1-\\alpha}\\sin(\\pi\\alpha)}\n{\\pi\\alpha(1-\\alpha)}.\n}\n\\]\nIf $A,B\\ge mI$ for some $m>0$, then the stronger linear estimate holds:\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\n\\alpha m^{\\alpha-1}\\norm{D}.\n}\n\\]\nIn particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\\alpha=B^\\alpha C$.\n\\end{theorem}","rm.  Moreover,\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\nK_\\alpha\\norm{C}^{1-\\alpha}\\norm{D}^{\\alpha},\n\\qquad\nK_\\alpha=\n\\frac{2^{1-\\alpha}\\sin(\\pi\\alpha)}\n{\\pi\\alpha(1-\\alpha)}.\n}\n\\]\nIf $A,B\\ge mI$ for some $m>0$, then the stronger linear estimate holds:\n\\[\n\\boxed{\n\\norm{D_\\alpha}\n\\le\n\\alpha m^{\\alpha-1}\\norm{D}.\n}\n\\]\nIn particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\\alpha=B^\\alpha C$.\n\\end{theorem}","\\begin{proof}\nFor $0<\\alpha<1$, the Balakrishnan formula gives\n\\[\nA^\\alpha\n=\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty t^{\\alpha-1}A(tI+A)^{-1}\\,dt,\n\\]\nand likewise for $B$.  Using\n\\[\nA(tI+A)^{-1}=I-t(tI+A)^{-1}\n\\]\nand the rectangular resolvent identity\n\\[\n(tI+B)^{-1}C-C(tI+A)^{-1}\n=\n(tI+B)^{-1}(CA-BC)(tI+A)^{-1},\n\\]\nthe difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.","For small $t$,\n\\[\n\\norm{CA(tI+A)^{-1}-B(tI+B)^{-1}C}\n\\le 2\\norm C,\n\\]\nbecause the two positive resolvent factors are contractions.  For large $t$, the resolvent identity gives the bound $\\norm D/t$.  Splitting the integral at\n\\[\nr=\\frac{\\norm D}{2\\norm C}\n\\]\nwhen $C$ and $D$ are nonzero yields\n\\[\n\\begin{aligned}\n\\norm{D_\\alpha}\n&\\le\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\left(\n2\\norm C\\int_0^r t^{\\alpha-1}\\,dt\n+\n\\norm D\\int_r^\\infty t^{\\alpha-2}\\,dt\n\\right)","For small $t$,\n\\[\n\\norm{CA(tI+A)^{-1}-B(tI+B)^{-1}C}\n\\le 2\\norm C,\n\\]\nbecause the two positive resolvent factors are contractions.  For large $t$, the resolvent identity gives the bound $\\norm D/t$.  Splitting the integral at\n\\[\nr=\\frac{\\norm D}{2\\norm C}\n\\]\nwhen $C$ and $D$ are nonzero yields\n\\[\n\\begin{aligned}\n\\norm{D_\\alpha}\n&\\le\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\left(\n2\\norm C\\int_0^r t^{\\alpha-1}\\,dt\n+\n\\norm D\\int_r^\\infty t^{\\alpha-2}\\,dt\n\\right)\\\\\n&=\nK_\\alpha\\norm C^{1-\\alpha}\\norm D^\\alpha.\n\\end{aligned}\n\\]\nThe zero cases are immediate.","If $A,B\\ge mI$, then both resolvents have norm at most $(t+m)^{-1}$, so\n\\[\n\\norm{D_\\alpha}\n\\le\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\\norm D\n\\int_0^\\infty\\frac{t^\\alpha}{(t+m)^2}\\,dt.\n\\]\nAfter $t=mu$, the beta integral equals\n\\[\nm^{\\alpha-1}\\frac{\\alpha\\pi}{\\sin(\\pi\\alpha)},\n\\]\nwhich gives the linear estimate.\n\\end{proof}","\\begin{corollary}[Fractional semigroup refinement]\\label{core:record:034}\\label{core:cor:fractional-semigroup-refinement}\nFor $s\\ge0$,\n\\[\n\\boxed{\nCe^{-sA^\\alpha}-e^{-sB^\\alpha}C\n=\n-\\int_0^s\n e^{-(s-r)B^\\alpha}D_\\alpha e^{-rA^\\alpha}\\,dr.\n}\n\\]\nHence\n\\[\n\\norm{Ce^{-sA^\\alpha}-e^{-sB^\\alpha}C}\n\\le s\\norm{D_\\alpha}.\n\\]\nUnder the common gap $A,B\\ge mI$,\n\\[\n\\boxed{\n\\norm{Ce^{-sA^\\alpha}-e^{-sB^\\alpha}C}\n\\le\ns e^{-sm^\\alpha}\\alpha m^{\\alpha-1}\\norm D.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nDifferentiate\n\\[\nr\\longmapsto e^{-(s-r)B^\\alpha}Ce^{-rA^\\alpha}\n\\]\nand integrate from $0$ to $s$.  Positive fractional semigroups are contractions. Under the gap, the product of their two norm bounds is $e^{-sm^\\alpha}$.\n\\end{proof}","\\begin{corollary}[Fractional refinement towers]\\label{core:record:035}\nLet $D_\\ell=C_\\ell A_{\\ell+1}-A_\\ell C_\\ell$ in a bounded nonnegative self-adjoint tower, with $\\sup_\\ell\\norm{C_\\ell}\\le M$.  If\n\\[\n\\sum_\\ell\\norm{D_\\ell}^{\\alpha}<\\infty,\n\\]\nthen the fractional defects\n\\[\nC_\\ell A_{\\ell+1}^{\\alpha}-A_\\ell^{\\alpha}C_\\ell\n\\]\nhave summable operator norms. If the tower has a uniform gap $m>0$, it is enough that $\\sum_\\ell\\norm{D_\\ell}<\\infty$. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces is asserted.\n\\end{corollary}","[Fractional refinement towers]\\label{core:record:035}\nLet $D_\\ell=C_\\ell A_{\\ell+1}-A_\\ell C_\\ell$ in a bounded nonnegative self-adjoint tower, with $\\sup_\\ell\\norm{C_\\ell}\\le M$.  If\n\\[\n\\sum_\\ell\\norm{D_\\ell}^{\\alpha}<\\infty,\n\\]\nthen the fractional defects\n\\[\nC_\\ell A_{\\ell+1}^{\\alpha}-A_\\ell^{\\alpha}C_\\ell\n\\]\nhave summable operator norms. If the tower has a uniform gap $m>0$, it is enough that $\\sum_\\ell\\norm{D_\\ell}<\\infty$. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces is asserted.\n\\end{corollary}","\\begin{boundary}[Unbounded and nonnormal generators]\\label{core:record:036}\nThe theorem resolves fractional naturality for bounded nonnegative self-adjoint refinement systems and gives the exact estimate needed for finite discretization towers.  For unbounded positive generators, the same resolvent argument requires a declared common-domain condition and a bounded extension of $CA-BC$.  Sectorial or nonnormal generators require their own contour calculus and resolvent bounds.  Those domain questions are not inferred from the finite certificates.\n\\end{boundary}","On the interval $(0,1)$ with Dirichlet boundary conditions, the Green kernel for $-u''=f$ is\n\\[\n\\boxed{G(x,y)=\\min(x,y)-xy.}\n\\]\nThus the point source $\\delta_y$ generates the solution kernel, and a general source is reconstructed by\n\\[\nu(x)=\\int_0^1G(x,y)f(y)\\,dy.\n\\]","For $n$ interior points with spacing $h=1/(n+1)$, define\n\\[\nL_h=\\frac1{h^2}\n\\begin{pmatrix}\n2&-1&&","For $n$ interior points with spacing $h=1/(n+1)$, define\n\\[\nL_h=\\frac1{h^2}\n\\begin{pmatrix}\n2&-1&&\\\\\n-1&2&\\ddots&\\\\\n&\\ddots&\\ddots&-1\\\\\n&&-1&2\n\\end{pmatrix}.\n\\]","\\begin{theorem}[Exact discrete Green reconstruction at grid nodes]\\label{core:record:038}\nLet $x_i=ih$. Then\n\\[\n\\boxed{\n\\frac1h(L_h^{-1})_{ij}\n=G(x_i,x_j).\n}\n\\]\nThus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.\n\\end{theorem}","\\begin{proof}\nThe inverse of the unscaled tridiagonal matrix has entries\n\\[\n\\frac{\\min(i,j)(n+1-\\max(i,j))}{n+1}.\n\\]\nMultiplying by $h^2$ and then dividing by $h$ gives\n\\[\nh\\min(i,j)\\left(1-\\frac{\\max(i,j)}{n+1}\\right)\n=\\min(x_i,x_j)-x_ix_j.\n\\]\n\\end{proof}","The $k$th discrete eigenvalue is\n\\[\n\\lambda_{k,h}=\\frac4{h^2}\\sin^2\\left(\\frac{k\\pi h}{2}\\right),\n\\]\nwhereas the continuum eigenvalue is $(k\\pi)^2$.","\\begin{theorem}[Second-order low-mode convergence]\\label{core:record:039}\nFor fixed $k$,\n\\[\n\\lambda_{k,h}=(k\\pi)^2+O(h^2).\n\\]\nMore precisely, for sufficiently small $h$,\n\\[\n0\\le (k\\pi)^2-\\lambda_{k,h}\n\\le \\frac{(k\\pi)^4}{12}h^2.\n\\]\n\\end{theorem}","\\begin{proof}\nFor $x\\ge0$, $0\\le x^2-\\sin^2x\\le x^4/3$. Set $x=k\\pi h/2$ and multiply by $4/h^2$.\n\\end{proof}","Let $K$ be a column-stochastic matrix, let $V$ be a potential on the states, and let $f$ be terminal data. Define\n\\[\nM_V=\\diag(e^{-V})K^T.\n\\]\nThe backward Feynman--Kac value after $n$ steps is\n\\[\nu_n=M_V^nf.\n\\]","\\begin{theorem}[Discrete Feynman--Kac path formula]\\label{core:record:040}\nFor an initial state $i$,\n\\[\n\\boxed{\n(u_n)_i\n=\n\\Exp_i\\left[\n\\exp\\left(-\\sum_{r=0}^{n-1}V(X_r)\\right)f(X_n)\n\\right].\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFor $n=1$,\n\\[\n(M_Vf)_i=e^{-V(i)}\\sum_jK_{j i}f_j,\n\\]\nwhich is the stated expectation. Iterating and expanding the matrix product sums over every path $(i=i_0,i_1,\\ldots,i_n)$ with probability $\\prod_rK_{i_{r+1},i_r}$ and weight $\\prod_re^{-V(i_r)}$.\n\\end{proof}","Let $X=P X\\oplus Q X$ and write\n\\[\nA=\n\\begin{pmatrix}\nA_{PP}&A_{PQ}","Let $X=P X\\oplus Q X$ and write\n\\[\nA=\n\\begin{pmatrix}\nA_{PP}&A_{PQ}\\\\\nA_{QP}&A_{QQ}\n\\end{pmatrix}.\n\\]","\\begin{theorem}[Schur--Feshbach projected resolvent]\\label{core:record:041}\nAssume both $zI-A$ and $zI-A_{QQ}$ are invertible. Define\n\\[\n\\boxed{\nF_P(z)\n=zI-A_{PP}-A_{PQ}(zI-A_{QQ})^{-1}A_{QP}.\n}\n\\]\nThen\n\\[\n\\boxed{\n\\left.P(zI-A)^{-1}\\right|_{\\Ran P}=F_P(z)^{-1}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nSolve the hidden block equation\n\\[\n(zI-A_{QQ})x_Q=A_{QP}x_P+y_Q\n\\]\nfor $x_Q$ and substitute it into the retained equation. The resulting retained coefficient is $F_P(z)$. Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting $y_Q=0$ gives the stated identity as an operator on $\\Ran P$. Equivalently, this is the block inverse formula.\n\\end{proof}","The self-energy\n\\[\n\\Sigma_P(z)=A_{PQ}(zI-A_{QQ})^{-1}A_{QP}\n\\]\ncontains the exact effect of the eliminated packets. A bare truncation $A_{PP}$ discards this term and is generally not exact.","For the homogeneous system $\\dot x=Ax$, variation of constants gives\n\\[\nx_Q(t)=e^{tA_{QQ}}x_Q(0)\n+\\int_0^te^{(t-r)A_{QQ}}A_{QP}x_P(r)\\,dr.\n\\]\nSubstitution yields\n\\[\n\\dot x_P(t)\n=A_{PP}x_P(t)+A_{PQ}e^{tA_{QQ}}x_Q(0)\n+\\int_0^t\\cK(t-r)x_P(r)\\,dr,\n\\]\nwhere\n\\[\n\\boxed{\n\\cK(t)=A_{PQ}e^{tA_{QQ}}A_{QP}.\n}\n\\]","\\begin{theorem}[Memory/self-energy correspondence]\\label{core:record:042}\nFor finite complex block matrices, assume $\\operatorname{Re}z>\\operatorname{Re}\\lambda$ for every $\\lambda\\in\\sigma(A_{QQ})$. Then the following improper integral is absolutely convergent and\n\\[\n\\boxed{\n\\int_0^\\infty e^{-zt}\\cK(t)\\,dt\n=A_{PQ}(zI-A_{QQ})^{-1}A_{QP}\n=\\Sigma_P(z).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nPut $B=A_{QQ}$ and $T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}$.\nThe generalized-eigenspace decomposition in finite dimensions writes $T_z$ as a\nfinite sum of constant matrix coefficients times $t^k e^{(\\lambda-z)t}$.\nThis follows by expanding the nilpotent part on each generalized eigenspace.\nEquivalently, apply the decomposition to left multiplication by $B$ on the matrix\nalgebra and evaluate the exponential at $I$.\nThe strict spectral inequality makes every such term absolutely integrable on\n$(0,\\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable\nand $T_z(t)\\to0$, including when $B$ is nonnormal.","_{QQ}$ and $T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}$.\nThe generalized-eigenspace decomposition in finite dimensions writes $T_z$ as a\nfinite sum of constant matrix coefficients times $t^k e^{(\\lambda-z)t}$.\nThis follows by expanding the nilpotent part on each generalized eigenspace.\nEquivalently, apply the decomposition to left multiplication by $B$ on the matrix\nalgebra and evaluate the exponential at $I$.\nThe strict spectral inequality makes every such term absolutely integrable on\n$(0,\\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable\nand $T_z(t)\\to0$, including when $B$ is nonnormal.","tion in finite dimensions writes $T_z$ as a\nfinite sum of constant matrix coefficients times $t^k e^{(\\lambda-z)t}$.\nThis follows by expanding the nilpotent part on each generalized eigenspace.\nEquivalently, apply the decomposition to left multiplication by $B$ on the matrix\nalgebra and evaluate the exponential at $I$.\nThe strict spectral inequality makes every such term absolutely integrable on\n$(0,\\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable\nand $T_z(t)\\to0$, including when $B$ is nonnormal.","sum of constant matrix coefficients times $t^k e^{(\\lambda-z)t}$.\nThis follows by expanding the nilpotent part on each generalized eigenspace.\nEquivalently, apply the decomposition to left multiplication by $B$ on the matrix\nalgebra and evaluate the exponential at $I$.\nThe strict spectral inequality makes every such term absolutely integrable on\n$(0,\\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable\nand $T_z(t)\\to0$, including when $B$ is nonnormal.","ents times $t^k e^{(\\lambda-z)t}$.\nThis follows by expanding the nilpotent part on each generalized eigenspace.\nEquivalently, apply the decomposition to left multiplication by $B$ on the matrix\nalgebra and evaluate the exponential at $I$.\nThe strict spectral inequality makes every such term absolutely integrable on\n$(0,\\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable\nand $T_z(t)\\to0$, including when $B$ is nonnormal.",")t}$.\nThis follows by expanding the nilpotent part on each generalized eigenspace.\nEquivalently, apply the decomposition to left multiplication by $B$ on the matrix\nalgebra and evaluate the exponential at $I$.\nThe strict spectral inequality makes every such term absolutely integrable on\n$(0,\\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable\nand $T_z(t)\\to0$, including when $B$ is nonnormal.","Let $J=\\int_0^\\infty T_z(t)\\,dt$. Integrating both derivative identities\n$T_z'=(B-zI)T_z=T_z(B-zI)$ and using $T_z(0)=I$ gives\n\\[\n(zI-B)J=J(zI-B)=I.\n\\]\nThus $J=(zI-B)^{-1}$. Continuous left and right multiplication carry the\nBochner integral through the two rectangular transfer blocks, yielding the claim.\n\\end{proof}","Let $A$ be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states. Eventual absorption is a separate hypothesis. Let $p_0$ be an initial transient-state distribution. Define\n\\[\np(t)=e^{tA}p_0,\n\\qquad\nS(t)=\\bm1^Tp(t).\n\\]\nThe exit-rate row vector is\n\\[\na^T=-\\bm1^TA\\ge0.\n\\]","\\begin{theorem}[Semigroup survival identities]\\label{core:record:043}\nFor $t\\ge0$, the finite-time exit density and, wherever $S(t)>0$, the hazard are\n\\[\n\\boxed{\nf(t)=a^Te^{tA}p_0=-S'(t),\n\\qquad\nh(t)=\\frac{f(t)}{S(t)}.\n}\n\\]\nConsequently, $S$ is nonincreasing and\n\\[\nS(t)=\\exp\\left(-\\int_0^th(r)\\,dr\\right)\n\\]\nwhenever $S(t)>0$. The density may be defective: its total mass is $1-\\lim_{t\\to\\infty}S(t)$. It is a probability density when absorption is eventual, so that this limit is zero.\n\\end{theorem}","\\begin{proof}\nDifferentiate:\n\\[\nS'(t)=\\bm1^TAe^{tA}p_0=-a^Te^{tA}p_0.\n\\]\nThe hazard identity is the definition of conditional instantaneous exit rate. Integrating $(\\log S)'=-h$ gives the final formula.\n\\end{proof}","\\midrule\nMarkov & point mass $\\delta_i$ & $K^n$ or $e^{tQ}$ & state lumping and $CK^n=\\bar K^nC$","Conditional & random variable & $P_{\\mathcal Q}$ & nested partitions and tower law","Bayesian & prior $p$ & $\\cB_\\lambda$ & packet evidences and normalized synthesis","Fractional & spectral source & $e^{-tL^\\alpha}$ & spectral projectors $P_j$","PDE & source $f$ or $\\delta_y$ & resolvent/Green operator & eigenmode or spatial packet synthesis","Feynman--Kac & terminal data $f$ & $(\\diag(e^{-V})K^T)^n$ & weighted path reconstruction","Survival & initial transient law & $e^{tA}$ & sum over transient packets","For clarity, the finite-dimensional holomorphic calculus used below has the\nfollowing canonical definition. For $A$ acting on $\\mathbb C^d$, its generalized\neigenspaces $E_\\mu=\\ker(A-\\mu I)^d$ form a direct-sum decomposition over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_","For clarity, the finite-dimensional holomorphic calculus used below has the\nfollowing canonical definition. For $A$ acting on $\\mathbb C^d$, its generalized\neigenspaces $E_\\mu=\\ker(A-\\mu I)^d$ form a direct-sum decomposition over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\m","For clarity, the finite-dimensional holomorphic calculus used below has the\nfollowing canonical definition. For $A$ acting on $\\mathbb C^d$, its generalized\neigenspaces $E_\\mu=\\ker(A-\\mu I)^d$ form a direct-sum decomposition over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical ope","For clarity, the finite-dimensional holomorphic calculus used below has the\nfollowing canonical definition. For $A$ acting on $\\mathbb C^d$, its generalized\neigenspaces $E_\\mu=\\ker(A-\\mu I)^d$ form a direct-sum decomposition over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is th","For clarity, the finite-dimensional holomorphic calculus used below has the\nfollowing canonical definition. For $A$ acting on $\\mathbb C^d$, its generalized\neigenspaces $E_\\mu=\\ker(A-\\mu I)^d$ form a direct-sum decomposition over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","eigenspaces $E_\\mu=\\ker(A-\\mu I)^d$ form a direct-sum decomposition over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","(A-\\mu I)^d$ form a direct-sum decomposition over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","n over the\neigenvalues. Define the unique linear operator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","ator $F(A)$ by\n\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","\\[\n\\left.F(A)\\right|_{E_\\mu}\n=\\left.\\sum_{k=0}^{d-1}\\frac{F^{(k)}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","}(\\mu)}{k!}(A-\\mu I)^k\\right|_{E_\\mu}.\n\\]\nThe formula is independent of a basis and of increasing the truncation index:\nthe extra nilpotent powers vanish. It depends only on the finite Taylor jets of\n$F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the\nspectrum, each eigenvalue admits a radius $r_\\mu>0$ with\n$\\overline{D(\\mu,r_\\mu)}\\subset U$, and Cauchy's formula gives\n\\[\n\\frac{F^{(k)}(\\mu)}{k!}\n=\\frac{1}{2\\pi i}\\oint_{|z-\\mu|=r_\\mu}\n  \\frac{F(z)}{(z-\\mu)^{k+1}}\\,dz.\n\\]\nConsequently the same canonical operator is obtained from any such local\nCauchy circles, even when $U$ is disconnected. This is the standard finite\nTaylor-jet definition of the holomorphic matrix function.","\\begin{theorem}[Naturality under exact intertwining]\\label{core:record:044}\nFor finite complex matrices, let $F$ be holomorphic on a neighborhood of the union of the spectra of $A_f$ and $A_c$. Real matrices are treated by complexification. If\n\\[\nCA_f=A_cC,\n\\]\nthen\n\\[\n\\boxed{CF(A_f)=F(A_c)C.}\n\\]\nIn particular, exact generator refinement propagates simultaneously to semigroups, resolvents, integer powers, and analytic transfer functions.\n\\end{theorem}","\\begin{proof}\nThe intertwining identity implies\n$C(A_f-\\mu I)^k=(A_c-\\mu I)^kC$ for every $k\\ge0$.\nIt therefore sends the generalized $\\mu$-eigenspace of $A_f$ into that of $A_c$.\nUse a common truncation index at least as large as both dimensions in the\ncanonical formula above. Multiplying its finite sum by $C$ now gives the desired\nidentity on every generalized eigenspace. The direct-sum decomposition gives\nit on the full space. The argument covers nilpotent Jordan parts and uses only\nlocal spectral Taylor jets. It does not require one Taylor series to converge\non both spectra. The zero-dimensional case is immediate.\n\\end{proof}","\\midrule\nPacket-block semigroup composition residual & $2.554\\times10^{-15}$","Random-walk mass error & $0$","Exact Markov generator-lumping defect & $5.551\\times10^{-17}$","Exact finite-horizon lumping error & $2.220\\times10^{-16}$","Perturbed ten-step lumping error & $6.523\\times10^{-3}$","Linear defect bound for that example & $2.000\\times10^{-1}$","Bayesian composition residual & $5.551\\times10^{-17}$","Bayesian packet reconstruction residual & $0$","Bayesian refinement commutation residual & $1.110\\times10^{-16}$","Bayesian defect-factorization residual & $7.286\\times10^{-17}$","Bayesian flow generator residual & $1.959\\times10^{-11}$","Conditional-expectation tower residual & $0$","Hermite Gram residual & $9.992\\times10^{-15}$","Fractional semigroup residual & $2.082\\times10^{-17}$","Maximum square-root subordination error & $1.665\\times10^{-16}$","Exact fractional-naturality residual & $0$","Fractional-power defect / gap certificate & $0.51059 / 0.95079$","Fractional-semigroup defect / certificate & $0.12969 / 0.44480$","Observed low-mode PDE convergence order & $1.999854$","Feynman--Kac enumeration residual & $1.665\\times10^{-16}$","Feshbach projected-resolvent residual & $5.551\\times10^{-17}$","Memory Laplace-transform residual & $6.939\\times10^{-18}$","Duhamel refinement identity residual & $2.108\\times10^{-16}$","Survival monotonicity violation & $0$","Hidden-Markov normalization error & $1.110\\times10^{-16}$","\\begin{certificate}[Test suite]\\label{core:record:046}\nThe September 6 release records the following automated-test result:\n\\[\n\\boxed{54/54\\text{ automated tests}.}\n\\]\nThe tests cover projector reconstruction, packet-block semigroup composition, exact and approximate Markov lumping, Bayesian composition and flow, the exact Bayesian refinement criterion, information-loss factorization, likelihood-profile partitions, conditional-expectation towers, Hermite orthogonality, fractional semigroups, square-root subordination, fractional intertwining and semigroup-defect bounds, discrete Green kernels, PDE convergence, Feynman--Kac enumeration, Schur--Feshbach elimination, memory kernels, refinement defects, hidden Markov filtering, and survival curves.\n\\end{certificate}","\\begin{conjecture}[Uniform refinement control]\\label{core:record:048}\nFor a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that\n\\[\n\\norm{C_\\ell e^{tA_{\\ell+1}}-e^{tA_\\ell}C_\\ell}\n\\le\nM\\,t\\,\\norm{C_\\ell A_{\\ell+1}-A_\\ell C_\\ell}\n\\]\non every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.\n\\end{conjecture}","The accompanying Python package implements the formulas and finite models used in the chapter.\n\\begin{center}\n\\texttt{projector\\_transfer\\_semigroups}\n\\end{center}\nThe computational organization mirrors the chapter:\n\\[\n\\begin{array}{c|l}\n\\texttt{packets.py} & \\text{projectors, blocks, reconstruction, block composition}\\\\\n\\texttt{semigroup.py} & \\text{matrix semigroups, defects, Feshbach operators, memory}\\\\\n\\texttt{markov.py} & \\text{random walks, exact lumping, survival}\\\\\n\\texttt{bayes.py} & \\text{normalized updates, packet evidence, conjugacy, flows, HMM}\\\\\n\\texttt{conditional.py} & \\text{conditional expectation and Gaussian chaos}\\\\\n\\texttt{fractional.py} & \\text{spectral powers, subordination, and fractional refinement certificates}\\\\\n\\texttt{pde.py} & \\text{Dirichlet Laplacians, Green kernels, eigenvalue error}\\\\\n\\texttt{feynman\\_kac.py} & \\text{weighted transfer and exhaustive path enumeration}.\n\\end{array}\n\\]","The central structural identity is\n\\[\n\\boxed{\n\\text{generator defect}\n\\longrightarrow\n\\text{Duhamel semigroup defect}\n\\longrightarrow\n\\text{controlled refinement error}.\n}\n\\]\nTogether with exact packet reconstruction, this makes the semigroup chapter the natural bridge from the foundational calculus to the quantum, computational, and arithmetic realizations that follow.","\\paragraph{General nonnormal memory (13.2).}\nThe actual damped exponential is decomposed into finitely many terms of the form\n$t^k e^{(\\mu-z)t}$. Integrability and convergence to zero follow from\n$\\operatorname{Re}\\mu<\\operatorname{Re}z$. The formal proof uses generalized\neigenspaces of left multiplication on the matrix algebra, so nilpotent Jordan\nparts are included. Both inverse identities follow by integrating the two\nexponential derivative formulas. A continuous linear sandwich map gives the\nliteral rectangular matrix memory integral and the actual inverse. No normality\nor externally supplied growth bound is required.","The first sections of this part treat bounded semigroups on Banach spaces. A finite packet system resolves every operator into input and output blocks. The semigroup law becomes block convolution. Splitting the generator into its packet diagonal and transfer part gives a convergent Dyson expansion over directed packet paths. For a coarse map \\(C\\), the generator defect\n\\[\n\\Delta=CA_f-A_cC\n\\]\nproduces the entire semigroup defect through a Duhamel integral. The same defect controls resolvents and holomorphic functions of the generators.","For a positive operator \\(L\\), spectral packets define \\(L^\\alpha\\) and \\(\\e^{-tL^\\alpha}\\). Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound \\(m\\), the Balakrishnan formula gives the quantitative estimate\n\\[\n\\norm{CL_f^\\alpha-L_c^\\alpha C}\n\\leq\n\\alpha m^{\\alpha-1}\n\\norm{CL_f-L_cC},\n\\qquad 0<\\alpha<1.\n\\]\nA Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.","h Poisson solutions. Finite state Feynman--Kac transfer is proved in discrete and continuous time. Schur--Feshbach elimination yields the exact projected resolvent, the retained memory equation, and the Laplace correspondence between memory and self-energy. Every stable finite hidden system gives a polynomial exponential memory expansion. Primary blocks can be ranked optimally for the certified \\(L^1\\) bound, and a Volterra estimate carries kernel error to the retained solution. Survival, event density, moments, and competing risks are then read directly from a transient semigroup.","\\midrule\nOperator evolution & $\\e^{tA}$ & generator intertwining","Markov chain & $K^n$ or $\\e^{tQ}$ & equality of aggregated transition masses","Conditional expectation & $P_{\\mathcal Q}$ & nesting of the underlying partitions","Bayesian update & $\\Bcal_\\lambda$ & likelihood constant inside each coarse packet","Fractional diffusion & $\\e^{-tL^\\alpha}$ & spectral or resolvent intertwining","Let \\(X\\) be a complex Banach space. A finite packet system is a family of bounded idempotents\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\n\\]\nsuch that\n\\[\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nNo orthogonality is needed in this section.","\\begin{definition}[Packet resolved transfer system]\\label{extended:record:001}\nA packet resolved transfer system is a triple\n\\[\n(X,A,\\Pcal)\n\\]\nconsisting of a Banach state space, a bounded generator, and a finite packet system. A marked transfer system also includes an initial state or source $s\\in X$. The topology and all limiting operations are supplied by the ambient Banach space structure.\n\\end{definition}","\\begin{definition}[Packet blocks]\\label{extended:record:002}\nFor \\(B\\in\\mathcal B(X)\\), define\n\\[\nB_{\\beta\\alpha}=P_\\beta BP_\\alpha.\n\\]\nIf \\(A\\in\\mathcal B(X)\\) and \\(T(t)=\\e^{tA}\\), write\n\\[\nT_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha.\n\\]\n\\end{definition}","\\begin{theorem}[Semigroup and packet convolution]\\label{extended:record:003}\\label{extended:thm:packet_convolution}\nFor every bounded operator \\(A\\), the family \\(T(t)=\\e^{tA}\\) is a uniformly continuous group and\n\\[\nT(s+t)=T(s)T(t).\n\\]\nFor every \\(\\alpha,\\beta\\in I\\),\n\\[\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n\\]\n\\end{theorem}","\\begin{proof}\nThe exponential series converges in operator norm. Since every power of \\(A\\) commutes with every other power, the Cauchy product gives the group law. Inserting \\(I_X=\\sum_\\gamma P_\\gamma\\) between the two factors gives the block formula.\n\\end{proof}","\\begin{proposition}[Block differential system]\\label{extended:record:004}\\label{extended:prop:block_differential}\nThe packet blocks are differentiable in operator norm and satisfy\n\\[\n\\frac{\\dd}{\\dd t}T_{\\beta\\alpha}(t)\n=\n\\sum_{\\gamma\\in I}\nA_{\\beta\\gamma}T_{\\gamma\\alpha}(t)\n=\n\\sum_{\\gamma\\in I}\nT_{\\beta\\gamma}(t)A_{\\gamma\\alpha}.\n\\]\nTheir initial values are\n\\[\nT_{\\beta\\alpha}(0)=\\delta_{\\beta\\alpha}P_\\alpha.\n\\]\n\\end{proposition}","\\begin{proof}\nDifferentiate \\(T(t)\\) in operator norm and insert the packet resolution between \\(A\\) and \\(T(t)\\), then between \\(T(t)\\) and \\(A\\).\n\\end{proof}","\\begin{theorem}[Laplace resolvent and packet transform]\\label{extended:record:005}\\label{extended:thm:laplace_packet_resolvent}\nLet \\(A\\in\\mathcal B(X)\\), let \\(T(t)=\\e^{tA}\\), and let\n\\[\n\\operatorname{Re}z>\\norm A.\n\\]\nThen the Bochner integral converges in operator norm and\n\\[\n(zI-A)^{-1}\n=\n\\int_0^\\infty \\e^{-zt}T(t)\\dd t.\n\\]\nFor every pair of packets,\n\\[\nP_\\beta(zI-A)^{-1}P_\\alpha\n=\n\\int_0^\\infty\n\\e^{-zt}T_{\\beta\\alpha}(t)\\dd t.\n\\]\n\\end{theorem}","\\begin{proof}\nThe estimate \\(\\norm{T(t)}\\leq\\e^{t\\norm A}\\) gives absolute convergence of the operator valued integral. Let\n\\[\nR(z)=\\int_0^\\infty\\e^{-zt}T(t)\\dd t.\n\\]\nSince \\(T'(t)=AT(t)=T(t)A\\), integration by parts gives\n\\[\n(zI-A)R(z)\n=\n-\\int_0^\\infty\n\\frac{\\dd}{\\dd t}\n\\bigl(\\e^{-zt}T(t)\\bigr)\n\\dd t\n=I.\n\\]\nThe same calculation on the other side gives\n\\(\nR(z)(zI-A)=I\n\\).\nMultiplication by \\(P_\\beta\\) and \\(P_\\alpha\\) may be passed under the norm convergent integral, which gives the packet formula.\n\\end{proof}","f}\nThe estimate \\(\\norm{T(t)}\\leq\\e^{t\\norm A}\\) gives absolute convergence of the operator valued integral. Let\n\\[\nR(z)=\\int_0^\\infty\\e^{-zt}T(t)\\dd t.\n\\]\nSince \\(T'(t)=AT(t)=T(t)A\\), integration by parts gives\n\\[\n(zI-A)R(z)\n=\n-\\int_0^\\infty\n\\frac{\\dd}{\\dd t}\n\\bigl(\\e^{-zt}T(t)\\bigr)\n\\dd t\n=I.\n\\]\nThe same calculation on the other side gives\n\\(\nR(z)(zI-A)=I\n\\).\nMultiplication by \\(P_\\beta\\) and \\(P_\\alpha\\) may be passed under the norm convergent integral, which gives the packet formula.\n\\end{proof}","\\begin{corollary}[Resolvent expansion over packet paths]\\label{extended:record:006}\\label{extended:cor:resolvent_packet_paths}\nWith \\(A=D+N\\) as in the directed path decomposition, let\n\\[\nR_D(z)=(zI-D)^{-1}.\n\\]\nIf\n\\[\n\\norm{R_D(z)N}<1,\n\\]\nthen\n\\[\n(zI-A)^{-1}\n=\n\\sum_{m=0}^\\infty\n\\bigl(R_D(z)N\\bigr)^mR_D(z).\n\\]\nAfter multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{corollary}","n over packet paths]\\label{extended:record:006}\\label{extended:cor:resolvent_packet_paths}\nWith \\(A=D+N\\) as in the directed path decomposition, let\n\\[\nR_D(z)=(zI-D)^{-1}.\n\\]\nIf\n\\[\n\\norm{R_D(z)N}<1,\n\\]\nthen\n\\[\n(zI-A)^{-1}\n=\n\\sum_{m=0}^\\infty\n\\bigl(R_D(z)N\\bigr)^mR_D(z).\n\\]\nAfter multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{corollary}","et paths]\\label{extended:record:006}\\label{extended:cor:resolvent_packet_paths}\nWith \\(A=D+N\\) as in the directed path decomposition, let\n\\[\nR_D(z)=(zI-D)^{-1}.\n\\]\nIf\n\\[\n\\norm{R_D(z)N}<1,\n\\]\nthen\n\\[\n(zI-A)^{-1}\n=\n\\sum_{m=0}^\\infty\n\\bigl(R_D(z)N\\bigr)^mR_D(z).\n\\]\nAfter multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{corollary}","l{extended:record:006}\\label{extended:cor:resolvent_packet_paths}\nWith \\(A=D+N\\) as in the directed path decomposition, let\n\\[\nR_D(z)=(zI-D)^{-1}.\n\\]\nIf\n\\[\n\\norm{R_D(z)N}<1,\n\\]\nthen\n\\[\n(zI-A)^{-1}\n=\n\\sum_{m=0}^\\infty\n\\bigl(R_D(z)N\\bigr)^mR_D(z).\n\\]\nAfter multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{corollary}","\\begin{proof}\nFactor\n\\[\nzI-A=(zI-D)(I-R_D(z)N)\n\\]\nand expand the second inverse by its norm convergent Neumann series. The diagonal resolvent preserves each packet and every factor \\(N\\) changes the packet along an active transfer edge.\n\\end{proof}","\\begin{theorem}[Reducing packet criterion]\\label{extended:record:007}\\label{extended:thm:reducing_packets}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item \\(AP_\\alpha=P_\\alpha A\\) for every \\(\\alpha\\).\n\\item \\(A_{\\beta\\alpha}=0\\) whenever \\(\\beta\\neq\\alpha\\).\n\\item \\(T_{\\beta\\alpha}(t)=0\\) whenever \\(\\beta\\neq\\alpha\\) and \\(t\\in\\RR\\).\n\\item Every packet space \\(P_\\alpha X\\) is invariant under \\(T(t)\\) for every \\(t\\).\n\\end{enumerate}\nWhen these conditions hold,\n\\[\nT(t)|_{P_\\alpha X}\n=\n\\e^{tA_{\\alpha\\alpha}}.\n\\]\n\\end{theorem}","[Reducing packet criterion]\\label{extended:record:007}\\label{extended:thm:reducing_packets}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item \\(AP_\\alpha=P_\\alpha A\\) for every \\(\\alpha\\).\n\\item \\(A_{\\beta\\alpha}=0\\) whenever \\(\\beta\\neq\\alpha\\).\n\\item \\(T_{\\beta\\alpha}(t)=0\\) whenever \\(\\beta\\neq\\alpha\\) and \\(t\\in\\RR\\).\n\\item Every packet space \\(P_\\alpha X\\) is invariant under \\(T(t)\\) for every \\(t\\).\n\\end{enumerate}\nWhen these conditions hold,\n\\[\nT(t)|_{P_\\alpha X}\n=\n\\e^{tA_{\\alpha\\alpha}}.\n\\]\n\\end{theorem}","iterion]\\label{extended:record:007}\\label{extended:thm:reducing_packets}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item \\(AP_\\alpha=P_\\alpha A\\) for every \\(\\alpha\\).\n\\item \\(A_{\\beta\\alpha}=0\\) whenever \\(\\beta\\neq\\alpha\\).\n\\item \\(T_{\\beta\\alpha}(t)=0\\) whenever \\(\\beta\\neq\\alpha\\) and \\(t\\in\\RR\\).\n\\item Every packet space \\(P_\\alpha X\\) is invariant under \\(T(t)\\) for every \\(t\\).\n\\end{enumerate}\nWhen these conditions hold,\n\\[\nT(t)|_{P_\\alpha X}\n=\n\\e^{tA_{\\alpha\\alpha}}.\n\\]\n\\end{theorem}","thm:reducing_packets}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item \\(AP_\\alpha=P_\\alpha A\\) for every \\(\\alpha\\).\n\\item \\(A_{\\beta\\alpha}=0\\) whenever \\(\\beta\\neq\\alpha\\).\n\\item \\(T_{\\beta\\alpha}(t)=0\\) whenever \\(\\beta\\neq\\alpha\\) and \\(t\\in\\RR\\).\n\\item Every packet space \\(P_\\alpha X\\) is invariant under \\(T(t)\\) for every \\(t\\).\n\\end{enumerate}\nWhen these conditions hold,\n\\[\nT(t)|_{P_\\alpha X}\n=\n\\e^{tA_{\\alpha\\alpha}}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe first two conditions are equivalent because the packet resolution gives\n\\[\nAP_\\alpha=\\sum_\\beta A_{\\beta\\alpha},\n\\qquad\nP_\\alpha A=\\sum_\\gamma A_{\\alpha\\gamma}.\n\\]\nA block diagonal generator has block diagonal powers, so the second condition implies the third. Differentiating the off-diagonal blocks at zero proves the converse. The equivalence with invariance follows from the definition of the blocks. The final identity follows by restricting the exponential series.\n\\end{proof}","Set\n\\[\nD=\\sum_{\\alpha\\in I}P_\\alpha AP_\\alpha,\n\\qquad\nN=A-D.\n\\]\nThe operator \\(D\\) preserves every packet and \\(N\\) contains only transfers between distinct packets.","\\begin{theorem}[Dyson expansion over packet paths]\\label{extended:record:008}\\label{extended:thm:dyson_paths}\nFor \\(t\\geq0\\),\n\\[\n\\begin{aligned}\n\\e^{tA}\n={}&\\e^{tD}\n+\n\\sum_{m=1}^{\\infty}\n\\int_{0<s_1<\\cdots<s_m<t}\n\\e^{(t-s_m)D}N\\e^{(s_m-s_{m-1})D}\n\\cdots\nN\\e^{s_1D}\n\\dd s_1\\cdots\\dd s_m.\n\\end{aligned}\n\\]\nThe series converges absolutely in operator norm. After multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{theorem}","ion over packet paths]\\label{extended:record:008}\\label{extended:thm:dyson_paths}\nFor \\(t\\geq0\\),\n\\[\n\\begin{aligned}\n\\e^{tA}\n={}&\\e^{tD}\n+\n\\sum_{m=1}^{\\infty}\n\\int_{0<s_1<\\cdots<s_m<t}\n\\e^{(t-s_m)D}N\\e^{(s_m-s_{m-1})D}\n\\cdots\nN\\e^{s_1D}\n\\dd s_1\\cdots\\dd s_m.\n\\end{aligned}\n\\]\nThe series converges absolutely in operator norm. After multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{theorem}","bel{extended:thm:dyson_paths}\nFor \\(t\\geq0\\),\n\\[\n\\begin{aligned}\n\\e^{tA}\n={}&\\e^{tD}\n+\n\\sum_{m=1}^{\\infty}\n\\int_{0<s_1<\\cdots<s_m<t}\n\\e^{(t-s_m)D}N\\e^{(s_m-s_{m-1})D}\n\\cdots\nN\\e^{s_1D}\n\\dd s_1\\cdots\\dd s_m.\n\\end{aligned}\n\\]\nThe series converges absolutely in operator norm. After multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{theorem}","d:thm:dyson_paths}\nFor \\(t\\geq0\\),\n\\[\n\\begin{aligned}\n\\e^{tA}\n={}&\\e^{tD}\n+\n\\sum_{m=1}^{\\infty}\n\\int_{0<s_1<\\cdots<s_m<t}\n\\e^{(t-s_m)D}N\\e^{(s_m-s_{m-1})D}\n\\cdots\nN\\e^{s_1D}\n\\dd s_1\\cdots\\dd s_m.\n\\end{aligned}\n\\]\nThe series converges absolutely in operator norm. After multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{theorem}","ths}\nFor \\(t\\geq0\\),\n\\[\n\\begin{aligned}\n\\e^{tA}\n={}&\\e^{tD}\n+\n\\sum_{m=1}^{\\infty}\n\\int_{0<s_1<\\cdots<s_m<t}\n\\e^{(t-s_m)D}N\\e^{(s_m-s_{m-1})D}\n\\cdots\nN\\e^{s_1D}\n\\dd s_1\\cdots\\dd s_m.\n\\end{aligned}\n\\]\nThe series converges absolutely in operator norm. After multiplication by \\(P_\\beta\\) and \\(P_\\alpha\\), the term of order \\(m\\) is a sum over directed packet paths of length \\(m\\) from \\(\\alpha\\) to \\(\\beta\\).\n\\end{theorem}","\\begin{proof}\nWrite the variation of constants identity\n\\[\n\\e^{tA}\n=\n\\e^{tD}\n+\n\\int_0^t\\e^{(t-s)D}N\\e^{sA}\\dd s.\n\\]\nIterating the identity gives the displayed series. The norm of the term of order \\(m\\) is at most\n\\[\n\\e^{t\\norm D}\\frac{(t\\norm N)^m}{m!}.\n\\]\nThe majorant is summable. Inserting the packet resolution around every factor \\(N\\) leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.\n\\end{proof}","\\begin{corollary}[Termination on an acyclic transfer graph]\\label{extended:record:009}\\label{extended:cor:acyclic_termination}\nSuppose the directed graph with an edge \\(\\alpha\\to\\beta\\) whenever \\(P_\\beta NP_\\alpha\\neq0\\) is acyclic. If its longest directed path has length \\(q\\), then every Dyson term of order greater than \\(q\\) vanishes.\n\\end{corollary}","\\begin{proof}\nEvery nonzero term of order \\(m\\) contains a directed transfer path of length \\(m\\). An acyclic graph has no path longer than \\(q\\).\n\\end{proof}","Let \\(X_f\\) and \\(X_c\\) be complex Banach spaces. Let \\(A_f\\in\\mathcal B(X_f)\\), \\(A_c\\in\\mathcal B(X_c)\\), and \\(C\\in\\mathcal B(X_f,X_c)\\). Define\n\\[\n\\Delta=CA_f-A_cC\n\\]\nand\n\\[\n\\mathfrak D(t)=C\\e^{tA_f}-\\e^{tA_c}C.\n\\]","\\begin{theorem}[Duhamel defect identity]\\label{extended:record:010}\\label{extended:thm:duhamel_defect}\nFor every \\(t\\geq0\\),\n\\[\n\\mathfrak D(t)\n=\n\\int_0^t\n\\e^{(t-s)A_c}\\Delta\\e^{sA_f}\n\\dd s.\n\\]\nIf\n\\[\n\\norm{\\e^{tA_f}}\\leq M_f\\e^{\\omega_f t},\n\\qquad\n\\norm{\\e^{tA_c}}\\leq M_c\\e^{\\omega_c t},\n\\]\nthen\n\\[\n\\norm{\\mathfrak D(t)}\n\\leq\nM_fM_c\\norm\\Delta\n\\int_0^t\n\\e^{\\omega_c(t-s)+\\omega_fs}\n\\dd s.\n\\]\nFor contraction semigroups,\n\\[\n\\norm{\\mathfrak D(t)}\\leq t\\norm\\Delta.\n\\]\n\\end{theorem}","l defect identity]\\label{extended:record:010}\\label{extended:thm:duhamel_defect}\nFor every \\(t\\geq0\\),\n\\[\n\\mathfrak D(t)\n=\n\\int_0^t\n\\e^{(t-s)A_c}\\Delta\\e^{sA_f}\n\\dd s.\n\\]\nIf\n\\[\n\\norm{\\e^{tA_f}}\\leq M_f\\e^{\\omega_f t},\n\\qquad\n\\norm{\\e^{tA_c}}\\leq M_c\\e^{\\omega_c t},\n\\]\nthen\n\\[\n\\norm{\\mathfrak D(t)}\n\\leq\nM_fM_c\\norm\\Delta\n\\int_0^t\n\\e^{\\omega_c(t-s)+\\omega_fs}\n\\dd s.\n\\]\nFor contraction semigroups,\n\\[\n\\norm{\\mathfrak D(t)}\\leq t\\norm\\Delta.\n\\]\n\\end{theorem}","\\begin{proof}\nFor fixed \\(t\\), set\n\\[\nF(s)=\\e^{(t-s)A_c}C\\e^{sA_f}.\n\\]\nThen\n\\[\nF'(s)=\\e^{(t-s)A_c}\\Delta\\e^{sA_f}.\n\\]\nIntegrating from zero to \\(t\\) gives the identity. The estimates follow by taking norms under the integral.\n\\end{proof}","\\begin{proposition}[Infinitesimal recovery of the defect]\\label{extended:record:011}\\label{extended:prop:infinitesimal_defect}\nThe generator defect is the first order semigroup defect:\n\\[\n\\lim_{t\\to0}\n\\frac{\\mathfrak D(t)}{t}\n=\n\\Delta\n\\]\nin operator norm. Consequently,\n\\[\nC\\e^{tA_f}=\\e^{tA_c}C\n\\quad\\hbox{for every }t\n\\]\nif and only if \\(CA_f=A_cC\\).\n\\end{proposition}","\\begin{proof}\nThe operator norm expansions\n\\[\n\\e^{tA_f}=I+tA_f+o(t),\n\\qquad\n\\e^{tA_c}=I+tA_c+o(t)\n\\]\ngive the limit. The equivalence follows from this limit and \\cref{extended:thm:duhamel_defect}.\n\\end{proof}","\\begin{theorem}[Resolvent defect identity]\\label{extended:record:012}\\label{extended:thm:resolvent_defect}\nLet \\(z\\) belong to the resolvent sets of \\(A_f\\) and \\(A_c\\). With\n\\[\nR_f(z)=(zI-A_f)^{-1},\n\\qquad\nR_c(z)=(zI-A_c)^{-1},\n\\]\none has\n\\[\nCR_f(z)-R_c(z)C\n=\nR_c(z)\\Delta R_f(z).\n\\]\n\\end{theorem}","\\begin{proof}\nMultiply the proposed identity on the left by \\(zI-A_c\\) and on the right by \\(zI-A_f\\). Both sides reduce to\n\\[\n(zI-A_c)C-C(zI-A_f)=\\Delta.\n\\]\n\\end{proof}","\\begin{theorem}[Holomorphic defect formula]\\label{extended:record:013}\\label{extended:thm:holomorphic_defect}\nLet \\(\\Gamma\\) be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that \\(\\Gamma\\) lies in the common resolvent set. Thus its winding number is one about every spectral point. If \\(g\\) is holomorphic on a neighborhood of the closure of the union, then\n\\[\nCg(A_f)-g(A_c)C\n=\n\\frac{1}{2\\pi\\ii}\n\\int_\\Gamma\n g(z)R_c(z)\\Delta R_f(z)\n\\dd z.\n\\]\nIn particular,\n\\[\n\\begin{aligned}\n\\norm{Cg(A_f)-g(A_c)C}\n\\leq{}&\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}\\abs{g(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_c(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_f(z)}\n\\norm\\Delta.\n\\end{aligned}\n\\]\n\\end{theorem}","begin{theorem}[Holomorphic defect formula]\\label{extended:record:013}\\label{extended:thm:holomorphic_defect}\nLet \\(\\Gamma\\) be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that \\(\\Gamma\\) lies in the common resolvent set. Thus its winding number is one about every spectral point. If \\(g\\) is holomorphic on a neighborhood of the closure of the union, then\n\\[\nCg(A_f)-g(A_c)C\n=\n\\frac{1}{2\\pi\\ii}\n\\int_\\Gamma\n g(z)R_c(z)\\Delta R_f(z)\n\\dd z.\n\\]\nIn particular,\n\\[\n\\begin{aligned}\n\\norm{Cg(A_f)-g(A_c)C}\n\\leq{}&\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}\\abs{g(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_c(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_f(z)}\n\\norm\\Delta.\n\\end{aligned}\n\\]\n\\end{theorem}","el{extended:thm:holomorphic_defect}\nLet \\(\\Gamma\\) be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that \\(\\Gamma\\) lies in the common resolvent set. Thus its winding number is one about every spectral point. If \\(g\\) is holomorphic on a neighborhood of the closure of the union, then\n\\[\nCg(A_f)-g(A_c)C\n=\n\\frac{1}{2\\pi\\ii}\n\\int_\\Gamma\n g(z)R_c(z)\\Delta R_f(z)\n\\dd z.\n\\]\nIn particular,\n\\[\n\\begin{aligned}\n\\norm{Cg(A_f)-g(A_c)C}\n\\leq{}&\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}\\abs{g(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_c(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_f(z)}\n\\norm\\Delta.\n\\end{aligned}\n\\]\n\\end{theorem}","a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that \\(\\Gamma\\) lies in the common resolvent set. Thus its winding number is one about every spectral point. If \\(g\\) is holomorphic on a neighborhood of the closure of the union, then\n\\[\nCg(A_f)-g(A_c)C\n=\n\\frac{1}{2\\pi\\ii}\n\\int_\\Gamma\n g(z)R_c(z)\\Delta R_f(z)\n\\dd z.\n\\]\nIn particular,\n\\[\n\\begin{aligned}\n\\norm{Cg(A_f)-g(A_c)C}\n\\leq{}&\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}\\abs{g(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_c(z)}\n\\sup_{z\\in\\Gamma}\\norm{R_f(z)}\n\\norm\\Delta.\n\\end{aligned}\n\\]\n\\end{theorem}","\\begin{corollary}[Exact naturality]\\label{extended:record:014}\\label{extended:cor:exact_naturality}\nIf \\(CA_f=A_cC\\), then\n\\[\nCg(A_f)=g(A_c)C\n\\]\nfor every holomorphic function covered by \\cref{extended:thm:holomorphic_defect}. The conclusion includes exponentials, resolvents, and polynomials.\n\\end{corollary}","\\begin{proof}\nSet \\(\\Delta=0\\) in \\cref{extended:thm:holomorphic_defect}.\n\\end{proof}","Let\n\\[\nX_0\\xleftarrow{C_0}X_1\\xleftarrow{C_1}X_2\\xleftarrow{}\\cdots\n\\]\nbe Banach spaces with bounded connecting maps. Define the bounded inverse limit\n\\[\nX_\\infty\n=\n\\left\\{\n(x_\\ell)_{\\ell\\geq0}:\nC_\\ell x_{\\ell+1}=x_\\ell,\n\\quad\n\\sup_\\ell\\norm{x_\\ell}<\\infty\n\\right\\}\n\\]\nwith the supremum norm.","\\begin{lemma}[Completeness of the bounded inverse limit]\\label{extended:record:015}\\label{extended:lem:inverse_limit_banach}\nThe space \\(X_\\infty\\) is Banach.\n\\end{lemma}","\\begin{proof}\nThe product of the spaces \\(X_\\ell\\), equipped with the bounded supremum norm, is Banach. The equations \\(C_\\ell x_{\\ell+1}=x_\\ell\\) define a closed linear subspace.\n\\end{proof}","\\begin{theorem}[Exact projective limit semigroup]\\label{extended:record:016}\\label{extended:thm:exact_projective_limit}\nFor each \\(\\ell\\), let \\(T_\\ell(t)\\) be a strongly continuous semigroup on \\(X_\\ell\\). Assume\n\\[\nC_\\ell T_{\\ell+1}(t)=T_\\ell(t)C_\\ell\n\\]\nfor every \\(\\ell\\) and \\(t\\geq0\\), and assume the uniform bound\n\\[\n\\sup_\\ell\\norm{T_\\ell(t)}\\leq M\\e^{\\omega t}.\n\\]\nDefine\n\\[\nT_\\infty(t)(x_\\ell)_\\ell\n=\n(T_\\ell(t)x_\\ell)_\\ell.\n\\]\nThen \\(T_\\infty(t)\\) is a bounded semigroup on \\(X_\\infty\\). It is strongly continuous if\n\\[\n\\lim_{t\\downarrow0}\n\\sup_\\ell\n\\norm{T_\\ell(t)x_\\ell-x_\\ell}=0\n\\]\nfor every \\((x_\\ell)_\\ell\\in X_\\infty\\).","roup]\\label{extended:record:016}\\label{extended:thm:exact_projective_limit}\nFor each \\(\\ell\\), let \\(T_\\ell(t)\\) be a strongly continuous semigroup on \\(X_\\ell\\). Assume\n\\[\nC_\\ell T_{\\ell+1}(t)=T_\\ell(t)C_\\ell\n\\]\nfor every \\(\\ell\\) and \\(t\\geq0\\), and assume the uniform bound\n\\[\n\\sup_\\ell\\norm{T_\\ell(t)}\\leq M\\e^{\\omega t}.\n\\]\nDefine\n\\[\nT_\\infty(t)(x_\\ell)_\\ell\n=\n(T_\\ell(t)x_\\ell)_\\ell.\n\\]\nThen \\(T_\\infty(t)\\) is a bounded semigroup on \\(X_\\infty\\). It is strongly continuous if\n\\[\n\\lim_{t\\downarrow0}\n\\sup_\\ell\n\\norm{T_\\ell(t)x_\\ell-x_\\ell}=0\n\\]\nfor every \\((x_\\ell)_\\ell\\in X_\\infty\\).","ended:thm:exact_projective_limit}\nFor each \\(\\ell\\), let \\(T_\\ell(t)\\) be a strongly continuous semigroup on \\(X_\\ell\\). Assume\n\\[\nC_\\ell T_{\\ell+1}(t)=T_\\ell(t)C_\\ell\n\\]\nfor every \\(\\ell\\) and \\(t\\geq0\\), and assume the uniform bound\n\\[\n\\sup_\\ell\\norm{T_\\ell(t)}\\leq M\\e^{\\omega t}.\n\\]\nDefine\n\\[\nT_\\infty(t)(x_\\ell)_\\ell\n=\n(T_\\ell(t)x_\\ell)_\\ell.\n\\]\nThen \\(T_\\infty(t)\\) is a bounded semigroup on \\(X_\\infty\\). It is strongly continuous if\n\\[\n\\lim_{t\\downarrow0}\n\\sup_\\ell\n\\norm{T_\\ell(t)x_\\ell-x_\\ell}=0\n\\]\nfor every \\((x_\\ell)_\\ell\\in X_\\infty\\).","\\(\\ell\\), let \\(T_\\ell(t)\\) be a strongly continuous semigroup on \\(X_\\ell\\). Assume\n\\[\nC_\\ell T_{\\ell+1}(t)=T_\\ell(t)C_\\ell\n\\]\nfor every \\(\\ell\\) and \\(t\\geq0\\), and assume the uniform bound\n\\[\n\\sup_\\ell\\norm{T_\\ell(t)}\\leq M\\e^{\\omega t}.\n\\]\nDefine\n\\[\nT_\\infty(t)(x_\\ell)_\\ell\n=\n(T_\\ell(t)x_\\ell)_\\ell.\n\\]\nThen \\(T_\\infty(t)\\) is a bounded semigroup on \\(X_\\infty\\). It is strongly continuous if\n\\[\n\\lim_{t\\downarrow0}\n\\sup_\\ell\n\\norm{T_\\ell(t)x_\\ell-x_\\ell}=0\n\\]\nfor every \\((x_\\ell)_\\ell\\in X_\\infty\\).","Assume\n\\[\nC_\\ell T_{\\ell+1}(t)=T_\\ell(t)C_\\ell\n\\]\nfor every \\(\\ell\\) and \\(t\\geq0\\), and assume the uniform bound\n\\[\n\\sup_\\ell\\norm{T_\\ell(t)}\\leq M\\e^{\\omega t}.\n\\]\nDefine\n\\[\nT_\\infty(t)(x_\\ell)_\\ell\n=\n(T_\\ell(t)x_\\ell)_\\ell.\n\\]\nThen \\(T_\\infty(t)\\) is a bounded semigroup on \\(X_\\infty\\). It is strongly continuous if\n\\[\n\\lim_{t\\downarrow0}\n\\sup_\\ell\n\\norm{T_\\ell(t)x_\\ell-x_\\ell}=0\n\\]\nfor every \\((x_\\ell)_\\ell\\in X_\\infty\\).","If \\(T_\\ell(t)=\\e^{tA_\\ell}\\), the generators are bounded, and\n\\[\n\\sup_\\ell\\norm{A_\\ell}<\\infty,\n\\qquad\nC_\\ell A_{\\ell+1}=A_\\ell C_\\ell,\n\\]\nthen the continuity condition holds and\n\\[\nA_\\infty(x_\\ell)_\\ell=(A_\\ell x_\\ell)_\\ell\n\\]\nis the bounded generator of \\(T_\\infty\\).\n\\end{theorem}","\\begin{proof}\nExact intertwining shows that \\((T_\\ell(t)x_\\ell)_\\ell\\) remains compatible. The uniform growth bound makes the coordinatewise operator bounded on the supremum norm. The semigroup law holds in every coordinate. The stated continuity condition is exactly strong continuity in the supremum norm.","For bounded generators with common bound \\(M_A\\),\n\\[\n\\sup_\\ell\n\\norm{(\\e^{tA_\\ell}-I)x_\\ell}\n\\leq\n(\\e^{tM_A}-1)\\sup_\\ell\\norm{x_\\ell}.\n\\]\nThis tends to zero with \\(t\\). Exact generator intertwining makes \\((A_\\ell x_\\ell)_\\ell\\) compatible, and coordinatewise differentiation identifies the generator.\n\\end{proof}","Assume now that each \\(C_\\ell:X_{\\ell+1}\\to X_\\ell\\) has a bounded right inverse\n\\[\nR_\\ell:X_\\ell\\to X_{\\ell+1},\n\\qquad\nC_\\ell R_\\ell=I_{X_\\ell}.\n\\]\nFor \\(m<n\\), set\n\\[\nC_{m,n}=C_mC_{m+1}\\cdots C_{n-1},\n\\qquad\nR_{n,m}=R_{n-1}\\cdots R_m.\n\\]\nThe empty products are\n\\[\nC_{m,m}=I_{X_m},\n\\qquad\nR_{m,m}=I_{X_m}.\n\\]","\\begin{definition}[Reconstructed fine evolution]\\label{extended:record:018}\nFor a semigroup \\(T_n(t)\\) on \\(X_n\\), define the evolution seen from level \\(m\\) by\n\\[\nU_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.\n\\]\n\\end{definition}","\\begin{theorem}[Weighted summable defect completion]\\label{extended:record:019}\\label{extended:thm:summable_defect_completion}\nDefine the adjacent semigroup defect\n\\[\nE_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.\n\\]\nFix a coarse level \\(m\\) and a time horizon \\(T>0\\). For \\(n\\geq m\\), set\n\\[\nw_{m,n}\n=\n\\norm{C_{m,n}}\\,\n\\norm{R_nR_{n,m}}.\n\\]\nIf\n\\[\n\\sum_{n=m}^{\\infty}\nw_{m,n}\n\\sup_{0\\leq t\\leq T}\\norm{E_n(t)}<\\infty,\n\\]\nthen \\(U_{m,n}(t)\\) converges in operator norm uniformly for \\(0\\leq t\\leq T\\). Its limit is denoted by \\(U_m(t)\\), and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\n\\sum_{j=n}^{\\infty}w_{m,j}\n\\sup_{0\\leq t\\leq T}\\norm{E_j(t)}.\n\\]\n\\end{theorem}","}[Weighted summable defect completion]\\label{extended:record:019}\\label{extended:thm:summable_defect_completion}\nDefine the adjacent semigroup defect\n\\[\nE_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.\n\\]\nFix a coarse level \\(m\\) and a time horizon \\(T>0\\). For \\(n\\geq m\\), set\n\\[\nw_{m,n}\n=\n\\norm{C_{m,n}}\\,\n\\norm{R_nR_{n,m}}.\n\\]\nIf\n\\[\n\\sum_{n=m}^{\\infty}\nw_{m,n}\n\\sup_{0\\leq t\\leq T}\\norm{E_n(t)}<\\infty,\n\\]\nthen \\(U_{m,n}(t)\\) converges in operator norm uniformly for \\(0\\leq t\\leq T\\). Its limit is denoted by \\(U_m(t)\\), and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\n\\sum_{j=n}^{\\infty}w_{m,j}\n\\sup_{0\\leq t\\leq T}\\norm{E_j(t)}.\n\\]\n\\end{theorem}","cord:019}\\label{extended:thm:summable_defect_completion}\nDefine the adjacent semigroup defect\n\\[\nE_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.\n\\]\nFix a coarse level \\(m\\) and a time horizon \\(T>0\\). For \\(n\\geq m\\), set\n\\[\nw_{m,n}\n=\n\\norm{C_{m,n}}\\,\n\\norm{R_nR_{n,m}}.\n\\]\nIf\n\\[\n\\sum_{n=m}^{\\infty}\nw_{m,n}\n\\sup_{0\\leq t\\leq T}\\norm{E_n(t)}<\\infty,\n\\]\nthen \\(U_{m,n}(t)\\) converges in operator norm uniformly for \\(0\\leq t\\leq T\\). Its limit is denoted by \\(U_m(t)\\), and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\n\\sum_{j=n}^{\\infty}w_{m,j}\n\\sup_{0\\leq t\\leq T}\\norm{E_j(t)}.\n\\]\n\\end{theorem}","t_completion}\nDefine the adjacent semigroup defect\n\\[\nE_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.\n\\]\nFix a coarse level \\(m\\) and a time horizon \\(T>0\\). For \\(n\\geq m\\), set\n\\[\nw_{m,n}\n=\n\\norm{C_{m,n}}\\,\n\\norm{R_nR_{n,m}}.\n\\]\nIf\n\\[\n\\sum_{n=m}^{\\infty}\nw_{m,n}\n\\sup_{0\\leq t\\leq T}\\norm{E_n(t)}<\\infty,\n\\]\nthen \\(U_{m,n}(t)\\) converges in operator norm uniformly for \\(0\\leq t\\leq T\\). Its limit is denoted by \\(U_m(t)\\), and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\n\\sum_{j=n}^{\\infty}w_{m,j}\n\\sup_{0\\leq t\\leq T}\\norm{E_j(t)}.\n\\]\n\\end{theorem}","fine the adjacent semigroup defect\n\\[\nE_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.\n\\]\nFix a coarse level \\(m\\) and a time horizon \\(T>0\\). For \\(n\\geq m\\), set\n\\[\nw_{m,n}\n=\n\\norm{C_{m,n}}\\,\n\\norm{R_nR_{n,m}}.\n\\]\nIf\n\\[\n\\sum_{n=m}^{\\infty}\nw_{m,n}\n\\sup_{0\\leq t\\leq T}\\norm{E_n(t)}<\\infty,\n\\]\nthen \\(U_{m,n}(t)\\) converges in operator norm uniformly for \\(0\\leq t\\leq T\\). Its limit is denoted by \\(U_m(t)\\), and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\n\\sum_{j=n}^{\\infty}w_{m,j}\n\\sup_{0\\leq t\\leq T}\\norm{E_j(t)}.\n\\]\n\\end{theorem}","\\begin{proof}\nSince \\(C_nR_n=I\\),\n\\[\n\\begin{aligned}\nU_{m,n+1}(t)-U_{m,n}(t)\n={}&\nC_{m,n}\n\\bigl(C_nT_{n+1}(t)-T_n(t)C_n\\bigr)\nR_nR_{n,m}.\n\\end{aligned}\n\\]\nTherefore\n\\[\n\\norm{U_{m,n+1}(t)-U_{m,n}(t)}\n\\leq\nw_{m,n}\\norm{E_n(t)}.\n\\]\nThe weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.\n\\end{proof}","\\begin{corollary}[Contractive summable defect completion]\\label{extended:record:020}\\label{extended:cor:contractive_defect_completion}\nAssume\n\\[\n\\norm{C_\\ell}\\leq1,\n\\qquad\n\\norm{R_\\ell}\\leq1,\n\\qquad\n\\norm{T_\\ell(t)}\\leq1\n\\]\nfor every \\(\\ell\\) and \\(t\\geq0\\). If\n\\[\n\\sum_{\\ell=0}^{\\infty}\n\\sup_{0\\leq t\\leq T}\\norm{E_\\ell(t)}<\\infty,\n\\]\nthen the conclusion of \\cref{extended:thm:summable_defect_completion} holds, and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\n\\sum_{\\ell=n}^{\\infty}\n\\sup_{0\\leq t\\leq T}\\norm{E_\\ell(t)}.\n\\]\n\\end{corollary}","ontractive summable defect completion]\\label{extended:record:020}\\label{extended:cor:contractive_defect_completion}\nAssume\n\\[\n\\norm{C_\\ell}\\leq1,\n\\qquad\n\\norm{R_\\ell}\\leq1,\n\\qquad\n\\norm{T_\\ell(t)}\\leq1\n\\]\nfor every \\(\\ell\\) and \\(t\\geq0\\). If\n\\[\n\\sum_{\\ell=0}^{\\infty}\n\\sup_{0\\leq t\\leq T}\\norm{E_\\ell(t)}<\\infty,\n\\]\nthen the conclusion of \\cref{extended:thm:summable_defect_completion} holds, and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\n\\sum_{\\ell=n}^{\\infty}\n\\sup_{0\\leq t\\leq T}\\norm{E_\\ell(t)}.\n\\]\n\\end{corollary}","\\begin{proof}\nEvery weight \\(w_{m,n}\\) is at most one.\n\\end{proof}","\\begin{corollary}[Generator defect criterion]\\label{extended:record:021}\\label{extended:cor:generator_defect_completion}\nAssume\n\\[\n\\norm{C_\\ell}\\leq1,\n\\qquad\n\\norm{R_\\ell}\\leq1,\n\\]\nand suppose \\(T_\\ell(t)=\\e^{tA_\\ell}\\) are contraction semigroups with bounded generators. Set\n\\[\n\\Delta_\\ell=C_\\ell A_{\\ell+1}-A_\\ell C_\\ell.\n\\]\nIf\n\\[\n\\sum_{\\ell=0}^{\\infty}\\norm{\\Delta_\\ell}<\\infty,\n\\]\nthen the conclusion of \\cref{extended:cor:contractive_defect_completion} holds on every bounded time interval, and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\nT\\sum_{\\ell=n}^{\\infty}\\norm{\\Delta_\\ell}.\n\\]\n\\end{corollary}","defect_completion}\nAssume\n\\[\n\\norm{C_\\ell}\\leq1,\n\\qquad\n\\norm{R_\\ell}\\leq1,\n\\]\nand suppose \\(T_\\ell(t)=\\e^{tA_\\ell}\\) are contraction semigroups with bounded generators. Set\n\\[\n\\Delta_\\ell=C_\\ell A_{\\ell+1}-A_\\ell C_\\ell.\n\\]\nIf\n\\[\n\\sum_{\\ell=0}^{\\infty}\\norm{\\Delta_\\ell}<\\infty,\n\\]\nthen the conclusion of \\cref{extended:cor:contractive_defect_completion} holds on every bounded time interval, and\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{U_m(t)-U_{m,n}(t)}\n\\leq\nT\\sum_{\\ell=n}^{\\infty}\\norm{\\Delta_\\ell}.\n\\]\n\\end{corollary}","\\begin{theorem}[Asymptotic closure criterion]\\label{extended:record:022}\\label{extended:thm:asymptotic_closure}\nAssume the weighted convergence in \\cref{extended:thm:summable_defect_completion} holds for every bounded time interval, and define\n\\[\nP_{m,n}=R_{n,m}C_{m,n}\n\\]\non \\(X_n\\). Suppose that for every \\(T>0\\),\n\\[\n\\rho_{m,n}(T)\n=\n\\sup_{0\\leq s,t\\leq T}\n\\norm{\nC_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}\n}\n\\longrightarrow0.\n\\]\nThen\n\\[\nU_m(s+t)=U_m(s)U_m(t)\n\\]\nfor all \\(s,t\\geq0\\). If the convergence in \\cref{extended:thm:summable_defect_completion} is locally uniform and every \\(T_n\\) is norm continuous, then \\(U_m\\) is a norm continuous semigroup.\n\\end{theorem}","criterion]\\label{extended:record:022}\\label{extended:thm:asymptotic_closure}\nAssume the weighted convergence in \\cref{extended:thm:summable_defect_completion} holds for every bounded time interval, and define\n\\[\nP_{m,n}=R_{n,m}C_{m,n}\n\\]\non \\(X_n\\). Suppose that for every \\(T>0\\),\n\\[\n\\rho_{m,n}(T)\n=\n\\sup_{0\\leq s,t\\leq T}\n\\norm{\nC_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}\n}\n\\longrightarrow0.\n\\]\nThen\n\\[\nU_m(s+t)=U_m(s)U_m(t)\n\\]\nfor all \\(s,t\\geq0\\). If the convergence in \\cref{extended:thm:summable_defect_completion} is locally uniform and every \\(T_n\\) is norm continuous, then \\(U_m\\) is a norm continuous semigroup.\n\\end{theorem}","2}\\label{extended:thm:asymptotic_closure}\nAssume the weighted convergence in \\cref{extended:thm:summable_defect_completion} holds for every bounded time interval, and define\n\\[\nP_{m,n}=R_{n,m}C_{m,n}\n\\]\non \\(X_n\\). Suppose that for every \\(T>0\\),\n\\[\n\\rho_{m,n}(T)\n=\n\\sup_{0\\leq s,t\\leq T}\n\\norm{\nC_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}\n}\n\\longrightarrow0.\n\\]\nThen\n\\[\nU_m(s+t)=U_m(s)U_m(t)\n\\]\nfor all \\(s,t\\geq0\\). If the convergence in \\cref{extended:thm:summable_defect_completion} is locally uniform and every \\(T_n\\) is norm continuous, then \\(U_m\\) is a norm continuous semigroup.\n\\end{theorem}","completion} holds for every bounded time interval, and define\n\\[\nP_{m,n}=R_{n,m}C_{m,n}\n\\]\non \\(X_n\\). Suppose that for every \\(T>0\\),\n\\[\n\\rho_{m,n}(T)\n=\n\\sup_{0\\leq s,t\\leq T}\n\\norm{\nC_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}\n}\n\\longrightarrow0.\n\\]\nThen\n\\[\nU_m(s+t)=U_m(s)U_m(t)\n\\]\nfor all \\(s,t\\geq0\\). If the convergence in \\cref{extended:thm:summable_defect_completion} is locally uniform and every \\(T_n\\) is norm continuous, then \\(U_m\\) is a norm continuous semigroup.\n\\end{theorem}","ed time interval, and define\n\\[\nP_{m,n}=R_{n,m}C_{m,n}\n\\]\non \\(X_n\\). Suppose that for every \\(T>0\\),\n\\[\n\\rho_{m,n}(T)\n=\n\\sup_{0\\leq s,t\\leq T}\n\\norm{\nC_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}\n}\n\\longrightarrow0.\n\\]\nThen\n\\[\nU_m(s+t)=U_m(s)U_m(t)\n\\]\nfor all \\(s,t\\geq0\\). If the convergence in \\cref{extended:thm:summable_defect_completion} is locally uniform and every \\(T_n\\) is norm continuous, then \\(U_m\\) is a norm continuous semigroup.\n\\end{theorem}","\\begin{proof}\nAt a finite level,\n\\[\n\\begin{aligned}\nU_{m,n}(s)U_{m,n}(t)-U_{m,n}(s+t)\n={}&\nC_{m,n}T_n(s)\n(P_{m,n}-I)\nT_n(t)R_{n,m}.\n\\end{aligned}\n\\]\nIts norm is at most \\(\\rho_{m,n}(T)\\) for \\(s,t\\leq T\\). Pass to the limit using uniform convergence. Norm continuity follows from a uniform limit of norm continuous functions on compact intervals.\n\\end{proof}","\\begin{remark}\\label{extended:record:023}\nSummable generator defects control consistency between adjacent levels. The quantity \\(\\rho_{m,n}(T)\\) controls closure of the reconstructed fine range under the fine evolution. These are different requirements.\n\\end{remark}","\\begin{example}[Summable defects do not force a semigroup limit]\\label{extended:record:024}\\label{extended:ex:summable_not_semigroup}\nLet \\(X_0=\\RR\\) and let \\(X_n=\\RR^2\\) for every \\(n\\geq1\\). Set\n\\[\nC_0(x,y)=x,\n\\qquad\nR_0x=(x,0),\n\\]\nand use identity connecting maps and reconstructions above level one. Let \\(T_0(t)=I\\) and let every higher level carry the rotation group\n\\[\nT_n(t)\n=\n\\begin{pmatrix}\n\\cos t&-\\sin t","\\begin{example}[Summable defects do not force a semigroup limit]\\label{extended:record:024}\\label{extended:ex:summable_not_semigroup}\nLet \\(X_0=\\RR\\) and let \\(X_n=\\RR^2\\) for every \\(n\\geq1\\). Set\n\\[\nC_0(x,y)=x,\n\\qquad\nR_0x=(x,0),\n\\]\nand use identity connecting maps and reconstructions above level one. Let \\(T_0(t)=I\\) and let every higher level carry the rotation group\n\\[\nT_n(t)\n=\n\\begin{pmatrix}\n\\cos t&-\\sin t\\\\\n\\sin t&\\cos t\n\\end{pmatrix}.\n\\]\nOnly the first adjacent defect is nonzero, so every defect sum is finite. Yet for \\(n\\geq1\\),\n\\[\nU_{0,n}(t)=C_0T_n(t)R_0=\\cos t.\n\\]\nThe limiting family is not a semigroup because in general\n\\[\n\\cos(s+t)\\neq\\cos s\\cos t.\n\\]\nThe failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.\n\\end{example}","\\sin t&\\cos t\n\\end{pmatrix}.\n\\]\nOnly the first adjacent defect is nonzero, so every defect sum is finite. Yet for \\(n\\geq1\\),\n\\[\nU_{0,n}(t)=C_0T_n(t)R_0=\\cos t.\n\\]\nThe limiting family is not a semigroup because in general\n\\[\n\\cos(s+t)\\neq\\cos s\\cos t.\n\\]\nThe failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.\n\\end{example}","\\begin{proposition}[Range leakage criterion]\\label{extended:record:025}\\label{extended:prop:range_leakage}\nUnder the contractive assumptions of \\cref{extended:cor:contractive_defect_completion}, define\n\\[\n\\eta_{m,n}(T)\n=\n\\sup_{0\\leq t\\leq T}\n\\norm{(I-P_{m,n})T_n(t)R_{n,m}}.\n\\]\nThen\n\\[\n\\rho_{m,n}(T)\\leq\\eta_{m,n}(T).\n\\]\nConsequently, \\(\\eta_{m,n}(T)\\to0\\) for every \\(T>0\\) is sufficient for the limiting family \\(U_m\\) to satisfy the semigroup law.\n\\end{proposition}","ion}[Range leakage criterion]\\label{extended:record:025}\\label{extended:prop:range_leakage}\nUnder the contractive assumptions of \\cref{extended:cor:contractive_defect_completion}, define\n\\[\n\\eta_{m,n}(T)\n=\n\\sup_{0\\leq t\\leq T}\n\\norm{(I-P_{m,n})T_n(t)R_{n,m}}.\n\\]\nThen\n\\[\n\\rho_{m,n}(T)\\leq\\eta_{m,n}(T).\n\\]\nConsequently, \\(\\eta_{m,n}(T)\\to0\\) for every \\(T>0\\) is sufficient for the limiting family \\(U_m\\) to satisfy the semigroup law.\n\\end{proposition}","\\begin{proof}\nFor \\(s,t\\leq T\\), the contraction assumptions give\n\\[\n\\begin{aligned}\n&\\norm{\nC_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}\n}","\\begin{proof}\nFor \\(s,t\\leq T\\), the contraction assumptions give\n\\[\n\\begin{aligned}\n&\\norm{\nC_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}\n}\\\\\n&\\qquad\\leq\n\\norm{(I-P_{m,n})T_n(t)R_{n,m}}.\n\\end{aligned}\n\\]\nTake the supremum and apply \\cref{extended:thm:asymptotic_closure}.\n\\end{proof}","\\begin{theorem}[Generator control of range leakage]\\label{extended:record:026}\\label{extended:thm:generator_range_leakage}\nLet \\(T(t)=\\e^{tA}\\) on a Banach space \\(X\\), let \\(P\\) be a bounded projection, and set \\(Q=I-P\\). Assume\n\\[\n\\norm{T(t)}\\leq M\\e^{\\omega t}\n\\]\nand\n\\[\n\\norm{\\e^{t(QAQ)|_{QX}}}\n\\leq M_Q\\e^{\\omega_Qt}.\n\\]\nThen\n\\[\n\\norm{QT(t)P}\n\\leq\nM M_Q\\norm P^2\\norm{QAP}\n\\int_0^t\n\\e^{\\omega_Q(t-s)+\\omega s}\n\\dd s.\n\\]\nIn particular, for orthogonal projections and contraction bounds,\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{QT(t)P}\n\\leq\nT\\norm{QAP}.\n\\]\n\\end{theorem}",":thm:generator_range_leakage}\nLet \\(T(t)=\\e^{tA}\\) on a Banach space \\(X\\), let \\(P\\) be a bounded projection, and set \\(Q=I-P\\). Assume\n\\[\n\\norm{T(t)}\\leq M\\e^{\\omega t}\n\\]\nand\n\\[\n\\norm{\\e^{t(QAQ)|_{QX}}}\n\\leq M_Q\\e^{\\omega_Qt}.\n\\]\nThen\n\\[\n\\norm{QT(t)P}\n\\leq\nM M_Q\\norm P^2\\norm{QAP}\n\\int_0^t\n\\e^{\\omega_Q(t-s)+\\omega s}\n\\dd s.\n\\]\nIn particular, for orthogonal projections and contraction bounds,\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{QT(t)P}\n\\leq\nT\\norm{QAP}.\n\\]\n\\end{theorem}","\\begin{proof}\nFor \\(x\\in PX\\), write\n\\[\np(t)=PT(t)x,\n\\qquad\nq(t)=QT(t)x.\n\\]\nThe hidden component satisfies\n\\[\nq'(t)=QAQq(t)+QAPp(t),\n\\qquad\nq(0)=0.\n\\]\nVariation of constants gives\n\\[\nq(t)=\n\\int_0^t\n\\e^{(t-s)(QAQ)|_{QX}}\nQAPp(s)\n\\dd s.\n\\]\nSince\n\\(\n\\norm{p(s)}\\leq\\norm P\\norm{T(s)}\\norm x\n\\)\nand \\(x=Px\\), the first estimate follows. The contraction case is immediate.\n\\end{proof}","\\begin{corollary}[A usable closure condition for refinement towers]\\label{extended:record:027}\\label{extended:cor:generator_closure_tower}\nAssume the contractive summability hypothesis of \\cref{extended:cor:contractive_defect_completion} holds on every bounded time interval. Suppose every \\(P_{m,n}\\) is an orthogonal projection on a Hilbert space and \\(T_n(t)=\\e^{tA_n}\\) is contractive. If, for every fixed \\(m\\),\n\\[\n\\norm{(I-P_{m,n})A_nP_{m,n}}\n\\longrightarrow0,\n\\]\nthen the limiting family \\(U_m\\) is a semigroup.\n\\end{corollary}","lary}[A usable closure condition for refinement towers]\\label{extended:record:027}\\label{extended:cor:generator_closure_tower}\nAssume the contractive summability hypothesis of \\cref{extended:cor:contractive_defect_completion} holds on every bounded time interval. Suppose every \\(P_{m,n}\\) is an orthogonal projection on a Hilbert space and \\(T_n(t)=\\e^{tA_n}\\) is contractive. If, for every fixed \\(m\\),\n\\[\n\\norm{(I-P_{m,n})A_nP_{m,n}}\n\\longrightarrow0,\n\\]\nthen the limiting family \\(U_m\\) is a semigroup.\n\\end{corollary}","usable closure condition for refinement towers]\\label{extended:record:027}\\label{extended:cor:generator_closure_tower}\nAssume the contractive summability hypothesis of \\cref{extended:cor:contractive_defect_completion} holds on every bounded time interval. Suppose every \\(P_{m,n}\\) is an orthogonal projection on a Hilbert space and \\(T_n(t)=\\e^{tA_n}\\) is contractive. If, for every fixed \\(m\\),\n\\[\n\\norm{(I-P_{m,n})A_nP_{m,n}}\n\\longrightarrow0,\n\\]\nthen the limiting family \\(U_m\\) is a semigroup.\n\\end{corollary}","bel{extended:cor:generator_closure_tower}\nAssume the contractive summability hypothesis of \\cref{extended:cor:contractive_defect_completion} holds on every bounded time interval. Suppose every \\(P_{m,n}\\) is an orthogonal projection on a Hilbert space and \\(T_n(t)=\\e^{tA_n}\\) is contractive. If, for every fixed \\(m\\),\n\\[\n\\norm{(I-P_{m,n})A_nP_{m,n}}\n\\longrightarrow0,\n\\]\nthen the limiting family \\(U_m\\) is a semigroup.\n\\end{corollary}","\\begin{proof}\nA bounded generator of a contraction semigroup on a Hilbert space is dissipative. If \\(Q_{m,n}=I-P_{m,n}\\) and \\(y\\in Q_{m,n}X_n\\), then\n\\[\n\\operatorname{Re}\\ip{Q_{m,n}A_nQ_{m,n}y}{y}\n=\n\\operatorname{Re}\\ip{A_ny}{y}\n\\leq0.\n\\]\nHence the compressed hidden semigroup is contractive. Apply \\cref{extended:thm:generator_range_leakage} to \\(P_{m,n}\\). Since \\(P_{m,n}R_{n,m}=R_{n,m}\\),\n\\[\n\\eta_{m,n}(T)\n\\leq\nT\\norm{(I-P_{m,n})A_nP_{m,n}}\n\\norm{R_{n,m}}.\n\\]\nThe right side tends to zero. Use \\cref{extended:prop:range_leakage}.\n\\end{proof}","Probability vectors are columns. A column stochastic matrix \\(K\\) has nonnegative entries and column sums one. A continuous time generator \\(Q\\) has nonnegative off-diagonal entries and column sums zero.","\\begin{theorem}[Finite Markov semigroup]\\label{extended:record:028}\\label{extended:thm:finite_markov}\nIf \\(Q\\) is a finite Markov generator, then \\(\\e^{tQ}\\) is column stochastic for every \\(t\\geq0\\).\n\\end{theorem}","\\begin{proof}\nChoose \\(a\\geq\\max_i(-Q_{ii})\\). The matrix \\(Q+aI\\) is nonnegative, so its exponential is nonnegative. Hence\n\\[\n\\e^{tQ}=\\e^{-at}\\e^{t(Q+aI)}\n\\]\nis nonnegative. Since \\(\\one^TQ=0\\), differentiation gives\n\\[\n\\one^T\\e^{tQ}=\\one^T.\n\\]\n\\end{proof}","Let \\(\\{B_1,\\ldots,B_m\\}\\) be a partition of the fine state set. Its aggregation matrix is\n\\[\nC_{\\alpha i}=\\one_{\\{i\\in B_\\alpha\\}}.\n\\]","\\begin{definition}[Canonical lumping obstruction]\\label{extended:record:029}\nFor a linear fine transition operator \\(K\\), define\n\\[\n\\Ob_C(K)\n=\n(CK)|_{\\Ker C}\n\\in\n\\mathcal B(\\Ker C,\\RR^m).\n\\]\n\\end{definition}","\\begin{theorem}[Descent through an aggregation map]\\label{extended:record:030}\\label{extended:thm:canonical_lumping_obstruction}\nLet \\(C:X_f\\to X_c\\) be a surjective linear map between finite dimensional spaces. A unique operator \\(\\overline K:X_c\\to X_c\\) satisfying\n\\[\nCK=\\overline KC\n\\]\nexists if and only if\n\\[\n\\Ob_C(K)=0.\n\\]\nFor any right inverse \\(R\\) of \\(C\\), the only possible coarse operator is\n\\[\n\\overline K=CKR,\n\\]\nand\n\\[\nCK-\\overline KC\n=\nCK(I-RC).\n\\]\n\\end{theorem}","theorem}[Descent through an aggregation map]\\label{extended:record:030}\\label{extended:thm:canonical_lumping_obstruction}\nLet \\(C:X_f\\to X_c\\) be a surjective linear map between finite dimensional spaces. A unique operator \\(\\overline K:X_c\\to X_c\\) satisfying\n\\[\nCK=\\overline KC\n\\]\nexists if and only if\n\\[\n\\Ob_C(K)=0.\n\\]\nFor any right inverse \\(R\\) of \\(C\\), the only possible coarse operator is\n\\[\n\\overline K=CKR,\n\\]\nand\n\\[\nCK-\\overline KC\n=\nCK(I-RC).\n\\]\n\\end{theorem}","aggregation map]\\label{extended:record:030}\\label{extended:thm:canonical_lumping_obstruction}\nLet \\(C:X_f\\to X_c\\) be a surjective linear map between finite dimensional spaces. A unique operator \\(\\overline K:X_c\\to X_c\\) satisfying\n\\[\nCK=\\overline KC\n\\]\nexists if and only if\n\\[\n\\Ob_C(K)=0.\n\\]\nFor any right inverse \\(R\\) of \\(C\\), the only possible coarse operator is\n\\[\n\\overline K=CKR,\n\\]\nand\n\\[\nCK-\\overline KC\n=\nCK(I-RC).\n\\]\n\\end{theorem}","\\begin{proof}\nIf \\(CK=\\overline KC\\), then \\(CK\\) vanishes on \\(\\Ker C\\). Conversely, suppose the obstruction vanishes and define\n\\[\n\\overline K(Cx)=CKx.\n\\]\nIf \\(Cx=Cy\\), then \\(x-y\\in\\Ker C\\), so the value does not depend on the representative. Surjectivity defines \\(\\overline K\\) on all of \\(X_c\\), and the intertwining relation follows from the definition. It also gives uniqueness. If \\(R\\) is a right inverse, then\n\\[\n\\overline K=\\overline KCR=CKR.\n\\]\nSubstitution gives the defect identity.\n\\end{proof}","overline KC\\), then \\(CK\\) vanishes on \\(\\Ker C\\). Conversely, suppose the obstruction vanishes and define\n\\[\n\\overline K(Cx)=CKx.\n\\]\nIf \\(Cx=Cy\\), then \\(x-y\\in\\Ker C\\), so the value does not depend on the representative. Surjectivity defines \\(\\overline K\\) on all of \\(X_c\\), and the intertwining relation follows from the definition. It also gives uniqueness. If \\(R\\) is a right inverse, then\n\\[\n\\overline K=\\overline KCR=CKR.\n\\]\nSubstitution gives the defect identity.\n\\end{proof}","\\begin{corollary}[Coordinate form of the lumping obstruction]\\label{extended:record:031}\\label{extended:cor:lumping_obstruction_coordinates}\nFor the partition aggregation above, \\(\\Ker C\\) is spanned by differences \\(e_i-e_j\\) with \\(i\\) and \\(j\\) in one source block. Also,\n\\[\n\\Ob_C(K)(e_i-e_j)\n=\n\\left(\n\\sum_{r\\in B_\\alpha}(K_{ri}-K_{rj})\n\\right)_{\\alpha=1}^m.\n\\]\n\\end{corollary}","\\begin{proof}\nInside each block, the vectors with coordinate sum zero are spanned by differences of coordinate vectors. Applying \\(CK\\) gives the displayed aggregated column differences.\n\\end{proof}","\\begin{theorem}[Characterization of exact discrete lumping]\\label{extended:record:032}\\label{extended:thm:lumping_characterization}\nLet \\(K\\) be column stochastic. A coarse column stochastic matrix \\(\\overline K\\) satisfying\n\\[\nCK=\\overline KC\n\\]\nexists if and only if, for every source block \\(B_\\beta\\), every pair \\(i,j\\in B_\\beta\\), and every target block \\(B_\\alpha\\),\n\\[\n\\sum_{r\\in B_\\alpha}K_{ri}\n=\n\\sum_{r\\in B_\\alpha}K_{rj}.\n\\]\nWhen it exists, \\(\\overline K\\) is unique and\n\\[\n\\overline K_{\\alpha\\beta}\n=\n\\sum_{r\\in B_\\alpha}K_{ri}\n\\]\nfor any \\(i\\in B_\\beta\\).\n\\end{theorem}","ing]\\label{extended:record:032}\\label{extended:thm:lumping_characterization}\nLet \\(K\\) be column stochastic. A coarse column stochastic matrix \\(\\overline K\\) satisfying\n\\[\nCK=\\overline KC\n\\]\nexists if and only if, for every source block \\(B_\\beta\\), every pair \\(i,j\\in B_\\beta\\), and every target block \\(B_\\alpha\\),\n\\[\n\\sum_{r\\in B_\\alpha}K_{ri}\n=\n\\sum_{r\\in B_\\alpha}K_{rj}.\n\\]\nWhen it exists, \\(\\overline K\\) is unique and\n\\[\n\\overline K_{\\alpha\\beta}\n=\n\\sum_{r\\in B_\\alpha}K_{ri}\n\\]\nfor any \\(i\\in B_\\beta\\).\n\\end{theorem}","}\\label{extended:thm:lumping_characterization}\nLet \\(K\\) be column stochastic. A coarse column stochastic matrix \\(\\overline K\\) satisfying\n\\[\nCK=\\overline KC\n\\]\nexists if and only if, for every source block \\(B_\\beta\\), every pair \\(i,j\\in B_\\beta\\), and every target block \\(B_\\alpha\\),\n\\[\n\\sum_{r\\in B_\\alpha}K_{ri}\n=\n\\sum_{r\\in B_\\alpha}K_{rj}.\n\\]\nWhen it exists, \\(\\overline K\\) is unique and\n\\[\n\\overline K_{\\alpha\\beta}\n=\n\\sum_{r\\in B_\\alpha}K_{ri}\n\\]\nfor any \\(i\\in B_\\beta\\).\n\\end{theorem}","ochastic. A coarse column stochastic matrix \\(\\overline K\\) satisfying\n\\[\nCK=\\overline KC\n\\]\nexists if and only if, for every source block \\(B_\\beta\\), every pair \\(i,j\\in B_\\beta\\), and every target block \\(B_\\alpha\\),\n\\[\n\\sum_{r\\in B_\\alpha}K_{ri}\n=\n\\sum_{r\\in B_\\alpha}K_{rj}.\n\\]\nWhen it exists, \\(\\overline K\\) is unique and\n\\[\n\\overline K_{\\alpha\\beta}\n=\n\\sum_{r\\in B_\\alpha}K_{ri}\n\\]\nfor any \\(i\\in B_\\beta\\).\n\\end{theorem}","\\begin{proof}\nThe \\(j\\)th column of \\(CK\\) records the transition mass from state \\(j\\) into every target block. The \\(j\\)th column of \\(\\overline KC\\) is the column of \\(\\overline K\\) indexed by the source block containing \\(j\\). Equality therefore holds exactly under the stated condition. The formula gives uniqueness. Summing over \\(\\alpha\\) gives one, so the coarse matrix is column stochastic.\n\\end{proof}","\\begin{corollary}[Continuous time lumping]\\label{extended:record:033}\\label{extended:cor:continuous_lumping}\nA coarse Markov generator \\(\\overline Q\\) satisfying\n\\[\nCQ=\\overline QC\n\\]\nexists if and only if the aggregated rates\n\\[\n\\sum_{r\\in B_\\alpha}Q_{ri}\n\\]\ndepend only on the source block of \\(i\\). In that case\n\\[\nC\\e^{tQ}=\\e^{t\\overline Q}C\n\\]\nfor every \\(t\\geq0\\).\n\\end{corollary}","\\begin{proof}\nThe characterization is the same column argument as in \\cref{extended:thm:lumping_characterization}. The off-diagonal entries of \\(\\overline Q\\) are nonnegative and its column sums are zero. Exact semigroup intertwining follows from \\cref{extended:prop:infinitesimal_defect}.\n\\end{proof}","\\begin{theorem}[Finite horizon lumping defect]\\label{extended:record:034}\\label{extended:thm:finite_horizon_lumping}\nFor arbitrary fine and coarse operators \\(K\\) and \\(\\overline K\\), with the fixed aggregation map \\(C\\), set\n\\[\n\\Delta_K=CK-\\overline KC.\n\\]\nThen\n\\[\nCK^n-\\overline K^nC\n=\n\\sum_{j=0}^{n-1}\n\\overline K^{n-1-j}\\Delta_KK^j.\n\\]\nIf \\(K\\) and \\(\\overline K\\) are column stochastic, then\n\\[\n\\norm{CK^n-\\overline K^nC}_{1\\to1}\n\\leq\nn\\norm{\\Delta_K}_{1\\to1}.\n\\]\nFor every probability vector \\(p\\), the total variation discrepancy is at most\n\\[\n\\frac n2\\norm{\\Delta_K}_{1\\to1}.\n\\]\n\\end{theorem}","horizon_lumping}\nFor arbitrary fine and coarse operators \\(K\\) and \\(\\overline K\\), with the fixed aggregation map \\(C\\), set\n\\[\n\\Delta_K=CK-\\overline KC.\n\\]\nThen\n\\[\nCK^n-\\overline K^nC\n=\n\\sum_{j=0}^{n-1}\n\\overline K^{n-1-j}\\Delta_KK^j.\n\\]\nIf \\(K\\) and \\(\\overline K\\) are column stochastic, then\n\\[\n\\norm{CK^n-\\overline K^nC}_{1\\to1}\n\\leq\nn\\norm{\\Delta_K}_{1\\to1}.\n\\]\nFor every probability vector \\(p\\), the total variation discrepancy is at most\n\\[\n\\frac n2\\norm{\\Delta_K}_{1\\to1}.\n\\]\n\\end{theorem}","rators \\(K\\) and \\(\\overline K\\), with the fixed aggregation map \\(C\\), set\n\\[\n\\Delta_K=CK-\\overline KC.\n\\]\nThen\n\\[\nCK^n-\\overline K^nC\n=\n\\sum_{j=0}^{n-1}\n\\overline K^{n-1-j}\\Delta_KK^j.\n\\]\nIf \\(K\\) and \\(\\overline K\\) are column stochastic, then\n\\[\n\\norm{CK^n-\\overline K^nC}_{1\\to1}\n\\leq\nn\\norm{\\Delta_K}_{1\\to1}.\n\\]\nFor every probability vector \\(p\\), the total variation discrepancy is at most\n\\[\n\\frac n2\\norm{\\Delta_K}_{1\\to1}.\n\\]\n\\end{theorem}","\\begin{proof}\nSum the telescoping identities\n\\[\n\\overline K^{n-1-j}CK^{j+1}\n-\n\\overline K^{n-j}CK^j\n=\n\\overline K^{n-1-j}\\Delta_KK^j.\n\\]\nStochastic matrices are contractions in the \\(\\ell^1\\) norm. The last statement uses total variation equal to one half of the \\(\\ell^1\\) distance.\n\\end{proof}","\\begin{corollary}[Stationary mass descends]\\label{extended:record:035}\\label{extended:cor:stationary_descends}\nIf \\(CK=\\overline KC\\) and \\(Kp=p\\), then\n\\[\n\\overline K(Cp)=Cp.\n\\]\n\\end{corollary}","\\begin{proof}\nApply the intertwining identity to \\(p\\).\n\\end{proof}","Let \\(\\Omega=\\{1,\\ldots,n\\}\\) have strictly positive probabilities \\(\\pi_i\\). For a partition \\(\\mathcal Q\\), let \\(B(i)\\) denote the block containing \\(i\\). Define\n\\[\n(P_\\mathcal Qf)_i\n=\n\\frac{\\sum_{j\\in B(i)}\\pi_jf_j}\n{\\sum_{j\\in B(i)}\\pi_j}.\n\\]\nEquip \\(\\CC^n\\) with\n\\[\n\\ip{f}{g}_\\pi\n=\n\\sum_i\\pi_if_i\\overline{g_i}.\n\\]","\\begin{theorem}[Finite conditional expectation]\\label{extended:record:036}\\label{extended:thm:finite_conditional_expectation}\nThe operator \\(P_\\mathcal Q\\) is the orthogonal projector onto the functions that are constant on every block of \\(\\mathcal Q\\). In particular,\n\\[\nP_\\mathcal Q^2=P_\\mathcal Q,\n\\qquad\nP_\\mathcal Q^*=P_\\mathcal Q.\n\\]\n\\end{theorem}","\\begin{proof}\nOne application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives\n\\[\n\\ip{P_\\mathcal Qf}{g}_\\pi\n=\n\\sum_{B\\in\\mathcal Q}\n\\frac{\n\\left(\\sum_{i\\in B}\\pi_if_i\\right)\n\\overline{\\left(\\sum_{j\\in B}\\pi_jg_j\\right)}\n}{\\pi(B)}\n=\n\\ip{f}{P_\\mathcal Qg}_\\pi.\n\\]\nThe range is exactly the block constant subspace.\n\\end{proof}","\\begin{theorem}[Tower law and martingale packets]\\label{extended:record:037}\\label{extended:thm:martingale_packets}\nLet $P_0,P_1,\\ldots,P_r$ come from increasingly fine partitions, so that\n\\[\n\\Ran P_0\\subseteq\\Ran P_1\\subseteq\\cdots\\subseteq\\Ran P_r.\n\\]\nDefine\n\\[\nD_0=P_0,\n\\qquad\nD_j=P_j-P_{j-1}\n\\quad(1\\leq j\\leq r).\n\\]\nThen\n\\[\nP_iP_j=P_{\\min(i,j)},\n\\qquad\nD_iD_j=\\delta_{ij}D_j,\n\\qquad\nP_r=\\sum_{j=0}^rD_j.\n\\]\nEvery \\(D_j\\) is an orthogonal projector, and\n\\[\n\\norm{P_rf}_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n\\]\nand\n\\[\n\\norm f_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n+\n\\norm{f-P_rf}_\\pi^2.\n\\]\n\\end{theorem}","r law and martingale packets]\\label{extended:record:037}\\label{extended:thm:martingale_packets}\nLet $P_0,P_1,\\ldots,P_r$ come from increasingly fine partitions, so that\n\\[\n\\Ran P_0\\subseteq\\Ran P_1\\subseteq\\cdots\\subseteq\\Ran P_r.\n\\]\nDefine\n\\[\nD_0=P_0,\n\\qquad\nD_j=P_j-P_{j-1}\n\\quad(1\\leq j\\leq r).\n\\]\nThen\n\\[\nP_iP_j=P_{\\min(i,j)},\n\\qquad\nD_iD_j=\\delta_{ij}D_j,\n\\qquad\nP_r=\\sum_{j=0}^rD_j.\n\\]\nEvery \\(D_j\\) is an orthogonal projector, and\n\\[\n\\norm{P_rf}_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n\\]\nand\n\\[\n\\norm f_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n+\n\\norm{f-P_rf}_\\pi^2.\n\\]\n\\end{theorem}","nded:record:037}\\label{extended:thm:martingale_packets}\nLet $P_0,P_1,\\ldots,P_r$ come from increasingly fine partitions, so that\n\\[\n\\Ran P_0\\subseteq\\Ran P_1\\subseteq\\cdots\\subseteq\\Ran P_r.\n\\]\nDefine\n\\[\nD_0=P_0,\n\\qquad\nD_j=P_j-P_{j-1}\n\\quad(1\\leq j\\leq r).\n\\]\nThen\n\\[\nP_iP_j=P_{\\min(i,j)},\n\\qquad\nD_iD_j=\\delta_{ij}D_j,\n\\qquad\nP_r=\\sum_{j=0}^rD_j.\n\\]\nEvery \\(D_j\\) is an orthogonal projector, and\n\\[\n\\norm{P_rf}_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n\\]\nand\n\\[\n\\norm f_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n+\n\\norm{f-P_rf}_\\pi^2.\n\\]\n\\end{theorem}","$P_0,P_1,\\ldots,P_r$ come from increasingly fine partitions, so that\n\\[\n\\Ran P_0\\subseteq\\Ran P_1\\subseteq\\cdots\\subseteq\\Ran P_r.\n\\]\nDefine\n\\[\nD_0=P_0,\n\\qquad\nD_j=P_j-P_{j-1}\n\\quad(1\\leq j\\leq r).\n\\]\nThen\n\\[\nP_iP_j=P_{\\min(i,j)},\n\\qquad\nD_iD_j=\\delta_{ij}D_j,\n\\qquad\nP_r=\\sum_{j=0}^rD_j.\n\\]\nEvery \\(D_j\\) is an orthogonal projector, and\n\\[\n\\norm{P_rf}_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n\\]\nand\n\\[\n\\norm f_\\pi^2\n=\n\\sum_{j=0}^r\\norm{D_jf}_\\pi^2\n+\n\\norm{f-P_rf}_\\pi^2.\n\\]\n\\end{theorem}","\\begin{proof}\nA coarse average of a fine average is the coarse average. A fine average leaves a coarse constant function unchanged. Hence \\(P_iP_j=P_{\\min(i,j)}\\). Expanding products of the differences gives the orthogonality relations. The sum telescopes to \\(P_r\\). Orthogonal Pythagorean identities give the norm formulas.\n\\end{proof}","\\begin{corollary}[Law of total variance]\\label{extended:record:038}\\label{extended:cor:total_variance}\nFor a real valued \\(f\\), let \\(P_0f=\\Exp_\\pi[f]\\one\\). Then\n\\[\n\\Var_\\pi(f)\n=\n\\Var_\\pi(P_\\mathcal Qf)\n+\n\\norm{f-P_\\mathcal Qf}_\\pi^2.\n\\]\n\\end{corollary}","\\begin{proof}\nSubtract the mean and apply the orthogonal decomposition into the range of \\(P_\\mathcal Q\\) and its orthogonal complement.\n\\end{proof}","\\begin{standardfact}[General conditional expectations]\\label{extended:record:039}\nOn a probability space, conditional expectation onto a sub sigma algebra is the orthogonal projector in \\(L^2\\). Nested sigma algebras satisfy the same tower law. Under the usual hypotheses, martingale convergence gives a limit in \\(L^2\\) and almost surely. Standard treatments include \\cite{Williams1991}.\n\\end{standardfact}","Let \\(\\gamma\\) be the standard Gaussian probability measure on \\(\\RR\\). For \\(t\\geq0\\), define the Ornstein--Uhlenbeck transfer\n\\[\n(\\Ocal_t f)(x)\n=\n\\int_\\RR\nf\\left(\n\\e^{-t}x+\\sqrt{1-\\e^{-2t}}\\,y\n\\right)\n\\gamma(\\dd y).\n\\]\nLet \\(H_n\\) be the probabilists' Hermite polynomial defined by\n\\[\n\\exp\\left(ux-\\frac{u^2}{2}\\right)\n=\n\\sum_{n=0}^\\infty H_n(x)\\frac{u^n}{n!},\n\\]\nand set \\(h_n=H_n/\\sqrt{n!}\\).","\\begin{theorem}[Ornstein--Uhlenbeck semigroup]\\label{extended:record:040}\\label{extended:thm:ou_semigroup}\nThe operators \\((\\Ocal_t)_{t\\geq0}\\) form a positive Markov contraction semigroup on \\(L^p(\\gamma)\\) for every \\(1\\leq p\\leq\\infty\\). On \\(L^2(\\gamma)\\), the semigroup is self-adjoint and\n\\[\n\\Ocal_t h_n=\\e^{-nt}h_n.\n\\]\nThe family \\((h_n)_{n\\geq0}\\) is an orthonormal basis of \\(L^2(\\gamma)\\).\n\\end{theorem}","\\begin{proof}\nPositivity and preservation of constants follow from the integral formula. If \\(X\\) and \\(Y\\) are independent standard Gaussian variables, then\n\\[\n\\e^{-s}\n\\left(\n\\e^{-t}x+\\sqrt{1-\\e^{-2t}}X\n\\right)\n+\n\\sqrt{1-\\e^{-2s}}Y\n\\]\nhas the same distribution as\n\\[\n\\e^{-(s+t)}x\n+\n\\sqrt{1-\\e^{-2(s+t)}}Z\n\\]\nwith \\(Z\\) standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of \\(\\gamma\\) give the \\(L^p\\) contraction. The joint Gaussian pair\n\\[\n\\left(\nX,\n\\e^{-t}X+\\sqrt{1-\\e^{-2t}}Y\n\\right)\n\\]\nis symmetric under interchange, which proves self-adjointness.","of}\nPositivity and preservation of constants follow from the integral formula. If \\(X\\) and \\(Y\\) are independent standard Gaussian variables, then\n\\[\n\\e^{-s}\n\\left(\n\\e^{-t}x+\\sqrt{1-\\e^{-2t}}X\n\\right)\n+\n\\sqrt{1-\\e^{-2s}}Y\n\\]\nhas the same distribution as\n\\[\n\\e^{-(s+t)}x\n+\n\\sqrt{1-\\e^{-2(s+t)}}Z\n\\]\nwith \\(Z\\) standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of \\(\\gamma\\) give the \\(L^p\\) contraction. The joint Gaussian pair\n\\[\n\\left(\nX,\n\\e^{-t}X+\\sqrt{1-\\e^{-2t}}Y\n\\right)\n\\]\nis symmetric under interchange, which proves self-adjointness.","reservation of constants follow from the integral formula. If \\(X\\) and \\(Y\\) are independent standard Gaussian variables, then\n\\[\n\\e^{-s}\n\\left(\n\\e^{-t}x+\\sqrt{1-\\e^{-2t}}X\n\\right)\n+\n\\sqrt{1-\\e^{-2s}}Y\n\\]\nhas the same distribution as\n\\[\n\\e^{-(s+t)}x\n+\n\\sqrt{1-\\e^{-2(s+t)}}Z\n\\]\nwith \\(Z\\) standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of \\(\\gamma\\) give the \\(L^p\\) contraction. The joint Gaussian pair\n\\[\n\\left(\nX,\n\\e^{-t}X+\\sqrt{1-\\e^{-2t}}Y\n\\right)\n\\]\nis symmetric under interchange, which proves self-adjointness.","gral formula. If \\(X\\) and \\(Y\\) are independent standard Gaussian variables, then\n\\[\n\\e^{-s}\n\\left(\n\\e^{-t}x+\\sqrt{1-\\e^{-2t}}X\n\\right)\n+\n\\sqrt{1-\\e^{-2s}}Y\n\\]\nhas the same distribution as\n\\[\n\\e^{-(s+t)}x\n+\n\\sqrt{1-\\e^{-2(s+t)}}Z\n\\]\nwith \\(Z\\) standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of \\(\\gamma\\) give the \\(L^p\\) contraction. The joint Gaussian pair\n\\[\n\\left(\nX,\n\\e^{-t}X+\\sqrt{1-\\e^{-2t}}Y\n\\right)\n\\]\nis symmetric under interchange, which proves self-adjointness.","Apply \\(\\Ocal_t\\) to the Hermite generating function. Gaussian integration gives\n\\[\n\\begin{aligned}\n\\Ocal_t\n\\exp\\left(ux-\\frac{u^2}{2}\\right)\n&=\n\\exp\\left(\n\\e^{-t}ux-\n\\frac{\\e^{-2t}u^2}{2}\n\\right)","Apply \\(\\Ocal_t\\) to the Hermite generating function. Gaussian integration gives\n\\[\n\\begin{aligned}\n\\Ocal_t\n\\exp\\left(ux-\\frac{u^2}{2}\\right)\n&=\n\\exp\\left(\n\\e^{-t}ux-\n\\frac{\\e^{-2t}u^2}{2}\n\\right)\\\\\n&=\n\\sum_{n=0}^\\infty\n\\e^{-nt}H_n(x)\\frac{u^n}{n!}.\n\\end{aligned}\n\\]\nComparison of coefficients proves the eigenvalue formula. The same generating function shows\n\\[\n\\int H_mH_n\\dd\\gamma=n!\\delta_{mn}.\n\\]\nThe Hermite polynomials span all polynomials, and polynomials are dense in \\(L^2(\\gamma)\\). Hence the normalized family is an orthonormal basis. See \\cite{Janson1997} for the general Gaussian Hilbert space setting.\n\\end{proof}","&=\n\\sum_{n=0}^\\infty\n\\e^{-nt}H_n(x)\\frac{u^n}{n!}.\n\\end{aligned}\n\\]\nComparison of coefficients proves the eigenvalue formula. The same generating function shows\n\\[\n\\int H_mH_n\\dd\\gamma=n!\\delta_{mn}.\n\\]\nThe Hermite polynomials span all polynomials, and polynomials are dense in \\(L^2(\\gamma)\\). Hence the normalized family is an orthonormal basis. See \\cite{Janson1997} for the general Gaussian Hilbert space setting.\n\\end{proof}","\\begin{corollary}[Hermite packet reconstruction]\\label{extended:record:041}\\label{extended:cor:hermite_packets}\nLet \\(P_n\\) be the orthogonal projection onto \\(\\CC h_n\\). Then\n\\[\nI=\\sum_{n=0}^\\infty P_n\n\\]\nin the strong operator topology and\n\\[\n\\Ocal_t\n=\n\\sum_{n=0}^\\infty\\e^{-nt}P_n.\n\\]\nFor \\(f\\in L^2(\\gamma)\\),\n\\[\n\\norm{\\Ocal_t f}_2^2\n=\n\\sum_{n=0}^\\infty\n\\e^{-2nt}\\norm{P_nf}_2^2.\n\\]\n\\end{corollary}","\\begin{proposition}[Generator and Dirichlet form]\\label{extended:record:042}\\label{extended:prop:ou_generator}\nOn the polynomial subspace, the infinitesimal generator of \\((\\Ocal_t)\\) is the differential operator\n\\[\n\\Lcal f=f''-xf'.\n\\]\nIt satisfies\n\\[\n\\Lcal h_n=-nh_n.\n\\]\nFor every polynomial \\(f\\),\n\\[\n\\ip{-\\Lcal f}{f}_{L^2(\\gamma)}\n=\n\\int_\\RR\\abs{f'(x)}^2\\gamma(\\dd x).\n\\]\n\\end{proposition}","\\begin{proof}\nWrite\n\\[\n\\rho_t=\\e^{-t}=1-t+O(t^2),\n\\qquad\n\\sigma_t^2=1-\\e^{-2t}=2t+O(t^2).\n\\]\nFor a polynomial \\(f\\), Taylor expansion at \\(x\\) and Gaussian integration give\n\\[\n\\begin{aligned}\n\\Ocal_tf(x)\n={}&f(x)\n+f'(x)(\\rho_t-1)x\n+\\frac12f''(x)\\sigma_t^2\n+o(t)","\\begin{proof}\nWrite\n\\[\n\\rho_t=\\e^{-t}=1-t+O(t^2),\n\\qquad\n\\sigma_t^2=1-\\e^{-2t}=2t+O(t^2).\n\\]\nFor a polynomial \\(f\\), Taylor expansion at \\(x\\) and Gaussian integration give\n\\[\n\\begin{aligned}\n\\Ocal_tf(x)\n={}&f(x)\n+f'(x)(\\rho_t-1)x\n+\\frac12f''(x)\\sigma_t^2\n+o(t)\\\\\n={}&f(x)+t\\bigl(f''(x)-xf'(x)\\bigr)+o(t).\n\\end{aligned}\n\\]\nEvery omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is \\(o(t)\\), so the expansion may be integrated term by term. Therefore\n\\[\n\\left.\\frac{\\dd}{\\dd t}\\Ocal_tf\\right|_{t=0}\n=f''-xf'.\n\\]\nThe Hermite eigenvalue identity also follows by differentiating\n\\(\n\\Ocal_th_n=\\e^{-nt}h_n\n\\)\nat zero. Gaussian integration by parts gives\n\\[\n\\int_\\RR f'(x)\\overline{g(x)}\\gamma(\\dd x)\n=\n\\int_\\RR f(x)\\overline{xg(x)-g'(x)}\\gamma(\\dd x).\n\\]\nSet \\(g=f'\\) and rearrange.\n\\end{proof}","={}&f(x)+t\\bigl(f''(x)-xf'(x)\\bigr)+o(t).\n\\end{aligned}\n\\]\nEvery omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is \\(o(t)\\), so the expansion may be integrated term by term. Therefore\n\\[\n\\left.\\frac{\\dd}{\\dd t}\\Ocal_tf\\right|_{t=0}\n=f''-xf'.\n\\]\nThe Hermite eigenvalue identity also follows by differentiating\n\\(\n\\Ocal_th_n=\\e^{-nt}h_n\n\\)\nat zero. Gaussian integration by parts gives\n\\[\n\\int_\\RR f'(x)\\overline{g(x)}\\gamma(\\dd x)\n=\n\\int_\\RR f(x)\\overline{xg(x)-g'(x)}\\gamma(\\dd x).\n\\]\nSet \\(g=f'\\) and rearrange.\n\\end{proof}","-xf'(x)\\bigr)+o(t).\n\\end{aligned}\n\\]\nEvery omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is \\(o(t)\\), so the expansion may be integrated term by term. Therefore\n\\[\n\\left.\\frac{\\dd}{\\dd t}\\Ocal_tf\\right|_{t=0}\n=f''-xf'.\n\\]\nThe Hermite eigenvalue identity also follows by differentiating\n\\(\n\\Ocal_th_n=\\e^{-nt}h_n\n\\)\nat zero. Gaussian integration by parts gives\n\\[\n\\int_\\RR f'(x)\\overline{g(x)}\\gamma(\\dd x)\n=\n\\int_\\RR f(x)\\overline{xg(x)-g'(x)}\\gamma(\\dd x).\n\\]\nSet \\(g=f'\\) and rearrange.\n\\end{proof}","ed by a quantity that is \\(o(t)\\), so the expansion may be integrated term by term. Therefore\n\\[\n\\left.\\frac{\\dd}{\\dd t}\\Ocal_tf\\right|_{t=0}\n=f''-xf'.\n\\]\nThe Hermite eigenvalue identity also follows by differentiating\n\\(\n\\Ocal_th_n=\\e^{-nt}h_n\n\\)\nat zero. Gaussian integration by parts gives\n\\[\n\\int_\\RR f'(x)\\overline{g(x)}\\gamma(\\dd x)\n=\n\\int_\\RR f(x)\\overline{xg(x)-g'(x)}\\gamma(\\dd x).\n\\]\nSet \\(g=f'\\) and rearrange.\n\\end{proof}","\\begin{corollary}[Gaussian spectral gap]\\label{extended:record:043}\\label{extended:cor:ou_spectral_gap}\nIf \\(f\\in L^2(\\gamma)\\) has mean zero, then\n\\[\n\\norm{\\Ocal_tf}_2\n\\leq\n\\e^{-t}\\norm f_2.\n\\]\nFor every polynomial \\(f\\),\n\\[\n\\Var_\\gamma(f)\n\\leq\n\\int_\\RR\\abs{f'(x)}^2\\gamma(\\dd x).\n\\]\n\\end{corollary}","\\begin{proof}\nA mean zero function has no \\(h_0\\) component, so every surviving Hermite multiplier is at most \\(\\e^{-t}\\). For a polynomial\n\\[\nf-\\Exp_\\gamma f=\\sum_{n\\geq1}c_nh_n.\n\\]\nThen\n\\[\n\\Var_\\gamma(f)=\\sum_{n\\geq1}\\abs{c_n}^2\n\\leq\n\\sum_{n\\geq1}n\\abs{c_n}^2\n=\n\\ip{-\\Lcal f}{f}.\n\\]\nApply \\cref{extended:prop:ou_generator}.\n\\end{proof}","\\begin{corollary}[Fractional Hermite transfer]\\label{extended:record:044}\\label{extended:cor:fractional_hermite}\nLet \\(-\\Lcal\\) be the nonnegative Ornstein--Uhlenbeck number operator with\n\\[\n-\\Lcal h_n=nh_n.\n\\]\nFor every \\(\\alpha>0\\),\n\\[\n\\e^{-t(-\\Lcal)^\\alpha}f\n=\n\\sum_{n=0}^\\infty\n\\e^{-tn^\\alpha}P_nf.\n\\]\nThis is a contraction semigroup on \\(L^2(\\gamma)\\).\n\\end{corollary}","Let\n\\[\n\\Delta_{n-1}\n=\n\\left\\{p\\in\\RR^n:\np_i\\geq0,\n\\quad\n\\sum_ip_i=1\n\\right\\}\n\\]\nbe the probability simplex. Let \\(\\lambda\\in[0,\\infty)^n\\) have positive evidence\n\\[\nZ_\\lambda(p)=\\sum_i\\lambda_ip_i.\n\\]\nThe Bayesian transfer is\n\\[\n\\Bcal_\\lambda(p)\n=\n\\frac{\\lambda\\odot p}{Z_\\lambda(p)}.\n\\]","\\begin{proposition}[Scale invariance and composition]\\label{extended:record:045}\\label{extended:prop:bayes_composition}\nFor every \\(c>0\\),\n\\[\n\\Bcal_{c\\lambda}=\\Bcal_\\lambda.\n\\]\nIf all relevant evidences are positive, then\n\\[\n\\Bcal_\\mu(\\Bcal_\\lambda(p))\n=\n\\Bcal_{\\lambda\\odot\\mu}(p).\n\\]\n\\end{proposition}","Let \\(I=I_1\\sqcup\\cdots\\sqcup I_m\\) be a partition. Define packet evidences\n\\[\nZ_\\alpha\n=\n\\sum_{i\\in I_\\alpha}\\lambda_ip_i.\n\\]\nFor \\(Z_\\alpha>0\\), define\n\\[\nm_\\alpha=\\frac{Z_\\alpha}{\\sum_\\beta Z_\\beta},\n\\qquad\nq_{\\alpha,i}=\\frac{\\lambda_ip_i}{Z_\\alpha}\n\\quad(i\\in I_\\alpha).\n\\]\nPackets with zero evidence receive posterior mass zero.","\\begin{theorem}[Bayesian packet reconstruction]\\label{extended:record:046}\\label{extended:thm:bayesian_packet_reconstruction}\nThe posterior is the disjoint packet synthesis\n\\[\n\\Bcal_\\lambda(p)\n=\n\\sum_{\\alpha:Z_\\alpha>0}\nm_\\alpha q_\\alpha.\n\\]\nThe local shape \\(q_\\alpha\\) depends only on the prior and likelihood inside packet \\(I_\\alpha\\). Coupling between different packets occurs through the normalization of the packet evidences.\n\\end{theorem}","\\begin{proof}\nFor \\(i\\in I_\\alpha\\),\n\\[\nm_\\alpha q_{\\alpha,i}\n=\n\\frac{Z_\\alpha}{\\sum_\\beta Z_\\beta}\n\\frac{\\lambda_ip_i}{Z_\\alpha}\n=\n\\frac{\\lambda_ip_i}{Z_\\lambda(p)}.\n\\]\nThe packet supports are disjoint.\n\\end{proof}","\\begin{theorem}[Likelihood perturbation]\\label{extended:record:047}\\label{extended:thm:likelihood_perturbation}\nLet \\(\\lambda,\\mu\\geq0\\) have positive evidences under a fixed prior \\(p\\). Set\n\\[\nd_p(\\lambda,\\mu)\n=\n\\sum_i\\abs{\\lambda_i-\\mu_i}p_i.\n\\]\nThen\n\\[\n\\norm{\\Bcal_\\lambda(p)-\\Bcal_\\mu(p)}_1\n\\leq\n\\frac{2d_p(\\lambda,\\mu)}\n{\\max\\{Z_\\lambda(p),Z_\\mu(p)\\}}.\n\\]\n\\end{theorem}","\\begin{proof}\nAdd and subtract \\(\\mu\\odot p/Z_\\lambda(p)\\). This gives\n\\[\n\\norm{\\Bcal_\\lambda(p)-\\Bcal_\\mu(p)}_1\n\\leq\n\\frac{2d_p(\\lambda,\\mu)}{Z_\\lambda(p)}.\n\\]\nInterchanging \\(\\lambda\\) and \\(\\mu\\) gives the same estimate with \\(Z_\\mu(p)\\). Take the smaller of the two upper bounds.\n\\end{proof}","\\begin{theorem}[Stability with respect to the prior]\\label{extended:record:048}\\label{extended:thm:prior_stability}\nAssume\n\\[\n0<a\\leq\\lambda_i\\leq b\n\\]\nfor every \\(i\\). Then\n\\[\n\\norm{\\Bcal_\\lambda(p)-\\Bcal_\\lambda(q)}_1\n\\leq\n\\frac{2b}{a}\\norm{p-q}_1\n\\]\nfor all priors \\(p,q\\in\\Delta_{n-1}\\).\n\\end{theorem}","\\begin{proof}\nThe evidences are at least \\(a\\). Add and subtract \\(\\lambda\\odot q/Z_\\lambda(p)\\). The numerator difference is at most \\(b\\norm{p-q}_1\\), and the evidence difference has the same bound. The two terms give the stated constant.\n\\end{proof}","Let \\(C\\) send a fine probability vector to its block masses\n\\[\n(Cp)_\\alpha\n=\n\\sum_{i\\in I_\\alpha}p_i.\n\\]\nAssume first that every block mass is positive. Write\n\\[\nr_{\\alpha,i}\n=\n\\frac{p_i}{(Cp)_\\alpha}\n\\]\nfor the prior conditional distribution inside packet \\(I_\\alpha\\). Define the prior dependent packet average likelihood\n\\[\n\\overline\\lambda_\\alpha(p)\n=\n\\sum_{i\\in I_\\alpha}\\lambda_ir_{\\alpha,i}.\n\\]","\\begin{theorem}[Exact two stage Bayesian update]\\label{extended:record:049}\\label{extended:thm:two_stage_bayes}\nFor every strictly positive prior,\n\\[\nC\\Bcal_\\lambda(p)\n=\n\\Bcal_{\\overline\\lambda(p)}(Cp).\n\\]\nFor every packet with \\(\\overline\\lambda_\\alpha(p)>0\\), the posterior conditional distribution is\n\\[\nq_{\\alpha,i}\n=\n\\frac{\\lambda_ir_{\\alpha,i}}\n{\\overline\\lambda_\\alpha(p)}.\n\\]\nA packet with zero average likelihood has zero posterior mass. Thus a fine Bayesian update consists of a coarse update by packet average likelihoods followed by independent local updates inside the packets that survive.\n\\end{theorem}","\\begin{proof}\nThe posterior mass of packet \\(I_\\alpha\\) is\n\\[\n\\frac{\\sum_{i\\in I_\\alpha}\\lambda_ip_i}\n{\\sum_j\\lambda_jp_j}\n=\n\\frac{(Cp)_\\alpha\\overline\\lambda_\\alpha(p)}\n{\\sum_\\beta(Cp)_\\beta\\overline\\lambda_\\beta(p)}.\n\\]\nThis is the coarse Bayesian formula. Dividing the fine posterior coordinates by the packet mass gives the local formula.\n\\end{proof}","\\begin{theorem}[Characterization of fixed coarse sufficiency]\\label{extended:record:050}\\label{extended:thm:bayesian_sufficiency}\nAssume \\(\\lambda_i>0\\) for every state. The following conditions are equivalent.\n\\begin{enumerate}\n\\item There is a positive coarse likelihood \\(\\ell\\in(0,\\infty)^m\\) such that\n\\[\nC\\Bcal_\\lambda(p)\n=\n\\Bcal_\\ell(Cp)\n\\]\nfor every strictly positive prior \\(p\\), and the posterior conditional distribution inside every packet equals its prior conditional distribution.\n\\item The likelihood is constant on every packet.\n\\end{enumerate}\nWhen these conditions hold, \\(\\ell_\\alpha\\) can be chosen as the common value of \\(\\lambda\\) on \\(I_\\alpha\\), and it is unique up to a common positive scale.\n\\end{theorem}","ons are equivalent.\n\\begin{enumerate}\n\\item There is a positive coarse likelihood \\(\\ell\\in(0,\\infty)^m\\) such that\n\\[\nC\\Bcal_\\lambda(p)\n=\n\\Bcal_\\ell(Cp)\n\\]\nfor every strictly positive prior \\(p\\), and the posterior conditional distribution inside every packet equals its prior conditional distribution.\n\\item The likelihood is constant on every packet.\n\\end{enumerate}\nWhen these conditions hold, \\(\\ell_\\alpha\\) can be chosen as the common value of \\(\\lambda\\) on \\(I_\\alpha\\), and it is unique up to a common positive scale.\n\\end{theorem}","a positive coarse likelihood \\(\\ell\\in(0,\\infty)^m\\) such that\n\\[\nC\\Bcal_\\lambda(p)\n=\n\\Bcal_\\ell(Cp)\n\\]\nfor every strictly positive prior \\(p\\), and the posterior conditional distribution inside every packet equals its prior conditional distribution.\n\\item The likelihood is constant on every packet.\n\\end{enumerate}\nWhen these conditions hold, \\(\\ell_\\alpha\\) can be chosen as the common value of \\(\\lambda\\) on \\(I_\\alpha\\), and it is unique up to a common positive scale.\n\\end{theorem}","se likelihood \\(\\ell\\in(0,\\infty)^m\\) such that\n\\[\nC\\Bcal_\\lambda(p)\n=\n\\Bcal_\\ell(Cp)\n\\]\nfor every strictly positive prior \\(p\\), and the posterior conditional distribution inside every packet equals its prior conditional distribution.\n\\item The likelihood is constant on every packet.\n\\end{enumerate}\nWhen these conditions hold, \\(\\ell_\\alpha\\) can be chosen as the common value of \\(\\lambda\\) on \\(I_\\alpha\\), and it is unique up to a common positive scale.\n\\end{theorem}","\\begin{proof}\nIf \\(\\lambda\\) is constant on each packet, \\cref{extended:thm:two_stage_bayes} has packet averages independent of the prior and its local posterior formula gives \\(q_\\alpha=r_\\alpha\\). This proves the first condition.","Conversely, fix any strictly positive prior. Equality of the local posterior and prior conditionals gives\n\\[\n\\frac{\\lambda_i r_{\\alpha,i}}\n{\\overline\\lambda_\\alpha(p)}\n=\nr_{\\alpha,i}\n\\]\nfor every \\(i\\in I_\\alpha\\). Positivity of the conditional prior implies\n\\[\n\\lambda_i=\\overline\\lambda_\\alpha(p)\n\\]\nfor every state in the packet. Thus the likelihood is constant on every packet. If two positive coarse likelihoods define the same update for every positive coarse prior, comparison of any two posterior odds shows that their coordinate ratios agree. Hence they differ only by a common positive factor.\n\\end{proof}","\\begin{theorem}[Coarsest fixed sufficient partition]\\label{extended:record:051}\\label{extended:thm:coarsest_bayesian_partition}\nLet \\(\\mathcal L\\) be a nonempty family of strictly positive likelihood vectors. Define\n\\[\ni\\sim_{\\mathcal L}j\n\\quad\\Longleftrightarrow\\quad\n\\lambda_i=\\lambda_j\n\\quad\\hbox{for every }\\lambda\\in\\mathcal L.\n\\]\nLet \\(\\mathcal Q_{\\mathcal L}\\) be the partition into equivalence classes. A partition \\(\\mathcal Q\\) gives a fixed coarse sufficient description in the sense of \\cref{extended:thm:bayesian_sufficiency} for every \\(\\lambda\\in\\mathcal L\\) if and only if every block of \\(\\mathcal Q\\) is contained in a block of \\(\\mathcal Q_{\\mathcal L}\\). Hence \\(\\mathcal Q_{\\mathcal L}\\) is the unique coarsest partition with this property.\n\\end{theorem}","fixed sufficient partition]\\label{extended:record:051}\\label{extended:thm:coarsest_bayesian_partition}\nLet \\(\\mathcal L\\) be a nonempty family of strictly positive likelihood vectors. Define\n\\[\ni\\sim_{\\mathcal L}j\n\\quad\\Longleftrightarrow\\quad\n\\lambda_i=\\lambda_j\n\\quad\\hbox{for every }\\lambda\\in\\mathcal L.\n\\]\nLet \\(\\mathcal Q_{\\mathcal L}\\) be the partition into equivalence classes. A partition \\(\\mathcal Q\\) gives a fixed coarse sufficient description in the sense of \\cref{extended:thm:bayesian_sufficiency} for every \\(\\lambda\\in\\mathcal L\\) if and only if every block of \\(\\mathcal Q\\) is contained in a block of \\(\\mathcal Q_{\\mathcal L}\\). Hence \\(\\mathcal Q_{\\mathcal L}\\) is the unique coarsest partition with this property.\n\\end{theorem}","nonempty family of strictly positive likelihood vectors. Define\n\\[\ni\\sim_{\\mathcal L}j\n\\quad\\Longleftrightarrow\\quad\n\\lambda_i=\\lambda_j\n\\quad\\hbox{for every }\\lambda\\in\\mathcal L.\n\\]\nLet \\(\\mathcal Q_{\\mathcal L}\\) be the partition into equivalence classes. A partition \\(\\mathcal Q\\) gives a fixed coarse sufficient description in the sense of \\cref{extended:thm:bayesian_sufficiency} for every \\(\\lambda\\in\\mathcal L\\) if and only if every block of \\(\\mathcal Q\\) is contained in a block of \\(\\mathcal Q_{\\mathcal L}\\). Hence \\(\\mathcal Q_{\\mathcal L}\\) is the unique coarsest partition with this property.\n\\end{theorem}","s. Define\n\\[\ni\\sim_{\\mathcal L}j\n\\quad\\Longleftrightarrow\\quad\n\\lambda_i=\\lambda_j\n\\quad\\hbox{for every }\\lambda\\in\\mathcal L.\n\\]\nLet \\(\\mathcal Q_{\\mathcal L}\\) be the partition into equivalence classes. A partition \\(\\mathcal Q\\) gives a fixed coarse sufficient description in the sense of \\cref{extended:thm:bayesian_sufficiency} for every \\(\\lambda\\in\\mathcal L\\) if and only if every block of \\(\\mathcal Q\\) is contained in a block of \\(\\mathcal Q_{\\mathcal L}\\). Hence \\(\\mathcal Q_{\\mathcal L}\\) is the unique coarsest partition with this property.\n\\end{theorem}","gleftrightarrow\\quad\n\\lambda_i=\\lambda_j\n\\quad\\hbox{for every }\\lambda\\in\\mathcal L.\n\\]\nLet \\(\\mathcal Q_{\\mathcal L}\\) be the partition into equivalence classes. A partition \\(\\mathcal Q\\) gives a fixed coarse sufficient description in the sense of \\cref{extended:thm:bayesian_sufficiency} for every \\(\\lambda\\in\\mathcal L\\) if and only if every block of \\(\\mathcal Q\\) is contained in a block of \\(\\mathcal Q_{\\mathcal L}\\). Hence \\(\\mathcal Q_{\\mathcal L}\\) is the unique coarsest partition with this property.\n\\end{theorem}","bda_j\n\\quad\\hbox{for every }\\lambda\\in\\mathcal L.\n\\]\nLet \\(\\mathcal Q_{\\mathcal L}\\) be the partition into equivalence classes. A partition \\(\\mathcal Q\\) gives a fixed coarse sufficient description in the sense of \\cref{extended:thm:bayesian_sufficiency} for every \\(\\lambda\\in\\mathcal L\\) if and only if every block of \\(\\mathcal Q\\) is contained in a block of \\(\\mathcal Q_{\\mathcal L}\\). Hence \\(\\mathcal Q_{\\mathcal L}\\) is the unique coarsest partition with this property.\n\\end{theorem}","\\begin{proof}\nApply \\cref{extended:thm:bayesian_sufficiency} to every likelihood in the family. A fixed sufficient description exists exactly when each likelihood is constant on every block of \\(\\mathcal Q\\). This is equivalent to every block of \\(\\mathcal Q\\) being contained in one common level set of the family. The equivalence class partition is itself admissible, and every other admissible partition refines it.\n\\end{proof}","\\begin{definition}[Within packet information loss]\\label{extended:record:052}\nFor a prior \\(p\\), define\n\\[\n\\eta_i(p)\n=\n\\lambda_i-\\overline\\lambda_\\alpha(p)\n\\quad\\hbox{when }i\\in I_\\alpha.\n\\]\nThen\n\\[\n\\sum_{i\\in I_\\alpha}r_{\\alpha,i}\\eta_i(p)=0.\n\\]\n\\end{definition}","\\begin{proposition}[Exact local distortion]\\label{extended:record:053}\\label{extended:prop:local_bayes_distortion}\nFor every packet with positive mass and evidence,\n\\[\nq_{\\alpha,i}-r_{\\alpha,i}\n=\n\\frac{r_{\\alpha,i}\\eta_i(p)}\n{\\overline\\lambda_\\alpha(p)}.\n\\]\nConsequently,\n\\[\n\\norm{q_\\alpha-r_\\alpha}_1\n=\n\\frac{\n\\sum_{i\\in I_\\alpha}r_{\\alpha,i}\\abs{\\eta_i(p)}\n}{\\overline\\lambda_\\alpha(p)}.\n\\]\nIf \\(\\lambda\\) is positive on the packet, then\n\\[\n\\norm{q_\\alpha-r_\\alpha}_1\n\\leq\n\\frac{\\osc_{I_\\alpha}(\\lambda)}\n{\\min_{i\\in I_\\alpha}\\lambda_i}.\n\\]\n\\end{proposition}","]\\label{extended:record:053}\\label{extended:prop:local_bayes_distortion}\nFor every packet with positive mass and evidence,\n\\[\nq_{\\alpha,i}-r_{\\alpha,i}\n=\n\\frac{r_{\\alpha,i}\\eta_i(p)}\n{\\overline\\lambda_\\alpha(p)}.\n\\]\nConsequently,\n\\[\n\\norm{q_\\alpha-r_\\alpha}_1\n=\n\\frac{\n\\sum_{i\\in I_\\alpha}r_{\\alpha,i}\\abs{\\eta_i(p)}\n}{\\overline\\lambda_\\alpha(p)}.\n\\]\nIf \\(\\lambda\\) is positive on the packet, then\n\\[\n\\norm{q_\\alpha-r_\\alpha}_1\n\\leq\n\\frac{\\osc_{I_\\alpha}(\\lambda)}\n{\\min_{i\\in I_\\alpha}\\lambda_i}.\n\\]\n\\end{proposition}","\\begin{proof}\nSubtract \\(r_{\\alpha,i}\\) from the local posterior formula in \\cref{extended:thm:two_stage_bayes}. The norm identity follows because the factor multiplying \\(\\eta_i\\) is nonnegative. The last estimate uses\n\\[\n\\sum_i r_{\\alpha,i}\\abs{\\eta_i}\n\\leq\n\\osc_{I_\\alpha}(\\lambda)\n\\]\nand \\(\\overline\\lambda_\\alpha(p)\\geq\\min_{I_\\alpha}\\lambda\\).\n\\end{proof}","\\begin{theorem}[Approximate coarse Bayesian rule]\\label{extended:record:054}\\label{extended:thm:approximate_coarse_bayes}\nChoose one positive number \\(\\ell_\\alpha\\) for every packet and define the block constant fine likelihood\n\\[\n\\widetilde\\lambda_i=\\ell_\\alpha\n\\quad(i\\in I_\\alpha).\n\\]\nThen\n\\[\n\\norm{\nC\\Bcal_\\lambda(p)-\\Bcal_\\ell(Cp)\n}_1\n\\leq\n\\frac{\n2\\sum_i p_i\\abs{\\lambda_i-\\widetilde\\lambda_i}\n}{\n\\max\\{Z_\\lambda(p),Z_{\\widetilde\\lambda}(p)\\}\n}.\n\\]\n\\end{theorem}","\\begin{proof}\nBlock constancy gives\n\\[\nC\\Bcal_{\\widetilde\\lambda}(p)\n=\n\\Bcal_\\ell(Cp).\n\\]\nAggregation is an \\(\\ell^1\\) contraction. Apply \\cref{extended:thm:likelihood_perturbation} to the two fine posteriors and then aggregate.\n\\end{proof}","\\begin{definition}[Relative information]\\label{extended:record:055}\nFor probability vectors \\(q\\) and \\(p\\), define\n\\[\nD(q\\Vert p)\n=\n\\sum_iq_i\\log\\frac{q_i}{p_i}.\n\\]\nThe convention is \\(0\\log0=0\\). A term with \\(q_i>0\\) and \\(p_i=0\\) has value \\(+\\infty\\).\n\\end{definition}","\\begin{theorem}[Exact information decomposition]\\label{extended:record:056}\\label{extended:thm:bayesian_information_chain}\nLet \\(p\\) and \\(q\\) be probability vectors on a finite partition \\((I_\\alpha)_\\alpha\\), and assume \\(q_i=0\\) whenever \\(p_i=0\\). Let\n\\[\nm=Cp,\n\\qquad\nn=Cq,\n\\]\nand let \\(p_\\alpha\\) and \\(q_\\alpha\\) be the conditional distributions inside packet \\(I_\\alpha\\) whenever the corresponding mass is positive. Then\n\\[\nD(q\\Vert p)\n=\nD(n\\Vert m)\n+\n\\sum_{\\alpha:n_\\alpha>0}\n n_\\alpha D(q_\\alpha\\Vert p_\\alpha).\n\\]\nThe first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.\n\\end{theorem}","on decomposition]\\label{extended:record:056}\\label{extended:thm:bayesian_information_chain}\nLet \\(p\\) and \\(q\\) be probability vectors on a finite partition \\((I_\\alpha)_\\alpha\\), and assume \\(q_i=0\\) whenever \\(p_i=0\\). Let\n\\[\nm=Cp,\n\\qquad\nn=Cq,\n\\]\nand let \\(p_\\alpha\\) and \\(q_\\alpha\\) be the conditional distributions inside packet \\(I_\\alpha\\) whenever the corresponding mass is positive. Then\n\\[\nD(q\\Vert p)\n=\nD(n\\Vert m)\n+\n\\sum_{\\alpha:n_\\alpha>0}\n n_\\alpha D(q_\\alpha\\Vert p_\\alpha).\n\\]\nThe first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.\n\\end{theorem}","\\begin{proof}\nFor \\(i\\in I_\\alpha\\), write\n\\[\np_i=m_\\alpha p_{\\alpha,i},\n\\qquad\nq_i=n_\\alpha q_{\\alpha,i}.\n\\]\nThen\n\\[\n\\log\\frac{q_i}{p_i}\n=\n\\log\\frac{n_\\alpha}{m_\\alpha}\n+\n\\log\\frac{q_{\\alpha,i}}{p_{\\alpha,i}}.\n\\]\nMultiply by \\(q_i\\), sum first inside each packet, and then sum over the packets. This is the finite relative entropy chain rule. Standard information theoretic treatments include \\cite{CoverThomas2006}.\n\\end{proof}","\\begin{corollary}[Coarse data processing and equality]\\label{extended:record:057}\\label{extended:cor:coarse_data_processing}\nUnder the assumptions of \\cref{extended:thm:bayesian_information_chain},\n\\[\nD(Cq\\Vert Cp)\n\\leq\nD(q\\Vert p).\n\\]\nEquality holds if and only if\n\\[\nq_\\alpha=p_\\alpha\n\\]\nfor every packet with positive posterior mass.\n\\end{corollary}","\\begin{corollary}[Information created by a Bayesian update]\\label{extended:record:058}\\label{extended:cor:bayesian_information_gain}\nLet \\(q=\\Bcal_\\lambda(p)\\) with positive likelihood and positive prior. Then\n\\[\nD(q\\Vert p)\n=\n\\sum_iq_i\\log\\lambda_i\n-\n\\log Z_\\lambda(p).\n\\]\nInside a packet,\n\\[\nD(q_\\alpha\\Vert r_\\alpha)\n=\n\\sum_{i\\in I_\\alpha}\nq_{\\alpha,i}\n\\log\\frac{\\lambda_i}{\\overline\\lambda_\\alpha(p)}.\n\\]\nHence\n\\[\nD(q_\\alpha\\Vert r_\\alpha)\n\\leq\n\\log\n\\frac{\\max_{i\\in I_\\alpha}\\lambda_i}\n{\\min_{i\\in I_\\alpha}\\lambda_i}.\n\\]\n\\end{corollary}","[Information created by a Bayesian update]\\label{extended:record:058}\\label{extended:cor:bayesian_information_gain}\nLet \\(q=\\Bcal_\\lambda(p)\\) with positive likelihood and positive prior. Then\n\\[\nD(q\\Vert p)\n=\n\\sum_iq_i\\log\\lambda_i\n-\n\\log Z_\\lambda(p).\n\\]\nInside a packet,\n\\[\nD(q_\\alpha\\Vert r_\\alpha)\n=\n\\sum_{i\\in I_\\alpha}\nq_{\\alpha,i}\n\\log\\frac{\\lambda_i}{\\overline\\lambda_\\alpha(p)}.\n\\]\nHence\n\\[\nD(q_\\alpha\\Vert r_\\alpha)\n\\leq\n\\log\n\\frac{\\max_{i\\in I_\\alpha}\\lambda_i}\n{\\min_{i\\in I_\\alpha}\\lambda_i}.\n\\]\n\\end{corollary}","\\begin{proof}\nThe posterior ratio is\n\\[\n\\frac{q_i}{p_i}\n=\n\\frac{\\lambda_i}{Z_\\lambda(p)}.\n\\]\nThe local ratio is\n\\[\n\\frac{q_{\\alpha,i}}{r_{\\alpha,i}}\n=\n\\frac{\\lambda_i}{\\overline\\lambda_\\alpha(p)}.\n\\]\nInsert these identities into the definition of relative entropy. The final estimate follows because both \\(\\lambda_i\\) and the packet average lie between the packet minimum and maximum.\n\\end{proof}","\\begin{proof}\nIf the likelihood is block constant, every local posterior equals its local prior. Conversely, choose any positive prior. Vanishing relative entropy gives \\(q_\\alpha=r_\\alpha\\). The local posterior formula then forces all likelihood values in that packet to equal their packet average. Repeat for every packet.\n\\end{proof}","Fix a score vector \\(g\\in\\RR^n\\). For \\(t\\geq0\\), define\n\\[\n\\Phi_t(p)_i\n=\n\\frac{\\e^{tg_i}p_i}\n{\\sum_j\\e^{tg_j}p_j}.\n\\]","\\begin{theorem}[Bayesian flow and replicator equation]\\label{extended:record:061}\\label{extended:thm:bayesian_flow}\nThe maps \\(\\Phi_t\\) form a nonlinear semigroup:\n\\[\n\\Phi_{s+t}=\\Phi_s\\circ\\Phi_t.\n\\]\nEvery coordinate satisfies\n\\[\n\\frac{\\dd}{\\dd t}p_i(t)\n=\n\\bigl(g_i-\\ip{g}{p(t)}\\bigr)p_i(t).\n\\]\nThe support of \\(p(t)\\) is independent of \\(t\\), and for any two coordinates in the support,\n\\[\n\\frac{p_i(t)}{p_j(t)}\n=\n\\e^{t(g_i-g_j)}\n\\frac{p_i(0)}{p_j(0)}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe semigroup law follows from multiplication of the likelihoods \\(\\e^{sg}\\) and \\(\\e^{tg}\\). Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.\n\\end{proof}","\\begin{theorem}[Evidence cumulants and score monotonicity]\\label{extended:record:062}\\label{extended:thm:evidence_cumulants}\nSet\n\\[\n\\psi(t)\n=\n\\log\\left(\n\\sum_i\\e^{tg_i}p_i(0)\n\\right).\n\\]\nThen\n\\[\n\\psi'(t)=\\ip{g}{p(t)},\n\\qquad\n\\psi''(t)=\\Var_{p(t)}(g)\\geq0.\n\\]\nConsequently,\n\\[\n\\frac{\\dd}{\\dd t}\\ip{g}{p(t)}\n=\n\\Var_{p(t)}(g).\n\\]\n\\end{theorem}","\\begin{corollary}[Concentration on the maximal score packet]\\label{extended:record:063}\\label{extended:cor:bayes_flow_concentration}\nLet\n\\[\nG=\\max\\{g_i:p_i(0)>0\\}\n\\]\nand let \\(M\\) be the set of supported indices with score \\(G\\). If \\(M\\) is the full support, then the conclusion below is immediate. Otherwise, suppose\n\\[\n\\delta\n=\n\\min\\{G-g_i:p_i(0)>0,\n\\quad i\\notin M\\}>0.\n\\]\nThen\n\\[\n1-\\sum_{i\\in M}p_i(t)\n\\leq\n\\frac{1-p_M(0)}{p_M(0)}\\e^{-\\delta t},\n\\]\nwhere \\(p_M(0)=\\sum_{i\\in M}p_i(0)\\).\n\\end{corollary}","abel{extended:record:063}\\label{extended:cor:bayes_flow_concentration}\nLet\n\\[\nG=\\max\\{g_i:p_i(0)>0\\}\n\\]\nand let \\(M\\) be the set of supported indices with score \\(G\\). If \\(M\\) is the full support, then the conclusion below is immediate. Otherwise, suppose\n\\[\n\\delta\n=\n\\min\\{G-g_i:p_i(0)>0,\n\\quad i\\notin M\\}>0.\n\\]\nThen\n\\[\n1-\\sum_{i\\in M}p_i(t)\n\\leq\n\\frac{1-p_M(0)}{p_M(0)}\\e^{-\\delta t},\n\\]\nwhere \\(p_M(0)=\\sum_{i\\in M}p_i(0)\\).\n\\end{corollary}","\\begin{proof}\nFactor \\(\\e^{tG}\\) from the normalizing constant. The total weight outside \\(M\\) is at most \\((1-p_M(0))\\e^{-\\delta t}\\), while the weight on \\(M\\) is \\(p_M(0)\\).\n\\end{proof}","Let \\(K\\) be column stochastic and let \\(\\lambda^{(r)}\\) be the likelihood of the \\(r\\)th observation. The hidden Markov filter is\n\\[\n\\widehat p_r=Kp_{r-1},\n\\qquad\np_r=\\Bcal_{\\lambda^{(r)}}(\\widehat p_r).\n\\]\nA standard account of finite hidden Markov models appears in \\cite{Rabiner1989}.","\\begin{proposition}[Simplex preservation]\\label{extended:record:064}\\label{extended:prop:hmm_preservation}\nSuppose \\(p_{r-1}\\) is a probability vector and the new evidence is positive. Prediction produces the probability vector \\(\\widehat p_r\\), and normalization produces the probability vector \\(p_r\\).\n\\end{proposition}","\\begin{theorem}[One step filter stability]\\label{extended:record:065}\\label{extended:thm:hmm_stability}\nLet \\(K\\) and \\(\\widetilde K\\) be column stochastic. Assume\n\\[\n0<a\\leq\\lambda_i\\leq b.\n\\]\nThen\n\\[\n\\norm{\n\\Bcal_\\lambda(Kp)-\\Bcal_\\lambda(\\widetilde Kq)\n}_1\n\\leq\n\\frac{2b}{a}\n\\left(\n\\norm{p-q}_1\n+\n\\norm{K-\\widetilde K}_{1\\to1}\n\\right).\n\\]\n\\end{theorem}","\\begin{proof}\nApply \\cref{extended:thm:prior_stability} to \\(Kp\\) and \\(\\widetilde Kq\\). Then use\n\\[\n\\norm{Kp-\\widetilde Kq}_1\n\\leq\n\\norm{K(p-q)}_1\n+\n\\norm{(K-\\widetilde K)q}_1\n\\]\nand the stochastic contraction property.\n\\end{proof}","\\begin{example}[Beta binomial transfer]\\label{extended:record:066}\nA beta prior with parameters \\((a,b)\\) and binomial data containing \\(s\\) successes in \\(n\\) trials gives the posterior parameters\n\\[\n(a+s,b+n-s).\n\\]\nThe update is additive in the two exponent coordinates. This is a finite dimensional conjugate realization of likelihood composition. Standard Bayesian references include \\cite{BernardoSmith1994}.\n\\end{example}","Let \\(H\\) be a finite dimensional Hilbert space and let \\(L=L^*\\geq0\\). Write its spectral resolution as\n\\[\nL=\\sum_j\\lambda_jP_j.\n\\]\nFor \\(\\alpha>0\\), define\n\\[\nL^\\alpha\n=\n\\sum_j\\lambda_j^\\alpha P_j\n\\]\nand\n\\[\nT_\\alpha(t)\n=\n\\e^{-tL^\\alpha}\n=\n\\sum_j\\e^{-t\\lambda_j^\\alpha}P_j.\n\\]","\\begin{theorem}[Fractional packet calculus]\\label{extended:record:067}\\label{extended:thm:fractional_packet_calculus}\nFor \\(\\alpha,\\beta>0\\),\n\\[\nL^\\alpha L^\\beta=L^{\\alpha+\\beta}.\n\\]\nFor \\(s,t\\geq0\\),\n\\[\nT_\\alpha(s)T_\\alpha(t)=T_\\alpha(s+t).\n\\]\nEvery spectral packet \\(P_jH\\) reduces \\(L^\\alpha\\) and \\(T_\\alpha(t)\\).\n\\end{theorem}","\\begin{theorem}[Balakrishnan formula in finite dimension]\\label{extended:record:068}\\label{extended:thm:balakrishnan}\nFor \\(0<\\alpha<1\\),\n\\[\nL^\\alpha\n=\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty\ns^{\\alpha-1}L(sI+L)^{-1}\n\\dd s.\n\\]\nThe integral converges in operator norm.\n\\end{theorem}","\\begin{proof}\nIt is enough to prove the formula on an eigenvector with eigenvalue \\(x\\geq0\\). For \\(x>0\\), the substitution \\(s=xu\\) gives\n\\[\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty\ns^{\\alpha-1}\\frac{x}{s+x}\\dd s\n=x^\\alpha\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\int_0^\\infty\\frac{u^{\\alpha-1}}{1+u}\\dd u\n=x^\\alpha.\n\\]\nThe value at \\(x=0\\) is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power construction in \\cite{Balakrishnan1960}.\n\\end{proof}","\\begin{theorem}[Exact fractional naturality]\\label{extended:record:069}\\label{extended:thm:exact_fractional_naturality}\nLet \\(L_f\\) and \\(L_c\\) be finite dimensional positive self-adjoint operators and let \\(C\\) satisfy\n\\[\nCL_f=L_cC.\n\\]\nThen, for every \\(\\alpha>0\\) and \\(t\\geq0\\),\n\\[\nCL_f^\\alpha=L_c^\\alpha C\n\\]\nand\n\\[\nC\\e^{-tL_f^\\alpha}\n=\n\\e^{-tL_c^\\alpha}C.\n\\]\n\\end{theorem}","\\begin{proof}\nThe union of the two finite spectra is a finite set. There is a polynomial that agrees with \\(x^\\alpha\\) on that set, and another polynomial that agrees with \\(\\e^{-tx^\\alpha}\\). Exact intertwining passes to every polynomial by induction.\n\\end{proof}","\\begin{theorem}[Fractional defect bound]\\label{extended:record:070}\\label{extended:thm:fractional_defect}\nLet \\(0<\\alpha<1\\). Suppose\n\\[\nL_f\\geq mI,\n\\qquad\nL_c\\geq mI\n\\]\nfor some \\(m>0\\). Define\n\\[\n\\Delta_L=CL_f-L_cC.\n\\]\nThen\n\\[\n\\norm{CL_f^\\alpha-L_c^\\alpha C}\n\\leq\n\\alpha m^{\\alpha-1}\\norm{\\Delta_L}.\n\\]\nConsequently,\n\\[\n\\norm{\nC\\e^{-tL_f^\\alpha}\n-\n\\e^{-tL_c^\\alpha}C\n}\n\\leq\n\\alpha t m^{\\alpha-1}\\norm{\\Delta_L}.\n\\]\n\\end{theorem}","\\begin{proof}\nSet\n\\[\nR_f(s)=(sI+L_f)^{-1},\n\\qquad\nR_c(s)=(sI+L_c)^{-1}.\n\\]\nThe resolvent identity gives\n\\[\nCR_f(s)-R_c(s)C\n=\n-R_c(s)\\Delta_LR_f(s).\n\\]\nSince \\(LR=I-sR\\),\n\\[\nCL_fR_f(s)-L_cR_c(s)C\n=\nsR_c(s)\\Delta_LR_f(s).\n\\]\nInsert this into \\cref{extended:thm:balakrishnan}. The spectral lower bound gives\n\\[\n\\norm{R_f(s)},\\norm{R_c(s)}\n\\leq\n\\frac{1}{s+m}.\n\\]\nTherefore\n\\[\n\\begin{aligned}\n\\norm{CL_f^\\alpha-L_c^\\alpha C}\n&\\leq\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\norm{\\Delta_L}\n\\int_0^\\infty\n\\frac{s^\\alpha}{(s+m)^2}\\dd s","\\begin{proof}\nSet\n\\[\nR_f(s)=(sI+L_f)^{-1},\n\\qquad\nR_c(s)=(sI+L_c)^{-1}.\n\\]\nThe resolvent identity gives\n\\[\nCR_f(s)-R_c(s)C\n=\n-R_c(s)\\Delta_LR_f(s).\n\\]\nSince \\(LR=I-sR\\),\n\\[\nCL_fR_f(s)-L_cR_c(s)C\n=\nsR_c(s)\\Delta_LR_f(s).\n\\]\nInsert this into \\cref{extended:thm:balakrishnan}. The spectral lower bound gives\n\\[\n\\norm{R_f(s)},\\norm{R_c(s)}\n\\leq\n\\frac{1}{s+m}.\n\\]\nTherefore\n\\[\n\\begin{aligned}\n\\norm{CL_f^\\alpha-L_c^\\alpha C}\n&\\leq\n\\frac{\\sin(\\pi\\alpha)}{\\pi}\n\\norm{\\Delta_L}\n\\int_0^\\infty\n\\frac{s^\\alpha}{(s+m)^2}\\dd s\\\\\n&=\n\\alpha m^{\\alpha-1}\\norm{\\Delta_L}.\n\\end{aligned}\n\\]\nThe semigroup estimate follows from \\cref{extended:thm:duhamel_defect}, applied to the contraction generators \\(-L_f^\\alpha\\) and \\(-L_c^\\alpha\\).\n\\end{proof}","&=\n\\alpha m^{\\alpha-1}\\norm{\\Delta_L}.\n\\end{aligned}\n\\]\nThe semigroup estimate follows from \\cref{extended:thm:duhamel_defect}, applied to the contraction generators \\(-L_f^\\alpha\\) and \\(-L_c^\\alpha\\).\n\\end{proof}","\\begin{remark}\\label{extended:record:071}\nThe factor \\(m^{\\alpha-1}\\) records the singular behavior of \\(x^\\alpha\\) near zero. Without a common positive lower bound, a uniform Lipschitz estimate of this form cannot be expected.\n\\end{remark}","\\begin{standardfact}[Stable subordination]\\label{extended:record:072}\\label{extended:fact:stable_subordination}\nLet \\(0<\\alpha<1\\) and \\(t>0\\). There is a probability measure \\(\\eta_{\\alpha,t}\\) on \\([0,\\infty)\\) such that\n\\[\n\\int_0^\\infty \\e^{-s\\lambda}\\eta_{\\alpha,t}(\\dd s)\n=\n\\e^{-t\\lambda^\\alpha}\n\\]\nfor every \\(\\lambda\\geq0\\). Consequently, for a finite dimensional positive self-adjoint operator,\n\\[\n\\e^{-tL^\\alpha}\n=\n\\int_0^\\infty\n\\e^{-sL}\\eta_{\\alpha,t}(\\dd s)\n\\]\nin operator norm. The measures satisfy the convolution law\n\\[\n\\eta_{\\alpha,s}*\\eta_{\\alpha,t}\n=\n\\eta_{\\alpha,s+t}.\n\\]\n\\end{standardfact}","n{standardfact}[Stable subordination]\\label{extended:record:072}\\label{extended:fact:stable_subordination}\nLet \\(0<\\alpha<1\\) and \\(t>0\\). There is a probability measure \\(\\eta_{\\alpha,t}\\) on \\([0,\\infty)\\) such that\n\\[\n\\int_0^\\infty \\e^{-s\\lambda}\\eta_{\\alpha,t}(\\dd s)\n=\n\\e^{-t\\lambda^\\alpha}\n\\]\nfor every \\(\\lambda\\geq0\\). Consequently, for a finite dimensional positive self-adjoint operator,\n\\[\n\\e^{-tL^\\alpha}\n=\n\\int_0^\\infty\n\\e^{-sL}\\eta_{\\alpha,t}(\\dd s)\n\\]\nin operator norm. The measures satisfy the convolution law\n\\[\n\\eta_{\\alpha,s}*\\eta_{\\alpha,t}\n=\n\\eta_{\\alpha,s+t}.\n\\]\n\\end{standardfact}","d \\(t>0\\). There is a probability measure \\(\\eta_{\\alpha,t}\\) on \\([0,\\infty)\\) such that\n\\[\n\\int_0^\\infty \\e^{-s\\lambda}\\eta_{\\alpha,t}(\\dd s)\n=\n\\e^{-t\\lambda^\\alpha}\n\\]\nfor every \\(\\lambda\\geq0\\). Consequently, for a finite dimensional positive self-adjoint operator,\n\\[\n\\e^{-tL^\\alpha}\n=\n\\int_0^\\infty\n\\e^{-sL}\\eta_{\\alpha,t}(\\dd s)\n\\]\nin operator norm. The measures satisfy the convolution law\n\\[\n\\eta_{\\alpha,s}*\\eta_{\\alpha,t}\n=\n\\eta_{\\alpha,s+t}.\n\\]\n\\end{standardfact}","The existence of \\(\\eta_{\\alpha,t}\\) follows from complete monotonicity of \\(\\e^{-t\\lambda^\\alpha}\\). The operator identity is obtained by applying the scalar Laplace transform on every spectral packet. This is the finite spectral form of Bochner subordination. See \\cite{SchillingSongVondracek2012}.","\\begin{theorem}[Poisson subordination]\\label{extended:record:073}\\label{extended:thm:poisson_subordination}\nFor \\(t>0\\),\n\\[\n\\e^{-t\\sqrt L}\n=\n\\frac{t}{2\\sqrt\\pi}\n\\int_0^\\infty\ns^{-3/2}\n\\e^{-t^2/(4s)}\n\\e^{-sL}\n\\dd s.\n\\]\nThe integral converges in operator norm.\n\\end{theorem}","\\begin{proof}\nOn a spectral packet with eigenvalue \\(\\lambda\\), the formula reduces to\n\\[\n\\e^{-t\\sqrt\\lambda}\n=\n\\frac{t}{2\\sqrt\\pi}\n\\int_0^\\infty\ns^{-3/2}\\e^{-t^2/(4s)-\\lambda s}\\dd s.\n\\]\nThis scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.\n\\end{proof}","Consider\n\\[\n-u''=f\n\\quad\\hbox{on }(0,1),\n\\qquad\nu(0)=u(1)=0.\n\\]\nThe continuum Green kernel is\n\\[\nG(x,y)=\\min(x,y)-xy.\n\\]\nThus\n\\[\nu(x)=\\int_0^1G(x,y)f(y)\\dd y.\n\\]","For \\(n\\) interior nodes, let\n\\[\nh=\\frac{1}{n+1},\n\\qquad\nx_i=ih,\n\\]\nand define\n\\[\nL_h\n=\n\\frac{1}{h^2}\n\\begin{pmatrix}\n2&-1&&","For \\(n\\) interior nodes, let\n\\[\nh=\\frac{1}{n+1},\n\\qquad\nx_i=ih,\n\\]\nand define\n\\[\nL_h\n=\n\\frac{1}{h^2}\n\\begin{pmatrix}\n2&-1&&\\\\\n-1&2&\\ddots&\\\\\n&\\ddots&\\ddots&-1\\\\\n&&-1&2\n\\end{pmatrix}.\n\\]","\\begin{theorem}[Exact nodal Green reconstruction]\\label{extended:record:075}\\label{extended:thm:nodal_green}\nFor \\(1\\leq i,j\\leq n\\),\n\\[\n\\frac{1}{h}(L_h^{-1})_{ij}\n=\nG(x_i,x_j).\n\\]\n\\end{theorem}","\\begin{proof}\nThe inverse of the unscaled tridiagonal matrix has entry\n\\[\n\\frac{\\min(i,j)(n+1-\\max(i,j))}{n+1}.\n\\]\nMultiplying by \\(h^2\\) and dividing by \\(h\\) gives\n\\[\nh\\min(i,j)\n\\left(1-\\frac{\\max(i,j)}{n+1}\\right)\n=\n\\min(x_i,x_j)-x_ix_j.\n\\]\n\\end{proof}","\\begin{corollary}[Discrete Green quadrature]\\label{extended:record:076}\\label{extended:cor:discrete_green_quadrature}\nIf \\(u^h\\) solves\n\\[\nL_hu^h=f^h,\n\\]\nthen every nodal value has the exact representation\n\\[\nu_i^h\n=\nh\\sum_{j=1}^nG(x_i,x_j)f_j^h.\n\\]\nThus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.\n\\end{corollary}","\\begin{proof}\nBy \\cref{extended:thm:nodal_green},\n\\[\n(L_h^{-1})_{ij}=hG(x_i,x_j).\n\\]\nMultiply the inverse matrix by the source vector.\n\\end{proof}","\\begin{proposition}[Discrete maximum bound]\\label{extended:record:077}\\label{extended:prop:discrete_inverse_bound}\nThe inverse \\(L_h^{-1}\\) has nonnegative entries and\n\\[\n\\norm{L_h^{-1}}_{\\infty\\to\\infty}\n=\n\\max_{1\\leq i\\leq n}\n\\frac{x_i(1-x_i)}{2}\n\\leq\n\\frac18.\n\\]\n\\end{proposition}","\\begin{proof}\nNonnegativity follows from the entry formula in \\cref{extended:thm:nodal_green}. Let \\(v_i=x_i(1-x_i)/2\\). Since the second difference is exact on quadratics,\n\\[\nL_hv=\\one.\n\\]\nHence the \\(i\\)th row sum of \\(L_h^{-1}\\) is \\(v_i\\). For a nonnegative matrix, the infinity operator norm is the largest row sum.\n\\end{proof}","\\begin{theorem}[Second order Poisson solution error]\\label{extended:record:078}\\label{extended:thm:poisson_error}\nSuppose \\(u\\in C^4[0,1]\\) solves the Dirichlet problem above. Let\n\\[\nf^h_i=f(x_i)\n\\]\nand let \\(u^h\\) solve\n\\[\nL_hu^h=f^h.\n\\]\nThen\n\\[\n\\max_i\\abs{u^h_i-u(x_i)}\n\\leq\n\\frac{h^2}{96}\n\\norm{u^{(4)}}_\\infty.\n\\]\n\\end{theorem}","\\begin{proof}\nTaylor expansion with remainder gives\n\\[\n\\norm{L_h(u(x_i))_i-f^h}_\\infty\n\\leq\n\\frac{h^2}{12}\\norm{u^{(4)}}_\\infty.\n\\]\nApply \\(L_h^{-1}\\) and use \\cref{extended:prop:discrete_inverse_bound}.\n\\end{proof}","The discrete eigenpairs are\n\\[\n\\lambda_{k,h}\n=\n\\frac{4}{h^2}\n\\sin^2\\left(\\frac{k\\pi h}{2}\\right),\n\\qquad\nv_{k,h}(i)=\\sin(k\\pi x_i).\n\\]","\\begin{theorem}[Low mode eigenvalue control]\\label{extended:record:079}\\label{extended:thm:low_mode}\nFor \\(kh\\leq1\\),\n\\[\n0\\leq(k\\pi)^2-\\lambda_{k,h}\n\\leq\n\\frac{(k\\pi)^4}{12}h^2.\n\\]\nConsequently, for every fixed \\(k\\),\n\\[\n\\lambda_{k,h}=(k\\pi)^2+O(h^2).\n\\]\n\\end{theorem}","\\begin{proof}\nSet \\(x=k\\pi h/2\\). The condition \\(kh\\leq1\\) gives \\(0\\leq x\\leq\\pi/2\\). On this interval,\n\\[\n\\sin x\\geq x-\\frac{x^3}{6},\n\\]\nso\n\\[\n0\\leq x^2-\\sin^2x\\leq\\frac{x^4}{3}.\n\\]\nMultiply by \\(4/h^2\\).\n\\end{proof}","\\begin{corollary}[Heat and fractional mode transfer]\\label{extended:record:080}\\label{extended:cor:mode_transfer}\nLet \\(\\lambda_k=(k\\pi)^2\\). For \\(t\\geq0\\),\n\\[\n0\\leq\n\\e^{-t\\lambda_{k,h}}-\\e^{-t\\lambda_k}\n\\leq\n\\frac{t(k\\pi)^4}{12}h^2\\e^{-t\\lambda_{k,h}}.\n\\]\nFor \\(0<\\alpha\\leq1\\) and fixed \\(k\\),\n\\[\n\\lambda_{k,h}^\\alpha\n=\n\\lambda_k^\\alpha+O(h^2)\n\\]\nand\n\\[\n\\e^{-t\\lambda_{k,h}^\\alpha}\n=\n\\e^{-t\\lambda_k^\\alpha}+O(th^2)\n\\]\nuniformly for \\(t\\) in bounded intervals.\n\\end{corollary}","mode transfer]\\label{extended:record:080}\\label{extended:cor:mode_transfer}\nLet \\(\\lambda_k=(k\\pi)^2\\). For \\(t\\geq0\\),\n\\[\n0\\leq\n\\e^{-t\\lambda_{k,h}}-\\e^{-t\\lambda_k}\n\\leq\n\\frac{t(k\\pi)^4}{12}h^2\\e^{-t\\lambda_{k,h}}.\n\\]\nFor \\(0<\\alpha\\leq1\\) and fixed \\(k\\),\n\\[\n\\lambda_{k,h}^\\alpha\n=\n\\lambda_k^\\alpha+O(h^2)\n\\]\nand\n\\[\n\\e^{-t\\lambda_{k,h}^\\alpha}\n=\n\\e^{-t\\lambda_k^\\alpha}+O(th^2)\n\\]\nuniformly for \\(t\\) in bounded intervals.\n\\end{corollary}","\\begin{proof}\nThe ordinary heat estimate follows from the mean value theorem and \\cref{extended:thm:low_mode}. For fixed \\(k\\), the eigenvalues stay uniformly away from zero once \\(h\\) is small. The function \\(x^\\alpha\\) is then Lipschitz on a common compact interval containing both eigenvalues. Apply the mean value theorem twice.\n\\end{proof}","Let \\(K\\) be column stochastic. Let \\(V\\) be a real potential on the states and let\n\\[\nD_V=\\diag(V).\n\\]\nFor functions written as columns, the backward Markov operator is \\(K^T\\).","\\begin{theorem}[Discrete Feynman--Kac formula]\\label{extended:record:081}\\label{extended:thm:discrete_feynman_kac}\nDefine\n\\[\nM_V=\\diag(\\e^{-V})K^T.\n\\]\nFor terminal data \\(f\\),\n\\[\n(M_V^nf)_i\n=\n\\Exp_i\\left[\n\\exp\\left(\n-\\sum_{r=0}^{n-1}V(X_r)\n\\right)\nf(X_n)\n\\right].\n\\]\n\\end{theorem}","\\begin{proof}\nFor one step,\n\\[\n(M_Vf)_i\n=\n\\e^{-V(i)}\\sum_jK_{ji}f_j.\n\\]\nIterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.\n\\end{proof}","\\begin{theorem}[Continuous time finite state formula]\\label{extended:record:082}\\label{extended:thm:continuous_feynman_kac}\nLet \\(Q\\) be a finite Markov generator and let \\((X_t)_{t\\geq0}\\) be the corresponding chain. Then\n\\[\n\\left(\n\\e^{t(Q^T-D_V)}f\n\\right)_i\n=\n\\Exp_i\\left[\n\\exp\\left(\n-\\int_0^tV(X_s)\\dd s\n\\right)\nf(X_t)\n\\right].\n\\]\nIf \\(V\\geq v_0\\), then\n\\[\n\\norm{\\e^{t(Q^T-D_V)}}_{\\infty\\to\\infty}\n\\leq\n\\e^{-v_0t}.\n\\]\n\\end{theorem}","\\begin{proof}\nConditioning over the first small time interval gives the backward equation\n\\[\n\\dot u=(Q^T-D_V)u,\n\\qquad\nu(0)=f.\n\\]\nThe finite system has a unique solution, namely the displayed matrix exponential. The expectation bound follows from\n\\[\n\\exp\\left(-\\int_0^tV(X_s)\\dd s\\right)\n\\leq\n\\e^{-v_0t}.\n\\]\nThis finite argument is the matrix form of the Feynman--Kac principle originating in \\cite{Kac1949}.\n\\end{proof}","\\begin{theorem}[Alternating propagation and killing]\\label{extended:record:083}\\label{extended:thm:feynman_kac_product}\nFor every \\(t\\geq0\\),\n\\[\n\\e^{t(Q^T-D_V)}\n=\n\\lim_{n\\to\\infty}\n\\left(\n\\e^{-tD_V/n}\\e^{tQ^T/n}\n\\right)^n\n\\]\nin operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.\n\\end{theorem}","\\begin{proof}\nSet \\(B=-D_V\\) and \\(C=Q^T\\). For bounded matrices,\n\\[\n\\e^{tB/n}\\e^{tC/n}\n=\nI+\\frac{t}{n}(B+C)+O(n^{-2})\n\\]\nuniformly in operator norm. The standard telescoping comparison with\n\\(\n(I+t(B+C)/n)^n\n\\)\nshows that the product converges to \\(\\e^{t(B+C)}\\). This is the displayed factor order, so the limit follows.\n\\end{proof}","\\begin{corollary}[Positivity and composition]\\label{extended:record:084}\\label{extended:cor:feynman_kac_semigroup}\nThe operators\n\\[\nF_V(t)=\\e^{t(Q^T-D_V)}\n\\]\nform a positive semigroup. If \\(f\\geq0\\), then \\(F_V(t)f\\geq0\\).\n\\end{corollary}","Let \\(X=X_P\\oplus X_Q\\) and write\n\\[\nA=\n\\begin{pmatrix}\nA_{PP}&A_{PQ}","Let \\(X=X_P\\oplus X_Q\\) and write\n\\[\nA=\n\\begin{pmatrix}\nA_{PP}&A_{PQ}\\\\\nA_{QP}&A_{QQ}\n\\end{pmatrix}.\n\\]\nFor \\(zI-A_{QQ}\\) invertible, define\n\\[\n\\Sigma_P(z)\n=\nA_{PQ}(zI-A_{QQ})^{-1}A_{QP}\n\\]\nand\n\\[\nF_P(z)\n=\nzI-A_{PP}-\\Sigma_P(z).\n\\]","A_{QP}&A_{QQ}\n\\end{pmatrix}.\n\\]\nFor \\(zI-A_{QQ}\\) invertible, define\n\\[\n\\Sigma_P(z)\n=\nA_{PQ}(zI-A_{QQ})^{-1}A_{QP}\n\\]\nand\n\\[\nF_P(z)\n=\nzI-A_{PP}-\\Sigma_P(z).\n\\]","\\begin{theorem}[Schur--Feshbach resolvent]\\label{extended:record:085}\\label{extended:thm:feshbach_resolvent}\nAssume \\(zI-A_{QQ}\\) is invertible. Then \\(zI-A\\) is invertible if and only if \\(F_P(z)\\) is invertible. In that case,\n\\[\nP(zI-A)^{-1}P\n=\nF_P(z)^{-1}.\n\\]\nIn finite dimension,\n\\[\n\\det(zI-A)\n=\n\\det(zI-A_{QQ})\\det F_P(z).\n\\]\n\\end{theorem}","\\begin{proof}\nSet\n\\[\nL_z=\n\\begin{pmatrix}\nI&-A_{PQ}(zI-A_{QQ})^{-1}\\\\\n0&I\n\\end{pmatrix}\n\\]\nand\n\\[\nS_z=\n\\begin{pmatrix}\nF_P(z)&0","0&I\n\\end{pmatrix}\n\\]\nand\n\\[\nS_z=\n\\begin{pmatrix}\nF_P(z)&0\\\\\n-A_{QP}&zI-A_{QQ}\n\\end{pmatrix}.\n\\]\nDirect multiplication gives \\(zI-A=L_zS_z\\). The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction \\cite{Feshbach1958}.\n\\end{proof}","-A_{QP}&zI-A_{QQ}\n\\end{pmatrix}.\n\\]\nDirect multiplication gives \\(zI-A=L_zS_z\\). The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction \\cite{Feshbach1958}.\n\\end{proof}","For the homogeneous equation \\(\\dot x=Ax\\), variation of constants gives\n\\[\nx_Q(t)\n=\n\\e^{tA_{QQ}}x_Q(0)\n+\n\\int_0^t\n\\e^{(t-s)A_{QQ}}A_{QP}x_P(s)\n\\dd s.\n\\]\nSubstitution into the retained equation gives\n\\[\n\\begin{aligned}\n\\dot x_P(t)\n={}&\nA_{PP}x_P(t)\n+\nA_{PQ}\\e^{tA_{QQ}}x_Q(0)","For the homogeneous equation \\(\\dot x=Ax\\), variation of constants gives\n\\[\nx_Q(t)\n=\n\\e^{tA_{QQ}}x_Q(0)\n+\n\\int_0^t\n\\e^{(t-s)A_{QQ}}A_{QP}x_P(s)\n\\dd s.\n\\]\nSubstitution into the retained equation gives\n\\[\n\\begin{aligned}\n\\dot x_P(t)\n={}&\nA_{PP}x_P(t)\n+\nA_{PQ}\\e^{tA_{QQ}}x_Q(0)\\\\\n&+\n\\int_0^t\n\\Kcal(t-s)x_P(s)\n\\dd s,\n\\end{aligned}\n\\]\nwhere\n\\[\n\\Kcal(t)\n=\nA_{PQ}\\e^{tA_{QQ}}A_{QP}.\n\\]","&+\n\\int_0^t\n\\Kcal(t-s)x_P(s)\n\\dd s,\n\\end{aligned}\n\\]\nwhere\n\\[\n\\Kcal(t)\n=\nA_{PQ}\\e^{tA_{QQ}}A_{QP}.\n\\]","\\begin{theorem}[Memory and self-energy]\\label{extended:record:086}\\label{extended:thm:memory_self_energy}\nIf \\(z\\) lies to the right of the spectrum of \\(A_{QQ}\\), then\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\Kcal(t)\\dd t\n=\n\\Sigma_P(z).\n\\]\nIf, for constants \\(M\\geq0\\) and \\(\\gamma>0\\),\n\\[\n\\norm{\\e^{tA_{QQ}}}\n\\leq\nM\\e^{-\\gamma t},\n\\]\nthen\n\\[\n\\norm{\\Kcal(t)}\n\\leq\nM\\norm{A_{PQ}}\\norm{A_{QP}}\\e^{-\\gamma t}\n\\]\nand\n\\[\n\\norm{\\Kcal}_{L^1(0,\\infty)}\n\\leq\n\\frac{M}{\\gamma}\n\\norm{A_{PQ}}\\norm{A_{QP}}.\n\\]\n\\end{theorem}","begin{theorem}[Memory and self-energy]\\label{extended:record:086}\\label{extended:thm:memory_self_energy}\nIf \\(z\\) lies to the right of the spectrum of \\(A_{QQ}\\), then\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\Kcal(t)\\dd t\n=\n\\Sigma_P(z).\n\\]\nIf, for constants \\(M\\geq0\\) and \\(\\gamma>0\\),\n\\[\n\\norm{\\e^{tA_{QQ}}}\n\\leq\nM\\e^{-\\gamma t},\n\\]\nthen\n\\[\n\\norm{\\Kcal(t)}\n\\leq\nM\\norm{A_{PQ}}\\norm{A_{QP}}\\e^{-\\gamma t}\n\\]\nand\n\\[\n\\norm{\\Kcal}_{L^1(0,\\infty)}\n\\leq\n\\frac{M}{\\gamma}\n\\norm{A_{PQ}}\\norm{A_{QP}}.\n\\]\n\\end{theorem}","\\begin{proof}\nIf \\(X_Q=\\{0\\}\\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero.","\\begin{proof}\nIf \\(X_Q=\\{0\\}\\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R","\\begin{proof}\nIf \\(X_Q=\\{0\\}\\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{Enge","\\begin{proof}\nIf \\(X_Q=\\{0\\}\\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplicati","\\begin{proof}\nIf \\(X_Q=\\{0\\}\\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the fir","\\begin{proof}\nIf \\(X_Q=\\{0\\}\\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay r","gin{proof}\nIf \\(X_Q=\\{0\\}\\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{pr…","y are both zero, so the identity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","ity is immediate. Otherwise write \\(B=A_{QQ}\\), which is a bounded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","unded operator on the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","the hidden Banach space, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","pace, and put \\(s(B)=\\max\\operatorname{Re}\\Spec(B)\\). Choose a real \\(\\beta\\) with \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","ith \\(s(B)<\\beta<\\operatorname{Re}z\\). Spectral mapping and the spectral-radius formula give \\(r(\\e^B)=\\e^{s(B)}<\\e^\\beta\\), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","), hence \\(\\norm{\\e^{nB}}\\leq M_\\beta\\e^{\\beta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","ta n}\\) for every integer \\(n\\geq0\\), after increasing \\(M_\\beta\\) to cover the finitely many initial values. Write \\(t=n+r\\), where \\(0\\leq r<1\\). Then\n\\[\n\\norm{\\e^{tB}}\\leq M_\\beta\\exp(\\norm B+\\max(0,-\\beta))\\e^{\\beta t}.\n\\]\nThus \\(\\e^{-zt}\\e^{tB}\\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \\(R=\\int_0^\\infty\\e^{-zt}\\e^{tB}\\dd t\\), so \\(R=(zI-B)^{-1}\\). This is the bounded-generator growth principle in \\cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.\n\\end{proof}","\\begin{theorem}[Criterion for exact local closure]\\label{extended:record:087}\\label{extended:thm:local_closure}\nAssume \\(x_Q(0)=0\\) and \\(d=\\dim X_Q<\\infty\\). The following conditions are equivalent.\n\\begin{enumerate}\n\\item For every retained initial state,\n\\[\nP\\e^{tA}P\n=\n\\e^{tA_{PP}}P\n\\]\nfor all \\(t\\geq0\\).\n\\item \\(\\Sigma_P(z)=0\\) for every sufficiently large \\(z\\).\n\\item \\(\\Kcal(t)=0\\) for every \\(t\\geq0\\).\n\\item If \\(d=\\dim X_Q\\), then\n\\[\nA_{PQ}A_{QQ}^kA_{QP}=0\n\\]\nfor \\(0\\leq k<d\\).\n\\end{enumerate}\n\\end{theorem}","gin{theorem}[Criterion for exact local closure]\\label{extended:record:087}\\label{extended:thm:local_closure}\nAssume \\(x_Q(0)=0\\) and \\(d=\\dim X_Q<\\infty\\). The following conditions are equivalent.\n\\begin{enumerate}\n\\item For every retained initial state,\n\\[\nP\\e^{tA}P\n=\n\\e^{tA_{PP}}P\n\\]\nfor all \\(t\\geq0\\).\n\\item \\(\\Sigma_P(z)=0\\) for every sufficiently large \\(z\\).\n\\item \\(\\Kcal(t)=0\\) for every \\(t\\geq0\\).\n\\item If \\(d=\\dim X_Q\\), then\n\\[\nA_{PQ}A_{QQ}^kA_{QP}=0\n\\]\nfor \\(0\\leq k<d\\).\n\\end{enumerate}\n\\end{theorem}","n for exact local closure]\\label{extended:record:087}\\label{extended:thm:local_closure}\nAssume \\(x_Q(0)=0\\) and \\(d=\\dim X_Q<\\infty\\). The following conditions are equivalent.\n\\begin{enumerate}\n\\item For every retained initial state,\n\\[\nP\\e^{tA}P\n=\n\\e^{tA_{PP}}P\n\\]\nfor all \\(t\\geq0\\).\n\\item \\(\\Sigma_P(z)=0\\) for every sufficiently large \\(z\\).\n\\item \\(\\Kcal(t)=0\\) for every \\(t\\geq0\\).\n\\item If \\(d=\\dim X_Q\\), then\n\\[\nA_{PQ}A_{QQ}^kA_{QP}=0\n\\]\nfor \\(0\\leq k<d\\).\n\\end{enumerate}\n\\end{theorem}","sure]\\label{extended:record:087}\\label{extended:thm:local_closure}\nAssume \\(x_Q(0)=0\\) and \\(d=\\dim X_Q<\\infty\\). The following conditions are equivalent.\n\\begin{enumerate}\n\\item For every retained initial state,\n\\[\nP\\e^{tA}P\n=\n\\e^{tA_{PP}}P\n\\]\nfor all \\(t\\geq0\\).\n\\item \\(\\Sigma_P(z)=0\\) for every sufficiently large \\(z\\).\n\\item \\(\\Kcal(t)=0\\) for every \\(t\\geq0\\).\n\\item If \\(d=\\dim X_Q\\), then\n\\[\nA_{PQ}A_{QQ}^kA_{QP}=0\n\\]\nfor \\(0\\leq k<d\\).\n\\end{enumerate}\n\\end{theorem}","label{extended:thm:local_closure}\nAssume \\(x_Q(0)=0\\) and \\(d=\\dim X_Q<\\infty\\). The following conditions are equivalent.\n\\begin{enumerate}\n\\item For every retained initial state,\n\\[\nP\\e^{tA}P\n=\n\\e^{tA_{PP}}P\n\\]\nfor all \\(t\\geq0\\).\n\\item \\(\\Sigma_P(z)=0\\) for every sufficiently large \\(z\\).\n\\item \\(\\Kcal(t)=0\\) for every \\(t\\geq0\\).\n\\item If \\(d=\\dim X_Q\\), then\n\\[\nA_{PQ}A_{QQ}^kA_{QP}=0\n\\]\nfor \\(0\\leq k<d\\).\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nBy \\cref{extended:thm:laplace_packet_resolvent}, the Laplace transform of the projected propagator is the projected resolvent. By \\cref{extended:thm:feshbach_resolvent}, the first condition is equivalent to\n\\[\n(zI-A_{PP}-\\Sigma_P(z))^{-1}\n=\n(zI-A_{PP})^{-1}\n\\]\nfor all sufficiently large \\(z\\), which is equivalent to the second condition. The second and third conditions are equivalent by \\cref{extended:thm:memory_self_energy} and uniqueness of the Laplace transform.","For large \\(z\\),\n\\[\n\\Sigma_P(z)\n=\n\\sum_{k=0}^{\\infty}\n\\frac{A_{PQ}A_{QQ}^kA_{QP}}{z^{k+1}}.\n\\]\nThus the second condition implies the moment identities. Conversely, the first \\(d\\) identities imply all later ones by the Cayley--Hamilton theorem, so the series vanishes.\n\\end{proof}","\\begin{remark}\\label{extended:record:088}\nA finite hidden block produces a strictly proper rational self-energy. It tends to zero as \\(z\\) tends to infinity. Hence a nonzero constant self-energy cannot arise from hidden dynamics alone. A constant correction must already be included in the retained generator.\n\\end{remark}","\\begin{theorem}[Polynomial exponential memory of a finite hidden system]\\label{extended:record:089}\\label{extended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then","\\begin{theorem}[Polynomial exponential memory of a finite hidden system]\\label{extended:record:089}\\label{extended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum","\\begin{theorem}[Polynomial exponential memory of a finite hidden system]\\label{extended:record:089}\\label{extended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-","\\begin{theorem}[Polynomial exponential memory of a finite hidden system]\\label{extended:record:089}\\label{extended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname","\\begin{theorem}[Polynomial exponential memory of a finite hidden system]\\label{extended:record:089}\\label{extended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","rem}[Polynomial exponential memory of a finite hidden system]\\label{extended:record:089}\\label{extended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","mory of a finite hidden system]\\label{extended:record:089}\\label{extended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","xtended:thm:modal_memory}\nLet \\(X_Q\\) be a finite dimensional complex space. Let \\(\\mu_1,\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","\\ldots,\\mu_r\\) be the distinct eigenvalues of \\(A_{QQ}\\), let \\(Q_j\\) be the corresponding primary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","ary spectral projections, and set\n\\[\nN_j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","j=(A_{QQ}-\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","\\mu_jI)Q_j.\n\\]\nLet \\(\\nu_j\\) be the least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","he least positive integer for which \\(N_j^{\\nu_j}=0\\), and define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","define\n\\[\nB_{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","{j,k}\n=\nA_{PQ}N_j^kQ_jA_{QP}\n\\qquad\n(0\\leq k<\\nu_j).\n\\]\nThen\n\\[\n\\Kcal(t)\n=\n\\sum_{j=1}^r\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}B_{j,k}\n\\]\nand\n\\[\n\\Sigma_P(z)\n=\n\\sum_{j=1}^r\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{B_{j,k}}{(z-\\mu_j)^{k+1}}.\n\\]\nSuppose \\(\\operatorname{Re}\\mu_j<0\\) for every \\(j\\). If \\(J\\subseteq\\{1,\\ldots,r\\}\\) and \\(\\Kcal_J\\) retains the complete primary blocks indexed by \\(J\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe primary decomposition gives\n\\[\nA_{QQ}Q_j=\\mu_jQ_j+N_j\n\\]\nwith commuting scalar and nilpotent parts on \\(Q_jX_Q\\). Therefore\n\\[\n\\e^{tA_{QQ}}Q_j\n=\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}N_j^kQ_j.\n\\]\nMultiplication by \\(A_{PQ}\\) and \\(A_{QP}\\) gives the memory formula. On the \\(j\\)th primary space, the nilpotent resolvent expansion is\n\\[\n(zI-A_{QQ})^{-1}Q_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{N_j^kQ_j}{(z-\\mu_j)^{k+1}}.\n\\]\nMultiplication by the coupling blocks gives the self-energy formula for every \\(z\\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\e^{\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(z-\\mu_j)^{k+1}}.\n\\]\nUnder stability,\n\\[\n\\int_0^\\infty\n\\e^{\\operatorname{R","\\begin{proof}\nThe primary decomposition gives\n\\[\nA_{QQ}Q_j=\\mu_jQ_j+N_j\n\\]\nwith commuting scalar and nilpotent parts on \\(Q_jX_Q\\). Therefore\n\\[\n\\e^{tA_{QQ}}Q_j\n=\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}N_j^kQ_j.\n\\]\nMultiplication by \\(A_{PQ}\\) and \\(A_{QP}\\) gives the memory formula. On the \\(j\\)th primary space, the nilpotent resolvent expansion is\n\\[\n(zI-A_{QQ})^{-1}Q_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{N_j^kQ_j}{(z-\\mu_j)^{k+1}}.\n\\]\nMultiplication by the coupling blocks gives the self-energy formula for every \\(z\\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\e^{\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(z-\\mu_j)^{k+1}}.\n\\]\nUnder stability,\n\\[\n\\int_0^\\infty\n\\e^{\\operatorname{Re}\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(-\\operatornam","\\begin{proof}\nThe primary decomposition gives\n\\[\nA_{QQ}Q_j=\\mu_jQ_j+N_j\n\\]\nwith commuting scalar and nilpotent parts on \\(Q_jX_Q\\). Therefore\n\\[\n\\e^{tA_{QQ}}Q_j\n=\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}N_j^kQ_j.\n\\]\nMultiplication by \\(A_{PQ}\\) and \\(A_{QP}\\) gives the memory formula. On the \\(j\\)th primary space, the nilpotent resolvent expansion is\n\\[\n(zI-A_{QQ})^{-1}Q_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{N_j^kQ_j}{(z-\\mu_j)^{k+1}}.\n\\]\nMultiplication by the coupling blocks gives the self-energy formula for every \\(z\\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\e^{\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(z-\\mu_j)^{k+1}}.\n\\]\nUnder stability,\n\\[\n\\int_0^\\infty\n\\e^{\\operatorname{Re}\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nSum the omitted terms.\n\\end{proof}","_jX_Q\\). Therefore\n\\[\n\\e^{tA_{QQ}}Q_j\n=\n\\e^{\\mu_jt}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{t^k}{k!}N_j^kQ_j.\n\\]\nMultiplication by \\(A_{PQ}\\) and \\(A_{QP}\\) gives the memory formula. On the \\(j\\)th primary space, the nilpotent resolvent expansion is\n\\[\n(zI-A_{QQ})^{-1}Q_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{N_j^kQ_j}{(z-\\mu_j)^{k+1}}.\n\\]\nMultiplication by the coupling blocks gives the self-energy formula for every \\(z\\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\e^{\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(z-\\mu_j)^{k+1}}.\n\\]\nUnder stability,\n\\[\n\\int_0^\\infty\n\\e^{\\operatorname{Re}\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nSum the omitted terms.\n\\end{proof}","cation by \\(A_{PQ}\\) and \\(A_{QP}\\) gives the memory formula. On the \\(j\\)th primary space, the nilpotent resolvent expansion is\n\\[\n(zI-A_{QQ})^{-1}Q_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{N_j^kQ_j}{(z-\\mu_j)^{k+1}}.\n\\]\nMultiplication by the coupling blocks gives the self-energy formula for every \\(z\\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\e^{\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(z-\\mu_j)^{k+1}}.\n\\]\nUnder stability,\n\\[\n\\int_0^\\infty\n\\e^{\\operatorname{Re}\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nSum the omitted terms.\n\\end{proof}","esolvent expansion is\n\\[\n(zI-A_{QQ})^{-1}Q_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{N_j^kQ_j}{(z-\\mu_j)^{k+1}}.\n\\]\nMultiplication by the coupling blocks gives the self-energy formula for every \\(z\\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity\n\\[\n\\int_0^\\infty\n\\e^{-zt}\\e^{\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(z-\\mu_j)^{k+1}}.\n\\]\nUnder stability,\n\\[\n\\int_0^\\infty\n\\e^{\\operatorname{Re}\\mu_jt}\n\\frac{t^k}{k!}\n\\dd t\n=\n\\frac{1}{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nSum the omitted terms.\n\\end{proof}","\\begin{corollary}[Optimal selection for the certified primary block bound]\\label{extended:record:090}\\label{extended:cor:optimal_primary_selection}\nUnder the stable assumptions of \\cref{extended:thm:modal_memory}, define the primary block weights\n\\[\nw_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","ptimal selection for the certified primary block bound]\\label{extended:record:090}\\label{extended:cor:optimal_primary_selection}\nUnder the stable assumptions of \\cref{extended:thm:modal_memory}, define the primary block weights\n\\[\nw_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","label{extended:record:090}\\label{extended:cor:optimal_primary_selection}\nUnder the stable assumptions of \\cref{extended:thm:modal_memory}, define the primary block weights\n\\[\nw_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","\\label{extended:cor:optimal_primary_selection}\nUnder the stable assumptions of \\cref{extended:thm:modal_memory}, define the primary block weights\n\\[\nw_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","ine the primary block weights\n\\[\nw_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","weights\n\\[\nw_j\n=\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\nAmong all sets \\(J\\) containing \\(k\\) primary blocks, the certified upper bound\n\\[\n\\sum_{j\\notin J}w_j\n\\]\nis minimized by retaining the \\(k\\) largest weights. If\n\\[\nw_{(1)}\\geq w_{(2)}\\geq\\cdots\\geq w_{(r)},\n\\]\nthen the least number of retained primary blocks certified to give \\(L^1\\) error at most \\(\\varepsilon\\) is the least \\(k\\) for which\n\\[\n\\sum_{j>k}w_{(j)}\\leq\\varepsilon.\n\\]\n\\end{corollary}","\\begin{proof}\nIf a retained set contains a block with smaller weight than an omitted block, exchanging them reduces the omitted weight sum. Repeating the exchange gives the set of the \\(k\\) largest weights. The tolerance statement is the corresponding sorted tail condition.\n\\end{proof}","\\begin{corollary}[Normal hidden dynamics]\\label{extended:record:091}\\label{extended:cor:normal_modal_memory}\nSuppose \\(X_Q\\) is a finite dimensional Hilbert space. If \\(A_{QQ}\\) is normal with spectral resolution\n\\[\nA_{QQ}=\\sum_{j=1}^r\\mu_jQ_j,\n\\]\nthen\n\\[\n\\Kcal(t)=\\sum_{j=1}^r\\e^{\\mu_jt}B_j,\n\\qquad\n\\Sigma_P(z)=\\sum_{j=1}^r\\frac{B_j}{z-\\mu_j},\n\\]\nwhere\n\\[\nB_j=A_{PQ}Q_jA_{QP}.\n\\]\nIf \\(\\operatorname{Re}\\mu_j<0\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\frac{\\norm{B_j}}{-\\operatorname{Re}\\mu_j}.\n\\]\n\\end{corollary}","mal hidden dynamics]\\label{extended:record:091}\\label{extended:cor:normal_modal_memory}\nSuppose \\(X_Q\\) is a finite dimensional Hilbert space. If \\(A_{QQ}\\) is normal with spectral resolution\n\\[\nA_{QQ}=\\sum_{j=1}^r\\mu_jQ_j,\n\\]\nthen\n\\[\n\\Kcal(t)=\\sum_{j=1}^r\\e^{\\mu_jt}B_j,\n\\qquad\n\\Sigma_P(z)=\\sum_{j=1}^r\\frac{B_j}{z-\\mu_j},\n\\]\nwhere\n\\[\nB_j=A_{PQ}Q_jA_{QP}.\n\\]\nIf \\(\\operatorname{Re}\\mu_j<0\\), then\n\\[\n\\norm{\\Kcal-\\Kcal_J}_{L^1(0,\\infty)}\n\\leq\n\\sum_{j\\notin J}\n\\frac{\\norm{B_j}}{-\\operatorname{Re}\\mu_j}.\n\\]\n\\end{corollary}","\\begin{proof}\nNormality removes every nilpotent part, so \\(\\nu_j=1\\). Apply \\cref{extended:thm:modal_memory}.\n\\end{proof}","\\begin{corollary}[Pole order and hidden Jordan depth]\\label{extended:record:092}\\label{extended:cor:pole_jordan_depth}\nEvery finite hidden block produces a strictly proper rational self-energy. At an eigenvalue \\(\\mu_j\\), its pole order is at most the size of the largest Jordan block of \\(A_{QQ}\\) at \\(\\mu_j\\). Equality holds when the corresponding highest coefficient \\(B_{j,\\nu_j-1}\\) is nonzero.\n\\end{corollary}","\\begin{corollary}[Finite auxiliary state realization]\\label{extended:record:093}\\label{extended:cor:finite_auxiliary_realization}\nEvery kernel of the form\n\\[\n\\Kcal(t)=B\\e^{tD}C\n\\]\ncan be represented exactly by adjoining the finite auxiliary state\n\\[\n\\dot y=Dy+Cx_P,\n\\qquad\n\\dot x_P=A_{PP}x_P+By,\n\\qquad\ny(0)=0.\n\\]\nA nonzero auxiliary initial state adds the explicit forcing \\(B\\e^{tD}y(0)\\). The self-energy of the augmented system is\n\\[\nB(zI-D)^{-1}C.\n\\]\n\\end{corollary}","ate realization]\\label{extended:record:093}\\label{extended:cor:finite_auxiliary_realization}\nEvery kernel of the form\n\\[\n\\Kcal(t)=B\\e^{tD}C\n\\]\ncan be represented exactly by adjoining the finite auxiliary state\n\\[\n\\dot y=Dy+Cx_P,\n\\qquad\n\\dot x_P=A_{PP}x_P+By,\n\\qquad\ny(0)=0.\n\\]\nA nonzero auxiliary initial state adds the explicit forcing \\(B\\e^{tD}y(0)\\). The self-energy of the augmented system is\n\\[\nB(zI-D)^{-1}C.\n\\]\n\\end{corollary}","\\begin{proof}\nEliminate \\(y\\) by variation of constants or apply \\cref{extended:thm:feshbach_resolvent} to the augmented block matrix.\n\\end{proof}","\\begin{theorem}[Volterra stability under kernel compression]\\label{extended:record:094}\\label{extended:thm:volterra_compression}\nLet \\(S(t)\\) be a strongly continuous semigroup on a Banach space \\(X\\), fix \\(T>0\\), and assume\n\\[\n\\norm{S(t)}\\leq M_T\n\\quad(0\\leq t\\leq T).\n\\]\nLet \\(K,\\widetilde K\\in L^1((0,T),\\mathcal B(X))\\), let \\(g\\in L^1((0,T),X)\\), and let \\(x,\\widetilde x\\in C([0,T],X)\\). Suppose these are mild solutions with the same initial state and forcing, except that \\(K\\) is replaced by \\(\\widetilde K\\):\n\\[\n\\begin{aligned}\nx(t)\n={}&S(t)x_0+\n\\int_0^tS(t-s)g(s)\\dd s","094}\\label{extended:thm:volterra_compression}\nLet \\(S(t)\\) be a strongly continuous semigroup on a Banach space \\(X\\), fix \\(T>0\\), and assume\n\\[\n\\norm{S(t)}\\leq M_T\n\\quad(0\\leq t\\leq T).\n\\]\nLet \\(K,\\widetilde K\\in L^1((0,T),\\mathcal B(X))\\), let \\(g\\in L^1((0,T),X)\\), and let \\(x,\\widetilde x\\in C([0,T],X)\\). Suppose these are mild solutions with the same initial state and forcing, except that \\(K\\) is replaced by \\(\\widetilde K\\):\n\\[\n\\begin{aligned}\nx(t)\n={}&S(t)x_0+\n\\int_0^tS(t-s)g(s)\\dd s","hm:volterra_compression}\nLet \\(S(t)\\) be a strongly continuous semigroup on a Banach space \\(X\\), fix \\(T>0\\), and assume\n\\[\n\\norm{S(t)}\\leq M_T\n\\quad(0\\leq t\\leq T).\n\\]\nLet \\(K,\\widetilde K\\in L^1((0,T),\\mathcal B(X))\\), let \\(g\\in L^1((0,T),X)\\), and let \\(x,\\widetilde x\\in C([0,T],X)\\). Suppose these are mild solutions with the same initial state and forcing, except that \\(K\\) is replaced by \\(\\widetilde K\\):\n\\[\n\\begin{aligned}\nx(t)\n={}&S(t)x_0+\n\\int_0^tS(t-s)g(s)\\dd s","ssion}\nLet \\(S(t)\\) be a strongly continuous semigroup on a Banach space \\(X\\), fix \\(T>0\\), and assume\n\\[\n\\norm{S(t)}\\leq M_T\n\\quad(0\\leq t\\leq T).\n\\]\nLet \\(K,\\widetilde K\\in L^1((0,T),\\mathcal B(X))\\), let \\(g\\in L^1((0,T),X)\\), and let \\(x,\\widetilde x\\in C([0,T],X)\\). Suppose these are mild solutions with the same initial state and forcing, except that \\(K\\) is replaced by \\(\\widetilde K\\):\n\\[\n\\begin{aligned}\nx(t)\n={}&S(t)x_0+\n\\int_0^tS(t-s)g(s)\\dd s\\\\\n&+\n\\int_0^tS(t-s)\n\\int_0^sK(s-r)x(r)\\dd r\\dd s.\n\\end{aligned}\n\\]\nAssume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq B.\n\\]\nSet\n\\[\nL=\\norm K_{L^1(0,T)},\n\\qquad\n\\varepsilon=\\norm{K-\\widetilde K}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nM_TBT\\varepsilon\n\\exp(M_TLT).\n\\]\n\\end{theorem}","&+\n\\int_0^tS(t-s)\n\\int_0^sK(s-r)x(r)\\dd r\\dd s.\n\\end{aligned}\n\\]\nAssume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq B.\n\\]\nSet\n\\[\nL=\\norm K_{L^1(0,T)},\n\\qquad\n\\varepsilon=\\norm{K-\\widetilde K}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nM_TBT\\varepsilon\n\\exp(M_TLT).\n\\]\n\\end{theorem}","\\begin{proof}\nLet\n\\[\nE(t)=\\sup_{0\\leq r\\leq t}\\norm{x(r)-\\widetilde x(r)}.\n\\]\nSubtract the two mild equations. Fubini's theorem and the triangle inequality give\n\\[\nE(t)\n\\leq\nM_TL\\int_0^tE(r)\\dd r\n+\nM_TBt\\varepsilon.\n\\]\nFor \\(t\\leq T\\), replace the last factor \\(t\\) by \\(T\\) and apply Gronwall's inequality.\n\\end{proof}","\\begin{corollary}[Trajectory bound for modal truncation]\\label{extended:record:095}\\label{extended:cor:modal_trajectory_bound}\nUnder the assumptions of \\cref{extended:thm:modal_memory,extended:thm:volterra_compression}, replacing \\(\\Kcal\\) by \\(\\Kcal_J\\) changes the retained trajectory by at most\n\\[\nM_TBT\n\\exp(M_TLT)\n\\sum_{j\\notin J}\n\\sum_{k=0}^{\\nu_j-1}\n\\frac{\\norm{B_{j,k}}}\n{(-\\operatorname{Re}\\mu_j)^{k+1}}.\n\\]\n\\end{corollary}","\\begin{proof}\nInsert the \\(L^1\\) error from \\cref{extended:thm:modal_memory} into \\cref{extended:thm:volterra_compression}.\n\\end{proof}","\\begin{remark}\\label{extended:record:096}\nA finite hidden realization is exact for rational self-energy. A general memory kernel may require infinitely many modes or a continuum of modes. The compression theorem gives a stability estimate once an \\(L^1\\) approximation has been supplied. The largest weight rule is optimal for the certified blockwise bound. It need not minimize the true kernel error when different modal terms cancel.\n\\end{remark}","Let \\(A\\) be a transient state subgenerator. Its off-diagonal entries are nonnegative and its column sums are nonpositive. Let \\(p_0\\) be a transient probability vector. Define\n\\[\np(t)=\\e^{tA}p_0,\n\\qquad\nS(t)=\\one^Tp(t),\n\\qquad\na^T=-\\one^TA.\n\\]\nThe row vector \\(a^T\\) contains the exit rates.","\\begin{theorem}[State dependent killing as survival]\\label{extended:record:097}\\label{extended:thm:killing_survival}\nLet \\(Q\\) be a finite Markov generator, let \\(V\\geq0\\), and set\n\\[\nA=Q-D_V.\n\\]\nFor an initial distribution \\(p_0\\), define\n\\[\nS_V(t)=\\one^T\\e^{tA}p_0.\n\\]\nThen\n\\[\nS_V(t)\n=\n\\Exp_{p_0}\n\\left[\n\\exp\\left(\n-\\int_0^tV(X_s)\\dd s\n\\right)\n\\right].\n\\]\nAlso,\n\\[\n-S_V'(t)\n=\n\\one^TD_V\\e^{tA}p_0.\n\\]\nThe matrix \\(A=Q-D_V\\) has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman--Kac formula is a state dependent killing rate in the forward sub-Markov semigroup. Almost-sure absorption follows under the additional spectral stability hypothesis used below.\n\\end{theorem}","[State dependent killing as survival]\\label{extended:record:097}\\label{extended:thm:killing_survival}\nLet \\(Q\\) be a finite Markov generator, let \\(V\\geq0\\), and set\n\\[\nA=Q-D_V.\n\\]\nFor an initial distribution \\(p_0\\), define\n\\[\nS_V(t)=\\one^T\\e^{tA}p_0.\n\\]\nThen\n\\[\nS_V(t)\n=\n\\Exp_{p_0}\n\\left[\n\\exp\\left(\n-\\int_0^tV(X_s)\\dd s\n\\right)\n\\right].\n\\]\nAlso,\n\\[\n-S_V'(t)\n=\n\\one^TD_V\\e^{tA}p_0.\n\\]\nThe matrix \\(A=Q-D_V\\) has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman--Kac formula is a state dependent killing rate in the forward sub-Markov semigroup. Almost-sure absorption follows under the additional spectral stability hypothesis used below.\n\\end{theorem}","\\begin{proof}\nAverage \\cref{extended:thm:continuous_feynman_kac} with terminal function \\(f=\\one\\) against the initial distribution. Transposition gives\n\\[\np_0^T\\e^{t(Q^T-D_V)}\\one\n=\n\\one^T\\e^{t(Q-D_V)}p_0.\n\\]\nSince \\(\\one^TQ=0\\), differentiation gives\n\\[\nS_V'(t)\n=\n-\\one^TD_V\\e^{tA}p_0.\n\\]\n\\end{proof}","\\begin{theorem}[Survival and hazard identities]\\label{extended:record:098}\\label{extended:thm:survival_hazard}\nThe event density and hazard are\n\\[\nf(t)=a^T\\e^{tA}p_0=-S'(t),\n\\qquad\nh(t)=\\frac{f(t)}{S(t)}\n\\]\nwhenever \\(S(t)>0\\). The survival function is nonincreasing and\n\\[\nS(t)\n=\n\\exp\\left(\n-\\int_0^th(s)\\dd s\n\\right).\n\\]\n\\end{theorem}","\\begin{proof}\nBecause \\(A\\) is a Metzler matrix, \\(\\e^{tA}\\) is nonnegative. The column sum assumption makes \\(a^T\\) nonnegative. Differentiate \\(S(t)\\):\n\\[\nS'(t)=\\one^TA\\e^{tA}p_0=-a^T\\e^{tA}p_0.\n\\]\nThe remaining identity follows by integrating \\((\\log S)'=-h\\).\n\\end{proof}","\\begin{theorem}[Laplace transform and moments]\\label{extended:record:099}\\label{extended:thm:survival_moments}\nAssume the spectrum of \\(A\\) lies in the open left half plane. Then absorption occurs with probability one. For \\(s\\geq0\\),\n\\[\n\\Exp[\\e^{-s\\tau}]\n=\na^T(sI-A)^{-1}p_0.\n\\] For every integer \\(m\\geq1\\),\n\\[\n\\Exp[\\tau^m]\n=\nm!\\one^T(-A)^{-m}p_0.\n\\]\n\\end{theorem}","\\begin{proof}\nThe Laplace transform follows by integrating the density in \\cref{extended:thm:survival_hazard}. Stability of \\(A\\) gives\n\\[\n\\int_0^\\infty\n\\e^{-st}\\e^{tA}\\dd t\n=\n(sI-A)^{-1}.\n\\]\nAt \\(s=0\\),\n\\[\na^T(-A)^{-1}p_0\n=\n\\one^Tp_0=1.\n\\]\nFor the moments,\n\\[\n\\int_0^\\infty t^m\\e^{tA}\\dd t\n=\nm!(-A)^{-m-1}.\n\\]\nMultiplying by \\(a^T=\\one^T(-A)\\) gives the result. These are the standard matrix formulas for phase type laws \\cite{Neuts1981,AsmussenNermanOlsson1996}.\n\\end{proof}","\\begin{proposition}[Residual life as a restarted semigroup]\\label{extended:record:100}\\label{extended:prop:residual_life}\nFor \\(S(t)>0\\), define the conditional transient distribution\n\\[\np_t\n=\n\\frac{\\e^{tA}p_0}{S(t)}.\n\\]\nThen\n\\[\n\\frac{S(t+s)}{S(t)}\n=\n\\one^T\\e^{sA}p_t.\n\\]\n\\end{proposition}","\\begin{proof}\nUse the semigroup law in the numerator \\(S(t+s)=\\one^T\\e^{sA}\\e^{tA}p_0\\).\n\\end{proof}","\\begin{theorem}[Competing risk packets]\\label{extended:record:101}\\label{extended:thm:competing_risks}\nSuppose the exit row splits into nonnegative cause rows\n\\[\na^T\n=\n\\sum_{r=1}^q a_r^T.\n\\]\nDefine\n\\[\nf_r(t)=a_r^T\\e^{tA}p_0.\n\\]\nThen\n\\[\nf(t)=\\sum_{r=1}^qf_r(t).\n\\]\nThe cause specific hazard is\n\\[\nh_r(t)=\\frac{f_r(t)}{S(t)},\n\\]\nand\n\\[\nh(t)=\\sum_{r=1}^qh_r(t).\n\\]\nIf the spectrum of \\(A\\) lies in the open left half-plane, the eventual probability of cause \\(r\\) is\n\\[\n\\int_0^\\infty f_r(t)\\dd t\n=\na_r^T(-A)^{-1}p_0.\n\\]\n\\end{theorem}","cord:101}\\label{extended:thm:competing_risks}\nSuppose the exit row splits into nonnegative cause rows\n\\[\na^T\n=\n\\sum_{r=1}^q a_r^T.\n\\]\nDefine\n\\[\nf_r(t)=a_r^T\\e^{tA}p_0.\n\\]\nThen\n\\[\nf(t)=\\sum_{r=1}^qf_r(t).\n\\]\nThe cause specific hazard is\n\\[\nh_r(t)=\\frac{f_r(t)}{S(t)},\n\\]\nand\n\\[\nh(t)=\\sum_{r=1}^qh_r(t).\n\\]\nIf the spectrum of \\(A\\) lies in the open left half-plane, the eventual probability of cause \\(r\\) is\n\\[\n\\int_0^\\infty f_r(t)\\dd t\n=\na_r^T(-A)^{-1}p_0.\n\\]\n\\end{theorem}","\\label{extended:thm:competing_risks}\nSuppose the exit row splits into nonnegative cause rows\n\\[\na^T\n=\n\\sum_{r=1}^q a_r^T.\n\\]\nDefine\n\\[\nf_r(t)=a_r^T\\e^{tA}p_0.\n\\]\nThen\n\\[\nf(t)=\\sum_{r=1}^qf_r(t).\n\\]\nThe cause specific hazard is\n\\[\nh_r(t)=\\frac{f_r(t)}{S(t)},\n\\]\nand\n\\[\nh(t)=\\sum_{r=1}^qh_r(t).\n\\]\nIf the spectrum of \\(A\\) lies in the open left half-plane, the eventual probability of cause \\(r\\) is\n\\[\n\\int_0^\\infty f_r(t)\\dd t\n=\na_r^T(-A)^{-1}p_0.\n\\]\n\\end{theorem}","\\midrule\nPacket block convolution & below \\(3\\times10^{-14}\\)","Duhamel defect & below \\(3\\times10^{-12}\\)","Markov lumping & below \\(3\\times10^{-14}\\)","Bayesian reconstruction & below \\(3\\times10^{-15}\\)","Fractional spectral law & below \\(3\\times10^{-12}\\)","Nodal Green identity & below \\(3\\times10^{-13}\\)","Feynman--Kac enumeration & below \\(3\\times10^{-14}\\)","Schur--Feshbach resolvent & below \\(3\\times10^{-12}\\)","Memory Laplace identity & below \\(3\\times10^{-11}\\)","Survival moment formulas & below \\(3\\times10^{-11}\\)","This part therefore separates four operations that are often compressed into one informal idea of coarse dynamics:\n\\[\n\\begin{aligned}\n\\text{intertwining}\n&\\longrightarrow\n\\text{refinement control}\\\\\n&\\longrightarrow\n\\text{statistical sufficiency}\\\\\n&\\longrightarrow\n\\text{memory closure}.\n\\end{aligned}\n\\]\nEach arrow has its own hypotheses and its own obstruction.","figure}{0}\\setcounter{table}{0}\n\\renewcommand{\\thesection}{R\\arabic{section}}\n\\renewcommand{\\theHsection}{R.\\arabic{section}}\n\\renewcommand{\\thefigure}{R\\arabic{figure}}\n\\renewcommand{\\theHfigure}{R.\\arabic{figure}}\n\\renewcommand{\\thetable}{R\\arabic{table}}\n\\renewcommand{\\theHtable}{R.\\arabic{table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theor…","renewcommand{\\theHsection}{R.\\arabic{section}}\n\\renewcommand{\\thefigure}{R\\arabic{figure}}\n\\renewcommand{\\theHfigure}{R.\\arabic{figure}}\n\\renewcommand{\\thetable}{R\\arabic{table}}\n\\renewcommand{\\theHtable}{R.\\arabic{table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_conv…","ction}{R.\\arabic{section}}\n\\renewcommand{\\thefigure}{R\\arabic{figure}}\n\\renewcommand{\\theHfigure}{R.\\arabic{figure}}\n\\renewcommand{\\thetable}{R\\arabic{table}}\n\\renewcommand{\\theHtable}{R.\\arabic{table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s…","section}}\n\\renewcommand{\\thefigure}{R\\arabic{figure}}\n\\renewcommand{\\theHfigure}{R.\\arabic{figure}}\n\\renewcommand{\\thetable}{R\\arabic{table}}\n\\renewcommand{\\theHtable}{R.\\arabic{table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all…","\\arabic{figure}}\n\\renewcommand{\\theHfigure}{R.\\arabic{figure}}\n\\renewcommand{\\thetable}{R\\arabic{table}}\n\\renewcommand{\\theHtable}{R.\\arabic{table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{…","e}}\n\\renewcommand{\\theHfigure}{R.\\arabic{figure}}\n\\renewcommand{\\thetable}{R\\arabic{table}}\n\\renewcommand{\\theHtable}{R.\\arabic{table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alp…","ure}{R.\\arabic{figure}}\n\\renewcommand{\\thetable}{R\\arabic{table}}\n\\renewcommand{\\theHtable}{R.\\arabic{table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in…","table}}\n\\FloatBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blo…","tBarrier\n\\section{Transfer paths and product-topology limits}\\label{earlier:section:01}\nLet $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t…","ace $X$, with generator $A$, and suppose $\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t)=\\sum_{\\alpha,\\beta\\in I_0}T_{\\beta\\alpha}(t).\n\\]\n\\end{theorem}","\\norm{T(t)}\\leq M e^{\\omega t}$ for $M\\geq1$ and $\\omega\\in\\mathbb R$. Let $(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t)=\\sum_{\\alpha,\\beta\\in I_0}T_{\\beta\\alpha}(t).\n\\]\n\\end{theorem}","(P_\\alpha)_{\\alpha\\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\\beta\\alpha}(t)=P_\\beta T(t)P_\\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t)=\\sum_{\\alpha,\\beta\\in I_0}T_{\\beta\\alpha}(t).\n\\]\n\\end{theorem}","ng resolvent statement the integral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t)=\\sum_{\\alpha,\\beta\\in I_0}T_{\\beta\\alpha}(t).\n\\]\n\\end{theorem}","tegral is strong: apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t)=\\sum_{\\alpha,\\beta\\in I_0}T_{\\beta\\alpha}(t).\n\\]\n\\end{theorem}","apply the integrand to each $x\\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\\mathcal B(X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t)=\\sum_{\\alpha,\\beta\\in I_0}T_{\\beta\\alpha}(t).\n\\]\n\\end{theorem}","X)$. The general Laplace formula is \\cite[Chapter II, Theorem 1.10]{EngelNagel2000}.\n\\begin{theorem}[Packet convolution]\\label{earlier:record:001}\\label{earlier:thm:packet_convolution}\nFor all \\(s,t\\geq0\\) and all packets \\(\\alpha,\\beta\\),\n\\[\n\\boxed{\nT_{\\beta\\alpha}(s+t)\n=\n\\sum_{\\gamma\\in I_0}\nT_{\\beta\\gamma}(s)T_{\\gamma\\alpha}(t).\n}\n\\]\nThe original semigroup is reconstructed from its blocks by\n\\[\nT(t)=\\sum_{\\alpha,\\beta\\in I_0}T_{\\beta\\alpha}(t).\n\\]\n\\end{theorem}","\\begin{proof}\nInsert the identity between the two semigroup factors:\n\\[\n\\begin{aligned}\nP_\\beta T(s+t)P_\\alpha\n&=P_\\beta T(s)\\left(\\sum_\\gamma P_\\gamma\\right)T(t)P_\\alpha\\\\\n&=\\sum_\\gamma\nP_\\beta T(s)P_\\gamma T(t)P_\\alpha.\n\\end{aligned}\n\\]\nThe reconstruction formula follows by inserting the packet identity on both sides of \\(T(t)\\).\n\\end{proof}","&=\\sum_\\gamma\nP_\\beta T(s)P_\\gamma T(t)P_\\alpha.\n\\end{aligned}\n\\]\nThe reconstruction formula follows by inserting the packet identity on both sides of \\(T(t)\\).\n\\end{proof}","\\begin{proposition}[Packet resolvent convolution]\\label{earlier:record:002}\\label{earlier:prop:packet_resolvent}\nLet \\(z\\) belong to the resolvent set of \\(A\\). If \\(\\operatorname{Re}z\\) is larger than a growth bound for the semigroup, then\n\\[\nP_\\beta(zI-A)^{-1}P_\\alpha\n=\n\\int_0^{\\infty}\\e^{-zt}T_{\\beta\\alpha}(t)\\dd t.\n\\]\n\\end{proposition}","\\begin{proof}\nThe standard Laplace representation of the resolvent is\n\\[\n(zI-A)^{-1}\n=\n\\int_0^{\\infty}\\e^{-zt}T(t)\\dd t.\n\\]\nMultiplication by the bounded packet projections may be passed through the Bochner integral.\n\\end{proof}","For the path-order statement, now assume $A$ is bounded as in Part II. A directed edge from $\\alpha$ to $\\beta$ means that $P_\\beta A P_\\alpha$ is nonzero, with $\\alpha\\ne\\beta$. A packet always has the length-zero path to itself.\n\\begin{corollary}[Minimum transfer order]\\label{earlier:record:003}\\label{earlier:cor:minimum_transfer_order}\nSuppose every directed packet path from \\(\\alpha\\) to \\(\\beta\\) uses at least \\(m_0\\) off-diagonal transfers. Then\n\\[\nP_\\beta\\e^{tA}P_\\alpha\n=O(t^{m_0})\n\\quad\\text{as }t\\downarrow0.\n\\]\nIf there is no directed path from \\(\\alpha\\) to \\(\\beta\\), the block vanishes for all \\(t\\).\n\\end{corollary}","atement, now assume $A$ is bounded as in Part II. A directed edge from $\\alpha$ to $\\beta$ means that $P_\\beta A P_\\alpha$ is nonzero, with $\\alpha\\ne\\beta$. A packet always has the length-zero path to itself.\n\\begin{corollary}[Minimum transfer order]\\label{earlier:record:003}\\label{earlier:cor:minimum_transfer_order}\nSuppose every directed packet path from \\(\\alpha\\) to \\(\\beta\\) uses at least \\(m_0\\) off-diagonal transfers. Then\n\\[\nP_\\beta\\e^{tA}P_\\alpha\n=O(t^{m_0})\n\\quad\\text{as }t\\downarrow0.\n\\]\nIf there is no directed path from \\(\\alpha\\) to \\(\\beta\\), the block vanishes for all \\(t\\).\n\\end{corollary}","in Part II. A directed edge from $\\alpha$ to $\\beta$ means that $P_\\beta A P_\\alpha$ is nonzero, with $\\alpha\\ne\\beta$. A packet always has the length-zero path to itself.\n\\begin{corollary}[Minimum transfer order]\\label{earlier:record:003}\\label{earlier:cor:minimum_transfer_order}\nSuppose every directed packet path from \\(\\alpha\\) to \\(\\beta\\) uses at least \\(m_0\\) off-diagonal transfers. Then\n\\[\nP_\\beta\\e^{tA}P_\\alpha\n=O(t^{m_0})\n\\quad\\text{as }t\\downarrow0.\n\\]\nIf there is no directed path from \\(\\alpha\\) to \\(\\beta\\), the block vanishes for all \\(t\\).\n\\end{corollary}","\\alpha\\ne\\beta$. A packet always has the length-zero path to itself.\n\\begin{corollary}[Minimum transfer order]\\label{earlier:record:003}\\label{earlier:cor:minimum_transfer_order}\nSuppose every directed packet path from \\(\\alpha\\) to \\(\\beta\\) uses at least \\(m_0\\) off-diagonal transfers. Then\n\\[\nP_\\beta\\e^{tA}P_\\alpha\n=O(t^{m_0})\n\\quad\\text{as }t\\downarrow0.\n\\]\nIf there is no directed path from \\(\\alpha\\) to \\(\\beta\\), the block vanishes for all \\(t\\).\n\\end{corollary}","$. A packet always has the length-zero path to itself.\n\\begin{corollary}[Minimum transfer order]\\label{earlier:record:003}\\label{earlier:cor:minimum_transfer_order}\nSuppose every directed packet path from \\(\\alpha\\) to \\(\\beta\\) uses at least \\(m_0\\) off-diagonal transfers. Then\n\\[\nP_\\beta\\e^{tA}P_\\alpha\n=O(t^{m_0})\n\\quad\\text{as }t\\downarrow0.\n\\]\nIf there is no directed path from \\(\\alpha\\) to \\(\\beta\\), the block vanishes for all \\(t\\).\n\\end{corollary}","path to itself.\n\\begin{corollary}[Minimum transfer order]\\label{earlier:record:003}\\label{earlier:cor:minimum_transfer_order}\nSuppose every directed packet path from \\(\\alpha\\) to \\(\\beta\\) uses at least \\(m_0\\) off-diagonal transfers. Then\n\\[\nP_\\beta\\e^{tA}P_\\alpha\n=O(t^{m_0})\n\\quad\\text{as }t\\downarrow0.\n\\]\nIf there is no directed path from \\(\\alpha\\) to \\(\\beta\\), the block vanishes for all \\(t\\).\n\\end{corollary}","\\begin{proof}\nEvery nonzero term in \\cref{extended:thm:dyson_paths} contains at least \\(m_0\\) time integrations. The simplex \\(0<r_1<\\cdots<r_m<t\\) has volume \\(t^m/m!\\), so the first possible order is \\(t^{m_0}\\). If there is no path, every packet term vanishes.\n\\end{proof}","Let $C_j:X_{j+1}\\to X_j$ be bounded linear maps, and put $C_{\\ell,m}=C_\\ell\\cdots C_{m-1}$ for $\\ell<m$, with $C_{j,j}=I$. No common ambient space is assumed.\n\\begin{theorem}[Multilevel telescoping identity]\\label{earlier:record:004}\\label{earlier:thm:multilevel_telescoping}\nLet \\(T_j\\) be bounded operators on \\(X_j\\), and set\n\\[\nE_j=C_jT_{j+1}-T_jC_j.\n\\]\nThen\n\\[\n\\boxed{\nC_{\\ell,m}T_m-T_\\ell C_{\\ell,m}\n=\n\\sum_{j=\\ell}^{m-1}\nC_{\\ell,j}E_jC_{j+1,m}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nExpand each summand as\n\\[\nC_{\\ell,j+1}T_{j+1}C_{j+1,m}\n-\nC_{\\ell,j}T_jC_{j,m}.\n\\]\nThe intermediate terms cancel. The first surviving term is \\(C_{\\ell,m}T_m\\), and the last is \\(-T_\\ell C_{\\ell,m}\\).\n\\end{proof}","\\begin{corollary}[Summable generator defects]\\label{earlier:record:005}\\label{earlier:cor:summable_defects}\nSuppose \\(C_j\\) are contractions and \\((T_j(t))\\) are contraction semigroups with bounded generators \\(A_j\\). Set\n\\[\n\\delta_j=\\norm{C_jA_{j+1}-A_jC_j}.\n\\]\nThen\n\\[\n\\norm{C_{\\ell,m}T_m(t)-T_\\ell(t)C_{\\ell,m}}\n\\leq\n t\\sum_{j=\\ell}^{m-1}\\delta_j.\n\\]\nIf \\(\\sum_{j\\geq0}\\delta_j<\\infty\\), the right side tends to zero as \\(\\ell\\to\\infty\\), uniformly in \\(m>\\ell\\) and uniformly for \\(t\\) in a fixed bounded interval.\n\\end{corollary}","e generator defects]\\label{earlier:record:005}\\label{earlier:cor:summable_defects}\nSuppose \\(C_j\\) are contractions and \\((T_j(t))\\) are contraction semigroups with bounded generators \\(A_j\\). Set\n\\[\n\\delta_j=\\norm{C_jA_{j+1}-A_jC_j}.\n\\]\nThen\n\\[\n\\norm{C_{\\ell,m}T_m(t)-T_\\ell(t)C_{\\ell,m}}\n\\leq\n t\\sum_{j=\\ell}^{m-1}\\delta_j.\n\\]\nIf \\(\\sum_{j\\geq0}\\delta_j<\\infty\\), the right side tends to zero as \\(\\ell\\to\\infty\\), uniformly in \\(m>\\ell\\) and uniformly for \\(t\\) in a fixed bounded interval.\n\\end{corollary}","er:record:005}\\label{earlier:cor:summable_defects}\nSuppose \\(C_j\\) are contractions and \\((T_j(t))\\) are contraction semigroups with bounded generators \\(A_j\\). Set\n\\[\n\\delta_j=\\norm{C_jA_{j+1}-A_jC_j}.\n\\]\nThen\n\\[\n\\norm{C_{\\ell,m}T_m(t)-T_\\ell(t)C_{\\ell,m}}\n\\leq\n t\\sum_{j=\\ell}^{m-1}\\delta_j.\n\\]\nIf \\(\\sum_{j\\geq0}\\delta_j<\\infty\\), the right side tends to zero as \\(\\ell\\to\\infty\\), uniformly in \\(m>\\ell\\) and uniformly for \\(t\\) in a fixed bounded interval.\n\\end{corollary}","cor:summable_defects}\nSuppose \\(C_j\\) are contractions and \\((T_j(t))\\) are contraction semigroups with bounded generators \\(A_j\\). Set\n\\[\n\\delta_j=\\norm{C_jA_{j+1}-A_jC_j}.\n\\]\nThen\n\\[\n\\norm{C_{\\ell,m}T_m(t)-T_\\ell(t)C_{\\ell,m}}\n\\leq\n t\\sum_{j=\\ell}^{m-1}\\delta_j.\n\\]\nIf \\(\\sum_{j\\geq0}\\delta_j<\\infty\\), the right side tends to zero as \\(\\ell\\to\\infty\\), uniformly in \\(m>\\ell\\) and uniformly for \\(t\\) in a fixed bounded interval.\n\\end{corollary}","\\begin{proof}\nBy \\cref{extended:thm:duhamel_defect}, the interlevel semigroup defect has norm at most \\(t\\delta_j\\). Apply \\cref{earlier:thm:multilevel_telescoping} and use the contraction bounds on all coarsening maps.\n\\end{proof}","\\begin{definition}[Projective limit space]\\label{earlier:record:006}\\label{earlier:def:projective_limit}\nLet \\((X_j,C_j)\\) be a sequence of Banach spaces and bounded linear maps \\(C_j:X_{j+1}\\to X_j\\). Its projective limit is\n\\[\nX_\\infty\n=\n\\left\\{\n(x_j)_{j\\geq0}\\in\\prod_{j\\geq0}X_j:\nC_jx_{j+1}=x_j\\text{ for every }j\n\\right\\}.\n\\]\nWe equip \\(X_\\infty\\) with the subspace topology inherited from the product topology.\n\\end{definition}","\\begin{theorem}[Exact projective limit semigroup]\\label{earlier:record:007}\\label{earlier:thm:projective_limit_semigroup}\nSuppose each \\(X_j\\) carries a strongly continuous semigroup \\((T_j(t))_{t\\geq0}\\), and\n\\[\nC_jT_{j+1}(t)=T_j(t)C_j\n\\]\nfor every \\(j\\) and every \\(t\\geq0\\). Then\n\\[\nT_\\infty(t)(x_j)_{j\\geq0}\n=\n(T_j(t)x_j)_{j\\geq0}\n\\]\ndefines a strongly continuous semigroup on \\(X_\\infty\\).\n\\end{theorem}","\\begin{proof}\nExact intertwining gives\n\\[\nC_jT_{j+1}(t)x_{j+1}\n=\nT_j(t)C_jx_{j+1}\n=\nT_j(t)x_j.\n\\]\nThus the coordinatewise evolution remains in the projective limit. The semigroup law holds in every coordinate. Strong continuity in the product topology follows because every coordinate map\n\\[\nt\\longmapsto T_j(t)x_j\n\\]\nis continuous.\n\\end{proof}","\\begin{corollary}[Coordinate generators]\\label{earlier:record:008}\\label{earlier:cor:projective_generators}\nAssume the setting of \\cref{earlier:thm:projective_limit_semigroup}. If \\(x=(x_j)\\) satisfies \\(x_j\\in\\operatorname{Dom}(A_j)\\) and the sequence \\((A_jx_j)\\) belongs to \\(X_\\infty\\), then \\(x\\) lies in the generator domain of \\(T_\\infty\\) and\n\\[\nA_\\infty x=(A_jx_j)_{j\\geq0}.\n\\]\n\\end{corollary}","\\begin{proof}\nThe difference quotients converge in every coordinate to \\(A_jx_j\\). The assumed compatibility places the coordinatewise limit in \\(X_\\infty\\), which is exactly convergence in the product topology.\n\\end{proof}","The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement.\n\\FloatBarrier\n\\section{Reversible coarse dynamics}\\label{earlier:section:02}\nLet $Q$ be a finite Markov generator acting on column probability vectors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\\one_{i\\in B_a}$. Strong lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q","The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement.\n\\FloatBarrier\n\\section{Reversible coarse dynamics}\\label{earlier:section:02}\nLet $Q$ be a finite Markov generator acting on column probability vectors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\\one_{i\\in B_a}$. Strong lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]","The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement.\n\\FloatBarrier\n\\section{Reversible coarse dynamics}\\label{earlier:section:02}\nLet $Q$ be a finite Markov generator acting on column probability vectors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\\one_{i\\in B_a}$. Strong lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","om the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement.\n\\FloatBarrier\n\\section{Reversible coarse dynamics}\\label{earlier:section:02}\nLet $Q$ be a finite Markov generator acting on column probability vectors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\\one_{i\\in B_a}$. Strong lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","is required by the coordinatewise statement.\n\\FloatBarrier\n\\section{Reversible coarse dynamics}\\label{earlier:section:02}\nLet $Q$ be a finite Markov generator acting on column probability vectors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\\one_{i\\in B_a}$. Strong lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","the coordinatewise statement.\n\\FloatBarrier\n\\section{Reversible coarse dynamics}\\label{earlier:section:02}\nLet $Q$ be a finite Markov generator acting on column probability vectors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\\one_{i\\in B_a}$. Strong lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","tors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\\one_{i\\in B_a}$. Strong lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","ng lumpability means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","ty means $CQ=\\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","se Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","rd generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","2$ inner products use the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","the stated stationary probability.\n\\begin{theorem}[Reversibility descends]\\label{earlier:record:009}\\label{earlier:thm:reversibility_descends}\nAssume \\(Q\\) is strongly lumpable and reversible with respect to a positive stationary distribution \\(\\pi\\), so\n\\[\nQ_{ij}\\pi_j=Q_{ji}\\pi_i.\n\\]\nDefine\n\\[\n\\overline\\pi_a=\\sum_{i\\in B_a}\\pi_i.\n\\]\nThen \\(\\overline\\pi\\) is stationary for \\(\\overline Q\\), and\n\\[\n\\overline Q_{ab}\\overline\\pi_b\n=\n\\overline Q_{ba}\\overline\\pi_a.\n\\]\nThus the coarse chain is reversible.\n\\end{theorem}","\\begin{proof}\nStationarity follows from\n\\[\n\\overline Q\\overline\\pi\n=\\overline QC\\pi\n=CQ\\pi=0.\n\\]\nFor reversibility, use the aggregate rate formula:\n\\[\n\\begin{aligned}\n\\overline Q_{ab}\\overline\\pi_b\n&=\\sum_{j\\in B_b}\\pi_j\\sum_{i\\in B_a}Q_{ij}","\\begin{proof}\nStationarity follows from\n\\[\n\\overline Q\\overline\\pi\n=\\overline QC\\pi\n=CQ\\pi=0.\n\\]\nFor reversibility, use the aggregate rate formula:\n\\[\n\\begin{aligned}\n\\overline Q_{ab}\\overline\\pi_b\n&=\\sum_{j\\in B_b}\\pi_j\\sum_{i\\in B_a}Q_{ij}\\\\\n&=\\sum_{i\\in B_a}\\sum_{j\\in B_b}\\pi_iQ_{ji}\\\\\n&=\\overline Q_{ba}\\overline\\pi_a.\n\\end{aligned}\n\\]\n\\end{proof}","\\begin{corollary}[Coarse Dirichlet form]\\label{earlier:record:010}\\label{earlier:cor:coarse_dirichlet_form}\nUnder the hypotheses of \\cref{earlier:thm:reversibility_descends}, let \\(Jg\\) be the fine function that is constant with value \\(g_a\\) on \\(B_a\\). Then\n\\[\n-\\ip{Jg}{Q^TJg}_{L^2(\\pi)}\n=\n-\\ip{g}{\\overline Q^Tg}_{L^2(\\overline\\pi)}.\n\\]\n\\end{corollary}","\\FloatBarrier\n\\section{Prior-restricted sufficiency and filtering}\\label{earlier:section:03}\nLet $L$ be a strictly positive finite likelihood and let $\\Bcal_L(p)=L\\odot p/\\ip{L}{p}$ for a probability vector $p$. Fix probability vectors $r^{(a)}$ supported on $B_a$, and restrict the priors to $p_i=q_a r_i^{(a)}$ for $i\\in B_a$, with $q$ a coarse probability vector.\n\\begin{proposition}[Sufficiency relative to a prior disintegration]\\label{earlier:record:011}\\label{earlier:prop:relative_sufficiency}\nDefine\n\\[\n\\overline L_a\n=\n\\sum_{i\\in B_a}L_ir_i^{(a)}.\n\\]\nFor every prior in the declared family,\n\\[\nC\\Bcal_L(p)\n=\n\\Bcal_{\\overline L}(q).\n\\]\n\\end{proposition}","tly positive finite likelihood and let $\\Bcal_L(p)=L\\odot p/\\ip{L}{p}$ for a probability vector $p$. Fix probability vectors $r^{(a)}$ supported on $B_a$, and restrict the priors to $p_i=q_a r_i^{(a)}$ for $i\\in B_a$, with $q$ a coarse probability vector.\n\\begin{proposition}[Sufficiency relative to a prior disintegration]\\label{earlier:record:011}\\label{earlier:prop:relative_sufficiency}\nDefine\n\\[\n\\overline L_a\n=\n\\sum_{i\\in B_a}L_ir_i^{(a)}.\n\\]\nFor every prior in the declared family,\n\\[\nC\\Bcal_L(p)\n=\n\\Bcal_{\\overline L}(q).\n\\]\n\\end{proposition}","ctor $p$. Fix probability vectors $r^{(a)}$ supported on $B_a$, and restrict the priors to $p_i=q_a r_i^{(a)}$ for $i\\in B_a$, with $q$ a coarse probability vector.\n\\begin{proposition}[Sufficiency relative to a prior disintegration]\\label{earlier:record:011}\\label{earlier:prop:relative_sufficiency}\nDefine\n\\[\n\\overline L_a\n=\n\\sum_{i\\in B_a}L_ir_i^{(a)}.\n\\]\nFor every prior in the declared family,\n\\[\nC\\Bcal_L(p)\n=\n\\Bcal_{\\overline L}(q).\n\\]\n\\end{proposition}","\\begin{proof}\nThe evidence in block \\(B_a\\) is\n\\[\nZ_a\n=q_a\\sum_{i\\in B_a}L_ir_i^{(a)}\n=q_a\\overline L_a.\n\\]\nApply \\cref{extended:thm:bayesian_packet_reconstruction}.\n\\end{proof}","\\begin{lemma}[Normalization bound]\\label{earlier:record:012}\\label{earlier:lem:normalization_bound}\nLet \\(a,b\\in[0,\\infty)^m\\) be nonzero, and set \\(A=\\norm a_1\\) and \\(B=\\norm b_1\\). Then\n\\[\n\\boxed{\n\\norm{a/A-b/B}_1\n\\leq\n\\frac{2\\norm{a-b}_1}{\\max\\{A,B\\}}.\n}\n\\]\n\\end{lemma}","\\begin{proof}\nUsing \\(A\\) in the denominator gives\n\\[\n\\norm{a/A-b/B}_1\n\\leq\n\\frac{\\norm{a-b}_1}{A}\n+\n\\norm b_1\\abs{1/A-1/B}.\n\\]\nThe second term equals \\(\\abs{B-A}/A\\), which is at most \\(\\norm{a-b}_1/A\\). Interchanging \\(a\\) and \\(b\\) gives the analogous bound with \\(B\\). Taking the smaller of the two bounds gives the result.\n\\end{proof}","For an arbitrary prior $p$, put $q=Cp$ and $Z(p,L)=\\ip{L}{p}$. Define $\\ell_a(p)=\\sum_{i\\in B_a}L_ip_i/q_a$ when $q_a>0$. Choose any positive value when $q_a=0$. Only the product $q_a\\ell_a(p)$ enters the formulas.\n\\begin{theorem}[Exact information loss formula]\\label{earlier:record:013}\\label{earlier:thm:information_loss}\nFor every prior,\n\\[\n\\boxed{\nC\\Bcal_L(p)\n=\n\\Bcal_{\\ell(p)}(q).\n}\n\\]\nLet \\(\\overline L\\) be any fixed positive coarse likelihood and set\n\\[\nd(p\\mid L,\\overline L)\n=\n\\sum_{a=1}^mq_a\\abs{\\ell_a(p)-\\overline L_a}.\n\\]\nThen\n\\[\n\\norm{\nC\\Bcal_L(p)-\\Bcal_{\\overline L}(q)\n}_1\n\\leq\n\\frac{2d(p\\mid L,\\overline L)}\n{\\max\\{Z(p,L),\\ip{\\overline L}{q}\\}}.\n\\]\n\\end{theorem}","Define $\\ell_a(p)=\\sum_{i\\in B_a}L_ip_i/q_a$ when $q_a>0$. Choose any positive value when $q_a=0$. Only the product $q_a\\ell_a(p)$ enters the formulas.\n\\begin{theorem}[Exact information loss formula]\\label{earlier:record:013}\\label{earlier:thm:information_loss}\nFor every prior,\n\\[\n\\boxed{\nC\\Bcal_L(p)\n=\n\\Bcal_{\\ell(p)}(q).\n}\n\\]\nLet \\(\\overline L\\) be any fixed positive coarse likelihood and set\n\\[\nd(p\\mid L,\\overline L)\n=\n\\sum_{a=1}^mq_a\\abs{\\ell_a(p)-\\overline L_a}.\n\\]\nThen\n\\[\n\\norm{\nC\\Bcal_L(p)-\\Bcal_{\\overline L}(q)\n}_1\n\\leq\n\\frac{2d(p\\mid L,\\overline L)}\n{\\max\\{Z(p,L),\\ip{\\overline L}{q}\\}}.\n\\]\n\\end{theorem}","lue when $q_a=0$. Only the product $q_a\\ell_a(p)$ enters the formulas.\n\\begin{theorem}[Exact information loss formula]\\label{earlier:record:013}\\label{earlier:thm:information_loss}\nFor every prior,\n\\[\n\\boxed{\nC\\Bcal_L(p)\n=\n\\Bcal_{\\ell(p)}(q).\n}\n\\]\nLet \\(\\overline L\\) be any fixed positive coarse likelihood and set\n\\[\nd(p\\mid L,\\overline L)\n=\n\\sum_{a=1}^mq_a\\abs{\\ell_a(p)-\\overline L_a}.\n\\]\nThen\n\\[\n\\norm{\nC\\Bcal_L(p)-\\Bcal_{\\overline L}(q)\n}_1\n\\leq\n\\frac{2d(p\\mid L,\\overline L)}\n{\\max\\{Z(p,L),\\ip{\\overline L}{q}\\}}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe packet evidence in block \\(a\\) is \\(q_a\\ell_a(p)\\), which proves the exact identity. Apply \\cref{earlier:lem:normalization_bound} to\n\\[\na=q\\odot\\ell(p),\n\\qquad\nb=q\\odot\\overline L.\n\\]\nTheir \\(\\ell^1\\) norms are the two evidences, and \\(\\norm{a-b}_1=d(p\\mid L,\\overline L)\\).\n\\end{proof}","\\begin{corollary}[Oscillation bound]\\label{earlier:record:014}\\label{earlier:cor:bayes_oscillation}\nSuppose \\(\\overline L_a\\) is any representative likelihood for block \\(B_a\\), and set\n\\[\n\\omega_a\n=\n\\max_{i\\in B_a}\\abs{L_i-\\overline L_a}.\n\\]\nThen\n\\[\nd(p\\mid L,\\overline L)\n\\leq\n\\sum_aq_a\\omega_a.\n\\]\nThus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.\n\\end{corollary}","\\begin{proof}\nThe conditional average \\(\\ell_a(p)\\) lies in the convex hull of the likelihood values inside \\(B_a\\). Hence\n\\[\n\\abs{\\ell_a(p)-\\overline L_a}\\leq\\omega_a.\n\\]\nMultiply by \\(q_a\\) and sum.\n\\end{proof}","Along the Bayesian flow $p_i(t)=e^{tg_i}p_i(0)/\\sum_j e^{tg_j}p_j(0)$, the following statement holds for every real observable $h$.\n\\begin{corollary}[Covariance identity]\\label{earlier:record:015}\\label{earlier:cor:bayesian_covariance}\nFor any observable \\(h\\in\\RR^n\\),\n\\[\n\\frac{\\dd}{\\dd t}\\EE_{p(t)}[h]\n=\n\\Cov_{p(t)}(h,g).\n\\]\nIn particular,\n\\[\n\\frac{\\dd}{\\dd t}\\EE_{p(t)}[g]\n=\n\\Var_{p(t)}(g)\\geq0.\n\\]\n\\end{corollary}","\\begin{proof}\nDifferentiate \\(\\sum_ih_ip_i(t)\\) and insert the replicator equation.\n\\end{proof}","For stochastic matrices $K,\\overline K$ and positive likelihoods $L_y,\\overline L_y$, put $\\Phi_y=\\Bcal_{L_y}\\circ K$ and $\\overline\\Phi_y=\\Bcal_{\\overline L_y}\\circ\\overline K$.\n\\begin{theorem}[Exact coarse hidden Markov filter]\\label{earlier:record:016}\\label{earlier:thm:exact_coarse_filter}\nAssume\n\\[\nCK=\\overline KC\n\\]\nand every likelihood \\(L_y\\) is constant on each coarse block with block values \\(\\overline L_y\\). Then\n\\[\nC\\Phi_y(p)=\\overline\\Phi_y(Cp)\n\\]\nfor every prior. For any observation sequence \\(y_1,\\ldots,y_r\\), the complete filtering recursion commutes with aggregation:\n\\[\nC\\Phi_{y_r}\\circ\\cdots\\circ\\Phi_{y_1}\n=\n\\overline\\Phi_{y_r}\\circ\\cdots\\circ\\overline\\Phi_{y_1}C.\n\\]\n\\end{theorem}","ochastic matrices $K,\\overline K$ and positive likelihoods $L_y,\\overline L_y$, put $\\Phi_y=\\Bcal_{L_y}\\circ K$ and $\\overline\\Phi_y=\\Bcal_{\\overline L_y}\\circ\\overline K$.\n\\begin{theorem}[Exact coarse hidden Markov filter]\\label{earlier:record:016}\\label{earlier:thm:exact_coarse_filter}\nAssume\n\\[\nCK=\\overline KC\n\\]\nand every likelihood \\(L_y\\) is constant on each coarse block with block values \\(\\overline L_y\\). Then\n\\[\nC\\Phi_y(p)=\\overline\\Phi_y(Cp)\n\\]\nfor every prior. For any observation sequence \\(y_1,\\ldots,y_r\\), the complete filtering recursion commutes with aggregation:\n\\[\nC\\Phi_{y_r}\\circ\\cdots\\circ\\Phi_{y_1}\n=\n\\overline\\Phi_{y_r}\\circ\\cdots\\circ\\overline\\Phi_{y_1}C.\n\\]\n\\end{theorem}","erline K$ and positive likelihoods $L_y,\\overline L_y$, put $\\Phi_y=\\Bcal_{L_y}\\circ K$ and $\\overline\\Phi_y=\\Bcal_{\\overline L_y}\\circ\\overline K$.\n\\begin{theorem}[Exact coarse hidden Markov filter]\\label{earlier:record:016}\\label{earlier:thm:exact_coarse_filter}\nAssume\n\\[\nCK=\\overline KC\n\\]\nand every likelihood \\(L_y\\) is constant on each coarse block with block values \\(\\overline L_y\\). Then\n\\[\nC\\Phi_y(p)=\\overline\\Phi_y(Cp)\n\\]\nfor every prior. For any observation sequence \\(y_1,\\ldots,y_r\\), the complete filtering recursion commutes with aggregation:\n\\[\nC\\Phi_{y_r}\\circ\\cdots\\circ\\Phi_{y_1}\n=\n\\overline\\Phi_{y_r}\\circ\\cdots\\circ\\overline\\Phi_{y_1}C.\n\\]\n\\end{theorem}","c K$ and $\\overline\\Phi_y=\\Bcal_{\\overline L_y}\\circ\\overline K$.\n\\begin{theorem}[Exact coarse hidden Markov filter]\\label{earlier:record:016}\\label{earlier:thm:exact_coarse_filter}\nAssume\n\\[\nCK=\\overline KC\n\\]\nand every likelihood \\(L_y\\) is constant on each coarse block with block values \\(\\overline L_y\\). Then\n\\[\nC\\Phi_y(p)=\\overline\\Phi_y(Cp)\n\\]\nfor every prior. For any observation sequence \\(y_1,\\ldots,y_r\\), the complete filtering recursion commutes with aggregation:\n\\[\nC\\Phi_{y_r}\\circ\\cdots\\circ\\Phi_{y_1}\n=\n\\overline\\Phi_{y_r}\\circ\\cdots\\circ\\overline\\Phi_{y_1}C.\n\\]\n\\end{theorem}","}[Exact coarse hidden Markov filter]\\label{earlier:record:016}\\label{earlier:thm:exact_coarse_filter}\nAssume\n\\[\nCK=\\overline KC\n\\]\nand every likelihood \\(L_y\\) is constant on each coarse block with block values \\(\\overline L_y\\). Then\n\\[\nC\\Phi_y(p)=\\overline\\Phi_y(Cp)\n\\]\nfor every prior. For any observation sequence \\(y_1,\\ldots,y_r\\), the complete filtering recursion commutes with aggregation:\n\\[\nC\\Phi_{y_r}\\circ\\cdots\\circ\\Phi_{y_1}\n=\n\\overline\\Phi_{y_r}\\circ\\cdots\\circ\\overline\\Phi_{y_1}C.\n\\]\n\\end{theorem}","\\begin{proof}\nExact Markov lumping gives \\(CKp=\\overline KCp\\). The Bayesian sufficiency criterion gives exact commutation of the update. Their composition proves the one step identity. Iterate it over the observation sequence.\n\\end{proof}","\\begin{theorem}[One step filter defect]\\label{earlier:record:017}\\label{earlier:thm:filter_defect}\nLet \\(r=Kp\\), \\(q=Cr\\), and let \\(\\ell_y(r)\\) be the exact coarse likelihood obtained by averaging \\(L_y\\) inside each block with respect to \\(r\\). Assume\n\\[\n0<\\overline m\\leq(\\overline L_y)_a\\leq\\overline M.\n\\]\nThen\n\\[\n\\begin{aligned}\n\\norm{C\\Phi_y(p)-\\overline\\Phi_y(Cp)}_1\n\\leq{}&\n\\frac{2d(r\\mid L_y,\\overline L_y)}\n{\\max\\{\\ip{L_y}{r},\\ip{\\overline L_y}{q}\\}}","\\begin{theorem}[One step filter defect]\\label{earlier:record:017}\\label{earlier:thm:filter_defect}\nLet \\(r=Kp\\), \\(q=Cr\\), and let \\(\\ell_y(r)\\) be the exact coarse likelihood obtained by averaging \\(L_y\\) inside each block with respect to \\(r\\). Assume\n\\[\n0<\\overline m\\leq(\\overline L_y)_a\\leq\\overline M.\n\\]\nThen\n\\[\n\\begin{aligned}\n\\norm{C\\Phi_y(p)-\\overline\\Phi_y(Cp)}_1\n\\leq{}&\n\\frac{2d(r\\mid L_y,\\overline L_y)}\n{\\max\\{\\ip{L_y}{r},\\ip{\\overline L_y}{q}\\}}\\\\\n&+\n\\frac{2\\overline M}{\\overline m}\n\\norm{CKp-\\overline KCp}_1.\n\\end{aligned}\n\\]\n\\end{theorem}","\\begin{proof}\nInsert the intermediate posterior \\(\\Bcal_{\\overline L_y}(CKp)\\). The first difference is bounded by \\cref{earlier:thm:information_loss}. The second is bounded by \\cref{extended:thm:prior_stability} applied to the coarse likelihood.\n\\end{proof}","\\FloatBarrier\n\\section{Fractional modes and killed transfer}\\label{earlier:section:04}\nFor the one-dimensional Dirichlet difference operator, let $h=1/M$ with an integer $M\\geq2$, let $1\\leq k\\leq M-1$, and set $\\lambda_{k,h}=4h^{-2}\\sin^2(k\\pi h/2)$. The retained explicit coefficient below is valid for every discrete mode.\n\\begin{corollary}[Fractional low mode dispersion]\\label{earlier:record:018}\\label{earlier:cor:fractional_low_mode_dispersion}\nLet \\(0<\\alpha\\leq1\\). Then\n\\[\n0\n\\leq\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\frac{\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\nFor every \\(t\\geq0\\),\n\\[\n\\abs{\n\\e^{-t\\lambda_{k,h}^\\alpha}\n-\n\\e^{-t(k\\pi)^{2\\alpha}}\n}\n\\leq\n\\frac{t\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\n\\end{corollary}","r}\\label{earlier:section:04}\nFor the one-dimensional Dirichlet difference operator, let $h=1/M$ with an integer $M\\geq2$, let $1\\leq k\\leq M-1$, and set $\\lambda_{k,h}=4h^{-2}\\sin^2(k\\pi h/2)$. The retained explicit coefficient below is valid for every discrete mode.\n\\begin{corollary}[Fractional low mode dispersion]\\label{earlier:record:018}\\label{earlier:cor:fractional_low_mode_dispersion}\nLet \\(0<\\alpha\\leq1\\). Then\n\\[\n0\n\\leq\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\frac{\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\nFor every \\(t\\geq0\\),\n\\[\n\\abs{\n\\e^{-t\\lambda_{k,h}^\\alpha}\n-\n\\e^{-t(k\\pi)^{2\\alpha}}\n}\n\\leq\n\\frac{t\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\n\\end{corollary}",":04}\nFor the one-dimensional Dirichlet difference operator, let $h=1/M$ with an integer $M\\geq2$, let $1\\leq k\\leq M-1$, and set $\\lambda_{k,h}=4h^{-2}\\sin^2(k\\pi h/2)$. The retained explicit coefficient below is valid for every discrete mode.\n\\begin{corollary}[Fractional low mode dispersion]\\label{earlier:record:018}\\label{earlier:cor:fractional_low_mode_dispersion}\nLet \\(0<\\alpha\\leq1\\). Then\n\\[\n0\n\\leq\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\frac{\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\nFor every \\(t\\geq0\\),\n\\[\n\\abs{\n\\e^{-t\\lambda_{k,h}^\\alpha}\n-\n\\e^{-t(k\\pi)^{2\\alpha}}\n}\n\\leq\n\\frac{t\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\n\\end{corollary}","q M-1$, and set $\\lambda_{k,h}=4h^{-2}\\sin^2(k\\pi h/2)$. The retained explicit coefficient below is valid for every discrete mode.\n\\begin{corollary}[Fractional low mode dispersion]\\label{earlier:record:018}\\label{earlier:cor:fractional_low_mode_dispersion}\nLet \\(0<\\alpha\\leq1\\). Then\n\\[\n0\n\\leq\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\frac{\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\nFor every \\(t\\geq0\\),\n\\[\n\\abs{\n\\e^{-t\\lambda_{k,h}^\\alpha}\n-\n\\e^{-t(k\\pi)^{2\\alpha}}\n}\n\\leq\n\\frac{t\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\n\\end{corollary}","set $\\lambda_{k,h}=4h^{-2}\\sin^2(k\\pi h/2)$. The retained explicit coefficient below is valid for every discrete mode.\n\\begin{corollary}[Fractional low mode dispersion]\\label{earlier:record:018}\\label{earlier:cor:fractional_low_mode_dispersion}\nLet \\(0<\\alpha\\leq1\\). Then\n\\[\n0\n\\leq\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\frac{\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\nFor every \\(t\\geq0\\),\n\\[\n\\abs{\n\\e^{-t\\lambda_{k,h}^\\alpha}\n-\n\\e^{-t(k\\pi)^{2\\alpha}}\n}\n\\leq\n\\frac{t\\alpha(4k^2)^{\\alpha-1}(k\\pi)^4}{12}h^2.\n\\]\n\\end{corollary}","\\begin{proof}\nThe chord bound \\(\\sin x\\geq2x/\\pi\\) on \\([0,\\pi/2]\\) gives \\(\\lambda_{k,h}\\geq4k^2\\). Since \\(x\\mapsto x^\\alpha\\) is increasing and has decreasing derivative,\n\\[\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\alpha\\lambda_{k,h}^{\\alpha-1}\n\\bigl((k\\pi)^2-\\lambda_{k,h}\\bigr).\n\\]\nUse \\cref{extended:thm:low_mode} and the lower bound for \\(\\lambda_{k,h}\\). The exponential estimate follows from\n\\[\n\\abs{\\e^{-ta}-\\e^{-tb}}\n\\leq t\\abs{a-b}\n\\]\nfor nonnegative \\(a\\) and \\(b\\).\n\\end{proof}","{proof}\nThe chord bound \\(\\sin x\\geq2x/\\pi\\) on \\([0,\\pi/2]\\) gives \\(\\lambda_{k,h}\\geq4k^2\\). Since \\(x\\mapsto x^\\alpha\\) is increasing and has decreasing derivative,\n\\[\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\alpha\\lambda_{k,h}^{\\alpha-1}\n\\bigl((k\\pi)^2-\\lambda_{k,h}\\bigr).\n\\]\nUse \\cref{extended:thm:low_mode} and the lower bound for \\(\\lambda_{k,h}\\). The exponential estimate follows from\n\\[\n\\abs{\\e^{-ta}-\\e^{-tb}}\n\\leq t\\abs{a-b}\n\\]\nfor nonnegative \\(a\\) and \\(b\\).\n\\end{proof}","gives \\(\\lambda_{k,h}\\geq4k^2\\). Since \\(x\\mapsto x^\\alpha\\) is increasing and has decreasing derivative,\n\\[\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\alpha\\lambda_{k,h}^{\\alpha-1}\n\\bigl((k\\pi)^2-\\lambda_{k,h}\\bigr).\n\\]\nUse \\cref{extended:thm:low_mode} and the lower bound for \\(\\lambda_{k,h}\\). The exponential estimate follows from\n\\[\n\\abs{\\e^{-ta}-\\e^{-tb}}\n\\leq t\\abs{a-b}\n\\]\nfor nonnegative \\(a\\) and \\(b\\).\n\\end{proof}","ambda_{k,h}\\geq4k^2\\). Since \\(x\\mapsto x^\\alpha\\) is increasing and has decreasing derivative,\n\\[\n(k\\pi)^{2\\alpha}-\\lambda_{k,h}^\\alpha\n\\leq\n\\alpha\\lambda_{k,h}^{\\alpha-1}\n\\bigl((k\\pi)^2-\\lambda_{k,h}\\bigr).\n\\]\nUse \\cref{extended:thm:low_mode} and the lower bound for \\(\\lambda_{k,h}\\). The exponential estimate follows from\n\\[\n\\abs{\\e^{-ta}-\\e^{-tb}}\n\\leq t\\abs{a-b}\n\\]\nfor nonnegative \\(a\\) and \\(b\\).\n\\end{proof}","Let $Q$ and $\\overline Q$ be finite Markov generators in the column convention. All times in this section satisfy $t\\geq0$.\n\\begin{theorem}[Potential perturbation bound]\\label{earlier:record:019}\\label{earlier:thm:feynman_kac_potential_bound}\nLet \\(V\\) and \\(W\\) be nonnegative diagonal matrices. In the supremum norm,\n\\[\n\\boxed{\n\\norm{\n\\e^{t(Q^T-V)}\n-\n\\e^{t(Q^T-W)}\n}_{\\infty\\to\\infty}\n\\leq\n t\\norm{V-W}_{\\infty\\to\\infty}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nBoth killed Markov semigroups are contractions in the supremum norm. Apply the Duhamel identity with generator defect \\(W-V\\).\n\\end{proof}","\\begin{theorem}[Coarse Feynman--Kac transfer]\\label{earlier:record:020}\\label{earlier:thm:coarse_feynman_kac}\nAssume \\(CQ=\\overline QC\\), and let \\(J=C^T\\) lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so\n\\[\nVJ=J\\overline V.\n\\]\nThen\n\\[\n\\boxed{\n\\e^{t(Q^T-V)}J\n=\nJ\\e^{t(\\overline Q^T-\\overline V)}\n}\n\\]\nfor every \\(t\\geq0\\).\n\\end{theorem}","\\begin{proof}\nTranspose the Markov intertwining relation to obtain\n\\[\nQ^TJ=J\\overline Q^T.\n\\]\nTogether with \\(VJ=J\\overline V\\), this gives\n\\[\n(Q^T-V)J=J(\\overline Q^T-\\overline V).\n\\]\nThe exponential intertwining follows from the power series.\n\\end{proof}","\\FloatBarrier\n\\section{Memory compression, rational realization, and phase-type consequences}\\label{earlier:section:05}\nFix $T>0$. For the next estimate let $A$ be a bounded operator on a Banach space $X$, let $\\Kcal,\\widetilde\\Kcal\\in L^1((0,T),\\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same","\\FloatBarrier\n\\section{Memory compression, rational realization, and phase-type consequences}\\label{earlier:section:05}\nFix $T>0$. For the next estimate let $A$ be a bounded operator on a Banach space $X$, let $\\Kcal,\\widetilde\\Kcal\\in L^1((0,T),\\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and","\\FloatBarrier\n\\section{Memory compression, rational realization, and phase-type consequences}\\label{earlier:section:05}\nFix $T>0$. For the next estimate let $A$ be a bounded operator on a Banach space $X$, let $\\Kcal,\\widetilde\\Kcal\\in L^1((0,T),\\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\wideti","\\FloatBarrier\n\\section{Memory compression, rational realization, and phase-type consequences}\\label{earlier:section:05}\nFix $T>0$. For the next estimate let $A$ be a bounded operator on a Banach space $X$, let $\\Kcal,\\widetilde\\Kcal\\in L^1((0,T),\\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kca","\\FloatBarrier\n\\section{Memory compression, rational realization, and phase-type consequences}\\label{earlier:section:05}\nFix $T>0$. For the next estimate let $A$ be a bounded operator on a Banach space $X$, let $\\Kcal,\\widetilde\\Kcal\\in L^1((0,T),\\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)}","etilde\\Kcal\\in L^1((0,T),\\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","l\\in L^1((0,T),\\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","entiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","eneral mild-semigroup estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","p estimate in Part II.\n\\begin{theorem}[Stability of memory compression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","ompression]\\label{earlier:record:021}\\label{earlier:thm:memory_compression_stability}\nLet \\(x\\) and \\(\\widetilde x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","e x\\) solve\n\\[\n\\dot x(t)\n=Ax(t)+\\int_0^t\\Kcal(t-s)x(s)\\dd s+f(t)\n\\]\nand\n\\[\n\\dot{\\widetilde x}(t)\n=A\\widetilde x(t)\n+\\int_0^t\\widetilde\\Kcal(t-s)\\widetilde x(s)\\dd s\n+f(t)\n\\]\nwith the same initial value. Fix \\(T>0\\), and assume\n\\[\n\\sup_{0\\leq t\\leq T}\\norm{\\widetilde x(t)}\\leq M.\n\\]\nSet\n\\[\na=\\norm A,\n\\qquad\nk=\\norm{\\Kcal}_{L^1(0,T)},\n\\qquad\n\\delta=\\norm{\\Kcal-\\widetilde\\Kcal}_{L^1(0,T)}.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{0\\leq t\\leq T}\n\\norm{x(t)-\\widetilde x(t)}\n\\leq\nMT\\delta\\e^{(a+k)T}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLet \\(e=x-\\widetilde x\\). Integrating the difference equation and changing the order of integration gives\n\\[\n\\norm{e(t)}\n\\leq\n(a+k)\\int_0^t\\norm{e(r)}\\dd r\n+Mt\\delta.\n\\]\nSince \\(t\\leq T\\), Gronwall's inequality gives\n\\[\n\\norm{e(t)}\n\\leq\nMT\\delta\\e^{(a+k)t}.\n\\]\nTake the supremum over \\([0,T]\\).\n\\end{proof}","For the exact closure criterion of Part II, allowing an arbitrary bounded local generator $B$ does not enlarge the class: differentiating $P e^{tA}P=e^{tB}P$ at $t=0$ forces $B=A_{PP}$ on the retained space. Thus its zero-memory criterion excludes every alternative bounded local generator.","A finite auxiliary realization of a matrix kernel means $\\Kcal(t)=B e^{tD}C$, with finite matrices of compatible sizes. Laplace transforms are taken in a right half-plane of absolute convergence. The rational characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies.\n\\begin{theorem}[Rational memory realization]\\label{earlier:record:022}\\label{earlier:thm:rational_memory_realization}\nA matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely,\n\\[\n\\boxed{\n\\widehat\\Kcal(z)=B(zI-D)^{-1}C\n}\n\\]\nif and only if\n\\[\n\\boxed{\n\\Kcal(t)=B\\e^{tD}C.\n}\n\\]\n\\end{theorem}","tional characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies.\n\\begin{theorem}[Rational memory realization]\\label{earlier:record:022}\\label{earlier:thm:rational_memory_realization}\nA matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely,\n\\[\n\\boxed{\n\\widehat\\Kcal(z)=B(zI-D)^{-1}C\n}\n\\]\nif and only if\n\\[\n\\boxed{\n\\Kcal(t)=B\\e^{tD}C.\n}\n\\]\n\\end{theorem}","tial order, so uniqueness of the ordinary Laplace transform applies.\n\\begin{theorem}[Rational memory realization]\\label{earlier:record:022}\\label{earlier:thm:rational_memory_realization}\nA matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely,\n\\[\n\\boxed{\n\\widehat\\Kcal(z)=B(zI-D)^{-1}C\n}\n\\]\nif and only if\n\\[\n\\boxed{\n\\Kcal(t)=B\\e^{tD}C.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe forward implication follows by taking the Laplace transform. The resulting matrix is rational and tends to zero as \\(z\\) tends to infinity.","Conversely, decompose every entry of a strictly proper rational matrix into partial fractions. A term\n\\[\n\\frac{c}{(z-\\lambda)^r}\n\\]\nis an entry of the resolvent of an \\(r\\) by \\(r\\) Jordan block with eigenvalue \\(\\lambda\\). Taking a direct sum of these blocks and choosing the input and output maps entry by entry gives matrices \\(B,D,C\\) with the required resolvent. Inverse Laplace transformation gives the time domain formula.\n\\end{proof}","\\begin{corollary}[Exponential sum memory]\\label{earlier:record:023}\\label{earlier:cor:exponential_sum_memory}\nIf\n\\[\n\\Kcal(t)=\\sum_{j=1}^{m}B_j\\e^{\\lambda_jt}C_j,\n\\]\nthen \\(\\Kcal\\) has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in \\(t\\) and are realized by Jordan blocks.\n\\end{corollary}","\\begin{proof}\nIf \\(B_j\\) and \\(C_j\\) have an intermediate space of dimension \\(d_j\\), use a block diagonal matrix \\(D\\) with blocks \\(\\lambda_j I_{d_j}\\), or the required Jordan blocks when poles repeat. Assemble the matrices \\(B_j\\) and \\(C_j\\) into the output and input maps.\n\\end{proof}","For the survival consequences, $A$ is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, $p_0$ is a probability vector, $S(t)=\\one^T e^{tA}p_0$, and $f(t)=-S'(t)$. The one-phase model has $\\lambda>0$. The survival Laplace identity of Part II also holds for every complex $s$ with $\\operatorname{Re}s\\geq0$: stability gives absolute convergence of $\\int_0^\\infty e^{-st}e^{tA}\\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\)","For the survival consequences, $A$ is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, $p_0$ is a probability vector, $S(t)=\\one^T e^{tA}p_0$, and $f(t)=-S'(t)$. The one-phase model has $\\lambda>0$. The survival Laplace identity of Part II also holds for every complex $s$ with $\\operatorname{Re}s\\geq0$: stability gives absolute convergence of $\\int_0^\\infty e^{-st}e^{tA}\\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\) and \\(f(t)\\) are finite linear combinations of exponential functions. Jordan blocks add polynomial facto","For the survival consequences, $A$ is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, $p_0$ is a probability vector, $S(t)=\\one^T e^{tA}p_0$, and $f(t)=-S'(t)$. The one-phase model has $\\lambda>0$. The survival Laplace identity of Part II also holds for every complex $s$ with $\\operatorname{Re}s\\geq0$: stability gives absolute convergence of $\\int_0^\\infty e^{-st}e^{tA}\\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\) and \\(f(t)\\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \\(t\\).\n\\end{corollary}","ive column sums and spectrum in the open left half-plane, $p_0$ is a probability vector, $S(t)=\\one^T e^{tA}p_0$, and $f(t)=-S'(t)$. The one-phase model has $\\lambda>0$. The survival Laplace identity of Part II also holds for every complex $s$ with $\\operatorname{Re}s\\geq0$: stability gives absolute convergence of $\\int_0^\\infty e^{-st}e^{tA}\\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\) and \\(f(t)\\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \\(t\\).\n\\end{corollary}","for every complex $s$ with $\\operatorname{Re}s\\geq0$: stability gives absolute convergence of $\\int_0^\\infty e^{-st}e^{tA}\\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\) and \\(f(t)\\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \\(t\\).\n\\end{corollary}","operatorname{Re}s\\geq0$: stability gives absolute convergence of $\\int_0^\\infty e^{-st}e^{tA}\\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\) and \\(f(t)\\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \\(t\\).\n\\end{corollary}","Re}s\\geq0$: stability gives absolute convergence of $\\int_0^\\infty e^{-st}e^{tA}\\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\) and \\(f(t)\\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \\(t\\).\n\\end{corollary}","tiplying by the exit row and $p_0$ gives the same transform.\n\\begin{corollary}[Phase type rationality]\\label{earlier:record:024}\\label{earlier:cor:phase_type_rationality}\nThe Laplace transform of every finite phase type distribution is rational. If \\(A\\) is diagonalizable, then \\(S(t)\\) and \\(f(t)\\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \\(t\\).\n\\end{corollary}","\\begin{proof}\nThe resolvent \\((sI-A)^{-1}\\) is rational. The time domain statement follows from the Jordan form of \\(A\\).\n\\end{proof}","\\begin{corollary}[One phase model]\\label{earlier:record:025}\\label{earlier:cor:one_phase_model}\nIf \\(A=(-\\lambda)\\) and \\(p_0=1\\), then\n\\[\nS(t)=\\e^{-\\lambda t},\n\\qquad\nf(t)=\\lambda\\e^{-\\lambda t},\n\\qquad\nh(t)=\\lambda.\n\\]\n\\end{corollary}","\\FloatBarrier\n\\section{Earlier exact examples}\\label{earlier:section:06}\nThese five short examples are retained from the earlier human edition. Their numerical parameters are independent of the later visual guide's worked models.\n\\begin{example}[One transfer between two packets]\\label{earlier:record:026}\\label{earlier:ex:two_packet_transfer}\nLet\n\\[\nA=\n\\begin{pmatrix}\na&0\\\\\nc&b\n\\end{pmatrix}\n\\]\nand let the two coordinate axes be the packet spaces. If \\(a\\neq b\\), then\n\\[\n\\e^{tA}\n=\n\\begin{pmatrix}\n\\e^{at}&0","c&b\n\\end{pmatrix}\n\\]\nand let the two coordinate axes be the packet spaces. If \\(a\\neq b\\), then\n\\[\n\\e^{tA}\n=\n\\begin{pmatrix}\n\\e^{at}&0","c&b\n\\end{pmatrix}\n\\]\nand let the two coordinate axes be the packet spaces. If \\(a\\neq b\\), then\n\\[\n\\e^{tA}\n=\n\\begin{pmatrix}\n\\e^{at}&0\\\\\n\\displaystyle c\\frac{\\e^{at}-\\e^{bt}}{a-b}&\\e^{bt}\n\\end{pmatrix}.\n\\]\nThe lower left block is produced by one transfer from the first packet to the second. Its expansion begins with \\(ct\\), in agreement with \\cref{earlier:cor:minimum_transfer_order}. When \\(a=b\\), the same block is \\(ct\\e^{at}\\).\n\\end{example}","\\displaystyle c\\frac{\\e^{at}-\\e^{bt}}{a-b}&\\e^{bt}\n\\end{pmatrix}.\n\\]\nThe lower left block is produced by one transfer from the first packet to the second. Its expansion begins with \\(ct\\), in agreement with \\cref{earlier:cor:minimum_transfer_order}. When \\(a=b\\), the same block is \\(ct\\e^{at}\\).\n\\end{example}","\\begin{example}[A four state lumpable chain]\\label{earlier:record:027}\\label{earlier:ex:four_state_lumping}\nPartition four states into \\(B_1=\\{1,2\\}\\) and \\(B_2=\\{3,4\\}\\). For positive rates \\(a,b,u,v\\), set\n\\[\nQ=\n\\begin{pmatrix}\n-u-a&u&b/2&b/2","\\begin{example}[A four state lumpable chain]\\label{earlier:record:027}\\label{earlier:ex:four_state_lumping}\nPartition four states into \\(B_1=\\{1,2\\}\\) and \\(B_2=\\{3,4\\}\\). For positive rates \\(a,b,u,v\\), set\n\\[\nQ=\n\\begin{pmatrix}\n-u-a&u&b/2&b/2\\\\\nu&-u-a&b/2&b/2\\\\\na/2&a/2&-v-b&v\\\\\na/2&a/2&v&-v-b\n\\end{pmatrix}.\n\\]\nThe rates \\(u\\) and \\(v\\) move inside the coarse blocks. Every state in \\(B_1\\) leaves for \\(B_2\\) at total rate \\(a\\), and every state in \\(B_2\\) leaves for \\(B_1\\) at total rate \\(b\\). Hence\n\\[\n\\overline Q=\n\\begin{pmatrix}\n-a&b","a/2&a/2&v&-v-b\n\\end{pmatrix}.\n\\]\nThe rates \\(u\\) and \\(v\\) move inside the coarse blocks. Every state in \\(B_1\\) leaves for \\(B_2\\) at total rate \\(a\\), and every state in \\(B_2\\) leaves for \\(B_1\\) at total rate \\(b\\). Hence\n\\[\n\\overline Q=\n\\begin{pmatrix}\n-a&b","a/2&a/2&v&-v-b\n\\end{pmatrix}.\n\\]\nThe rates \\(u\\) and \\(v\\) move inside the coarse blocks. Every state in \\(B_1\\) leaves for \\(B_2\\) at total rate \\(a\\), and every state in \\(B_2\\) leaves for \\(B_1\\) at total rate \\(b\\). Hence\n\\[\n\\overline Q=\n\\begin{pmatrix}\n-a&b\\\\\na&-b\n\\end{pmatrix}\n\\]\nand \\(CQ=\\overline QC\\). The internal rates disappear from the coarse generator even though they remain present in the fine evolution.\n\\end{example}","a&-b\n\\end{pmatrix}\n\\]\nand \\(CQ=\\overline QC\\). The internal rates disappear from the coarse generator even though they remain present in the fine evolution.\n\\end{example}","\\begin{example}[A visible likelihood loss]\\label{earlier:record:028}\\label{earlier:ex:bayes_information_loss}\nLet \\(B_1=\\{1,2\\}\\), let \\(B_2=\\{3\\}\\), and take\n\\[\np=(3/10,2/10,5/10),\n\\qquad\nL=(1,3,2).\n\\]\nThe coarse prior is \\((1/2,1/2)\\). The exact block likelihoods are\n\\[\n\\ell_1(p)=\\frac{(3/10)+3(2/10)}{1/2}=\\frac95,\n\\qquad\n\\ell_2(p)=2.\n\\]\nThus the exact coarse posterior is \\((9/19,10/19)\\). If the first block is replaced by the fixed representative value \\(2\\), the approximate coarse posterior is \\((1/2,1/2)\\). The \\(\\ell^1\\) defect is \\(1/19\\), while \\cref{earlier:thm:information_loss} gives the valid upper bound \\(1/10\\).\n\\end{example}","d:028}\\label{earlier:ex:bayes_information_loss}\nLet \\(B_1=\\{1,2\\}\\), let \\(B_2=\\{3\\}\\), and take\n\\[\np=(3/10,2/10,5/10),\n\\qquad\nL=(1,3,2).\n\\]\nThe coarse prior is \\((1/2,1/2)\\). The exact block likelihoods are\n\\[\n\\ell_1(p)=\\frac{(3/10)+3(2/10)}{1/2}=\\frac95,\n\\qquad\n\\ell_2(p)=2.\n\\]\nThus the exact coarse posterior is \\((9/19,10/19)\\). If the first block is replaced by the fixed representative value \\(2\\), the approximate coarse posterior is \\((1/2,1/2)\\). The \\(\\ell^1\\) defect is \\(1/19\\), while \\cref{earlier:thm:information_loss} gives the valid upper bound \\(1/10\\).\n\\end{example}","oss}\nLet \\(B_1=\\{1,2\\}\\), let \\(B_2=\\{3\\}\\), and take\n\\[\np=(3/10,2/10,5/10),\n\\qquad\nL=(1,3,2).\n\\]\nThe coarse prior is \\((1/2,1/2)\\). The exact block likelihoods are\n\\[\n\\ell_1(p)=\\frac{(3/10)+3(2/10)}{1/2}=\\frac95,\n\\qquad\n\\ell_2(p)=2.\n\\]\nThus the exact coarse posterior is \\((9/19,10/19)\\). If the first block is replaced by the fixed representative value \\(2\\), the approximate coarse posterior is \\((1/2,1/2)\\). The \\(\\ell^1\\) defect is \\(1/19\\), while \\cref{earlier:thm:information_loss} gives the valid upper bound \\(1/10\\).\n\\end{example}","2\\}\\), let \\(B_2=\\{3\\}\\), and take\n\\[\np=(3/10,2/10,5/10),\n\\qquad\nL=(1,3,2).\n\\]\nThe coarse prior is \\((1/2,1/2)\\). The exact block likelihoods are\n\\[\n\\ell_1(p)=\\frac{(3/10)+3(2/10)}{1/2}=\\frac95,\n\\qquad\n\\ell_2(p)=2.\n\\]\nThus the exact coarse posterior is \\((9/19,10/19)\\). If the first block is replaced by the fixed representative value \\(2\\), the approximate coarse posterior is \\((1/2,1/2)\\). The \\(\\ell^1\\) defect is \\(1/19\\), while \\cref{earlier:thm:information_loss} gives the valid upper bound \\(1/10\\).\n\\end{example}","}\\), and take\n\\[\np=(3/10,2/10,5/10),\n\\qquad\nL=(1,3,2).\n\\]\nThe coarse prior is \\((1/2,1/2)\\). The exact block likelihoods are\n\\[\n\\ell_1(p)=\\frac{(3/10)+3(2/10)}{1/2}=\\frac95,\n\\qquad\n\\ell_2(p)=2.\n\\]\nThus the exact coarse posterior is \\((9/19,10/19)\\). If the first block is replaced by the fixed representative value \\(2\\), the approximate coarse posterior is \\((1/2,1/2)\\). The \\(\\ell^1\\) defect is \\(1/19\\), while \\cref{earlier:thm:information_loss} gives the valid upper bound \\(1/10\\).\n\\end{example}","(1/2,1/2)\\). The exact block likelihoods are\n\\[\n\\ell_1(p)=\\frac{(3/10)+3(2/10)}{1/2}=\\frac95,\n\\qquad\n\\ell_2(p)=2.\n\\]\nThus the exact coarse posterior is \\((9/19,10/19)\\). If the first block is replaced by the fixed representative value \\(2\\), the approximate coarse posterior is \\((1/2,1/2)\\). The \\(\\ell^1\\) defect is \\(1/19\\), while \\cref{earlier:thm:information_loss} gives the valid upper bound \\(1/10\\).\n\\end{example}","\\begin{example}[One hidden mode]\\label{earlier:record:029}\\label{earlier:ex:one_hidden_mode}\nConsider\n\\[\nA=\n\\begin{pmatrix}\n-a&c\\\\\nd&-b\n\\end{pmatrix},\n\\qquad\na,b>0,\n\\]\nand retain the first coordinate. The memory kernel and self-energy are\n\\[\n\\Kcal(t)=cd\\e^{-bt},\n\\qquad\n\\Sigma(z)=\\frac{cd}{z+b}.\n\\]\nThe retained coordinate has an exact time local closure for every retained initial value precisely when \\(cd=0\\). When \\(cd\\neq0\\), one auxiliary scalar state realizes the memory exactly.\n\\end{example}","d&-b\n\\end{pmatrix},\n\\qquad\na,b>0,\n\\]\nand retain the first coordinate. The memory kernel and self-energy are\n\\[\n\\Kcal(t)=cd\\e^{-bt},\n\\qquad\n\\Sigma(z)=\\frac{cd}{z+b}.\n\\]\nThe retained coordinate has an exact time local closure for every retained initial value precisely when \\(cd=0\\). When \\(cd\\neq0\\), one auxiliary scalar state realizes the memory exactly.\n\\end{example}","\\begin{example}[Two sequential phases]\\label{earlier:record:030}\\label{earlier:ex:two_phase_survival}\nLet\n\\[\nA=\n\\begin{pmatrix}\n-\\lambda&0\\\\\n\\lambda&-\\mu\n\\end{pmatrix},\n\\qquad\np_0=\n\\begin{pmatrix}1\\\\0\\end{pmatrix},\n\\]\nwhere \\(\\lambda\\neq\\mu\\) are positive. The first phase moves to the second at rate \\(\\lambda\\), and the second is absorbed at rate \\(\\mu\\). Direct calculation gives\n\\[\nS(t)=\n\\frac{\\mu\\e^{-\\lambda t}-\\lambda\\e^{-\\mu t}}{\\mu-\\lambda},\n\\]\n\\[\nf(t)=\n\\frac{\\lambda\\mu}{\\mu-\\lambda}\n\\left(\\e^{-\\lambda t}-\\e^{-\\mu t}\\right).\n\\]\nThe ratio \\(f(t)/S(t)\\) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after the phase coordinate is hidden.\n\\end{example}","0\\end{pmatrix},\n\\]\nwhere \\(\\lambda\\neq\\mu\\) are positive. The first phase moves to the second at rate \\(\\lambda\\), and the second is absorbed at rate \\(\\mu\\). Direct calculation gives\n\\[\nS(t)=\n\\frac{\\mu\\e^{-\\lambda t}-\\lambda\\e^{-\\mu t}}{\\mu-\\lambda},\n\\]\n\\[\nf(t)=\n\\frac{\\lambda\\mu}{\\mu-\\lambda}\n\\left(\\e^{-\\lambda t}-\\e^{-\\mu t}\\right).\n\\]\nThe ratio \\(f(t)/S(t)\\) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after the phase coordinate is hidden.\n\\end{example}","This section presents the operator-substitution result transferred from Chapter IV. Fix an integer $N\\geq1$ and the entire scalar function\n\\[\n\\Dfun_N(z)=\\sum_{m=0}^{\\infty}\\frac{(-1)^m z^{Nm}}{(Nm)!}.\n\\]\nMore generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain $U$, choose $g=\\log\\Dfun_N$ once and define $\\Sfun_N^{(r)}(z)=\\exp(-r g(z))$ for complex $r$. The bounded Banach-space result below uses classical holomorphic functional calculus. It is separate from the finite canonical-jet calculus formalized in Part I.","tion presents the operator-substitution result transferred from Chapter IV. Fix an integer $N\\geq1$ and the entire scalar function\n\\[\n\\Dfun_N(z)=\\sum_{m=0}^{\\infty}\\frac{(-1)^m z^{Nm}}{(Nm)!}.\n\\]\nMore generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain $U$, choose $g=\\log\\Dfun_N$ once and define $\\Sfun_N^{(r)}(z)=\\exp(-r g(z))$ for complex $r$. The bounded Banach-space result below uses classical holomorphic functional calculus. It is separate from the finite canonical-jet calculus formalized in Part I.","t transferred from Chapter IV. Fix an integer $N\\geq1$ and the entire scalar function\n\\[\n\\Dfun_N(z)=\\sum_{m=0}^{\\infty}\\frac{(-1)^m z^{Nm}}{(Nm)!}.\n\\]\nMore generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain $U$, choose $g=\\log\\Dfun_N$ once and define $\\Sfun_N^{(r)}(z)=\\exp(-r g(z))$ for complex $r$. The bounded Banach-space result below uses classical holomorphic functional calculus. It is separate from the finite canonical-jet calculus formalized in Part I.","\\begin{standardfact}[Holomorphic functional calculus]\\label{migration:record:001}\\label{migration:sf:holomorphic_calculus}\nLet $A$ be a bounded operator on a complex Banach space.  If $f$ is holomorphic on a neighborhood of $\\Spec(A)$, then\n\\[\nf(A)\n=\n\\frac1{2\\pi i}\\int_\\Gamma f(z)(zI-A)^{-1}\\,dz\n\\]\nfor an admissible positively oriented contour $\\Gamma$ lying in the holomorphy domain and the resolvent set, with winding number one about the spectrum and zero outside the domain. The calculus preserves sums, products, and composition.  We use the standard form from \\cite{DunfordSchwartz1958}. An accessible proof is given in \\cite[Appendix A]{TaylorBanachAlgebras}.\n\\end{standardfact}","\\begin{proposition}[Operator reciprocal family]\\label{migration:record:002}\\label{migration:prop:operator_reciprocal}\nLet $A$ be bounded and suppose $\\Spec(A)$ lies in a simply connected region on which $\\Dfun_N$ has no zeros.  Fix a holomorphic branch $g=\\log\\Dfun_N$ there.  Then\n\\[\n\\Sfun_N^{(r)}(A)\n=\n\\exp\\bigl(-r g(A)\\bigr)\n=\n\\bigl(\\Dfun_N^{-r}\\bigr)(A)\n\\]\nis well defined and\n\\[\n\\Sfun_N^{(r)}(A)\\Sfun_N^{(s)}(A)\n=\n\\Sfun_N^{(r+s)}(A).\n\\]\nIf $(P_i)$ is a finite complete family of bounded idempotents, then\n\\[\nP_i f(A)P_j\n=\n\\frac1{2\\pi i}\n\\int_\\Gamma\nf(z)P_i(zI-A)^{-1}P_j\\,dz\n\\]\nfor every holomorphic $f$ on a neighborhood of the spectrum.\n\\end{proposition}","]\\label{migration:record:002}\\label{migration:prop:operator_reciprocal}\nLet $A$ be bounded and suppose $\\Spec(A)$ lies in a simply connected region on which $\\Dfun_N$ has no zeros.  Fix a holomorphic branch $g=\\log\\Dfun_N$ there.  Then\n\\[\n\\Sfun_N^{(r)}(A)\n=\n\\exp\\bigl(-r g(A)\\bigr)\n=\n\\bigl(\\Dfun_N^{-r}\\bigr)(A)\n\\]\nis well defined and\n\\[\n\\Sfun_N^{(r)}(A)\\Sfun_N^{(s)}(A)\n=\n\\Sfun_N^{(r+s)}(A).\n\\]\nIf $(P_i)$ is a finite complete family of bounded idempotents, then\n\\[\nP_i f(A)P_j\n=\n\\frac1{2\\pi i}\n\\int_\\Gamma\nf(z)P_i(zI-A)^{-1}P_j\\,dz\n\\]\nfor every holomorphic $f$ on a neighborhood of the spectrum.\n\\end{proposition}","ocal}\nLet $A$ be bounded and suppose $\\Spec(A)$ lies in a simply connected region on which $\\Dfun_N$ has no zeros.  Fix a holomorphic branch $g=\\log\\Dfun_N$ there.  Then\n\\[\n\\Sfun_N^{(r)}(A)\n=\n\\exp\\bigl(-r g(A)\\bigr)\n=\n\\bigl(\\Dfun_N^{-r}\\bigr)(A)\n\\]\nis well defined and\n\\[\n\\Sfun_N^{(r)}(A)\\Sfun_N^{(s)}(A)\n=\n\\Sfun_N^{(r+s)}(A).\n\\]\nIf $(P_i)$ is a finite complete family of bounded idempotents, then\n\\[\nP_i f(A)P_j\n=\n\\frac1{2\\pi i}\n\\int_\\Gamma\nf(z)P_i(zI-A)^{-1}P_j\\,dz\n\\]\nfor every holomorphic $f$ on a neighborhood of the spectrum.\n\\end{proposition}","fun_N$ has no zeros.  Fix a holomorphic branch $g=\\log\\Dfun_N$ there.  Then\n\\[\n\\Sfun_N^{(r)}(A)\n=\n\\exp\\bigl(-r g(A)\\bigr)\n=\n\\bigl(\\Dfun_N^{-r}\\bigr)(A)\n\\]\nis well defined and\n\\[\n\\Sfun_N^{(r)}(A)\\Sfun_N^{(s)}(A)\n=\n\\Sfun_N^{(r+s)}(A).\n\\]\nIf $(P_i)$ is a finite complete family of bounded idempotents, then\n\\[\nP_i f(A)P_j\n=\n\\frac1{2\\pi i}\n\\int_\\Gamma\nf(z)P_i(zI-A)^{-1}P_j\\,dz\n\\]\nfor every holomorphic $f$ on a neighborhood of the spectrum.\n\\end{proposition}","\\begin{proof}\nThe spectral mapping theorem makes $\\Dfun_N(A)$ invertible.  The product identity follows from multiplicativity of the functional calculus and the fixed logarithm.  Applying $P_i$ and $P_j$ to the Cauchy integral formula gives the packet formula.\n\\end{proof}","'s language, not by pretending that their state spaces are identical.\n\\vfig{three_views}\n\\platecaption{Left and center use one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on \\pagerefV{22}. Every curve is drawn from an explicit formula.}\n\\head{Coordinates retain, aggregation forgets, and elimination remembers}\nKeeping every component $P_a x$ is an exact change of representation when $\\sum_aP_a=I$. Reconstruct by summing the components. In contrast, replacing $(p_0,p_1,p_2,p_3)$ by $(p_0+p_1,p_2+p_3)$ forgets how mass was distributed inside either pair. That information can matter for later observations.","aces are identical.\n\\vfig{three_views}\n\\platecaption{Left and center use one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on \\pagerefV{22}. Every curve is drawn from an explicit formula.}\n\\head{Coordinates retain, aggregation forgets, and elimination remembers}\nKeeping every component $P_a x$ is an exact change of representation when $\\sum_aP_a=I$. Reconstruct by summing the components. In contrast, replacing $(p_0,p_1,p_2,p_3)$ by $(p_0+p_1,p_2+p_3)$ forgets how mass was distributed inside either pair. That information can matter for later observations.","probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on \\pagerefV{22}. Every curve is drawn from an explicit formula.}\n\\head{Coordinates retain, aggregation forgets, and elimination remembers}\nKeeping every component $P_a x$ is an exact change of representation when $\\sum_aP_a=I$. Reconstruct by summing the components. In contrast, replacing $(p_0,p_1,p_2,p_3)$ by $(p_0+p_1,p_2+p_3)$ forgets how mass was distributed inside either pair. That information can matter for later observations.","its two block masses. The right panel is a separate retained/hidden model, developed on \\pagerefV{22}. Every curve is drawn from an explicit formula.}\n\\head{Coordinates retain, aggregation forgets, and elimination remembers}\nKeeping every component $P_a x$ is an exact change of representation when $\\sum_aP_a=I$. Reconstruct by summing the components. In contrast, replacing $(p_0,p_1,p_2,p_3)$ by $(p_0+p_1,p_2+p_3)$ forgets how mass was distributed inside either pair. That information can matter for later observations.","\\vpage{2}{A dictionary with orientations}{READ BEFORE MULTIPLYING / CONVENTIONS}{Chapter VI, Part I: Sections 2, 4, 6, 7, and 12. Column probabilities and backward observables.}\nAn apparent disagreement between two transfer formulas often comes from acting on different objects. Throughout this guide, probabilities are columns and observables are functions written as columns. Their pairing is $f^{\\mathsf T}p$. The transpose distinguishes the forward action on distributions from the backward action on observations.\n\\begin{center}\\small\n\\begin{tabularx}{\\linewidth}{@{}lYY@{}}\\toprule\nObject & Exact meaning & What to check","midrule\n$p$ & A nonnegative column with $\\ones^{\\mathsf T}p=1$. & It is not a vector of function values.","$Q$ & Continuous Markov generator. & Off-diagonal entries are nonnegative and columns sum to zero.","$K$ & One discrete stochastic step. & $\\ones^{\\mathsf T}K=\\ones^{\\mathsf T}$ and $K_{ij}\\ge0$.","$K^{\\mathsf T}f$ & Expected next observable given the current state. & The same dynamics, with the opposite action on the dual object.","$P_a$ & A coordinate packet projector. & All packets together reconstruct the full state.","$C$ & Sums fine probability masses into coarse blocks. & Usually rectangular and not invertible.","$E_\\pi$ & Conditional expectation of observables. & A weighted average, self-adjoint for the $\\pi$ inner product.","$\\mathcal B_\\lambda$ & $\\operatorname{diag}(\\lambda)p/(\\lambda^{\\mathsf T}p)$. & Nonnegative likelihood and positive total evidence.","$L$ & A nonnegative self-adjoint operator. & Its evolution is $e^{-tL}$, not $e^{tL}$.","$R(z)$ & $(zI-A)^{-1}$. & $z$ belongs to the resolvent set.","$\\mathcal K(t)$ & The memory kernel after hidden-state elimination. & It is not the transition matrix $K$.","bottomrule\n\\end{tabularx}\\end{center}\n\\head{One convention test}\nFor\n\\[\nK_2=\\begin{pmatrix}3/4&1/4\\\\1/4&3/4\\end{pmatrix},\n\\qquad p=e_0,\\qquad f=(2,1)^{\\mathsf T},\n\\]\nwe obtain $K_2p=(3/4,1/4)^{\\mathsf T}$ and\n\\[\nf^{\\mathsf T}K_2p=7/4=(K_2^{\\mathsf T}f)_0.\n\\]\nWith departure weights $w_i=e^{-V(i)}$, the backward Feynman--Kac operator is $\\operatorname{diag}(w)K_2^{\\mathsf T}$. Its forward dual is $K_2\\operatorname{diag}(w)$. Interchanging these factors changes when the weight is applied.\n\\takeaway{Always name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.}","1/4&3/4\\end{pmatrix},\n\\qquad p=e_0,\\qquad f=(2,1)^{\\mathsf T},\n\\]\nwe obtain $K_2p=(3/4,1/4)^{\\mathsf T}$ and\n\\[\nf^{\\mathsf T}K_2p=7/4=(K_2^{\\mathsf T}f)_0.\n\\]\nWith departure weights $w_i=e^{-V(i)}$, the backward Feynman--Kac operator is $\\operatorname{diag}(w)K_2^{\\mathsf T}$. Its forward dual is $K_2\\operatorname{diag}(w)$. Interchanging these factors changes when the weight is applied.\n\\takeaway{Always name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.}","\\vpage{3}{A clock generated by one matrix}{CONTINUOUS TRANSFER / THE ENTIRE SOLUTION}{Chapter VI, Part I: Theorems 1.2 and 4.1. Finite matrix exponentials and Markov positivity.}\nTake the generator\n\\[\nG=\\begin{pmatrix}-1&1\\\\1&-1\\end{pmatrix}.\n\\]\nThe total-mass direction $(1,1)$ has eigenvalue zero. The contrast $(1,-1)$ has eigenvalue $-2$. Resolving those two directions gives the full transfer, not only its first-order approximation:\n\\[\nT(t)=e^{tG}=\\frac12\\begin{pmatrix}1+e^{-2t}&1-e^{-2t}","1&-1\\end{pmatrix}.\n\\]\nThe total-mass direction $(1,1)$ has eigenvalue zero. The contrast $(1,-1)$ has eigenvalue $-2$. Resolving those two directions gives the full transfer, not only its first-order approximation:\n\\[\nT(t)=e^{tG}=\\frac12\\begin{pmatrix}1+e^{-2t}&1-e^{-2t}","1&-1\\end{pmatrix}.\n\\]\nThe total-mass direction $(1,1)$ has eigenvalue zero. The contrast $(1,-1)$ has eigenvalue $-2$. Resolving those two directions gives the full transfer, not only its first-order approximation:\n\\[\nT(t)=e^{tG}=\\frac12\\begin{pmatrix}1+e^{-2t}&1-e^{-2t}\\\\1-e^{-2t}&1+e^{-2t}\\end{pmatrix}.\n\\]\n\\vfig{clock}\n\\platecaption{A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory.}\n\\head{Two checks explain the mechanism}\nThe constant mode is unchanged, while each elapsed interval multiplies the contrast by $e^{-2t}$. Consequently two intervals multiply their contrast factors:\n\\[\ne^{-2s}e^{-2t}=e^{-2(s+t)},\\qquad T(s)T(t)=T(s+t).\n\\]\nFor $t\\ge0$, the entries are nonnegative and every column sums to one. Differentiation gives $T'(t)=GT(t)$. These three facts separately check the initial condition, the evolution equation, and preservation of probab","1-e^{-2t}&1+e^{-2t}\\end{pmatrix}.\n\\]\n\\vfig{clock}\n\\platecaption{A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory.}\n\\head{Two checks explain the mechanism}\nThe constant mode is unchanged, while each elapsed interval multiplies the contrast by $e^{-2t}$. Consequently two intervals multiply their contrast factors:\n\\[\ne^{-2s}e^{-2t}=e^{-2(s+t)},\\qquad T(s)T(t)=T(s+t).\n\\]\nFor $t\\ge0$, the entries are nonnegative and every column sums to one. Differentiation gives $T'(t)=GT(t)$. These three facts separately check the initial condition, the evolution equation, and preservation of probability.","{A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory.}\n\\head{Two checks explain the mechanism}\nThe constant mode is unchanged, while each elapsed interval multiplies the contrast by $e^{-2t}$. Consequently two intervals multiply their contrast factors:\n\\[\ne^{-2s}e^{-2t}=e^{-2(s+t)},\\qquad T(s)T(t)=T(s+t).\n\\]\nFor $t\\ge0$, the entries are nonnegative and every column sums to one. Differentiation gives $T'(t)=GT(t)$. These three facts separately check the initial condition, the evolution equation, and preservation of probability.","ion. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory.}\n\\head{Two checks explain the mechanism}\nThe constant mode is unchanged, while each elapsed interval multiplies the contrast by $e^{-2t}$. Consequently two intervals multiply their contrast factors:\n\\[\ne^{-2s}e^{-2t}=e^{-2(s+t)},\\qquad T(s)T(t)=T(s+t).\n\\]\nFor $t\\ge0$, the entries are nonnegative and every column sums to one. Differentiation gives $T'(t)=GT(t)$. These three facts separately check the initial condition, the evolution equation, and preservation of probability.","nuous-time evolution, not a simulated trajectory.}\n\\head{Two checks explain the mechanism}\nThe constant mode is unchanged, while each elapsed interval multiplies the contrast by $e^{-2t}$. Consequently two intervals multiply their contrast factors:\n\\[\ne^{-2s}e^{-2t}=e^{-2(s+t)},\\qquad T(s)T(t)=T(s+t).\n\\]\nFor $t\\ge0$, the entries are nonnegative and every column sums to one. Differentiation gives $T'(t)=GT(t)$. These three facts separately check the initial condition, the evolution equation, and preservation of probability.","At $t=\\log 2$, the retained first coordinate is $5/8$. It would instead be $1/2$ under the isolated scalar equation $\\dot x=-x$. The discrepancy is the mass that can leave state zero and return. On \\pagerefV{22}, we remove the second coordinate correctly and recover exactly the same $5/8$ using a memory integral.\n\\takeaway{The semigroup lives on the complete state. One diagonal entry of a semigroup need not itself satisfy a scalar semigroup law.}","\\vpage{4}{One evolution, two packet choices}{FOUR-STATE MODEL / COORDINATES AND MODES}{Chapter VI, Part I: Theorem 2.1, Corollary 2.2, and the packet path expansion.}\nOur main finite model has generator\n\\[\nQ=\\begin{pmatrix}-3&1&2&0\\\\1&-3&0&2\\\\2&0&-3&1\\\\0&2&1&-3\\end{pmatrix}.\n\\]\nThe graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis $u_j$. Their projectors $P_j=u_ju_j^{\\mathsf T}$ sum to the identity and satisfy $P_iP_j=0$ for $i\\ne j$.\n\\vfig{generator_modes}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without nume","0&2&1&-3\\end{pmatrix}.\n\\]\nThe graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis $u_j$. Their projectors $P_j=u_ju_j^{\\mathsf T}$ sum to the identity and satisfy $P_iP_j=0$ for $i\\ne j$.\n\\vfig{generator_modes}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without numerical matrix exponentiation.","orm an orthonormal basis $u_j$. Their projectors $P_j=u_ju_j^{\\mathsf T}$ sum to the identity and satisfy $P_iP_j=0$ for $i\\ne j$.\n\\vfig{generator_modes}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without numerical matrix exponentiation.","basis $u_j$. Their projectors $P_j=u_ju_j^{\\mathsf T}$ sum to the identity and satisfy $P_iP_j=0$ for $i\\ne j$.\n\\vfig{generator_modes}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without numerical matrix exponentiation.","_j^{\\mathsf T}$ sum to the identity and satisfy $P_iP_j=0$ for $i\\ne j$.\n\\vfig{generator_modes}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without numerical matrix exponentiation.","he identity and satisfy $P_iP_j=0$ for $i\\ne j$.\n\\vfig{generator_modes}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without numerical matrix exponentiation.","d satisfy $P_iP_j=0$ for $i\\ne j$.\n\\vfig{generator_modes}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without numerical matrix exponentiation.","des}\n\\platecaption{State coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next.}\n\\head{The same evolution has a closed formula}\nWriting $q=e^{-2t}$ gives\n\\[\ne^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.\n\\]\nAt $q=1/2$, the point seed $e_0$ becomes $(15,5,9,3)^{\\mathsf T}/32$. The polynomial identity $T(q)T(r)=T(qr)$ checks composition exactly, without numerical matrix exponentiation.","Any other complete packet family still has exact block convolution. For the coordinate groups A and B,\n\\[\nP_b e^{(s+t)Q}P_a=\\sum_{c\\in\\{\\mathrm A,\\mathrm B\\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).\n\\]\nThe intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","For later discrete calculations define $K=I+Q/8$. This matrix is stochastic, but \\emph{it is not} $e^{Q/8}$. It is the step kernel in the uniformization identity $e^{tQ}=e^{-8t}\\sum_{n\\ge0}(8t)^nK^n/n!$.\n\\takeaway{Changing packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.}","\\vpage{5}{When two states are truly equivalent}{EXACT LUMPABILITY / AGGREGATE COLUMNS}{Chapter VI, Part I: Theorem 5.1. The hypothesis is an intertwining identity.}\nGroup states $0,1$ into A and states $2,3$ into B. The aggregation map and the candidate coarse kernel are\n\\[\nC=\\begin{pmatrix}1&1&0&0","\\vpage{5}{When two states are truly equivalent}{EXACT LUMPABILITY / AGGREGATE COLUMNS}{Chapter VI, Part I: Theorem 5.1. The hypothesis is an intertwining identity.}\nGroup states $0,1$ into A and states $2,3$ into B. The aggregation map and the candidate coarse kernel are\n\\[\nC=\\begin{pmatrix}1&1&0&0\\\\0&0&1&1\\end{pmatrix},\\qquad\n\\bar K=\\begin{pmatrix}3/4&1/4\\\\1/4&3/4\\end{pmatrix}.\n\\]\nA fine state in A has probability $1/4$ of entering B in one step, regardless of which fine state in A was occupied. The analogous statement holds for B.\n\\vfig{lumpability}\n\\platecaption{The fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works.}\n\\head{A one-step calculation controls every discrete horizon}\nThe displayed columns give $CK=\\bar KC$. Multiplication then yields\n\\[\nCK^n=\\bar K^nC\\qquad(n\\ge0).\n\\]\nFor example, $e_0$ and $e_1$ have the same coarse initial vector $(1,0)^{\\mathsf T}$. Their fine evolutions differ, but their coarse vectors agree at every step. No choice of th","1/4&3/4\\end{pmatrix}.\n\\]\nA fine state in A has probability $1/4$ of entering B in one step, regardless of which fine state in A was occupied. The analogous statement holds for B.\n\\vfig{lumpability}\n\\platecaption{The fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works.}\n\\head{A one-step calculation controls every discrete horizon}\nThe displayed columns give $CK=\\bar KC$. Multiplication then yields\n\\[\nCK^n=\\bar K^nC\\qquad(n\\ge0).\n\\]\nFor example, $e_0$ and $e_1$ have the same coarse initial vector $(1,0)^{\\mathsf T}$. Their fine evolutions differ, but their coarse vectors agree at every step. No choice of the within-A conditional distribution is ne","tering B in one step, regardless of which fine state in A was occupied. The analogous statement holds for B.\n\\vfig{lumpability}\n\\platecaption{The fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works.}\n\\head{A one-step calculation controls every discrete horizon}\nThe displayed columns give $CK=\\bar KC$. Multiplication then yields\n\\[\nCK^n=\\bar K^nC\\qquad(n\\ge0).\n\\]\nFor example, $e_0$ and $e_1$ have the same coarse initial vector $(1,0)^{\\mathsf T}$. Their fine evolutions differ, but their coarse vectors agree at every step. No choice of the within-A conditional distribution is needed for the prediction.","fine state in A was occupied. The analogous statement holds for B.\n\\vfig{lumpability}\n\\platecaption{The fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works.}\n\\head{A one-step calculation controls every discrete horizon}\nThe displayed columns give $CK=\\bar KC$. Multiplication then yields\n\\[\nCK^n=\\bar K^nC\\qquad(n\\ge0).\n\\]\nFor example, $e_0$ and $e_1$ have the same coarse initial vector $(1,0)^{\\mathsf T}$. Their fine evolutions differ, but their coarse vectors agree at every step. No choice of the within-A conditional distribution is needed for the prediction.","tement holds for B.\n\\vfig{lumpability}\n\\platecaption{The fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works.}\n\\head{A one-step calculation controls every discrete horizon}\nThe displayed columns give $CK=\\bar KC$. Multiplication then yields\n\\[\nCK^n=\\bar K^nC\\qquad(n\\ge0).\n\\]\nFor example, $e_0$ and $e_1$ have the same coarse initial vector $(1,0)^{\\mathsf T}$. Their fine evolutions differ, but their coarse vectors agree at every step. No choice of the within-A conditional distribution is needed for the prediction.","ds for B.\n\\vfig{lumpability}\n\\platecaption{The fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works.}\n\\head{A one-step calculation controls every discrete horizon}\nThe displayed columns give $CK=\\bar KC$. Multiplication then yields\n\\[\nCK^n=\\bar K^nC\\qquad(n\\ge0).\n\\]\nFor example, $e_0$ and $e_1$ have the same coarse initial vector $(1,0)^{\\mathsf T}$. Their fine evolutions differ, but their coarse vectors agree at every step. No choice of the within-A conditional distribution is needed for the prediction.","{The fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works.}\n\\head{A one-step calculation controls every discrete horizon}\nThe displayed columns give $CK=\\bar KC$. Multiplication then yields\n\\[\nCK^n=\\bar K^nC\\qquad(n\\ge0).\n\\]\nFor example, $e_0$ and $e_1$ have the same coarse initial vector $(1,0)^{\\mathsf T}$. Their fine evolutions differ, but their coarse vectors agree at every step. No choice of the within-A conditional distribution is needed for the prediction.","The continuous generator also closes:\n\\[\nCQ=\\bar Q C,\\qquad\n\\bar Q=\\begin{pmatrix}-2&2\\\\2&-2\\end{pmatrix}.\n\\]\nThe Duhamel identity therefore gives $Ce^{tQ}=e^{t\\bar Q}C$. This is a second consequence of the corresponding generator identity, not an inference that $K$ is a fixed-time sample of $Q$.","2&-2\\end{pmatrix}.\n\\]\nThe Duhamel identity therefore gives $Ce^{tQ}=e^{t\\bar Q}C$. This is a second consequence of the corresponding generator identity, not an inference that $K$ is a fixed-time sample of $Q$.","\\vpage{6}{A broken lump and its error budget}{DEFECTS / A GUARANTEE WITHOUT A FIT}{Chapter VI, Part I: Theorems 3.1, 3.3, and 5.2. Continuous and discrete defect identities.}\nModify only column zero of $K$: move $1/16$ of its mass from target state zero to target state two. The perturbed matrix $K_\\varepsilon$ remains stochastic, but its aggregate column now differs from column one. Relative to the unchanged coarse kernel,\n\\[\n\\Delta=CK_\\varepsilon-\\bar KC\n=\\begin{pmatrix}-1/16&0&0&0","ts error budget}{DEFECTS / A GUARANTEE WITHOUT A FIT}{Chapter VI, Part I: Theorems 3.1, 3.3, and 5.2. Continuous and discrete defect identities.}\nModify only column zero of $K$: move $1/16$ of its mass from target state zero to target state two. The perturbed matrix $K_\\varepsilon$ remains stochastic, but its aggregate column now differs from column one. Relative to the unchanged coarse kernel,\n\\[\n\\Delta=CK_\\varepsilon-\\bar KC\n=\\begin{pmatrix}-1/16&0&0&0\\\\1/16&0&0&0\\end{pmatrix},\n\\qquad \\|\\Delta\\|_1=1/8.\n\\]\n\\vfig{defect_budget}\n\\platecaption{The solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp.}\n\\head{Telescoping tells us where every error originates}\nFor $n\\ge0$,\n\\[\nCK_\\varepsilon^n-\\bar K^nC\n=\\sum_{j=0}^{n-1}\\bar K^{n-1-j}\\Delta K_\\varepsilon^j.\n\\]\nEach summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced $\\ell^1$ norm one, so the norm is at most $n/8$. The empty sum at $n=0$ is zero.","1/16&0&0&0\\end{pmatrix},\n\\qquad \\|\\Delta\\|_1=1/8.\n\\]\n\\vfig{defect_budget}\n\\platecaption{The solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp.}\n\\head{Telescoping tells us where every error originates}\nFor $n\\ge0$,\n\\[\nCK_\\varepsilon^n-\\bar K^nC\n=\\sum_{j=0}^{n-1}\\bar K^{n-1-j}\\Delta K_\\varepsilon^j.\n\\]\nEach summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced $\\ell^1$ norm one, so the norm is at most $n/8$. The empty sum at $n=0$ is zero.","The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp.}\n\\head{Telescoping tells us where every error originates}\nFor $n\\ge0$,\n\\[\nCK_\\varepsilon^n-\\bar K^nC\n=\\sum_{j=0}^{n-1}\\bar K^{n-1-j}\\Delta K_\\varepsilon^j.\n\\]\nEach summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced $\\ell^1$ norm one, so the norm is at most $n/8$. The empty sum at $n=0$ is zero.","icity and the exact one-step defect. It need not be sharp.}\n\\head{Telescoping tells us where every error originates}\nFor $n\\ge0$,\n\\[\nCK_\\varepsilon^n-\\bar K^nC\n=\\sum_{j=0}^{n-1}\\bar K^{n-1-j}\\Delta K_\\varepsilon^j.\n\\]\nEach summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced $\\ell^1$ norm one, so the norm is at most $n/8$. The empty sum at $n=0$ is zero.","step defect. It need not be sharp.}\n\\head{Telescoping tells us where every error originates}\nFor $n\\ge0$,\n\\[\nCK_\\varepsilon^n-\\bar K^nC\n=\\sum_{j=0}^{n-1}\\bar K^{n-1-j}\\Delta K_\\varepsilon^j.\n\\]\nEach summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced $\\ell^1$ norm one, so the norm is at most $n/8$. The empty sum at $n=0$ is zero.","The continuous counterpart is equally explicit. With fine generator $-1$, coarse generator $-2$, and $C=1$, the generator defect is $1$ and\n\\[\ne^{-t}-e^{-2t}=\\int_0^t e^{-2(t-r)}e^{-r}\\,dr.\n\\]\nIts resolvent version is $1/(z+1)-1/(z+2)=1/((z+1)(z+2))$ for $z>0$.\n\\takeaway{A small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn that identity into a finite-horizon certificate.}","\\vpage{7}{Averaging is not summing}{OBSERVABLES / WEIGHTED ORTHOGONALITY}{Chapter VI, Part I: Theorem 6.1. Strictly positive weights are part of the setting.}\nFor this observable example, choose the fixed probability weights $\\pi=(1,2,3,4)/10$. They are not being identified with the stationary law of the earlier symmetric chain. On the same two-block partition, conditional expectation is\n\\[\nE=\\begin{pmatrix}\n1/3&2/3&0&0","\\vpage{7}{Averaging is not summing}{OBSERVABLES / WEIGHTED ORTHOGONALITY}{Chapter VI, Part I: Theorem 6.1. Strictly positive weights are part of the setting.}\nFor this observable example, choose the fixed probability weights $\\pi=(1,2,3,4)/10$. They are not being identified with the stationary law of the earlier symmetric chain. On the same two-block partition, conditional expectation is\n\\[\nE=\\begin{pmatrix}\n1/3&2/3&0&0\\\\1/3&2/3&0&0\\\\0&0&3/7&4/7\\\\0&0&3/7&4/7\n\\end{pmatrix}.\n\\]\nIt replaces function values by their \\emph{weighted block average}. In particular, $f=(0,3,2,5)^{\\mathsf T}$ becomes $Ef=(2,2,26/7,26/7)^{\\mathsf T}$.\n\\vfig{weighted_projection}\n\\platecaption{The bars are function values, not state probabilities. Their total need not be one. The block masses $3/10$ and $7/10$ determine the denominators in the averaging formula.}\n\\head{Which inner product makes this an orthogonal projection?}\nPut $W=\\operatorname{diag}(\\pi)$. Direct multiplication checks\n\\[\nE^2=E,\\qquad WE=E^{\\mathsf T}W,\\qquad E\\ones=\\ones.\n\\]\nThe middle equation is weighted self-adjointness. Ordinary symmetry f…","0&0&3/7&4/7\n\\end{pmatrix}.\n\\]\nIt replaces function values by their \\emph{weighted block average}. In particular, $f=(0,3,2,5)^{\\mathsf T}$ becomes $Ef=(2,2,26/7,26/7)^{\\mathsf T}$.\n\\vfig{weighted_projection}\n\\platecaption{The bars are function values, not state probabilities. Their total need not be one. The block masses $3/10$ and $7/10$ determine the denominators in the averaging formula.}\n\\head{Which inner product makes this an orthogonal projection?}\nPut $W=\\operatorname{diag}(\\pi)$. Direct multiplication checks\n\\[\nE^2=E,\\qquad WE=E^{\\mathsf T}W,\\qquad E\\ones=\\ones.\n\\]\nThe middle equation is weighted self-adjointness. Ordinary symmetry fails: $E_{01}=2/3$ whereas $E_{10}=1/3$. Testing $E=E^{\\mathsf T}$ would therefore reject a correct conditional-expectation projector.","ir \\emph{weighted block average}. In particular, $f=(0,3,2,5)^{\\mathsf T}$ becomes $Ef=(2,2,26/7,26/7)^{\\mathsf T}$.\n\\vfig{weighted_projection}\n\\platecaption{The bars are function values, not state probabilities. Their total need not be one. The block masses $3/10$ and $7/10$ determine the denominators in the averaging formula.}\n\\head{Which inner product makes this an orthogonal projection?}\nPut $W=\\operatorname{diag}(\\pi)$. Direct multiplication checks\n\\[\nE^2=E,\\qquad WE=E^{\\mathsf T}W,\\qquad E\\ones=\\ones.\n\\]\nThe middle equation is weighted self-adjointness. Ordinary symmetry fails: $E_{01}=2/3$ whereas $E_{10}=1/3$. Testing $E=E^{\\mathsf T}$ would therefore reject a correct conditional-expectation projector.",")^{\\mathsf T}$ becomes $Ef=(2,2,26/7,26/7)^{\\mathsf T}$.\n\\vfig{weighted_projection}\n\\platecaption{The bars are function values, not state probabilities. Their total need not be one. The block masses $3/10$ and $7/10$ determine the denominators in the averaging formula.}\n\\head{Which inner product makes this an orthogonal projection?}\nPut $W=\\operatorname{diag}(\\pi)$. Direct multiplication checks\n\\[\nE^2=E,\\qquad WE=E^{\\mathsf T}W,\\qquad E\\ones=\\ones.\n\\]\nThe middle equation is weighted self-adjointness. Ordinary symmetry fails: $E_{01}=2/3$ whereas $E_{10}=1/3$. Testing $E=E^{\\mathsf T}$ would therefore reject a correct conditional-expectation projector.","state probabilities. Their total need not be one. The block masses $3/10$ and $7/10$ determine the denominators in the averaging formula.}\n\\head{Which inner product makes this an orthogonal projection?}\nPut $W=\\operatorname{diag}(\\pi)$. Direct multiplication checks\n\\[\nE^2=E,\\qquad WE=E^{\\mathsf T}W,\\qquad E\\ones=\\ones.\n\\]\nThe middle equation is weighted self-adjointness. Ordinary symmetry fails: $E_{01}=2/3$ whereas $E_{10}=1/3$. Testing $E=E^{\\mathsf T}$ would therefore reject a correct conditional-expectation projector.","Their total need not be one. The block masses $3/10$ and $7/10$ determine the denominators in the averaging formula.}\n\\head{Which inner product makes this an orthogonal projection?}\nPut $W=\\operatorname{diag}(\\pi)$. Direct multiplication checks\n\\[\nE^2=E,\\qquad WE=E^{\\mathsf T}W,\\qquad E\\ones=\\ones.\n\\]\nThe middle equation is weighted self-adjointness. Ordinary symmetry fails: $E_{01}=2/3$ whereas $E_{10}=1/3$. Testing $E=E^{\\mathsf T}$ would therefore reject a correct conditional-expectation projector.","e one. The block masses $3/10$ and $7/10$ determine the denominators in the averaging formula.}\n\\head{Which inner product makes this an orthogonal projection?}\nPut $W=\\operatorname{diag}(\\pi)$. Direct multiplication checks\n\\[\nE^2=E,\\qquad WE=E^{\\mathsf T}W,\\qquad E\\ones=\\ones.\n\\]\nThe middle equation is weighted self-adjointness. Ordinary symmetry fails: $E_{01}=2/3$ whereas $E_{10}=1/3$. Testing $E=E^{\\mathsf T}$ would therefore reject a correct conditional-expectation projector.","The range consists exactly of functions constant on each block. The error $f-Ef$ has weighted mean zero inside every block, so it is weighted-orthogonal to every block-constant function. This is the finite projection interpretation of conditional expectation.","In contrast, $C$ acts on probability columns by summing masses. The shapes of $C$ and $E$ differ, and their domains have different interpretations. The notation ``coarsening'' alone does not specify which operation is intended.\n\\takeaway{An exact weighted projector can be oblique in the unweighted Euclidean picture. Keep the measure and the inner product in the model.}","\\vpage{8}{Where the variance goes}{MARTINGALE PACKETS / THREE EXACT PIECES}{Chapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences.}\nKeep the weights and observable from \\pagerefV{7}. Let $E_0f$ be the constant global mean, let $E_1=E$ average within A and B, and let $E_2=I$ retain every value. Define\n\\[\nD_0=E_0,\\qquad D_1=E_1-E_0,\\qquad D_2=I-E_1.\n\\]\nThen $D_0f$ is the mean, $D_1f$ describes the contrast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i","\\vpage{8}{Where the variance goes}{MARTINGALE PACKETS / THREE EXACT PIECES}{Chapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences.}\nKeep the weights and observable from \\pagerefV{7}. Let $E_0f$ be the constant global mean, let $E_1=E$ average within A and B, and let $E_2=I$ retain every value. Define\n\\[\nD_0=E_0,\\qquad D_1=E_1-E_0,\\qquad D_2=I-E_1.\n\\]\nThen $D_0f$ is the mean, $D_1f$ describes the contrast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor","\\vpage{8}{Where the variance goes}{MARTINGALE PACKETS / THREE EXACT PIECES}{Chapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences.}\nKeep the weights and observable from \\pagerefV{7}. Let $E_0f$ be the constant global mean, let $E_1=E$ average within A and B, and let $E_2=I$ retain every value. Define\n\\[\nD_0=E_0,\\qquad D_1=E_1-E_0,\\qquad D_2=I-E_1.\n\\]\nThen $D_0f$ is the mean, $D_1f$ describes the contrast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energ","\\vpage{8}{Where the variance goes}{MARTINGALE PACKETS / THREE EXACT PIECES}{Chapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences.}\nKeep the weights and observable from \\pagerefV{7}. Let $E_0f$ be the constant global mean, let $E_1=E$ average within A and B, and let $E_2=I$ retain every value. Define\n\\[\nD_0=E_0,\\qquad D_1=E_1-E_0,\\qquad D_2=I-E_1.\n\\]\nThen $D_0f$ is the mean, $D_1f$ describes the contrast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{V","\\vpage{8}{Where the variance goes}{MARTINGALE PACKETS / THREE EXACT PIECES}{Chapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences.}\nKeep the weights and observable from \\pagerefV{7}. Let $E_0f$ be the constant global mean, let $E_1=E$ average within A and B, and let $E_2=I$ retain every value. Define\n\\[\nD_0=E_0,\\qquad D_1=E_1-E_0,\\qquad D_2=I-E_1.\n\\]\nThen $D_0f$ is the mean, $D_1f$ describes the contrast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=1","}{Where the variance goes}{MARTINGALE PACKETS / THREE EXACT PIECES}{Chapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences.}\nKeep the weights and observable from \\pagerefV{7}. Let $E_0f$ be the constant global mean, let $E_1=E$ average within A and B, and let $E_2=I$ retain every value. Define\n\\[\nD_0=E_0,\\qquad D_1=E_1-E_0,\\qquad D_2=I-E_1.\n\\]\nThen $D_0f$ is the mean, $D_1f$ describes the contrast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16…","XACT PIECES}{Chapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences.}\nKeep the weights and observable from \\pagerefV{7}. Let $E_0f$ be the constant global mean, let $E_1=E$ average within A and B, and let $E_2=I$ retain every value. Define\n\\[\nD_0=E_0,\\qquad D_1=E_1-E_0,\\qquad D_2=I-E_1.\n\\]\nThen $D_0f$ is the mean, $D_1f$ describes the contrast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16}{5}\\right)^2\n=\\frac{69}{25}\n=\\frac{108}{175}+\\frac{15}…","ast between block means, and $D_2f$ is the fluctuation left inside each block.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16}{5}\\right)^2\n=\\frac{69}{25}\n=\\frac{108}{175}+\\frac{15}{7}.\n\\]\nThe first term is the energy of the between-block contrast, and the second is the weighted within-block variance. This is a decomposition of one observable in one fixed weighted space.","ck.\n\\vfig{variance}\n\\platecaption{The strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16}{5}\\right)^2\n=\\frac{69}{25}\n=\\frac{108}{175}+\\frac{15}{7}.\n\\]\nThe first term is the energy of the between-block contrast, and the second is the weighted within-block variance. This is a decomposition of one observable in one fixed weighted space.","al to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16}{5}\\right)^2\n=\\frac{69}{25}\n=\\frac{108}{175}+\\frac{15}{7}.\n\\]\nThe first term is the energy of the between-block contrast, and the second is the weighted within-block variance. This is a decomposition of one observable in one fixed weighted space.","rgies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16}{5}\\right)^2\n=\\frac{69}{25}\n=\\frac{108}{175}+\\frac{15}{7}.\n\\]\nThe first term is the energy of the between-block contrast, and the second is the weighted within-block variance. This is a decomposition of one observable in one fixed weighted space.","t is small but not zero. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16}{5}\\right)^2\n=\\frac{69}{25}\n=\\frac{108}{175}+\\frac{15}{7}.\n\\]\nThe first term is the energy of the between-block contrast, and the second is the weighted within-block variance. This is a decomposition of one observable in one fixed weighted space.","o. The vectors below specify the corresponding directions, not only their sizes.}\n\\head{The energy identity follows from actual orthogonality}\nThe tower identities give $E_0E_1=E_1E_0=E_0$. Together with weighted self-adjointness, these imply\n\\[\nD_iD_j=0\\ (i\\ne j),\\qquad \\sum_{j=0}^2 D_j=I.\n\\]\nFor the chosen $f$, the total weighted energy is $\\|f\\|_\\pi^2=13$ and the mean is $16/5$. Consequently\n\\[\n\\operatorname{Var}_\\pi(f)\n=13-\\left(\\frac{16}{5}\\right)^2\n=\\frac{69}{25}\n=\\frac{108}{175}+\\frac{15}{7}.\n\\]\nThe first term is the energy of the between-block contrast, and the second is the weighted within-block variance. This is a decomposition of one observable in one fixed weighted space.","Changing the probability weights between levels would change the inner products and the averaging operators. The displayed orthogonal decomposition would no longer follow merely from nesting the sets. Likewise, choosing unrelated projections can preserve the telescope $\\sum D_j=I$ while losing orthogonality and the energy sum.\n\\takeaway{The differences between nested conditional expectations are not just bookkeeping. They are orthogonal, separately measurable parts of the original signal.}","\\vpage{9}{Evidence changes shape and mass}{BAYESIAN TRANSFER / AN EXACT POSTERIOR}{Chapter VI, Part I: Theorems 7.1--7.3. Normalization, packet reconstruction, and positive evidence.}\nStart with $p=(1,1,1,1)^{\\mathsf T}/4$ and likelihood weights $\\lambda=(4,1,1,1)^{\\mathsf T}$. The unnormalized vector is $(1,1/4,1/4,1/4)^{\\mathsf T}$, its evidence is $Z=7/4$, and\n\\[\n\\mathcal B_\\lambda(p)=\\frac{(4,1,1,1)^{\\mathsf T}}7.\n\\]\nLikelihood weights may be rescaled, so an evidence larger than one is not a probability claim. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged.\n\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\m","\\vpage{9}{Evidence changes shape and mass}{BAYESIAN TRANSFER / AN EXACT POSTERIOR}{Chapter VI, Part I: Theorems 7.1--7.3. Normalization, packet reconstruction, and positive evidence.}\nStart with $p=(1,1,1,1)^{\\mathsf T}/4$ and likelihood weights $\\lambda=(4,1,1,1)^{\\mathsf T}$. The unnormalized vector is $(1,1/4,1/4,1/4)^{\\mathsf T}$, its evidence is $Z=7/4$, and\n\\[\n\\mathcal B_\\lambda(p)=\\frac{(4,1,1,1)^{\\mathsf T}}7.\n\\]\nLikelihood weights may be rescaled, so an evidence larger than one is not a probability claim. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged.\n\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting ma","\\vpage{9}{Evidence changes shape and mass}{BAYESIAN TRANSFER / AN EXACT POSTERIOR}{Chapter VI, Part I: Theorems 7.1--7.3. Normalization, packet reconstruction, and positive evidence.}\nStart with $p=(1,1,1,1)^{\\mathsf T}/4$ and likelihood weights $\\lambda=(4,1,1,1)^{\\mathsf T}$. The unnormalized vector is $(1,1/4,1/4,1/4)^{\\mathsf T}$, its evidence is $Z=7/4$, and\n\\[\n\\mathcal B_\\lambda(p)=\\frac{(4,1,1,1)^{\\mathsf T}}7.\n\\]\nLikelihood weights may be rescaled, so an evidence larger than one is not a probability claim. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged.\n\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting masses and conditional shapes are\n\\[\nm_{\\mathrm A}=5/7,\\quad","\\vpage{9}{Evidence changes shape and mass}{BAYESIAN TRANSFER / AN EXACT POSTERIOR}{Chapter VI, Part I: Theorems 7.1--7.3. Normalization, packet reconstruction, and positive evidence.}\nStart with $p=(1,1,1,1)^{\\mathsf T}/4$ and likelihood weights $\\lambda=(4,1,1,1)^{\\mathsf T}$. The unnormalized vector is $(1,1/4,1/4,1/4)^{\\mathsf T}$, its evidence is $Z=7/4$, and\n\\[\n\\mathcal B_\\lambda(p)=\\frac{(4,1,1,1)^{\\mathsf T}}7.\n\\]\nLikelihood weights may be rescaled, so an evidence larger than one is not a probability claim. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged.\n\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting masses and conditional shapes are\n\\[\nm_{\\mathrm A}=5/7,\\quad q_{\\mathrm A}=(4/5,1/5,0","\\vpage{9}{Evidence changes shape and mass}{BAYESIAN TRANSFER / AN EXACT POSTERIOR}{Chapter VI, Part I: Theorems 7.1--7.3. Normalization, packet reconstruction, and positive evidence.}\nStart with $p=(1,1,1,1)^{\\mathsf T}/4$ and likelihood weights $\\lambda=(4,1,1,1)^{\\mathsf T}$. The unnormalized vector is $(1,1/4,1/4,1/4)^{\\mathsf T}$, its evidence is $Z=7/4$, and\n\\[\n\\mathcal B_\\lambda(p)=\\frac{(4,1,1,1)^{\\mathsf T}}7.\n\\]\nLikelihood weights may be rescaled, so an evidence larger than one is not a probability claim. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged.\n\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting masses and conditional shapes are\n\\[\nm_{\\mathrm A}=5/7,\\quad q_{\\mathrm A}=(4/5,1/5,0,0)^{\\mathsf T},\n\\qquad m_{\\mathrm B}=2/7…","im. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged.\n\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting masses and conditional shapes are\n\\[\nm_{\\mathrm A}=5/7,\\quad q_{\\mathrm A}=(4/5,1/5,0,0)^{\\mathsf T},\n\\qquad m_{\\mathrm B}=2/7,\\quad q_{\\mathrm B}=(0,0,1/2,1/2)^{\\mathsf T}.\n\\]\nThus $m_{\\mathrm A}q_{\\mathrm A}+m_{\\mathrm B}q_{\\mathrm B}$ is exactly the displayed posterior. Keeping only the masses would not reconstruct the fine state.","lihoods by one common positive factor leaves this posterior unchanged.\n\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting masses and conditional shapes are\n\\[\nm_{\\mathrm A}=5/7,\\quad q_{\\mathrm A}=(4/5,1/5,0,0)^{\\mathsf T},\n\\qquad m_{\\mathrm B}=2/7,\\quad q_{\\mathrm B}=(0,0,1/2,1/2)^{\\mathsf T}.\n\\]\nThus $m_{\\mathrm A}q_{\\mathrm A}+m_{\\mathrm B}q_{\\mathrm B}$ is exactly the displayed posterior. Keeping only the masses would not reconstruct the fine state.","\\vfig{evidence}\n\\platecaption{The A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting masses and conditional shapes are\n\\[\nm_{\\mathrm A}=5/7,\\quad q_{\\mathrm A}=(4/5,1/5,0,0)^{\\mathsf T},\n\\qquad m_{\\mathrm B}=2/7,\\quad q_{\\mathrm B}=(0,0,1/2,1/2)^{\\mathsf T}.\n\\]\nThus $m_{\\mathrm A}q_{\\mathrm A}+m_{\\mathrm B}q_{\\mathrm B}$ is exactly the displayed posterior. Keeping only the masses would not reconstruct the fine state.","total mass. These are different effects of the same likelihood.}\n\\head{Reconstruct from packet mass times packet shape}\nThe unnormalized packet evidences are $Z_{\\mathrm A}=5/4$ and $Z_{\\mathrm B}=1/2$. The resulting masses and conditional shapes are\n\\[\nm_{\\mathrm A}=5/7,\\quad q_{\\mathrm A}=(4/5,1/5,0,0)^{\\mathsf T},\n\\qquad m_{\\mathrm B}=2/7,\\quad q_{\\mathrm B}=(0,0,1/2,1/2)^{\\mathsf T}.\n\\]\nThus $m_{\\mathrm A}q_{\\mathrm A}+m_{\\mathrm B}q_{\\mathrm B}$ is exactly the displayed posterior. Keeping only the masses would not reconstruct the fine state.","There is no need to divide by a zero packet evidence. For likelihood $(0,0,1,1)$, packet A has mass zero and contributes the zero vector. Only the occupied B packet has a conditional probability vector. If the \\emph{total} evidence is zero, however, the Bayesian update is not defined and the implementation rejects the input.\n\\takeaway{Bayesian reconstruction uses two kinds of data: how much posterior mass a packet carries and how that mass is distributed inside the packet.}","\\vpage{10}{Two priors that fool the coarse model}{SUFFICIENCY / A COMPUTABLE FAILURE WITNESS}{Chapter VI, Part I: Theorem 8.2 and the one-packet exception.}\nKeep the same likelihood $\\lambda=(4,1,1,1)^{\\mathsf T}$ and partition A$=\\{0,1\\}$, B$=\\{2,3\\}$. Compare\n\\[\np^{(a)}=(1/2,0,1/2,0)^{\\mathsf T},\\qquad\np^{(b)}=(0,1/2,1/2,0)^{\\mathsf T}.\n\\]\nBoth have coarse prior $(1/2,1/2)^{\\mathsf T}$. A procedure given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distr","\\vpage{10}{Two priors that fool the coarse model}{SUFFICIENCY / A COMPUTABLE FAILURE WITNESS}{Chapter VI, Part I: Theorem 8.2 and the one-packet exception.}\nKeep the same likelihood $\\lambda=(4,1,1,1)^{\\mathsf T}$ and partition A$=\\{0,1\\}$, B$=\\{2,3\\}$. Compare\n\\[\np^{(a)}=(1/2,0,1/2,0)^{\\mathsf T},\\qquad\np^{(b)}=(0,1/2,1/2,0)^{\\mathsf T}.\n\\]\nBoth have coarse prior $(1/2,1/2)^{\\mathsf T}$. A procedure given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","ikelihood $\\lambda=(4,1,1,1)^{\\mathsf T}$ and partition A$=\\{0,1\\}$, B$=\\{2,3\\}$. Compare\n\\[\np^{(a)}=(1/2,0,1/2,0)^{\\mathsf T},\\qquad\np^{(b)}=(0,1/2,1/2,0)^{\\mathsf T}.\n\\]\nBoth have coarse prior $(1/2,1/2)^{\\mathsf T}$. A procedure given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","da=(4,1,1,1)^{\\mathsf T}$ and partition A$=\\{0,1\\}$, B$=\\{2,3\\}$. Compare\n\\[\np^{(a)}=(1/2,0,1/2,0)^{\\mathsf T},\\qquad\np^{(b)}=(0,1/2,1/2,0)^{\\mathsf T}.\n\\]\nBoth have coarse prior $(1/2,1/2)^{\\mathsf T}$. A procedure given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","rior $(1/2,1/2)^{\\mathsf T}$. A procedure given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","2)^{\\mathsf T}$. A procedure given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","procedure given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","given only that two-entry vector cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","r cannot distinguish them.\n\\vfig{insufficiency}\n\\platecaption{After the identical fine likelihood is applied, the coarse posteriors are $(4/5,1/5)$ and $(1/2,1/2)$. A single deterministic update of the common coarse prior cannot produce both.}\n\\head{Why this witnesses the necessity of blockwise constancy}\nThe A evidence for $p^{(a)}$ is $2$, whereas it is $1/2$ for $p^{(b)}$. Their B evidence is $1/2$ in both cases. The posterior odds therefore distinguish the hidden distribution inside A.","For strictly positive fine and coarse likelihoods, and at least two coarse blocks, the manuscript proves the all-prior criterion\n\\[\nC\\mathcal B_\\lambda(p)=\\mathcal B_{\\bar\\lambda}(Cp)\\quad\\hbox{for every }p\n\\quad\\Longleftrightarrow\\quad\n\\lambda_i=c\\bar\\lambda_a\\quad(i\\in I_a)\n\\]\nfor one global $c>0$. The global scale is invisible after normalization. A different arbitrary scale for each packet is not invisible: it changes posterior odds.","If all states are merged into a single packet, every normalized posterior coarsens to $(1)$. That degenerate commutation holds for every likelihood and cannot identify any within-packet information. It is explicitly excluded from the nontrivial criterion.\n\\takeaway{A prior-dependent effective likelihood can always summarize the update. A fixed likelihood that works for every prior is a much stronger requirement.}","\\vpage{11}{The exact information-loss vector}{BAYESIAN DEFECT / MORE THAN AN ERROR BAR}{Chapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6.}\nFor a prior $p$, let $P=Cp$ and let $r_a(p)$ be the conditional average of the fine likelihood inside packet $a$. For the uniform prior and $\\lambda=(4,1,1,1)$,\n\\[\nP=(1/2,1/2)^{\\mathsf T},\\qquad r=(5/2,1)^{\\mathsf T}.\n\\]\nThe exact coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and","\\vpage{11}{The exact information-loss vector}{BAYESIAN DEFECT / MORE THAN AN ERROR BAR}{Chapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6.}\nFor a prior $p$, let $P=Cp$ and let $r_a(p)$ be the conditional average of the fine likelihood inside packet $a$. For the uniform prior and $\\lambda=(4,1,1,1)$,\n\\[\nP=(1/2,1/2)^{\\mathsf T},\\qquad r=(5/2,1)^{\\mathsf T}.\n\\]\nThe exact coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. H","\\vpage{11}{The exact information-loss vector}{BAYESIAN DEFECT / MORE THAN AN ERROR BAR}{Chapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6.}\nFor a prior $p$, let $P=Cp$ and let $r_a(p)$ be the conditional average of the fine likelihood inside packet $a$. For the uniform prior and $\\lambda=(4,1,1,1)$,\n\\[\nP=(1/2,1/2)^{\\mathsf T},\\qquad r=(5/2,1)^{\\mathsf T}.\n\\]\nThe exact coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_","\\vpage{11}{The exact information-loss vector}{BAYESIAN DEFECT / MORE THAN AN ERROR BAR}{Chapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6.}\nFor a prior $p$, let $P=Cp$ and let $r_a(p)$ be the conditional average of the fine likelihood inside packet $a$. For the uniform prior and $\\lambda=(4,1,1,1)$,\n\\[\nP=(1/2,1/2)^{\\mathsf T},\\qquad r=(5/2,1)^{\\mathsf T}.\n\\]\nThe exact coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{p","\\vpage{11}{The exact information-loss vector}{BAYESIAN DEFECT / MORE THAN AN ERROR BAR}{Chapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6.}\nFor a prior $p$, let $P=Cp$ and let $r_a(p)$ be the conditional average of the fine likelihood inside packet $a$. For the uniform prior and $\\lambda=(4,1,1,1)$,\n\\[\nP=(1/2,1/2)^{\\mathsf T},\\qquad r=(5/2,1)^{\\mathsf T}.\n\\]\nThe exact coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{pmatrix}1/4&-1/4","page{11}{The exact information-loss vector}{BAYESIAN DEFECT / MORE THAN AN ERROR BAR}{Chapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6.}\nFor a prior $p$, let $P=Cp$ and let $r_a(p)$ be the conditional average of the fine likelihood inside packet $a$. For the uniform prior and $\\lambda=(4,1,1,1)$,\n\\[\nP=(1/2,1/2)^{\\mathsf T},\\qquad r=(5/2,1)^{\\mathsf T}.\n\\]\nThe exact coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{pmatrix}1/4&-1/4",", Part I: Theorems 8.1 and 8.4, and Corollary 8.6.}\nFor a prior $p$, let $P=Cp$ and let $r_a(p)$ be the conditional average of the fine likelihood inside packet $a$. For the uniform prior and $\\lambda=(4,1,1,1)$,\n\\[\nP=(1/2,1/2)^{\\mathsf T},\\qquad r=(5/2,1)^{\\mathsf T}.\n\\]\nThe exact coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{pmatrix}1/4&-1/4","act coarse posterior is $\\mathcal B_r(P)=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{pmatrix}1/4&-1/4","=(5/7,2/7)^{\\mathsf T}$. Suppose instead that the declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{pmatrix}1/4&-1/4","e declared coarse likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{pmatrix}1/4&-1/4","likelihood is $\\bar\\lambda=(1,1)$.\n\\vfig{loss_vector}\n\\platecaption{The posterior error lies along the mass-preserving direction of the two-state simplex. Its entries are opposite because both endpoints have total mass one.}\n\\head{Compute the factorization, including the null direction}\nHere $\\bar b=\\mathcal B_{\\bar\\lambda}(P)=P$, $\\delta=r-\\bar\\lambda=(3/2,0)^{\\mathsf T}$, and $Z_r=7/4$. Hence\n\\[\n\\frac1{Z_r}\\left(\\diag(P)-\\bar bP^{\\mathsf T}\\right)\\delta\n=\\frac4{7}\\begin{pmatrix}1/4&-1/4\\\\-1/4&1/4\\end{pmatrix}\n\\begin{pmatrix}3/2\\\\0\\end{pmatrix}\n=\\begin{pmatrix}3/14\\\\-3/14\\end{pmatrix}.\n\\]\nThis is the actual fine-to-coarse discrepancy, not a fitted approximation. The matrix kills $\\bar\\lambda$, recording the common-scale null direction. Its $\\ell^1$ error is $3/7$.","-3/14\\end{pmatrix}.\n\\]\nThis is the actual fine-to-coarse discrepancy, not a fitted approximation. The matrix kills $\\bar\\lambda$, recording the common-scale null direction. Its $\\ell^1$ error is $3/7$.","For a family of likelihoods, states may be grouped by equality of their \\emph{whole likelihood profile}. The family $(3,3,1,1)$ and $(1,2,1,2)$ separates all four states even though either likelihood alone permits a coarser grouping. The common refinement is therefore the discrete partition. The theorem's one-packet exception still remains.\n\\takeaway{The defect factors through exactly the conditional evidence information missing from the declared coarse model. A new observation family can require a finer partition.}","\\vpage{12}{A nonlinear semigroup of odds}{BAYESIAN FLOW / COMPOSITION WITHOUT LINEARITY}{Chapter VI, Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field.}\nLet the score vector be $g=(1,0)$ and start at $(1/2,1/2)$. The likelihood at time $t$ is $(e^t,1)$, so the first posterior coordinate is\n\\[\np_0(t)=\\frac{e^t}{1+e^t},\\qquad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ giv","\\vpage{12}{A nonlinear semigroup of odds}{BAYESIAN FLOW / COMPOSITION WITHOUT LINEARITY}{Chapter VI, Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field.}\nLet the score vector be $g=(1,0)$ and start at $(1/2,1/2)$. The likelihood at time $t$ is $(e^t,1)$, so the first posterior coordinate is\n\\[\np_0(t)=\\frac{e^t}{1+e^t},\\qquad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup.","\\vpage{12}{A nonlinear semigroup of odds}{BAYESIAN FLOW / COMPOSITION WITHOUT LINEARITY}{Chapter VI, Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field.}\nLet the score vector be $g=(1,0)$ and start at $(1/2,1/2)$. The likelihood at time $t$ is $(e^t,1)$, so the first posterior coordinate is\n\\[\np_0(t)=\\frac{e^t}{1+e^t},\\qquad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=","\\vpage{12}{A nonlinear semigroup of odds}{BAYESIAN FLOW / COMPOSITION WITHOUT LINEARITY}{Chapter VI, Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field.}\nLet the score vector be $g=(1,0)$ and start at $(1/2,1/2)$. The likelihood at time $t$ is $(e^t,1)$, so the first posterior coordinate is\n\\[\np_0(t)=\\frac{e^t}{1+e^t},\\qquad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while t","\\vpage{12}{A nonlinear semigroup of odds}{BAYESIAN FLOW / COMPOSITION WITHOUT LINEARITY}{Chapter VI, Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field.}\nLet the score vector be $g=(1,0)$ and start at $(1/2,1/2)$. The likelihood at time $t$ is $(e^t,1)$, so the first posterior coordinate is\n\\[\np_0(t)=\\frac{e^t}{1+e^t},\\qquad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","HOUT LINEARITY}{Chapter VI, Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field.}\nLet the score vector be $g=(1,0)$ and start at $(1/2,1/2)$. The likelihood at time $t$ is $(e^t,1)$, so the first posterior coordinate is\n\\[\np_0(t)=\\frac{e^t}{1+e^t},\\qquad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","Theorems 7.1 and 9.1. Likelihood composition and the replicator field.}\nLet the score vector be $g=(1,0)$ and start at $(1/2,1/2)$. The likelihood at time $t$ is $(e^t,1)$, so the first posterior coordinate is\n\\[\np_0(t)=\\frac{e^t}{1+e^t},\\qquad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","ad p_1(t)=\\frac1{1+e^t}.\n\\]\nThe odds $p_0/p_1$ grow exponentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","nentially, while the probabilities remain inside the simplex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","lex. At $t=\\log 2$ the posterior is exactly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","ctly $(2/3,1/3)$.\n\\vfig{odds}\n\\platecaption{The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","id and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","sed by the shared evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","ed evidence denominator, not by a nonlinear likelihood multiplication law.}\n\\head{Two properties that must not be conflated}\nIf $a>0$ multiplies the odds, the first-coordinate map is\n\\[\n\\phi_a(x)=\\frac{ax}{1+(a-1)x}.\n\\]\nDirect algebra gives $\\phi_a(\\phi_b(x))=\\phi_{ab}(x)$. Thus $a=e^t$ gives a nonlinear semigroup. Nevertheless, $\\phi_3(1/2)=3/4$, while the mean of $\\phi_3(0)$ and $\\phi_3(1)$ is $1/2$. Superposition fails.","Differentiation of the normalized flow yields the centered score equation\n\\[\n\\dot p_i=\\bigl(g_i-\\langle g,p\\rangle\\bigr)p_i.\n\\]\nFor the displayed model, $\\dot p_0=p_0(1-p_0)$. The negative average-score term is what keeps the total derivative zero. Omitting the derivative of the normalizer would not preserve mass.","\\vpage{13}{Prediction and observation do not commute}{HIDDEN MARKOV FILTERING / TIME ORDER}{Chapter VI, Part I: Proposition 9.3. Linear prediction followed by a positive-evidence update.}\nUse the two-state kernel $K_2$ from \\pagerefV{2}, the initial distribution $e_0$, and likelihood $(1,1/2)$. Predicting first gives $(3/4,1/4)$. Its likelihood-weighted vector is $(3/4,1/8)$, with evidence $7/8$, so the posterior is $(6/7,1/7)$.","and observation do not commute}{HIDDEN MARKOV FILTERING / TIME ORDER}{Chapter VI, Part I: Proposition 9.3. Linear prediction followed by a positive-evidence update.}\nUse the two-state kernel $K_2$ from \\pagerefV{2}, the initial distribution $e_0$, and likelihood $(1,1/2)$. Predicting first gives $(3/4,1/4)$. Its likelihood-weighted vector is $(3/4,1/8)$, with evidence $7/8$, so the posterior is $(6/7,1/7)$.","Updating first leaves $e_0$ unchanged. Predicting afterward gives $(3/4,1/4)$ instead. The order of operations expresses which time the observation concerns.\n\\vfig{filter_order}\n\\platecaption{Both routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem.}\n\\head{Keep an unnormalized path alongside the filter}\nFor a sequence of observations, let $D_r=\\diag(\\lambda^{(r)})$. The forward recursion is\n\\[\n\\widehat p_r=Kp_{r-1},\\qquad p_r=D_r\\widehat p_r/Z_r.\n\\]\nIts unnormalized companion is\n\\[\nv_r=D_rK\\cdots D_1Kp_0,\n\\qquad p_r=\\frac{v_r}{\\ones^{\\mathsf T}v_r},\n\\qquad \\ones^{\\mathsf T}v_r=\\prod_{j=1}^r Z_j.\n\\]\nThe product of evidences records the","Updating first leaves $e_0$ unchanged. Predicting afterward gives $(3/4,1/4)$ instead. The order of operations expresses which time the observation concerns.\n\\vfig{filter_order}\n\\platecaption{Both routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem.}\n\\head{Keep an unnormalized path alongside the filter}\nFor a sequence of observations, let $D_r=\\diag(\\lambda^{(r)})$. The forward recursion is\n\\[\n\\widehat p_r=Kp_{r-1},\\qquad p_r=D_r\\widehat p_r/Z_r.\n\\]\nIts unnormalized companion is\n\\[\nv_r=D_rK\\cdots D_1Kp_0,\n\\qquad p_r=\\frac{v_r}{\\ones^{\\mathsf T}v_r},\n\\qquad \\ones^{\\mathsf T}v_r=\\prod_{j=1}^r Z_j.\n\\]\nThe product of evidences records the normalization lost at each individual step. The f","$ unchanged. Predicting afterward gives $(3/4,1/4)$ instead. The order of operations expresses which time the observation concerns.\n\\vfig{filter_order}\n\\platecaption{Both routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem.}\n\\head{Keep an unnormalized path alongside the filter}\nFor a sequence of observations, let $D_r=\\diag(\\lambda^{(r)})$. The forward recursion is\n\\[\n\\widehat p_r=Kp_{r-1},\\qquad p_r=D_r\\widehat p_r/Z_r.\n\\]\nIts unnormalized companion is\n\\[\nv_r=D_rK\\cdots D_1Kp_0,\n\\qquad p_r=\\frac{v_r}{\\ones^{\\mathsf T}v_r},\n\\qquad \\ones^{\\mathsf T}v_r=\\prod_{j=1}^r Z_j.\n\\]\nThe product of evidences records the normalization lost at each individual step. The four-state computation on \\pagerefV{26} checks both paths independently, including their coarse versions.","stead. The order of operations expresses which time the observation concerns.\n\\vfig{filter_order}\n\\platecaption{Both routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem.}\n\\head{Keep an unnormalized path alongside the filter}\nFor a sequence of observations, let $D_r=\\diag(\\lambda^{(r)})$. The forward recursion is\n\\[\n\\widehat p_r=Kp_{r-1},\\qquad p_r=D_r\\widehat p_r/Z_r.\n\\]\nIts unnormalized companion is\n\\[\nv_r=D_rK\\cdots D_1Kp_0,\n\\qquad p_r=\\frac{v_r}{\\ones^{\\mathsf T}v_r},\n\\qquad \\ones^{\\mathsf T}v_r=\\prod_{j=1}^r Z_j.\n\\]\nThe product of evidences records the normalization lost at each individual step. The four-state computation on \\pagerefV{26} checks both paths independently, including their coarse versions.","_order}\n\\platecaption{Both routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem.}\n\\head{Keep an unnormalized path alongside the filter}\nFor a sequence of observations, let $D_r=\\diag(\\lambda^{(r)})$. The forward recursion is\n\\[\n\\widehat p_r=Kp_{r-1},\\qquad p_r=D_r\\widehat p_r/Z_r.\n\\]\nIts unnormalized companion is\n\\[\nv_r=D_rK\\cdots D_1Kp_0,\n\\qquad p_r=\\frac{v_r}{\\ones^{\\mathsf T}v_r},\n\\qquad \\ones^{\\mathsf T}v_r=\\prod_{j=1}^r Z_j.\n\\]\nThe product of evidences records the normalization lost at each individual step. The four-state computation on \\pagerefV{26} checks both paths independently, including their coarse versions.","Here $D_rK$ means an observation after prediction. In the Feynman--Kac example on \\pagerefV{20}, departure-state weighting has forward operator $K D_w$. Both conventions are legitimate once their meanings are fixed, but they cannot be substituted for one another.\n\\takeaway{Write the chronological sequence before multiplying matrices. A product can be normalized correctly and still answer the wrong time-ordered question.}","\\vpage{14}{Conjugacy tracks exponents}{BETA MODEL / POSTERIOR AND EVIDENCE}{Chapter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model.}\nFor a small exact example, take the $\\mathrm{Beta}(2,3)$ density\n\\[\n\\rho(\\theta)=12\\theta(1-\\theta)^2,\\qquad0\\le\\theta\\le1.\n\\]\nObserve a specified ordered sequence with two successes and one failure. Its likelihood is $\\theta^2(1-\\theta)$. Multiplication adds exponents, so the posterior belongs to $\\mathrm{Beta}(4,4)$ with density $140\\theta^3(1-\\theta)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/","\\vpage{14}{Conjugacy tracks exponents}{BETA MODEL / POSTERIOR AND EVIDENCE}{Chapter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model.}\nFor a small exact example, take the $\\mathrm{Beta}(2,3)$ density\n\\[\n\\rho(\\theta)=12\\theta(1-\\theta)^2,\\qquad0\\le\\theta\\le1.\n\\]\nObserve a specified ordered sequence with two successes and one failure. Its likelihood is $\\theta^2(1-\\theta)$. Multiplication adds exponents, so the posterior belongs to $\\mathrm{Beta}(4,4)$ with density $140\\theta^3(1-\\theta)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is ins","\\vpage{14}{Conjugacy tracks exponents}{BETA MODEL / POSTERIOR AND EVIDENCE}{Chapter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model.}\nFor a small exact example, take the $\\mathrm{Beta}(2,3)$ density\n\\[\n\\rho(\\theta)=12\\theta(1-\\theta)^2,\\qquad0\\le\\theta\\le1.\n\\]\nObserve a specified ordered sequence with two successes and one failure. Its likelihood is $\\theta^2(1-\\theta)$. Multiplication adds exponents, so the posterior belongs to $\\mathrm{Beta}(4,4)$ with density $140\\theta^3(1-\\theta)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordere…","ter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model.}\nFor a small exact example, take the $\\mathrm{Beta}(2,3)$ density\n\\[\n\\rho(\\theta)=12\\theta(1-\\theta)^2,\\qquad0\\le\\theta\\le1.\n\\]\nObserve a specified ordered sequence with two successes and one failure. Its likelihood is $\\theta^2(1-\\theta)$. Multiplication adds exponents, so the posterior belongs to $\\mathrm{Beta}(4,4)$ with density $140\\theta^3(1-\\theta)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evi…","ite-parameter conjugate family, not a universal posterior model.}\nFor a small exact example, take the $\\mathrm{Beta}(2,3)$ density\n\\[\n\\rho(\\theta)=12\\theta(1-\\theta)^2,\\qquad0\\le\\theta\\le1.\n\\]\nObserve a specified ordered sequence with two successes and one failure. Its likelihood is $\\theta^2(1-\\theta)$. Multiplication adds exponents, so the posterior belongs to $\\mathrm{Beta}(4,4)$ with density $140\\theta^3(1-\\theta)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evidence is $9/35$.","ty\n\\[\n\\rho(\\theta)=12\\theta(1-\\theta)^2,\\qquad0\\le\\theta\\le1.\n\\]\nObserve a specified ordered sequence with two successes and one failure. Its likelihood is $\\theta^2(1-\\theta)$. Multiplication adds exponents, so the posterior belongs to $\\mathrm{Beta}(4,4)$ with density $140\\theta^3(1-\\theta)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evidence is $9/35$.","rior belongs to $\\mathrm{Beta}(4,4)$ with density $140\\theta^3(1-\\theta)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evidence is $9/35$.","a)^3$.\n\\vfig{beta}\n\\platecaption{The dashed prior is asymmetric. The solid posterior is symmetric with mean $1/2$. Polynomial integration verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evidence is $9/35$.","verifies the normalizers exactly. The curves are sampled drawings of those polynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evidence is $9/35$.","lynomials.}\n\\head{The observation convention changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evidence is $9/35$.","on changes the evidence}\nThe integral of the unnormalized posterior is\n\\[\n\\int_0^1 12\\theta^3(1-\\theta)^3\\,d\\theta=\\frac3{35}.\n\\]\nDividing by $3/35$ gives the displayed normalized density. If the observation is instead the \\emph{count} of two successes in three trials, there are $\\binom32=3$ compatible ordered sequences. The count likelihood is therefore $3\\theta^2(1-\\theta)$ and its evidence is $9/35$.","The extra factor does not depend on $\\theta$, so it cancels from the posterior. The two experiments thus have the same posterior distribution but different data probabilities. Evidence is not determined by the final normalized density alone.","The exact check also integrates both densities to one and integrates $\\theta$ against the posterior to obtain $1/2$. This demonstrates one conjugate family in elementary terms. It does not show that arbitrary likelihoods preserve that family or that all posterior information can be compressed into two exponents.\n\\takeaway{Conjugacy gives a closed parameter update when a specified likelihood respects a specified family. Preserve the evidence as well as the normalized posterior.}","\\vpage{15}{The modes stay as their clocks change}{FRACTIONAL DIFFUSION / NEW COUPLINGS}{Chapter VI, Part I: Theorem 10.1. A positive self-adjoint finite model and its spectral powers.}\nReturn to the four-state generator on \\pagerefV{4} and put $L=-Q$. It is real symmetric and nonnegative, with orthogonal projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral squa","\\vpage{15}{The modes stay as their clocks change}{FRACTIONAL DIFFUSION / NEW COUPLINGS}{Chapter VI, Part I: Theorem 10.1. A positive self-adjoint finite model and its spectral powers.}\nReturn to the four-state generator on \\pagerefV{4} and put $L=-Q$. It is real symmetric and nonnegative, with orthogonal projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$","\\vpage{15}{The modes stay as their clocks change}{FRACTIONAL DIFFUSION / NEW COUPLINGS}{Chapter VI, Part I: Theorem 10.1. A positive self-adjoint finite model and its spectral powers.}\nReturn to the four-state generator on \\pagerefV{4} and put $L=-Q$. It is real symmetric and nonnegative, with orthogonal projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transitio","\\vpage{15}{The modes stay as their clocks change}{FRACTIONAL DIFFUSION / NEW COUPLINGS}{Chapter VI, Part I: Theorem 10.1. A positive self-adjoint finite model and its spectral powers.}\nReturn to the four-state generator on \\pagerefV{4} and put $L=-Q$. It is real symmetric and nonnegative, with orthogonal projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were …",". A positive self-adjoint finite model and its spectral powers.}\nReturn to the four-state generator on \\pagerefV{4} and put $L=-Q$. It is real symmetric and nonnegative, with orthogonal projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $…","itive self-adjoint finite model and its spectral powers.}\nReturn to the four-state generator on \\pagerefV{4} and put $L=-Q$. It is real symmetric and nonnegative, with orthogonal projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","wers.}\nReturn to the four-state generator on \\pagerefV{4} and put $L=-Q$. It is real symmetric and nonnegative, with orthogonal projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","l projectors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","tors $P_0,P_1,P_2,P_3$ and eigenvalues $0,2,4,6$. Its square root is\n\\[\nL^{1/2}=\\sqrt2P_1+2P_2+\\sqrt6P_3,\n\\qquad (L^{1/2})^2=L.\n\\]\nBoth heat evolutions retain these modes. Ordinary diffusion multiplies mode $j$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","$ by $e^{-2jt}$. Square-root diffusion multiplies it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","s it by $e^{-\\sqrt{2j}\\,t}$. The constant mode is unchanged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","anged in both.\n\\vfig{fractional}\n\\platecaption{The highest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","hest mode decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","e decays at rates $6$ and $\\sqrt6$. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.",". The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","ay the explicit formulas, not an empirical fit.}\n\\head{Fractional does not mean entrywise}\nAlthough $L_{03}=0$, its spectral square root satisfies\n\\[\n(L^{1/2})_{03}=\\frac{\\sqrt6-\\sqrt2-2}{4}<0.\n\\]\nThus $-L^{1/2}$ permits a direct transition between states $0$ and $3$, which were not neighbors for $Q$. Taking the square root of each matrix entry would neither produce this value nor square back to $L$.","The exact block aggregation remains compatible in this example:\n\\[\nC L^{1/2}=\\mat{1&-1\\\\-1&1}C.\n\\]\nThe coarse matrix is the positive square root of $-Q_c$. This is a concrete naturality identity, checked on the full matrices. It is not a claim that an arbitrary partition commutes with every fractional construction.","-1&1}C.\n\\]\nThe coarse matrix is the positive square root of $-Q_c$. This is a concrete naturality identity, checked on the full matrices. It is not a claim that an arbitrary partition commutes with every fractional construction.","\\vpage{16}{A gap changes the error scale}{FRACTIONAL DEFECTS / NEAR ZERO VERSUS AWAY FROM ZERO}{Chapter VI, Part I: Theorem 10.3 and Corollary 10.4. Separate generator error from fractional error.}\nThe function $a\\mapsto\\sqrt a$ changes most rapidly near zero. A scalar model already shows why a spectral gap appears in fractional-defect estimates:\n\\[\n|\\sqrt\\varepsilon-\\sqrt0|=\\sqrt\\varepsilon,\n\\qquad\n\\sqrt{1+\\varepsilon}-1\n=\\frac{\\varepsilon}{\\sqrt{1+\\varepsilon}+1}\\le\\frac\\varepsilon2.\n\\]\nThe same input difference has a square-root scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite","\\vpage{16}{A gap changes the error scale}{FRACTIONAL DEFECTS / NEAR ZERO VERSUS AWAY FROM ZERO}{Chapter VI, Part I: Theorem 10.3 and Corollary 10.4. Separate generator error from fractional error.}\nThe function $a\\mapsto\\sqrt a$ changes most rapidly near zero. A scalar model already shows why a spectral gap appears in fractional-defect estimates:\n\\[\n|\\sqrt\\varepsilon-\\sqrt0|=\\sqrt\\varepsilon,\n\\qquad\n\\sqrt{1+\\varepsilon}-1\n=\\frac{\\varepsilon}{\\sqrt{1+\\varepsilon}+1}\\le\\frac\\varepsilon2.\n\\]\nThe same input difference has a square-root scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an in…","o. A scalar model already shows why a spectral gap appears in fractional-defect estimates:\n\\[\n|\\sqrt\\varepsilon-\\sqrt0|=\\sqrt\\varepsilon,\n\\qquad\n\\sqrt{1+\\varepsilon}-1\n=\\frac{\\varepsilon}{\\sqrt{1+\\varepsilon}+1}\\le\\frac\\varepsilon2.\n\\]\nThe same input difference has a square-root scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","ar model already shows why a spectral gap appears in fractional-defect estimates:\n\\[\n|\\sqrt\\varepsilon-\\sqrt0|=\\sqrt\\varepsilon,\n\\qquad\n\\sqrt{1+\\varepsilon}-1\n=\\frac{\\varepsilon}{\\sqrt{1+\\varepsilon}+1}\\le\\frac\\varepsilon2.\n\\]\nThe same input difference has a square-root scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","p appears in fractional-defect estimates:\n\\[\n|\\sqrt\\varepsilon-\\sqrt0|=\\sqrt\\varepsilon,\n\\qquad\n\\sqrt{1+\\varepsilon}-1\n=\\frac{\\varepsilon}{\\sqrt{1+\\varepsilon}+1}\\le\\frac\\varepsilon2.\n\\]\nThe same input difference has a square-root scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","in fractional-defect estimates:\n\\[\n|\\sqrt\\varepsilon-\\sqrt0|=\\sqrt\\varepsilon,\n\\qquad\n\\sqrt{1+\\varepsilon}-1\n=\\frac{\\varepsilon}{\\sqrt{1+\\varepsilon}+1}\\le\\frac\\varepsilon2.\n\\]\nThe same input difference has a square-root scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","\\[\n|\\sqrt\\varepsilon-\\sqrt0|=\\sqrt\\varepsilon,\n\\qquad\n\\sqrt{1+\\varepsilon}-1\n=\\frac{\\varepsilon}{\\sqrt{1+\\varepsilon}+1}\\le\\frac\\varepsilon2.\n\\]\nThe same input difference has a square-root scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","scale at zero and a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","nd a linear bound above a fixed positive lower bound.\n\\vfig{gap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","ap}\n\\platecaption{The upper curve compares $0$ with $\\varepsilon$. The lower curve compares $1$ with $1+\\varepsilon$. The dotted line $\\varepsilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","ilon/2$ bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","bounds the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","the latter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","tter. All plotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","lotted values come from the displayed scalar formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","formulas.}\n\\head{Read an operator bound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","ound with its hypotheses}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","es}\nFor positive self-adjoint finite matrices $A,B\\ge mI$ and $0<\\alpha<1$, the manuscript's gapped estimate has the form\n\\[\n\\norm{CA^\\alpha-B^\\alpha C}\n\\le \\alpha m^{\\alpha-1}\\norm{CA-BC}.\n\\]\nIt concerns an intertwining defect, not just two eigenvalue lists. With $C=1$, $A=4$, $B=9$, $m=4$, and $\\alpha=1/2$, the generator defect has size $5$, the fractional defect has size $1$, and the bound is $5/4$.","On \\pagerefV{27}, the sequences $1/n^2$ and $1/n$ turn this change in error scale into a concrete distinction between summable and nonsummable defects.\n\\takeaway{A fixed gap is quantitative information. It prevents the derivative of the fractional function from becoming singular along the comparison.}","\\vpage{17}{A random diffusion clock}{SUBORDINATION / A DISTRIBUTION OF ORDINARY TIMES}{Chapter VI, Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model.}\nSquare-root evolution can be written as an average of ordinary diffusion over a positive clock variable $s$:\n\\[\ne^{-t\\sqrt L}=\\int_0^\\infty \\eta_t(s)e^{-sL}\\,ds,\n\\qquad\n\\eta_t(s)=\\frac{t}{2\\sqrt\\pi}s^{-3/2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$,","\\vpage{17}{A random diffusion clock}{SUBORDINATION / A DISTRIBUTION OF ORDINARY TIMES}{Chapter VI, Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model.}\nSquare-root evolution can be written as an average of ordinary diffusion over a positive clock variable $s$:\n\\[\ne^{-t\\sqrt L}=\\int_0^\\infty \\eta_t(s)e^{-sL}\\,ds,\n\\qquad\n\\eta_t(s)=\\frac{t}{2\\sqrt\\pi}s^{-3/2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{…","on clock}{SUBORDINATION / A DISTRIBUTION OF ORDINARY TIMES}{Chapter VI, Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model.}\nSquare-root evolution can be written as an average of ordinary diffusion over a positive clock variable $s$:\n\\[\ne^{-t\\sqrt L}=\\int_0^\\infty \\eta_t(s)e^{-sL}\\,ds,\n\\qquad\n\\eta_t(s)=\\frac{t}{2\\sqrt\\pi}s^{-3/2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe man…","STRIBUTION OF ORDINARY TIMES}{Chapter VI, Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model.}\nSquare-root evolution can be written as an average of ordinary diffusion over a positive clock variable $s$:\n\\[\ne^{-t\\sqrt L}=\\int_0^\\infty \\eta_t(s)e^{-sL}\\,ds,\n\\qquad\n\\eta_t(s)=\\frac{t}{2\\sqrt\\pi}s^{-3/2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar i…","re-root evolution can be written as an average of ordinary diffusion over a positive clock variable $s$:\n\\[\ne^{-t\\sqrt L}=\\int_0^\\infty \\eta_t(s)e^{-sL}\\,ds,\n\\qquad\n\\eta_t(s)=\\frac{t}{2\\sqrt\\pi}s^{-3/2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not us…","can be written as an average of ordinary diffusion over a positive clock variable $s$:\n\\[\ne^{-t\\sqrt L}=\\int_0^\\infty \\eta_t(s)e^{-sL}\\,ds,\n\\qquad\n\\eta_t(s)=\\frac{t}{2\\sqrt\\pi}s^{-3/2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an i…","a positive clock variable $s$:\n\\[\ne^{-t\\sqrt L}=\\int_0^\\infty \\eta_t(s)e^{-sL}\\,ds,\n\\qquad\n\\eta_t(s)=\\frac{t}{2\\sqrt\\pi}s^{-3/2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an improper integral.","2}e^{-t^2/(4s)},\\quad t>0.\n\\]\nHere $t$ is the external time, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an improper integral.","ime, and $s$ is the ordinary diffusion time being mixed. They are not interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an improper integral.","interchangeable variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an improper integral.","variables.\n\\vfig{random_clock}\n\\platecaption{The density $\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an improper integral.","$\\eta_1$ peaks at $s=1/6$ but has a long right tail. The drawing ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an improper integral.","ends at $s=2$. It does not claim that the missing tail has zero mass.}\n\\head{Check the normalizer before interpreting a density}\nUnder $s=t^2/(4u^2)$, reversing the integration limits gives\n\\[\n\\eta_t(s)\\,|ds|=\\frac2{\\sqrt\\pi}e^{-u^2}\\,du.\n\\]\nIts integral over $u>0$ is one, so $\\eta_t$ is a probability density. On each eigenmode of $L$, the operator formula reduces to the scalar Laplace identity\n\\[\n\\int_0^\\infty\\eta_t(s)e^{-\\lambda s}\\,ds=e^{-t\\sqrt\\lambda}.\n\\]\nThe manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the change of variables and normalization. Sampled curves are not used to certify an improper integral.","A normalized density need not have a finite mean. For large $s$, the exponential factor tends to one and $\\eta_t(s)$ is proportional to $s^{-3/2}$. Consequently $s\\eta_t(s)$ has a nonintegrable $s^{-1/2}$ tail. The clock has infinite mean even though its total mass is one.\n\\takeaway{Subordination is a weighted mixture of diffusion times. Replacing the clock distribution by a single average time is not justified, and here that average is infinite.}","\\vpage{18}{A Green kernel is an inverse with units}{DIRICHLET GRID / THE MISSING FACTOR OF h}{Chapter VI, Part I: Theorem 11.1. Exact nodal Green reconstruction for this one-dimensional stencil.}\nTake three interior nodes $x_i=i/4$ on $[0,1]$, with zero boundary values and $h=1/4$. The discrete negative second derivative is\n\\[\nL_h=\\frac1{h^2}\\mat{2&-1&0","\\vpage{18}{A Green kernel is an inverse with units}{DIRICHLET GRID / THE MISSING FACTOR OF h}{Chapter VI, Part I: Theorem 11.1. Exact nodal Green reconstruction for this one-dimensional stencil.}\nTake three interior nodes $x_i=i/4$ on $[0,1]$, with zero boundary values and $h=1/4$. The discrete negative second derivative is\n\\[\nL_h=\\frac1{h^2}\\mat{2&-1&0\\\\-1&2&-1\\\\0&-1&2}.\n\\]\nFor the continuous Dirichlet problem, the Green function is $G(x,y)=\\min(x,y)-xy$. At these nodes the exact comparison is\n\\[\nL_h^{-1}=h\\,[G(x_i,x_j)]_{i,j=1}^3\n=\\frac1{64}\\mat{3&2&1","0&-1&2}.\n\\]\nFor the continuous Dirichlet problem, the Green function is $G(x,y)=\\min(x,y)-xy$. At these nodes the exact comparison is\n\\[\nL_h^{-1}=h\\,[G(x_i,x_j)]_{i,j=1}^3\n=\\frac1{64}\\mat{3&2&1","0&-1&2}.\n\\]\nFor the continuous Dirichlet problem, the Green function is $G(x,y)=\\min(x,y)-xy$. At these nodes the exact comparison is\n\\[\nL_h^{-1}=h\\,[G(x_i,x_j)]_{i,j=1}^3\n=\\frac1{64}\\mat{3&2&1\\\\2&4&2\\\\1&2&3}.\n\\]\n\\vfig{green}\n\\platecaption{The tent is the central column of the continuous Green kernel. It has nodal values $(1,2,1)/8$. Its discrete inverse column is $(1,2,1)/32$, smaller by the factor $h$.}\n\\head{A forcing vector is not a quadrature-weighted measure}\nFor sampled forcing $f_j$, the finite solution is\n\\[\nu_i=\\sum_j(L_h^{-1})_{ij}f_j\n=h\\sum_jG(x_i,x_j)f_j.\n\\]\nThe factor $h$ converts samples into the quadrature weights appropriate to this formula. Dropping it multiplies the computed solution by four on the displayed grid.","1&2&3}.\n\\]\n\\vfig{green}\n\\platecaption{The tent is the central column of the continuous Green kernel. It has nodal values $(1,2,1)/8$. Its discrete inverse column is $(1,2,1)/32$, smaller by the factor $h$.}\n\\head{A forcing vector is not a quadrature-weighted measure}\nFor sampled forcing $f_j$, the finite solution is\n\\[\nu_i=\\sum_j(L_h^{-1})_{ij}f_j\n=h\\sum_jG(x_i,x_j)f_j.\n\\]\nThe factor $h$ converts samples into the quadrature weights appropriate to this formula. Dropping it multiplies the computed solution by four on the displayed grid.","A discrete unit coordinate vector and a discretized unit point mass use different normalizations. For $f=e_2$, the result is the second inverse column. For $f=e_2/h$, it is the unscaled tent sampled at the nodes. Either convention can be useful, but it must be specified before comparing answers.","Exact checks multiply $L_h$ by the proposed inverse at $1,3,7,$ and $15$ interior nodes. The general identity is the manuscript theorem. These finite checks do not extend it to arbitrary meshes, dimensions, or boundary conditions.\n\\takeaway{An inverse kernel carries a measure convention. A tent that looks correct can give a wrong solution if the grid weight is omitted.}","\\vpage{19}{Convergence with a nonzero exact error}{FINITE DIFFERENCES / A QUARTIC TEST PROBLEM}{Chapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors.}\nUse $u(x)=x-x^4$, which vanishes at both endpoints and solves $-u''=12x^2$. Its centered second difference is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nA","\\vpage{19}{Convergence with a nonzero exact error}{FINITE DIFFERENCES / A QUARTIC TEST PROBLEM}{Chapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors.}\nUse $u(x)=x-x^4$, which vanishes at both endpoints and solves $-u''=12x^2$. Its centered second difference is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute","\\vpage{19}{Convergence with a nonzero exact error}{FINITE DIFFERENCES / A QUARTIC TEST PROBLEM}{Chapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors.}\nUse $u(x)=x-x^4$, which vanishes at both endpoints and solves $-u''=12x^2$. Its centered second difference is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exa","vpage{19}{Convergence with a nonzero exact error}{FINITE DIFFERENCES / A QUARTIC TEST PROBLEM}{Chapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors.}\nUse $u(x)=x-x^4$, which vanishes at both endpoints and solves $-u''=12x^2$. Its centered second difference is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values …","nzero exact error}{FINITE DIFFERENCES / A QUARTIC TEST PROBLEM}{Chapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors.}\nUse $u(x)=x-x^4$, which vanishes at both endpoints and solves $-u''=12x^2$. Its centered second difference is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed err…","FERENCES / A QUARTIC TEST PROBLEM}{Chapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors.}\nUse $u(x)=x-x^4$, which vanishes at both endpoints and solves $-u''=12x^2$. Its centered second difference is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","TEST PROBLEM}{Chapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors.}\nUse $u(x)=x-x^4$, which vanishes at both endpoints and solves $-u''=12x^2$. Its centered second difference is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","is not exact:\n\\[\nL_hu|_{\\rm nodes}=12x_i^2+2h^2.\n\\]\nSince the stencil differentiates the quadratic $x(1-x)$ exactly, the solution $u_h$ to the sampled forcing $12x_i^2$ satisfies\n\\[\n\\boxed{u_h(x_i)-u(x_i)=-h^2x_i(1-x_i).}\n\\]\nThis gives both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","ves both the sign and size of the error, rather than merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","n merely a log--log slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","slope.\n\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","\\vfig{mesh}\n\\platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","platecaption{The three curves show magnitudes of the exact nodal errors for $h=1/4,1/8,1/16$. Lines between grid points are visual guides, not an additional continuous numerical solution. The error is negative on every interior node.}\n\\head{One refinement divides this error by four}\nAll three grids include $x=1/2$, where the maximum absolute nodal error is $h^2/4$. It equals $1/64$, $1/256$, and $1/1024$, respectively. Exact rational matrix inversion verifies these values independently of the closed error formula.","The manuscript also compares the discrete eigenvalues\n\\[\n\\lambda_{k,h}=4h^{-2}\\sin^2(k\\pi h/2)\n\\]\nwith $(k\\pi)^2$, with error at most $(k\\pi)^4h^2/12$. For a fixed mode $k$, the bound tends to zero quadratically. It does not claim a uniform small error for modes whose index grows like $1/h$.","\\vpage{20}{Eight paths, one weighted expectation}{FEYNMAN--KAC / DEPARTURE-STATE WEIGHTS}{Chapter VI, Part I: Theorem 12.1. A finite path sum with its time convention displayed.}\nLet $K_2=\\frac14\\mat{3&1\\\\1&3}$, start at state $0$, use weights $w=(1,1/2)$, and terminal observable $f=(2,1)$. For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate\n\\[\n\\EE_0\\left[\\prod_{r=0}^{2}w(X_r)\\,f(X_3)\\right].\n\\]\nEach path contributes its transition probability times its three departure weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth","1&3}$, start at state $0$, use weights $w=(1,1/2)$, and terminal observable $f=(2,1)$. For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate\n\\[\n\\EE_0\\left[\\prod_{r=0}^{2}w(X_r)\\,f(X_3)\\right].\n\\]\nEach path contributes its transition probability times its three departure weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D","1&3}$, start at state $0$, use weights $w=(1,1/2)$, and terminal observable $f=(2,1)$. For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate\n\\[\n\\EE_0\\left[\\prod_{r=0}^{2}w(X_r)\\,f(X_3)\\right].\n\\]\nEach path contributes its transition probability times its three departure weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]","1&3}$, start at state $0$, use weights $w=(1,1/2)$, and terminal observable $f=(2,1)$. For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate\n\\[\n\\EE_0\\left[\\prod_{r=0}^{2}w(X_r)\\,f(X_3)\\right].\n\\]\nEach path contributes its transition probability times its three departure weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]\nIts forward dual is $K_2D_w$: weig","1&3}$, start at state $0$, use weights $w=(1,1/2)$, and terminal observable $f=(2,1)$. For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate\n\\[\n\\EE_0\\left[\\prod_{r=0}^{2}w(X_r)\\,f(X_3)\\right].\n\\]\nEach path contributes its transition probability times its three departure weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]\nIts forward dual is $K_2D_w$: weight the mass at the departing state, then move it. Weighting arrival states instead gives $D_wK_2$, a different expression. The exact tests deliberately check that","For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate\n\\[\n\\EE_0\\left[\\prod_{r=0}^{2}w(X_r)\\,f(X_3)\\right].\n\\]\nEach path contributes its transition probability times its three departure weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]\nIts forward dual is $K_2D_w$: weight the mass at the departing state, then move it. Weighting arrival states instead gives $D_wK_2$, a different expression. The exact tests deliberately check that the two conventions do not agree in this example.","].\n\\]\nEach path contributes its transition probability times its three departure weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]\nIts forward dual is $K_2D_w$: weight the mass at the departing state, then move it. Weighting arrival states instead gives $D_wK_2$, a different expression. The exact tests deliberately check that the two conventions do not agree in this example.","re weights times its terminal value.\n\\vfig{paths}\n\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]\nIts forward dual is $K_2D_w$: weight the mass at the departing state, then move it. Weighting arrival states instead gives $D_wK_2$, a different expression. The exact tests deliberately check that the two conventions do not agree in this example.","\\platecaption{Every possible path is included. The probability column sums to one. The contribution column sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]\nIts forward dual is $K_2D_w$: weight the mass at the departing state, then move it. Weighting arrival states instead gives $D_wK_2$, a different expression. The exact tests deliberately check that the two conventions do not agree in this example.","umn sums to $311/256$, agreeing exactly with the transfer calculation.}\n\\head{Check which side of the transition gets weighted}\nThe backward operator on observables is $D_w K_2^{\\mathsf T}$. Therefore the expectation is the zeroth coordinate of\n\\[\n(D_wK_2^{\\mathsf T})^3f.\n\\]\nIts forward dual is $K_2D_w$: weight the mass at the departing state, then move it. Weighting arrival states instead gives $D_wK_2$, a different expression. The exact tests deliberately check that the two conventions do not agree in this example.","The answer $311/256$ exceeds one, but it is not a probability. The terminal observable can equal two. With $0\\le w_i\\le1$, the weighted expectation is bounded by the maximum terminal value, not necessarily by one.","\\vpage{21}{Elimination makes a frequency-dependent term}{SCHUR--FESHBACH / RETAINED RESOLVENT}{Chapter VI, Part I: Theorem 13.1. A two-state Schur complement with the inverse checked directly.}\nThe two-state evolution from \\pagerefV{3} has generator\n\\[\nA=\\mat{-1&1\\\\1&-1}.\n\\]\nRetain the first coordinate and eliminate the second. For $z$ away from the relevant poles, the retained entry of $(zI-A)^{-1}$ is\n\\[\nR_x(z)=\\left[z+1-\\frac1{z+1}\\right]^{-1}\n=\\frac{z+1}{z(z+2)}.\n\\]\nThe term $\\Sigma(z)=1/(z+1)$ is the hidden coordinate's self-energy.\n\\vfig{resolvent}\n\\platecaption{The full inverse gives a rational response with two poles, at $0$ and $-2$. Its partial fractions are $R_x(z)=\\frac12z^{-1}+\\frac12(z+2)^{-1}$. The hidden intermediate inverse has its own excluded point $z=-1$.}\n\\head{An eliminated variable has not disappeared}\nIf the coupling to the second coordinate were simply deleted, the retained resolvent would be $(z+1)^{-1}$. That has a different pole an","1&-1}.\n\\]\nRetain the first coordinate and eliminate the second. For $z$ away from the relevant poles, the retained entry of $(zI-A)^{-1}$ is\n\\[\nR_x(z)=\\left[z+1-\\frac1{z+1}\\right]^{-1}\n=\\frac{z+1}{z(z+2)}.\n\\]\nThe term $\\Sigma(z)=1/(z+1)$ is the hidden coordinate's self-energy.\n\\vfig{resolvent}\n\\platecaption{The full inverse gives a rational response with two poles, at $0$ and $-2$. Its partial fractions are $R_x(z)=\\frac12z^{-1}+\\frac12(z+2)^{-1}$. The hidden intermediate inverse has its own excluded point $z=-1$.}\n\\head{An eliminated variable has not disappeared}\nIf the coupling to the second coordinate were simply deleted, the retained resolvent would be $(z+1)^{-1}$. That has a different pole and a different long-time limit. The correction is frequency depe","1&-1}.\n\\]\nRetain the first coordinate and eliminate the second. For $z$ away from the relevant poles, the retained entry of $(zI-A)^{-1}$ is\n\\[\nR_x(z)=\\left[z+1-\\frac1{z+1}\\right]^{-1}\n=\\frac{z+1}{z(z+2)}.\n\\]\nThe term $\\Sigma(z)=1/(z+1)$ is the hidden coordinate's self-energy.\n\\vfig{resolvent}\n\\platecaption{The full inverse gives a rational response with two poles, at $0$ and $-2$. Its partial fractions are $R_x(z)=\\frac12z^{-1}+\\frac12(z+2)^{-1}$. The hidden intermediate inverse has its own excluded point $z=-1$.}\n\\head{An eliminated variable has not disappeared}\nIf the coupling to the second coordinate were simply deleted, the retained resolvent would be $(z+1)^{-1}$. That has a different pole and a different long-time limit. The correction is frequency dependent, so it cannot be represented by one constant scalar generato","1&-1}.\n\\]\nRetain the first coordinate and eliminate the second. For $z$ away from the relevant poles, the retained entry of $(zI-A)^{-1}$ is\n\\[\nR_x(z)=\\left[z+1-\\frac1{z+1}\\right]^{-1}\n=\\frac{z+1}{z(z+2)}.\n\\]\nThe term $\\Sigma(z)=1/(z+1)$ is the hidden coordinate's self-energy.\n\\vfig{resolvent}\n\\platecaption{The full inverse gives a rational response with two poles, at $0$ and $-2$. Its partial fractions are $R_x(z)=\\frac12z^{-1}+\\frac12(z+2)^{-1}$. The hidden intermediate inverse has its own excluded point $z=-1$.}\n\\head{An eliminated variable has not disappeared}\nIf the coupling to the second coordinate were simply deleted, the retained resolvent would be $(z+1)^{-1}$. That has a different pole and a different long-time limit. The correction is frequency dependent, so it cannot be represented by one constant scalar generator on the retained coordinate.","tain the first coordinate and eliminate the second. For $z$ away from the relevant poles, the retained entry of $(zI-A)^{-1}$ is\n\\[\nR_x(z)=\\left[z+1-\\frac1{z+1}\\right]^{-1}\n=\\frac{z+1}{z(z+2)}.\n\\]\nThe term $\\Sigma(z)=1/(z+1)$ is the hidden coordinate's self-energy.\n\\vfig{resolvent}\n\\platecaption{The full inverse gives a rational response with two poles, at $0$ and $-2$. Its partial fractions are $R_x(z)=\\frac12z^{-1}+\\frac12(z+2)^{-1}$. The hidden intermediate inverse has its own excluded point $z=-1$.}\n\\head{An eliminated variable has not disappeared}\nIf the coupling to the second coordinate were simply deleted, the retained resolvent would be $(z+1)^{-1}$. That has a different pole and a different long-time limit. The correction is frequency dependent, so it cannot be represented by one constant scalar generator on the retained coordinate.","d entry of $(zI-A)^{-1}$ is\n\\[\nR_x(z)=\\left[z+1-\\frac1{z+1}\\right]^{-1}\n=\\frac{z+1}{z(z+2)}.\n\\]\nThe term $\\Sigma(z)=1/(z+1)$ is the hidden coordinate's self-energy.\n\\vfig{resolvent}\n\\platecaption{The full inverse gives a rational response with two poles, at $0$ and $-2$. Its partial fractions are $R_x(z)=\\frac12z^{-1}+\\frac12(z+2)^{-1}$. The hidden intermediate inverse has its own excluded point $z=-1$.}\n\\head{An eliminated variable has not disappeared}\nIf the coupling to the second coordinate were simply deleted, the retained resolvent would be $(z+1)^{-1}$. That has a different pole and a different long-time limit. The correction is frequency dependent, so it cannot be represented by one constant scalar generator on the retained coordinate.","elf-energy.\n\\vfig{resolvent}\n\\platecaption{The full inverse gives a rational response with two poles, at $0$ and $-2$. Its partial fractions are $R_x(z)=\\frac12z^{-1}+\\frac12(z+2)^{-1}$. The hidden intermediate inverse has its own excluded point $z=-1$.}\n\\head{An eliminated variable has not disappeared}\nIf the coupling to the second coordinate were simply deleted, the retained resolvent would be $(z+1)^{-1}$. That has a different pole and a different long-time limit. The correction is frequency dependent, so it cannot be represented by one constant scalar generator on the retained coordinate.","The Schur expression first requires $z+1\\ne0$, as well as invertibility of the full matrix. Its simplified rational form has a removable intermediate singularity at $z=-1$, where the retained entry of the full inverse is well-defined and equals zero. Simplification does not retroactively make the eliminated block invertible there.","For $z>0$, the Laplace transform of $(1+e^{-2t})/2$ is exactly $R_x(z)$. The next page derives the time-domain equation producing that response.\n\\takeaway{The Schur complement is exact elimination, not independent dynamics on a smaller state space. Hidden evolution returns as a $z$-dependent self-energy.}","\\vpage{22}{Hidden state returns as memory}{EXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUP}{Chapter VI, Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data.}\nWrite the same system as\n\\[\nx'=-x+y,\\qquad y'=x-y.\n\\]\nSolving the second equation first gives\n\\[\ny(t)=e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nSubstitution is exact:\n\\[\nx'(t)=-x(t)+e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nThe hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad","\\vpage{22}{Hidden state returns as memory}{EXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUP}{Chapter VI, Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data.}\nWrite the same system as\n\\[\nx'=-x+y,\\qquad y'=x-y.\n\\]\nSolving the second equation first gives\n\\[\ny(t)=e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nSubstitution is exact:\n\\[\nx'(t)=-x(t)+e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nThe hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained","\\vpage{22}{Hidden state returns as memory}{EXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUP}{Chapter VI, Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data.}\nWrite the same system as\n\\[\nx'=-x+y,\\qquad y'=x-y.\n\\]\nSolving the second equation first gives\n\\[\ny(t)=e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nSubstitution is exact:\n\\[\nx'(t)=-x(t)+e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nThe hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained scalar response is not. Restarting it as a scalar model silent…","g hidden initial data.}\nWrite the same system as\n\\[\nx'=-x+y,\\qquad y'=x-y.\n\\]\nSolving the second equation first gives\n\\[\ny(t)=e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nSubstitution is exact:\n\\[\nx'(t)=-x(t)+e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nThe hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained scalar response is not. Restarting it as a scalar model silently resets the hidden coordinate, which generally carries information from the first interval.","=-x+y,\\qquad y'=x-y.\n\\]\nSolving the second equation first gives\n\\[\ny(t)=e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nSubstitution is exact:\n\\[\nx'(t)=-x(t)+e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nThe hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained scalar response is not. Restarting it as a scalar model silently resets the hidden coordinate, which generally carries information from the first interval.","e second equation first gives\n\\[\ny(t)=e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nSubstitution is exact:\n\\[\nx'(t)=-x(t)+e^{-t}y(0)+\\int_0^t e^{-(t-r)}x(r)\\,dr.\n\\]\nThe hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained scalar response is not. Restarting it as a scalar model silently resets the hidden coordinate, which generally carries information from the first interval.","\\,dr.\n\\]\nThe hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained scalar response is not. Restarting it as a scalar model silently resets the hidden coordinate, which generally carries information from the first interval.","st be retained even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained scalar response is not. Restarting it as a scalar model silently resets the hidden coordinate, which generally carries information from the first interval.","ed even when the memory kernel is known.\n\\vfig{memory}\n\\platecaption{For $(x(0),y(0))=(1,0)$, the exact retained coordinate is $(1+e^{-2t})/2$. The dashed curve $e^{-t}$ solves the equation obtained by deleting the hidden coordinate rather than eliminating it.}\n\\head{Projection can destroy the semigroup law}\nLet $R(t)=(1+e^{-2t})/2$, the response to this fixed initial preparation. At $t=\\log2$,\n\\[\nR(t)=\\frac58,\\qquad R(2t)=\\frac{17}{32},\n\\qquad R(2t)-R(t)^2=\\frac9{64}.\n\\]\nThe full two-state propagator is a semigroup, but this retained scalar response is not. Restarting it as a scalar model silently resets the hidden coordinate, which generally carries information from the first interval.","The appendix verifies the full differential equation, initial condition, integral memory equation, and Laplace resolvent in this model. Its kernel is $e^{-t}$, whose transform is the self-energy on \\pagerefV{21}.\n\\takeaway{Exact reduction retains a record of the eliminated state through convolution and initial forcing. A memoryless approximation must justify the information it removes.}","\\vpage{23}{A repeated pole remembers a Jordan chain}{NONDIAGONAL HIDDEN DYNAMICS / POLYNOMIAL EXPONENTIALS}{Chapter VI, Part I: Theorem 13.2 and Section 15. Functional calculus must retain nilpotent data.}\nNow use a two-dimensional hidden block\n\\[\nJ=-2I+N,\\qquad N=\\mat{0&1\\\\0&0},\\qquad N^2=0.\n\\]\nChoose the entrance column $V=(0,1)^{\\mathsf T}$ and exit row $U=(1,0)$. The hidden exponential is not $e^{-2t}I$:\n\\[\ne^{tJ}=e^{-2t}(I+tN),\n\\qquad K_{\\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.\n\\]\nThe nilpotent link carries an input from the second hidden coordinate to the first.\n\\vfig{jordan}\n\\platecaption{The memory kernel starts at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and","0&0},\\qquad N^2=0.\n\\]\nChoose the entrance column $V=(0,1)^{\\mathsf T}$ and exit row $U=(1,0)$. The hidden exponential is not $e^{-2t}I$:\n\\[\ne^{tJ}=e^{-2t}(I+tN),\n\\qquad K_{\\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.\n\\]\nThe nilpotent link carries an input from the second hidden coordinate to the first.\n\\vfig{jordan}\n\\platecaption{The memory kernel starts at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same informatio","0&0},\\qquad N^2=0.\n\\]\nChoose the entrance column $V=(0,1)^{\\mathsf T}$ and exit row $U=(1,0)$. The hidden exponential is not $e^{-2t}I$:\n\\[\ne^{tJ}=e^{-2t}(I+tN),\n\\qquad K_{\\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.\n\\]\nThe nilpotent link carries an input from the second hidden coordinate to the first.\n\\vfig{jordan}\n\\platecaption{The memory kernel starts at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same information in two domains. For a","0&0},\\qquad N^2=0.\n\\]\nChoose the entrance column $V=(0,1)^{\\mathsf T}$ and exit row $U=(1,0)$. The hidden exponential is not $e^{-2t}I$:\n\\[\ne^{tJ}=e^{-2t}(I+tN),\n\\qquad K_{\\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.\n\\]\nThe nilpotent link carries an input from the second hidden coordinate to the first.\n\\vfig{jordan}\n\\platecaption{The memory kernel starts at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same information in two domains. For a longer nilpotent chain, higher powers of","0&0},\\qquad N^2=0.\n\\]\nChoose the entrance column $V=(0,1)^{\\mathsf T}$ and exit row $U=(1,0)$. The hidden exponential is not $e^{-2t}I$:\n\\[\ne^{tJ}=e^{-2t}(I+tN),\n\\qquad K_{\\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.\n\\]\nThe nilpotent link carries an input from the second hidden coordinate to the first.\n\\vfig{jordan}\n\\platecaption{The memory kernel starts at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same information in two domains. For a longer nilpotent chain, higher powers of $t$ occur with factorial denominators.","he entrance column $V=(0,1)^{\\mathsf T}$ and exit row $U=(1,0)$. The hidden exponential is not $e^{-2t}I$:\n\\[\ne^{tJ}=e^{-2t}(I+tN),\n\\qquad K_{\\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.\n\\]\nThe nilpotent link carries an input from the second hidden coordinate to the first.\n\\vfig{jordan}\n\\platecaption{The memory kernel starts at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same information in two domains. For a longer nilpotent chain, higher powers of $t$ occur with factorial denominators.","t}.\n\\]\nThe nilpotent link carries an input from the second hidden coordinate to the first.\n\\vfig{jordan}\n\\platecaption{The memory kernel starts at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same information in two domains. For a longer nilpotent chain, higher powers of $t$ occur with factorial denominators.","at zero, becomes positive, and peaks at $t=1/2$. A rule that retained the eigenvalue $-2$ but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same information in two domains. For a longer nilpotent chain, higher powers of $t$ occur with factorial denominators.","ut discarded the Jordan chain would predict the zero kernel for these entrance and exit maps.}\n\\head{The transfer function sees pole order}\nThe exact transform is\n\\[\nU(zI-J)^{-1}V=\\frac1{(z+2)^2}\n=\\int_0^\\infty t e^{-(z+2)t}\\,dt,\n\\qquad \\operatorname{Re}z>-2.\n\\]\nThe repeated pole and the polynomial factor $t$ are the same information in two domains. For a longer nilpotent chain, higher powers of $t$ occur with factorial denominators.","A zero value $K_{\\rm mem}(0)$ does not make the entire kernel zero. Nor do stable eigenvalues alone specify every transient contribution: entrance maps, exit maps, and nilpotent structure matter. Here the full kernel is nevertheless elementary, so its peak and transform can be checked without numerical fitting.","\\vpage{24}{A zero initial hazard can still lead to certain exit}{PHASE-TYPE SURVIVAL / TWO SUCCESSIVE STAGES}{Chapter VI, Part I: Theorem 14.1. Survival, density, hazard, and moments have distinct meanings.}\nLet a transient process move through two successive rate-one stages, then leave the system. With column probabilities, the transient generator and starting vector are\n\\[\nT=\\mat{-1&0\\\\1&-1},\\qquad p(0)=e_0.\n\\]\nThus $p(t)=e^{-t}(1,t)^{\\mathsf T}$. Its total remaining mass, exit density, and hazard are\n\\[\nS(t)=(1+t)e^{-t},\\qquad f(t)=t e^{-t},\n\\qquad h(t)=\\frac{t}{1+t}.\n\\]\nThe initial hazard is zero because an exit needs two stages, not one.\n\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restore…","1&-1},\\qquad p(0)=e_0.\n\\]\nThus $p(t)=e^{-t}(1,t)^{\\mathsf T}$. Its total remaining mass, exit density, and hazard are\n\\[\nS(t)=(1+t)e^{-t},\\qquad f(t)=t e^{-t},\n\\qquad h(t)=\\frac{t}{1+t}.\n\\]\nThe initial hazard is zero because an exit needs two stages, not one.\n\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restores a three-state probability vector with total mass","1&-1},\\qquad p(0)=e_0.\n\\]\nThus $p(t)=e^{-t}(1,t)^{\\mathsf T}$. Its total remaining mass, exit density, and hazard are\n\\[\nS(t)=(1+t)e^{-t},\\qquad f(t)=t e^{-t},\n\\qquad h(t)=\\frac{t}{1+t}.\n\\]\nThe initial hazard is zero because an exit needs two stages, not one.\n\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restores a three-state probability vector with total mass one.","initial hazard is zero because an exit needs two stages, not one.\n\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restores a three-state probability vector with total mass one.","hazard is zero because an exit needs two stages, not one.\n\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restores a three-state probability vector with total mass one.","is zero because an exit needs two stages, not one.\n\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restores a three-state probability vector with total mass one.","ges, not one.\n\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restores a three-state probability vector with total mass one.","\\vfig{survival}\n\\platecaption{Survival falls from one to zero, while absorbed mass $1-S$ rises to one. The dotted hazard is an instantaneous rate, not a probability, even though it shares the displayed numerical scale. Time is measured in inverse stage-rate units.}\n\\head{Missing transient mass is absorbed mass}\nThe column sums of $T$ are $0$ and $-1$. The exit-rate row is therefore $(0,1)$, and $f(t)=(0,1)p(t)$. Adding an absorbing state restores a three-state probability vector with total mass one.","Exact integration gives\n\\[\n\\int_0^\\infty f(t)\\,dt=1,\\qquad\n\\EE[\\tau]=2,\\qquad \\EE[\\tau^2]=6,\n\\qquad \\Var(\\tau)=2.\n\\]\nEquivalently, the mean is $\\ones^{\\mathsf T}(-T)^{-1}e_0$. A single exponential waiting time with mean two would have constant hazard $1/2$ and variance four, so matching the mean alone would not reproduce this distribution.","The hazard formula is used where $S(t)>0$. The model has that property for every finite $t\\ge0$. No division by an already vanished survival mass is required.\n\\takeaway{Survival tracks mass, density tracks exits per unit time, and hazard conditions on survival. Equal means or equal initial rates do not identify a waiting-time law.}","\\vpage{25}{Naturality also transports derivatives}{HOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISON}{Chapter VI, Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus.}\nConsider a fine operator, a coarse operator, and a rectangular map\n\\[\nA_f=\\mat{2&1&0\\\\0&2&0\\\\0&0&9},\\qquad\nA_c=\\mat{2&1\\\\0&2},\\qquad\nC_r=\\mat{1&0&0\\\\0&1&0}.\n\\]\nDirect multiplication gives $C_rA_f=A_cC_r$. For $F(z)=1/(1+z)$, which is defined at both spectra, compute the matrices by inversion:\n\\[\nF(A_f)=\\mat{1/3&-1/9&0","0&1&0}.\n\\]\nDirect multiplication gives $C_rA_f=A_cC_r$. For $F(z)=1/(1+z)$, which is defined at both spectra, compute the matrices by inversion:\n\\[\nF(A_f)=\\mat{1/3&-1/9&0","0&1&0}.\n\\]\nDirect multiplication gives $C_rA_f=A_cC_r$. For $F(z)=1/(1+z)$, which is defined at both spectra, compute the matrices by inversion:\n\\[\nF(A_f)=\\mat{1/3&-1/9&0\\\\0&1/3&0\\\\0&0&1/10}.\n\\]\n\\vfig{naturality}\n\\platecaption{The discarded eigenvalue $9$ contributes $1/10$ only on the fine side. On the retained Jordan block, the off-diagonal entry is $F'(2)=-1/9$, not zero. The rectangular comparison preserves that entry.}\n\\head{Why spectral values are not enough}\nOn the retained block $A_c=2I+N$, with $N^2=0$, one has\n\\[\nF(A_c)=F(2)I+F'(2)N=\\tfrac13I-\\tfrac19N.\n\\]\nHence $C_rF(A_f)=F(A_c)C_r$. Replacing the block by $F(2)I$ would lose the nonzero derivative term and no longer give the actual inverse of $I+A_c$.","0&0&1/10}.\n\\]\n\\vfig{naturality}\n\\platecaption{The discarded eigenvalue $9$ contributes $1/10$ only on the fine side. On the retained Jordan block, the off-diagonal entry is $F'(2)=-1/9$, not zero. The rectangular comparison preserves that entry.}\n\\head{Why spectral values are not enough}\nOn the retained block $A_c=2I+N$, with $N^2=0$, one has\n\\[\nF(A_c)=F(2)I+F'(2)N=\\tfrac13I-\\tfrac19N.\n\\]\nHence $C_rF(A_f)=F(A_c)C_r$. Replacing the block by $F(2)I$ would lose the nonzero derivative term and no longer give the actual inverse of $I+A_c$.","{The discarded eigenvalue $9$ contributes $1/10$ only on the fine side. On the retained Jordan block, the off-diagonal entry is $F'(2)=-1/9$, not zero. The rectangular comparison preserves that entry.}\n\\head{Why spectral values are not enough}\nOn the retained block $A_c=2I+N$, with $N^2=0$, one has\n\\[\nF(A_c)=F(2)I+F'(2)N=\\tfrac13I-\\tfrac19N.\n\\]\nHence $C_rF(A_f)=F(A_c)C_r$. Replacing the block by $F(2)I$ would lose the nonzero derivative term and no longer give the actual inverse of $I+A_c$.","Jordan block, the off-diagonal entry is $F'(2)=-1/9$, not zero. The rectangular comparison preserves that entry.}\n\\head{Why spectral values are not enough}\nOn the retained block $A_c=2I+N$, with $N^2=0$, one has\n\\[\nF(A_c)=F(2)I+F'(2)N=\\tfrac13I-\\tfrac19N.\n\\]\nHence $C_rF(A_f)=F(A_c)C_r$. Replacing the block by $F(2)I$ would lose the nonzero derivative term and no longer give the actual inverse of $I+A_c$.","The same mechanism explains why the Jordan memory kernel on \\pagerefV{23} contains $t e^{-2t}$. Exponentiation also has nonzero derivatives at repeated spectral values.\n\\takeaway{A functional calculus acts on operators, including their nilpotent parts. Exact intertwining transports that full action, not merely a list of scalar eigenvalue images.}","\\vpage{26}{One complete coarse filter}{END-TO-END MODEL / TWO OBSERVATIONS}{Chapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate.}\nUse the four-state kernel $K$ and aggregation $C$ from \\pagerefV{5}, starting at $p_0=e_0$. Observe two likelihood vectors constant inside both packets:\n\\[\n\\lambda^{(1)}=(3/4,3/4,1/4,1/4),\\qquad\n\\lambda^{(2)}=(1/2,1/2,1,1).\n\\]\nPredict, then update at each observation. The fine calculation gives\n\\[\np_1=\\tfrac1{20}(15,3,2,0)^{\\mathsf T},\\qquad\np_2=\\tfrac1{104}(41,15,40,8)^{\\mathsf T}.\n\\]\n\\vfig{integrated_filter}\n\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac5","\\vpage{26}{One complete coarse filter}{END-TO-END MODEL / TWO OBSERVATIONS}{Chapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate.}\nUse the four-state kernel $K$ and aggregation $C$ from \\pagerefV{5}, starting at $p_0=e_0$. Observe two likelihood vectors constant inside both packets:\n\\[\n\\lambda^{(1)}=(3/4,3/4,1/4,1/4),\\qquad\n\\lambda^{(2)}=(1/2,1/2,1,1).\n\\]\nPredict, then update at each observation. The fine calculation gives\n\\[\np_1=\\tfrac1{20}(15,3,2,0)^{\\mathsf T},\\qquad\np_2=\\tfrac1{104}(41,15,40,8)^{\\mathsf T}.\n\\]\n\\vfig{integrated_filter}\n\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{1","\\vpage{26}{One complete coarse filter}{END-TO-END MODEL / TWO OBSERVATIONS}{Chapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate.}\nUse the four-state kernel $K$ and aggregation $C$ from \\pagerefV{5}, starting at $p_0=e_0$. Observe two likelihood vectors constant inside both packets:\n\\[\n\\lambda^{(1)}=(3/4,3/4,1/4,1/4),\\qquad\n\\lambda^{(2)}=(1/2,1/2,1,1).\n\\]\nPredict, then update at each observation. The fine calculation gives\n\\[\np_1=\\tfrac1{20}(15,3,2,0)^{\\mathsf T},\\qquad\np_2=\\tfrac1{104}(41,15,40,8)^{\\mathsf T}.\n\\]\n\\vfig{integrated_filter}\n\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nT","\\vpage{26}{One complete coarse filter}{END-TO-END MODEL / TWO OBSERVATIONS}{Chapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate.}\nUse the four-state kernel $K$ and aggregation $C$ from \\pagerefV{5}, starting at $p_0=e_0$. Observe two likelihood vectors constant inside both packets:\n\\[\n\\lambda^{(1)}=(3/4,3/4,1/4,1/4),\\qquad\n\\lambda^{(2)}=(1/2,1/2,1,1).\n\\]\nPredict, then update at each observation. The fine calculation gives\n\\[\np_1=\\tfrac1{20}(15,3,2,0)^{\\mathsf T},\\qquad\np_2=\\tfrac1{104}(41,15,40,8)^{\\mathsf T}.\n\\]\n\\vfig{integrated_filter}\n\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here…",": Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate.}\nUse the four-state kernel $K$ and aggregation $C$ from \\pagerefV{5}, starting at $p_0=e_0$. Observe two likelihood vectors constant inside both packets:\n\\[\n\\lambda^{(1)}=(3/4,3/4,1/4,1/4),\\qquad\n\\lambda^{(2)}=(1/2,1/2,1,1).\n\\]\nPredict, then update at each observation. The fine calculation gives\n\\[\np_1=\\tfrac1{20}(15,3,2,0)^{\\mathsf T},\\qquad\np_2=\\tfrac1{104}(41,15,40,8)^{\\mathsf T}.\n\\]\n\\vfig{integrated_filter}\n\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here lie in $[0,1]$, they can serve as probabilities of the specified observations. This makes the…","mbda^{(2)}=(1/2,1/2,1,1).\n\\]\nPredict, then update at each observation. The fine calculation gives\n\\[\np_1=\\tfrac1{20}(15,3,2,0)^{\\mathsf T},\\qquad\np_2=\\tfrac1{104}(41,15,40,8)^{\\mathsf T}.\n\\]\n\\vfig{integrated_filter}\n\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here lie in $[0,1]$, they can serve as probabilities of the specified observations. This makes the product their joint evidence in the stated model.",",3,2,0)^{\\mathsf T},\\qquad\np_2=\\tfrac1{104}(41,15,40,8)^{\\mathsf T}.\n\\]\n\\vfig{integrated_filter}\n\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here lie in $[0,1]$, they can serve as probabilities of the specified observations. This makes the product their joint evidence in the stated model.","\\platecaption{Within-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here lie in $[0,1]$, they can serve as probabilities of the specified observations. This makes the product their joint evidence in the stated model.","cket probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here lie in $[0,1]$, they can serve as probabilities of the specified observations. This makes the product their joint evidence in the stated model.","during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here lie in $[0,1]$, they can serve as probabilities of the specified observations. This makes the product their joint evidence in the stated model.","agree exactly with an independent two-state filter. The second coarse posterior is $(7/13,6/13)$.}\n\\head{Check the normalizers, not only the final vector}\nThe first and second predictive evidences are\n\\[\nZ_1=\\frac58,\\qquad Z_2=\\frac{13}{20},\n\\qquad Z_1Z_2=\\frac{13}{32}.\n\\]\nThe unnormalized product $D_2KD_1Kp_0$ has mass $13/32$ and normalizes to $p_2$. Because the likelihood entries here lie in $[0,1]$, they can serve as probabilities of the specified observations. This makes the product their joint evidence in the stated model.","Two distinct equalities justify the coarse computation. First, $CK=K_cC$ transports prediction. Second, the likelihood is constant on each packet, so $CD_r=\\overline D_rC$ transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.","\\vpage{27}{Small at each level is not summable across levels}{REFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVE}{Chapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data.}\nFor each $n\\ge1$, compare the two scalar operators $A_n=0$ and $B_n=1/n^2$ through the identity. Their ordinary defects have sizes $1/n^2$, while their square-root defects have sizes $1/n$:\n\\[\n\\sum_n|A_n-B_n|<\\infty,\n\\qquad\n\\sum_n|A_n^{1/2}-B_n^{1/2}|=\\infty.\n\\]\nThese are independent scalar comparisons, not a constructed inverse system. They isolate an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change th","\\vpage{27}{Small at each level is not summable across levels}{REFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVE}{Chapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data.}\nFor each $n\\ge1$, compare the two scalar operators $A_n=0$ and $B_n=1/n^2$ through the identity. Their ordinary defects have sizes $1/n^2$, while their square-root defects have sizes $1/n$:\n\\[\n\\sum_n|A_n-B_n|<\\infty,\n\\qquad\n\\sum_n|A_n^{1/2}-B_n^{1/2}|=\\infty.\n\\]\nThese are independent scalar comparisons, not a constructed inverse system. They isolate an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$","\\vpage{27}{Small at each level is not summable across levels}{REFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVE}{Chapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data.}\nFor each $n\\ge1$, compare the two scalar operators $A_n=0$ and $B_n=1/n^2$ through the identity. Their ordinary defects have sizes $1/n^2$, while their square-root defects have sizes $1/n$:\n\\[\n\\sum_n|A_n-B_n|<\\infty,\n\\qquad\n\\sum_n|A_n^{1/2}-B_n^{1/2}|=\\infty.\n\\]\nThese are independent scalar comparisons, not a constructed inverse system. They isolate an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2","\\vpage{27}{Small at each level is not summable across levels}{REFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVE}{Chapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data.}\nFor each $n\\ge1$, compare the two scalar operators $A_n=0$ and $B_n=1/n^2$ through the identity. Their ordinary defects have sizes $1/n^2$, while their square-root defects have sizes $1/n$:\n\\[\n\\sum_n|A_n-B_n|<\\infty,\n\\qquad\n\\sum_n|A_n^{1/2}-B_n^{1/2}|=\\infty.\n\\]\nThese are independent scalar comparisons, not a constructed inverse system. They isolate an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2$. Then\n\\[\n\\left|B_n^{1/2}-A_n^{1/2}\\right|\n=\\frac{1/n^2}{\\sqrt{","7}{Small at each level is not summable across levels}{REFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVE}{Chapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data.}\nFor each $n\\ge1$, compare the two scalar operators $A_n=0$ and $B_n=1/n^2$ through the identity. Their ordinary defects have sizes $1/n^2$, while their square-root defects have sizes $1/n$:\n\\[\n\\sum_n|A_n-B_n|<\\infty,\n\\qquad\n\\sum_n|A_n^{1/2}-B_n^{1/2}|=\\infty.\n\\]\nThese are independent scalar comparisons, not a constructed inverse system. They isolate an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2$. Then\n\\[\n\\left|B_n^{1/2}-A_n^{1/2}\\right|\n=\\frac{1/n^2}{\\sqrt{1+1/n^2}+1}\n\\le\\frac1{2n^2}.\n\\]\nThis is the ele…","at each level is not summable across levels}{REFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVE}{Chapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data.}\nFor each $n\\ge1$, compare the two scalar operators $A_n=0$ and $B_n=1/n^2$ through the identity. Their ordinary defects have sizes $1/n^2$, while their square-root defects have sizes $1/n$:\n\\[\n\\sum_n|A_n-B_n|<\\infty,\n\\qquad\n\\sum_n|A_n^{1/2}-B_n^{1/2}|=\\infty.\n\\]\nThese are independent scalar comparisons, not a constructed inverse system. They isolate an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2$. Then\n\\[\n\\left|B_n^{1/2}-A_n^{1/2}\\right|\n=\\frac{1/n^2}{\\sqrt{1+1/n^2}+1}\n\\le\\frac1{2n^2}.\n\\]\nThis is the elementary f…","gh the identity. Their ordinary defects have sizes $1/n^2$, while their square-root defects have sizes $1/n$:\n\\[\n\\sum_n|A_n-B_n|<\\infty,\n\\qquad\n\\sum_n|A_n^{1/2}-B_n^{1/2}|=\\infty.\n\\]\nThese are independent scalar comparisons, not a constructed inverse system. They isolate an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2$. Then\n\\[\n\\left|B_n^{1/2}-A_n^{1/2}\\right|\n=\\frac{1/n^2}{\\sqrt{1+1/n^2}+1}\n\\le\\frac1{2n^2}.\n\\]\nThis is the elementary form of the uniform-gap mechanism on \\pagerefV{16}. Without such information, transferring a summable generator-defect budget through a fractional function can fail.","e an error-scale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2$. Then\n\\[\n\\left|B_n^{1/2}-A_n^{1/2}\\right|\n=\\frac{1/n^2}{\\sqrt{1+1/n^2}+1}\n\\le\\frac1{2n^2}.\n\\]\nThis is the elementary form of the uniform-gap mechanism on \\pagerefV{16}. Without such information, transferring a summable generator-defect budget through a fractional function can fail.","cale obstruction without claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2$. Then\n\\[\n\\left|B_n^{1/2}-A_n^{1/2}\\right|\n=\\frac{1/n^2}{\\sqrt{1+1/n^2}+1}\n\\le\\frac1{2n^2}.\n\\]\nThis is the elementary form of the uniform-gap mechanism on \\pagerefV{16}. Without such information, transferring a summable generator-defect budget through a fractional function can fail.","out claiming a coherent refinement tower.\n\\vfig{summability}\n\\platecaption{Partial sums through $n=16$ illustrate the contrast. The convergence and divergence statements follow from the displayed series, not from the finite plot. A common positive gap gives the third, summable upper bound.}\n\\head{A uniform hypothesis can change the conclusion}\nReplace the pair by $A_n=1$ and $B_n=1+1/n^2$. Then\n\\[\n\\left|B_n^{1/2}-A_n^{1/2}\\right|\n=\\frac{1/n^2}{\\sqrt{1+1/n^2}+1}\n\\le\\frac1{2n^2}.\n\\]\nThis is the elementary form of the uniform-gap mechanism on \\pagerefV{16}. Without such information, transferring a summable generator-defect budget through a fractional function can fail.","\\vpage{29}{Acyclic paths stop after finitely many transfers}{DIRECTED TRANSFER / A FINITE DYSON EXAMPLE}{Chapter VI, Part II: Theorem E2.8 and Corollary E2.9. Acyclicity concerns the transfer graph.}\nLet three scalar packets form the directed chain $0\\to1\\to2$, with transfer coefficients $2$ and $3$. The generator and its powers are\n\\[\nA=\\mat{0&0&0","\\vpage{29}{Acyclic paths stop after finitely many transfers}{DIRECTED TRANSFER / A FINITE DYSON EXAMPLE}{Chapter VI, Part II: Theorem E2.8 and Corollary E2.9. Acyclicity concerns the transfer graph.}\nLet three scalar packets form the directed chain $0\\to1\\to2$, with transfer coefficients $2$ and $3$. The generator and its powers are\n\\[\nA=\\mat{0&0&0\\\\2&0&0\\\\0&3&0},\\qquad\nA^2=\\mat{0&0&0\\\\0&0&0\\\\6&0&0},\\qquad A^3=0.\n\\]\nTherefore the exponential is the exact polynomial\n\\[\ne^{tA}=I+tA+\\frac{t^2}{2}A^2.\n\\]\n\\vfig{ext_paths}\n\\platecaption{Only the empty path, one-edge paths, and the two-edge path exist. The coefficient $3t^2$ includes both the edge product $2\\cdot3$ and the ordered-time simplex area $t^2/2$.}\n\\head{Path length counts transfers, not elapsed time}\nStarting at $e_0$, the complete state is $(1,2t,3t^2)^{\\mathsf T}$. No fourth packet or repeated cycle can contribute. Direct differentiation gives $x'(t)=Ax(t)$, including the relation $x_2'(t)=3x_1(t)$.","6&0&0},\\qquad A^3=0.\n\\]\nTherefore the exponential is the exact polynomial\n\\[\ne^{tA}=I+tA+\\frac{t^2}{2}A^2.\n\\]\n\\vfig{ext_paths}\n\\platecaption{Only the empty path, one-edge paths, and the two-edge path exist. The coefficient $3t^2$ includes both the edge product $2\\cdot3$ and the ordered-time simplex area $t^2/2$.}\n\\head{Path length counts transfers, not elapsed time}\nStarting at $e_0$, the complete state is $(1,2t,3t^2)^{\\mathsf T}$. No fourth packet or repeated cycle can contribute. Direct differentiation gives $x'(t)=Ax(t)$, including the relation $x_2'(t)=3x_1(t)$.","ad A^3=0.\n\\]\nTherefore the exponential is the exact polynomial\n\\[\ne^{tA}=I+tA+\\frac{t^2}{2}A^2.\n\\]\n\\vfig{ext_paths}\n\\platecaption{Only the empty path, one-edge paths, and the two-edge path exist. The coefficient $3t^2$ includes both the edge product $2\\cdot3$ and the ordered-time simplex area $t^2/2$.}\n\\head{Path length counts transfers, not elapsed time}\nStarting at $e_0$, the complete state is $(1,2t,3t^2)^{\\mathsf T}$. No fourth packet or repeated cycle can contribute. Direct differentiation gives $x'(t)=Ax(t)$, including the relation $x_2'(t)=3x_1(t)$.","t_paths}\n\\platecaption{Only the empty path, one-edge paths, and the two-edge path exist. The coefficient $3t^2$ includes both the edge product $2\\cdot3$ and the ordered-time simplex area $t^2/2$.}\n\\head{Path length counts transfers, not elapsed time}\nStarting at $e_0$, the complete state is $(1,2t,3t^2)^{\\mathsf T}$. No fourth packet or repeated cycle can contribute. Direct differentiation gives $x'(t)=Ax(t)$, including the relation $x_2'(t)=3x_1(t)$.","path, one-edge paths, and the two-edge path exist. The coefficient $3t^2$ includes both the edge product $2\\cdot3$ and the ordered-time simplex area $t^2/2$.}\n\\head{Path length counts transfers, not elapsed time}\nStarting at $e_0$, the complete state is $(1,2t,3t^2)^{\\mathsf T}$. No fourth packet or repeated cycle can contribute. Direct differentiation gives $x'(t)=Ax(t)$, including the relation $x_2'(t)=3x_1(t)$.","This is a linear transfer example. The columns of $A$ do not sum to zero. The total of the displayed state is $1+2t+3t^2$. It is consequently not a Markov probability model. Positivity of off-diagonal coefficients alone does not imply conservation of probability.\n\\takeaway{An acyclic packet graph terminates the transfer expansion. Nilpotence makes this particular full exponential polynomial. The two statements have different hypotheses.}","\\vpage{30}{Convergence does not close the dynamics}{REFINEMENT / A PERSISTENT RANGE EXCURSION}{Chapter VI, Part II: Theorem E4.8 and Example E4.10. Summability and closure are different tests.}\nLet the coarse space be $\\RR$ and every finer space be $\\RR^2$. The first connecting maps are $C(x,y)=x$ and $Rx=(x,0)$. All later maps are identities. Let the coarse evolution be the identity and all fine evolutions be rotations:\n\\[\nT(t)=\\mat{\\cos t&-\\sin t","not close the dynamics}{REFINEMENT / A PERSISTENT RANGE EXCURSION}{Chapter VI, Part II: Theorem E4.8 and Example E4.10. Summability and closure are different tests.}\nLet the coarse space be $\\RR$ and every finer space be $\\RR^2$. The first connecting maps are $C(x,y)=x$ and $Rx=(x,0)$. All later maps are identities. Let the coarse evolution be the identity and all fine evolutions be rotations:\n\\[\nT(t)=\\mat{\\cos t&-\\sin t\\\\\\sin t&\\cos t},\\qquad\nU_n(t)=CT(t)R=\\cos t\\quad(n\\ge1).\n\\]\nThe reconstructed sequence is already constant above its first level. Only one adjacent defect is nonzero, so its defect sum is finite on every bounded time interval.\n\\vfig{ext_closure}\n\\platecaption{The full state returns to the coarse line with negative sign after two quarter turns. Resetting the hidden coordinate after the first quarter turn loses the returning contribution.}\n\\head{A limit can fail the semigroup identity}\nAt $s=t=\\pi/2$,\n\\[\nU(s)U(t)=0,\\qquad U(s+t)=-1.\n\\]\nWrite $P=RC=\\diag(1,0)$. The exact algebraic difference is\n\\[\nU(s)U(t)-U(s+t)=CT(s)(P-I)T(t)R.\n\\]\nIts value is $1$ at these times, matc…","eady constant above its first level. Only one adjacent defect is nonzero, so its defect sum is finite on every bounded time interval.\n\\vfig{ext_closure}\n\\platecaption{The full state returns to the coarse line with negative sign after two quarter turns. Resetting the hidden coordinate after the first quarter turn loses the returning contribution.}\n\\head{A limit can fail the semigroup identity}\nAt $s=t=\\pi/2$,\n\\[\nU(s)U(t)=0,\\qquad U(s+t)=-1.\n\\]\nWrite $P=RC=\\diag(1,0)$. The exact algebraic difference is\n\\[\nU(s)U(t)-U(s+t)=CT(s)(P-I)T(t)R.\n\\]\nIts value is $1$ at these times, matching the missing hidden excursion. This is a failure of closure, not a failure of convergence.","above its first level. Only one adjacent defect is nonzero, so its defect sum is finite on every bounded time interval.\n\\vfig{ext_closure}\n\\platecaption{The full state returns to the coarse line with negative sign after two quarter turns. Resetting the hidden coordinate after the first quarter turn loses the returning contribution.}\n\\head{A limit can fail the semigroup identity}\nAt $s=t=\\pi/2$,\n\\[\nU(s)U(t)=0,\\qquad U(s+t)=-1.\n\\]\nWrite $P=RC=\\diag(1,0)$. The exact algebraic difference is\n\\[\nU(s)U(t)-U(s+t)=CT(s)(P-I)T(t)R.\n\\]\nIts value is $1$ at these times, matching the missing hidden excursion. This is a failure of closure, not a failure of convergence.","defect is nonzero, so its defect sum is finite on every bounded time interval.\n\\vfig{ext_closure}\n\\platecaption{The full state returns to the coarse line with negative sign after two quarter turns. Resetting the hidden coordinate after the first quarter turn loses the returning contribution.}\n\\head{A limit can fail the semigroup identity}\nAt $s=t=\\pi/2$,\n\\[\nU(s)U(t)=0,\\qquad U(s+t)=-1.\n\\]\nWrite $P=RC=\\diag(1,0)$. The exact algebraic difference is\n\\[\nU(s)U(t)-U(s+t)=CT(s)(P-I)T(t)R.\n\\]\nIts value is $1$ at these times, matching the missing hidden excursion. This is a failure of closure, not a failure of convergence.","ry bounded time interval.\n\\vfig{ext_closure}\n\\platecaption{The full state returns to the coarse line with negative sign after two quarter turns. Resetting the hidden coordinate after the first quarter turn loses the returning contribution.}\n\\head{A limit can fail the semigroup identity}\nAt $s=t=\\pi/2$,\n\\[\nU(s)U(t)=0,\\qquad U(s+t)=-1.\n\\]\nWrite $P=RC=\\diag(1,0)$. The exact algebraic difference is\n\\[\nU(s)U(t)-U(s+t)=CT(s)(P-I)T(t)R.\n\\]\nIts value is $1$ at these times, matching the missing hidden excursion. This is a failure of closure, not a failure of convergence.","caption{The full state returns to the coarse line with negative sign after two quarter turns. Resetting the hidden coordinate after the first quarter turn loses the returning contribution.}\n\\head{A limit can fail the semigroup identity}\nAt $s=t=\\pi/2$,\n\\[\nU(s)U(t)=0,\\qquad U(s+t)=-1.\n\\]\nWrite $P=RC=\\diag(1,0)$. The exact algebraic difference is\n\\[\nU(s)U(t)-U(s+t)=CT(s)(P-I)T(t)R.\n\\]\nIts value is $1$ at these times, matching the missing hidden excursion. This is a failure of closure, not a failure of convergence.","\\vpage{31}{A tail bound states what refinement buys}{SUMMABLE DEFECTS / A SCALAR COMPLETION}{Chapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps.}\nTake $X_n=\\RR$, $C_n=R_n=1$, and\n\\[\na_n=1-2^{-n},\\qquad A_n=-a_n,\\qquad T_n(t)=e^{-a_nt}\\quad(n\\ge0).\n\\]\nEach semigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigr","\\vpage{31}{A tail bound states what refinement buys}{SUMMABLE DEFECTS / A SCALAR COMPLETION}{Chapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps.}\nTake $X_n=\\RR$, $C_n=R_n=1$, and\n\\[\na_n=1-2^{-n},\\qquad A_n=-a_n,\\qquad T_n(t)=e^{-a_nt}\\quad(n\\ge0).\n\\]\nEach semigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here","\\vpage{31}{A tail bound states what refinement buys}{SUMMABLE DEFECTS / A SCALAR COMPLETION}{Chapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps.}\nTake $X_n=\\RR$, $C_n=R_n=1$, and\n\\[\na_n=1-2^{-n},\\qquad A_n=-a_n,\\qquad T_n(t)=e^{-a_nt}\\quad(n\\ge0).\n\\]\nEach semigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is iden","\\vpage{31}{A tail bound states what refinement buys}{SUMMABLE DEFECTS / A SCALAR COMPLETION}{Chapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps.}\nTake $X_n=\\RR$, $C_n=R_n=1$, and\n\\[\na_n=1-2^{-n},\\qquad A_n=-a_n,\\qquad T_n(t)=e^{-a_nt}\\quad(n\\ge0).\n\\]\nEach semigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is addition","\\vpage{31}{A tail bound states what refinement buys}{SUMMABLE DEFECTS / A SCALAR COMPLETION}{Chapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps.}\nTake $X_n=\\RR$, $C_n=R_n=1$, and\n\\[\na_n=1-2^{-n},\\qquad A_n=-a_n,\\qquad T_n(t)=e^{-a_nt}\\quad(n\\ge0).\n\\]\nEach semigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","orollary E4.7. This example has identity connecting maps.}\nTake $X_n=\\RR$, $C_n=R_n=1$, and\n\\[\na_n=1-2^{-n},\\qquad A_n=-a_n,\\qquad T_n(t)=e^{-a_nt}\\quad(n\\ge0).\n\\]\nEach semigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","example has identity connecting maps.}\nTake $X_n=\\RR$, $C_n=R_n=1$, and\n\\[\na_n=1-2^{-n},\\qquad A_n=-a_n,\\qquad T_n(t)=e^{-a_nt}\\quad(n\\ge0).\n\\]\nEach semigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","emigroup is contractive for $t\\ge0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","0$. The exact adjacent generator-defect magnitude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","ude is\n\\[\n|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","|A_{n+1}-A_n|=2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","2^{-(n+1)},\\qquad\n\\sum_{j=n}^{\\infty}|A_{j+1}-A_j|=2^{-n}.\n\\]\n\\vfig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","fig{ext_tails}\n\\platecaption{The bars show the certified tail $2^{-n}$ at horizon $T=1$. They are bounds on a whole time interval, not errors sampled at the right endpoint.}\n\\head{The estimate follows before any graph is drawn}\nFor $0\\le t\\le T$, the derivative of $a\\mapsto e^{-at}$ has absolute value at most $T$ when $a\\ge0$. Thus\n\\[\n\\sup_{0\\le t\\le T}|T_n(t)-e^{-t}|\\le T2^{-n}.\n\\]\nThe limit is $e^{-t}$, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is identically zero. That closure fact is additional to the tail estimate.","\\vpage{32}{The obstruction lives in the forgotten directions}{MARKOV DESCENT / TEST THE AGGREGATION KERNEL}{Chapter VI, Part II: Theorem E5.3 and Corollary E5.4. A quotient criterion with a concrete witness.}\nReturn to the four-state model of Plate VI.4 and the perturbed stochastic matrix $K_\\varepsilon$ of Plate VI.6. Let\n\\[\nC=\\mat{1&1&0&0","\\vpage{32}{The obstruction lives in the forgotten directions}{MARKOV DESCENT / TEST THE AGGREGATION KERNEL}{Chapter VI, Part II: Theorem E5.3 and Corollary E5.4. A quotient criterion with a concrete witness.}\nReturn to the four-state model of Plate VI.4 and the perturbed stochastic matrix $K_\\varepsilon$ of Plate VI.6. Let\n\\[\nC=\\mat{1&1&0&0\\\\0&0&1&1},\\qquad w=e_0-e_1.\n\\]\nThe vector $w$ changes the fine distribution inside the first packet without changing its mass: $Cw=0$. Yet\n\\[\nCK_\\varepsilon w=\\mat{-1/16","0&0&1&1},\\qquad w=e_0-e_1.\n\\]\nThe vector $w$ changes the fine distribution inside the first packet without changing its mass: $Cw=0$. Yet\n\\[\nCK_\\varepsilon w=\\mat{-1/16","0&0&1&1},\\qquad w=e_0-e_1.\n\\]\nThe vector $w$ changes the fine distribution inside the first packet without changing its mass: $Cw=0$. Yet\n\\[\nCK_\\varepsilon w=\\mat{-1/16\\\\1/16}\\ne0.\n\\]\n\\vfig{ext_obstruction}\n\\platecaption{The coarse input cannot see $w$. The next coarse output can. Therefore no linear coarse operator can reproduce this fine step for every input.}\n\\head{Why one witness rules out every coarse matrix}\nIf some $\\bar K$ satisfied $CK_\\varepsilon=\\bar KC$, then $CK_\\varepsilon w=\\bar KCw=0$, contradicting the displayed vector. The failure is not repaired by fitting a different coarse transition matrix to selected priors.","1/16}\\ne0.\n\\]\n\\vfig{ext_obstruction}\n\\platecaption{The coarse input cannot see $w$. The next coarse output can. Therefore no linear coarse operator can reproduce this fine step for every input.}\n\\head{Why one witness rules out every coarse matrix}\nIf some $\\bar K$ satisfied $CK_\\varepsilon=\\bar KC$, then $CK_\\varepsilon w=\\bar KCw=0$, contradicting the displayed vector. The failure is not repaired by fitting a different coarse transition matrix to selected priors.","More generally, for a surjective aggregation map $C$, a linear map $M$ descends exactly if $CM$ vanishes on $\\ker C$. The descended map is defined by $\\bar M(Cx)=CMx$. The kernel condition makes this independent of the representative $x$.","For a deterministic partition of a finite stochastic system, differences $e_i-e_j$ within the same source block span the forgotten directions. Testing them gives the familiar equality of aggregate columns. Nonnegativity and column sums of the descended stochastic matrix then follow from the fine stochastic matrix and the partition structure.\n\\takeaway{The canonical obstruction is the action on information that aggregation discards. A nonzero witness in that kernel rules out exact descent at once.}","\\vpage{33}{Gaussian packets have separate clocks}{ORNSTEIN--UHLENBECK / HERMITE ENERGY}{Chapter VI, Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure.}\nIn $L^2(\\gamma)$ for the standard Gaussian law on $\\RR$, use the probabilists' Hermite polynomials\n\\[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(","\\vpage{33}{Gaussian packets have separate clocks}{ORNSTEIN--UHLENBECK / HERMITE ENERGY}{Chapter VI, Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure.}\nIn $L^2(\\gamma)$ for the standard Gaussian law on $\\RR$, use the probabilists' Hermite polynomials\n\\[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean","\\vpage{33}{Gaussian packets have separate clocks}{ORNSTEIN--UHLENBECK / HERMITE ENERGY}{Chapter VI, Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure.}\nIn $L^2(\\gamma)$ for the standard Gaussian law on $\\RR$, use the probabilists' Hermite polynomials\n\\[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when","\\vpage{33}{Gaussian packets have separate clocks}{ORNSTEIN--UHLENBECK / HERMITE ENERGY}{Chapter VI, Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure.}\nIn $L^2(\\gamma)$ for the standard Gaussian law on $\\RR$, use the probabilists' Hermite polynomials\n\\[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","{ORNSTEIN--UHLENBECK / HERMITE ENERGY}{Chapter VI, Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure.}\nIn $L^2(\\gamma)$ for the standard Gaussian law on $\\RR$, use the probabilists' Hermite polynomials\n\\[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","d Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure.}\nIn $L^2(\\gamma)$ for the standard Gaussian law on $\\RR$, use the probabilists' Hermite polynomials\n\\[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","standard Gaussian law on $\\RR$, use the probabilists' Hermite polynomials\n\\[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","[\nH_1(x)=x,\\qquad H_2(x)=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","=x^2-1,\\qquad\nH_4(x)=x^4-6x^2+3.\n\\]\nThe Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","The Ornstein--Uhlenbeck generator is $\\mathcal Lf=f''-xf'$. Direct polynomial differentiation gives $\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","$\\mathcal LH_n=-nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","nH_n$ for these three polynomials. Their squared norms are $n!$ and different Hermite degrees are orthogonal.\n\\vfig{ext_hermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","ermite}\n\\platecaption{For $f=H_1+H_2$, the two bars are its separate squared-energy contributions. At $t=\\log2$, their sum is $3/8$, below the spectral-gap bound $3/4$.}\n\\head{Reconstruction preserves the modes as time changes their weights}\nThe semigroup sends $f=H_1+H_2$ to\n\\[\nT(t)f=e^{-t}H_1+e^{-2t}H_2,\\qquad\n\\|T(t)f\\|_2^2=e^{-2t}+2e^{-4t}.\n\\]\nIts mean is zero. The Gaussian gap bounds this by $e^{-2t}\\|f\\|_2^2=3e^{-2t}$. It need not be an equality when multiple degrees are present.","For the nonnegative operator $-\\mathcal L$, a square-root evolution replaces rate $n$ by $\\sqrt n$. On $g=H_1+H_4$, the coefficients at $t=\\log2$ are $1/2$ and $1/4$, whereas ordinary OU evolution gives $1/2$ and $1/16$. The eigenfunctions agree. The rates differ.","These calculations concern polynomial elements of Gaussian $L^2$. The general expansion and generator-domain statements belong to the traditional proof. A finite list of Hermite checks does not establish completeness of the basis.\n\\takeaway{Orthogonal packets make the energy accounting exact. A fractional generator alters the spectral clock while keeping the same Gaussian modes.}","\\vpage{34}{Several observations determine one sufficient partition}{FIXED LIKELIHOODS / JOINT PROFILES}{Chapter VI, Part II: Theorem E9.3. A finite family of strictly positive likelihoods.}\nOn four states, consider two likelihoods\n\\[\n\\lambda^{(1)}=(2,2,1,1),\\qquad\n\\lambda^{(2)}=(1,1,3,1).\n\\]\nTwo states may share a packet in the coarsest partition on which every listed likelihood is constant exactly when their entire likelihood profiles agree. The profiles are\n\\[\n(2,1),\\ (2,1),\\ (1,3),\\ (1,1),\n\\]\nso the partition is $\\{0,1\\},\\{2\\},\\{3\\}$.\n\\vfig{ext_profiles}\n\\platecaption{The first observation alone allows the last two states to be merged. The second observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the l","}{FIXED LIKELIHOODS / JOINT PROFILES}{Chapter VI, Part II: Theorem E9.3. A finite family of strictly positive likelihoods.}\nOn four states, consider two likelihoods\n\\[\n\\lambda^{(1)}=(2,2,1,1),\\qquad\n\\lambda^{(2)}=(1,1,3,1).\n\\]\nTwo states may share a packet in the coarsest partition on which every listed likelihood is constant exactly when their entire likelihood profiles agree. The profiles are\n\\[\n(2,1),\\ (2,1),\\ (1,3),\\ (1,1),\n\\]\nso the partition is $\\{0,1\\},\\{2\\},\\{3\\}$.\n\\vfig{ext_profiles}\n\\platecaption{The first observation alone allows the last two states to be merged. The second observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the likelihoods componentwise, giving weights $(2,2,3,1)$ and posterior $(2,2,3,1)/8$. Its packet masses are precisely the displayed coarse posterior.","I: Theorem E9.3. A finite family of strictly positive likelihoods.}\nOn four states, consider two likelihoods\n\\[\n\\lambda^{(1)}=(2,2,1,1),\\qquad\n\\lambda^{(2)}=(1,1,3,1).\n\\]\nTwo states may share a packet in the coarsest partition on which every listed likelihood is constant exactly when their entire likelihood profiles agree. The profiles are\n\\[\n(2,1),\\ (2,1),\\ (1,3),\\ (1,1),\n\\]\nso the partition is $\\{0,1\\},\\{2\\},\\{3\\}$.\n\\vfig{ext_profiles}\n\\platecaption{The first observation alone allows the last two states to be merged. The second observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the likelihoods componentwise, giving weights $(2,2,3,1)$ and posterior $(2,2,3,1)/8$. Its packet masses are precisely the displayed coarse posterior.","files agree. The profiles are\n\\[\n(2,1),\\ (2,1),\\ (1,3),\\ (1,1),\n\\]\nso the partition is $\\{0,1\\},\\{2\\},\\{3\\}$.\n\\vfig{ext_profiles}\n\\platecaption{The first observation alone allows the last two states to be merged. The second observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the likelihoods componentwise, giving weights $(2,2,3,1)$ and posterior $(2,2,3,1)/8$. Its packet masses are precisely the displayed coarse posterior.","(1,1),\n\\]\nso the partition is $\\{0,1\\},\\{2\\},\\{3\\}$.\n\\vfig{ext_profiles}\n\\platecaption{The first observation alone allows the last two states to be merged. The second observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the likelihoods componentwise, giving weights $(2,2,3,1)$ and posterior $(2,2,3,1)/8$. Its packet masses are precisely the displayed coarse posterior.","2\\},\\{3\\}$.\n\\vfig{ext_profiles}\n\\platecaption{The first observation alone allows the last two states to be merged. The second observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the likelihoods componentwise, giving weights $(2,2,3,1)$ and posterior $(2,2,3,1)/8$. Its packet masses are precisely the displayed coarse posterior.","to be merged. The second observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the likelihoods componentwise, giving weights $(2,2,3,1)$ and posterior $(2,2,3,1)/8$. Its packet masses are precisely the displayed coarse posterior.","observation distinguishes them. The shared partition must respect both likelihoods.}\n\\head{A complete two-stage calculation}\nStart from the uniform prior. The coarse prior is $(1/2,1/4,1/4)$. After the first likelihood, it becomes $(2/3,1/6,1/6)$. After the second, it is\n\\[\n(1/2,3/8,1/8).\n\\]\nThe direct fine update multiplies the likelihoods componentwise, giving weights $(2,2,3,1)$ and posterior $(2,2,3,1)/8$. Its packet masses are precisely the displayed coarse posterior.","The individual evidences are $Z_1=3/2$ and $Z_2=4/3$, and their product is $2$, the direct evidence of the product likelihood. These are relative likelihood weights, so evidence larger than one is allowed.","This partition supplies fixed coarse likelihoods that work for every prior. A single-packet aggregation always gives posterior $(1)$. That mass-only commutation does not identify likelihood profiles. A prior-dependent conditional average can summarize one update on a coarser partition, but requires the within-packet information that a fixed sufficient rule avoids.\n\\takeaway{Equal full likelihood profiles determine the coarsest partition on which every listed likelihood is constant. This partition admits fixed sufficient update rules.}","\\vpage{35}{Information loss has an exact location}{RELATIVE ENTROPY / COARSE AND CONDITIONAL TERMS}{Chapter VI, Part II: Theorem E9.8 and Corollary E9.9. Positive reference masses in this example.}\nTake\n\\[\np=(1/2,1/4,1/8,1/8),\\qquad q=(1/4,1/4,1/4,1/4)\n\\]\nand group the first two and last two states. Their coarse laws are $P=(3/4,1/4)$ and $Q=(1/2,1/2)$. Define $D(p\\Vert q)=\\sum_i p_i\\log(p_i/q_i)$ with natural logarithms.\n\\vfig{ext_information}\n\\platecaption{The bar lengths use the displayed exact formulas evaluated for illustration. The split is an identity: coarse information plus posterior-weighted conditional information. It is not a fitted allocation.}\n\\head{The chain rule identifies exactly what aggregation removes}\nInside packet A, the conditional laws are $(2/3,1/3)$ and $(1/2,1/2)$. Inside B they agree. Hence\n\\[\n\\begin{aligned}\nD(p\\Vert q)&=\\tfrac14\\log2,","=\\sum_i p_i\\log(p_i/q_i)$ with natural logarithms.\n\\vfig{ext_information}\n\\platecaption{The bar lengths use the displayed exact formulas evaluated for illustration. The split is an identity: coarse information plus posterior-weighted conditional information. It is not a fitted allocation.}\n\\head{The chain rule identifies exactly what aggregation removes}\nInside packet A, the conditional laws are $(2/3,1/3)$ and $(1/2,1/2)$. Inside B they agree. Hence\n\\[\n\\begin{aligned}\nD(p\\Vert q)&=\\tfrac14\\log2,","p_i/q_i)$ with natural logarithms.\n\\vfig{ext_information}\n\\platecaption{The bar lengths use the displayed exact formulas evaluated for illustration. The split is an identity: coarse information plus posterior-weighted conditional information. It is not a fitted allocation.}\n\\head{The chain rule identifies exactly what aggregation removes}\nInside packet A, the conditional laws are $(2/3,1/3)$ and $(1/2,1/2)$. Inside B they agree. Hence\n\\[\n\\begin{aligned}\nD(p\\Vert q)&=\\tfrac14\\log2,","ext_information}\n\\platecaption{The bar lengths use the displayed exact formulas evaluated for illustration. The split is an identity: coarse information plus posterior-weighted conditional information. It is not a fitted allocation.}\n\\head{The chain rule identifies exactly what aggregation removes}\nInside packet A, the conditional laws are $(2/3,1/3)$ and $(1/2,1/2)$. Inside B they agree. Hence\n\\[\n\\begin{aligned}\nD(p\\Vert q)&=\\tfrac14\\log2,\\\\\nD(P\\Vert Q)&=\\tfrac34\\log3-\\log2,\\\\\n\\sum_aP_aD(p(\\cdot\\mid a)\\Vert q(\\cdot\\mid a))\n&=\\tfrac54\\log2-\\tfrac34\\log3.\n\\end{aligned}\n\\]\nAdding the last two lines recovers the first. Every conditional term is nonnegative, so aggregation cannot increase relative entropy. Equality holds precisely when the occupied packets have the same conditional laws, under the stated support convention.","The weighting is by the packet mass $P_a$ from the first argument of $D(p\\Vert q)$. Replacing it by $Q_a$ generally breaks the identity. Likewise, forgetting zero-mass conventions or allowing a positive $p_i$ over a zero $q_i$ changes which expressions are finite.\n\\takeaway{Coarse relative entropy measures distinguishability that survives aggregation. The conditional term records exactly the distinguishability removed inside the packets.}","\\vpage{36}{The score rises because it has variance}{BAYESIAN FLOW / EVIDENCE CUMULANTS}{Chapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior.}\nStart with equal prior masses on scores $u=(0,1,2)$ and likelihood $e^{tu}$. Put $q=e^t$. The evidence and tilted distribution are\n\\[\nZ(t)=\\frac{1+q+q^2}{3},\\qquad\np(t)=\\frac{(1,q,q^2)}{1+q+q^2}.\n\\]\nAt $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad","\\vpage{36}{The score rises because it has variance}{BAYESIAN FLOW / EVIDENCE CUMULANTS}{Chapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior.}\nStart with equal prior masses on scores $u=(0,1,2)$ and likelihood $e^{tu}$. Put $q=e^t$. The evidence and tilted distribution are\n\\[\nZ(t)=\\frac{1+q+q^2}{3},\\qquad\np(t)=\\frac{(1,q,q^2)}{1+q+q^2}.\n\\]\nAt $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\q","\\vpage{36}{The score rises because it has variance}{BAYESIAN FLOW / EVIDENCE CUMULANTS}{Chapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior.}\nStart with equal prior masses on scores $u=(0,1,2)$ and likelihood $e^{tu}$. Put $q=e^t$. The evidence and tilted distribution are\n\\[\nZ(t)=\\frac{1+q+q^2}{3},\\qquad\np(t)=\\frac{(1,q,q^2)}{1+q+q^2}.\n\\]\nAt $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=","\\vpage{36}{The score rises because it has variance}{BAYESIAN FLOW / EVIDENCE CUMULANTS}{Chapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior.}\nStart with equal prior masses on scores $u=(0,1,2)$ and likelihood $e^{tu}$. Put $q=e^t$. The evidence and tilted distribution are\n\\[\nZ(t)=\\frac{1+q+q^2}{3},\\qquad\np(t)=\\frac{(1,q,q^2)}{1+q+q^2}.\n\\]\nAt $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u","\\vpage{36}{The score rises because it has variance}{BAYESIAN FLOW / EVIDENCE CUMULANTS}{Chapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior.}\nStart with equal prior masses on scores $u=(0,1,2)$ and likelihood $e^{tu}$. Put $q=e^t$. The evidence and tilted distribution are\n\\[\nZ(t)=\\frac{1+q+q^2}{3},\\qquad\np(t)=\\frac{(1,q,q^2)}{1+q+q^2}.\n\\]\nAt $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives de","ge{36}{The score rises because it has variance}{BAYESIAN FLOW / EVIDENCE CUMULANTS}{Chapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior.}\nStart with equal prior masses on scores $u=(0,1,2)$ and likelihood $e^{tu}$. Put $q=e^t$. The evidence and tilted distribution are\n\\[\nZ(t)=\\frac{1+q+q^2}{3},\\qquad\np(t)=\\frac{(1,q,q^2)}{1+q+q^2}.\n\\]\nAt $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-…","core rises because it has variance}{BAYESIAN FLOW / EVIDENCE CUMULANTS}{Chapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior.}\nStart with equal prior masses on scores $u=(0,1,2)$ and likelihood $e^{tu}$. Put $q=e^t$. The evidence and tilted distribution are\n\\[\nZ(t)=\\frac{1+q+q^2}{3},\\qquad\np(t)=\\frac{(1,q,q^2)}{1+q+q^2}.\n\\]\nAt $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-10,-6,16)/49…","t $t=\\log2$, they are $Z=7/3$ and $p=(1,2,4)/7$.\n\\vfig{ext_cumulants}\n\\platecaption{The score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-10,-6,16)/49$, which sum to zero.","lues stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-10,-6,16)/49$, which sum to zero.","bilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores.}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-10,-6,16)/49$, which sum to zero.","}\n\\head{Two derivatives have different statistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-10,-6,16)/49$, which sum to zero.","tatistical meanings}\nThe first derivative of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-10,-6,16)/49$, which sum to zero.","e of log evidence is the current mean score, and the second is its variance:\n\\[\n\\frac{d}{dt}\\log Z(t)=\\EE_{p(t)}u,\n\\qquad\n\\frac{d^2}{dt^2}\\log Z(t)=\\Var_{p(t)}(u)\\ge0.\n\\]\nAt $q=2$ the exact values are\n\\[\n\\EE u=\\frac{10}{7},\\qquad\n\\EE u^2=\\frac{18}{7},\\qquad\n\\Var(u)=\\frac{26}{49}.\n\\]\nThus the derivative of the mean is $26/49$ at this time. The flow equation $p_i'=(u_i-\\EE u)p_i$ gives derivatives $(-10,-6,16)/49$, which sum to zero.","As $t\\to\\infty$, mass concentrates on the largest score present in the initial support. A score with zero initial prior mass is never created by multiplication. If several occupied states tie for the maximum, their relative prior proportions remain unchanged within that maximal packet.\n\\takeaway{Log evidence is a cumulant generator for the tilted law. Its curvature is variance, so the mean score is monotone even though the flow is nonlinear.}","\\vpage{37}{Keep blocks by their certified contribution}{MEMORY COMPRESSION / AN OPTIMIZATION WITH A DEFINED COST}{Chapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget.}\nConsider the scalar memory kernel\n\\[\n\\mathcal K(t)=\\tfrac14e^{-t}+(3+2t)e^{-2t}+\\tfrac13e^{-3t}.\n\\]\nIt has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are\n\\[\nw_1=\\frac14,\\qquad w_2=\\frac32+\\frac{2}{2^2}=2,\n\\qquad w_3=\\frac{1/3}{3}=\\frac19.\n\\]\n\\vfig{ext_selection}\n\\platecaption{With one primary block retained, choose the middle block and certify omitted weight $13/36$. With two retained, add the rate-one block and the remaining certificate is $1/9$.}\n\\head{The slowest eigenvalue is not necessarily the largest effect}\nThe rate-two block has the greatest weight because its couplings and nilpotent contribution dominate. Retaining the slowest rate alone would omit weight $2","A DEFINED COST}{Chapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget.}\nConsider the scalar memory kernel\n\\[\n\\mathcal K(t)=\\tfrac14e^{-t}+(3+2t)e^{-2t}+\\tfrac13e^{-3t}.\n\\]\nIt has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are\n\\[\nw_1=\\frac14,\\qquad w_2=\\frac32+\\frac{2}{2^2}=2,\n\\qquad w_3=\\frac{1/3}{3}=\\frac19.\n\\]\n\\vfig{ext_selection}\n\\platecaption{With one primary block retained, choose the middle block and certify omitted weight $13/36$. With two retained, add the rate-one block and the remaining certificate is $1/9$.}\n\\head{The slowest eigenvalue is not necessarily the largest effect}\nThe rate-two block has the greatest weight because its couplings and nilpotent contribution dominate. Retaining the slowest rate alone would omit weight $2+1/9$, a much larger certificate.","rac13e^{-3t}.\n\\]\nIt has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are\n\\[\nw_1=\\frac14,\\qquad w_2=\\frac32+\\frac{2}{2^2}=2,\n\\qquad w_3=\\frac{1/3}{3}=\\frac19.\n\\]\n\\vfig{ext_selection}\n\\platecaption{With one primary block retained, choose the middle block and certify omitted weight $13/36$. With two retained, add the rate-one block and the remaining certificate is $1/9$.}\n\\head{The slowest eigenvalue is not necessarily the largest effect}\nThe rate-two block has the greatest weight because its couplings and nilpotent contribution dominate. Retaining the slowest rate alone would omit weight $2+1/9$, a much larger certificate.","and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are\n\\[\nw_1=\\frac14,\\qquad w_2=\\frac32+\\frac{2}{2^2}=2,\n\\qquad w_3=\\frac{1/3}{3}=\\frac19.\n\\]\n\\vfig{ext_selection}\n\\platecaption{With one primary block retained, choose the middle block and certify omitted weight $13/36$. With two retained, add the rate-one block and the remaining certificate is $1/9$.}\n\\head{The slowest eigenvalue is not necessarily the largest effect}\nThe rate-two block has the greatest weight because its couplings and nilpotent contribution dominate. Retaining the slowest rate alone would omit weight $2+1/9$, a much larger certificate.","n{With one primary block retained, choose the middle block and certify omitted weight $13/36$. With two retained, add the rate-one block and the remaining certificate is $1/9$.}\n\\head{The slowest eigenvalue is not necessarily the largest effect}\nThe rate-two block has the greatest weight because its couplings and nilpotent contribution dominate. Retaining the slowest rate alone would omit weight $2+1/9$, a much larger certificate.","The descending order is $2,1/4,1/9$. A target tolerance $1/8$ needs two blocks according to this bound: one block leaves $13/36>1/8$, while two leave $1/9\\le1/8$. Every term in this particular scalar kernel is nonnegative, so its omitted integral equals the certificate.","\\vpage{38}{A kernel certificate becomes a trajectory certificate}{VOLTERRA STABILITY / SAME INITIAL DATA AND FORCING}{Chapter VI, Part II: Theorem E15.6. Norms and a finite time horizon are explicit.}\nLet $N=\\mat{0&1\\\\0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\vare","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\wide","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial.","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general th","0&0}$, so $N^2=0$ and $\\|N\\|_2=1$. On $[0,T]$, compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.",", compare the memory kernels $K(t)=N$ and $\\widetilde K(t)=0$, with identity free semigroup, no forcing, and initial state $x_0=e_1=(0,1)^{\\mathsf T}$. The equations are\n\\[\nx'(t)=\\int_0^t N x(r)\\,dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.",",dr,\\qquad\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.","d\n\\widetilde x'(t)=0.\n\\]\nTheir exact solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.","act solutions are $x(t)=(t^2/2,1)^{\\mathsf T}$ and $\\widetilde x(t)=(0,1)^{\\mathsf T}$.\n\\vfig{ext_trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.","trajectory}\n\\platecaption{The first-coordinate error is exactly $t^2/2$. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.","ificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.","n the entire interval. The bound is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.","und is deliberately shown separately from the trajectory.}\n\\head{Track every constant before applying the estimate}\nHere $M_T=1$, $B=\\sup\\|\\widetilde x\\|=1$, and\n\\[\nL=\\|K\\|_{L^1(0,T)}=T,\\qquad\n\\varepsilon=\\|K-\\widetilde K\\|_{L^1(0,T)}=T.\n\\]\nThe theorem yields\n\\[\n\\sup_{0\\le t\\le T}\\|x(t)-\\widetilde x(t)\\|\n=\\frac{T^2}{2}\\le T^2e^{T^2}.\n\\]\nAt $T=1/2$, the exact error is $1/8$ and the certificate is $e^{1/4}/4$. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.","The two solutions share initial state, free semigroup, and forcing. If those are changed as well, their errors need additional terms. An integrated kernel error alone does not automatically control a trajectory on an infinite horizon.\n\\takeaway{The compression estimate transports an $L^1$ kernel budget into a finite-horizon state bound, using an explicit bound on the comparison trajectory.}","\\vpage{39}{Two auxiliary states realize a repeated pole}{FINITE REALIZATION / AN EXACT DIFFERENTIAL MODEL}{Chapter VI, Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization.}\nThe kernel $K(t)=te^{-2t}$ from Plate VI.23 has transform $1/(z+2)^2$. It is realized by two auxiliary states:\n\\[\n\\begin{aligned}\n\\dot y_1&=-2y_1+y_2,& y_1(0)&=0,","\\vpage{39}{Two auxiliary states realize a repeated pole}{FINITE REALIZATION / AN EXACT DIFFERENTIAL MODEL}{Chapter VI, Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization.}\nThe kernel $K(t)=te^{-2t}$ from Plate VI.23 has transform $1/(z+2)^2$. It is realized by two auxiliary states:\n\\[\n\\begin{aligned}\n\\dot y_1&=-2y_1+y_2,& y_1(0)&=0,\\\\\n\\dot y_2&=-2y_2+x,& y_2(0)&=0,\\\\\n\\dot x&=a x+y_1.&\n\\end{aligned}\n\\]\nWith $D=\\mat{-2&1","\\dot x&=a x+y_1.&\n\\end{aligned}\n\\]\nWith $D=\\mat{-2&1\\\\0&-2}$, $B=(1,0)$, and $C=(0,1)^{\\mathsf T}$, the feedback kernel is $Be^{tD}C=te^{-2t}$.\n\\vfig{ext_auxiliary}\n\\platecaption{An input first enters $y_2$, transfers to $y_1$, and then returns to $x$. The two equal decay rates produce a repeated pole through the intervening link.}\n\\head{Elimination recovers the convolution, including its preparation}\nVariation of constants gives\n\\[\ny(t)=e^{tD}y(0)+\\int_0^t e^{(t-r)D}Cx(r)\\,dr,\n\\]\nso the retained equation is\n\\[\n\\dot x(t)=a x(t)+Be^{tD}y(0)\n+\\int_0^t (t-r)e^{-2(t-r)}x(r)\\,dr.\n\\]\nThe convolution-only version is exact for $y(0)=0$. For a nonzero preparation, $Be^{tD}y(0)=e^{-2t}(y_1(0)+t y_2(0))$ is an additional forcing.","0&-2}$, $B=(1,0)$, and $C=(0,1)^{\\mathsf T}$, the feedback kernel is $Be^{tD}C=te^{-2t}$.\n\\vfig{ext_auxiliary}\n\\platecaption{An input first enters $y_2$, transfers to $y_1$, and then returns to $x$. The two equal decay rates produce a repeated pole through the intervening link.}\n\\head{Elimination recovers the convolution, including its preparation}\nVariation of constants gives\n\\[\ny(t)=e^{tD}y(0)+\\int_0^t e^{(t-r)D}Cx(r)\\,dr,\n\\]\nso the retained equation is\n\\[\n\\dot x(t)=a x(t)+Be^{tD}y(0)\n+\\int_0^t (t-r)e^{-2(t-r)}x(r)\\,dr.\n\\]\nThe convolution-only version is exact for $y(0)=0$. For a nonzero preparation, $Be^{tD}y(0)=e^{-2t}(y_1(0)+t y_2(0))$ is an additional forcing.","nel is $Be^{tD}C=te^{-2t}$.\n\\vfig{ext_auxiliary}\n\\platecaption{An input first enters $y_2$, transfers to $y_1$, and then returns to $x$. The two equal decay rates produce a repeated pole through the intervening link.}\n\\head{Elimination recovers the convolution, including its preparation}\nVariation of constants gives\n\\[\ny(t)=e^{tD}y(0)+\\int_0^t e^{(t-r)D}Cx(r)\\,dr,\n\\]\nso the retained equation is\n\\[\n\\dot x(t)=a x(t)+Be^{tD}y(0)\n+\\int_0^t (t-r)e^{-2(t-r)}x(r)\\,dr.\n\\]\nThe convolution-only version is exact for $y(0)=0$. For a nonzero preparation, $Be^{tD}y(0)=e^{-2t}(y_1(0)+t y_2(0))$ is an additional forcing.","and then returns to $x$. The two equal decay rates produce a repeated pole through the intervening link.}\n\\head{Elimination recovers the convolution, including its preparation}\nVariation of constants gives\n\\[\ny(t)=e^{tD}y(0)+\\int_0^t e^{(t-r)D}Cx(r)\\,dr,\n\\]\nso the retained equation is\n\\[\n\\dot x(t)=a x(t)+Be^{tD}y(0)\n+\\int_0^t (t-r)e^{-2(t-r)}x(r)\\,dr.\n\\]\nThe convolution-only version is exact for $y(0)=0$. For a nonzero preparation, $Be^{tD}y(0)=e^{-2t}(y_1(0)+t y_2(0))$ is an additional forcing.","l decay rates produce a repeated pole through the intervening link.}\n\\head{Elimination recovers the convolution, including its preparation}\nVariation of constants gives\n\\[\ny(t)=e^{tD}y(0)+\\int_0^t e^{(t-r)D}Cx(r)\\,dr,\n\\]\nso the retained equation is\n\\[\n\\dot x(t)=a x(t)+Be^{tD}y(0)\n+\\int_0^t (t-r)e^{-2(t-r)}x(r)\\,dr.\n\\]\nThe convolution-only version is exact for $y(0)=0$. For a nonzero preparation, $Be^{tD}y(0)=e^{-2t}(y_1(0)+t y_2(0))$ is an additional forcing.","\\vpage{40}{Survivors have a different mixture of causes}{COMPETING RISKS / RESIDUAL LIFE}{Chapter VI, Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.","\\vpage{40}{Survivors have a different mixture of causes}{COMPETING RISKS / RESIDUAL LIFE}{Chapter VI, Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe","\\vpage{40}{Survivors have a different mixture of causes}{COMPETING RISKS / RESIDUAL LIFE}{Chapter VI, Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\","\\vpage{40}{Survivors have a different mixture of causes}{COMPETING RISKS / RESIDUAL LIFE}{Chapter VI, Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower …","auses}{COMPETING RISKS / RESIDUAL LIFE}{Chapter VI, Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","/ RESIDUAL LIFE}{Chapter VI, Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","16.4 and Theorem E16.5. This example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","example is stable and absorbs surely.}\nTwo transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","transient states have no internal transitions. Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","Their exit rates are $1$ and $2$, their initial masses are both $1/2$, and the two states exit by distinct causes. Thus\n\\[\nA=\\diag(-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","-1,-2),\\qquad\nS(t)=\\tfrac12e^{-t}+\\tfrac12e^{-2t},\n\\]\nwith cause densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","use densities $f_1(t)=e^{-t}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","}/2$ and $f_2(t)=e^{-2t}$.\n\\vfig{ext_risks}\n\\platecaption{At $t=\\log2$, surviving mass is $3/8$. Cause 1 has accumulated probability $1/4$ and cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","nd cause 2 has accumulated probability $3/8$. The three pieces sum to one.}\n\\head{Condition on survival before restarting the clock}\nAt this time the transient masses are $(1/4,1/8)$, so the conditional distribution of survivors is $(2/3,1/3)$. Consequently\n\\[\n\\frac{S(t+s)}{S(t)}=\\frac23e^{-s}+\\frac13e^{-2s},\n\\qquad\n\\EE[\\tau-t\\mid\\tau>t]=\\frac23+\\frac16=\\frac56.\n\\]\nThe original mean lifetime was $\\tfrac12\\cdot1+\\tfrac12\\cdot\\tfrac12=3/4$. Surviving longer changes the mixture toward the slower state. It does not restart the original mixture.","The cause densities at $t=\\log2$ both equal $1/4$, so the cause-specific hazards are both $2/3$ and total hazard is $4/3$. Hazards are rates and need not lie in $[0,1]$. Each eventual cause probability is $1/2$, obtained by integrating its density. Stability of the subgenerator is part of that eventual-absorption statement.\n\\takeaway{Residual life restarts the same transient semigroup from the conditional surviving distribution. Cause probabilities, densities, and hazards answer different questions.}","\\vpage{41}{An operator substitution transports the defect too}{FUNCTIONAL CALCULUS / AN EXACT RATIONAL TEST}{Chapter VI, Part II: Theorem E3.4; compare Chapter VI, Part I, Theorem 15.1, and Plate VI.25.}\nUse the operators from Plate VI.25, but perturb the rectangular comparison map:\n\\[\nA=\\mat{2&1&0\\\\0&2&0\\\\0&0&9},\\quad\nB=\\mat{2&1\\\\0&2},\\quad\nC_\\eta=\\mat{1&0&\\eta\\\\0&1&0}.\n\\]\nThe generator defect is $\\Delta=C_\\eta A-BC_\\eta$, with only one nonzero entry, $\\Delta_{13}=7\\eta$ (matrix indices begin at one).\n\\vfig{ext_substitution}\n\\platecaption{The perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at $\\eta=0$. A nonzero defect is transformed, not erased, by the rational calculus.}\n\\head{A resolvent identity supplies the sign and both inverse factors}\nFor $F(z)=1/(1+z)$, all required inverses exist, and\n\\[\nC_\\eta F(A)-F(B)C_\\eta\n=-(I+B)^{-1}\\Delta(I+A)^{-1}\n=\\mat{0&0&-7\\eta/30","0&1&0}.\n\\]\nThe generator defect is $\\Delta=C_\\eta A-BC_\\eta$, with only one nonzero entry, $\\Delta_{13}=7\\eta$ (matrix indices begin at one).\n\\vfig{ext_substitution}\n\\platecaption{The perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at $\\eta=0$. A nonzero defect is transformed, not erased, by the rational calculus.}\n\\head{A resolvent identity supplies the sign and both inverse factors}\nFor $F(z)=1/(1+z)$, all required inverses exist, and\n\\[\nC_\\eta F(A)-F(B)C_\\eta\n=-(I+B)^{-1}\\Delta(I+A)^{-1}\n=\\mat{0&0&-7\\eta/30","s $\\Delta=C_\\eta A-BC_\\eta$, with only one nonzero entry, $\\Delta_{13}=7\\eta$ (matrix indices begin at one).\n\\vfig{ext_substitution}\n\\platecaption{The perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at $\\eta=0$. A nonzero defect is transformed, not erased, by the rational calculus.}\n\\head{A resolvent identity supplies the sign and both inverse factors}\nFor $F(z)=1/(1+z)$, all required inverses exist, and\n\\[\nC_\\eta F(A)-F(B)C_\\eta\n=-(I+B)^{-1}\\Delta(I+A)^{-1}\n=\\mat{0&0&-7\\eta/30","entry, $\\Delta_{13}=7\\eta$ (matrix indices begin at one).\n\\vfig{ext_substitution}\n\\platecaption{The perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at $\\eta=0$. A nonzero defect is transformed, not erased, by the rational calculus.}\n\\head{A resolvent identity supplies the sign and both inverse factors}\nFor $F(z)=1/(1+z)$, all required inverses exist, and\n\\[\nC_\\eta F(A)-F(B)C_\\eta\n=-(I+B)^{-1}\\Delta(I+A)^{-1}\n=\\mat{0&0&-7\\eta/30\\\\0&0&0}.\n\\]\nThis follows by multiplying the difference by $I+B$ on the left and $I+A$ on the right. The middle numerator is $(I+B)C_\\eta-C_\\eta(I+A)=-\\Delta$.","0&0&0}.\n\\]\nThis follows by multiplying the difference by $I+B$ on the left and $I+A$ on the right. The middle numerator is $(I+B)C_\\eta-C_\\eta(I+A)=-\\Delta$.","The retained Jordan block still contributes $F'(2)=-1/9$, as on Plate VI.25. The new third-column discrepancy compares $F(9)=1/10$ with $F(2)=1/3$, multiplied by $\\eta$.","}{Chapter VII: Section II.1, Section II.7, Section II.9.}\nA quantum calculation becomes easier to follow when its objects remain visible. This appendix follows a state vector, a density matrix, and a retained part of a dynamical system. They may describe the same experiment, but they do not answer the same question.\n\\vfig{three_objects}\n\\head{Amplitudes retain relative phase}\nFor the Bell vector $\\psi=(\\ket{00}+\\ket{11})/\\sqrt2$, the two nonzero entries are amplitudes. The density matrix $\\rho=\\psi\\psi^*$ also records coherence in its off-diagonal entries. Measurements that compare the two addresses can detect this coherence. Deleting those entries applies a physical channel and changes the state.\n\\head{How hidden dynamics returns}\nWhen a Hamiltonian couples a retained state to a hidden state, the retained amplitude can leave and later return. Its exact equation then remembers the hidden coupling. A smaller matrix obtained by deleting rows and columns does not generally reproduce that equation.\n\\head{A route through the paper}\n\\pagerefV{3}--\\pagerefV{10} develop seeds, symmetry, and …","to follow when its objects remain visible. This appendix follows a state vector, a density matrix, and a retained part of a dynamical system. They may describe the same experiment, but they do not answer the same question.\n\\vfig{three_objects}\n\\head{Amplitudes retain relative phase}\nFor the Bell vector $\\psi=(\\ket{00}+\\ket{11})/\\sqrt2$, the two nonzero entries are amplitudes. The density matrix $\\rho=\\psi\\psi^*$ also records coherence in its off-diagonal entries. Measurements that compare the two addresses can detect this coherence. Deleting those entries applies a physical channel and changes the state.\n\\head{How hidden dynamics returns}\nWhen a Hamiltonian couples a retained state to a hidden state, the retained amplitude can leave and later return. Its exact equation then remembers the hidden coupling. A smaller matrix obtained by deleting rows and columns does not generally reproduce that equation.\n\\head{A route through the paper}\n\\pagerefV{3}--\\pagerefV{10} develop seeds, symmetry, and coherent circuit paths. \\pagerefV{11}--\\pagerefV{15} track leakage and moving subspaces. \\pager…","\\vpage{2}{The objects on the page}{AN ATLAS OF TYPES AND CONVENTIONS}{Chapter VII: Definition II.2.1, Definition II.7.1.}\nThe same letter can stand for different kinds of data in different calculations. Keep the following types fixed while reading a formula. In this appendix, vectors are columns, operators compose from right to left, and $A^*$ denotes the conjugate transpose.\n\\par\\vspace{4pt}\n\\begin{tabularx}{\\linewidth}{@{}p{.13\\linewidth}YY@{}}\n\\toprule\nSymbol & Object & What it records","midrule\n$\\HH,\\psi$ & Hilbert space and state vector & They specify the space of addresses and its coherent amplitudes.","$\\AA$ & Declared operator algebra & It specifies the available linear operations.","$P_a$ & Orthogonal packet projector & It selects an entire subspace.","$\\rho$ & Positive matrix of trace one & It records populations and coherences.","$E_k$ & Positive measurement effect & It gives the outcome probability $\\Tr(E_k\\rho)$.","$K_a$ & Kraus operator & It describes one channel branch and need not be a projector.","$H$ & Self-adjoint Hamiltonian & It generates evolution in real time through $e^{-\\ii tH}$.","$C,\\nabla$ & Packet transport generator and connection & They describe moving subspaces and differentiation within their ranges.","$P,Q$ & Retained and hidden projectors & They define a decomposition with $Q=I-P$.","$\\Sigma(z)$ & Self-energy & It records the spectral influence of the hidden sector.","$L,T(\\tau)$ & Nonnegative operator and Euclidean transfer & They determine decay factors and finite heat trace diagnostics.","bottomrule\n\\end{tabularx}\n\\head{Normalization and indexing}\nThe tensor factors are ordered from left to right, so $X_1=X\\otimes I\\otimes I$. A normalized pure state gives $\\rho=\\psi\\psi^*$. A coefficient matrix used to count a seed orbit need not be normalized. Scaling a nonzero seed does not change its generated linear span.","A Hamiltonian and a Hadamard gate are both traditionally denoted $H$. Where both appear, the gate is $H_{\\rm Had}$ and the Hamiltonian keeps the letter $H$. Numerical labels on a drawing are coordinates or exact example values, not observations of a physical device.\n\\takeaway{Check the type before interpreting the picture. An idempotent projector, a density matrix, and a channel matrix have different defining equations.}","\\vpage{3}{One seed, eight addresses}{CYCLICITY DEPENDS ON THE GRAMMAR}{Chapter VII: Proposition II.2.5.}\nStart with $\\psi_0=\\ket{000}$ and allow the three bit flip operators $X_1,X_2,X_3$. For a binary address $a=(a_1,a_2,a_3)$, put\n\\[\nW_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}.\n\\qquad W_a\\psi_0=\\ket{a_1a_2a_3}.\n\\]\nThe drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits.\n\\vfig{addresses}\n\\head{Two different meanings of ``reach''}\nThe eight vectors are orthonormal, so their \\emph{linear span} is all of $\\CC^8$. This proves cyclicity for the algebra generated by the flips. Preparing every normalized vector requires more than products of these flips alone.","2,a_3)$, put\n\\[\nW_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}.\n\\qquad W_a\\psi_0=\\ket{a_1a_2a_3}.\n\\]\nThe drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits.\n\\vfig{addresses}\n\\head{Two different meanings of ``reach''}\nThe eight vectors are orthonormal, so their \\emph{linear span} is all of $\\CC^8$. This proves cyclicity for the algebra generated by the flips. Preparing every normalized vector requires more than products of these flips alone.","For example, no such word sends $\\ket{000}$ to $(\\ket{000}+\\ket{001})/\\sqrt2$. A Hadamard gate, a suitable linear combination in the observable algebra, or a larger preparation grammar is required for that operation. Counting linearly independent addressed vectors and synthesizing a physical gate sequence are separate tasks.","The exact check constructs each $8\\times8$ word matrix, applies it to the seed, and verifies that the matrix of all eight outputs is the identity. It also tests the displayed superposition against every word output. No search over an unspecified gate set is involved.\n\\takeaway{A cyclic seed generates a vector space under an algebra. That conclusion does not automatically provide a unitary preparation procedure for every vector in it.}","\\vpage{4}{Entanglement changes the local span}{SCHMIDT RANK AS ACCESSIBLE MULTIPLICITY}{Chapter VII: Theorem II.2.2, Corollary II.2.4.}\nWrite a bipartite vector as a coefficient matrix $C$, with the first subsystem indexing rows. Acting only on that subsystem transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It","\\vpage{4}{Entanglement changes the local span}{SCHMIDT RANK AS ACCESSIBLE MULTIPLICITY}{Chapter VII: Theorem II.2.2, Corollary II.2.4.}\nWrite a bipartite vector as a coefficient matrix $C$, with the first subsystem indexing rows. Acting only on that subsystem transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve S","\\vpage{4}{Entanglement changes the local span}{SCHMIDT RANK AS ACCESSIBLE MULTIPLICITY}{Chapter VII: Theorem II.2.2, Corollary II.2.4.}\nWrite a bipartite vector as a coefficient matrix $C$, with the first subsystem indexing rows. Acting only on that subsystem transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve Schmidt c","\\vpage{4}{Entanglement changes the local span}{SCHMIDT RANK AS ACCESSIBLE MULTIPLICITY}{Chapter VII: Theorem II.2.2, Corollary II.2.4.}\nWrite a bipartite vector as a coefficient matrix $C$, with the first subsystem indexing rows. Acting only on that subsystem transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve Schmidt coefficients. They therefore cannot turn the Bell vect","Entanglement changes the local span}{SCHMIDT RANK AS ACCESSIBLE MULTIPLICITY}{Chapter VII: Theorem II.2.2, Corollary II.2.4.}\nWrite a bipartite vector as a coefficient matrix $C$, with the first subsystem indexing rows. Acting only on that subsystem transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve Schmidt coefficients. They therefore cannot turn the Bell vector into an arbitrary product vector by a deterministic transformation betwee…",".2.2, Corollary II.2.4.}\nWrite a bipartite vector as a coefficient matrix $C$, with the first subsystem indexing rows. Acting only on that subsystem transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve Schmidt coefficients. They therefore cannot turn the Bell vector into an arbitrary product vector by a deterministic transformation between pure states.",".}\nWrite a bipartite vector as a coefficient matrix $C$, with the first subsystem indexing rows. Acting only on that subsystem transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve Schmidt coefficients. They therefore cannot turn the Bell vector into an arbitrary product vector by a deterministic transformation between pure states.","m transforms $C$ to $AC$. Every resulting row remains in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve Schmidt coefficients. They therefore cannot turn the Bell vector into an arbitrary product vector by a deterministic transformation between pure states.","in the row space of $C$, and every independent choice of such rows can be obtained in this way.\n\\[\n\\dim\\Span\\{(A\\otimes I)\\psi:A\\in M_d(\\CC)\\}\n=d\\,\\rank C.\n\\]\n\\vfig{local_orbits}\nThe product vector $\\ket{00}$ has matrix $C=\\diag(1,0)$. Its local orbit span allows arbitrary vectors in the first subsystem but only the second subsystem direction $\\ket0$, so its dimension is two. The Bell vector has $C=I/\\sqrt2$, rank two, and a four-dimensional local orbit span.\n\\head{What the formula does not say}\nThe action here allows arbitrary matrices and then takes a linear span. It is not the orbit under local unitary gates alone. Local unitary operations preserve Schmidt coefficients. They therefore cannot turn the Bell vector into an arbitrary product vector by a deterministic transformation between pure states.","For the exact calculation, apply the four matrix units $E_{ab}$ to each seed matrix, flatten the outputs, and compute their column rank. The observed dimensions are two and four, respectively. The calculation directly exhibits the independent rows counted by the theorem.\n\\takeaway{Schmidt rank measures the remote multiplicity directions accessible to the local operator span. It is not permission for unrestricted deterministic local state preparation.}","\\vpage{5}{Why one seed is not always enough}{AN EXACT MULTIPLICITY LOWER BOUND}{Chapter VII: Theorem II.2.3.}\nConsider $M_2(\\CC)\\otimes I_3$ acting on $\\CC^2\\otimes\\CC^3$. A seed is a $2\\times3$ coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra.\n\\[\nC_1=\\mat{1&0&0","\\vpage{5}{Why one seed is not always enough}{AN EXACT MULTIPLICITY LOWER BOUND}{Chapter VII: Theorem II.2.3.}\nConsider $M_2(\\CC)\\otimes I_3$ acting on $\\CC^2\\otimes\\CC^3$. A seed is a $2\\times3$ coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra.\n\\[\nC_1=\\mat{1&0&0\\\\0&1&0},\\qquad\nC_2=\\mat{0&0&1\\\\0&0&0}.\n\\]\nThese two matrices provide a complete set of row directions. Their individual orbit dimensions are four and two, and together those spans fill the six-dimensional state space.\n\\vfig{seed_rank}\n\\head{From the example to the finite formula}\nFor a block $M_d(\\CC)\\otimes I_m$, $k$ seed matrices contribute at most $kd$ independent multiplicity directions. Hence $k\\ge\\lceil m/d\\rceil$. Partitioning a multiplicity basis into groups of size at most $d$ attains this bound. For a direct sum of such blocks, the worst block determines the required count:\n\\[\n\\operatorname{srank}_{\\AA}(\\HH)=\\max_i\\lceil m_i/d_i\\rceil.\n\\]\nThis is an invariant of the declared representation of the algebra, an…","0&0&0}.\n\\]\nThese two matrices provide a complete set of row directions. Their individual orbit dimensions are four and two, and together those spans fill the six-dimensional state space.\n\\vfig{seed_rank}\n\\head{From the example to the finite formula}\nFor a block $M_d(\\CC)\\otimes I_m$, $k$ seed matrices contribute at most $kd$ independent multiplicity directions. Hence $k\\ge\\lceil m/d\\rceil$. Partitioning a multiplicity basis into groups of size at most $d$ attains this bound. For a direct sum of such blocks, the worst block determines the required count:\n\\[\n\\operatorname{srank}_{\\AA}(\\HH)=\\max_i\\lceil m_i/d_i\\rceil.\n\\]\nThis is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full matrix algebra on $\\CC^6$, every nonzero seed would instead be cyclic.","ow directions. Their individual orbit dimensions are four and two, and together those spans fill the six-dimensional state space.\n\\vfig{seed_rank}\n\\head{From the example to the finite formula}\nFor a block $M_d(\\CC)\\otimes I_m$, $k$ seed matrices contribute at most $kd$ independent multiplicity directions. Hence $k\\ge\\lceil m/d\\rceil$. Partitioning a multiplicity basis into groups of size at most $d$ attains this bound. For a direct sum of such blocks, the worst block determines the required count:\n\\[\n\\operatorname{srank}_{\\AA}(\\HH)=\\max_i\\lceil m_i/d_i\\rceil.\n\\]\nThis is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full matrix algebra on $\\CC^6$, every nonzero seed would instead be cyclic.","dimensional state space.\n\\vfig{seed_rank}\n\\head{From the example to the finite formula}\nFor a block $M_d(\\CC)\\otimes I_m$, $k$ seed matrices contribute at most $kd$ independent multiplicity directions. Hence $k\\ge\\lceil m/d\\rceil$. Partitioning a multiplicity basis into groups of size at most $d$ attains this bound. For a direct sum of such blocks, the worst block determines the required count:\n\\[\n\\operatorname{srank}_{\\AA}(\\HH)=\\max_i\\lceil m_i/d_i\\rceil.\n\\]\nThis is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full matrix algebra on $\\CC^6$, every nonzero seed would instead be cyclic.","o groups of size at most $d$ attains this bound. For a direct sum of such blocks, the worst block determines the required count:\n\\[\n\\operatorname{srank}_{\\AA}(\\HH)=\\max_i\\lceil m_i/d_i\\rceil.\n\\]\nThis is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full matrix algebra on $\\CC^6$, every nonzero seed would instead be cyclic.","\\vpage{6}{Four phases resolve a cyclic shift}{CHARACTER PROJECTORS YOU CAN MULTIPLY}{Chapter VII: Corollary II.3.3.}\nOn four sites, let $S\\ket k=\\ket{k+1\\bmod4}$ and let $\\omega=\\ii$. The character projectors are the exact matrices\n\\[\nP_j=\\frac14\\sum_{r=0}^3\\ii^{-jr}S^r,\n\\qquad v_j(k)=\\frac12\\ii^{-jk},\\qquad P_j=v_jv_j^*.\n\\]\nAll entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model.\n\\vfig{cyclic_four}\n\\head{Reading the columns}\nEach column $v_j$ is a normalized wave around the four sites. Advancing the address multiplies it by $\\ii^j$. The entries show phase differences rather than four different probability distributions: every column has uniform site probability $1/4$.","Corollary II.3.3.}\nOn four sites, let $S\\ket k=\\ket{k+1\\bmod4}$ and let $\\omega=\\ii$. The character projectors are the exact matrices\n\\[\nP_j=\\frac14\\sum_{r=0}^3\\ii^{-jr}S^r,\n\\qquad v_j(k)=\\frac12\\ii^{-jk},\\qquad P_j=v_jv_j^*.\n\\]\nAll entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model.\n\\vfig{cyclic_four}\n\\head{Reading the columns}\nEach column $v_j$ is a normalized wave around the four sites. Advancing the address multiplies it by $\\ii^j$. The entries show phase differences rather than four different probability distributions: every column has uniform site probability $1/4$.","character projectors are the exact matrices\n\\[\nP_j=\\frac14\\sum_{r=0}^3\\ii^{-jr}S^r,\n\\qquad v_j(k)=\\frac12\\ii^{-jk},\\qquad P_j=v_jv_j^*.\n\\]\nAll entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model.\n\\vfig{cyclic_four}\n\\head{Reading the columns}\nEach column $v_j$ is a normalized wave around the four sites. Advancing the address multiplies it by $\\ii^j$. The entries show phase differences rather than four different probability distributions: every column has uniform site probability $1/4$.","All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model.\n\\vfig{cyclic_four}\n\\head{Reading the columns}\nEach column $v_j$ is a normalized wave around the four sites. Advancing the address multiplies it by $\\ii^j$. The entries show phase differences rather than four different probability distributions: every column has uniform site probability $1/4$.","Multiplying the actual projectors gives $P_jP_k=0$ for $j\\ne k$, $P_j^2=P_j$, and $\\sum_jP_j=I$. In particular, the complete packet tuple $(P_j\\psi)_j$ retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.","The same shift on four sites returns on \\pagerefV{27}, where it diagonalizes the Hamiltonian $L=2I-S-S^*$, and its four character phases explain perfect transfer between opposite sites. The same fixed projectors serve as coordinates, symmetry sectors, and a complete interference ledger.\n\\takeaway{A character projector retains a complex subspace component. Its norm summarizes the size of that component while leaving its phase and direction unspecified.}","\\vpage{7}{A coarse packet can still contain coherence}{TENSOR REFINEMENT BY TOTAL PARITY}{Chapter VII: Theorem II.3.7, Corollary II.3.8.}\nFor three qubits, the eight fine projectors are $P_a=\\ket a\\bra a$. Grouping addresses according to the parity of their Hamming weight gives\n\\[\nP_{\\rm even}=P_{000}+P_{011}+P_{101}+P_{110}\n=\\frac{I+Z\\otimes Z\\otimes Z}{2},\n\\qquad P_{\\rm odd}=I-P_{\\rm even}.\n\\]\nThis is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children.\n\\vfig{parity}\n\\head{Refinement is not measurement}\nThe fine packet description and the coarse packet description each reconstruct the full vector when every component is retained. Replacing a state by the mixture $\\sum_aP_a\\rho P_a$ is instead a dephasing channel. That operation removes coherence between the fine packets and can differ from dephasing only","\\vpage{7}{A coarse packet can still contain coherence}{TENSOR REFINEMENT BY TOTAL PARITY}{Chapter VII: Theorem II.3.7, Corollary II.3.8.}\nFor three qubits, the eight fine projectors are $P_a=\\ket a\\bra a$. Grouping addresses according to the parity of their Hamming weight gives\n\\[\nP_{\\rm even}=P_{000}+P_{011}+P_{101}+P_{110}\n=\\frac{I+Z\\otimes Z\\otimes Z}{2},\n\\qquad P_{\\rm odd}=I-P_{\\rm even}.\n\\]\nThis is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children.\n\\vfig{parity}\n\\head{Refinement is not measurement}\nThe fine packet description and the coarse packet description each reconstruct the full vector when every component is retained. Replacing a state by the mixture $\\sum_aP_a\\rho P_a$ is instead a dephasing channel. That operation removes coherence between the fine packets and can differ from dephasing only between the even and odd sectors.",",\n\\qquad P_{\\rm odd}=I-P_{\\rm even}.\n\\]\nThis is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children.\n\\vfig{parity}\n\\head{Refinement is not measurement}\nThe fine packet description and the coarse packet description each reconstruct the full vector when every component is retained. Replacing a state by the mixture $\\sum_aP_a\\rho P_a$ is instead a dephasing channel. That operation removes coherence between the fine packets and can differ from dephasing only between the even and odd sectors.","For instance, a superposition of $\\ket{000}$ and $\\ket{011}$ lies entirely in the even packet. Its internal relative phase survives a parity measurement but is lost by a complete measurement in the computational basis if outcomes are forgotten. The two measurements use projectors from the same refinement hierarchy but expose different information.","The exact check verifies all eight addresses, their parity grouping, the rank of both coarse projectors, and equality with the tensor expression above. The general residue version replaces parity by total label modulo $N$ and groups the tensor descendants by their summed residue.\n\\takeaway{A coarse packet is a whole subspace. Internal coherence survives a change of address description. It is lost only when an operation discards it.}","\\vpage{8}{Track the Bell state through the circuit}{INTERMEDIATE PACKETS ARE NOT FINAL PACKETS}{Chapter VII: Theorem II.4.1, Example II.4.6.}\nThe input is $\\ket{00}$. First apply a Hadamard to the first qubit, then a controlled flip:\n\\[\n\\ket{00}\\ \\xmapsto{H_{\\rm Had}\\otimes I}\\\n\\frac{\\ket{00}+\\ket{10}}{\\sqrt2}\n\\ \\xmapsto{\\rm CNOT}\\\n\\frac{\\ket{00}+\\ket{11}}{\\sqrt2}.\n\\]\nThe displayed columns keep amplitudes, including their normalization, at every step. The middle vector occupies both parity packets. The final vector occupies only the even packet.\n\\vfig{bell_circuit}\n\\head{How the exact path expansion works}\nInsert $P_{\\rm even}+P_{\\rm odd}=I$ between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original $4\\times4$ circuit matrix:\n\\[","\\vpage{8}{Track the Bell state through the circuit}{INTERMEDIATE PACKETS ARE NOT FINAL PACKETS}{Chapter VII: Theorem II.4.1, Example II.4.6.}\nThe input is $\\ket{00}$. First apply a Hadamard to the first qubit, then a controlled flip:\n\\[\n\\ket{00}\\ \\xmapsto{H_{\\rm Had}\\otimes I}\\\n\\frac{\\ket{00}+\\ket{10}}{\\sqrt2}\n\\ \\xmapsto{\\rm CNOT}\\\n\\frac{\\ket{00}+\\ket{11}}{\\sqrt2}.\n\\]\nThe displayed columns keep amplitudes, including their normalization, at every step. The middle vector occupies both parity packets. The final vector occupies only the even packet.\n\\vfig{bell_circuit}\n\\head{How the exact path expansion works}\nInsert $P_{\\rm even}+P_{\\rm odd}=I$ between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original $4\\times4$ circuit matrix:\n\\[\n\\sum_{a,b,c}P_b\\,\\mathrm{CNOT}\\,P_a\n(H_{\\rm Had}\\otimes I)P_c\n=\\mathrm{CNOT}(H_{\\rm Had}\\otimes I).\n\\]\nThis expansion is an identity even though the intermediate parity is not measured. Physically insert","olled flip:\n\\[\n\\ket{00}\\ \\xmapsto{H_{\\rm Had}\\otimes I}\\\n\\frac{\\ket{00}+\\ket{10}}{\\sqrt2}\n\\ \\xmapsto{\\rm CNOT}\\\n\\frac{\\ket{00}+\\ket{11}}{\\sqrt2}.\n\\]\nThe displayed columns keep amplitudes, including their normalization, at every step. The middle vector occupies both parity packets. The final vector occupies only the even packet.\n\\vfig{bell_circuit}\n\\head{How the exact path expansion works}\nInsert $P_{\\rm even}+P_{\\rm odd}=I$ between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original $4\\times4$ circuit matrix:\n\\[\n\\sum_{a,b,c}P_b\\,\\mathrm{CNOT}\\,P_a\n(H_{\\rm Had}\\otimes I)P_c\n=\\mathrm{CNOT}(H_{\\rm Had}\\otimes I).\n\\]\nThis expansion is an identity even though the intermediate parity is not measured. Physically inserting a measurement would alter the problem by removing interference between its alternatives. The derivation resolves intermediate coordinates without physically measuring them.","ion, at every step. The middle vector occupies both parity packets. The final vector occupies only the even packet.\n\\vfig{bell_circuit}\n\\head{How the exact path expansion works}\nInsert $P_{\\rm even}+P_{\\rm odd}=I$ between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original $4\\times4$ circuit matrix:\n\\[\n\\sum_{a,b,c}P_b\\,\\mathrm{CNOT}\\,P_a\n(H_{\\rm Had}\\otimes I)P_c\n=\\mathrm{CNOT}(H_{\\rm Had}\\otimes I).\n\\]\nThis expansion is an identity even though the intermediate parity is not measured. Physically inserting a measurement would alter the problem by removing interference between its alternatives. The derivation resolves intermediate coordinates without physically measuring them.","le vector occupies both parity packets. The final vector occupies only the even packet.\n\\vfig{bell_circuit}\n\\head{How the exact path expansion works}\nInsert $P_{\\rm even}+P_{\\rm odd}=I$ between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original $4\\times4$ circuit matrix:\n\\[\n\\sum_{a,b,c}P_b\\,\\mathrm{CNOT}\\,P_a\n(H_{\\rm Had}\\otimes I)P_c\n=\\mathrm{CNOT}(H_{\\rm Had}\\otimes I).\n\\]\nThis expansion is an identity even though the intermediate parity is not measured. Physically inserting a measurement would alter the problem by removing interference between its alternatives. The derivation resolves intermediate coordinates without physically measuring them.","\\vpage{9}{Why path probabilities do not add}{HOW TWO PATH AMPLITUDES CANCEL}{Chapter VII: Theorem II.4.2, Theorem II.4.4.}\nTwo consecutive Hadamard gates multiply to the identity. Starting from $\\ket0$, there are two resolved paths to output $\\ket1$. Their signed products are\n\\[\n\\bra1H_{\\rm Had}\\ket0\\bra0H_{\\rm Had}\\ket0=\\frac12,\n\\qquad\n\\bra1H_{\\rm Had}\\ket1\\bra1H_{\\rm Had}\\ket0=-\\frac12.\n\\]\nTheir sum is zero. Squaring each separately gives a different answer.\n\\vfig{interference}\n\\head{Where the missing term goes}\nFor path amplitudes $a,b$, the outcome probability is $|a+b|^2=|a|^2+|b|^2+2\\Re(a\\bar b)$. Here the last term is $-1/2$, canceling the first two terms. The path picture retains this information only if its edges carry complex amplitudes rather than unsigned reachability or probabilities.","damard gates multiply to the identity. Starting from $\\ket0$, there are two resolved paths to output $\\ket1$. Their signed products are\n\\[\n\\bra1H_{\\rm Had}\\ket0\\bra0H_{\\rm Had}\\ket0=\\frac12,\n\\qquad\n\\bra1H_{\\rm Had}\\ket1\\bra1H_{\\rm Had}\\ket0=-\\frac12.\n\\]\nTheir sum is zero. Squaring each separately gives a different answer.\n\\vfig{interference}\n\\head{Where the missing term goes}\nFor path amplitudes $a,b$, the outcome probability is $|a+b|^2=|a|^2+|b|^2+2\\Re(a\\bar b)$. Here the last term is $-1/2$, canceling the first two terms. The path picture retains this information only if its edges carry complex amplitudes rather than unsigned reachability or probabilities.","ity. Starting from $\\ket0$, there are two resolved paths to output $\\ket1$. Their signed products are\n\\[\n\\bra1H_{\\rm Had}\\ket0\\bra0H_{\\rm Had}\\ket0=\\frac12,\n\\qquad\n\\bra1H_{\\rm Had}\\ket1\\bra1H_{\\rm Had}\\ket0=-\\frac12.\n\\]\nTheir sum is zero. Squaring each separately gives a different answer.\n\\vfig{interference}\n\\head{Where the missing term goes}\nFor path amplitudes $a,b$, the outcome probability is $|a+b|^2=|a|^2+|b|^2+2\\Re(a\\bar b)$. Here the last term is $-1/2$, canceling the first two terms. The path picture retains this information only if its edges carry complex amplitudes rather than unsigned reachability or probabilities.","output $\\ket1$. Their signed products are\n\\[\n\\bra1H_{\\rm Had}\\ket0\\bra0H_{\\rm Had}\\ket0=\\frac12,\n\\qquad\n\\bra1H_{\\rm Had}\\ket1\\bra1H_{\\rm Had}\\ket0=-\\frac12.\n\\]\nTheir sum is zero. Squaring each separately gives a different answer.\n\\vfig{interference}\n\\head{Where the missing term goes}\nFor path amplitudes $a,b$, the outcome probability is $|a+b|^2=|a|^2+|b|^2+2\\Re(a\\bar b)$. Here the last term is $-1/2$, canceling the first two terms. The path picture retains this information only if its edges carry complex amplitudes rather than unsigned reachability or probabilities.","The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations:\n\\[\n\\sum_b U_{ba}^*U_{bc}=\\delta_{ac}P_a,\n\\qquad U_{ba}=P_bUP_a.\n\\]\nIt cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of $a,c$ for the two rank-one packets, as well as the two path products leading to the opposite output.","A measurement at the intermediate stage would remove the cross term. In that modified experiment the two alternatives really would contribute total probability $1/2$ to output one. Adding the two path probabilities would therefore describe the experiment with an intermediate measurement.\n\\takeaway{Insert projectors algebraically to resolve paths, then add amplitudes. Add probabilities only when the physical setup has actually made the alternatives incoherent.}","\\vpage{10}{A symmetry reveals blocks, not approximations}{ONE HAMILTONIAN IN TWO ORDERS}{Chapter VII: Theorem II.5.1.}\nTake the exact rational Hamiltonian\n\\[\nH=X\\otimes X+\\frac38(Z\\otimes I+I\\otimes Z).\n\\]\nIt commutes with total parity $Z\\otimes Z$. Reordering the computational basis into even addresses $(00,11)$ and odd addresses $(01,10)$ displays the invariant subspaces explicitly.\n\\vfig{hamiltonian}\n\\[\nH_{\\rm even}=\\mat{3/4&1","}{A symmetry reveals blocks, not approximations}{ONE HAMILTONIAN IN TWO ORDERS}{Chapter VII: Theorem II.5.1.}\nTake the exact rational Hamiltonian\n\\[\nH=X\\otimes X+\\frac38(Z\\otimes I+I\\otimes Z).\n\\]\nIt commutes with total parity $Z\\otimes Z$. Reordering the computational basis into even addresses $(00,11)$ and odd addresses $(01,10)$ displays the invariant subspaces explicitly.\n\\vfig{hamiltonian}\n\\[\nH_{\\rm even}=\\mat{3/4&1\\\\1&-3/4},\\qquad\nH_{\\rm odd}=\\mat{0&1\\\\1&0}.\n\\]\nThe even block squares to $(25/16)I$, so its eigenvalues are $\\pm5/4$. The odd block squares to $I$, so its eigenvalues are $\\pm1$. Exponentiating either block gives the corresponding part of the exact evolution. Transfer between the packets remains zero for every real time.\n\\head{What has changed?}\nOnly the order of basis vectors has changed. No matrix entries, amplitudes, or energy levels were discarded. This is why the decomposition is exact. Keeping just the even block would be a separate restriction of the state space, justified for initial data known to be even and for dynamics that preserve it.","1&0}.\n\\]\nThe even block squares to $(25/16)I$, so its eigenvalues are $\\pm5/4$. The odd block squares to $I$, so its eigenvalues are $\\pm1$. Exponentiating either block gives the corresponding part of the exact evolution. Transfer between the packets remains zero for every real time.\n\\head{What has changed?}\nOnly the order of basis vectors has changed. No matrix entries, amplitudes, or energy levels were discarded. This is why the decomposition is exact. Keeping just the even block would be a separate restriction of the state space, justified for initial data known to be even and for dynamics that preserve it.","\\vpage{11}{Quantify the amplitude that escapes}{COMMUTATORS AND PROBABILITY BOUNDS}{Chapter VII: Theorem II.5.3, Corollary II.5.4.}\nFor a two-level system, choose $H=\\varepsilon X$, $P=\\ket0\\bra0$, and $Q=I-P$. Direct exponentiation gives\n\\[\ne^{-\\ii tH}=\\cos(\\varepsilon t)I-\\ii\\sin(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape prob","\\vpage{11}{Quantify the amplitude that escapes}{COMMUTATORS AND PROBABILITY BOUNDS}{Chapter VII: Theorem II.5.3, Corollary II.5.4.}\nFor a two-level system, choose $H=\\varepsilon X$, $P=\\ket0\\bra0$, and $Q=I-P$. Direct exponentiation gives\n\\[\ne^{-\\ii tH}=\\cos(\\varepsilon t)I-\\ii\\sin(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are se","\\vpage{11}{Quantify the amplitude that escapes}{COMMUTATORS AND PROBABILITY BOUNDS}{Chapter VII: Theorem II.5.3, Corollary II.5.4.}\nFor a two-level system, choose $H=\\varepsilon X$, $P=\\ket0\\bra0$, and $Q=I-P$. Direct exponentiation gives\n\\[\ne^{-\\ii tH}=\\cos(\\varepsilon t)I-\\ii\\sin(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. M","\\vpage{11}{Quantify the amplitude that escapes}{COMMUTATORS AND PROBABILITY BOUNDS}{Chapter VII: Theorem II.5.3, Corollary II.5.4.}\nFor a two-level system, choose $H=\\varepsilon X$, $P=\\ket0\\bra0$, and $Q=I-P$. Direct exponentiation gives\n\\[\ne^{-\\ii tH}=\\cos(\\varepsilon t)I-\\ii\\sin(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","COMMUTATORS AND PROBABILITY BOUNDS}{Chapter VII: Theorem II.5.3, Corollary II.5.4.}\nFor a two-level system, choose $H=\\varepsilon X$, $P=\\ket0\\bra0$, and $Q=I-P$. Direct exponentiation gives\n\\[\ne^{-\\ii tH}=\\cos(\\varepsilon t)I-\\ii\\sin(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","irect exponentiation gives\n\\[\ne^{-\\ii tH}=\\cos(\\varepsilon t)I-\\ii\\sin(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","e^{-\\ii tH}=\\cos(\\varepsilon t)I-\\ii\\sin(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","in(\\varepsilon t)X,\n\\quad\n\\norm{Qe^{-\\ii tH}P}=|\\sin(\\varepsilon t)|,\n\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","\\quad\\norm{[H,P]}=|\\varepsilon|.\n\\]\nThe general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","general commutator estimate becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","becomes the elementary bound $|\\sin u|\\le |u|$. Unitarity also bounds the amplitude by one.\n\\vfig{leakage}\n\\head{From a norm to a probability}\nFor this initial state, the escape probability is $\\sin^2(\\varepsilon t)$, not $|\\sin(\\varepsilon t)|$. Consequently\n\\[\n\\Pr(\\text{hidden packet at }t)\\le\\min\\{1,\\varepsilon^2t^2\\}.\n\\]\nAt $\\varepsilon t=\\pi/6$, the amplitude is $1/2$ and the probability is $1/4$. Near zero, amplitudes are first order while escape probabilities are second order. Mixing those two quantities produces an incorrect interpretation of a leakage plot.","\\vpage{12}{A moving frame creates a connection term}{NOT ALL MOTION IS PHYSICAL EVOLUTION}{Chapter VII: Theorem II.6.3.}\nSet the laboratory Hamiltonian to zero and keep $\\psi(t)=\\ket0$ fixed. Describe it in rotating coordinates,\n\\[\nU(t)=\\mat{\\cos t&-\\sin t","\\vpage{12}{A moving frame creates a connection term}{NOT ALL MOTION IS PHYSICAL EVOLUTION}{Chapter VII: Theorem II.6.3.}\nSet the laboratory Hamiltonian to zero and keep $\\psi(t)=\\ket0$ fixed. Describe it in rotating coordinates,\n\\[\nU(t)=\\mat{\\cos t&-\\sin t\\\\\\sin t&\\cos t},\n\\qquad y(t)=U(t)^*\\psi(t)=\\mat{\\cos t\\\\-\\sin t}.\n\\]\nThe vector has not moved in the laboratory, but its coordinate entries have changed.\n\\vfig{moving_frame}\n\\head{The missing term can be computed exactly}\nLet $J=\\mat{0&-1","-\\sin t}.\n\\]\nThe vector has not moved in the laboratory, but its coordinate entries have changed.\n\\vfig{moving_frame}\n\\head{The missing term can be computed exactly}\nLet $J=\\mat{0&-1\\\\1&0}$. Since $U^*\\dot U=J$, the coordinate Hamiltonian is\n\\[\nH_{\\rm frame}=-\\ii J=\\mat{0&\\ii","1&0}$. Since $U^*\\dot U=J$, the coordinate Hamiltonian is\n\\[\nH_{\\rm frame}=-\\ii J=\\mat{0&\\ii","1&0}$. Since $U^*\\dot U=J$, the coordinate Hamiltonian is\n\\[\nH_{\\rm frame}=-\\ii J=\\mat{0&\\ii\\\\-\\ii&0},\n\\qquad \\ii\\dot y=H_{\\rm frame}y.\n\\]\nThis matrix is Hermitian. Omitting it would predict a constant $y$, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.","-\\ii&0},\n\\qquad \\ii\\dot y=H_{\\rm frame}y.\n\\]\nThis matrix is Hermitian. Omitting it would predict a constant $y$, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.","For a nonzero laboratory Hamiltonian, the same product rule adds the transformed physical term $U^*HU$. Thus $U^*HU-\\ii U^*\\dot U$ separates the physical generator from the coordinate correction. Both pieces belong to the same equation but have different origins.","\\vpage{13}{Differentiate the subspace itself}{THE CANONICAL PACKET CONNECTION}{Chapter VII: Theorem II.6.1, Theorem II.6.2.}\nContinue with $P(t)=U(t)\\diag(1,0)U(t)^*$ and $Q(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[","\\vpage{13}{Differentiate the subspace itself}{THE CANONICAL PACKET CONNECTION}{Chapter VII: Theorem II.6.1, Theorem II.6.2.}\nContinue with $P(t)=U(t)\\diag(1,0)U(t)^*$ and $Q(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]","\\vpage{13}{Differentiate the subspace itself}{THE CANONICAL PACKET CONNECTION}{Chapter VII: Theorem II.6.1, Theorem II.6.2.}\nContinue with $P(t)=U(t)\\diag(1,0)U(t)^*$ and $Q(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection re","\\vpage{13}{Differentiate the subspace itself}{THE CANONICAL PACKET CONNECTION}{Chapter VII: Theorem II.6.1, Theorem II.6.2.}\nContinue with $P(t)=U(t)\\diag(1,0)U(t)^*$ and $Q(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No ba","}{Chapter VII: Theorem II.6.1, Theorem II.6.2.}\nContinue with $P(t)=U(t)\\diag(1,0)U(t)^*$ and $Q(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No basis within a packet is needed for this formula. A basis choice would add internal coordinate data without changing which subspaces are selected.",")\\diag(1,0)U(t)^*$ and $Q(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No basis within a packet is needed for this formula. A basis choice would add internal coordinate data without changing which subspaces are selected.","(t)^*$ and $Q(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No basis within a packet is needed for this formula. A basis choice would add internal coordinate data without changing which subspaces are selected.","(t)=I-P(t)$. Differentiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No basis within a packet is needed for this formula. A basis choice would add internal coordinate data without changing which subspaces are selected.","entiating $P^2=P$ gives\n\\[\n\\dot P P+P\\dot P=\\dot P,\n\\qquad P\\dot P P=0.\n\\]\nThe second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No basis within a packet is needed for this formula. A basis choice would add internal coordinate data without changing which subspaces are selected.","subspace relative to its complement.\n\\vfig{connection}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No basis within a packet is needed for this formula. A basis choice would add internal coordinate data without changing which subspaces are selected.","n}\nFor this complete pair of packets, define\n\\[\nC=\\dot P P+\\dot Q Q.\n\\]\nExact calculation yields the constant matrix $C=J$, with $C^*=-C$ and $[C,P]=\\dot P$. At $t=0$, the displayed small matrices allow these identities to be checked by direct multiplication.\n\\head{From two packets to a complete family}\nFor differentiable orthogonal projectors $P_a$ summing to the identity, the paper constructs\n\\[\nC=\\sum_a\\dot P_aP_a=\\frac12\\sum_a[\\dot P_a,P_a].\n\\]\nThis single connection reproduces the motion of every packet in the family. No basis within a packet is needed for this formula. A basis choice would add internal coordinate data without changing which subspaces are selected.","The completeness of the family matters in deriving skew-Hermiticity and the simultaneous transport identities. Using only one summand of an incomplete list is not automatically the same connection. The example retains both $P$ and $Q$ throughout, even when the picture emphasizes only the first line.\n\\takeaway{The projector derivative describes motion between subspaces. The canonical connection reconstructs that motion from the complete family without choosing internal packet bases.}","\\vpage{14}{A chart failure is not a singular fiber}{THE RANGE BUNDLE IS CONSTRUCTED LOCALLY}{Chapter VII: Theorem II.6.7. Details of the range atlas: Chapter VII, Part III, Sections III.24--III.26 (range-bundle construction).}\nThe fiber of a projector field is the actual fixed subspace $E_x=\\{v:P(x)v=v\\}$. To compare it with a nearby anchor fiber $E_a$, the completion proof constructs\n\\[\nT_a(x)=P(x)P(a)+(I-P(x))(I-P(a)).\n\\]\nIt equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates.\n\\vfig{range_atlas}\n\\head{An exact example of the domain restriction}\nFor the rotating line and anchor $P(0)$,\n\\[\nT_0(t)=\\cos t\\,U(t),\\qquad \\det T_0(t)=\\cos^2t.\n\\]\nThe chart fails at $t=\\pi/2$, but $P(\\pi/2)=\\diag(0,1)$ is a perfectly regular rank-one projector. Choosing a vertical anchor supplies a new chart there. What failed was the comparison with one reference fiber, not the fiber itself.","subspace $E_x=\\{v:P(x)v=v\\}$. To compare it with a nearby anchor fiber $E_a$, the completion proof constructs\n\\[\nT_a(x)=P(x)P(a)+(I-P(x))(I-P(a)).\n\\]\nIt equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates.\n\\vfig{range_atlas}\n\\head{An exact example of the domain restriction}\nFor the rotating line and anchor $P(0)$,\n\\[\nT_0(t)=\\cos t\\,U(t),\\qquad \\det T_0(t)=\\cos^2t.\n\\]\nThe chart fails at $t=\\pi/2$, but $P(\\pi/2)=\\diag(0,1)$ is a perfectly regular rank-one projector. Choosing a vertical anchor supplies a new chart there. What failed was the comparison with one reference fiber, not the fiber itself.","$E_x=\\{v:P(x)v=v\\}$. To compare it with a nearby anchor fiber $E_a$, the completion proof constructs\n\\[\nT_a(x)=P(x)P(a)+(I-P(x))(I-P(a)).\n\\]\nIt equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates.\n\\vfig{range_atlas}\n\\head{An exact example of the domain restriction}\nFor the rotating line and anchor $P(0)$,\n\\[\nT_0(t)=\\cos t\\,U(t),\\qquad \\det T_0(t)=\\cos^2t.\n\\]\nThe chart fails at $t=\\pi/2$, but $P(\\pi/2)=\\diag(0,1)$ is a perfectly regular rank-one projector. Choosing a vertical anchor supplies a new chart there. What failed was the comparison with one reference fiber, not the fiber itself.","completion proof constructs\n\\[\nT_a(x)=P(x)P(a)+(I-P(x))(I-P(a)).\n\\]\nIt equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates.\n\\vfig{range_atlas}\n\\head{An exact example of the domain restriction}\nFor the rotating line and anchor $P(0)$,\n\\[\nT_0(t)=\\cos t\\,U(t),\\qquad \\det T_0(t)=\\cos^2t.\n\\]\nThe chart fails at $t=\\pi/2$, but $P(\\pi/2)=\\diag(0,1)$ is a perfectly regular rank-one projector. Choosing a vertical anchor supplies a new chart there. What failed was the comparison with one reference fiber, not the fiber itself.","of constructs\n\\[\nT_a(x)=P(x)P(a)+(I-P(x))(I-P(a)).\n\\]\nIt equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates.\n\\vfig{range_atlas}\n\\head{An exact example of the domain restriction}\nFor the rotating line and anchor $P(0)$,\n\\[\nT_0(t)=\\cos t\\,U(t),\\qquad \\det T_0(t)=\\cos^2t.\n\\]\nThe chart fails at $t=\\pi/2$, but $P(\\pi/2)=\\diag(0,1)$ is a perfectly regular rank-one projector. Choosing a vertical anchor supplies a new chart there. What failed was the comparison with one reference fiber, not the fiber itself.","In general, the open sets where $T_a(x)$ is invertible cover the base. The inverse chart maps and smooth overlap maps are constructed from the actual matrices. Their cocycle identities express consistent coordinate changes, rather than assuming a global basis exists.","\\vpage{15}{Curvature measures a local area response}{AN EXACT BLOCH PROJECTOR CALCULATION}{Chapter VII: Theorem II.6.7, Corollary II.6.8.}\nOn the unit sphere, set $n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta)$ and\n\\[\nP(\\theta,\\phi)=\\frac12(I+n\\cdot\\sigma).\n\\]\nThis is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form $P(\\dd P\\wedge\\dd P)P$. With the paper's scalar convention, its component is\n\\[\nF_{\\theta\\phi}=-\\ii\\Tr\\!\\left(P[\\partial_\\theta P,\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta","\\vpage{15}{Curvature measures a local area response}{AN EXACT BLOCH PROJECTOR CALCULATION}{Chapter VII: Theorem II.6.7, Corollary II.6.8.}\nOn the unit sphere, set $n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta)$ and\n\\[\nP(\\theta,\\phi)=\\frac12(I+n\\cdot\\sigma).\n\\]\nThis is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form $P(\\dd P\\wedge\\dd P)P$. With the paper's scalar convention, its component is\n\\[\nF_{\\theta\\phi}=-\\ii\\Tr\\!\\left(P[\\partial_\\theta P,\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta","\\vpage{15}{Curvature measures a local area response}{AN EXACT BLOCH PROJECTOR CALCULATION}{Chapter VII: Theorem II.6.7, Corollary II.6.8.}\nOn the unit sphere, set $n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta)$ and\n\\[\nP(\\theta,\\phi)=\\frac12(I+n\\cdot\\sigma).\n\\]\nThis is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form $P(\\dd P\\wedge\\dd P)P$. With the paper's scalar convention, its component is\n\\[\nF_{\\theta\\phi}=-\\ii\\Tr\\!\\left(P[\\partial_\\theta P,\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","EXACT BLOCH PROJECTOR CALCULATION}{Chapter VII: Theorem II.6.7, Corollary II.6.8.}\nOn the unit sphere, set $n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta)$ and\n\\[\nP(\\theta,\\phi)=\\frac12(I+n\\cdot\\sigma).\n\\]\nThis is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form $P(\\dd P\\wedge\\dd P)P$. With the paper's scalar convention, its component is\n\\[\nF_{\\theta\\phi}=-\\ii\\Tr\\!\\left(P[\\partial_\\theta P,\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","n\\cdot\\sigma).\n\\]\nThis is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form $P(\\dd P\\wedge\\dd P)P$. With the paper's scalar convention, its component is\n\\[\nF_{\\theta\\phi}=-\\ii\\Tr\\!\\left(P[\\partial_\\theta P,\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","ojector. The intrinsic curvature is the endomorphism-valued two-form $P(\\dd P\\wedge\\dd P)P$. With the paper's scalar convention, its component is\n\\[\nF_{\\theta\\phi}=-\\ii\\Tr\\!\\left(P[\\partial_\\theta P,\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","ndomorphism-valued two-form $P(\\dd P\\wedge\\dd P)P$. With the paper's scalar convention, its component is\n\\[\nF_{\\theta\\phi}=-\\ii\\Tr\\!\\left(P[\\partial_\\theta P,\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","\\partial_\\phi P]\\right)\n=\\frac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","rac12\\sin\\theta.\n\\]\n\\vfig{bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","bloch_curvature}\n\\head{A local frame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","rame gives the same answer}\nAway from the south pole, use $u_N=(\\cos(\\theta/2),e^{\\ii\\phi}\\sin(\\theta/2))^T$. Then $P=u_Nu_N^*$ and the scalar one-form is\n\\[\nA_N=-\\ii u_N^*\\dd u_N=\\frac{1-\\cos\\theta}{2}\\,\\dd\\phi,\n\\qquad \\dd A_N=\\frac12\\sin\\theta\\,\\dd\\theta\\wedge\\dd\\phi.\n\\]\nThus the curvature flux through the cap $0\\le\\theta\\le\\theta_0$ is $\\pi(1-\\cos\\theta_0)$. At $\\theta_0=\\pi/3$, it is exactly $\\pi/2$. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.","The vanishing of the coordinate density at a pole does not mean the geometric curvature disappears there. The spherical area form also carries the factor $\\sin\\theta$. Assigning a phase to parallel transport also requires an orientation and a sign convention for the connection. These choices determine how flux relates to phase.\n\\takeaway{The projector formula and a local frame calculation agree. Curvature is intrinsic even when no single smooth frame covers the whole sphere.}","\\vpage{16}{Which coherence can a measurement see?}{THE ANSWER DEPENDS ON ITS EFFECTS}{Chapter VII: Theorem II.7.2, Theorem II.7.4, Corollary II.7.5.}\nFor the Bell density matrix $\\rho=\\ket{\\Phi^+}\\bra{\\Phi^+}$, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses $00$ and $11$.\n\\vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is no","\\vpage{16}{Which coherence can a measurement see?}{THE ANSWER DEPENDS ON ITS EFFECTS}{Chapter VII: Theorem II.7.2, Theorem II.7.4, Corollary II.7.5.}\nFor the Bell density matrix $\\rho=\\ket{\\Phi^+}\\bra{\\Phi^+}$, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses $00$ and $11$.\n\\vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","?}{THE ANSWER DEPENDS ON ITS EFFECTS}{Chapter VII: Theorem II.7.2, Theorem II.7.4, Corollary II.7.5.}\nFor the Bell density matrix $\\rho=\\ket{\\Phi^+}\\bra{\\Phi^+}$, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses $00$ and $11$.\n\\vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","eorem II.7.4, Corollary II.7.5.}\nFor the Bell density matrix $\\rho=\\ket{\\Phi^+}\\bra{\\Phi^+}$, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses $00$ and $11$.\n\\vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","7.4, Corollary II.7.5.}\nFor the Bell density matrix $\\rho=\\ket{\\Phi^+}\\bra{\\Phi^+}$, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses $00$ and $11$.\n\\vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","llary II.7.5.}\nFor the Bell density matrix $\\rho=\\ket{\\Phi^+}\\bra{\\Phi^+}$, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses $00$ and $11$.\n\\vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","$ and $11$.\n\\vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","vfig{dephasing}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","g}\nEvery effect in the computational basis $E_a=\\ket a\\bra a$ has the same expectation on both matrices, namely $1/2$ at $00$ and $11$ and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture.\n\\head{A measurement that resolves the difference}\nUse the valid two-outcome measurement with effects $E=\\rho$ and $I-E$. Then\n\\[\n\\Tr(E\\rho)=1,\\qquad\n\\Tr\\!\\left(E D_{\\rm computational}(\\rho)\\right)=\\frac12.\n\\]\nThe effect itself is not diagonal in the fine computational packets. It compares their coherent combination, so it falls outside the paper's invariance result for block-diagonal effects.","\\vpage{17}{A channel acts on four matrix entries}{POPULATIONS AND COHERENCES TRANSFORM DIFFERENTLY}{Chapter VII: Theorem II.7.6, Example II.7.12.}\nChoose amplitude damping with $\\gamma=9/25$:\n\\[\nK_0=\\mat{1&0","\\vpage{17}{A channel acts on four matrix entries}{POPULATIONS AND COHERENCES TRANSFORM DIFFERENTLY}{Chapter VII: Theorem II.7.6, Example II.7.12.}\nChoose amplitude damping with $\\gamma=9/25$:\n\\[\nK_0=\\mat{1&0\\\\0&4/5},\\qquad\nK_1=\\mat{0&3/5\\\\0&0},\\qquad K_0^*K_0+K_1^*K_1=I.\n\\]\nThe channel $\\Phi(\\rho)=\\sum_aK_a\\rho K_a^*$ is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational.\n\\vfig{damping}\nThe superoperator uses the entry order $(\\rho_{00},\\rho_{01},\\rho_{10},\\rho_{11})$. Population leaves the excited state with factor $9/25$, while coherences contract by $4/5$. These are not the same rate or the same coordinate.","0&0},\\qquad K_0^*K_0+K_1^*K_1=I.\n\\]\nThe channel $\\Phi(\\rho)=\\sum_aK_a\\rho K_a^*$ is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational.\n\\vfig{damping}\nThe superoperator uses the entry order $(\\rho_{00},\\rho_{01},\\rho_{10},\\rho_{11})$. Population leaves the excited state with factor $9/25$, while coherences contract by $4/5$. These are not the same rate or the same coordinate.","For $\\rho_+=\\frac12\\mat{1&1\\\\1&1}$, the exact output is\n\\[\n\\Phi(\\rho_+)=\\mat{17/25&2/5","1&1}$, the exact output is\n\\[\n\\Phi(\\rho_+)=\\mat{17/25&2/5\\\\2/5&8/25}.\n\\]\nIts trace is one and determinant is $36/625>0$, so it is a mixed positive state. Arrows that show population transfer alone would leave its off-diagonal entries unspecified.\n\\head{How complete positivity is certified}\nThe Choi matrix is an explicit sum of two outer products of vectorized Kraus operators. Its eigenvalues are $41/25,9/25,0,0$, all nonnegative. This gives an independent finite certificate in addition to Kraus normalization. The Choi matrix has trace two under the convention used by the check, which leaves the maximally entangled vector unnormalized.\n\\takeaway{A packet channel acts on pairs of packet addresses, because density matrices have two indices. Population transfer al","2/5&8/25}.\n\\]\nIts trace is one and determinant is $36/625>0$, so it is a mixed positive state. Arrows that show population transfer alone would leave its off-diagonal entries unspecified.\n\\head{How complete positivity is certified}\nThe Choi matrix is an explicit sum of two outer products of vectorized Kraus operators. Its eigenvalues are $41/25,9/25,0,0$, all nonnegative. This gives an independent finite certificate in addition to Kraus normalization. The Choi matrix has trace two under the convention used by the check, which leaves the maximally entangled vector unnormalized.\n\\takeaway{A packet channel acts on pairs of packet addresses, because density matrices have two indices. Population transfer alone does not reconstruct a quantum channel.}","\\vpage{18}{An ancilla tests more than positivity}{WHY COMPLETE POSITIVITY IS THE RELEVANT CONDITION}{Chapter VII: Classical input II.7.7.}\nOrdinary transpose $\\mathsf T$ sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The test must include $\\id\\otimes\\mathsf T$ on the larger system.\n\\vfig{complete_positivity}\nApply this extended map to the Bell density matrix. The result is\n\\[\n(\\id\\otimes\\mathsf T)(\\rho_{\\rm Bell})\n=\\frac12\\mat{1&0&0&0","lassical input II.7.7.}\nOrdinary transpose $\\mathsf T$ sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The test must include $\\id\\otimes\\mathsf T$ on the larger system.\n\\vfig{complete_positivity}\nApply this extended map to the Bell density matrix. The result is\n\\[\n(\\id\\otimes\\mathsf T)(\\rho_{\\rm Bell})\n=\\frac12\\mat{1&0&0&0\\\\0&0&1&0\\\\0&1&0&0\\\\0&0&0&1}.\n\\]\nIts middle block has one negative eigenvalue. More concretely, the normalized vector $w=(\\ket{01}-\\ket{10})/\\sqrt2$ satisfies\n\\[\nw^*(\\id\\otimes\\mathsf T)(\\rho_{\\rm Bell})w=-\\frac12.\n\\]\nThis exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed.\n\\head{Why the Kraus form avoids this problem}\nFor every auxiliary dimension and positive $X$, each term $(I\\otimes K_a)X(I\\otimes K_a)^*$ is positive. Their sum is positive as well. This argument explains the word \\emph{completely}. The conclusion holds for every auxiliar…","0&0&0&1}.\n\\]\nIts middle block has one negative eigenvalue. More concretely, the normalized vector $w=(\\ket{01}-\\ket{10})/\\sqrt2$ satisfies\n\\[\nw^*(\\id\\otimes\\mathsf T)(\\rho_{\\rm Bell})w=-\\frac12.\n\\]\nThis exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed.\n\\head{Why the Kraus form avoids this problem}\nFor every auxiliary dimension and positive $X$, each term $(I\\otimes K_a)X(I\\otimes K_a)^*$ is positive. Their sum is positive as well. This argument explains the word \\emph{completely}. The conclusion holds for every auxiliary dimension, which a test of one input density matrix on the original system cannot establish.","negative eigenvalue. More concretely, the normalized vector $w=(\\ket{01}-\\ket{10})/\\sqrt2$ satisfies\n\\[\nw^*(\\id\\otimes\\mathsf T)(\\rho_{\\rm Bell})w=-\\frac12.\n\\]\nThis exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed.\n\\head{Why the Kraus form avoids this problem}\nFor every auxiliary dimension and positive $X$, each term $(I\\otimes K_a)X(I\\otimes K_a)^*$ is positive. Their sum is positive as well. This argument explains the word \\emph{completely}. The conclusion holds for every auxiliary dimension, which a test of one input density matrix on the original system cannot establish.","alue. More concretely, the normalized vector $w=(\\ket{01}-\\ket{10})/\\sqrt2$ satisfies\n\\[\nw^*(\\id\\otimes\\mathsf T)(\\rho_{\\rm Bell})w=-\\frac12.\n\\]\nThis exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed.\n\\head{Why the Kraus form avoids this problem}\nFor every auxiliary dimension and positive $X$, each term $(I\\otimes K_a)X(I\\otimes K_a)^*$ is positive. Their sum is positive as well. This argument explains the word \\emph{completely}. The conclusion holds for every auxiliary dimension, which a test of one input density matrix on the original system cannot establish.","\\vpage{19}{Find the error without reading the logical state}{FOUR EXACT SYNDROME PACKETS}{Chapter VII: Theorem II.8.2, Theorem II.8.4, Example II.8.8.}\nEncode a qubit by $\\ket{0_L}=\\ket{000}$ and $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical cohere","\\vpage{19}{Find the error without reading the logical state}{FOUR EXACT SYNDROME PACKETS}{Chapter VII: Theorem II.8.2, Theorem II.8.4, Example II.8.8.}\nEncode a qubit by $\\ket{0_L}=\\ket{000}$ and $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered a","\\vpage{19}{Find the error without reading the logical state}{FOUR EXACT SYNDROME PACKETS}{Chapter VII: Theorem II.8.2, Theorem II.8.4, Example II.8.8.}\nEncode a qubit by $\\ket{0_L}=\\ket{000}$ and $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as","\\vpage{19}{Find the error without reading the logical state}{FOUR EXACT SYNDROME PACKETS}{Chapter VII: Theorem II.8.2, Theorem II.8.4, Example II.8.8.}\nEncode a qubit by $\\ket{0_L}=\\ket{000}$ and $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","out reading the logical state}{FOUR EXACT SYNDROME PACKETS}{Chapter VII: Theorem II.8.2, Theorem II.8.4, Example II.8.8.}\nEncode a qubit by $\\ket{0_L}=\\ket{000}$ and $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","{Chapter VII: Theorem II.8.2, Theorem II.8.4, Example II.8.8.}\nEncode a qubit by $\\ket{0_L}=\\ket{000}$ and $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","II.8.4, Example II.8.8.}\nEncode a qubit by $\\ket{0_L}=\\ket{000}$ and $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","d $\\ket{1_L}=\\ket{111}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","}$. Let $P$ project onto this code and take errors $E_a\\in\\{I,X_1,X_2,X_3\\}$. Their images are four orthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","rthogonal rank-two packets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","ets $Q_a=E_aPE_a^*$ that sum to the full identity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","dentity.\n\\vfig{syndromes}\nUse the logical state $\\psi=(3\\ket{0_L}+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","+4\\ket{1_L})/5$ and channel probabilities $w=(1,2,3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.",",3,4)/10$. The noisy state is $\\sum_aw_aE_a\\psi\\psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","psi^*E_a^*$. A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.",". A syndrome measurement returns label $a$ with probability $w_a$, independent of the logical amplitudes.\n\\head{The recovery is an actual channel}\nThe operators $R_a=E_a^*Q_a$ satisfy $\\sum_aR_a^*R_a=I$. Applying the recovery $\\mathcal R(X)=\\sum_aR_aXR_a^*$ to the noisy state gives exactly $\\psi\\psi^*$. The off-diagonal logical coherence $12/25$ is recovered as well as the populations $9/25$ and $16/25$.","The Knill--Laflamme compressions are $PE_a^*E_bP=\\delta_{ab}P$. Their scalar form says that the error labels do not resolve the encoded vector. The exact checks evaluate all sixteen compressions, all sixteen syndrome products, the four probabilities, the recovery normalization, and the full recovered matrix.","\\vpage{20}{An unchanged syndrome can hide a logical error}{THE SCALAR COMPRESSION TEST MATTERS}{Chapter VII: Theorem II.8.2, Proposition II.8.7.}\nApply $Z_1=Z\\otimes I\\otimes I$ to the same repetition code. It sends $\\ket{000}$ to itself and $\\ket{111}$ to its negative. Both states remain inside the original code, so the previous syndrome partition reports no bit flip.\n\\[\n\\frac{3\\ket{000}+4\\ket{111}}5\n\\ \\longmapsto\\\n\\frac{3\\ket{000}-4\\ket{111}}5.\n\\]\nThe populations are unchanged, but the relative phase is not.\n\\vfig{logical_error}\nLet $V$ be the code isometry with columns $\\ket{000},\\ket{111}$. For the proposed error family $\\{I,Z_1\\}$, its off-diagonal error compression is\n\\[\nV^*I^*Z_1V=\\mat{1&0","7.}\nApply $Z_1=Z\\otimes I\\otimes I$ to the same repetition code. It sends $\\ket{000}$ to itself and $\\ket{111}$ to its negative. Both states remain inside the original code, so the previous syndrome partition reports no bit flip.\n\\[\n\\frac{3\\ket{000}+4\\ket{111}}5\n\\ \\longmapsto\\\n\\frac{3\\ket{000}-4\\ket{111}}5.\n\\]\nThe populations are unchanged, but the relative phase is not.\n\\vfig{logical_error}\nLet $V$ be the code isometry with columns $\\ket{000},\\ket{111}$. For the proposed error family $\\{I,Z_1\\}$, its off-diagonal error compression is\n\\[\nV^*I^*Z_1V=\\mat{1&0","the same repetition code. It sends $\\ket{000}$ to itself and $\\ket{111}$ to its negative. Both states remain inside the original code, so the previous syndrome partition reports no bit flip.\n\\[\n\\frac{3\\ket{000}+4\\ket{111}}5\n\\ \\longmapsto\\\n\\frac{3\\ket{000}-4\\ket{111}}5.\n\\]\nThe populations are unchanged, but the relative phase is not.\n\\vfig{logical_error}\nLet $V$ be the code isometry with columns $\\ket{000},\\ket{111}$. For the proposed error family $\\{I,Z_1\\}$, its off-diagonal error compression is\n\\[\nV^*I^*Z_1V=\\mat{1&0","self and $\\ket{111}$ to its negative. Both states remain inside the original code, so the previous syndrome partition reports no bit flip.\n\\[\n\\frac{3\\ket{000}+4\\ket{111}}5\n\\ \\longmapsto\\\n\\frac{3\\ket{000}-4\\ket{111}}5.\n\\]\nThe populations are unchanged, but the relative phase is not.\n\\vfig{logical_error}\nLet $V$ be the code isometry with columns $\\ket{000},\\ket{111}$. For the proposed error family $\\{I,Z_1\\}$, its off-diagonal error compression is\n\\[\nV^*I^*Z_1V=\\mat{1&0","main inside the original code, so the previous syndrome partition reports no bit flip.\n\\[\n\\frac{3\\ket{000}+4\\ket{111}}5\n\\ \\longmapsto\\\n\\frac{3\\ket{000}-4\\ket{111}}5.\n\\]\nThe populations are unchanged, but the relative phase is not.\n\\vfig{logical_error}\nLet $V$ be the code isometry with columns $\\ket{000},\\ket{111}$. For the proposed error family $\\{I,Z_1\\}$, its off-diagonal error compression is\n\\[\nV^*I^*Z_1V=\\mat{1&0\\\\0&-1}.\n\\]\nThis matrix is not $cI$ for any scalar $c$. Thus the Knill--Laflamme condition fails for that family. Its trace is zero even though its action on the code is nonzero. A trace test alone would miss the obstruction.\n\\head{Known correction versus family recovery}\nIf it is known with certainty that $Z_1$ occurred, applying $Z_1$ again reverses it. The failed criterion concerns a \\emph{single recovery valid for every unknown error in the declared family}, not invertibility of each error considered separately. This distinction prevents a misleading conclusion that no phase error can ever be undone.","0&-1}.\n\\]\nThis matrix is not $cI$ for any scalar $c$. Thus the Knill--Laflamme condition fails for that family. Its trace is zero even though its action on the code is nonzero. A trace test alone would miss the obstruction.\n\\head{Known correction versus family recovery}\nIf it is known with certainty that $Z_1$ occurred, applying $Z_1$ again reverses it. The failed criterion concerns a \\emph{single recovery valid for every unknown error in the declared family}, not invertibility of each error considered separately. This distinction prevents a misleading conclusion that no phase error can ever be undone.","\\vpage{21}{Hidden states return as spectral dependence}{TWO INVERTIBILITY CONDITIONS, NOT ONE}{Chapter VII: Theorem II.9.2, Theorem II.9.3.}\nTake $H=\\mat{0&1\\\\1&2}$ and retain its first coordinate. The eliminated block is $H_{QQ}=2$. Whenever $z-2$ and the full $zI-H$ are invertible,\n\\[\nH_{\\rm eff}(z)=\\frac1{z-2},\\qquad\n\\bigl[(zI-H)^{-1}\\bigr]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Neverth","1&2}$ and retain its first coordinate. The eliminated block is $H_{QQ}=2$. Whenever $z-2$ and the full $zI-H$ are invertible,\n\\[\nH_{\\rm eff}(z)=\\frac1{z-2},\\qquad\n\\bigl[(zI-H)^{-1}\\bigr]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The","1&2}$ and retain its first coordinate. The eliminated block is $H_{QQ}=2$. Whenever $z-2$ and the full $zI-H$ are invertible,\n\\[\nH_{\\rm eff}(z)=\\frac1{z-2},\\qquad\n\\bigl[(zI-H)^{-1}\\bigr]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The rational expres","1&2}$ and retain its first coordinate. The eliminated block is $H_{QQ}=2$. Whenever $z-2$ and the full $zI-H$ are invertible,\n\\[\nH_{\\rm eff}(z)=\\frac1{z-2},\\qquad\n\\bigl[(zI-H)^{-1}\\bigr]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The rational expression has a continuat","1&2}$ and retain its first coordinate. The eliminated block is $H_{QQ}=2$. Whenever $z-2$ and the full $zI-H$ are invertible,\n\\[\nH_{\\rm eff}(z)=\\frac1{z-2},\\qquad\n\\bigl[(zI-H)^{-1}\\bigr]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The rational expression has a continuation to this value. By contrast, $1\\pm\\sqrt2$ are poles of the full resolvent.","$ and retain its first coordinate. The eliminated block is $H_{QQ}=2$. Whenever $z-2$ and the full $zI-H$ are invertible,\n\\[\nH_{\\rm eff}(z)=\\frac1{z-2},\\qquad\n\\bigl[(zI-H)^{-1}\\bigr]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The rational expression has a continuation to this value. By contrast, $1\\pm\\sqrt2$ are poles of the full resolvent.","rst coordinate. The eliminated block is $H_{QQ}=2$. Whenever $z-2$ and the full $zI-H$ are invertible,\n\\[\nH_{\\rm eff}(z)=\\frac1{z-2},\\qquad\n\\bigl[(zI-H)^{-1}\\bigr]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The rational expression has a continuation to this value. By contrast, $1\\pm\\sqrt2$ are poles of the full resolvent.","]_{00}\n=\\frac1{z-1/(z-2)}=\\frac{z-2}{z^2-2z-1}.\n\\]\nThese are the hypotheses used in the formal companion.\n\\vfig{feshbach}\nAt $z=3$, the exact retained resolvent is $1/2$. Bare truncation uses only $H_{PP}=0$ and gives $1/3$, an error of $1/6$. The omitted coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The rational expression has a continuation to this value. By contrast, $1\\pm\\sqrt2$ are poles of the full resolvent.","coupling has changed the response even though the retained diagonal entry did not change.\n\\head{A pole can belong to the elimination chart}\nAt $z=2$, the eliminated inverse does not exist, so the displayed Schur formula cannot be applied there as written. Nevertheless, the full resolvent exists and its retained entry is zero. The rational expression has a continuation to this value. By contrast, $1\\pm\\sqrt2$ are poles of the full resolvent.","Invertibility of the full matrix is an independent requirement. At $z=0$, $H=\\diag(0,-1)$ has an invertible eliminated block in $zI-H$, but the full matrix is singular. The original prose omitted this condition. The preserved amendments restore it. This appendix uses the corrected form throughout.\n\\takeaway{An exact elimination formula has a spectral domain. Check both inverses before evaluating it, and distinguish poles of an intermediate inverse from poles of the full physical resolvent.}","\\vpage{22}{The smaller system remembers the discarded one}{AN EXACT QUANTUM MEMORY EQUATION}{Chapter VII: Theorem II.9.8, Theorem II.9.11.}\nChoose $H=X$, initial state $(1,0)^T$, and retain the first coordinate. The coupled equations and their solution are\n\\[\n\\ii\\dot x=y,\\qquad \\ii\\dot y=x,\\qquad\nx(t)=\\cos t,\\quad y(t)=-\\ii\\sin t.\n\\]\nThe hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidde","\\vpage{22}{The smaller system remembers the discarded one}{AN EXACT QUANTUM MEMORY EQUATION}{Chapter VII: Theorem II.9.8, Theorem II.9.11.}\nChoose $H=X$, initial state $(1,0)^T$, and retain the first coordinate. The coupled equations and their solution are\n\\[\n\\ii\\dot x=y,\\qquad \\ii\\dot y=x,\\qquad\nx(t)=\\cos t,\\quad y(t)=-\\ii\\sin t.\n\\]\nThe hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add","\\vpage{22}{The smaller system remembers the discarded one}{AN EXACT QUANTUM MEMORY EQUATION}{Chapter VII: Theorem II.9.8, Theorem II.9.11.}\nChoose $H=X$, initial state $(1,0)^T$, and retain the first coordinate. The coupled equations and their solution are\n\\[\n\\ii\\dot x=y,\\qquad \\ii\\dot y=x,\\qquad\nx(t)=\\cos t,\\quad y(t)=-\\ii\\sin t.\n\\]\nThe hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","ANTUM MEMORY EQUATION}{Chapter VII: Theorem II.9.8, Theorem II.9.11.}\nChoose $H=X$, initial state $(1,0)^T$, and retain the first coordinate. The coupled equations and their solution are\n\\[\n\\ii\\dot x=y,\\qquad \\ii\\dot y=x,\\qquad\nx(t)=\\cos t,\\quad y(t)=-\\ii\\sin t.\n\\]\nThe hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","T$, and retain the first coordinate. The coupled equations and their solution are\n\\[\n\\ii\\dot x=y,\\qquad \\ii\\dot y=x,\\qquad\nx(t)=\\cos t,\\quad y(t)=-\\ii\\sin t.\n\\]\nThe hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","the first coordinate. The coupled equations and their solution are\n\\[\n\\ii\\dot x=y,\\qquad \\ii\\dot y=x,\\qquad\nx(t)=\\cos t,\\quad y(t)=-\\ii\\sin t.\n\\]\nThe hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","r solution are\n\\[\n\\ii\\dot x=y,\\qquad \\ii\\dot y=x,\\qquad\nx(t)=\\cos t,\\quad y(t)=-\\ii\\sin t.\n\\]\nThe hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","ly. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","the hidden row and column would instead leave the zero Hamiltonian and predict $x(t)=1$.\n\\vfig{memory}\n\\head{Eliminate exactly, not approximately}\nIntegrating the second equation with $y(0)=0$ gives $y(t)=-\\ii\\int_0^tx(s)\\,ds$. Substitution produces\n\\[\n\\dot x(t)=-\\int_0^tx(s)\\,ds.\n\\]\nThe memory kernel is the constant function one. Differentiating once more recovers $\\ddot x=-x$ with $x(0)=1$ and $\\dot x(0)=0$, explaining the cosine solution. A nonzero hidden initial state would add a forcing term that must also be retained.","The full propagator is a unitary group, but its retained scalar entry is not a semigroup. At $t=s=\\pi/4$,\n\\[\nx(t+s)-x(t)x(s)=0-\\frac12=-\\frac12.\n\\]\nThere is no contradiction: compression of a product need not equal the product of the compressed factors when the hidden subspace carries intermediate amplitude.","\\vpage{23}{Positive transfer is not a stochastic matrix}{SPECTRAL POSITIVITY IS BASIS INDEPENDENT}{Chapter VII: Theorem II.10.1, Theorem II.10.3.}\nFor $H=I-X$, the two eigenvalues are zero and two. Its Euclidean transfer is\n\\[\nT(\\tau)=e^{-\\tau H}\n=\\frac12\\mat{1+q&1-q","\\vpage{23}{Positive transfer is not a stochastic matrix}{SPECTRAL POSITIVITY IS BASIS INDEPENDENT}{Chapter VII: Theorem II.10.1, Theorem II.10.3.}\nFor $H=I-X$, the two eigenvalues are zero and two. Its Euclidean transfer is\n\\[\nT(\\tau)=e^{-\\tau H}\n=\\frac12\\mat{1+q&1-q\\\\1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\rig","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe fact","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite opera","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator iden","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure,","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure, limits, positivity","1-q&1+q},\\qquad q=e^{-2\\tau}.\n\\]\nFor $\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure, limits, positivity hypotheses, and the other assumptions of the chosen reconstruction framework. Each of these","\\tau\\ge0$, the eigenvalues $1,q$ lie in $(0,1]$, and multiplying two such matrices replaces $q,r$ by $qr$. This is the semigroup law in spectral coordinates.\n\\vfig{euclidean}\nAt $q=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure, limits, positivity hypotheses, and the other assumptions of the chosen reconstruction framework. Each of these assumptions requires its own justification beyond the two positive eigenvalues.","=1/2$, the displayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure, limits, positivity hypotheses, and the other assumptions of the chosen reconstruction framework. Each of these assumptions requires its own justification beyond the two positive eigenvalues.","splayed $T$ happens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure, limits, positivity hypotheses, and the other assumptions of the chosen reconstruction framework. Each of these assumptions requires its own justification beyond the two positive eigenvalues.","ens also to have nonnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure, limits, positivity hypotheses, and the other assumptions of the chosen reconstruction framework. Each of these assumptions requires its own justification beyond the two positive eigenvalues.","nnegative entries and column sums one. Conjugate by the unitary $Z$. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become $-1/4$. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation.\n\\head{A useful finite positive kernel}\nFor this nonnegative $H$, vectors $u_a$ and times $t_a\\ge0$ satisfy\n\\[\n\\sum_{a,b}\\langle u_a,e^{-(t_a+t_b)H}u_b\\rangle\n=\\left\\|\\sum_a e^{-t_aH}u_a\\right\\|^2\\ge0.\n\\]\nThe factorization makes positivity transparent. It remains a finite operator identity. A continuum reconstruction theorem requires a compatible field structure, limits, positivity hypotheses, and the other assumptions of the chosen reconstruction framework. Each of these assumptions requires its own justification beyond the two positive eigenvalues.","The exact check verifies the matrix semigroup law as a polynomial identity in $q,r$ and verifies the unchanged spectrum after conjugation. It rejects entrywise nonnegativity for the conjugated matrix.\n\\takeaway{Euclidean positivity refers to quadratic forms and spectral values. A stochastic interpretation or a continuum physical reconstruction needs additional structure.}","\\vpage{24}{A dimension curve is a scale diagnostic}{THE EXACT HEAT TRACE ON FOUR SITES}{Chapter VII: Definition II.11.1, Proposition II.11.2.}\nFor the Laplacian on a cycle with four sites $L=2I-S-S^*$, the eigenvalues are $0,2,2,4$. The normalized heat trace and its diagnostic of spectral dimension are therefore\n\\[\nK(\\tau)=\\frac{1+2e^{-2\\tau}+e^{-4\\tau}}4\n=\\frac{(1+e^{-2\\tau})^2}4,\n\\qquad\n d_s(\\tau)=-2\\tau\\frac{K'(\\tau)}{K(\\tau)}\n=\\frac{8\\tau}{e^{2\\tau}+1}.\n\\]\nNo numerical differentiation is needed for this finite example.\n\\vfig{spectral_dimension}\n\\head{Both ends reflect the finite size of the system}\nAs $\\tau\\to0$, every eigenmode contributes approximately one and the dimension diagnostic tends to zero. As $\\tau\\to\\infty$, the zero mode dominates, so it again tends to zero. A finite spectrum cannot sustain a nonzero plateau at all scales.","ues are $0,2,2,4$. The normalized heat trace and its diagnostic of spectral dimension are therefore\n\\[\nK(\\tau)=\\frac{1+2e^{-2\\tau}+e^{-4\\tau}}4\n=\\frac{(1+e^{-2\\tau})^2}4,\n\\qquad\n d_s(\\tau)=-2\\tau\\frac{K'(\\tau)}{K(\\tau)}\n=\\frac{8\\tau}{e^{2\\tau}+1}.\n\\]\nNo numerical differentiation is needed for this finite example.\n\\vfig{spectral_dimension}\n\\head{Both ends reflect the finite size of the system}\nAs $\\tau\\to0$, every eigenmode contributes approximately one and the dimension diagnostic tends to zero. As $\\tau\\to\\infty$, the zero mode dominates, so it again tends to zero. A finite spectrum cannot sustain a nonzero plateau at all scales.","\\tau)=\\frac{1+2e^{-2\\tau}+e^{-4\\tau}}4\n=\\frac{(1+e^{-2\\tau})^2}4,\n\\qquad\n d_s(\\tau)=-2\\tau\\frac{K'(\\tau)}{K(\\tau)}\n=\\frac{8\\tau}{e^{2\\tau}+1}.\n\\]\nNo numerical differentiation is needed for this finite example.\n\\vfig{spectral_dimension}\n\\head{Both ends reflect the finite size of the system}\nAs $\\tau\\to0$, every eigenmode contributes approximately one and the dimension diagnostic tends to zero. As $\\tau\\to\\infty$, the zero mode dominates, so it again tends to zero. A finite spectrum cannot sustain a nonzero plateau at all scales.","For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: $K_2=K^2$. Consequently $d_{s,2}=2d_s$ exactly. The product identity determines the second curve exactly from the first.","At $\\tau=\\log2/2$, the first curve equals $4\\log2/3$. This noninteger value is a spectral quantity that depends on scale, not a claim that the graph has a fractional topological dimension. Four sites are too few to promise a broad geometric plateau. Experiments on larger grids remain finite diagnostics. Convergence requires the analytic hypotheses developed on \\pagerefV{37}.\n\\takeaway{A convincing continuum dimension needs a stable scale window that persists under controlled refinement. One appealing finite curve is not that theorem.}","\\vpage{25}{Refine the grid while holding the mode fixed}{AN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPE}{Chapter VII: Theorem II.11.4.}\nFor the circle symbol, the continuum value is $k^2$ and the grid value is\n\\[\n\\lambda_{k,h}=\\frac4{h^2}\\sin^2\\!\\left(\\frac{kh}{2}\\right),\\qquad h\\ne0.\n\\]\nFor a fixed low mode, the paper bounds the error by $0\\le k^2-\\lambda_{k,h}\\le k^4h^2/12$. The spacing condition matters because the displayed expression divides by $h^2$.\n\\vfig{dispersion}\n\\head{What is certified in the displayed table?}\nTake $k=1$ and $h=1/4,1/8,1/16$. For $x=h/2\\in[0,1]$, alternating Taylor sums give rational lower and upper bounds for $\\sin x$. Squaring these positive endpoints and multiplying by $4/h^2$ yields exact rational enclosures for each error. The displayed decimals only locate those much narrower rational intervals.","ed}{AN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPE}{Chapter VII: Theorem II.11.4.}\nFor the circle symbol, the continuum value is $k^2$ and the grid value is\n\\[\n\\lambda_{k,h}=\\frac4{h^2}\\sin^2\\!\\left(\\frac{kh}{2}\\right),\\qquad h\\ne0.\n\\]\nFor a fixed low mode, the paper bounds the error by $0\\le k^2-\\lambda_{k,h}\\le k^4h^2/12$. The spacing condition matters because the displayed expression divides by $h^2$.\n\\vfig{dispersion}\n\\head{What is certified in the displayed table?}\nTake $k=1$ and $h=1/4,1/8,1/16$. For $x=h/2\\in[0,1]$, alternating Taylor sums give rational lower and upper bounds for $\\sin x$. Squaring these positive endpoints and multiplying by $4/h^2$ yields exact rational enclosures for each error. The displayed decimals only locate those much narrower rational intervals.","}\nFor the circle symbol, the continuum value is $k^2$ and the grid value is\n\\[\n\\lambda_{k,h}=\\frac4{h^2}\\sin^2\\!\\left(\\frac{kh}{2}\\right),\\qquad h\\ne0.\n\\]\nFor a fixed low mode, the paper bounds the error by $0\\le k^2-\\lambda_{k,h}\\le k^4h^2/12$. The spacing condition matters because the displayed expression divides by $h^2$.\n\\vfig{dispersion}\n\\head{What is certified in the displayed table?}\nTake $k=1$ and $h=1/4,1/8,1/16$. For $x=h/2\\in[0,1]$, alternating Taylor sums give rational lower and upper bounds for $\\sin x$. Squaring these positive endpoints and multiplying by $4/h^2$ yields exact rational enclosures for each error. The displayed decimals only locate those much narrower rational intervals.","circle symbol, the continuum value is $k^2$ and the grid value is\n\\[\n\\lambda_{k,h}=\\frac4{h^2}\\sin^2\\!\\left(\\frac{kh}{2}\\right),\\qquad h\\ne0.\n\\]\nFor a fixed low mode, the paper bounds the error by $0\\le k^2-\\lambda_{k,h}\\le k^4h^2/12$. The spacing condition matters because the displayed expression divides by $h^2$.\n\\vfig{dispersion}\n\\head{What is certified in the displayed table?}\nTake $k=1$ and $h=1/4,1/8,1/16$. For $x=h/2\\in[0,1]$, alternating Taylor sums give rational lower and upper bounds for $\\sin x$. Squaring these positive endpoints and multiplying by $4/h^2$ yields exact rational enclosures for each error. The displayed decimals only locate those much narrower rational intervals.","tinuum value is $k^2$ and the grid value is\n\\[\n\\lambda_{k,h}=\\frac4{h^2}\\sin^2\\!\\left(\\frac{kh}{2}\\right),\\qquad h\\ne0.\n\\]\nFor a fixed low mode, the paper bounds the error by $0\\le k^2-\\lambda_{k,h}\\le k^4h^2/12$. The spacing condition matters because the displayed expression divides by $h^2$.\n\\vfig{dispersion}\n\\head{What is certified in the displayed table?}\nTake $k=1$ and $h=1/4,1/8,1/16$. For $x=h/2\\in[0,1]$, alternating Taylor sums give rational lower and upper bounds for $\\sin x$. Squaring these positive endpoints and multiplying by $4/h^2$ yields exact rational enclosures for each error. The displayed decimals only locate those much narrower rational intervals.","eft(\\frac{kh}{2}\\right),\\qquad h\\ne0.\n\\]\nFor a fixed low mode, the paper bounds the error by $0\\le k^2-\\lambda_{k,h}\\le k^4h^2/12$. The spacing condition matters because the displayed expression divides by $h^2$.\n\\vfig{dispersion}\n\\head{What is certified in the displayed table?}\nTake $k=1$ and $h=1/4,1/8,1/16$. For $x=h/2\\in[0,1]$, alternating Taylor sums give rational lower and upper bounds for $\\sin x$. Squaring these positive endpoints and multiplying by $4/h^2$ yields exact rational enclosures for each error. The displayed decimals only locate those much narrower rational intervals.","the paper bounds the error by $0\\le k^2-\\lambda_{k,h}\\le k^4h^2/12$. The spacing condition matters because the displayed expression divides by $h^2$.\n\\vfig{dispersion}\n\\head{What is certified in the displayed table?}\nTake $k=1$ and $h=1/4,1/8,1/16$. For $x=h/2\\in[0,1]$, alternating Taylor sums give rational lower and upper bounds for $\\sin x$. Squaring these positive endpoints and multiplying by $4/h^2$ yields exact rational enclosures for each error. The displayed decimals only locate those much narrower rational intervals.","Each interval lies strictly inside $(0,h^2/12)$. Dividing successive interval endpoints proves that both error ratios are between $3.99$ and $4$. The check verifies these inequalities by rational arithmetic, without relying on approximate sine values or a fitted logarithmic slope.\n\\head{Why the mode regime matters}\nIf instead $k=\\pi/h$ moves with the grid, the ratio $\\lambda_{k,h}/k^2=4/\\pi^2$ stays constant, not one. The bound's factor $k^4$ warns that it is not uniform in arbitrary moving high modes. The displayed rational test evaluates the analytic symbol. It does not assert that every chosen spacing defines an integer periodic grid.\n\\takeaway{Second-order convergence is a quantified statement with a specified mode regime. Keep that","Each interval lies strictly inside $(0,h^2/12)$. Dividing successive interval endpoints proves that both error ratios are between $3.99$ and $4$. The check verifies these inequalities by rational arithmetic, without relying on approximate sine values or a fitted logarithmic slope.\n\\head{Why the mode regime matters}\nIf instead $k=\\pi/h$ moves with the grid, the ratio $\\lambda_{k,h}/k^2=4/\\pi^2$ stays constant, not one. The bound's factor $k^4$ warns that it is not uniform in arbitrary moving high modes. The displayed rational test evaluates the analytic symbol. It does not assert that every chosen spacing defines an integer periodic grid.\n\\takeaway{Second-order convergence is a quantified statement with a specified mode regime. Keep that regime fixed when reading an error curve or comparing refinements.}","/12)$. Dividing successive interval endpoints proves that both error ratios are between $3.99$ and $4$. The check verifies these inequalities by rational arithmetic, without relying on approximate sine values or a fitted logarithmic slope.\n\\head{Why the mode regime matters}\nIf instead $k=\\pi/h$ moves with the grid, the ratio $\\lambda_{k,h}/k^2=4/\\pi^2$ stays constant, not one. The bound's factor $k^4$ warns that it is not uniform in arbitrary moving high modes. The displayed rational test evaluates the analytic symbol. It does not assert that every chosen spacing defines an integer periodic grid.\n\\takeaway{Second-order convergence is a quantified statement with a specified mode regime. Keep that regime fixed when reading an error curve or comparing refinements.}","\\vpage{26}{A zero compressed anomaly can be automatic}{TEST THE CONSTRAINTS BEFORE THE PROJECTION}{Chapter VII: Theorem II.12.2, Proposition II.12.3.}\nTake the self-adjoint matrices\n\\[\nC_1=\\diag(0,1,-1),\\qquad\nC_2=\\mat{0&0&0\\\\0&0&1\\\\0&1&0},\\qquad\nP=\\diag(1,0,0).\n\\]\nBoth constraints annihilate $P\\CC^3$, and their common kernel is exactly that one-dimensional space. Nevertheless, their full commutator is nonzero.\n\\vfig{constraints}\n\\head{Why the compressed test cannot decide closure}\nSince $C_1P=C_2P=0$, each product $C_1C_2P$ and $C_2C_1P$ vanishes. Therefore $P[C_1,C_2]P=0$ without any requirement that the unprojected commutator close in the chosen constraint span. In this example the skew off-diagonal commutator is not a linear combination of the two displayed symmetric constraints.","0&1&0},\\qquad\nP=\\diag(1,0,0).\n\\]\nBoth constraints annihilate $P\\CC^3$, and their common kernel is exactly that one-dimensional space. Nevertheless, their full commutator is nonzero.\n\\vfig{constraints}\n\\head{Why the compressed test cannot decide closure}\nSince $C_1P=C_2P=0$, each product $C_1C_2P$ and $C_2C_1P$ vanishes. Therefore $P[C_1,C_2]P=0$ without any requirement that the unprojected commutator close in the chosen constraint span. In this example the skew off-diagonal commutator is not a linear combination of the two displayed symmetric constraints.","\\vpage{27}{One calculation of transfer to the opposite site}{SEEDS, CHARACTERS, INTERFERENCE, AND DEPHASING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephas","\\vpage{27}{One calculation of transfer to the opposite site}{SEEDS, CHARACTERS, INTERFERENCE, AND DEPHASING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\","\\vpage{27}{One calculation of transfer to the opposite site}{SEEDS, CHARACTERS, INTERFERENCE, AND DEPHASING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+","\\vpage{27}{One calculation of transfer to the opposite site}{SEEDS, CHARACTERS, INTERFERENCE, AND DEPHASING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le","\\vpage{27}{One calculation of transfer to the opposite site}{SEEDS, CHARACTERS, INTERFERENCE, AND DEPHASING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probabi","\\vpage{27}{One calculation of transfer to the opposite site}{SEEDS, CHARACTERS, INTERFERENCE, AND DEPHASING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probability at the opposite site is exactly $1-3\\et","}{One calculation of transfer to the opposite site}{SEEDS, CHARACTERS, INTERFERENCE, AND DEPHASING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probability at the opposite site is exactly $1-3\\eta/4$. Compl…","ING}{Chapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probability at the opposite site is exactly $1-3\\eta/4$. Complete momentum dephasing reduces it to $1/4$. This calculation isolates how discarded coherence c…","rem II.4.4, Theorem II.7.4.}\nReturn to the shift on four sites $S$ and Laplacian $L=2I-S-S^*$. The same character projectors have energies $(0,2,4,2)$. At time $t=\\pi/2$, their evolution phases are $(1,-1,1,-1)$, exactly the character eigenvalues of $S^2$. Hence\n\\[\ne^{-\\ii\\pi L/2}=S^2,\\qquad\n\\ket0\\longmapsto\\ket2.\n\\]\n\\vfig{perfect_transfer}\nAt halfway time $\\pi/4$, the actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probability at the opposite site is exactly $1-3\\eta/4$. Complete momentum dephasing reduces it to $1/4$. This calculation isolates how discarded coherence changes the interference result.","actual amplitude vector is $(-\\ii,1,\\ii,1)^T/2$, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probability at the opposite site is exactly $1-3\\eta/4$. Complete momentum dephasing reduces it to $1/4$. This calculation isolates how discarded coherence changes the interference result.","ite. Uniform intermediate probabilities do not imply an incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probability at the opposite site is exactly $1-3\\eta/4$. Complete momentum dephasing reduces it to $1/4$. This calculation isolates how discarded coherence changes the interference result.","incoherent intermediate state.\n\\head{A channel changes the conclusion predictably}\nBefore evolving, mix the seed density matrix with its complete momentum dephasing:\n\\[\n\\rho_\\eta=(1-\\eta)\\ket0\\bra0+\\eta\\sum_jP_j\\ket0\\bra0P_j\n=(1-\\eta)\\ket0\\bra0+\\eta I/4,\n\\quad0\\le\\eta\\le1.\n\\]\nAt transfer time, the probability at the opposite site is exactly $1-3\\eta/4$. Complete momentum dephasing reduces it to $1/4$. This calculation isolates how discarded coherence changes the interference result.","Finally, the shift grammar generates four independent vectors from $\\ket0$, while the Hamiltonian powers generate only three: its repeated energy links two momentum directions. The same seed can therefore be cyclic for the shift algebra and noncyclic for the Hamiltonian algebra, even though perfect transfer occurs at one time.\n\\takeaway{Perfect transfer is an exact finite interference statement. It neither implies cyclicity for every operator grammar nor survives an unaccounted dephasing operation.}","ily on \\pagerefV{20}, and the limitation of testing only the common kernel on \\pagerefV{26}.\n\\takeaway{The finite constructions have explicit scope. The appendix adds understanding and reproducibility, not unproved continuum claims or a new theorem count.}\n\\vpage{29}{Nonabelian packets carry multiplicity}{A SIX-DIMENSIONAL REGULAR REPRESENTATION}{Chapter VII: Theorem II.3.1, Theorem II.3.2.}\nLet $\\HH=\\CC[S_3]$ have the six orthonormal basis vectors $\\ket g$, with left action $U(h)\\ket g=\\ket{hg}$. The irreducible characters on the identity, transposition, and three-cycle classes are\n\\[\n\\begin{array}{c|rrr|r}\n & e & (12)&(123)&d_\\lambda","ly the common kernel on \\pagerefV{26}.\n\\takeaway{The finite constructions have explicit scope. The appendix adds understanding and reproducibility, not unproved continuum claims or a new theorem count.}\n\\vpage{29}{Nonabelian packets carry multiplicity}{A SIX-DIMENSIONAL REGULAR REPRESENTATION}{Chapter VII: Theorem II.3.1, Theorem II.3.2.}\nLet $\\HH=\\CC[S_3]$ have the six orthonormal basis vectors $\\ket g$, with left action $U(h)\\ket g=\\ket{hg}$. The irreducible characters on the identity, transposition, and three-cycle classes are\n\\[\n\\begin{array}{c|rrr|r}\n & e & (12)&(123)&d_\\lambda","gerefV{26}.\n\\takeaway{The finite constructions have explicit scope. The appendix adds understanding and reproducibility, not unproved continuum claims or a new theorem count.}\n\\vpage{29}{Nonabelian packets carry multiplicity}{A SIX-DIMENSIONAL REGULAR REPRESENTATION}{Chapter VII: Theorem II.3.1, Theorem II.3.2.}\nLet $\\HH=\\CC[S_3]$ have the six orthonormal basis vectors $\\ket g$, with left action $U(h)\\ket g=\\ket{hg}$. The irreducible characters on the identity, transposition, and three-cycle classes are\n\\[\n\\begin{array}{c|rrr|r}\n & e & (12)&(123)&d_\\lambda","cibility, not unproved continuum claims or a new theorem count.}\n\\vpage{29}{Nonabelian packets carry multiplicity}{A SIX-DIMENSIONAL REGULAR REPRESENTATION}{Chapter VII: Theorem II.3.1, Theorem II.3.2.}\nLet $\\HH=\\CC[S_3]$ have the six orthonormal basis vectors $\\ket g$, with left action $U(h)\\ket g=\\ket{hg}$. The irreducible characters on the identity, transposition, and three-cycle classes are\n\\[\n\\begin{array}{c|rrr|r}\n & e & (12)&(123)&d_\\lambda\\\\\\hline\n\\text{trivial}&1&1&1&1\\\\\n\\text{sign}&1&-1&1&1\\\\\n\\text{standard}&2&0&-1&2\n\\end{array}\n\\qquad\nP_\\lambda=\\frac{d_\\lambda}{6}\\sum_{g\\in S_3}\\chi_\\lambda(g^{-1})U(g).\n\\]\nAll matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks $1,1,4$, with pairwise zero products and sum $I_6$.\n\\xfig{s3}\n\\head{The standard packet has dimension four}\nThe standard representation occurs twice in the regular representation. Its isotypic component therefore has dimension $2\\cdot2=4$: one factor carries the two-dimensional irreducible action and the other records multiplicity. A character projector selects the whole i…","\\text{standard}&2&0&-1&2\n\\end{array}\n\\qquad\nP_\\lambda=\\frac{d_\\lambda}{6}\\sum_{g\\in S_3}\\chi_\\lambda(g^{-1})U(g).\n\\]\nAll matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks $1,1,4$, with pairwise zero products and sum $I_6$.\n\\xfig{s3}\n\\head{The standard packet has dimension four}\nThe standard representation occurs twice in the regular representation. Its isotypic component therefore has dimension $2\\cdot2=4$: one factor carries the two-dimensional irreducible action and the other records multiplicity. A character projector selects the whole isotypic component. It does not select a particular copy or a particular internal vector.","_\\lambda=\\frac{d_\\lambda}{6}\\sum_{g\\in S_3}\\chi_\\lambda(g^{-1})U(g).\n\\]\nAll matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks $1,1,4$, with pairwise zero products and sum $I_6$.\n\\xfig{s3}\n\\head{The standard packet has dimension four}\nThe standard representation occurs twice in the regular representation. Its isotypic component therefore has dimension $2\\cdot2=4$: one factor carries the two-dimensional irreducible action and the other records multiplicity. A character projector selects the whole isotypic component. It does not select a particular copy or a particular internal vector.","cible factor, while leaving the multiplicity factor untouched. Their multiplication law is stronger information than a list of packet ranks. A basis chosen inside this packet supplies coordinates. It does not canonically distinguish the two copies of the standard representation.\n\\head{What the retained check establishes}\nThe retained exact check constructs all six permutation matrices, tests all $36$ multiplication relations, computes the character averages, and checks ranks independently by Gaussian elimination and traces. It rejects the claim that the standard isotypic projector has rank two. This is one specified representation of $S_3$, not a computational classification of all finite group representations.\n\\takeaway{For a nonabelian symmetry, distinguish irreducible dimension from multiplicity. A central character projector retains both.}","pies of the standard representation.\n\\head{What the retained check establishes}\nThe retained exact check constructs all six permutation matrices, tests all $36$ multiplication relations, computes the character averages, and checks ranks independently by Gaussian elimination and traces. It rejects the claim that the standard isotypic projector has rank two. This is one specified representation of $S_3$, not a computational classification of all finite group representations.\n\\takeaway{For a nonabelian symmetry, distinguish irreducible dimension from multiplicity. A central character projector retains both.}","\\vpage{30}{Rotate whole packets by a polar factor}{AN EXACT LOCAL CHART AND ITS FAILURE}{Chapter VII: Theorem II.6.5, Theorem II.6.6.}\nOn $\\HH=\\CC^2$, put $P_0=\\diag(1,0)$, $P_1=I-P_0$, and\n\\[\nR=\\mat{3/5&-4/5","\\vpage{30}{Rotate whole packets by a polar factor}{AN EXACT LOCAL CHART AND ITS FAILURE}{Chapter VII: Theorem II.6.5, Theorem II.6.6.}\nOn $\\HH=\\CC^2$, put $P_0=\\diag(1,0)$, $P_1=I-P_0$, and\n\\[\nR=\\mat{3/5&-4/5\\\\4/5&3/5},\\qquad Q_a=RP_aR^*,\n\\qquad S=\\sum_{a=0}^1Q_aP_a=\\frac35R.\n\\]\nThe exact Gram matrices give\n\\[\n(S-I)^*(S-I)=\\frac{16}{25}I,\\quad S^*S=\\frac9{25}I,\n\\quad\\norm{S-I}=\\frac45<1.\n\\]\nConsequently the unitary polar factor is $S(S^*S)^{-1/2}=R$, and $Q_aR=RP_a$ for both complete packets.\n\\xfig{polar}\n\\head{The norm condition selects a chart}\nIf the target rank-one packet is orthogonal to the initial one, the two labels are exchanged and $S=0$. Its polar inverse square root is undefined. This is a failure of this direct local construction, although a unitary carrying one packet system to the other still exists. The strict norm assumption is sufficient, not a global existence criterion for all packet rotations.","4/5&3/5},\\qquad Q_a=RP_aR^*,\n\\qquad S=\\sum_{a=0}^1Q_aP_a=\\frac35R.\n\\]\nThe exact Gram matrices give\n\\[\n(S-I)^*(S-I)=\\frac{16}{25}I,\\quad S^*S=\\frac9{25}I,\n\\quad\\norm{S-I}=\\frac45<1.\n\\]\nConsequently the unitary polar factor is $S(S^*S)^{-1/2}=R$, and $Q_aR=RP_a$ for both complete packets.\n\\xfig{polar}\n\\head{The norm condition selects a chart}\nIf the target rank-one packet is orthogonal to the initial one, the two labels are exchanged and $S=0$. Its polar inverse square root is undefined. This is a failure of this direct local construction, although a unitary carrying one packet system to the other still exists. The strict norm assumption is sufficient, not a global existence criterion for all packet rotations.","m_{a=0}^1Q_aP_a=\\frac35R.\n\\]\nThe exact Gram matrices give\n\\[\n(S-I)^*(S-I)=\\frac{16}{25}I,\\quad S^*S=\\frac9{25}I,\n\\quad\\norm{S-I}=\\frac45<1.\n\\]\nConsequently the unitary polar factor is $S(S^*S)^{-1/2}=R$, and $Q_aR=RP_a$ for both complete packets.\n\\xfig{polar}\n\\head{The norm condition selects a chart}\nIf the target rank-one packet is orthogonal to the initial one, the two labels are exchanged and $S=0$. Its polar inverse square root is undefined. This is a failure of this direct local construction, although a unitary carrying one packet system to the other still exists. The strict norm assumption is sufficient, not a global existence criterion for all packet rotations.","Short steps can be composed before reaching a distant endpoint. For example, the rational step $R_1=\\mat{4/5&-3/5\\\\3/5&4/5}$ obeys\n\\[\nR_1^2=\\mat{7/25&-24/25","3/5&4/5}$ obeys\n\\[\nR_1^2=\\mat{7/25&-24/25\\\\24/25&7/25}.\n\\]\nThe multiplication is exact and each step has a valid local overlap. The convergence theorem has a separate analytic contract: a twice continuously differentiable complete projector path on a compact time interval, sufficiently fine partitions, and a constant depending on the path in an $O(|\\pi|)$ estimate in operator norm.\n\\head{Evidence boundary}\nThe retained run checks the Gram identities, polar factor, intertwining, and product of two rational steps. It also checks that the overlap has rank zero when the endpoint packets are orthogonal. It does not certify the analytic convergence theorem by sampling a path.\n\\takeaway{A local polar chart can fail while the subspaces remain regular. Us","24/25&7/25}.\n\\]\nThe multiplication is exact and each step has a valid local overlap. The convergence theorem has a separate analytic contract: a twice continuously differentiable complete projector path on a compact time interval, sufficiently fine partitions, and a constant depending on the path in an $O(|\\pi|)$ estimate in operator norm.\n\\head{Evidence boundary}\nThe retained run checks the Gram identities, polar factor, intertwining, and product of two rational steps. It also checks that the overlap has rank zero when the endpoint packets are orthogonal. It does not certify the analytic convergence theorem by sampling a path.\n\\takeaway{A local polar chart can fail while the subspaces remain regular. Use the stated chart hypothesis and the separate convergence theorem when composing small rotations.}","\\vpage{31}{Coherence order survives a covariant channel}{CHANNEL, CHOI, AND GENERATOR CHECKS}{Chapter VII: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14.}\nTake $S=Z=\\diag(1,-1)$ on $\\CC^2$. Under conjugation $X\\mapsto ZXZ$, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are $\\Lcal_0(X)=(X+ZXZ)/2$ and $\\Lcal_1(X)=(X-ZXZ)/2$.","ence order survives a covariant channel}{CHANNEL, CHOI, AND GENERATOR CHECKS}{Chapter VII: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14.}\nTake $S=Z=\\diag(1,-1)$ on $\\CC^2$. Under conjugation $X\\mapsto ZXZ$, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are $\\Lcal_0(X)=(X+ZXZ)/2$ and $\\Lcal_1(X)=(X-ZXZ)/2$.","Choose amplitude damping with $K_0=\\diag(1,4/5)$ and $K_1=(3/5)\\ket0\\bra1$. The channel is trace preserving, population flows from state one to state zero, and each coherence contracts by $4/5$.\n\\xfig{choi}\nWith the unnormalized Choi convention $J=\\sum_{a,b}\\Phi(\\ket a\\bra b)\\otimes\\ket a\\bra b$ and basis $(00,01,10,11)$, the exact matrix is\n\\[\nJ=\\mat{1&0&0&4/5","Choose amplitude damping with $K_0=\\diag(1,4/5)$ and $K_1=(3/5)\\ket0\\bra1$. The channel is trace preserving, population flows from state one to state zero, and each coherence contracts by $4/5$.\n\\xfig{choi}\nWith the unnormalized Choi convention $J=\\sum_{a,b}\\Phi(\\ket a\\bra b)\\otimes\\ket a\\bra b$ and basis $(00,01,10,11)$, the exact matrix is\n\\[\nJ=\\mat{1&0&0&4/5\\\\0&9/25&0&0\\\\0&0&0&0\\\\4/5&0&0&16/25},\n\\qquad [J,Z\\otimes Z]=0.\n\\]\nThe $00,11$ block is positive and has rank one. The other nonzero eigenvalue is $9/25$. The retained check also verifies that every principal minor is nonnegative.\n\\head{Symmetry in continuous time has its own hypotheses}\nFor the Lindblad generator with Hamiltonian $H=\\diag(0,2)$ and jump $L=(3/5)\\ket0\\bra1$, $H$ has charge zero and $L$ has charge one. Each matrix unit stays within its coherence order under the generator. Replacing $H$ by $X$ breaks that conclusion: the generator then sends a diagonal matrix to off-diagonal entries.","4/5&0&0&16/25},\n\\qquad [J,Z\\otimes Z]=0.\n\\]\nThe $00,11$ block is positive and has rank one. The other nonzero eigenvalue is $9/25$. The retained check also verifies that every principal minor is nonnegative.\n\\head{Symmetry in continuous time has its own hypotheses}\nFor the Lindblad generator with Hamiltonian $H=\\diag(0,2)$ and jump $L=(3/5)\\ket0\\bra1$, $H$ has charge zero and $L$ has charge one. Each matrix unit stays within its coherence order under the generator. Replacing $H$ by $X$ breaks that conclusion: the generator then sends a diagonal matrix to off-diagonal entries.","rank one. The other nonzero eigenvalue is $9/25$. The retained check also verifies that every principal minor is nonnegative.\n\\head{Symmetry in continuous time has its own hypotheses}\nFor the Lindblad generator with Hamiltonian $H=\\diag(0,2)$ and jump $L=(3/5)\\ket0\\bra1$, $H$ has charge zero and $L$ has charge one. Each matrix unit stays within its coherence order under the generator. Replacing $H$ by $X$ breaks that conclusion: the generator then sends a diagonal matrix to off-diagonal entries.","ne. The other nonzero eigenvalue is $9/25$. The retained check also verifies that every principal minor is nonnegative.\n\\head{Symmetry in continuous time has its own hypotheses}\nFor the Lindblad generator with Hamiltonian $H=\\diag(0,2)$ and jump $L=(3/5)\\ket0\\bra1$, $H$ has charge zero and $L$ has charge one. Each matrix unit stays within its coherence order under the generator. Replacing $H$ by $X$ breaks that conclusion: the generator then sends a diagonal matrix to off-diagonal entries.","\\vpage{32}{Null Gram directions are removed before recovery}{A TOTAL CHANNEL FROM SINGULAR ERROR DATA}{Chapter VII: Theorem II.8.3, Theorem II.8.4.}\nLet $\\HH=\\CC^3$, with nonzero code projector $P=\\diag(1,1,0)$. On the code use the error family $E_1=(3/5)P$, $E_2=(4/5)P$, and $E_3=0$. Its Knill--Laflamme Gram matrix is\n\\[\nc=\\frac1{25}\\mat{9&12&0","\\vpage{32}{Null Gram directions are removed before recovery}{A TOTAL CHANNEL FROM SINGULAR ERROR DATA}{Chapter VII: Theorem II.8.3, Theorem II.8.4.}\nLet $\\HH=\\CC^3$, with nonzero code projector $P=\\diag(1,1,0)$. On the code use the error family $E_1=(3/5)P$, $E_2=(4/5)P$, and $E_3=0$. Its Knill--Laflamme Gram matrix is\n\\[\nc=\\frac1{25}\\mat{9&12&0\\\\12&16&0\\\\0&0&0},\n\\qquad PE_a^*E_bP=c_{ab}P.\n\\]\nThe errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives\n\\[\nW=\\mat{3/5&4/5&0","0&0&0},\n\\qquad PE_a^*E_bP=c_{ab}P.\n\\]\nThe errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives\n\\[\nW=\\mat{3/5&4/5&0\\\\-4/5&3/5&0\\\\0&0&1},\n\\quad WcW^*=\\diag(1,0,0),\n\\quad(F_1,F_2,F_3)=(P,0,0).\n\\]\n\\xfig{gram}\n\\head{Normalize only positive eigenvalues}\nThe positive branch has $V_1=P$ as an isometry \\emph{on the code}, and syndrome support $Q=P$. No factor $\\lambda^{-1/2}$ is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.","0&0&1},\n\\quad WcW^*=\\diag(1,0,0),\n\\quad(F_1,F_2,F_3)=(P,0,0).\n\\]\n\\xfig{gram}\n\\head{Normalize only positive eigenvalues}\nThe positive branch has $V_1=P$ as an isometry \\emph{on the code}, and syndrome support $Q=P$. No factor $\\lambda^{-1/2}$ is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.","The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators\n\\[\nR_1=P,\\qquad R_2=\\ket0\\bra2,\n\\qquad R_1^*R_1+R_2^*R_2=I_3.\n\\]\nIt acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery returns every code matrix unit, including both off-diagonal units, exactly. It need not preserve states on the unused complement.\n\\head{Exceptional cases and scope}\nThis construction assumes a nonzero code so that a complement replacement state exists. The case of a zero code is vacuous and has its own formal statement in the saved ledger. The retained exact run verifies all nine Gram compressions, the diagonalization, both null branches, total channel normalization, and all four code matrix u","\\vpage{33}{An approximate defect has a precise norm meaning}{THE OPTIMAL SCALAR FOR TWO SPECTRAL POINTS}{Chapter VII: Definition II.8.6, Proposition II.8.7.}\nOn a two-dimensional code, consider the overlap compression\n\\[\nA=PE_a^*E_bP=\\diag(11/10,9/10).\n\\]\nFor instance, within the code take $E_a=I$ and $E_b=A$. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,","\\vpage{33}{An approximate defect has a precise norm meaning}{THE OPTIMAL SCALAR FOR TWO SPECTRAL POINTS}{Chapter VII: Definition II.8.6, Proposition II.8.7.}\nOn a two-dimensional code, consider the overlap compression\n\\[\nA=PE_a^*E_bP=\\diag(11/10,9/10).\n\\]\nFor instance, within the code take $E_a=I$ and $E_b=A$. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bo","\\vpage{33}{An approximate defect has a precise norm meaning}{THE OPTIMAL SCALAR FOR TWO SPECTRAL POINTS}{Chapter VII: Definition II.8.6, Proposition II.8.7.}\nOn a two-dimensional code, consider the overlap compression\n\\[\nA=PE_a^*E_bP=\\diag(11/10,9/10).\n\\]\nFor instance, within the code take $E_a=I$ and $E_b=A$. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attai","ate defect has a precise norm meaning}{THE OPTIMAL SCALAR FOR TWO SPECTRAL POINTS}{Chapter VII: Definition II.8.6, Proposition II.8.7.}\nOn a two-dimensional code, consider the overlap compression\n\\[\nA=PE_a^*E_bP=\\diag(11/10,9/10).\n\\]\nFor instance, within the code take $E_a=I$ and $E_b=A$. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also ente…","inition II.8.6, Proposition II.8.7.}\nOn a two-dimensional code, consider the overlap compression\n\\[\nA=PE_a^*E_bP=\\diag(11/10,9/10).\n\\]\nFor instance, within the code take $E_a=I$ and $E_b=A$. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full","On a two-dimensional code, consider the overlap compression\n\\[\nA=PE_a^*E_bP=\\diag(11/10,9/10).\n\\]\nFor instance, within the code take $E_a=I$ and $E_b=A$. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL…","declares an error family, not by itself a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","self a normalized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","malized noise channel. The optimal scalar compression has\n\\[\n\\inf_{c\\in\\CC}\\norm{A-cI}=\\frac1{10},\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.",",\\qquad c=1\\ \\text{attains it}.\n\\]\nIndeed, for every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","or every complex $c$, the triangle inequality gives\n\\[\n\\frac15=\\left|\\frac{11}{10}-\\frac9{10}\\right|\n\\leq\\left|\\frac{11}{10}-c\\right|+\\left|\\frac9{10}-c\\right|.\n\\]\nThus the larger of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","er of the two distances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","istances is at least $1/10$, while both distances equal $1/10$ at $c=1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","1$.\n\\xfig{defect}\n\\head{What one pair contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","contributes to the family defect}\nFor unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","unit code vectors $\\psi,\\phi$, this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","this pair therefore satisfies\n\\[\n\\left|\\langle E_a\\psi,E_b\\phi\\rangle-\\langle\\psi,\\phi\\rangle\\right|\n\\leq\\frac1{10}.\n\\]\nThe bound is attained on either coordinate vector. For the \\emph{whole} declared family $\\{I,A\\}$, the $A^*A$ compression also enters the maximum: its diagonal entries are $121/100,81/100$, giving pair defect $1/5$. Hence the full family value is $\\varepsilon_{\\rm KL}=1/5$, not $1/10$.","This distinction prevents a tempting shortcut from one favorable overlap to an approximate correction claim. The quantity measures how close all overlap operators are to scalar action on the code. A recovery fidelity bound requires a declared normalized channel, a metric, and a separate continuity result.\n\\head{What is checked exactly}\nFor $\\varepsilon\\in\\{0,1/10,1/4,1\\}$ the script checks $(A-I)^*(A-I)=\\varepsilon^2I$ for $A=\\diag(1+\\varepsilon,1-\\varepsilon)$ and several rational scalar controls. The triangle argument proves the lower bound for every complex scalar. Checking a finite list of candidates would not establish that conclusion. The family maximum above is also recorded separately.\n\\takeaway{An overlap norm has a useful operational interpretation. Evaluate every error pair, then keep that statement separate from an unproved recovery fidelity estimate.}","tempting shortcut from one favorable overlap to an approximate correction claim. The quantity measures how close all overlap operators are to scalar action on the code. A recovery fidelity bound requires a declared normalized channel, a metric, and a separate continuity result.\n\\head{What is checked exactly}\nFor $\\varepsilon\\in\\{0,1/10,1/4,1\\}$ the script checks $(A-I)^*(A-I)=\\varepsilon^2I$ for $A=\\diag(1+\\varepsilon,1-\\varepsilon)$ and several rational scalar controls. The triangle argument proves the lower bound for every complex scalar. Checking a finite list of candidates would not establish that conclusion. The family maximum above is also recorded separately.\n\\takeaway{An overlap norm has a useful operational interpretation. Evaluate every error pair, then keep that statement separate from an unproved recovery fidelity estimate.}","\\vpage{34}{Control the hidden spectral response}{SIGN, STABILITY, AND A CONVERGENT EXPANSION}{Chapter VII: Theorem II.9.4, Proposition II.9.5, Theorem II.9.12.}\nRetain one coordinate in $H=\\mat{0&1\\\\1&2}$, so $A=0$, $B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|","1&2}$, so $A=0$, $B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^","1&2}$, so $A=0$, $B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\del","1&2}$, so $A=0$, $B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\wi","1&2}$, so $A=0$, $B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds","1&2}$, so $A=0$, $B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference","1&2}$, so $A=0$, $B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires unifo","$B=1$, and $D=2$. In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four t…","In its elimination domain, $\\Sigma(z)=1/(z-2)$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact re…",")$. At $z=3+\\ii$,\n\\[\n\\Sigma(3+\\ii)=\\frac{1-\\ii}{2},\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric…","\\qquad\n\\Im\\Sigma(z)=-(\\Im z)\\left|(z-2)^{-1}\\right|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric series controls the infinite tail. The retained ch…","ht|^2\\leq0.\n\\]\nThe sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric series controls the infinite tail. The retained check verifies that remainder, the complex sign id…","The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix.\n\\xfig{stability}\n\\head{A stability estimate with a common spectral gap}\nCompare $(B,D)=(1,2)$ with $(\\widetilde B,\\widetilde D)=(9/8,17/8)$ at $z=3$. Both hidden spectral distances are at least $\\delta=3/4$, and both couplings are bounded by $g=9/8$. Then\n\\[\n|\\Sigma-\\widetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric series controls the infinite tail. The retained check verifies that remainder, the complex sign identity, and the…","idetilde\\Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric series controls the infinite tail. The retained check verifies that remainder, the complex sign identity, and the identities for the projected resolvent and bare truncation at four specified real spectral parameters.","Sigma|=\\left|1-\\frac{81}{56}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric series controls the infinite tail. The retained check verifies that remainder, the complex sign identity, and the identities for the projected resolvent and bare truncation at four specified real spectral parameters.","}\\right|=\\frac{25}{56},\n\\qquad\n\\frac{2g}{\\delta}|B-\\widetilde B|\n+\\frac{g^2}{\\delta^2}|D-\\widetilde D|=\\frac{21}{32}.\n\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric series controls the infinite tail. The retained check verifies that remainder, the complex sign identity, and the identities for the projected resolvent and bare truncation at four specified real spectral parameters.","\\]\nThe theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings.\n\\head{A series near zero energy and its exact remainder}\nAt $z=1/2$, $|z|\\,|D^{-1}|=1/4<1$, so\n\\[\n\\Sigma(z)=-\\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}.\n\\]\nAfter the first four terms, the exact remainder has magnitude $1/384$. The geometric series controls the infinite tail. The retained check verifies that remainder, the complex sign identity, and the identities for the projected resolvent and bare truncation at four specified real spectral parameters.","At $z=2$ the hidden inverse fails. At an eigenvalue of the full Hamiltonian, the retained Feshbach inverse fails. Neither point belongs to the resolvent identity's domain. Analytic continuation of a scalar expression does not erase its theorem hypotheses.\n\\takeaway{A uniform effective response requires a uniform gap and coupling bound. Exact small examples show how those hypotheses enter each estimate.}","\\vpage{35}{Memory positivity is a Gram identity}{TWO HIDDEN FREQUENCIES AND A TEST OF LOCAL CLOSURE}{Chapter VII: Theorem II.9.9, Theorem II.9.11.}\nUse retained space $\\CC$ and hidden space $\\CC^2$, with $A=0$, $B=(3/5,4/5)$, and $D=\\diag(1,2)$. The full Hamiltonian is $\\mat{0&3/5&4/5","\\vpage{35}{Memory positivity is a Gram identity}{TWO HIDDEN FREQUENCIES AND A TEST OF LOCAL CLOSURE}{Chapter VII: Theorem II.9.9, Theorem II.9.11.}\nUse retained space $\\CC$ and hidden space $\\CC^2$, with $A=0$, $B=(3/5,4/5)$, and $D=\\diag(1,2)$. The full Hamiltonian is $\\mat{0&3/5&4/5\\\\3/5&1&0\\\\4/5&0&2}$. Its memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the","4/5&0&2}$. Its memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\","4/5&0&2}$. Its memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWit","4/5&0&2}$. Its memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot","4/5&0&2}$. Its memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$.","4/5&0&2}$. Its memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for","4/5&0&2}$. Its memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initia","ts memory kernel is\n\\[\nK(t)=\\frac9{25}e^{-\\ii t}+\\frac{16}{25}e^{-2\\ii t},\n\\qquad K(0)=1.\n\\]\nEach oscillatory coefficient is positive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, e…","ive although $K(t)$ need not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, equality of these accelerations forces $BB^*=0$ and hence $B=0$.","ed not be real or pointwise nonnegative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, equality of these accelerations forces $BB^*=0$ and hence $B=0$.","egative. At $t=\\pi/2$, it equals $-16/25-9\\ii/25$.\n\\xfig{memory_positive}\n\\head{Positive definite means a matrix test at arbitrary times}\nFor retained scalars $v_j$ and real times $t_j$, the structural identity is\n\\[\n\\sum_{j,k}\\overline v_jK(t_j-t_k)v_k\n=\\left\\|\\sum_k e^{\\ii t_kD}B^*v_k\\right\\|^2\\geq0.\n\\]\nThe exact worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, equality of these accelerations forces $BB^*=0$ and hence $B=0$.","ct worked check specializes to $t=(0,\\pi/2,\\pi)$ and $v=(1,2,-1)$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, equality of these accelerations forces $BB^*=0$ and hence $B=0$.",")$. Its phases lie in $\\QQ(\\ii)$, so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, equality of these accelerations forces $BB^*=0$ and hence $B=0$.","positivity for all choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, equality of these accelerations forces $BB^*=0$ and hence $B=0$.","choices of times and vectors.\n\\head{Why this does not close to a bare local equation}\nWith hidden initial state zero, the full system has $\\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)$. The bare retained equation $\\ii\\dot x=Ax$ would instead give zero acceleration. Therefore exact local closure for every retained initial state fails. More generally, equality of these accelerations forces $BB^*=0$ and hence $B=0$.","A nonzero initial hidden vector also supplies the forcing term $Be^{-\\ii tD}y(0)$. It is additional data, not part of the convolution kernel. The retained checks cover the displayed kernel values, exact Gram factorization, and acceleration defect. The integral equation follows from the written elimination argument.\n\\takeaway{Memory is positive definite through a Gram factorization, not through pointwise signs. Exact bare local closure requires the coupling to vanish.}","\\vpage{36}{A positive transfer determines its Hamiltonian}{STRICT POSITIVITY AND THE GROUND GAP}{Chapter VII: Theorem II.10.2, Proposition II.10.5.}\nOn $\\HH=\\CC^3$, take the strictly positive contraction\n\\[\nT=\\diag(1,1/2,1/4),\\qquad H=-\\log T=\\diag(0,\\log2,2\\log2).\n\\]\nAt unit Euclidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to","\\vpage{36}{A positive transfer determines its Hamiltonian}{STRICT POSITIVITY AND THE GROUND GAP}{Chapter VII: Theorem II.10.2, Proposition II.10.5.}\nOn $\\HH=\\CC^3$, take the strictly positive contraction\n\\[\nT=\\diag(1,1/2,1/4),\\qquad H=-\\log T=\\diag(0,\\log2,2\\log2).\n\\]\nAt unit Euclidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which","\\vpage{36}{A positive transfer determines its Hamiltonian}{STRICT POSITIVITY AND THE GROUND GAP}{Chapter VII: Theorem II.10.2, Proposition II.10.5.}\nOn $\\HH=\\CC^3$, take the strictly positive contraction\n\\[\nT=\\diag(1,1/2,1/4),\\qquad H=-\\log T=\\diag(0,\\log2,2\\log2).\n\\]\nAt unit Euclidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","sitive transfer determines its Hamiltonian}{STRICT POSITIVITY AND THE GROUND GAP}{Chapter VII: Theorem II.10.2, Proposition II.10.5.}\nOn $\\HH=\\CC^3$, take the strictly positive contraction\n\\[\nT=\\diag(1,1/2,1/4),\\qquad H=-\\log T=\\diag(0,\\log2,2\\log2).\n\\]\nAt unit Euclidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","sfer determines its Hamiltonian}{STRICT POSITIVITY AND THE GROUND GAP}{Chapter VII: Theorem II.10.2, Proposition II.10.5.}\nOn $\\HH=\\CC^3$, take the strictly positive contraction\n\\[\nT=\\diag(1,1/2,1/4),\\qquad H=-\\log T=\\diag(0,\\log2,2\\log2).\n\\]\nAt unit Euclidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","I.10.2, Proposition II.10.5.}\nOn $\\HH=\\CC^3$, take the strictly positive contraction\n\\[\nT=\\diag(1,1/2,1/4),\\qquad H=-\\log T=\\diag(0,\\log2,2\\log2).\n\\]\nAt unit Euclidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","HH=\\CC^3$, take the strictly positive contraction\n\\[\nT=\\diag(1,1/2,1/4),\\qquad H=-\\log T=\\diag(0,\\log2,2\\log2).\n\\]\nAt unit Euclidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","lidean time, $T=e^{-H}$. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","positive spectral values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","al values. At integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","integer time $n\\geq0$,\n\\[\nT^n=\\diag(1,2^{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","{-n},4^{-n}),\\qquad\nP_0=\\diag(1,0,0),\\qquad\\norm{T^n-P_0}=2^{-n}.\n\\]\nThe gap $\\Delta=\\log2$ determines the rate $e^{-n\\Delta}$ exactly in this example.\n\\xfig{transfer_ground}\n\\head{Strict positivity is an essential finite hypothesis}\nFor $T'=\\diag(1,0)$, the operator is positive semidefinite, but there is no finite self-adjoint $H'$ on all of $\\CC^2$ with $T'=e^{-H'}$: every finite eigenvalue of $H'$ exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the logarithm is defined.","For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: $P_0$ selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is $E_0\\ne0$, the normalized transfer is $e^{-t(H-E_0I)}$ before taking the limit.\n\\head{Positivity is retained at several Euclidean times}\nFor arbitrary vectors $u_j$ and nonnegative integer times $n_j$, the reflected kernel obeys\n\\[\n\\sum_{j,k}\\langle u_j,T^{n_j+n_k}u_k\\rangle\n=\\left\\|\\sum_jT^{n_j}u_j\\right\\|^2.\n\\]\nThis is a finite Gram factorization. The retained run checks the integer semigroup and diagonal ground error for $n=0,\\ldots,12$, and rejects a logarithm on the full space for the singular control. The written spectral argument justifies the logarithms and extends the functional calculus statem","For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: $P_0$ selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is $E_0\\ne0$, the normalized transfer is $e^{-t(H-E_0I)}$ before taking the limit.\n\\head{Positivity is retained at several Euclidean times}\nFor arbitrary vectors $u_j$ and nonnegative integer times $n_j$, the reflected kernel obeys\n\\[\n\\sum_{j,k}\\langle u_j,T^{n_j+n_k}u_k\\rangle\n=\\left\\|\\sum_jT^{n_j}u_j\\right\\|^2.\n\\]\nThis is a finite Gram factorization. The retained run checks the integer semigroup and diagonal ground error for $n=0,\\ldots,12$, and rejects a logarithm on the full space for the singular control. The written spectral argument justifies the logarithms and extends the functional calculus statement to all times.\n\\takeaway{A positive definite transfer can be logarithmically reconstructed. Gro","For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: $P_0$ selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is $E_0\\ne0$, the normalized transfer is $e^{-t(H-E_0I)}$ before taking the limit.\n\\head{Positivity is retained at several Euclidean times}\nFor arbitrary vectors $u_j$ and nonnegative integer times $n_j$, the reflected kernel obeys\n\\[\n\\sum_{j,k}\\langle u_j,T^{n_j+n_k}u_k\\rangle\n=\\left\\|\\sum_jT^{n_j}u_j\\right\\|^2.\n\\]\nThis is a finite Gram factorization. The retained run checks the integer semigroup and diagonal ground error for $n=0,\\ldots,12$, and rejects a logarithm on the full space for the singular control. The written spectral argument justifies the logarithms and extends the functional calculus statement to all times.\n\\takeaway{A positive definite transfer can be logarithmically reconstructed. Ground convergence needs a declared normalizatio","trivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: $P_0$ selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is $E_0\\ne0$, the normalized transfer is $e^{-t(H-E_0I)}$ before taking the limit.\n\\head{Positivity is retained at several Euclidean times}\nFor arbitrary vectors $u_j$ and nonnegative integer times $n_j$, the reflected kernel obeys\n\\[\n\\sum_{j,k}\\langle u_j,T^{n_j+n_k}u_k\\rangle\n=\\left\\|\\sum_jT^{n_j}u_j\\right\\|^2.\n\\]\nThis is a finite Gram factorization. The retained run checks the integer semigroup and diagonal ground error for $n=0,\\ldots,12$, and rejects a logarithm on the full space for the singular control. The written spectral argument justifies the logarithms and extends the functional calculus statement to all times.\n\\takeaway{A positive definite transfer can be logarithmically reconstructed. Ground convergence needs a declared normalization and gap, while continuum reconstruction still needs its own structure.}","xponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: $P_0$ selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is $E_0\\ne0$, the normalized transfer is $e^{-t(H-E_0I)}$ before taking the limit.\n\\head{Positivity is retained at several Euclidean times}\nFor arbitrary vectors $u_j$ and nonnegative integer times $n_j$, the reflected kernel obeys\n\\[\n\\sum_{j,k}\\langle u_j,T^{n_j+n_k}u_k\\rangle\n=\\left\\|\\sum_jT^{n_j}u_j\\right\\|^2.\n\\]\nThis is a finite Gram factorization. The retained run checks the integer semigroup and diagonal ground error for $n=0,\\ldots,12$, and rejects a logarithm on the full space for the singular control. The written spectral argument justifies the logarithms and extends the functional calculus statement to all times.\n\\takeaway{A positive definite transfer can be logarithmically reconstructed. Ground convergence needs a declared normalization and gap, while continuum reconstruction still needs its own structure.}","tive lower bound on the gap. Ground multiplicity does not cause a problem: $P_0$ selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is $E_0\\ne0$, the normalized transfer is $e^{-t(H-E_0I)}$ before taking the limit.\n\\head{Positivity is retained at several Euclidean times}\nFor arbitrary vectors $u_j$ and nonnegative integer times $n_j$, the reflected kernel obeys\n\\[\n\\sum_{j,k}\\langle u_j,T^{n_j+n_k}u_k\\rangle\n=\\left\\|\\sum_jT^{n_j}u_j\\right\\|^2.\n\\]\nThis is a finite Gram factorization. The retained run checks the integer semigroup and diagonal ground error for $n=0,\\ldots,12$, and rejects a logarithm on the full space for the singular control. The written spectral argument justifies the logarithms and extends the functional calculus statement to all times.\n\\takeaway{A positive definite transfer can be logarithmically reconstructed. Ground convergence needs a declared normalization and gap, while continuum reconstruction still needs its own structure.}","nsfer is $e^{-t(H-E_0I)}$ before taking the limit.\n\\head{Positivity is retained at several Euclidean times}\nFor arbitrary vectors $u_j$ and nonnegative integer times $n_j$, the reflected kernel obeys\n\\[\n\\sum_{j,k}\\langle u_j,T^{n_j+n_k}u_k\\rangle\n=\\left\\|\\sum_jT^{n_j}u_j\\right\\|^2.\n\\]\nThis is a finite Gram factorization. The retained run checks the integer semigroup and diagonal ground error for $n=0,\\ldots,12$, and rejects a logarithm on the full space for the singular control. The written spectral argument justifies the logarithms and extends the functional calculus statement to all times.\n\\takeaway{A positive definite transfer can be logarithmically reconstructed. Ground convergence needs a declared normalization and gap, while continuum reconstruction still needs its own structure.}","\\vpage{37}{The dimension window has an order of limits}{FINITE HEAT DATA AND THE QUANTIFIERS}{Chapter VII: Theorem II.11.5, Corollary II.11.8.}\nFor the unit circle grid with $N$ sites, the scaled eigenvalues are $\\lambda_N(k)=4N^2\\sin^2(\\pi k/N)$. Put $Z_N(t)=\\sum_ke^{-t\\lambda_N(k)}$ and $d_N(t)=2t\\sum_k\\lambda_N(k)e^{-t\\lambda_N(k)}/Z_N(t)$.","At the single time $t=\\log2/16$, two small grids admit exact rational heat weights:\n\\[\n\\begin{array}{c|c|c|c|c}\nN&\\text{spectrum}&Z_N&\\sum\\lambda e^{-t\\lambda}&d_N/\\log2","At the single time $t=\\log2/16$, two small grids admit exact rational heat weights:\n\\[\n\\begin{array}{c|c|c|c|c}\nN&\\text{spectrum}&Z_N&\\sum\\lambda e^{-t\\lambda}&d_N/\\log2\\\\\\hline\n2&(0,16)&3/2&8&2/3\\\\\n4&(0,32,32,64)&25/16&20&8/5\n\\end{array}\n\\]\nThese are unnormalized traces. Normalizing by $N$ changes $Z_N$ but leaves the logarithmic derivative at that fixed grid unchanged.\n\\xfig{heat_limits}\n\\head{The theorem uses two separate choices}\nFor each $\\varepsilon>0$, first choose a fixed positive time $t_\\varepsilon$ so that the continuum torus dimension differs from one by less than $\\varepsilon/2$. Then choose $N_\\varepsilon$ so that every $N\\geq N_\\varepsilon$ differs from that continuum value by less than $\\varepsilon/2$ at the \\emph{same time}. This proves one controlled intermediate time for all sufficiently fine grids.","4&(0,32,32,64)&25/16&20&8/5\n\\end{array}\n\\]\nThese are unnormalized traces. Normalizing by $N$ changes $Z_N$ but leaves the logarithmic derivative at that fixed grid unchanged.\n\\xfig{heat_limits}\n\\head{The theorem uses two separate choices}\nFor each $\\varepsilon>0$, first choose a fixed positive time $t_\\varepsilon$ so that the continuum torus dimension differs from one by less than $\\varepsilon/2$. Then choose $N_\\varepsilon$ so that every $N\\geq N_\\varepsilon$ differs from that continuum value by less than $\\varepsilon/2$ at the \\emph{same time}. This proves one controlled intermediate time for all sufficiently fine grids.","&25/16&20&8/5\n\\end{array}\n\\]\nThese are unnormalized traces. Normalizing by $N$ changes $Z_N$ but leaves the logarithmic derivative at that fixed grid unchanged.\n\\xfig{heat_limits}\n\\head{The theorem uses two separate choices}\nFor each $\\varepsilon>0$, first choose a fixed positive time $t_\\varepsilon$ so that the continuum torus dimension differs from one by less than $\\varepsilon/2$. Then choose $N_\\varepsilon$ so that every $N\\geq N_\\varepsilon$ differs from that continuum value by less than $\\varepsilon/2$ at the \\emph{same time}. This proves one controlled intermediate time for all sufficiently fine grids.","hese are unnormalized traces. Normalizing by $N$ changes $Z_N$ but leaves the logarithmic derivative at that fixed grid unchanged.\n\\xfig{heat_limits}\n\\head{The theorem uses two separate choices}\nFor each $\\varepsilon>0$, first choose a fixed positive time $t_\\varepsilon$ so that the continuum torus dimension differs from one by less than $\\varepsilon/2$. Then choose $N_\\varepsilon$ so that every $N\\geq N_\\varepsilon$ differs from that continuum value by less than $\\varepsilon/2$ at the \\emph{same time}. This proves one controlled intermediate time for all sufficiently fine grids.","aves the logarithmic derivative at that fixed grid unchanged.\n\\xfig{heat_limits}\n\\head{The theorem uses two separate choices}\nFor each $\\varepsilon>0$, first choose a fixed positive time $t_\\varepsilon$ so that the continuum torus dimension differs from one by less than $\\varepsilon/2$. Then choose $N_\\varepsilon$ so that every $N\\geq N_\\varepsilon$ differs from that continuum value by less than $\\varepsilon/2$ at the \\emph{same time}. This proves one controlled intermediate time for all sufficiently fine grids.","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle wit","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window res","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves con","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves convergence after fix","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves convergence after fixing a suitable time.","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves convergence after fixing a suitable time. The exact table is a worked calculation","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves convergence after fixing a suitable time. The exact table is a worked calculation, not its proof.","It does not assert a uniform interval width from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves convergence after fixing a suitable time. The exact table is a worked calculation, not its proof.}","from that argument, a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves convergence after fixing a suitable time. The exact table is a worked calculation, not its proof.}","a moving time $t_N$ with no extra estimates, or a dimension plateau at every scale. At fixed finite $N$, $d_N(t)\\to0$ as $t\\downarrow0$ and as $t\\to\\infty$. By contrast, first taking $N\\to\\infty$ at positive $t$ and then taking $t\\downarrow0$ yields the continuum dimension.\n\\head{Where the infinite tail is controlled}\nAt a fixed time, the proof bounds centered modes below by $16k^2$ and above by $4\\pi^2k^2$. The summable Gaussian dominators control both the heat trace and its first moment. A finite grid table supplies neither those tails nor the quantifiers. The retained check verifies the two displayed spectra, rational weights, traces, and dimension coefficients. It also distinguishes the unscaled cycle with four sites from the scaled unit circle grid.\n\\takeaway{The strengthened window result proves convergence after fixing a suitable time. The exact table is a worked calculation, not its proof.}","\\vpage{38}{Transport constraints without hiding closure}{THE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVEL}{Chapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6.}\nOn $\\CC^3=\\CC\\ket0\\oplus\\CC^2$, use $C_a=0\\oplus\\sigma_a$ for $a=x,y,z$. The common kernel is $\\CC\\ket0$, and the full algebra closes:\n\\[\n[C_x,C_y]=2\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structu","\\vpage{38}{Transport constraints without hiding closure}{THE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVEL}{Chapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6.}\nOn $\\CC^3=\\CC\\ket0\\oplus\\CC^2$, use $C_a=0\\oplus\\sigma_a$ for $a=x,y,z$. The common kernel is $\\CC\\ket0$, and the full algebra closes:\n\\[\n[C_x,C_y]=2\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on","\\vpage{38}{Transport constraints without hiding closure}{THE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVEL}{Chapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6.}\nOn $\\CC^3=\\CC\\ket0\\oplus\\CC^2$, use $C_a=0\\oplus\\sigma_a$ for $a=x,y,z$. The common kernel is $\\CC\\ket0$, and the full algebra closes:\n\\[\n[C_x,C_y]=2\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fin","\\vpage{38}{Transport constraints without hiding closure}{THE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVEL}{Chapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6.}\nOn $\\CC^3=\\CC\\ket0\\oplus\\CC^2$, use $C_a=0\\oplus\\sigma_a$ for $a=x,y,z$. The common kernel is $\\CC\\ket0$, and the full algebra closes:\n\\[\n[C_x,C_y]=2\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate","\\vpage{38}{Transport constraints without hiding closure}{THE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVEL}{Chapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6.}\nOn $\\CC^3=\\CC\\ket0\\oplus\\CC^2$, use $C_a=0\\oplus\\sigma_a$ for $a=x,y,z$. The common kernel is $\\CC\\ket0$, and the full algebra closes:\n\\[\n[C_x,C_y]=2\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $","l is $\\CC\\ket0$, and the full algebra closes:\n\\[\n[C_x,C_y]=2\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same …","ull algebra closes:\n\\[\n[C_x,C_y]=2\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result f…","\\ii C_z,\\qquad[C_y,C_z]=2\\ii C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls …","C_x,\\qquad[C_z,C_x]=2\\ii C_y.\n\\]\nThese are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel.\n\\xfig{constraint_image}\n\\head{Exact transport holds on the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls these residuals and commutato…","the coarse image}\nEmbed by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls these residuals and commutator errors using the defect sizes and a common bound on the constraint norms. A nonzero anomaly remains an error that must be controlled.","by the coordinate isometry $R:\\CC^3\\to\\CC^4$ and set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls these residuals and commutator errors using the defect sizes and a common bound on the constraint norms. A nonzero anomaly remains an error that must be controlled.","d set $\\widetilde C_a=C_a\\oplus1$. Then $\\widetilde C_aR=RC_a$, so\n\\[\n[\\widetilde C_a,\\widetilde C_b]R=R[C_a,C_b].\n\\]\nThe coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls these residuals and commutator errors using the defect sizes and a common bound on the constraint norms. A nonzero anomaly remains an error that must be controlled.","e coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls these residuals and commutator errors using the defect sizes and a common bound on the constraint norms. A nonzero anomaly remains an error that must be controlled.","ctor maps into the fine common kernel. However, the same prescribed Pauli structure does \\emph{not} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls these residuals and commutator errors using the defect sizes and a common bound on the constraint norms. A nonzero anomaly remains an error that must be controlled.","t} close on the whole fine level: on the added coordinate the commutator is zero, whereas $2\\ii\\widetilde C_z$ is $2\\ii$. Exact transport constrains the image of $R$ and leaves its orthogonal complement as separate data.\n\\head{An approximate residual is also explicit}\nPerturb only $\\widetilde C_x$ by $(1/10)\\ket0\\bra0$. For the unit coarse physical vector, the fine constraint residual is exactly $1/10$. The intertwining defect has the same norm. The general result for approximate transport controls these residuals and commutator errors using the defect sizes and a common bound on the constraint norms. A nonzero anomaly remains an error that must be controlled.","\\vpage{39}{Two routes to a bounded Hamiltonian limit}{EXACT EMBEDDINGS AND A SUMMABLE TAIL}{Chapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5.}\nFor the exact route, let $\\HH_n=\\CC^n$ with coordinate inclusions and\n\\[\nH_n=\\diag(0,1,1/2,\\ldots,1/(n-1)),\\qquad H_1=(0).\n\\]\nThe old energies are retained at every level, $H_{n+1}R_n=R_nH_n$, and $\\norm{H_n}\\leq1$. On the Hilbert space of the direct limit $\\ell^2(\\NN_0)$, the diagonal sequence defines a bounded self-adjoint $H_\\infty$. Projectors onto even addresses also intertwine and give a fixed complete system of two packets in the limit.\n\\xfig{tower_tail}\n\\head{Compatibility and boundedness are different assumptions}\nThe equally compatible tower $\\diag(0,1,2,\\ldots,n-1)$ has growing norms. It does not satisfy the bounded direct limit theorem. Its natural diagonal operator needs a separately specified domain. Dropping the","\\vpage{39}{Two routes to a bounded Hamiltonian limit}{EXACT EMBEDDINGS AND A SUMMABLE TAIL}{Chapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5.}\nFor the exact route, let $\\HH_n=\\CC^n$ with coordinate inclusions and\n\\[\nH_n=\\diag(0,1,1/2,\\ldots,1/(n-1)),\\qquad H_1=(0).\n\\]\nThe old energies are retained at every level, $H_{n+1}R_n=R_nH_n$, and $\\norm{H_n}\\leq1$. On the Hilbert space of the direct limit $\\ell^2(\\NN_0)$, the diagonal sequence defines a bounded self-adjoint $H_\\infty$. Projectors onto even addresses also intertwine and give a fixed complete system of two packets in the limit.\n\\xfig{tower_tail}\n\\head{Compatibility and boundedness are different assumptions}\nThe equally compatible tower $\\diag(0,1,2,\\ldots,n-1)$ has growing norms. It does not satisfy the bounded direct limit theorem. Its natural diagonal operator needs a separately specified domain. Dropping the uniform norm hypothesis does not supply that domain argument.","utes to a bounded Hamiltonian limit}{EXACT EMBEDDINGS AND A SUMMABLE TAIL}{Chapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5.}\nFor the exact route, let $\\HH_n=\\CC^n$ with coordinate inclusions and\n\\[\nH_n=\\diag(0,1,1/2,\\ldots,1/(n-1)),\\qquad H_1=(0).\n\\]\nThe old energies are retained at every level, $H_{n+1}R_n=R_nH_n$, and $\\norm{H_n}\\leq1$. On the Hilbert space of the direct limit $\\ell^2(\\NN_0)$, the diagonal sequence defines a bounded self-adjoint $H_\\infty$. Projectors onto even addresses also intertwine and give a fixed complete system of two packets in the limit.\n\\xfig{tower_tail}\n\\head{Compatibility and boundedness are different assumptions}\nThe equally compatible tower $\\diag(0,1,2,\\ldots,n-1)$ has growing norms. It does not satisfy the bounded direct limit theorem. Its natural diagonal operator needs a separately specified domain. Dropping the uniform norm hypothesis does not supply that domain argument.","AIL}{Chapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5.}\nFor the exact route, let $\\HH_n=\\CC^n$ with coordinate inclusions and\n\\[\nH_n=\\diag(0,1,1/2,\\ldots,1/(n-1)),\\qquad H_1=(0).\n\\]\nThe old energies are retained at every level, $H_{n+1}R_n=R_nH_n$, and $\\norm{H_n}\\leq1$. On the Hilbert space of the direct limit $\\ell^2(\\NN_0)$, the diagonal sequence defines a bounded self-adjoint $H_\\infty$. Projectors onto even addresses also intertwine and give a fixed complete system of two packets in the limit.\n\\xfig{tower_tail}\n\\head{Compatibility and boundedness are different assumptions}\nThe equally compatible tower $\\diag(0,1,2,\\ldots,n-1)$ has growing norms. It does not satisfy the bounded direct limit theorem. Its natural diagonal operator needs a separately specified domain. Dropping the uniform norm hypothesis does not supply that domain argument.","every level, $H_{n+1}R_n=R_nH_n$, and $\\norm{H_n}\\leq1$. On the Hilbert space of the direct limit $\\ell^2(\\NN_0)$, the diagonal sequence defines a bounded self-adjoint $H_\\infty$. Projectors onto even addresses also intertwine and give a fixed complete system of two packets in the limit.\n\\xfig{tower_tail}\n\\head{Compatibility and boundedness are different assumptions}\nThe equally compatible tower $\\diag(0,1,2,\\ldots,n-1)$ has growing norms. It does not satisfy the bounded direct limit theorem. Its natural diagonal operator needs a separately specified domain. Dropping the uniform norm hypothesis does not supply that domain argument.","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed g","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A f","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A finite prefix cannot certify an unspeci","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A finite prefix cannot certify an unspecified continuation of a tow","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A finite prefix cannot certify an unspecified continuation of a tower.\n\\takeaway{Exact embedded compatibility uses a uniform","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A finite prefix cannot certify an unspecified continuation of a tower.\n\\takeaway{Exact embedded compatibility uses a uniform norm bound. Appr","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A finite prefix cannot certify an unspecified continuation of a tower.\n\\takeaway{Exact embedded compatibility uses a uniform norm bound. Approximate convergence uses a common space and summable defects. Keep the hypotheses of th","For the norm route, work instead on the \\emph{single fixed space} $\\CC^2$ and set\n\\[\nH_n=\\diag(0,1+2^{-n}),\\qquad H=\\diag(0,1).\n\\]\nHere $\\norm{H_{n+1}-H_n}=2^{-(n+1)}$ and $\\norm{H_n-H}=2^{-n}$. The summable tail supplies an error bound for every $n$. For $|t|\\leq T$, the paper's Duhamel estimate gives\n\\[\n\\norm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A finite prefix cannot certify an unspecified continuation of a tower.\n\\takeaway{Exact embedded compatibility uses a uniform norm bound. Approximate convergence uses a common space and summable defects. Keep the hypotheses of these two routes explicit.}","rm{e^{-\\ii tH_n}-e^{-\\ii tH}}\\leq T2^{-n}.\n\\]\nThe Euclidean counterpart holds on $0\\leq t\\leq T$ because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals.\n\\head{Execution scope}\nExact inclusions and projector compatibility are checked through level eight. The two-dimensional tail identities are checked for $n=0,\\ldots,12$. Their infinite conclusion uses the displayed geometric series and the theorem's analytic estimate. A finite prefix cannot certify an unspecified continuation of a tower.\n\\takeaway{Exact embedded compatibility uses a uniform norm bound. Approximate convergence uses a common space and summable defects. Keep the hypotheses of these two routes explicit.}","\\vpage{40}{Choose the topology that preserves the quantity}{REFLECTION GRAMS, TRACES, AND FINAL SCOPE}{Chapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14.}\nLet $q_n=1/2+2^{-(n+2)}$ and define the scalar reflected kernel $C_n(j,k)=1+q_n^{j+k}$ on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their ze","\\vpage{40}{Choose the topology that preserves the quantity}{REFLECTION GRAMS, TRACES, AND FINAL SCOPE}{Chapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14.}\nLet $q_n=1/2+2^{-(n+2)}$ and define the scalar reflected kernel $C_n(j,k)=1+q_n^{j+k}$ on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","oose the topology that preserves the quantity}{REFLECTION GRAMS, TRACES, AND FINAL SCOPE}{Chapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14.}\nLet $q_n=1/2+2^{-(n+2)}$ and define the scalar reflected kernel $C_n(j,k)=1+q_n^{j+k}$ on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","erves the quantity}{REFLECTION GRAMS, TRACES, AND FINAL SCOPE}{Chapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14.}\nLet $q_n=1/2+2^{-(n+2)}$ and define the scalar reflected kernel $C_n(j,k)=1+q_n^{j+k}$ on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","ntity}{REFLECTION GRAMS, TRACES, AND FINAL SCOPE}{Chapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14.}\nLet $q_n=1/2+2^{-(n+2)}$ and define the scalar reflected kernel $C_n(j,k)=1+q_n^{j+k}$ on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","egers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","nite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","um of two positive rank-one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","one matrices, and pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","pointwise convergence gives $C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","$C(j,k)=1+(1/2)^{j+k}$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","$. For times $(0,1,2)$ and coefficients $(1,-2,1)$,\n\\[\n\\sum_{j,k}v_jC_n(j,k)v_k=(1-q_n)^4\\geq0.\n\\]\nThis exact check illustrates how finite quadratic forms pass positivity to the limit.\n\\xfig{trace_topology}\n\\head{Operator norm does not preserve trace in growing rank}\nOn $\\ell^2$, let $P_n$ select the first $n$ coordinates and set $A_n=P_n/n$. Then $\\norm{A_n}=1/n\\to0$ while $\\Tr A_n=1$ for every $n$. These positive trace-class operators show why the choice of topology matters. Their zero limit is not the heat operator of a finite Hamiltonian.","For transfers on two states $T_n=\\diag(1,q_n)=e^{-H_n}$, with $H_n=\\diag(0,-\\log q_n)$ at unit time, the error in trace norm is $2^{-(n+2)}$ and the heat traces converge to $3/2$. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat trace alone does not determine that quotient.\n\\head{The retained evidence and the remaining open targets}\nThe extension includes retained exact checks over specified finite ranges in $\\QQ(\\ii)$ using \\texttt{Fraction} from Python's standard library. The original appendix's record of 180 checks, including symbolic derivatives, is also retained. The formal companion records the provenance","For transfers on two states $T_n=\\diag(1,q_n)=e^{-H_n}$, with $H_n=\\diag(0,-\\log q_n)$ at unit time, the error in trace norm is $2^{-(n+2)}$ and the heat traces converge to $3/2$. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat trace alone does not determine that quotient.\n\\head{The retained evidence and the remaining open targets}\nThe extension includes retained exact checks over specified finite ranges in $\\QQ(\\ii)$ using \\texttt{Fraction} from Python's standard library. The original appendix's record of 180 checks, including symbolic derivatives, is also retained. The formal companion records the provenance of these checks. All model and","For transfers on two states $T_n=\\diag(1,q_n)=e^{-H_n}$, with $H_n=\\diag(0,-\\log q_n)$ at unit time, the error in trace norm is $2^{-(n+2)}$ and the heat traces converge to $3/2$. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat trace alone does not determine that quotient.\n\\head{The retained evidence and the remaining open targets}\nThe extension includes retained exact checks over specified finite ranges in $\\QQ(\\ii)$ using \\texttt{Fraction} from Python's standard library. The original appendix's record of 180 checks, including symbolic derivatives, is also retained. The formal companion records the provenance of these checks. All model and vector sources remain supplied.","heat traces converge to $3/2$. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat trace alone does not determine that quotient.\n\\head{The retained evidence and the remaining open targets}\nThe extension includes retained exact checks over specified finite ranges in $\\QQ(\\ii)$ using \\texttt{Fraction} from Python's standard library. The original appendix's record of 180 checks, including symbolic derivatives, is also retained. The formal companion records the provenance of these checks. All model and vector sources remain supplied.","\\section*{Three exact finite models}\\addcontentsline{toc}{section}{Three exact finite models}\n\\subsection*{A seed remembers multiplicity support}\nFor $M_2(\\mathbb C)\\otimes I_3$, a coefficient matrix with rank one generates an orbit of dimension two. A matrix of rank two generates an orbit of dimension four. Neither single seed spans the full space of dimension six, because a $2\\times3$ matrix cannot have rank three. Two seeds can nevertheless cover all three multiplicity coordinates together. The example illustrates the finite seed rank formula with $d=2$ and $m=3$.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_seed_support.pdf}\n\\caption{The figure displays exact coefficient matrices for $d=2$, $m=3$. The combined row support of two seeds has rank three. The bar heights record algebraic orbit dimensions and should not b","\\section*{Three exact finite models}\\addcontentsline{toc}{section}{Three exact finite models}\n\\subsection*{A seed remembers multiplicity support}\nFor $M_2(\\mathbb C)\\otimes I_3$, a coefficient matrix with rank one generates an orbit of dimension two. A matrix of rank two generates an orbit of dimension four. Neither single seed spans the full space of dimension six, because a $2\\times3$ matrix cannot have rank three. Two seeds can nevertheless cover all three multiplicity coordinates together. The example illustrates the finite seed rank formula with $d=2$ and $m=3$.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_seed_support.pdf}\n\\caption{The figure displays exact coefficient matrices for $d=2$, $m=3$. The combined row support of two seeds has rank three. The bar heights record algebraic orbit dimensions and should not be read as probabilities of unitary state preparation.}\\label{monograph:fig:seed_support}\n\\end{figure}\n\\clearpage\n\\subsection*{An eliminated coordinate still contributes}\nTake $H=\\left(\\begin{smallmatrix}2&1","thbb C)\\otimes I_3$, a coefficient matrix with rank one generates an orbit of dimension two. A matrix of rank two generates an orbit of dimension four. Neither single seed spans the full space of dimension six, because a $2\\times3$ matrix cannot have rank three. Two seeds can nevertheless cover all three multiplicity coordinates together. The example illustrates the finite seed rank formula with $d=2$ and $m=3$.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_seed_support.pdf}\n\\caption{The figure displays exact coefficient matrices for $d=2$, $m=3$. The combined row support of two seeds has rank three. The bar heights record algebraic orbit dimensions and should not be read as probabilities of unitary state preparation.}\\label{monograph:fig:seed_support}\n\\end{figure}\n\\clearpage\n\\subsection*{An eliminated coordinate still contributes}\nTake $H=\\left(\\begin{smallmatrix}2&1","imes I_3$, a coefficient matrix with rank one generates an orbit of dimension two. A matrix of rank two generates an orbit of dimension four. Neither single seed spans the full space of dimension six, because a $2\\times3$ matrix cannot have rank three. Two seeds can nevertheless cover all three multiplicity coordinates together. The example illustrates the finite seed rank formula with $d=2$ and $m=3$.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_seed_support.pdf}\n\\caption{The figure displays exact coefficient matrices for $d=2$, $m=3$. The combined row support of two seeds has rank three. The bar heights record algebraic orbit dimensions and should not be read as probabilities of unitary state preparation.}\\label{monograph:fig:seed_support}\n\\end{figure}\n\\clearpage\n\\subsection*{An eliminated coordinate still contributes}\nTake $H=\\left(\\begin{smallmatrix}2&1","l space of dimension six, because a $2\\times3$ matrix cannot have rank three. Two seeds can nevertheless cover all three multiplicity coordinates together. The example illustrates the finite seed rank formula with $d=2$ and $m=3$.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_seed_support.pdf}\n\\caption{The figure displays exact coefficient matrices for $d=2$, $m=3$. The combined row support of two seeds has rank three. The bar heights record algebraic orbit dimensions and should not be read as probabilities of unitary state preparation.}\\label{monograph:fig:seed_support}\n\\end{figure}\n\\clearpage\n\\subsection*{An eliminated coordinate still contributes}\nTake $H=\\left(\\begin{smallmatrix}2&1","of dimension six, because a $2\\times3$ matrix cannot have rank three. Two seeds can nevertheless cover all three multiplicity coordinates together. The example illustrates the finite seed rank formula with $d=2$ and $m=3$.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_seed_support.pdf}\n\\caption{The figure displays exact coefficient matrices for $d=2$, $m=3$. The combined row support of two seeds has rank three. The bar heights record algebraic orbit dimensions and should not be read as probabilities of unitary state preparation.}\\label{monograph:fig:seed_support}\n\\end{figure}\n\\clearpage\n\\subsection*{An eliminated coordinate still contributes}\nTake $H=\\left(\\begin{smallmatrix}2&1","gures/exact_seed_support.pdf}\n\\caption{The figure displays exact coefficient matrices for $d=2$, $m=3$. The combined row support of two seeds has rank three. The bar heights record algebraic orbit dimensions and should not be read as probabilities of unitary state preparation.}\\label{monograph:fig:seed_support}\n\\end{figure}\n\\clearpage\n\\subsection*{An eliminated coordinate still contributes}\nTake $H=\\left(\\begin{smallmatrix}2&1\\\\1&3\\end{smallmatrix}\\right)$ and retain its first coordinate. The eliminated block is invertible and the Schur complement of $H$ is $2-1/3=5/3$. Its inverse is $3/5$, exactly the retained entry of $H^{-1}$. Inverting the compressed entry alone gives $1/2$. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution, while block inversion retains it.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_feshbach_compress","1&3\\end{smallmatrix}\\right)$ and retain its first coordinate. The eliminated block is invertible and the Schur complement of $H$ is $2-1/3=5/3$. Its inverse is $3/5$, exactly the retained entry of $H^{-1}$. Inverting the compressed entry alone gives $1/2$. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution, while block inversion retains it.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_feshbach_compression.pdf}\n\\caption{This rational model uses two coordinates to illustrate block inversion. Both the e","1&3\\end{smallmatrix}\\right)$ and retain its first coordinate. The eliminated block is invertible and the Schur complement of $H$ is $2-1/3=5/3$. Its inverse is $3/5$, exactly the retained entry of $H^{-1}$. Inverting the compressed entry alone gives $1/2$. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution, while block inversion retains it.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_feshbach_compression.pdf}\n\\caption{This rational model uses two coordinates to illustrate block inversion. Both the eliminated block","1&3\\end{smallmatrix}\\right)$ and retain its first coordinate. The eliminated block is invertible and the Schur complement of $H$ is $2-1/3=5/3$. Its inverse is $3/5$, exactly the retained entry of $H^{-1}$. Inverting the compressed entry alone gives $1/2$. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution, while block inversion retains it.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_feshbach_compression.pdf}\n\\caption{This rational model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator","1&3\\end{smallmatrix}\\right)$ and retain its first coordinate. The eliminated block is invertible and the Schur complement of $H$ is $2-1/3=5/3$. Its inverse is $3/5$, exactly the retained entry of $H^{-1}$. Inverting the compressed entry alone gives $1/2$. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution, while block inversion retains it.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_feshbach_compression.pdf}\n\\caption{This rational model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values","1&3\\end{smallmatrix}\\right)$ and retain its first coordinate. The eliminated block is invertible and the Schur complement of $H$ is $2-1/3=5/3$. Its inverse is $3/5$, exactly the retained entry of $H^{-1}$. Inverting the compressed entry alone gives $1/2$. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution, while block inversion retains it.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_feshbach_compression.pdf}\n\\caption{This rational model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct","df}\n\\caption{This rational model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograp","model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_c","ordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The fig","d block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported cod","ull operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The norm","e, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one","exact fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one, while the u","fractions. The example shows why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one, while the unnormal","why direct compression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one, while the unnormalized positive map has output trace fo","mpression and resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one, while the unnormalized positive map has output trace four. T","resolvent elimination give different results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one, while the unnormalized positive map has output trace four. The illustration records this fi","ferent results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one, while the unnormalized positive map has output trace four. The illustration records this finite scaling","results.}\\label{monograph:fig:feshbach_compression}\n\\end{figure}\n\\clearpage\n\\subsection*{Normalization changes the map}\nLet $P=|0\\rangle\\langle0|$ on $\\mathbb C^2$ and $Z=\\operatorname{diag}(1,-1)$. The normalized Kraus family $\\sqrt{3/4}\\,I$, $\\sqrt{1/4}\\,Z$ satisfies $\\sum_j K_j^*K_j=I$ and maps $P$ to $P$. The unnormalized family $\\sqrt3\\,I$, $Z$ satisfies $\\sum_j E_j^*E_j=4I$ and maps $P$ to $4P$. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ.\n\\begin{figure}[H]\\centering\n\\includegraphics[width=.94\\textwidth]{monograph_figures/exact_channel_scale.pdf}\n\\caption{The figure shows exact weights for one supported code state. The normalized map preserves trace one, while the unnormalized positive map has output trace four. The illustration records this finite scaling distinct","\\paragraph{Conventions.}\nUnless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying\n\\[\nP_\\alpha^*=P_\\alpha,\\qquad\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\\qquad\n\\sum_\\alpha P_\\alpha=I.\n\\]\nAll operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write $[A,B]=AB-BA$ and use units with $\\hbar=1$. Parameter derivatives are real derivatives. A matrix $X$ is supported on a code projector $P$ when $PXP=X$.","re nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying\n\\[\nP_\\alpha^*=P_\\alpha,\\qquad\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\\qquad\n\\sum_\\alpha P_\\alpha=I.\n\\]\nAll operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write $[A,B]=AB-BA$ and use units with $\\hbar=1$. Parameter derivatives are real derivatives. A matrix $X$ is supported on a code projector $P$ when $PXP=X$.","of finite dimension. A complete packet family is a finite collection satisfying\n\\[\nP_\\alpha^*=P_\\alpha,\\qquad\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\\qquad\n\\sum_\\alpha P_\\alpha=I.\n\\]\nAll operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write $[A,B]=AB-BA$ and use units with $\\hbar=1$. Parameter derivatives are real derivatives. A matrix $X$ is supported on a code projector $P$ when $PXP=X$.","ction satisfying\n\\[\nP_\\alpha^*=P_\\alpha,\\qquad\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\\qquad\n\\sum_\\alpha P_\\alpha=I.\n\\]\nAll operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write $[A,B]=AB-BA$ and use units with $\\hbar=1$. Parameter derivatives are real derivatives. A matrix $X$ is supported on a code projector $P$ when $PXP=X$.","ha,\\qquad\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\\qquad\n\\sum_\\alpha P_\\alpha=I.\n\\]\nAll operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write $[A,B]=AB-BA$ and use units with $\\hbar=1$. Parameter derivatives are real derivatives. A matrix $X$ is supported on a code projector $P$ when $PXP=X$.","P_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\\qquad\n\\sum_\\alpha P_\\alpha=I.\n\\]\nAll operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write $[A,B]=AB-BA$ and use units with $\\hbar=1$. Parameter derivatives are real derivatives. A matrix $X$ is supported on a code projector $P$ when $PXP=X$.","The common operator sequence of the series becomes\n\\[\n\\boxed{\n\\begin{gathered}\n\\text{quantum seed}\n\\longrightarrow\n\\text{gate or observable address}\n\\longrightarrow\n\\text{symmetry projection}\\\\\n\\longrightarrow\n\\text{packet transfer}\n\\longrightarrow\n\\text{constraint}\n\\longrightarrow\n\\text{refinement}\n\\longrightarrow\n\\text{reconstruction}\n\\end{gathered}\n}\n\\]","\\begin{definition}[Finite quantum seed system]\\label{core:definition:001}\nA finite quantum seed system is a tuple\n\\[\n\\mathfrak Q=\n\\left(\n\\cH,\n\\psi_0,\n\\cO,\n\\cR,\n\\{P_{\\ell,\\alpha}\\},\nH,\n\\tau\n\\right),\n\\]\nwhere $\\cH$ is a finite-dimensional Hilbert space, $\\psi_0\\in\\cH$ is a nonzero seed, $\\cO$ is the declared operator grammar, $\\cR$ is a collection of algebraic or physical constraints, $\\{P_{\\ell,\\alpha}\\}$ is an exact packet system at each refinement level, $H=H^*$ is a Hamiltonian when dynamics are present, and $\\tau$ specifies the topology used in a later completion.\n\\end{definition}","inition:001}\nA finite quantum seed system is a tuple\n\\[\n\\mathfrak Q=\n\\left(\n\\cH,\n\\psi_0,\n\\cO,\n\\cR,\n\\{P_{\\ell,\\alpha}\\},\nH,\n\\tau\n\\right),\n\\]\nwhere $\\cH$ is a finite-dimensional Hilbert space, $\\psi_0\\in\\cH$ is a nonzero seed, $\\cO$ is the declared operator grammar, $\\cR$ is a collection of algebraic or physical constraints, $\\{P_{\\ell,\\alpha}\\}$ is an exact packet system at each refinement level, $H=H^*$ is a Hamiltonian when dynamics are present, and $\\tau$ specifies the topology used in a later completion.\n\\end{definition}","\\mathfrak Q=\n\\left(\n\\cH,\n\\psi_0,\n\\cO,\n\\cR,\n\\{P_{\\ell,\\alpha}\\},\nH,\n\\tau\n\\right),\n\\]\nwhere $\\cH$ is a finite-dimensional Hilbert space, $\\psi_0\\in\\cH$ is a nonzero seed, $\\cO$ is the declared operator grammar, $\\cR$ is a collection of algebraic or physical constraints, $\\{P_{\\ell,\\alpha}\\}$ is an exact packet system at each refinement level, $H=H^*$ is a Hamiltonian when dynamics are present, and $\\tau$ specifies the topology used in a later completion.\n\\end{definition}","The generated state space is\n\\[\n\\cH(\\psi_0;\\cO)\n=\n\\operatorname{span}\n\\left\\{\nO_r\\cdots O_1\\psi_0:\nO_j\\in\\cO,\n\\ r\\ge0\n\\right\\}.\n\\]\nThe quantum seed is cyclic for the grammar when\n\\[\n\\cH(\\psi_0;\\cO)=\\cH.\n\\]\nThis is a statement about the pair $(\\psi_0,\\cO)$, not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.","Let a finite-dimensional unital $*$-subalgebra of operators act in its standard multiplicity form\n\\[\n\\cA\n\\cong\n\\bigoplus_{i=1}^r\n\\left(M_{d_i}(\\bbC)\\otimes I_{m_i}\\right)\n\\]\non\n\\[\n\\cH\n\\cong\n\\bigoplus_{i=1}^r\n\\left(\\bbC^{d_i}\\otimes\\bbC^{m_i}\\right).\n\\]\nHere every $d_i$ is a positive integer and every multiplicity $m_i$ is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector $\\psi_i\\in\\bbC^{d_i}\\otimes\\bbC^{m_i}$ is identified with a $d_i\\times m_i$ coefficient matrix $C_i$.","\\cA\n\\cong\n\\bigoplus_{i=1}^r\n\\left(M_{d_i}(\\bbC)\\otimes I_{m_i}\\right)\n\\]\non\n\\[\n\\cH\n\\cong\n\\bigoplus_{i=1}^r\n\\left(\\bbC^{d_i}\\otimes\\bbC^{m_i}\\right).\n\\]\nHere every $d_i$ is a positive integer and every multiplicity $m_i$ is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector $\\psi_i\\in\\bbC^{d_i}\\otimes\\bbC^{m_i}$ is identified with a $d_i\\times m_i$ coefficient matrix $C_i$.","i}\\right)\n\\]\non\n\\[\n\\cH\n\\cong\n\\bigoplus_{i=1}^r\n\\left(\\bbC^{d_i}\\otimes\\bbC^{m_i}\\right).\n\\]\nHere every $d_i$ is a positive integer and every multiplicity $m_i$ is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector $\\psi_i\\in\\bbC^{d_i}\\otimes\\bbC^{m_i}$ is identified with a $d_i\\times m_i$ coefficient matrix $C_i$.","lus_{i=1}^r\n\\left(\\bbC^{d_i}\\otimes\\bbC^{m_i}\\right).\n\\]\nHere every $d_i$ is a positive integer and every multiplicity $m_i$ is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector $\\psi_i\\in\\bbC^{d_i}\\otimes\\bbC^{m_i}$ is identified with a $d_i\\times m_i$ coefficient matrix $C_i$.","\\begin{theorem}[Orbit dimension of one quantum seed]\\label{core:theorem:001}\\label{core:thm:seed-orbit}\nFor integers $d\\ge1$ and $m\\ge0$, consider the action of $M_d(\\bbC)\\otimes I_m$ on $\\bbC^d\\otimes\\bbC^m$. Let $C$ be the coefficient matrix of a vector $\\psi$. Then\n\\[\n\\boxed{\n\\dim\\operatorname{span}\\{(A\\otimes I_m)\\psi:A\\in M_d(\\bbC)\\}\n=d\\,\\rank C.\n}\n\\]\nThe seed is cyclic if and only if $\\rank C=m$, which is possible only when $m\\le d$.\n\\end{theorem}","one quantum seed]\\label{core:theorem:001}\\label{core:thm:seed-orbit}\nFor integers $d\\ge1$ and $m\\ge0$, consider the action of $M_d(\\bbC)\\otimes I_m$ on $\\bbC^d\\otimes\\bbC^m$. Let $C$ be the coefficient matrix of a vector $\\psi$. Then\n\\[\n\\boxed{\n\\dim\\operatorname{span}\\{(A\\otimes I_m)\\psi:A\\in M_d(\\bbC)\\}\n=d\\,\\rank C.\n}\n\\]\nThe seed is cyclic if and only if $\\rank C=m$, which is possible only when $m\\le d$.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:001}\nUnder the identification by coefficient matrices, $(A\\otimes I_m)\\psi$ corresponds to $AC$. Every row of $AC$ lies in the row space of $C$. Conversely, any $d\\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\\rank C$-dimensional row space of $C$, and has dimension $d\\rank C$. It equals the full $dm$-dimensional space exactly when $\\rank C=m$.\n\\phantomsection\\label{core:proof:001:end}\\end{proof}","f}\\phantomsection\\label{core:proof:001}\nUnder the identification by coefficient matrices, $(A\\otimes I_m)\\psi$ corresponds to $AC$. Every row of $AC$ lies in the row space of $C$. Conversely, any $d\\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\\rank C$-dimensional row space of $C$, and has dimension $d\\rank C$. It equals the full $dm$-dimensional space exactly when $\\rank C=m$.\n\\phantomsection\\label{core:proof:001:end}\\end{proof}","1}\nUnder the identification by coefficient matrices, $(A\\otimes I_m)\\psi$ corresponds to $AC$. Every row of $AC$ lies in the row space of $C$. Conversely, any $d\\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\\rank C$-dimensional row space of $C$, and has dimension $d\\rank C$. It equals the full $dm$-dimensional space exactly when $\\rank C=m$.\n\\phantomsection\\label{core:proof:001:end}\\end{proof}","ficient matrices, $(A\\otimes I_m)\\psi$ corresponds to $AC$. Every row of $AC$ lies in the row space of $C$. Conversely, any $d\\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\\rank C$-dimensional row space of $C$, and has dimension $d\\rank C$. It equals the full $dm$-dimensional space exactly when $\\rank C=m$.\n\\phantomsection\\label{core:proof:001:end}\\end{proof}","times I_m)\\psi$ corresponds to $AC$. Every row of $AC$ lies in the row space of $C$. Conversely, any $d\\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\\rank C$-dimensional row space of $C$, and has dimension $d\\rank C$. It equals the full $dm$-dimensional space exactly when $\\rank C=m$.\n\\phantomsection\\label{core:proof:001:end}\\end{proof}","$AC$. Every row of $AC$ lies in the row space of $C$. Conversely, any $d\\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\\rank C$-dimensional row space of $C$, and has dimension $d\\rank C$. It equals the full $dm$-dimensional space exactly when $\\rank C=m$.\n\\phantomsection\\label{core:proof:001:end}\\end{proof}","row space of $C$. Conversely, any $d\\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\\rank C$-dimensional row space of $C$, and has dimension $d\\rank C$. It equals the full $dm$-dimensional space exactly when $\\rank C=m$.\n\\phantomsection\\label{core:proof:001:end}\\end{proof}","\\begin{theorem}[Quantum seed-rank formula]\\label{core:theorem:002}\\label{core:thm:seed-rank}\nFor\n\\[\n\\cA=\\bigoplus_i(M_{d_i}(\\bbC)\\otimes I_{m_i})\n\\quad\\text{on}\\quad\n\\cH=\\bigoplus_i(\\bbC^{d_i}\\otimes\\bbC^{m_i}),\n\\]\nthe minimum number of seeds required to generate $\\cH$ is\n\\[\n\\boxed{\n\\operatorname{srank}_{\\cA}(\\cH)\n=\n\\max_i\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:002}\nEach seed contributes a coefficient matrix of rank at most $d_i$ in the $i$th block. Thus $k$ seeds can cover at most a $kd_i$-dimensional subspace of the multiplicity factor $\\bbC^{m_i}$, so $k\\ge\\lceil m_i/d_i\\rceil$ for every $i$.","For the converse, take\n\\[\nk=\\max_i\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n\\]\nIn each block, partition a basis of $\\bbC^{m_i}$ into at most $k$ groups of size at most $d_i$. Choose the coefficient matrix of seed $a$ so that its nonzero rows span the $a$th group. The combined row spaces of the $k$ seeds then span $\\bbC^{m_i}$ in every block. The preceding theorem shows that their $\\cA$-orbits generate the whole direct sum.\n\\phantomsection\\label{core:proof:002:end}\\end{proof}","\\begin{corollary}[Local orbit and Schmidt rank]\\label{core:corollary:001}\\label{core:thm:schmidt}\nLet $\\psi\\in\\cH_A\\otimes\\cH_B$, with $\\dim\\cH_A=d_A\\ge1$, and let $r$ be its Schmidt rank. The reduced positive operator on $B$ is $\\rho_B=\\Tr_A(\\ket\\psi\\bra\\psi)$. Normalization of $\\psi$ does not affect its support. Then\n\\[\n\\boxed{\n\\dim\\operatorname{span}\\{(A\\otimes I_B)\\psi:A\\in M_{d_A}(\\bbC)\\}\n=d_A r.\n}\n\\]\nIn particular, local operators on $A$ generate the complete bipartite Hilbert space exactly when the reduced state on $B$ has full support.\n\\end{corollary}","k]\\label{core:corollary:001}\\label{core:thm:schmidt}\nLet $\\psi\\in\\cH_A\\otimes\\cH_B$, with $\\dim\\cH_A=d_A\\ge1$, and let $r$ be its Schmidt rank. The reduced positive operator on $B$ is $\\rho_B=\\Tr_A(\\ket\\psi\\bra\\psi)$. Normalization of $\\psi$ does not affect its support. Then\n\\[\n\\boxed{\n\\dim\\operatorname{span}\\{(A\\otimes I_B)\\psi:A\\in M_{d_A}(\\bbC)\\}\n=d_A r.\n}\n\\]\nIn particular, local operators on $A$ generate the complete bipartite Hilbert space exactly when the reduced state on $B$ has full support.\n\\end{corollary}","with $\\dim\\cH_A=d_A\\ge1$, and let $r$ be its Schmidt rank. The reduced positive operator on $B$ is $\\rho_B=\\Tr_A(\\ket\\psi\\bra\\psi)$. Normalization of $\\psi$ does not affect its support. Then\n\\[\n\\boxed{\n\\dim\\operatorname{span}\\{(A\\otimes I_B)\\psi:A\\in M_{d_A}(\\bbC)\\}\n=d_A r.\n}\n\\]\nIn particular, local operators on $A$ generate the complete bipartite Hilbert space exactly when the reduced state on $B$ has full support.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{core:proof:003}\nA Schmidt decomposition of length $r$ gives a coefficient matrix of rank $r$. Applying Theorem~\\ref{core:thm:seed-orbit} gives the orbit dimension $d_A r$. The range of $\\rho_B$ is the span of the $r$ Schmidt vectors on $B$, so full support is equivalent to $r=\\dim\\cH_B$. Because $d_A>0$, this is equivalent to the orbit having the full bipartite dimension.\n\\phantomsection\\label{core:proof:003:end}\\end{proof}","\\begin{theorem}[One seed for the computational basis]\\label{core:theorem:003}\\label{core:thm:computational}\nFor an integer $n\\ge0$, let\n\\[\n\\psi_0=\\ket{0}^{\\otimes n}\n\\]\nand let $X_j$ denote the Pauli $X$ operator on qubit $j$. Then\n\\[\n\\left\\{\nX_1^{a_1}\\cdots X_n^{a_n}\\psi_0:\n(a_1,\\ldots,a_n)\\in\\{0,1\\}^n\n\\right\\}\n\\]\nis the complete computational basis. Hence the seed is cyclic under the algebra generated by the $X_j$.\n\\end{theorem}","[One seed for the computational basis]\\label{core:theorem:003}\\label{core:thm:computational}\nFor an integer $n\\ge0$, let\n\\[\n\\psi_0=\\ket{0}^{\\otimes n}\n\\]\nand let $X_j$ denote the Pauli $X$ operator on qubit $j$. Then\n\\[\n\\left\\{\nX_1^{a_1}\\cdots X_n^{a_n}\\psi_0:\n(a_1,\\ldots,a_n)\\in\\{0,1\\}^n\n\\right\\}\n\\]\nis the complete computational basis. Hence the seed is cyclic under the algebra generated by the $X_j$.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:004}\nFor a binary address $a=(a_1,\\ldots,a_n)$,\n\\[\nX_1^{a_1}\\cdots X_n^{a_n}\\ket{0\\cdots0}\n=\n\\ket{a_1\\cdots a_n}.\n\\]\nDistinct binary addresses produce distinct orthonormal basis vectors. There are $2^n$ of them, equal to the dimension of the Hilbert space.\n\\phantomsection\\label{core:proof:004:end}\\end{proof}","\\begin{certificate}[Three-qubit seed orbit]\\label{core:certificate:001}\nThe original numerical implementation generated all eight computational basis states from $\\ket{000}$ using the three local $X$ address operators. It reported an orbit rank of $8/8$.\n\\end{certificate}","Let $N\\ge1$ be an integer, and let $S$ be a unitary symmetry whose order divides $N$:\n\\[\nS^N=I.\n\\]\nSet $\\omega=e^{2\\pi i/N}$.","\\begin{theorem}[Cyclic quantum projectors]\\label{core:theorem:004}\\label{core:thm:cyclic}\nThe operators\n\\[\n\\boxed{\nP_j\n=\n\\frac1N\\sum_{r=0}^{N-1}\\omega^{-jr}S^r,\n\\qquad\n0\\le j<N,\n}\n\\]\nare self-adjoint, pairwise orthogonal projectors satisfying\n\\[\nP_jP_k=\\delta_{jk}P_j,\n\\qquad\n\\sum_{j=0}^{N-1}P_j=I,\n\\qquad\nSP_j=\\omega^jP_j.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:005}\nUsing $S^{r+s}=S^{r+s\\bmod N}$ and the finite character orthogonality relation,\n\\[\n\\frac1N\\sum_{r=0}^{N-1}\\omega^{(k-j)r}=\\delta_{jk},\n\\]\nwe obtain $P_jP_k=\\delta_{jk}P_j$. Summing over $j$ cancels every term indexed by $r\\ne0$ and retains the $r=0$ term, which proves completeness. Unitarity and the change of residue $r\\mapsto-r$ give\n\\[\nP_j^*=\\frac1N\\sum_{r=0}^{N-1}\\omega^{jr}S^{-r}=P_j.\n\\]\nFinally, multiplication by $S$ shifts the summation index and contributes the factor $\\omega^j$.\n\\phantomsection\\label{core:proof:005:end}\\end{proof}","mod N}$ and the finite character orthogonality relation,\n\\[\n\\frac1N\\sum_{r=0}^{N-1}\\omega^{(k-j)r}=\\delta_{jk},\n\\]\nwe obtain $P_jP_k=\\delta_{jk}P_j$. Summing over $j$ cancels every term indexed by $r\\ne0$ and retains the $r=0$ term, which proves completeness. Unitarity and the change of residue $r\\mapsto-r$ give\n\\[\nP_j^*=\\frac1N\\sum_{r=0}^{N-1}\\omega^{jr}S^{-r}=P_j.\n\\]\nFinally, multiplication by $S$ shifts the summation index and contributes the factor $\\omega^j$.\n\\phantomsection\\label{core:proof:005:end}\\end{proof}","[\n\\frac1N\\sum_{r=0}^{N-1}\\omega^{(k-j)r}=\\delta_{jk},\n\\]\nwe obtain $P_jP_k=\\delta_{jk}P_j$. Summing over $j$ cancels every term indexed by $r\\ne0$ and retains the $r=0$ term, which proves completeness. Unitarity and the change of residue $r\\mapsto-r$ give\n\\[\nP_j^*=\\frac1N\\sum_{r=0}^{N-1}\\omega^{jr}S^{-r}=P_j.\n\\]\nFinally, multiplication by $S$ shifts the summation index and contributes the factor $\\omega^j$.\n\\phantomsection\\label{core:proof:005:end}\\end{proof}","For parity,\n\\[\nS=Z^{\\otimes n},\n\\qquad\nP_{\\mathrm{even}}=\\frac{I+S}{2},\n\\qquad\nP_{\\mathrm{odd}}=\\frac{I-S}{2}.\n\\]","Fix $N\\ge1$. Suppose every factor $a$ has a complete orthogonal family $P^{(a)}_0,\\ldots,P^{(a)}_{N-1}$. The fine tensor packets are\n\\[\nP_{\\mathbf j}^{\\mathrm{fine}}\n=\nP^{(1)}_{j_1}\\otimes\\cdots\\otimes P^{(m)}_{j_m}.\n\\]","\\begin{theorem}[Tensor/polyphase refinement]\\label{core:theorem:005}\\label{core:thm:tensor}\nFor each residue $r\\in\\bbZ/N\\bbZ$, define\n\\[\n\\boxed{\nP_r^{\\mathrm{coarse}}\n=\n\\sum_{j_1+\\cdots+j_m\\equiv r\\pmod N}\nP^{(1)}_{j_1}\\otimes\\cdots\\otimes P^{(m)}_{j_m}.\n}\n\\]\nThen the $P_r^{\\mathrm{coarse}}$ are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.\n\\end{theorem}","\\begin{certificate}[Exact parity refinement]\\label{core:certificate:002}\nFor three qubits, the local computational projectors were grouped by the parity of their Hamming weight. The resulting coarse projectors agreed with $(I\\pm Z^{\\otimes3})/2$ with residual $0$.\n\\end{certificate}","Let\n\\[\nU=U_mU_{m-1}\\cdots U_1\n\\]\nbe a circuit with $m\\ge1$, and let $\\{P_\\alpha\\}$ be a complete packet system. For the empty circuit $m=0$, set $U=I$. Its boundary block is $P_\\beta P_\\alpha=\\delta_{\\beta\\alpha}P_\\alpha$.","\\begin{theorem}[Exact circuit path expansion]\\label{core:theorem:006}\\label{core:thm:paths}\nFor source packet $\\alpha_0$ and target packet $\\alpha_m$,\n\\[\n\\boxed{\nP_{\\alpha_m}UP_{\\alpha_0}\n=\n\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}\nP_{\\alpha_m}U_mP_{\\alpha_{m-1}}\n\\cdots\nP_{\\alpha_1}U_1P_{\\alpha_0}.\n}\n\\]\nSumming over the source and target packets reconstructs the complete circuit operator.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:007}\nInsert the identity resolution $I=\\sum_\\alpha P_\\alpha$ between every adjacent pair of gates. Distributing the finite sums gives the path expansion. Summing the left and right boundary projectors reconstructs $U$.\n\\phantomsection\\label{core:proof:007:end}\\end{proof}","\\begin{theorem}[Packetwise unitarity conservation]\\label{core:theorem:007}\\label{core:thm:unitarity}\nIf $U$ is unitary and $U_{\\beta\\alpha}=P_\\beta UP_\\alpha$, then\n\\[\n\\boxed{\n\\sum_\\beta U_{\\beta\\alpha}^*U_{\\beta\\gamma}\n=\n\\delta_{\\alpha\\gamma}P_\\alpha.\n}\n\\]\nSimilarly,\n\\[\n\\sum_\\alpha U_{\\beta\\alpha}U_{\\delta\\alpha}^*\n=\n\\delta_{\\beta\\delta}P_\\beta.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:008}\nUsing completeness and unitarity,\n\\[\n\\sum_\\beta U_{\\beta\\alpha}^*U_{\\beta\\gamma}\n=\nP_\\alpha U^*\\left(\\sum_\\beta P_\\beta\\right)UP_\\gamma\n=\nP_\\alpha P_\\gamma.\n\\]\nFor the second identity, sum over the source packets and use unitarity in the reverse order.\n\\phantomsection\\label{core:proof:008:end}\\end{proof}","\\begin{example}[Bell seed circuit]\\label{core:example:001}\nIn this example only, $H$ denotes the Hadamard gate, not a Hamiltonian. Starting from $\\ket{00}$,\n\\[\n(H\\otimes I)\\ket{00}\n=\n\\frac{\\ket{00}+\\ket{10}}{\\sqrt2},\n\\]\nand then\n\\[\n\\operatorname{CNOT}(H\\otimes I)\\ket{00}\n=\n\\frac{\\ket{00}+\\ket{11}}{\\sqrt2}.\n\\]\nThe parity packets show that the final state lies entirely in the even sector, while the intermediate circuit path passes through both even and odd addresses.\n\\end{example}","\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}[t]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{02_bell_seed_probabilities.pdf}\n\\captionof{figure}{The Bell state generated from the seed $\\ket{00}$.}\n\\label{core:fig:bell}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{04_circuit_packet_paths.pdf}\n\\captionof{figure}{Circuit reconstruction as a sum over packet paths.}\n\\label{core:fig:circuit-paths}\n\\end{minipage}\n\\end{figure}","\\begin{certificate}[Bell and path reconstruction]\\label{core:certificate:003}\nThe original numerical implementation reported residuals of $0$ for both the Bell state and its complete reconstruction from parity paths.\n\\end{certificate}","Let $H=H^*$ and let $P$ be a packet projector. If $[H,P]=0$, then $P\\cH$ is invariant under $e^{-itH}$.","\\begin{theorem}[Exact symmetry packets]\\label{core:theorem:008}\\label{core:thm:symmetry}\nIf $[H,P_\\alpha]=0$ for every packet, then\n\\[\nH=\\sum_\\alpha P_\\alpha HP_\\alpha\n\\]\nand\n\\[\ne^{-itH}\n=\n\\sum_\\alpha P_\\alpha e^{-itH}P_\\alpha.\n\\]\nNo amplitude transfers between distinct packets.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:009}\nThe commutation relation gives $P_\\beta HP_\\alpha=0$ for $\\beta\\ne\\alpha$. Every polynomial in $H$ has the same block form, and the finite-dimensional exponential is the norm limit of its polynomial series.\n\\phantomsection\\label{core:proof:009:end}\\end{proof}","For two qubits and real parameters $J,h$, consider\n\\[\nH\n=\nJX\\otimes X\n+h(Z\\otimes I+I\\otimes Z).\n\\]\nIn the ordered even basis $\\{\\ket{00},\\ket{11}\\}$ and odd basis $\\{\\ket{01},\\ket{10}\\}$,\n\\[\nH_{\\mathrm{even}}\n=\n\\begin{pmatrix}\n2h&J","For two qubits and real parameters $J,h$, consider\n\\[\nH\n=\nJX\\otimes X\n+h(Z\\otimes I+I\\otimes Z).\n\\]\nIn the ordered even basis $\\{\\ket{00},\\ket{11}\\}$ and odd basis $\\{\\ket{01},\\ket{10}\\}$,\n\\[\nH_{\\mathrm{even}}\n=\n\\begin{pmatrix}\n2h&J\\\\\nJ&-2h\n\\end{pmatrix},\n\\qquad\nH_{\\mathrm{odd}}\n=\n\\begin{pmatrix}\n0&J\\\\\nJ&0\n\\end{pmatrix}.\n\\]\nTherefore\n\\[\n\\Spec H_{\\mathrm{even}}\n=\\{\\pm\\sqrt{4h^2+J^2}\\},\n\\qquad\n\\Spec H_{\\mathrm{odd}}=\\{\\pm J\\}.\n\\]","J&0\n\\end{pmatrix}.\n\\]\nTherefore\n\\[\n\\Spec H_{\\mathrm{even}}\n=\\{\\pm\\sqrt{4h^2+J^2}\\},\n\\qquad\n\\Spec H_{\\mathrm{odd}}=\\{\\pm J\\}.\n\\]","Set $Q=I-P$.","\\begin{theorem}[Commutator leakage bound]\\label{core:theorem:009}\\label{core:thm:leakage}\nFor every self-adjoint $H$, orthogonal projector $P$, and real time $t$,\n\\[\n\\boxed{\n\\norm{Qe^{-itH}P}\n\\le\n|t|\\,\\norm{[H,P]}.\n}\n\\]\nIn particular, exact commutation gives zero leakage for all time.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:010}\nSet $F(s)=e^{isH}Pe^{-isH}$. Differentiating the exponential gives\n\\[\nF'(s)=i\\,e^{isH}[H,P]e^{-isH},\\qquad F(0)=P.\n\\]\nUnitary conjugation preserves the operator norm, so $\\|F'(s)\\|=\\|[H,P]\\|$. The mean-value estimate on the interval between $0$ and $t$ yields\n\\[\n\\|F(t)-P\\|\\le |t|\\,\\|[H,P]\\|.\n\\]\nFinally, $e^{itH}Qe^{-itH}P=(P-F(t))P$ and $\\|P\\|\\le1$. Left multiplication by $e^{itH}$ preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.\n\\phantomsection\\label{core:proof:010:end}\\end{proof}","n\\label{core:proof:010}\nSet $F(s)=e^{isH}Pe^{-isH}$. Differentiating the exponential gives\n\\[\nF'(s)=i\\,e^{isH}[H,P]e^{-isH},\\qquad F(0)=P.\n\\]\nUnitary conjugation preserves the operator norm, so $\\|F'(s)\\|=\\|[H,P]\\|$. The mean-value estimate on the interval between $0$ and $t$ yields\n\\[\n\\|F(t)-P\\|\\le |t|\\,\\|[H,P]\\|.\n\\]\nFinally, $e^{itH}Qe^{-itH}P=(P-F(t))P$ and $\\|P\\|\\le1$. Left multiplication by $e^{itH}$ preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.\n\\phantomsection\\label{core:proof:010:end}\\end{proof}","\\begin{certificate}[Symmetry and negative control]\\label{core:certificate:004}\nFor the Hamiltonian that preserves parity, the norm of the block between distinct packets and the maximum measured leakage were both $0$. Adding an $X\\otimes I$ perturbation broke parity and produced a maximum leakage of approximately $0.79196$. The maximum violation of the proved bound remained $0$.\n\\end{certificate}","\\begin{theorem}[Moving-frame Hamiltonian]\\label{core:theorem:010}\\label{core:thm:moving}\nLet $U(t)$ and $\\psi(t)$ be differentiable on an open real time interval, let $U(t)$ be unitary, and let $H(t)=H(t)^*$. Suppose\n\\[\ni\\dot\\psi(t)=H(t)\\psi(t).\n\\]\nWriting\n\\[\n\\psi(t)=U(t)y(t)\n\\]\ngives\n\\[\n\\boxed{\ni\\dot y(t)\n=\n\\left(U(t)^*H(t)U(t)-iU(t)^*\\dot U(t)\\right)y(t).\n}\n\\]\nBoth the connection term $-iU^*\\dot U$ and the effective Hamiltonian $U^*HU-iU^*\\dot U$ are Hermitian.\n\\end{theorem}","onian]\\label{core:theorem:010}\\label{core:thm:moving}\nLet $U(t)$ and $\\psi(t)$ be differentiable on an open real time interval, let $U(t)$ be unitary, and let $H(t)=H(t)^*$. Suppose\n\\[\ni\\dot\\psi(t)=H(t)\\psi(t).\n\\]\nWriting\n\\[\n\\psi(t)=U(t)y(t)\n\\]\ngives\n\\[\n\\boxed{\ni\\dot y(t)\n=\n\\left(U(t)^*H(t)U(t)-iU(t)^*\\dot U(t)\\right)y(t).\n}\n\\]\nBoth the connection term $-iU^*\\dot U$ and the effective Hamiltonian $U^*HU-iU^*\\dot U$ are Hermitian.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:011}\nDifferentiate $\\psi=Uy$:\n\\[\ni\\dot U y+iU\\dot y=HUy.\n\\]\nMultiplying by $U^*$ gives the formula. Differentiating $U^*U=I$ shows $U^*\\dot U$ is skew-Hermitian, so $-iU^*\\dot U$ is Hermitian.\n\\phantomsection\\label{core:proof:011:end}\\end{proof}","\\begin{certificate}[Moving-frame correction]\\label{core:certificate:005}\nThe reported residual for the complete equation in the moving frame was $1.0645\\times10^{-10}$. Omitting the connection increased the residual to $0.7000$.\n\\end{certificate}","Let $P_1(t),\\ldots,P_m(t)$ be differentiable on an open real time interval, with $P_a(t)^*=P_a(t)$ and\n\\[\nP_aP_b=\\delta_{ab}P_a,\n\\qquad\n\\sum_aP_a=I.\n\\]","\\begin{theorem}[Canonical packet connection]\\label{core:theorem:011}\\label{core:thm:canonical}\nDefine\n\\[\n\\boxed{\nC(t)\n=\n\\frac12\\sum_a[\\dot P_a(t),P_a(t)]\n=\n\\sum_a\\dot P_a(t)P_a(t).\n}\n\\]\nThen\n\\[\nC(t)^*=-C(t)\n\\]\nand\n\\[\n\\boxed{\n[C(t),P_b(t)]=\\dot P_b(t)\n}\n\\]\nfor every packet $b$.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:012}\nDifferentiating $P_a^2=P_a$ gives\n\\[\n\\dot P_aP_a+P_a\\dot P_a=\\dot P_a.\n\\]\nSumming over $a$ and using $\\sum_a\\dot P_a=0$ yields\n\\[\n\\sum_aP_a\\dot P_a=-\\sum_a\\dot P_aP_a,\n\\]\nwhich proves both displayed formulas for $C$ and its skew-Hermiticity.","For fixed $b$,\n\\[\nCP_b=\\dot P_bP_b.\n\\]\nWhen $a\\ne b$, differentiating $P_bP_a=0$ gives\n\\[\nP_b\\dot P_a=-\\dot P_bP_a.\n\\]\nHence\n\\[\nP_bC\n=\\sum_aP_b\\dot P_aP_a\n=-\\dot P_b(I-P_b),\n\\]\nand subtraction gives $[C,P_b]=\\dot P_b$.\n\\phantomsection\\label{core:proof:012:end}\\end{proof}","\\begin{classical}[Projected connection]\\label{core:classical:001}\\label{core:thm:projected}\nLet $M$ be a smooth finite-dimensional real manifold without boundary, let $V$ be a finite-dimensional real normed vector space, and let $P:M\\to\\operatorname{End}_{\\bbR}(V)$ be smooth with $P(x)^2=P(x)$. In the quantum application, $V$ is the underlying real space of a finite complex Hilbert space and $P$ is an orthogonal complex-linear projector.","The rank of $P$ is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section $s$, so that $Ps=s$, define\n\\[\n\\nabla_Xs=P\\,ds(X).\n\\]\nThis is a connection on smooth range sections. For smooth vector fields $X,Y$, its curvature satisfies\n\\[\n\\boxed{\n(\\nabla_X\\nabla_Y-\\nabla_Y\\nabla_X-\\nabla_{[X,Y]})s\n=P[dP(X),dP(Y)]P\\,s.\n}\n\\]\nEquivalently, its endomorphism-valued curvature is\n\\[\n\\boxed{F=P(dP\\wedge dP)P.}\n\\]\nNo global frame or globally constant rank is required, and the vector fields need not commute. The geometric formalization concerns real smooth structures. It does not cover complex analytic manifolds or extensions to a boundary.\n\\end{classical}","nstant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section $s$, so that $Ps=s$, define\n\\[\n\\nabla_Xs=P\\,ds(X).\n\\]\nThis is a connection on smooth range sections. For smooth vector fields $X,Y$, its curvature satisfies\n\\[\n\\boxed{\n(\\nabla_X\\nabla_Y-\\nabla_Y\\nabla_X-\\nabla_{[X,Y]})s\n=P[dP(X),dP(Y)]P\\,s.\n}\n\\]\nEquivalently, its endomorphism-valued curvature is\n\\[\n\\boxed{F=P(dP\\wedge dP)P.}\n\\]\nNo global frame or globally constant rank is required, and the vector fields need not commute. The geometric formalization concerns real smooth structures. It does not cover complex analytic manifolds or extensions to a boundary.\n\\end{classical}","\\begin{proof}\\phantomsection\\label{core:proof:013}\n\\textit{Range coordinates.} Fix $a\\in M$ and put\n\\[\nT_a(x)=P(x)P(a)+(I-P(x))(I-P(a)).\n\\]\nThen $T_a(a)=I$ and $P(x)T_a(x)=T_a(x)P(a)$. Invertibility is open, so on a neighborhood of $a$, $T_a(x)$ restricts to an isomorphism from $\\Ran P(a)$ to $\\Ran P(x)$. This proves local constancy of rank. The local coordinates and their transitions are\n\\[\n(x,v)\\longmapsto (x,T_a(x)^{-1}v),\\qquad\nu\\longmapsto T_b(x)^{-1}T_a(x)u.\n\\]\nTheir smoothness follows from smoothness of $P$ and inversion. The transition maps satisfy the cocycle identity by cancellation. These charts construct the smooth range bundles on the open rank regions, including when different connected components have different ranks.",")+(I-P(x))(I-P(a)).\n\\]\nThen $T_a(a)=I$ and $P(x)T_a(x)=T_a(x)P(a)$. Invertibility is open, so on a neighborhood of $a$, $T_a(x)$ restricts to an isomorphism from $\\Ran P(a)$ to $\\Ran P(x)$. This proves local constancy of rank. The local coordinates and their transitions are\n\\[\n(x,v)\\longmapsto (x,T_a(x)^{-1}v),\\qquad\nu\\longmapsto T_b(x)^{-1}T_a(x)u.\n\\]\nTheir smoothness follows from smoothness of $P$ and inversion. The transition maps satisfy the cocycle identity by cancellation. These charts construct the smooth range bundles on the open rank regions, including when different connected components have different ranks.","\\textit{Connection rules.} The smoothness of $P\\,ds(X)$ and the identity $P^2=P$ show that $\\nabla_Xs$ is a smooth range section. Differentiation and projection give\n\\[\n\\begin{aligned}\n\\nabla_{X+Y}s&=\\nabla_Xs+\\nabla_Ys,&\n\\nabla_{fX}s&=f\\nabla_Xs,","\\textit{Connection rules.} The smoothness of $P\\,ds(X)$ and the identity $P^2=P$ show that $\\nabla_Xs$ is a smooth range section. Differentiation and projection give\n\\[\n\\begin{aligned}\n\\nabla_{X+Y}s&=\\nabla_Xs+\\nabla_Ys,&\n\\nabla_{fX}s&=f\\nabla_Xs,\\\\\n\\nabla_X(s+t)&=\\nabla_Xs+\\nabla_Xt,&\n\\nabla_X(fs)&=f\\nabla_Xs+(Xf)s\n\\end{aligned}\n\\]\nfor smooth real $f$. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bundle.","\\nabla_X(s+t)&=\\nabla_Xs+\\nabla_Xt,&\n\\nabla_X(fs)&=f\\nabla_Xs+(Xf)s\n\\end{aligned}\n\\]\nfor smooth real $f$. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bundle.","\\textit{Curvature.} Differentiating $P^2=P$ gives $P(dP(X))P=0$. Write $D_Xs=ds(X)$. For ambient smooth sections,\n\\[\nD_XD_Ys-D_YD_Xs-D_{[X,Y]}s=0.\n\\]\nExpansion of the projected derivatives therefore leaves\n\\[\nR(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs.\n\\]\nSince $Ps=s$, differentiation gives $D_Ys=(dP(Y))s+PD_Ys$. The terms with a middle $P$ vanish. Thus\n\\[\nR(X,Y)s=P\\bigl(dP(X)dP(Y)-dP(Y)dP(X)\\bigr)P\\,s.\n\\]\nThis expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation law for the Lie bracket make the local calculations agree on chart overlaps, which gives the global formula.\n\\phantomsection\\label{core:proof:013:end}\\end{proof}","For an orthogonal rank-one complex projector, use the scalar connection convention $A=-i\\langle u,du\\rangle$ in a local unit frame. The associated scalar curvature density is\n\\[\n\\cF_{\\mu\\nu}\n=-i\\Tr\\left(P[\\partial_\\mu P,\\partial_\\nu P]\\right).\n\\]\nOn the Bloch sphere,\n\\[\nP(\\theta,\\phi)\n=\n\\frac12\\left(I+\\mathbf n(\\theta,\\phi)\\cdot\\boldsymbol\\sigma\\right),\n\\]\nwhere\n\\[\n\\mathbf n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nThe Pauli multiplication law gives\n\\[\n[\\partial_\\theta P,\\partial_\\phi P]\n=\\frac{i}{2}(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\\cdot\\boldsymbol\\sigma.\n\\]\nUsing $\\Tr(P\\,\\mathbf a\\cdot\\boldsymbol\\sigma)=\\mathbf n\\cdot\\mathbf a$ therefore yields\n\\[\n\\boxed{\\cF_{\\theta\\phi}\n=\\frac12\\mathbf n\\cdot(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\math","For an orthogonal rank-one complex projector, use the scalar connection convention $A=-i\\langle u,du\\rangle$ in a local unit frame. The associated scalar curvature density is\n\\[\n\\cF_{\\mu\\nu}\n=-i\\Tr\\left(P[\\partial_\\mu P,\\partial_\\nu P]\\right).\n\\]\nOn the Bloch sphere,\n\\[\nP(\\theta,\\phi)\n=\n\\frac12\\left(I+\\mathbf n(\\theta,\\phi)\\cdot\\boldsymbol\\sigma\\right),\n\\]\nwhere\n\\[\n\\mathbf n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nThe Pauli multiplication law gives\n\\[\n[\\partial_\\theta P,\\partial_\\phi P]\n=\\frac{i}{2}(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\\cdot\\boldsymbol\\sigma.\n\\]\nUsing $\\Tr(P\\,\\mathbf a\\cdot\\boldsymbol\\sigma)=\\mathbf n\\cdot\\mathbf a$ therefore yields\n\\[\n\\boxed{\\cF_{\\theta\\phi}\n=\\frac12\\mathbf n\\cdot(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\n=\\frac12\\sin\\theta.}\n\\]\nThe scalar sign changes if the opposite convention for $A$ is chosen. This explicit calculation is not a se","For an orthogonal rank-one complex projector, use the scalar connection convention $A=-i\\langle u,du\\rangle$ in a local unit frame. The associated scalar curvature density is\n\\[\n\\cF_{\\mu\\nu}\n=-i\\Tr\\left(P[\\partial_\\mu P,\\partial_\\nu P]\\right).\n\\]\nOn the Bloch sphere,\n\\[\nP(\\theta,\\phi)\n=\n\\frac12\\left(I+\\mathbf n(\\theta,\\phi)\\cdot\\boldsymbol\\sigma\\right),\n\\]\nwhere\n\\[\n\\mathbf n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nThe Pauli multiplication law gives\n\\[\n[\\partial_\\theta P,\\partial_\\phi P]\n=\\frac{i}{2}(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\\cdot\\boldsymbol\\sigma.\n\\]\nUsing $\\Tr(P\\,\\mathbf a\\cdot\\boldsymbol\\sigma)=\\mathbf n\\cdot\\mathbf a$ therefore yields\n\\[\n\\boxed{\\cF_{\\theta\\phi}\n=\\frac12\\mathbf n\\cdot(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\n=\\frac12\\sin\\theta.}\n\\]\nThe scalar sign changes if the opposite convention for $A$ is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.","n convention $A=-i\\langle u,du\\rangle$ in a local unit frame. The associated scalar curvature density is\n\\[\n\\cF_{\\mu\\nu}\n=-i\\Tr\\left(P[\\partial_\\mu P,\\partial_\\nu P]\\right).\n\\]\nOn the Bloch sphere,\n\\[\nP(\\theta,\\phi)\n=\n\\frac12\\left(I+\\mathbf n(\\theta,\\phi)\\cdot\\boldsymbol\\sigma\\right),\n\\]\nwhere\n\\[\n\\mathbf n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nThe Pauli multiplication law gives\n\\[\n[\\partial_\\theta P,\\partial_\\phi P]\n=\\frac{i}{2}(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\\cdot\\boldsymbol\\sigma.\n\\]\nUsing $\\Tr(P\\,\\mathbf a\\cdot\\boldsymbol\\sigma)=\\mathbf n\\cdot\\mathbf a$ therefore yields\n\\[\n\\boxed{\\cF_{\\theta\\phi}\n=\\frac12\\mathbf n\\cdot(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\n=\\frac12\\sin\\theta.}\n\\]\nThe scalar sign changes if the opposite convention for $A$ is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.","P,\\partial_\\nu P]\\right).\n\\]\nOn the Bloch sphere,\n\\[\nP(\\theta,\\phi)\n=\n\\frac12\\left(I+\\mathbf n(\\theta,\\phi)\\cdot\\boldsymbol\\sigma\\right),\n\\]\nwhere\n\\[\n\\mathbf n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nThe Pauli multiplication law gives\n\\[\n[\\partial_\\theta P,\\partial_\\phi P]\n=\\frac{i}{2}(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\\cdot\\boldsymbol\\sigma.\n\\]\nUsing $\\Tr(P\\,\\mathbf a\\cdot\\boldsymbol\\sigma)=\\mathbf n\\cdot\\mathbf a$ therefore yields\n\\[\n\\boxed{\\cF_{\\theta\\phi}\n=\\frac12\\mathbf n\\cdot(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\n=\\frac12\\sin\\theta.}\n\\]\nThe scalar sign changes if the opposite convention for $A$ is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.","I+\\mathbf n(\\theta,\\phi)\\cdot\\boldsymbol\\sigma\\right),\n\\]\nwhere\n\\[\n\\mathbf n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nThe Pauli multiplication law gives\n\\[\n[\\partial_\\theta P,\\partial_\\phi P]\n=\\frac{i}{2}(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\\cdot\\boldsymbol\\sigma.\n\\]\nUsing $\\Tr(P\\,\\mathbf a\\cdot\\boldsymbol\\sigma)=\\mathbf n\\cdot\\mathbf a$ therefore yields\n\\[\n\\boxed{\\cF_{\\theta\\phi}\n=\\frac12\\mathbf n\\cdot(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\n=\\frac12\\sin\\theta.}\n\\]\nThe scalar sign changes if the opposite convention for $A$ is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.","rtial_\\phi P]\n=\\frac{i}{2}(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\\cdot\\boldsymbol\\sigma.\n\\]\nUsing $\\Tr(P\\,\\mathbf a\\cdot\\boldsymbol\\sigma)=\\mathbf n\\cdot\\mathbf a$ therefore yields\n\\[\n\\boxed{\\cF_{\\theta\\phi}\n=\\frac12\\mathbf n\\cdot(\\partial_\\theta\\mathbf n\\mathbin{\\times}\\partial_\\phi\\mathbf n)\n=\\frac12\\sin\\theta.}\n\\]\nThe scalar sign changes if the opposite convention for $A$ is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.76\\textwidth]{09_bloch_curvature.pdf}\n\\caption{The curvature of the Bloch projector, evaluated by finite differences, is compared with the exact density $\\frac12\\sin\\theta$.}\n\\label{core:fig:bloch}\n\\end{figure}","\\begin{certificate}[Geometric packet identities]\\label{core:certificate:006}\nThe canonical connection reconstructed every packet derivative with residual $0$. The maximum residual for the Bloch curvature was $2.0803\\times10^{-11}$.\n\\end{certificate}","Let $\\rho$ be a density matrix and let $\\{E_k\\}$ be a POVM:\n\\[\nE_k\\ge0,\n\\qquad\n\\sum_kE_k=I.\n\\]\nThe outcome probabilities are\n\\[\np_k=\\Tr(E_k\\rho).\n\\]","\\begin{theorem}[Block-diagonal measurement invariance]\\label{core:theorem:012}\\label{core:thm:measurement}\nLet $\\{P_\\alpha\\}$ be a complete packet family, and suppose every measurement effect is packet diagonal:\n\\[\nE_k=\\sum_\\alpha P_\\alpha E_kP_\\alpha.\n\\]\nDefine the packet-dephased state\n\\[\n\\cD_P(\\rho)=\\sum_\\alpha P_\\alpha\\rho P_\\alpha.\n\\]\nThen\n\\[\n\\boxed{\n\\Tr(E_k\\rho)\n=\n\\Tr(E_k\\cD_P(\\rho))\n}\n\\]\nfor every outcome $k$.\n\\end{theorem}","lock-diagonal measurement invariance]\\label{core:theorem:012}\\label{core:thm:measurement}\nLet $\\{P_\\alpha\\}$ be a complete packet family, and suppose every measurement effect is packet diagonal:\n\\[\nE_k=\\sum_\\alpha P_\\alpha E_kP_\\alpha.\n\\]\nDefine the packet-dephased state\n\\[\n\\cD_P(\\rho)=\\sum_\\alpha P_\\alpha\\rho P_\\alpha.\n\\]\nThen\n\\[\n\\boxed{\n\\Tr(E_k\\rho)\n=\n\\Tr(E_k\\cD_P(\\rho))\n}\n\\]\nfor every outcome $k$.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:014}\nInsert the packet decomposition of $E_k$ and use cyclicity of the trace:\n\\[\n\\Tr(E_k\\rho)\n=\n\\sum_\\alpha\\Tr(P_\\alpha E_kP_\\alpha\\rho)\n=\n\\sum_\\alpha\\Tr(E_kP_\\alpha\\rho P_\\alpha).\n\\]\n\\phantomsection\\label{core:proof:014:end}\\end{proof}","\\begin{classical}[Finite Kraus channel]\\label{core:classical:002}\\label{core:thm:kraus}\nLet $K_1,\\ldots,K_r$ satisfy\n\\[\n\\sum_aK_a^*K_a=I.\n\\]\nThen\n\\[\n\\boxed{\n\\Phi(\\rho)=\\sum_aK_a\\rho K_a^*\n}\n\\]\nis completely positive and trace preserving.\n\\end{classical}","\\begin{proof}\\phantomsection\\label{core:proof:015}\nTrace preservation follows from\n\\[\n\\Tr\\Phi(\\rho)\n=\\Tr\\left(\\rho\\sum_aK_a^*K_a\\right)\n=\\Tr\\rho.\n\\]\nFor any auxiliary dimension and positive matrix $X$, each term\n\\[\n(I\\otimes K_a)X(I\\otimes K_a)^*\n\\]\nis positive, so their sum is positive. This is complete positivity.\n\\phantomsection\\label{core:proof:015:end}\\end{proof}","Packet superoperator blocks are defined by\n\\[\n\\Phi_{\\alpha\\beta;\\gamma\\delta}(X)\n=\nP_\\alpha\n\\Phi(P_\\gamma XP_\\delta)\nP_\\beta.\n\\]\nSumming all four packet indices reconstructs $\\Phi(X)$ exactly.","For amplitude damping with a real parameter $0\\le\\gamma\\le1$,\n\\[\nK_0=\n\\begin{pmatrix}\n1&0","For amplitude damping with a real parameter $0\\le\\gamma\\le1$,\n\\[\nK_0=\n\\begin{pmatrix}\n1&0\\\\0&\\sqrt{1-\\gamma}\n\\end{pmatrix},\n\\qquad\nK_1=\n\\begin{pmatrix}\n0&\\sqrt\\gamma\\\\0&0\n\\end{pmatrix}.\n\\]\nThe excited state becomes\n\\[\n\\Phi(\\ket1\\bra1)\n=\n\\gamma\\ket0\\bra0+(1-\\gamma)\\ket1\\bra1.\n\\]","0&0\n\\end{pmatrix}.\n\\]\nThe excited state becomes\n\\[\n\\Phi(\\ket1\\bra1)\n=\n\\gamma\\ket0\\bra0+(1-\\gamma)\\ket1\\bra1.\n\\]","\\begin{certificate}[Channel reconstruction]\\label{core:certificate:007}\nFor $\\gamma=0.23$, the residuals for Kraus completeness and the output trace were $0$. The computed output from $\\ket1\\bra1$ was $\\diag(0.23,0.77)$. In the Bell state test, packet dephasing left every measurement probability in the computational basis unchanged.\n\\end{certificate}","Let $P=P^*=P^2$ project onto a code space $\\cC\\subset\\cH$, and let $E_1,\\ldots,E_r$ be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators\n\\[\nP E_a^*E_bP\n\\]\nare the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of $P$, the environment can distinguish logical states and exact correction is impossible.","We call a recovery complete when it is a Kraus map\n\\[\n\\cR(X)=\\sum_jK_jXK_j^*,\\qquad \\sum_jK_j^*K_j=I.\n\\]\nThe family is \\emph{exactly correctable} when one such recovery works for every coherent error combination $A_w=\\sum_aw_aE_a$: for every $w$ there is a real $t_w\\ge0$, independent of the logical input, such that\n\\[\nPXP=X\\quad\\Longrightarrow\\quad\n\\cR(A_wXA_w^*)=t_wX.\n\\]\nThe coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix $A_w$.","quad \\sum_jK_j^*K_j=I.\n\\]\nThe family is \\emph{exactly correctable} when one such recovery works for every coherent error combination $A_w=\\sum_aw_aE_a$: for every $w$ there is a real $t_w\\ge0$, independent of the logical input, such that\n\\[\nPXP=X\\quad\\Longrightarrow\\quad\n\\cR(A_wXA_w^*)=t_wX.\n\\]\nThe coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix $A_w$.","\\begin{classical}[Finite Knill--Laflamme criterion]\\label{core:classical:003}\\label{core:thm:kl}\nThe finite error family is exactly correctable in the sense just defined if and only if there is a positive semidefinite complex matrix $C=(c_{ab})$ such that\n\\[\n\\boxed{PE_a^*E_bP=c_{ab}P\\qquad\\text{for every }a,b.}\n\\]\nIf $\\sum_aE_a^*E_a=I$, the equivalent normalized-channel formulation is the existence of a complete recovery satisfying\n\\[\n\\cR\\!\\left(\\sum_aE_aXE_a^*\\right)=X\n\\quad\\text{for every }X\\text{ with }PXP=X.\n\\]\nThe zero code and an empty error family are included. An empty family cannot define a channel that preserves trace on a nonzero ambient space.\n\\end{classical}","me criterion]\\label{core:classical:003}\\label{core:thm:kl}\nThe finite error family is exactly correctable in the sense just defined if and only if there is a positive semidefinite complex matrix $C=(c_{ab})$ such that\n\\[\n\\boxed{PE_a^*E_bP=c_{ab}P\\qquad\\text{for every }a,b.}\n\\]\nIf $\\sum_aE_a^*E_a=I$, the equivalent normalized-channel formulation is the existence of a complete recovery satisfying\n\\[\n\\cR\\!\\left(\\sum_aE_aXE_a^*\\right)=X\n\\quad\\text{for every }X\\text{ with }PXP=X.\n\\]\nThe zero code and an empty error family are included. An empty family cannot define a channel that preserves trace on a nonzero ambient space.\n\\end{classical}","\\begin{proof}\\phantomsection\\label{core:proof:016}\n\\textit{Necessity.} Fix an error $E_a$ and let $L_j=K_jE_aP$. Perfect recovery on rank-one code operators gives\n\\[\n\\sum_j\\ket{L_j\\psi}\\bra{L_j\\psi}\n=t_a\\ket\\psi\\bra\\psi\\qquad(\\psi\\in\\cC).\n\\]\nFor every vector orthogonal to $\\psi$, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each $L_j\\psi$ therefore lies in the span of $\\psi$. A linear operator that preserves every line in $\\cC$ is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently\n\\[\nK_jE_aP=d_{ja}P.\n\\]\nRecovery completeness now gives\n\\[\n\\begin{aligned}\nPE_a^*E_bP\n&=\\sum_j (K_jE_aP)^*(K_jE_bP)","ction\\label{core:proof:016}\n\\textit{Necessity.} Fix an error $E_a$ and let $L_j=K_jE_aP$. Perfect recovery on rank-one code operators gives\n\\[\n\\sum_j\\ket{L_j\\psi}\\bra{L_j\\psi}\n=t_a\\ket\\psi\\bra\\psi\\qquad(\\psi\\in\\cC).\n\\]\nFor every vector orthogonal to $\\psi$, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each $L_j\\psi$ therefore lies in the span of $\\psi$. A linear operator that preserves every line in $\\cC$ is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently\n\\[\nK_jE_aP=d_{ja}P.\n\\]\nRecovery completeness now gives\n\\[\n\\begin{aligned}\nPE_a^*E_bP\n&=\\sum_j (K_jE_aP)^*(K_jE_bP)","error $E_a$ and let $L_j=K_jE_aP$. Perfect recovery on rank-one code operators gives\n\\[\n\\sum_j\\ket{L_j\\psi}\\bra{L_j\\psi}\n=t_a\\ket\\psi\\bra\\psi\\qquad(\\psi\\in\\cC).\n\\]\nFor every vector orthogonal to $\\psi$, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each $L_j\\psi$ therefore lies in the span of $\\psi$. A linear operator that preserves every line in $\\cC$ is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently\n\\[\nK_jE_aP=d_{ja}P.\n\\]\nRecovery completeness now gives\n\\[\n\\begin{aligned}\nPE_a^*E_bP\n&=\\sum_j (K_jE_aP)^*(K_jE_bP)","ry vector orthogonal to $\\psi$, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each $L_j\\psi$ therefore lies in the span of $\\psi$. A linear operator that preserves every line in $\\cC$ is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently\n\\[\nK_jE_aP=d_{ja}P.\n\\]\nRecovery completeness now gives\n\\[\n\\begin{aligned}\nPE_a^*E_bP\n&=\\sum_j (K_jE_aP)^*(K_jE_bP)","he left is a sum of nonnegative squared absolute values and equals zero. Each $L_j\\psi$ therefore lies in the span of $\\psi$. A linear operator that preserves every line in $\\cC$ is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently\n\\[\nK_jE_aP=d_{ja}P.\n\\]\nRecovery completeness now gives\n\\[\n\\begin{aligned}\nPE_a^*E_bP\n&=\\sum_j (K_jE_aP)^*(K_jE_bP)\\\\\n&=\\left(\\sum_j\\overline{d_{ja}}d_{jb}\\right)P.\n\\end{aligned}\n\\]\nThe coefficients form a Gram matrix, hence are positive semidefinite. For a normalized channel, apply the same rank-one argument to all composite branches $K_jE_aP$ at once to obtain the necessary condition.","&=\\left(\\sum_j\\overline{d_{ja}}d_{jb}\\right)P.\n\\end{aligned}\n\\]\nThe coefficients form a Gram matrix, hence are positive semidefinite. For a normalized channel, apply the same rank-one argument to all composite branches $K_jE_aP$ at once to obtain the necessary condition.","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. E","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every reco","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent co","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for t","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space v","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices,","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting reco","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families","\\textit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ a","tit{Sufficiency.} Diagonalize $C=U\\operatorname{diag}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne…","}(\\lambda_r)U^*$ with $U$ unitary and $\\lambda_r\\ge0$. Define compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1…","efine compressed transformed errors\n\\[\nF_r=\\sum_aU_{ar}E_aP.\n\\]\nThen $F_rP=F_r$ and $F_r^*F_s=\\lambda_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, …","_r\\delta_{rs}P$. If $\\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\\sqrt{\\lambda_r}$. These satisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by comp…","tisfy\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n…","r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nLet $S=\\sum_{\\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches\n\\[\nK_r=V_r^*,\\qquad K_{\\mathrm{unused}}=I-S.\n\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\…","\\]\nThey are complete on the whole ambient space because\n\\[\n\\sum_rK_r^*K_r+K_{\\mathrm{unused}}^*K_{\\mathrm{unused}}\n=S+(I-S)^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\label{core:proof:016:end}\\end{proof}",")^2=I.\n\\]\nThe inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\label{core:proof:016:end}\\end{proof}","nitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\label{core:proof:016:end}\\end{proof}","_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\label{core:proof:016:end}\\end{proof}","coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\label{core:proof:016:end}\\end{proof}","combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\label{core:proof:016:end}\\end{proof}","e scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is\n\\[\n\\cR(A_wXA_w^*)=(w^*Cw)X.\n\\]\nThis establishes the statement for arbitrary families without assuming normalization. Under $\\sum_aE_a^*E_a=I$ and $P\\ne0$, summing the diagonal KL equations gives $\\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\\ref{core:thm:kraus}.\n\\phantomsection\\label{core:proof:016:end}\\end{proof}","For the three-qubit bit-flip code,\n\\[\n\\ket{0_L}=\\ket{000},\n\\qquad\n\\ket{1_L}=\\ket{111},\n\\]\nand the error family is\n\\[\nI,\n\\quad X\\otimes I\\otimes I,\n\\quad I\\otimes X\\otimes I,\n\\quad I\\otimes I\\otimes X.\n\\]\nThe four error images of the code are pairwise orthogonal two-dimensional syndrome packets whose projectors sum to the identity on the eight-dimensional Hilbert space.","\\begin{certificate}[Exact bit-flip correction]\\label{core:certificate:008}\nThe residuals for the Knill--Laflamme condition and completeness of the syndrome packets were both $0$. A mixed channel of single bit flip errors, applied to a nontrivial logical superposition, was reversed with a residual of $1.0385\\times10^{-16}$ in the density matrix.\n\\end{certificate}","Let $P$ be a retained packet projector and $Q=I-P$. In packet coordinates,\n\\[\nH=\n\\begin{pmatrix}\nH_{PP}&H_{PQ}","Let $P$ be a retained packet projector and $Q=I-P$. In packet coordinates,\n\\[\nH=\n\\begin{pmatrix}\nH_{PP}&H_{PQ}\\\\\nH_{QP}&H_{QQ}\n\\end{pmatrix}.\n\\]","\\begin{theorem}[Exact Feshbach resolvent]\\label{core:theorem:013}\\label{core:thm:feshbach}\nAssume that both $zI_Q-H_{QQ}$ and $zI-H$ are invertible. Define\n\\[\n\\boxed{\nH_{\\mathrm{eff}}(z)\n=\nH_{PP}\n+H_{PQ}(z-H_{QQ})^{-1}H_{QP}.\n}\n\\]\nThen\n\\[\n\\boxed{\nP(z-H)^{-1}P\n=\n\\left(z-H_{\\mathrm{eff}}(z)\\right)^{-1}\n}\n\\]\nwhere the left side is restricted to $P\\cH$, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here $z-H_{QQ}$ and $z-H_{\\mathrm{eff}}$ abbreviate the appropriately sized scalar matrices minus the indicated blocks.\n\\end{theorem}","3}\\label{core:thm:feshbach}\nAssume that both $zI_Q-H_{QQ}$ and $zI-H$ are invertible. Define\n\\[\n\\boxed{\nH_{\\mathrm{eff}}(z)\n=\nH_{PP}\n+H_{PQ}(z-H_{QQ})^{-1}H_{QP}.\n}\n\\]\nThen\n\\[\n\\boxed{\nP(z-H)^{-1}P\n=\n\\left(z-H_{\\mathrm{eff}}(z)\\right)^{-1}\n}\n\\]\nwhere the left side is restricted to $P\\cH$, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here $z-H_{QQ}$ and $z-H_{\\mathrm{eff}}$ abbreviate the appropriately sized scalar matrices minus the indicated blocks.\n\\end{theorem}","hm:feshbach}\nAssume that both $zI_Q-H_{QQ}$ and $zI-H$ are invertible. Define\n\\[\n\\boxed{\nH_{\\mathrm{eff}}(z)\n=\nH_{PP}\n+H_{PQ}(z-H_{QQ})^{-1}H_{QP}.\n}\n\\]\nThen\n\\[\n\\boxed{\nP(z-H)^{-1}P\n=\n\\left(z-H_{\\mathrm{eff}}(z)\\right)^{-1}\n}\n\\]\nwhere the left side is restricted to $P\\cH$, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here $z-H_{QQ}$ and $z-H_{\\mathrm{eff}}$ abbreviate the appropriately sized scalar matrices minus the indicated blocks.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:017}\nThe block in the lower right corner of $z-H$ is $z-H_{QQ}$. Gaussian block elimination produces the Schur complement\n\\[\nz-H_{PP}-H_{PQ}(z-H_{QQ})^{-1}H_{QP}.\n\\]\nThe assumed invertibility of this block and of the full matrix implies that the Schur complement is invertible. The block in the upper left corner of the full inverse is the inverse of that Schur complement, which is exactly the displayed operator.\n\\phantomsection\\label{core:proof:017:end}\\end{proof}","The second premise cannot be dropped: at $z=0$ and $H=\\operatorname{diag}(0,-1)$, the eliminated block is $1$ but $zI-H$ is singular. This counterexample is also included in the Lean module.","The self-energy\n\\[\n\\Sigma_P(z)\n=H_{PQ}(z-H_{QQ})^{-1}H_{QP}\n\\]\ncontains the complete influence of the eliminated packet. Bare truncation drops $\\Sigma_P(z)$ and is therefore exact only when the coupling between packets vanishes or the self-energy cancels for another stated reason.","\\begin{certificate}[Effective resolvent]\\label{core:certificate:009}\nIn the reported test with a random Hermitian matrix of dimension six, the exact Feshbach residual was $3.2291\\times10^{-17}$. The corresponding residual for bare truncation was $1.3091\\times10^{-2}$.\n\\end{certificate}","For a time-independent finite Hamiltonian $H$ and fixed projectors $P,Q$, write $\\psi=x+y$, with $x=P\\psi$ and $y=Q\\psi$. Then\n\\[\n\\begin{aligned}\ni\\dot x&=H_{PP}x+H_{PQ}y,","For a time-independent finite Hamiltonian $H$ and fixed projectors $P,Q$, write $\\psi=x+y$, with $x=P\\psi$ and $y=Q\\psi$. Then\n\\[\n\\begin{aligned}\ni\\dot x&=H_{PP}x+H_{PQ}y,\\\\\ni\\dot y&=H_{QP}x+H_{QQ}y.\n\\end{aligned}\n\\]\nSolving the second equation gives\n\\[\ny(t)\n=e^{-itH_{QQ}}y(0)\n-i\\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\\,ds.\n\\]\nSubstitution yields the exact Volterra equation\n\\[\n\\boxed{\n\\begin{aligned}\ni\\dot x(t)\n={}&H_{PP}x(t)\n+H_{PQ}e^{-itH_{QQ}}y(0)","i\\dot y&=H_{QP}x+H_{QQ}y.\n\\end{aligned}\n\\]\nSolving the second equation gives\n\\[\ny(t)\n=e^{-itH_{QQ}}y(0)\n-i\\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\\,ds.\n\\]\nSubstitution yields the exact Volterra equation\n\\[\n\\boxed{\n\\begin{aligned}\ni\\dot x(t)\n={}&H_{PP}x(t)\n+H_{PQ}e^{-itH_{QQ}}y(0)","i\\dot y&=H_{QP}x+H_{QQ}y.\n\\end{aligned}\n\\]\nSolving the second equation gives\n\\[\ny(t)\n=e^{-itH_{QQ}}y(0)\n-i\\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\\,ds.\n\\]\nSubstitution yields the exact Volterra equation\n\\[\n\\boxed{\n\\begin{aligned}\ni\\dot x(t)\n={}&H_{PP}x(t)\n+H_{PQ}e^{-itH_{QQ}}y(0)\\\\\n&-i\\int_0^t\nK(t-s)x(s)\\,ds,\n\\end{aligned}\n}\n\\]\nwith memory kernel\n\\[\n\\boxed{\nK(t)=H_{PQ}e^{-itH_{QQ}}H_{QP}.\n}\n\\]\nThe influence of the eliminated degrees of freedom is therefore retained in a self-energy that depends on frequency, or equivalently in an exact memory term. For forward evolution the integral is causal. When the Hamiltonian depends on time, the corresponding propagator between two times replaces the stationary exponential used here. This Volterra derivation is not a separate entry in the finite Lean register.","&-i\\int_0^t\nK(t-s)x(s)\\,ds,\n\\end{aligned}\n}\n\\]\nwith memory kernel\n\\[\n\\boxed{\nK(t)=H_{PQ}e^{-itH_{QQ}}H_{QP}.\n}\n\\]\nThe influence of the eliminated degrees of freedom is therefore retained in a self-energy that depends on frequency, or equivalently in an exact memory term. For forward evolution the integral is causal. When the Hamiltonian depends on time, the corresponding propagator between two times replaces the stationary exponential used here. This Volterra derivation is not a separate entry in the finite Lean register.","At resolution $\\ell$, define\n\\[\n\\boxed{\n\\mathfrak Q_\\ell\n=\n\\left(\n\\cA_\\ell,\n\\cH_\\ell,\nD_\\ell,\n\\{P_{\\ell,\\alpha}\\},\nE_\\ell,\nC_\\ell,\n\\Pi_\\ell^{\\mathrm{phys}},\nT_\\ell\n\\right).\n}\n\\]\nIn order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types $E_\\ell:\\cH_\\ell\\to\\cH_{\\ell+1}$ and $C_\\ell:\\cH_{\\ell+1}\\to\\cH_\\ell$. The left inverse relation $C_\\ell E_\\ell=I$ alone does not imply $C_\\ell=E_\\ell^*$ or make the embedding isometric.","hfrak Q_\\ell\n=\n\\left(\n\\cA_\\ell,\n\\cH_\\ell,\nD_\\ell,\n\\{P_{\\ell,\\alpha}\\},\nE_\\ell,\nC_\\ell,\n\\Pi_\\ell^{\\mathrm{phys}},\nT_\\ell\n\\right).\n}\n\\]\nIn order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types $E_\\ell:\\cH_\\ell\\to\\cH_{\\ell+1}$ and $C_\\ell:\\cH_{\\ell+1}\\to\\cH_\\ell$. The left inverse relation $C_\\ell E_\\ell=I$ alone does not imply $C_\\ell=E_\\ell^*$ or make the embedding isometric.","ell,\nD_\\ell,\n\\{P_{\\ell,\\alpha}\\},\nE_\\ell,\nC_\\ell,\n\\Pi_\\ell^{\\mathrm{phys}},\nT_\\ell\n\\right).\n}\n\\]\nIn order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types $E_\\ell:\\cH_\\ell\\to\\cH_{\\ell+1}$ and $C_\\ell:\\cH_{\\ell+1}\\to\\cH_\\ell$. The left inverse relation $C_\\ell E_\\ell=I$ alone does not imply $C_\\ell=E_\\ell^*$ or make the embedding isometric.","\\mathrm{phys}},\nT_\\ell\n\\right).\n}\n\\]\nIn order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types $E_\\ell:\\cH_\\ell\\to\\cH_{\\ell+1}$ and $C_\\ell:\\cH_{\\ell+1}\\to\\cH_\\ell$. The left inverse relation $C_\\ell E_\\ell=I$ alone does not imply $C_\\ell=E_\\ell^*$ or make the embedding isometric.","order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types $E_\\ell:\\cH_\\ell\\to\\cH_{\\ell+1}$ and $C_\\ell:\\cH_{\\ell+1}\\to\\cH_\\ell$. The left inverse relation $C_\\ell E_\\ell=I$ alone does not imply $C_\\ell=E_\\ell^*$ or make the embedding isometric.","A coherent refinement should satisfy\n\\[\nC_\\ell E_\\ell=I\n\\]\nand\n\\[\n\\boxed{\nP_{\\ell,\\alpha}\n=\nC_\\ell\n\\left(\n\\sum_{\\beta\\succ\\alpha}P_{\\ell+1,\\beta}\n\\right)\nE_\\ell.\n}\n\\]\nFor an operator $D$, define the refinement defect\n\\[\n\\Delta_\\ell(D)\n=C_\\ell D_{\\ell+1}E_\\ell-D_\\ell.\n\\]\nExact resolution independence is $\\Delta_\\ell(D)=0$. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.","\\begin{theorem}[Finite Euclidean transfer]\\label{core:theorem:014}\\label{core:thm:euclidean}\nLet $H$ be Hermitian on a nonzero finite-dimensional Hilbert space. For all real $\\tau,\\sigma\\ge0$,\n\\[\nT(\\tau)=e^{-\\tau(H-E_0I)},\n\\qquad\nE_0=\\min\\Spec H,\n\\]\nis positive definite and satisfies\n\\[\n\\boxed{\nT(\\tau+\\sigma)=T(\\tau)T(\\sigma).\n}\n\\]\nIts spectrum is contained in $(0,1]$.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:018}\nDiagonalize $H=\\sum_jE_jP_j$. Then\n\\[\nT(\\tau)=\\sum_j e^{-\\tau(E_j-E_0)}P_j.\n\\]\nEvery eigenvalue is strictly positive and at most one. Multiplication adds the exponents, proving the semigroup law and $T(0)=I$. Positivity here is operator positivity, not an assertion that every matrix entry is nonnegative.\n\\phantomsection\\label{core:proof:018:end}\\end{proof}","The paper tests a controlled volume-state model\n\\[\nH_{nn}=2\\kappa+V(n),\n\\qquad\nH_{n,n+1}=H_{n+1,n}=-\\kappa,\n\\]\nwith $\\kappa>0$ and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.","\\begin{certificate}[Finite transfer model]\\label{core:certificate:010}\nThe minimum transfer eigenvalue was $0.0767054$, the semigroup residual was $0$, and reconstruction from four volume-residue packets had residual $0$.\n\\end{certificate}","Let $L\\ge0$ be a finite Laplacian on $N\\ge1$ states, with eigenvalues $\\lambda_j$ and at least one zero eigenvalue. For diffusion time $\\tau>0$, define the normalized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior a","Let $L\\ge0$ be a finite Laplacian on $N\\ge1$ states, with eigenvalues $\\lambda_j$ and at least one zero eigenvalue. For diffusion time $\\tau>0$, define the normalized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptot","Let $L\\ge0$ be a finite Laplacian on $N\\ge1$ states, with eigenvalues $\\lambda_j$ and at least one zero eigenvalue. For diffusion time $\\tau>0$, define the normalized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conc","Let $L\\ge0$ be a finite Laplacian on $N\\ge1$ states, with eigenvalues $\\lambda_j$ and at least one zero eigenvalue. For diffusion time $\\tau>0$, define the normalized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conclusion for operators with a zero mode therefore does not apply","Let $L\\ge0$ be a finite Laplacian on $N\\ge1$ states, with eigenvalues $\\lambda_j$ and at least one zero eigenvalue. For diffusion time $\\tau>0$, define the normalized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate wi","Let $L\\ge0$ be a finite Laplacian on $N\\ge1$ states, with eigenvalues $\\lambda_j$ and at least one zero eigenvalue. For diffusion time $\\tau>0$, define the normalized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controlled refinement limit.","tes, with eigenvalues $\\lambda_j$ and at least one zero eigenvalue. For diffusion time $\\tau>0$, define the normalized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controlled refinement limit.","lized heat trace\n\\[\nK(\\tau)=\\frac1N\\sum_j e^{-\\tau\\lambda_j}\n\\]\nand the scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controlled refinement limit.","e scale-dependent spectral dimension\n\\[\n\\boxed{\nd_s(\\tau)\n=-2\\frac{d\\log K(\\tau)}{d\\log\\tau}.\n}\n\\]\nFor a finite system, $d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controlled refinement limit.","$d_s(\\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives\n\\[\nd_s(\\tau)=2\\tau\\,\n\\frac{\\sum_j\\lambda_j e^{-\\tau\\lambda_j}}{\\sum_j e^{-\\tau\\lambda_j}}.\n\\]\nIt tends to zero as $\\tau\\downarrow0$ and, under the stated assumption of a zero mode, as $\\tau\\to\\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\\tau\\lambda_{\\min}$. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controlled refinement limit.","For a periodic grid with spacing $h>0$ and physical wave number $k\\in\\bbR$,\n\\[\n\\lambda_{k,h}\n=\n\\frac4{h^2}\\sin^2\\left(\\frac{kh}{2}\\right).\n\\]","\\begin{theorem}[Low-mode second-order bound]\\label{core:theorem:015}\\label{core:thm:lowmode}\nFor every real $k$ and every nonzero real $h$, define $\\lambda_{k,h}=4h^{-2}\\sin^2(kh/2)$. Then\n\\[\n\\boxed{\n0\\le k^2-\\lambda_{k,h}\n\\le\n\\frac{k^4h^2}{12}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{core:proof:019}\nPut $x=kh/2$. The inequalities $|\\sin x|\\le|x|$ and $|x-\\sin x|\\le |x|^3/6$, which hold for every real argument, give\n\\[\n0\\le x^2-\\sin^2x\n=|x-\\sin x|\\,|x+\\sin x|\n\\le\\frac{|x|^3}{6}\\,2|x|\n=\\frac{x^4}{3}.\n\\]\nMultiplying by $4/h^2$ proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps $k$ fixed while $h\\to0$, so it does not assert a uniformly small relative error across all grid modes.\n\\phantomsection\\label{core:proof:019:end}\\end{proof}","\\phantomsection\\label{core:proof:019}\nPut $x=kh/2$. The inequalities $|\\sin x|\\le|x|$ and $|x-\\sin x|\\le |x|^3/6$, which hold for every real argument, give\n\\[\n0\\le x^2-\\sin^2x\n=|x-\\sin x|\\,|x+\\sin x|\n\\le\\frac{|x|^3}{6}\\,2|x|\n=\\frac{x^4}{3}.\n\\]\nMultiplying by $4/h^2$ proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps $k$ fixed while $h\\to0$, so it does not assert a uniformly small relative error across all grid modes.\n\\phantomsection\\label{core:proof:019:end}\\end{proof}","\\begin{certificate}[Continuum dispersion]\\label{core:certificate:011}\nThe reported convergence order was $1.99936$. For every tested mode, the error was below the bound, so none of those tests violated the theorem.\n\\end{certificate}","\\begin{certificate}[Spectral dimension]\\label{core:certificate:012}\nThe median intermediate plateaus were\n\\[\nd_s^{(1)}=1.00062,\n\\qquad\nd_s^{(2)}=2.03781.\n\\]\nThese tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.\n\\end{certificate}","At one finite level, let $\\widehat C_A$ be a finite family of matrix constraints, with fixed scalar structure coefficients, and let $\\Pi^{\\mathrm{phys}}$ be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This","At one finite level, let $\\widehat C_A$ be a finite family of matrix constraints, with fixed scalar structure coefficients, and let $\\Pi^{\\mathrm{phys}}$ be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression","At one finite level, let $\\widehat C_A$ be a finite family of matrix constraints, with fixed scalar structure coefficients, and let $\\Pi^{\\mathrm{phys}}$ be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0","At one finite level, let $\\widehat C_A$ be a finite family of matrix constraints, with fixed scalar structure coefficients, and let $\\Pi^{\\mathrm{phys}}$ be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is n…","ite family of matrix constraints, with fixed scalar structure coefficients, and let $\\Pi^{\\mathrm{phys}}$ be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficient…","mathrm{phys}}$ be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression vanishes.","o their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression vanishes.","l. If the classical constraint algebra has structure coefficients $f_{AB}^{\\ \\ C}$, define the anomaly operator\n\\[\n\\boxed{\n\\mathfrak A_{AB}\n=\n[\\widehat C_A,\\widehat C_B]\n-i\\sum_C f_{AB}^{\\ \\ C}\\widehat C_C.\n}\n\\]\nIts physical packet is\n\\[\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}.\n\\]\nIn this exact finite setting, $\\widehat C_C\\Pi^{\\mathrm{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression vanishes.","{phys}}=0$ for every $C$. Associativity gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression vanishes.","ty gives\n\\[\n\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0,\n\\qquad\n\\Pi^{\\mathrm{phys}}\\mathfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression vanishes.","hfrak A_{AB}\\Pi^{\\mathrm{phys}}=0\n\\]\nautomatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression vanishes.","\\begin{conjecture}[Anomaly-free refinement]\\label{core:conjecture:001}\nFor a specified nontrivial quantum-geometric constraint model, there exists a coherent packet tower with nonzero physical spaces, compatible physical projectors and exact operator closure\n\\[\n\\mathfrak A^{(\\ell)}_{AB}=0\n\\]\nat every finite level. The compatible physical compression then vanishes as a consequence, not as a substitute for closure.\n\\end{conjecture}\nThe substantive research target is the operator condition. The earlier condition stated only in terms of compression was too weak. A choice of model, dynamics, and refinement maps is still needed before this program becomes a fully specified physical existence problem.","\\begin{conjecture}[Einstein infrared limit]\\label{core:conjecture:004}\nThere exist semiclassical packet states $\\Psi_{g,\\ell}$ such that the effective action converges to\n\\[\n\\frac1{16\\pi G}\n\\int\\sqrt{-g}\\,(R-2\\Lambda)\\,d^4x\n\\]\nwith controlled higher-order corrections.\n\\end{conjecture}","\\midrule\nBell-state residual & $0$","Three-qubit seed-orbit rank & $8/8$","Parity off-block norm & $0$","Circuit path reconstruction residual & $0$","Exact-symmetry maximum leakage & $0$","Maximum leakage-bound violation & $0$","Feshbach projected-resolvent residual & $3.2291\\times10^{-17}$","Bare-truncation projected-resolvent residual & $1.3091\\times10^{-2}$","Moving-frame residual with connection & $1.0645\\times10^{-10}$","Moving-frame residual without connection & $7.0000\\times10^{-1}$","Canonical packet-connection residual & $0$","Bloch-curvature residual & $2.0803\\times10^{-11}$","Kraus completeness residual & $0$","Output trace residual & $0$","Knill--Laflamme residual & $0$","Bit-flip recovery residual & $1.0385\\times10^{-16}$","Tensor parity-refinement residual & $0$","Circle low-mode convergence order & $1.99936$","One-dimensional spectral plateau & $1.00062$","Two-dimensional spectral plateau & $2.03781$","Minimum Euclidean-transfer eigenvalue & $0.0767054$","Transfer semigroup residual & $0$","Volume-packet reconstruction residual & $0$","The original reported count is\n\\[\n\\boxed{22/22\\text{ historical numerical tests, not rerun here}.}\n\\]\nThe 19 original figures are retained from the recovered numerical companion. The separate visual guide supplies a reading route through these constructions and the later strengthened results.","The finite theory therefore provides a coherent language for\n\\[\n\\boxed{\n\\begin{gathered}\n\\text{seed reachability}\n+\n\\text{symmetry resolution}\n+\n\\text{circuit paths}\\\\\n+\n\\text{effective dynamics}\n+\n\\text{geometric transport}\n+\n\\text{constraint diagnostics}\n\\end{gathered}\n}\n\\]\nA physical continuum theorem remains open. The next required results are anomaly-free refinement, uniform control of effective operators, Lorentzian reconstruction, the Einstein infrared limit, and a positive massless spin-two sector. Each requires its own hypotheses and proof. Agreement in a finite numerical model cannot replace those arguments.","The elementary coordinate object is a complete family of orthogonal projectors\n\\[\nP_aP_b=\\delta_{ab}P_a,\n\\qquad\n\\sum_{a\\in I}P_a=I.\n\\]\nEvery vector has packet coordinates \\(P_a\\psi\\). Every operator has transfer blocks\n\\[\nA_{ba}=P_bAP_a.\n\\]\nThese identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction, effective dynamics, and geometric limits.","ative to a declared observable algebra. For an algebra in multiplicity form, the local orbit dimension is controlled by the rank of a coefficient matrix. This recovers the relation between Schmidt rank and the dimension reached by local operations. The finite group section then constructs central isotypic projectors. Cyclic symmetry gives a sharper charge selection rule. An operator with charge \\(q\\) can move a state only from packet \\(j\\) to packet \\(j+q\\).","For an algebra in multiplicity form, the local orbit dimension is controlled by the rank of a coefficient matrix. This recovers the relation between Schmidt rank and the dimension reached by local operations. The finite group section then constructs central isotypic projectors. Cyclic symmetry gives a sharper charge selection rule. An operator with charge \\(q\\) can move a state only from packet \\(j\\) to packet \\(j+q\\).","n multiplicity form, the local orbit dimension is controlled by the rank of a coefficient matrix. This recovers the relation between Schmidt rank and the dimension reached by local operations. The finite group section then constructs central isotypic projectors. Cyclic symmetry gives a sharper charge selection rule. An operator with charge \\(q\\) can move a state only from packet \\(j\\) to packet \\(j+q\\).","The moving packet section treats time dependent projectors. The canonical generator\n\\[\nK(t)=\\sum_a\\dot P_a(t)P_a(t)\n\\]\nis skew adjoint and transports every packet. In the moving frame, the physical Hamiltonian acquires the geometric term \\(-\\ii W(t)^*\\dot W(t)\\). The leakage estimate must account for the motion of the packet itself. The physical coupling \\(QHP\\) is therefore replaced by\n\\[\nQ(t)\\bigl(H(t)-\\ii K(t)\\bigr)P(t).\n\\]\nThis is the exact finite distinction between following a moving subspace and remaining in a fixed one.","Density matrices require separate packet labels for their output and input coordinates. The superoperators\n\\[\n\\Pcal_{ab}(X)=P_aXP_b\n\\]\nform a complete orthogonal packet system on Hilbert--Schmidt operator space. Dephasing is the diagonal projection. Every channel has input and output superblocks, with the same composition law as ordinary matrix blocks. A covariant channel preserves the isotypic packets of the conjugation action. In the cyclic case these are coherence orders. The Choi matrix carries the equivalent symmetry under \\(U\\otimes\\overline U\\).","orm a complete orthogonal packet system on Hilbert--Schmidt operator space. Dephasing is the diagonal projection. Every channel has input and output superblocks, with the same composition law as ordinary matrix blocks. A covariant channel preserves the isotypic packets of the conjugation action. In the cyclic case these are coherence orders. The Choi matrix carries the equivalent symmetry under \\(U\\otimes\\overline U\\).","Error correction is expressed through compression to a code. Let \\(P\\) be the code projector. The class of \\(PAP\\) modulo scalar multiples of \\(P\\) measures the obstruction to making \\(A\\) undetectable within the code. The Knill--Laflamme criterion says that this obstruction vanishes for every product \\(E_a^*E_b\\) from the error family. Diagonalizing the resulting positive coefficient matrix produces orthogonal syndrome packets and an explicit recovery on the error span.","Constraint closure needs a different obstruction. If \\(C_a\\) are self adjoint constraints and \\(P_{\\phys}\\) projects onto their common kernel, then\n\\[\nP_{\\phys}[C_a,C_b]P_{\\phys}=0\n\\]\nholds automatically. This compression cannot test closure. The correct finite obstruction is the class of \\([C_a,C_b]\\) in the quotient of \\(\\End(\\Hcal)\\) by the span of the constraints. Vanishing means that the constraint span is closed under commutators.","\\midrule\nState & \\(P_a\\psi\\) & completeness of the projector family","Operator & \\(P_bAP_a\\) & block multiplication","Density matrix & \\(P_a\\rho P_b\\) & two sided operator packets","\\begin{definition}[Finite quantum seed system]\\label{strong:definition:001}\\label{strong:def:quantum_seed}\nA finite quantum seed system is a triple\n\\[\n(\\Hcal,\\Acal,S),\n\\]\nwhere \\(\\Hcal\\) is a finite dimensional Hilbert space, \\(\\Acal\\subseteq\\End(\\Hcal)\\) is a unital star subalgebra, and \\(S\\subseteq\\Hcal\\) is a finite set. Its generated hull is\n\\[\n\\Acal S\n=\n\\operatorname{span}\\{A\\psi:A\\in\\Acal,\\ \\psi\\in S\\}.\n\\]\nThe seed set is cyclic when \\(\\Acal S=\\Hcal\\). The quantum seed rank is the least cardinality of a cyclic seed set.\n\\end{definition}","\\label{strong:definition:001}\\label{strong:def:quantum_seed}\nA finite quantum seed system is a triple\n\\[\n(\\Hcal,\\Acal,S),\n\\]\nwhere \\(\\Hcal\\) is a finite dimensional Hilbert space, \\(\\Acal\\subseteq\\End(\\Hcal)\\) is a unital star subalgebra, and \\(S\\subseteq\\Hcal\\) is a finite set. Its generated hull is\n\\[\n\\Acal S\n=\n\\operatorname{span}\\{A\\psi:A\\in\\Acal,\\ \\psi\\in S\\}.\n\\]\nThe seed set is cyclic when \\(\\Acal S=\\Hcal\\). The quantum seed rank is the least cardinality of a cyclic seed set.\n\\end{definition}","The algebra is part of the data. A vector may be cyclic for \\(\\End(\\Hcal)\\) and fail to be cyclic for a restricted observable or gate algebra.","Let\n\\[\n\\Acal\n\\cong\n\\bigoplus_{r=1}^{s}\n\\bigl(M_{d_r}(\\CC)\\otimes I_{m_r}\\bigr)\n\\]\nact on\n\\[\n\\Hcal\n\\cong\n\\bigoplus_{r=1}^{s}\n\\bigl(\\CC^{d_r}\\otimes\\CC^{m_r}\\bigr).\n\\]\nAssume \\(d_r,m_r\\ge1\\) for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a \\(d_r\\) by \\(m_r\\) coefficient matrix.","\\begin{theorem}[Orbit dimension of one seed]\\label{strong:theorem:001}\\label{strong:thm:orbit_dimension}\nFor integers \\(d\\ge1\\), \\(m\\ge0\\), and the action of \\(M_d(\\CC)\\otimes I_m\\) on \\(\\CC^d\\otimes\\CC^m\\), let \\(C_\\psi\\) be the coefficient matrix of \\(\\psi\\). Then\n\\[\n\\dim\\operatorname{span}\\{(A\\otimes I_m)\\psi:A\\in M_d(\\CC)\\}\n=\n d\\rank C_\\psi.\n\\]\nThe vector is cyclic exactly when \\(\\rank C_\\psi=m\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:001}\nUnder the coefficient matrix identification, \\((A\\otimes I_m)\\psi\\) corresponds to \\(AC_\\psi\\). Every row of \\(AC_\\psi\\) lies in the row space of \\(C_\\psi\\). Conversely, a matrix whose rows lie in that row space equals \\(AC_\\psi\\) for a suitable \\(A\\). There are \\(d\\) independent row choices in a space of dimension \\(\\rank C_\\psi\\). The orbit dimension is therefore \\(d\\rank C_\\psi\\). It equals \\(dm\\) exactly when \\(C_\\psi\\) has full column rank.\n\\phantomsection\\label{strong:proof:001:end}\\end{proof}","ction\\label{strong:proof:001}\nUnder the coefficient matrix identification, \\((A\\otimes I_m)\\psi\\) corresponds to \\(AC_\\psi\\). Every row of \\(AC_\\psi\\) lies in the row space of \\(C_\\psi\\). Conversely, a matrix whose rows lie in that row space equals \\(AC_\\psi\\) for a suitable \\(A\\). There are \\(d\\) independent row choices in a space of dimension \\(\\rank C_\\psi\\). The orbit dimension is therefore \\(d\\rank C_\\psi\\). It equals \\(dm\\) exactly when \\(C_\\psi\\) has full column rank.\n\\phantomsection\\label{strong:proof:001:end}\\end{proof}","Under the coefficient matrix identification, \\((A\\otimes I_m)\\psi\\) corresponds to \\(AC_\\psi\\). Every row of \\(AC_\\psi\\) lies in the row space of \\(C_\\psi\\). Conversely, a matrix whose rows lie in that row space equals \\(AC_\\psi\\) for a suitable \\(A\\). There are \\(d\\) independent row choices in a space of dimension \\(\\rank C_\\psi\\). The orbit dimension is therefore \\(d\\rank C_\\psi\\). It equals \\(dm\\) exactly when \\(C_\\psi\\) has full column rank.\n\\phantomsection\\label{strong:proof:001:end}\\end{proof}","nt matrix identification, \\((A\\otimes I_m)\\psi\\) corresponds to \\(AC_\\psi\\). Every row of \\(AC_\\psi\\) lies in the row space of \\(C_\\psi\\). Conversely, a matrix whose rows lie in that row space equals \\(AC_\\psi\\) for a suitable \\(A\\). There are \\(d\\) independent row choices in a space of dimension \\(\\rank C_\\psi\\). The orbit dimension is therefore \\(d\\rank C_\\psi\\). It equals \\(dm\\) exactly when \\(C_\\psi\\) has full column rank.\n\\phantomsection\\label{strong:proof:001:end}\\end{proof}","\\begin{theorem}[Quantum seed rank]\\label{strong:theorem:002}\\label{strong:thm:quantum_seed_rank}\nFor the displayed multiplicity decomposition,\n\\[\n\\operatorname{srank}_{\\Acal}(\\Hcal)\n=\n\\max_{1\\le r\\le s}\n\\left\\lceil\\frac{m_r}{d_r}\\right\\rceil.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:002}\nA single seed contributes a row space of dimension at most \\(d_r\\) to the \\(r\\)th multiplicity factor. Hence \\(k\\) seeds cover at most \\(kd_r\\) multiplicity directions, which gives the lower bound.","Set\n\\[\nk=\\max_r\\left\\lceil\\frac{m_r}{d_r}\\right\\rceil.\n\\]\nFor each \\(r\\), partition a basis of \\(\\CC^{m_r}\\) into at most \\(k\\) groups of size at most \\(d_r\\). Choose the coefficient matrix of the \\(j\\)th seed so that its row space spans the \\(j\\)th group. The combined row spaces span every multiplicity factor. The preceding theorem then shows that the \\(k\\) seeds generate every summand.\n\\phantomsection\\label{strong:proof:002:end}\\end{proof}","\\begin{corollary}[Local orbit and Schmidt rank]\\label{strong:corollary:001}\\label{strong:cor:schmidt_orbit}\nLet \\(\\psi\\in\\Hcal_A\\otimes\\Hcal_B\\), let \\(d_A=\\dim\\Hcal_A\\ge1\\), and let \\(r\\) be the Schmidt rank of \\(\\psi\\). Then\n\\[\n\\dim\\operatorname{span}\\{(A\\otimes I_B)\\psi:A\\in\\End(\\Hcal_A)\\}\n=\n d_A r.\n\\]\nLocal operators on \\(A\\) generate the whole bipartite Hilbert space exactly when the reduced state on \\(B\\) has full support.\n\\end{corollary}","llary}[Local orbit and Schmidt rank]\\label{strong:corollary:001}\\label{strong:cor:schmidt_orbit}\nLet \\(\\psi\\in\\Hcal_A\\otimes\\Hcal_B\\), let \\(d_A=\\dim\\Hcal_A\\ge1\\), and let \\(r\\) be the Schmidt rank of \\(\\psi\\). Then\n\\[\n\\dim\\operatorname{span}\\{(A\\otimes I_B)\\psi:A\\in\\End(\\Hcal_A)\\}\n=\n d_A r.\n\\]\nLocal operators on \\(A\\) generate the whole bipartite Hilbert space exactly when the reduced state on \\(B\\) has full support.\n\\end{corollary}","\\begin{proposition}[Computational address orbit]\\label{strong:proposition:001}\\label{strong:prop:computational_orbit}\nFor \\(n\\) qubits, let\n\\[\n\\psi_0=\\ket{0}^{\\otimes n}\n\\]\nand let \\(X_j\\) be the Pauli \\(X\\) operator on the \\(j\\)th qubit. Then\n\\[\n\\left\\{\nX_1^{a_1}\\cdots X_n^{a_n}\\psi_0:\n(a_1,\\ldots,a_n)\\in\\{0,1\\}^n\n\\right\\}\n\\]\nis the computational basis.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:004}\nThe addressed vector is \\(\\ket{a_1\\cdots a_n}\\). Distinct binary addresses produce the \\(2^n\\) distinct basis vectors.\n\\phantomsection\\label{strong:proof:004:end}\\end{proof}","Let \\(G\\) be a finite group and let\n\\[\nU:G\\longrightarrow\\Ucal(\\Hcal)\n\\]\nbe a unitary representation. Write \\(\\widehat G\\) for the set of irreducible unitary representations. For \\(\\lambda\\in\\widehat G\\), let \\(d_\\lambda\\) and \\(\\chi_\\lambda\\) denote its dimension and character.","\\begin{theorem}[Finite group isotypic projectors]\\label{strong:theorem:003}\\label{strong:thm:isotypic_projectors}\nThe operators\n\\[\nP_\\lambda\n=\n\\frac{d_\\lambda}{\\abs{G}}\n\\sum_{g\\in G}\n\\overline{\\chi_\\lambda(g)}U(g)\n\\]\nare orthogonal projectors. They satisfy\n\\[\nP_\\lambda P_\\mu=\\delta_{\\lambda\\mu}P_\\lambda,\n\\qquad\n\\sum_{\\lambda\\in\\widehat G}P_\\lambda=I_{\\Hcal}.\n\\]\nThe range of \\(P_\\lambda\\) is the \\(\\lambda\\) isotypic component of \\(\\Hcal\\).\n\\end{theorem}","pic projectors]\\label{strong:theorem:003}\\label{strong:thm:isotypic_projectors}\nThe operators\n\\[\nP_\\lambda\n=\n\\frac{d_\\lambda}{\\abs{G}}\n\\sum_{g\\in G}\n\\overline{\\chi_\\lambda(g)}U(g)\n\\]\nare orthogonal projectors. They satisfy\n\\[\nP_\\lambda P_\\mu=\\delta_{\\lambda\\mu}P_\\lambda,\n\\qquad\n\\sum_{\\lambda\\in\\widehat G}P_\\lambda=I_{\\Hcal}.\n\\]\nThe range of \\(P_\\lambda\\) is the \\(\\lambda\\) isotypic component of \\(\\Hcal\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:005}\nComplete reducibility gives a unitary decomposition\n\\[\n\\Hcal\n\\cong\n\\bigoplus_{\\mu\\in\\widehat G}\nV_\\mu\\otimes M_\\mu,\n\\qquad\nU(g)\n\\cong\n\\bigoplus_\\mu\n\\pi_\\mu(g)\\otimes I_{M_\\mu}.\n\\]\nOn the \\(\\mu\\) summand, the displayed average is central in the irreducible representation \\(\\pi_\\mu\\), so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when \\(\\mu=\\lambda\\) and zero otherwise. The asserted identities follow on the direct sum.\n\\phantomsection\\label{strong:proof:005:end}\\end{proof}","educibility gives a unitary decomposition\n\\[\n\\Hcal\n\\cong\n\\bigoplus_{\\mu\\in\\widehat G}\nV_\\mu\\otimes M_\\mu,\n\\qquad\nU(g)\n\\cong\n\\bigoplus_\\mu\n\\pi_\\mu(g)\\otimes I_{M_\\mu}.\n\\]\nOn the \\(\\mu\\) summand, the displayed average is central in the irreducible representation \\(\\pi_\\mu\\), so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when \\(\\mu=\\lambda\\) and zero otherwise. The asserted identities follow on the direct sum.\n\\phantomsection\\label{strong:proof:005:end}\\end{proof}","\\begin{theorem}[Representation matrix units]\\label{strong:theorem:004}\\label{strong:thm:representation_matrix_units}\nFix an irreducible unitary representation \\(\\pi_\\lambda:G\\to U(d_\\lambda)\\) and an orthonormal basis of its representation space. Define\n\\[\nE_{ij}^{\\lambda}\n=\n\\frac{d_\\lambda}{\\abs{G}}\n\\sum_{g\\in G}\n\\bigl[\\pi_\\lambda(g^{-1})\\bigr]_{ji}U(g).\n\\]\nThen\n\\[\nE_{ij}^{\\lambda}E_{k\\ell}^{\\mu}\n=\n\\delta_{\\lambda\\mu}\\delta_{jk}E_{i\\ell}^{\\lambda},\n\\qquad\n\\bigl(E_{ij}^{\\lambda}\\bigr)^*=E_{ji}^{\\lambda},\n\\]\nand\n\\[\nP_\\lambda=\\sum_{i=1}^{d_\\lambda}E_{ii}^{\\lambda}.\n\\]\n\\end{theorem}","x an irreducible unitary representation \\(\\pi_\\lambda:G\\to U(d_\\lambda)\\) and an orthonormal basis of its representation space. Define\n\\[\nE_{ij}^{\\lambda}\n=\n\\frac{d_\\lambda}{\\abs{G}}\n\\sum_{g\\in G}\n\\bigl[\\pi_\\lambda(g^{-1})\\bigr]_{ji}U(g).\n\\]\nThen\n\\[\nE_{ij}^{\\lambda}E_{k\\ell}^{\\mu}\n=\n\\delta_{\\lambda\\mu}\\delta_{jk}E_{i\\ell}^{\\lambda},\n\\qquad\n\\bigl(E_{ij}^{\\lambda}\\bigr)^*=E_{ji}^{\\lambda},\n\\]\nand\n\\[\nP_\\lambda=\\sum_{i=1}^{d_\\lambda}E_{ii}^{\\lambda}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:006}\nApply the matrix coefficient form of Schur orthogonality to the two group averages. It gives the matrix unit multiplication law. Unitarity gives the adjoint relation after replacing \\(g\\) by \\(g^{-1}\\). Summing the diagonal coefficients gives the character average in \\cref{strong:thm:isotypic_projectors}.\n\\phantomsection\\label{strong:proof:006:end}\\end{proof}","Let \\(N\\ge1\\) be an integer, \\(S\\) unitary with \\(S^N=I\\), and \\(\\omega=\\e^{2\\pi\\ii/N}\\).","\\begin{corollary}[Cyclic charge projectors]\\label{strong:corollary:002}\\label{strong:cor:cyclic_projectors}\nThe operators\n\\[\nP_j\n=\n\\frac1N\n\\sum_{r=0}^{N-1}\n\\omega^{-jr}S^r,\n\\qquad\nj\\in\\ZZ/N\\ZZ,\n\\]\nform a complete orthogonal packet system and satisfy\n\\[\nSP_j=\\omega^jP_j.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:007}\nApply \\cref{strong:thm:isotypic_projectors} to the cyclic group. Its irreducible representations are the one dimensional characters \\(r\\mapsto\\omega^{jr}\\).\n\\phantomsection\\label{strong:proof:007:end}\\end{proof}","\\begin{definition}[Operator charge]\\label{strong:definition:002}\\label{strong:def:operator_charge}\nAn operator \\(A\\in\\End(\\Hcal)\\) has cyclic charge \\(q\\in\\ZZ/N\\ZZ\\) when\n\\[\nSAS^*=\\omega^qA.\n\\]\n\\end{definition}","\\begin{theorem}[Charge selection rule]\\label{strong:theorem:005}\\label{strong:thm:charge_selection}\nIf \\(A\\) has charge \\(q\\), then\n\\[\nP_kAP_j=0\n\\]\nunless \\(k=j+q\\) in \\(\\ZZ/N\\ZZ\\). In fact,\n\\[\nAP_j=P_{j+q}AP_j.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:008}\nFor \\(x\\in P_j\\Hcal\\),\n\\[\nS(Ax)\n=(SAS^*)Sx\n=\\omega^{q+j}Ax.\n\\]\nThus \\(Ax\\in P_{j+q}\\Hcal\\). The block statement follows.\n\\phantomsection\\label{strong:proof:008:end}\\end{proof}","\\begin{corollary}[Charge addition]\\label{strong:corollary:003}\\label{strong:cor:charge_addition}\nIf \\(A\\) and \\(B\\) have charges \\(q\\) and \\(r\\), then \\(AB\\) has charge \\(q+r\\). The adjoint \\(A^*\\) has charge \\(-q\\).\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:009}\nConjugation by \\(S\\) is a star automorphism. Applying it to \\(AB\\) adds the two charges, while applying it to \\(A^*\\) changes the sign of the charge.\n\\phantomsection\\label{strong:proof:009:end}\\end{proof}","Suppose the \\(a\\)th tensor factor carries a cyclic packet family\n\\[\n\\bigl(P_j^{(a)}\\bigr)_{j\\in\\ZZ/N\\ZZ}.\n\\]\nThe fine tensor packets are\n\\[\nP_{\\bm j}^{\\rm fine}\n=\nP_{j_1}^{(1)}\\otimes\\cdots\\otimes P_{j_m}^{(m)}.\n\\]","\\begin{theorem}[Tensor charge refinement]\\label{strong:theorem:006}\\label{strong:thm:tensor_refinement}\nFor \\(r\\in\\ZZ/N\\ZZ\\), define\n\\[\nP_r^{\\rm total}\n=\n\\sum_{j_1+\\cdots+j_m=r}\nP_{j_1}^{(1)}\\otimes\\cdots\\otimes P_{j_m}^{(m)}.\n\\]\nThen the \\(P_r^{\\rm total}\\) form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.\n\\end{theorem}","\\begin{corollary}[Tensor selection rule]\\label{strong:corollary:004}\\label{strong:cor:tensor_selection}\nIf \\(A_a\\) has local charge \\(q_a\\), then\n\\[\nA_1\\otimes\\cdots\\otimes A_m\n\\]\nhas total charge \\(q_1+\\cdots+q_m\\).\n\\end{corollary}","Let\n\\[\nU=U_mU_{m-1}\\cdots U_1\n\\]\nbe a finite circuit with \\(m\\ge1\\), and let \\((P_a)_{a\\in I}\\) be a complete orthogonal packet family. For the empty circuit, \\(U=I\\) and its transfer block is \\(P_bP_a=\\delta_{ba}P_a\\), whose endpoint sum reconstructs \\(I\\).","\\begin{theorem}[Exact circuit path expansion]\\label{strong:theorem:007}\\label{strong:thm:circuit_path}\nFor source packet \\(a_0\\) and target packet \\(a_m\\),\n\\[\nP_{a_m}UP_{a_0}\n=\n\\sum_{a_1,\\ldots,a_{m-1}}\nP_{a_m}U_mP_{a_{m-1}}\n\\cdots\nP_{a_1}U_1P_{a_0}.\n\\]\nSumming over \\(a_0\\) and \\(a_m\\) reconstructs \\(U\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:012}\nInsert \\(I=\\sum_aP_a\\) between every adjacent pair of gates and distribute the finite sums. Completeness at the two endpoints gives the full operator.\n\\phantomsection\\label{strong:proof:012:end}\\end{proof}","For a packet path\n\\[\n\\gamma=(a_0,a_1,\\ldots,a_m),\n\\]\nwrite\n\\[\nU_\\gamma\n=\nP_{a_m}U_mP_{a_{m-1}}\n\\cdots\nP_{a_1}U_1P_{a_0}.\n\\]","\\begin{theorem}[Packet unitarity identities]\\label{strong:theorem:008}\\label{strong:thm:packet_unitarity}\nIf \\(U\\) is unitary and \\(U_{ba}=P_bUP_a\\), then\n\\[\n\\sum_bU_{ba}^*U_{bc}=\\delta_{ac}P_a\n\\]\nand\n\\[\n\\sum_aU_{ba}U_{da}^*=\\delta_{bd}P_b.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:013}\nUse completeness and unitarity:\n\\[\n\\sum_bU_{ba}^*U_{bc}\n=\nP_aU^*\\left(\\sum_bP_b\\right)UP_c\n=P_aP_c.\n\\]\nSumming instead over the source packets and using unitarity in the reverse order gives the second identity.\n\\phantomsection\\label{strong:proof:013:end}\\end{proof}","\\begin{corollary}[Probability conservation from one packet]\\label{strong:corollary:005}\\label{strong:cor:packet_probability}\nIf \\(\\psi\\in P_a\\Hcal\\), then\n\\[\n\\sum_b\\norm{P_bU\\psi}^2=\\norm{\\psi}^2.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:014}\nTake the inner product of the first identity in \\cref{strong:thm:packet_unitarity} with \\(\\psi\\) and set \\(c=a\\).\n\\phantomsection\\label{strong:proof:014:end}\\end{proof}","\\begin{theorem}[Path interference formula]\\label{strong:theorem:009}\\label{strong:thm:path_interference}\nFix a source vector \\(\\psi\\in P_{a_0}\\Hcal\\) and a target packet \\(b\\). Let \\(\\Gamma_b\\) be the set of packet paths from \\(a_0\\) to \\(b\\), and set\n\\[\n\\eta_\\gamma=U_\\gamma\\psi.\n\\]\nChoose any total order on this finite path set. Then\n\\[\n\\norm{P_bU\\psi}^2\n=\n\\sum_{\\gamma\\in\\Gamma_b}\\norm{\\eta_\\gamma}^2\n+\n2\\operatorname{Re}\n\\sum_{\\substack{\\gamma,\\gamma'\\in\\Gamma_b","\\begin{theorem}[Path interference formula]\\label{strong:theorem:009}\\label{strong:thm:path_interference}\nFix a source vector \\(\\psi\\in P_{a_0}\\Hcal\\) and a target packet \\(b\\). Let \\(\\Gamma_b\\) be the set of packet paths from \\(a_0\\) to \\(b\\), and set\n\\[\n\\eta_\\gamma=U_\\gamma\\psi.\n\\]\nChoose any total order on this finite path set. Then\n\\[\n\\norm{P_bU\\psi}^2\n=\n\\sum_{\\gamma\\in\\Gamma_b}\\norm{\\eta_\\gamma}^2\n+\n2\\operatorname{Re}\n\\sum_{\\substack{\\gamma,\\gamma'\\in\\Gamma_b\\\\ \\gamma<\\gamma'}}\n\\ip{\\eta_\\gamma}{\\eta_{\\gamma'}}.\n\\]\nConsequently,\n\\[\n\\norm{P_bU\\psi}\n\\leq\n\\sum_{\\gamma\\in\\Gamma_b}\\norm{\\eta_\\gamma}.\n\\]\nIf the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.\n\\end{theorem}","\\gamma<\\gamma'}}\n\\ip{\\eta_\\gamma}{\\eta_{\\gamma'}}.\n\\]\nConsequently,\n\\[\n\\norm{P_bU\\psi}\n\\leq\n\\sum_{\\gamma\\in\\Gamma_b}\\norm{\\eta_\\gamma}.\n\\]\nIf the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:015}\nThe path expansion gives\n\\[\nP_bU\\psi=\\sum_{\\gamma\\in\\Gamma_b}\\eta_\\gamma.\n\\]\nExpanding the squared norm and grouping conjugate cross terms gives the interference formula. The triangle inequality gives the norm bound. When the path outputs are orthogonal, every cross term vanishes and only the individual path probabilities remain.\n\\phantomsection\\label{strong:proof:015:end}\\end{proof}","\\begin{proposition}[Charge pruning of circuit paths]\\label{strong:proposition:002}\\label{strong:prop:charge_pruning}\nSuppose every gate has a finite charge decomposition\n\\[\nU_r=\\sum_qU_r^{(q)},\n\\qquad\nSU_r^{(q)}S^*=\\omega^qU_r^{(q)}.\n\\]\nA term with gate charges \\(q_1,\\ldots,q_m\\) can contribute from source charge \\(a_0\\) only to target charge\n\\[\na_m=a_0+q_1+\\cdots+q_m.\n\\]\nEvery packet path violating one of the intermediate charge updates vanishes.\n\\end{proposition}","\\begin{example}[Bell circuit]\\label{strong:example:001}\\label{strong:ex:bell_circuit}\nStarting from \\(\\ket{00}\\), apply a Hadamard gate to the first qubit and then a controlled NOT gate. The final state is\n\\[\n\\frac{\\ket{00}+\\ket{11}}{\\sqrt2}.\n\\]\nIt lies in the even parity packet. The intermediate state has one even and one odd computational contribution, so the final even amplitude is reconstructed by more than one packet path.\n\\end{example}","Let \\(H=H^*\\) and let \\((P_a)_{a\\in I}\\) be a complete orthogonal packet system.","\\begin{theorem}[Exact invariant packets]\\label{strong:theorem:010}\\label{strong:thm:invariant_packets}\nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item The Hamiltonian commutes with each projector: \\([H,P_a]=0\\) for every \\(a\\).\n\\item Transfer blocks between distinct packets vanish: \\(P_bHP_a=0\\) whenever \\(a\\neq b\\).\n\\item Every packet \\(P_a\\Hcal\\) is invariant under \\(\\e^{-\\ii tH}\\) for every real \\(t\\).\n\\end{enumerate}\nWhen they hold,\n\\[\nH=\\sum_aP_aHP_a\n\\]\nand\n\\[\n\\e^{-\\ii tH}=\\sum_aP_a\\e^{-\\ii tH}P_a.\n\\]\n\\end{theorem}","invariant packets]\\label{strong:theorem:010}\\label{strong:thm:invariant_packets}\nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item The Hamiltonian commutes with each projector: \\([H,P_a]=0\\) for every \\(a\\).\n\\item Transfer blocks between distinct packets vanish: \\(P_bHP_a=0\\) whenever \\(a\\neq b\\).\n\\item Every packet \\(P_a\\Hcal\\) is invariant under \\(\\e^{-\\ii tH}\\) for every real \\(t\\).\n\\end{enumerate}\nWhen they hold,\n\\[\nH=\\sum_aP_aHP_a\n\\]\nand\n\\[\n\\e^{-\\ii tH}=\\sum_aP_a\\e^{-\\ii tH}P_a.\n\\]\n\\end{theorem}","m:010}\\label{strong:thm:invariant_packets}\nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item The Hamiltonian commutes with each projector: \\([H,P_a]=0\\) for every \\(a\\).\n\\item Transfer blocks between distinct packets vanish: \\(P_bHP_a=0\\) whenever \\(a\\neq b\\).\n\\item Every packet \\(P_a\\Hcal\\) is invariant under \\(\\e^{-\\ii tH}\\) for every real \\(t\\).\n\\end{enumerate}\nWhen they hold,\n\\[\nH=\\sum_aP_aHP_a\n\\]\nand\n\\[\n\\e^{-\\ii tH}=\\sum_aP_a\\e^{-\\ii tH}P_a.\n\\]\n\\end{theorem}","variant_packets}\nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item The Hamiltonian commutes with each projector: \\([H,P_a]=0\\) for every \\(a\\).\n\\item Transfer blocks between distinct packets vanish: \\(P_bHP_a=0\\) whenever \\(a\\neq b\\).\n\\item Every packet \\(P_a\\Hcal\\) is invariant under \\(\\e^{-\\ii tH}\\) for every real \\(t\\).\n\\end{enumerate}\nWhen they hold,\n\\[\nH=\\sum_aP_aHP_a\n\\]\nand\n\\[\n\\e^{-\\ii tH}=\\sum_aP_a\\e^{-\\ii tH}P_a.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:017}\nCommutation is equivalent to vanishing of every off diagonal block. The same block decomposition passes to polynomials in \\(H\\), then to the exponential by norm convergence. Invariance of every packet implies invariance under both \\(\\e^{-\\ii tH}\\) and its adjoint. Differentiation at \\(t=0\\) gives commutation with \\(H\\).\n\\phantomsection\\label{strong:proof:017:end}\\end{proof}","Fix one packet projector \\(P\\) and set \\(Q=I-P\\).","\\begin{lemma}[Off diagonal norm]\\label{strong:lemma:001}\\label{strong:lem:off_diagonal_norm}\nFor self adjoint \\(H\\),\n\\[\n\\norm{[H,P]}=\\norm{QHP}=\\norm{PHQ}.\n\\]\n\\end{lemma}","\\begin{proof}\\phantomsection\\label{strong:proof:018}\nRelative to \\(P\\Hcal\\oplus Q\\Hcal\\),\n\\[\n[H,P]\n=\n\\begin{pmatrix}\n0&-PHQ","\\begin{proof}\\phantomsection\\label{strong:proof:018}\nRelative to \\(P\\Hcal\\oplus Q\\Hcal\\),\n\\[\n[H,P]\n=\n\\begin{pmatrix}\n0&-PHQ\\\\\nQHP&0\n\\end{pmatrix}.\n\\]\nIts square has diagonal blocks \\(-PHQHP\\) and \\(-QHPHQ\\). The norm is therefore the largest singular value of \\(QHP\\).\n\\phantomsection\\label{strong:proof:018:end}\\end{proof}","QHP&0\n\\end{pmatrix}.\n\\]\nIts square has diagonal blocks \\(-PHQHP\\) and \\(-QHPHQ\\). The norm is therefore the largest singular value of \\(QHP\\).\n\\phantomsection\\label{strong:proof:018:end}\\end{proof}","\\begin{theorem}[Fixed packet leakage]\\label{strong:theorem:011}\\label{strong:thm:fixed_leakage}\nFor every real \\(t\\),\n\\[\n\\norm{Q\\e^{-\\ii tH}P}\n\\leq\n\\abs{t}\\norm{QHP}\n=\n\\abs{t}\\norm{[H,P]}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:019}\nLet\n\\[\nH_{\\rm d}=PHP+QHQ,\n\\qquad\nH_{\\rm o}=PHQ+QHP.\n\\]\nThe propagator generated by \\(H_{\\rm d}\\) preserves \\(P\\Hcal\\) and \\(Q\\Hcal\\). Duhamel's formula relative to \\(H_{\\rm d}\\) gives\n\\[\nQ\\e^{-\\ii tH}P\n=\n-\\ii\n\\int_0^t\nQ\\e^{-\\ii(t-s)H}H_{\\rm o}\\e^{-\\ii sH_{\\rm d}}P\n\\dd s.\n\\]\nSince \\(H_{\\rm o}P=QHP\\) and both propagators are unitary, the norm is at most \\(\\abs{t}\\norm{QHP}\\). Use \\cref{strong:lem:off_diagonal_norm} for the last equality.\n\\phantomsection\\label{strong:proof:019:end}\\end{proof}","oof}\\phantomsection\\label{strong:proof:019}\nLet\n\\[\nH_{\\rm d}=PHP+QHQ,\n\\qquad\nH_{\\rm o}=PHQ+QHP.\n\\]\nThe propagator generated by \\(H_{\\rm d}\\) preserves \\(P\\Hcal\\) and \\(Q\\Hcal\\). Duhamel's formula relative to \\(H_{\\rm d}\\) gives\n\\[\nQ\\e^{-\\ii tH}P\n=\n-\\ii\n\\int_0^t\nQ\\e^{-\\ii(t-s)H}H_{\\rm o}\\e^{-\\ii sH_{\\rm d}}P\n\\dd s.\n\\]\nSince \\(H_{\\rm o}P=QHP\\) and both propagators are unitary, the norm is at most \\(\\abs{t}\\norm{QHP}\\). Use \\cref{strong:lem:off_diagonal_norm} for the last equality.\n\\phantomsection\\label{strong:proof:019:end}\\end{proof}","\\begin{corollary}[Leakage probability]\\label{strong:corollary:006}\\label{strong:cor:leakage_probability}\nIf \\(\\psi\\in P\\Hcal\\) is a unit vector, then\n\\[\n\\norm{Q\\e^{-\\ii tH}\\psi}^2\n\\leq\n t^2\\norm{QHP}^2.\n\\]\n\\end{corollary}","Let \\((P_a(t))_{a\\in I}\\) be a twice continuously differentiable family of complete orthogonal packet systems on a finite dimensional Hilbert space.","\\begin{theorem}[Canonical packet connection]\\label{strong:theorem:012}\\label{strong:thm:kato_connection}\nDefine\n\\[\nK(t)\n=\n\\sum_a\\dot P_a(t)P_a(t).\n\\]\nThen\n\\[\nK(t)^*=-K(t)\n\\]\nand\n\\[\n[K(t),P_a(t)]=\\dot P_a(t)\n\\]\nfor every \\(a\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:021}\nDifferentiating \\(P_a^2=P_a\\) gives\n\\[\n\\dot P_aP_a+P_a\\dot P_a=\\dot P_a.\n\\]\nSumming over \\(a\\) and using \\(\\sum_a\\dot P_a=0\\) yields\n\\[\n\\sum_aP_a\\dot P_a=-\\sum_a\\dot P_aP_a.\n\\]\nThis proves skew adjointness. For fixed \\(b\\),\n\\[\nKP_b=\\dot P_bP_b.\n\\]\nDifferentiating \\(P_bP_a=0\\) for \\(a\\neq b\\) gives\n\\[\nP_b\\dot P_a=-\\dot P_bP_a.\n\\]\nHence\n\\[\nP_bK=-\\dot P_b(I-P_b).\n\\]\nSubtracting gives \\([K,P_b]=\\dot P_b\\).\n\\phantomsection\\label{strong:proof:021:end}\\end{proof}","ng:proof:021}\nDifferentiating \\(P_a^2=P_a\\) gives\n\\[\n\\dot P_aP_a+P_a\\dot P_a=\\dot P_a.\n\\]\nSumming over \\(a\\) and using \\(\\sum_a\\dot P_a=0\\) yields\n\\[\n\\sum_aP_a\\dot P_a=-\\sum_a\\dot P_aP_a.\n\\]\nThis proves skew adjointness. For fixed \\(b\\),\n\\[\nKP_b=\\dot P_bP_b.\n\\]\nDifferentiating \\(P_bP_a=0\\) for \\(a\\neq b\\) gives\n\\[\nP_b\\dot P_a=-\\dot P_bP_a.\n\\]\nHence\n\\[\nP_bK=-\\dot P_b(I-P_b).\n\\]\nSubtracting gives \\([K,P_b]=\\dot P_b\\).\n\\phantomsection\\label{strong:proof:021:end}\\end{proof}","\\begin{theorem}[Exact Kato transport]\\label{strong:theorem:013}\\label{strong:thm:kato_transport}\nLet \\(W(t)\\) solve\n\\[\n\\dot W(t)=K(t)W(t),\n\\qquad\nW(0)=I.\n\\]\nThen \\(W(t)\\) is unitary and\n\\[\nP_a(t)W(t)=W(t)P_a(0)\n\\]\nfor every packet \\(a\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:022}\nSkew adjointness gives\n\\[\n\\frac{\\dd}{\\dd t}W(t)^*W(t)=0.\n\\]\nThus \\(W(t)\\) is unitary. Next,\n\\[\n\\frac{\\dd}{\\dd t}\n\\left(W(t)^*P_a(t)W(t)\\right)\n=\nW(t)^*\n\\left(\n\\dot P_a-[K,P_a]\n\\right)\nW(t)=0.\n\\]\nThe expression equals \\(P_a(0)\\) at \\(t=0\\), which proves the intertwining identity.\n\\phantomsection\\label{strong:proof:022:end}\\end{proof}","\\begin{theorem}[Moving frame Hamiltonian]\\label{strong:theorem:014}\\label{strong:thm:moving_frame}\nLet \\(U(t)\\) solve\n\\[\n\\ii\\dot U(t)=H(t)U(t),\n\\qquad\nU(0)=I,\n\\]\nwhere \\(H(t)=H(t)^*\\) is continuous on the compact time interval under consideration. Set\n\\[\nV(t)=W(t)^*U(t).\n\\]\nThen\n\\[\n\\ii\\dot V(t)=H_{\\rm mov}(t)V(t),\n\\]\nwhere\n\\[\nH_{\\rm mov}(t)\n=\nW(t)^*H(t)W(t)-\\ii W(t)^*\\dot W(t)\n=\nW(t)^*\\bigl(H(t)-\\ii K(t)\\bigr)W(t).\n\\]\nThe moving frame Hamiltonian is self adjoint.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:023}\nDifferentiate \\(V=W^*U\\). Since \\(\\dot W=KW\\) and \\(K^*=-K\\),\n\\[\n\\dot W^*=-W^*K.\n\\]\nSubstitution gives the equation. The first term in \\(H_{\\rm mov}\\) is self adjoint. The operator \\(-\\ii W^*\\dot W\\) is self adjoint because \\(W^*\\dot W\\) is skew adjoint.\n\\phantomsection\\label{strong:proof:023:end}\\end{proof}","\\begin{theorem}[Moving packet leakage]\\label{strong:theorem:015}\\label{strong:thm:moving_leakage}\nFix a packet \\(P(t)=P_a(t)\\) and put \\(Q(t)=I-P(t)\\). For \\(0\\leq t\\leq T\\),\n\\[\n\\norm{Q(t)U(t)P(0)}\n\\leq\n\\int_0^t\n\\norm{Q(s)\\bigl(H(s)-\\ii K(s)\\bigr)P(s)}\n\\dd s.\n\\]\nIn particular, the physical evolution follows the moving packet exactly on \\([0,T]\\) when the displayed off packet block vanishes for every time in that interval.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:024}\nIn the Kato frame, \\(P(0)\\) is fixed and\n\\[\nQ(t)U(t)P(0)\n=\nW(t)Q(0)V(t)P(0).\n\\]\nApply the time dependent version of the Duhamel argument in \\cref{strong:thm:fixed_leakage} to \\(H_{\\rm mov}(t)\\). The off diagonal block is\n\\[\nQ(0)H_{\\rm mov}(s)P(0)\n=\nW(s)^*Q(s)\\bigl(H(s)-\\ii K(s)\\bigr)P(s)W(s).\n\\]\nUnitary conjugation preserves the norm.\n\\phantomsection\\label{strong:proof:024:end}\\end{proof}","Let \\((P_a)\\) and \\((Q_a)\\) be two complete packet systems with the same index set. Define\n\\[\nS(\\Qcal,\\Pcal)=\\sum_aQ_aP_a.\n\\]","\\begin{theorem}[Direct packet rotation]\\label{strong:theorem:016}\\label{strong:thm:direct_rotation}\nAssume\n\\[\n\\norm{S(\\Qcal,\\Pcal)-I}<1.\n\\]\nThen \\(S(\\Qcal,\\Pcal)\\) is invertible. Its unitary polar factor\n\\[\n\\Ucal(\\Qcal,\\Pcal)\n=\nS(S^*S)^{-1/2}\n\\]\nsatisfies\n\\[\nQ_a\\Ucal(\\Qcal,\\Pcal)\n=\n\\Ucal(\\Qcal,\\Pcal)P_a\n\\]\nfor every \\(a\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:025}\nThe norm condition gives invertibility by the Neumann series. The identity\n\\[\nQ_aS=SP_a=Q_aP_a\n\\]\nshows that \\(S^*S\\) commutes with every \\(P_a\\). Its inverse square root also commutes with every \\(P_a\\). Hence\n\\[\nQ_a\\Ucal\n=Q_aS(S^*S)^{-1/2}\n=SP_a(S^*S)^{-1/2}\n=\\Ucal P_a.\n\\]\nThe polar factor is unitary because \\(S\\) is invertible.\n\\phantomsection\\label{strong:proof:025:end}\\end{proof}","\\begin{theorem}[Convergence of short packet rotations]\\label{strong:theorem:017}\\label{strong:thm:short_rotation_convergence}\nLet \\((P_a(t))\\) be twice continuously differentiable on \\([0,T]\\). For a partition\n\\[\n0=t_0<t_1<\\cdots<t_n=T,\n\\]\nlet\n\\[\nW_\\pi\n=\n\\Ucal\\bigl(\\Pcal(t_n),\\Pcal(t_{n-1})\\bigr)\n\\cdots\n\\Ucal\\bigl(\\Pcal(t_1),\\Pcal(t_0)\\bigr).\n\\]\nFor sufficiently fine partitions the factors are defined, and\n\\[\n\\norm{W_\\pi-W(T)}\n\\leq\nC_T\\abs{\\pi}\n\\]\nfor a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.\n\\end{theorem}","em}[Convergence of short packet rotations]\\label{strong:theorem:017}\\label{strong:thm:short_rotation_convergence}\nLet \\((P_a(t))\\) be twice continuously differentiable on \\([0,T]\\). For a partition\n\\[\n0=t_0<t_1<\\cdots<t_n=T,\n\\]\nlet\n\\[\nW_\\pi\n=\n\\Ucal\\bigl(\\Pcal(t_n),\\Pcal(t_{n-1})\\bigr)\n\\cdots\n\\Ucal\\bigl(\\Pcal(t_1),\\Pcal(t_0)\\bigr).\n\\]\nFor sufficiently fine partitions the factors are defined, and\n\\[\n\\norm{W_\\pi-W(T)}\n\\leq\nC_T\\abs{\\pi}\n\\]\nfor a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:026}\nTaylor expansion gives\n\\[\nS\\bigl(\\Pcal(t+h),\\Pcal(t)\\bigr)\n=\nI+hK(t)+O(h^2)\n\\]\nuniformly on \\([0,T]\\). Since \\(K(t)^*=-K(t)\\),\n\\[\nS^*S=I+O(h^2),\n\\]\nso the polar factor satisfies\n\\[\n\\Ucal\\bigl(\\Pcal(t+h),\\Pcal(t)\\bigr)\n=\nI+hK(t)+O(h^2).\n\\]\nThis gives a first order approximation to \\(\\dot W=KW\\) over a single time step. A telescoping product estimate over the partition gives a total error bounded by a constant times \\(\\sum_j(t_{j+1}-t_j)^2\\), which is at most \\(T\\abs{\\pi}\\).\n\\phantomsection\\label{strong:proof:026:end}\\end{proof}","es\n\\[\nS\\bigl(\\Pcal(t+h),\\Pcal(t)\\bigr)\n=\nI+hK(t)+O(h^2)\n\\]\nuniformly on \\([0,T]\\). Since \\(K(t)^*=-K(t)\\),\n\\[\nS^*S=I+O(h^2),\n\\]\nso the polar factor satisfies\n\\[\n\\Ucal\\bigl(\\Pcal(t+h),\\Pcal(t)\\bigr)\n=\nI+hK(t)+O(h^2).\n\\]\nThis gives a first order approximation to \\(\\dot W=KW\\) over a single time step. A telescoping product estimate over the partition gives a total error bounded by a constant times \\(\\sum_j(t_{j+1}-t_j)^2\\), which is at most \\(T\\abs{\\pi}\\).\n\\phantomsection\\label{strong:proof:026:end}\\end{proof}","K(t)+O(h^2)\n\\]\nuniformly on \\([0,T]\\). Since \\(K(t)^*=-K(t)\\),\n\\[\nS^*S=I+O(h^2),\n\\]\nso the polar factor satisfies\n\\[\n\\Ucal\\bigl(\\Pcal(t+h),\\Pcal(t)\\bigr)\n=\nI+hK(t)+O(h^2).\n\\]\nThis gives a first order approximation to \\(\\dot W=KW\\) over a single time step. A telescoping product estimate over the partition gives a total error bounded by a constant times \\(\\sum_j(t_{j+1}-t_j)^2\\), which is at most \\(T\\abs{\\pi}\\).\n\\phantomsection\\label{strong:proof:026:end}\\end{proof}","Let \\(P(x)\\) be a smooth rank \\(r\\) orthogonal projector over a parameter manifold. Choose a local isometry\n\\[\nV(x):\\CC^r\\longrightarrow\\Hcal,\n\\qquad\nV(x)V(x)^*=P(x).\n\\]","\\begin{theorem}[Nonabelian packet connection]\\label{strong:theorem:018}\\label{strong:thm:nonabelian_connection}\nThe one form with matrix values\n\\[\n\\Acal=V^*\\dd V\n\\]\nis skew adjoint. Under a frame change \\(V'=VG\\), where \\(G(x)\\in U(r)\\),\n\\[\n\\Acal'\n=\nG^*\\Acal G+G^*\\dd G.\n\\]\nIts curvature\n\\[\n\\Fcal=\\dd\\Acal+\\Acal\\wedge\\Acal\n\\]\nsatisfies\n\\[\n\\Fcal=V^*(\\dd P\\wedge\\dd P)V.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:027}\nDifferentiate \\(V^*V=I\\) to obtain \\(\\Acal^*=-\\Acal\\). The gauge transformation law follows by differentiating \\(VG\\). Since \\(P=VV^*\\) and \\((I-P)\\dd V=\\dd P\\,V\\), direct expansion gives\n\\[\n\\dd\\Acal+\\Acal\\wedge\\Acal\n=\n\\dd V^*(I-P)\\wedge\\dd V\n=\nV^*(\\dd P\\wedge\\dd P)V.\n\\]\n\\phantomsection\\label{strong:proof:027:end}\\end{proof}","\\begin{corollary}[Bloch sphere curvature]\\label{strong:corollary:007}\\label{strong:cor:bloch_curvature}\nLet\n\\[\nP(\\theta,\\phi)\n=\n\\frac12\\left(I+\\bm n(\\theta,\\phi)\\cdot\\bm\\sigma\\right),\n\\]\nwhere\n\\[\n\\bm n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nWith the convention\n\\[\n\\Fcal_{\\theta\\phi}\n=-\\ii\\Tr\\bigl(P[\\partial_\\theta P,\\partial_\\phi P]\\bigr),\n\\]\none has\n\\[\n\\Fcal_{\\theta\\phi}=\\frac12\\sin\\theta\n\\]\nand\n\\[\n\\frac1{2\\pi}\n\\int_0^{2\\pi}\\int_0^\\pi\n\\Fcal_{\\theta\\phi}\\dd\\theta\\dd\\phi\n=1.\n\\]\n\\end{corollary}","[Bloch sphere curvature]\\label{strong:corollary:007}\\label{strong:cor:bloch_curvature}\nLet\n\\[\nP(\\theta,\\phi)\n=\n\\frac12\\left(I+\\bm n(\\theta,\\phi)\\cdot\\bm\\sigma\\right),\n\\]\nwhere\n\\[\n\\bm n=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta).\n\\]\nWith the convention\n\\[\n\\Fcal_{\\theta\\phi}\n=-\\ii\\Tr\\bigl(P[\\partial_\\theta P,\\partial_\\phi P]\\bigr),\n\\]\none has\n\\[\n\\Fcal_{\\theta\\phi}=\\frac12\\sin\\theta\n\\]\nand\n\\[\n\\frac1{2\\pi}\n\\int_0^{2\\pi}\\int_0^\\pi\n\\Fcal_{\\theta\\phi}\\dd\\theta\\dd\\phi\n=1.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:028}\nThe Pauli identity\n\\[\n(\\bm a\\cdot\\bm\\sigma)(\\bm b\\cdot\\bm\\sigma)\n=(\\bm a\\cdot\\bm b)I\n+\\ii(\\bm a\\times\\bm b)\\cdot\\bm\\sigma\n\\]\ngives\n\\[\n-\\ii\\Tr\\bigl(P[\\partial_\\theta P,\\partial_\\phi P]\\bigr)\n=\\frac12\n\\bm n\\cdot\n(\\partial_\\theta\\bm n\\times\\partial_\\phi\\bm n)\n=\\frac12\\sin\\theta.\n\\]\nIntegrating the displayed curvature over the sphere and applying the stated normalization gives one.\n\\phantomsection\\label{strong:proof:028:end}\\end{proof}","Density matrices and observables carry both an input packet and an output packet. Equip \\(\\End(\\Hcal)\\) with the Hilbert--Schmidt inner product\n\\[\n\\ip{X}{Y}_{\\HS}=\\Tr(X^*Y).\n\\]","\\begin{definition}[Two sided operator packet]\\label{strong:definition:003}\\label{strong:def:operator_packet}\nFor a complete orthogonal family \\((P_a)\\), define\n\\[\n\\Pcal_{ab}(X)=P_aXP_b.\n\\]\nThe range\n\\[\n\\Bcal_{ab}=P_a\\End(\\Hcal)P_b\n\\]\nis the operator packet with output address \\(a\\) and input address \\(b\\).\n\\end{definition}","\\begin{theorem}[Complete operator packet system]\\label{strong:theorem:019}\\label{strong:thm:operator_packets}\nThe superoperators \\((\\Pcal_{ab})_{a,b\\in I}\\) are mutually orthogonal projections on Hilbert--Schmidt operator space. They satisfy\n\\[\nX=\\sum_{a,b}P_aXP_b\n\\]\nand\n\\[\n\\norm{X}_{\\HS}^2\n=\n\\sum_{a,b}\\norm{P_aXP_b}_{\\HS}^2.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:029}\nIdempotence and completeness follow from the projector identities. For Hilbert--Schmidt orthogonality,\n\\[\n\\begin{aligned}\n\\ip{P_aXP_b}{P_cYP_d}_{\\HS}\n&=\n\\Tr(P_bX^*P_aP_cYP_d)\\\\\n&=\n\\delta_{ac}\\delta_{bd}\n\\Tr(P_bX^*P_aYP_b).\n\\end{aligned}\n\\]\nSince the packets are orthogonal and sum to the original operator, their squared norms add to its squared norm.\n\\phantomsection\\label{strong:proof:029:end}\\end{proof}","\\begin{definition}[Packet dephasing]\\label{strong:definition:004}\\label{strong:def:dephasing}\nThe packet dephasing map is\n\\[\n\\Dcal_{\\Pcal}(X)=\\sum_aP_aXP_a.\n\\]\n\\end{definition}","\\begin{theorem}[Dephasing as conditional expectation]\\label{strong:theorem:020}\\label{strong:thm:dephasing_expectation}\nThe map \\(\\Dcal_{\\Pcal}\\) is completely positive, trace preserving, unital, and idempotent. It is the Hilbert--Schmidt orthogonal projection onto the block diagonal algebra\n\\[\n\\Acal_{\\Pcal}\n=\n\\{X:[X,P_a]=0\\text{ for every }a\\}.\n\\]\nIn particular,\n\\[\n\\norm{\\Dcal_{\\Pcal}(X)}_{\\HS}\n\\leq\n\\norm{X}_{\\HS}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:030}\nThe projectors \\(P_a\\) form a Kraus family because \\(\\sum_aP_a^*P_a=I\\). Thus the map is completely positive and trace preserving. It is unital for the same reason. Orthogonality and idempotence follow from \\cref{strong:thm:operator_packets}. Its range is the sum of the diagonal operator packets. This is exactly the commutant of the projector family.\n\\phantomsection\\label{strong:proof:030:end}\\end{proof}","\\begin{corollary}[Block diagonal measurement invariance]\\label{strong:corollary:008}\\label{strong:cor:measurement_invariance}\nLet \\((E_k)\\) be a positive operator valued measurement, and suppose\n\\[\nE_k=\\Dcal_{\\Pcal}(E_k)\n\\]\nfor every \\(k\\). Then every density matrix \\(\\rho\\) satisfies\n\\[\n\\Tr(E_k\\rho)\n=\n\\Tr\\bigl(E_k\\Dcal_{\\Pcal}(\\rho)\\bigr).\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:031}\nSince \\(\\Dcal_{\\Pcal}\\) is self adjoint for the Hilbert--Schmidt inner product,\n\\[\n\\Tr(E_k\\rho)\n=\n\\Tr\\bigl(\\Dcal_{\\Pcal}(E_k)\\rho\\bigr)\n=\n\\Tr\\bigl(E_k\\Dcal_{\\Pcal}(\\rho)\\bigr).\n\\]\n\\phantomsection\\label{strong:proof:031:end}\\end{proof}","Let \\(\\Phi:\\End(\\Hcal_X)\\to\\End(\\Hcal_Y)\\) be linear. Choose packet systems \\((P_a)\\) on \\(\\Hcal_X\\) and \\((Q_c)\\) on \\(\\Hcal_Y\\). Define\n\\[\n\\Phi_{cd,ab}\n=\n\\Qcal_{cd}\\Phi\\Pcal_{ab}.\n\\]","\\begin{theorem}[Exact channel block calculus]\\label{strong:theorem:021}\\label{strong:thm:channel_blocks}\nEvery linear map has the exact reconstruction\n\\[\n\\Phi=\\sum_{a,b,c,d}\\Phi_{cd,ab}.\n\\]\nIf \\(\\Psi:\\End(\\Hcal_Y)\\to\\End(\\Hcal_Z)\\), then\n\\[\n(\\Psi\\Phi)_{ef,ab}\n=\n\\sum_{c,d}\n\\Psi_{ef,cd}\\Phi_{cd,ab}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:032}\nInsert the complete operator packet resolutions before and after \\(\\Phi\\). For composition, insert the middle resolution between \\(\\Psi\\) and \\(\\Phi\\) and collect the output and input packets.\n\\phantomsection\\label{strong:proof:032:end}\\end{proof}","\\begin{standardfact}[Finite Kraus and Choi criteria]\\label{strong:standardfact:001}\\label{strong:fact:kraus_choi}\nA linear map \\(\\Phi:M_n(\\CC)\\to M_m(\\CC)\\) is completely positive exactly when it has a Kraus representation\n\\[\n\\Phi(X)=\\sum_rK_rXK_r^*.\n\\]\nIt is trace preserving exactly when\n\\[\n\\sum_rK_r^*K_r=I.\n\\]\nEquivalently, its Choi matrix\n\\[\nJ(\\Phi)\n=\n\\sum_{i,j}\\Phi(\\ket{i}\\bra{j})\\otimes\\ket{i}\\bra{j}\n\\]\nis positive, and trace preservation is\n\\[\n\\Tr_{\\rm out}J(\\Phi)=I_{\\rm in}.\n\\]\n\\end{standardfact}","label{strong:standardfact:001}\\label{strong:fact:kraus_choi}\nA linear map \\(\\Phi:M_n(\\CC)\\to M_m(\\CC)\\) is completely positive exactly when it has a Kraus representation\n\\[\n\\Phi(X)=\\sum_rK_rXK_r^*.\n\\]\nIt is trace preserving exactly when\n\\[\n\\sum_rK_r^*K_r=I.\n\\]\nEquivalently, its Choi matrix\n\\[\nJ(\\Phi)\n=\n\\sum_{i,j}\\Phi(\\ket{i}\\bra{j})\\otimes\\ket{i}\\bra{j}\n\\]\nis positive, and trace preservation is\n\\[\n\\Tr_{\\rm out}J(\\Phi)=I_{\\rm in}.\n\\]\n\\end{standardfact}","Let \\(S\\) be unitary and \\(N\\ge1\\) an integer with \\(S^N=I\\), and define the conjugation action\n\\[\n\\Ad_S(X)=SXS^*.\n\\]","\\begin{theorem}[Coherence order projectors]\\label{strong:theorem:022}\\label{strong:thm:coherence_projectors}\nFor \\(q\\in\\ZZ/N\\ZZ\\), define\n\\[\n\\Lcal_q(X)\n=\n\\frac1N\n\\sum_{r=0}^{N-1}\n\\omega^{-qr}S^rXS^{-r}.\n\\]\nThen the \\(\\Lcal_q\\) form a complete orthogonal packet system on Hilbert--Schmidt operator space. Their ranges are the operators satisfying\n\\[\nSXS^*=\\omega^qX.\n\\]\nIf \\(P_j\\) are the state charge projectors, then\n\\[\n\\Ran\\Lcal_q\n=\n\\bigoplus_jP_{j+q}\\End(\\Hcal)P_j.\n\\]\n\\end{theorem}","rence order projectors]\\label{strong:theorem:022}\\label{strong:thm:coherence_projectors}\nFor \\(q\\in\\ZZ/N\\ZZ\\), define\n\\[\n\\Lcal_q(X)\n=\n\\frac1N\n\\sum_{r=0}^{N-1}\n\\omega^{-qr}S^rXS^{-r}.\n\\]\nThen the \\(\\Lcal_q\\) form a complete orthogonal packet system on Hilbert--Schmidt operator space. Their ranges are the operators satisfying\n\\[\nSXS^*=\\omega^qX.\n\\]\nIf \\(P_j\\) are the state charge projectors, then\n\\[\n\\Ran\\Lcal_q\n=\n\\bigoplus_jP_{j+q}\\End(\\Hcal)P_j.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:033}\nApply the cyclic character projector formula to the unitary representation \\(\\Ad_S\\) on operator space. The last identity follows from \\cref{strong:thm:charge_selection} applied to each operator packet.\n\\phantomsection\\label{strong:proof:033:end}\\end{proof}","\\begin{theorem}[Covariant channels preserve coherence order]\\label{strong:theorem:023}\\label{strong:thm:covariant_channel}\nSuppose\n\\[\n\\Phi(SXS^*)=S\\Phi(X)S^*\n\\]\nfor every \\(X\\). Then\n\\[\n\\Phi\\Lcal_q=\\Lcal_q\\Phi\n\\]\nfor every \\(q\\). Thus \\(\\Phi\\) cannot mix distinct coherence orders.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:034}\nThe covariance identity says that \\(\\Phi\\) commutes with every power of \\(\\Ad_S\\). It therefore commutes with each finite character average \\(\\Lcal_q\\).\n\\phantomsection\\label{strong:proof:034:end}\\end{proof}","\\begin{theorem}[Choi symmetry criterion]\\label{strong:theorem:024}\\label{strong:thm:choi_covariance}\nLet \\(U:G\\to\\Ucal(\\Hcal)\\) be a finite group representation. A channel \\(\\Phi\\) is covariant,\n\\[\n\\Phi\\bigl(U(g)XU(g)^*\\bigr)\n=\nU(g)\\Phi(X)U(g)^*,\n\\]\nif and only if\n\\[\n\\bigl(U(g)\\otimes\\overline{U(g)}\\bigr)\nJ(\\Phi)\n\\bigl(U(g)^*\\otimes U(g)^{\\mathsf T}\\bigr)\n=\nJ(\\Phi)\n\\]\nfor every \\(g\\in G\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:035}\nLet\n\\[\n\\Omega=\\sum_i\\ket{i}\\otimes\\ket{i}.\n\\]\nThen\n\\[\nJ(\\Phi)=(\\Phi\\otimes\\id)(\\ket{\\Omega}\\bra{\\Omega}).\n\\]\nThe vector \\(\\Omega\\) is invariant under \\(U\\otimes\\overline U\\). Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of\n\\[\n\\Ad_{U(g)}\\circ\\Phi\\circ\\Ad_{U(g)^*}.\n\\]\nThe Choi correspondence is injective, so invariance implies that this map equals \\(\\Phi\\), which is covariance.\n\\phantomsection\\label{strong:proof:035:end}\\end{proof}","i}\\otimes\\ket{i}.\n\\]\nThen\n\\[\nJ(\\Phi)=(\\Phi\\otimes\\id)(\\ket{\\Omega}\\bra{\\Omega}).\n\\]\nThe vector \\(\\Omega\\) is invariant under \\(U\\otimes\\overline U\\). Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of\n\\[\n\\Ad_{U(g)}\\circ\\Phi\\circ\\Ad_{U(g)^*}.\n\\]\nThe Choi correspondence is injective, so invariance implies that this map equals \\(\\Phi\\), which is covariance.\n\\phantomsection\\label{strong:proof:035:end}\\end{proof}","\\begin{corollary}[Choi packet decomposition]\\label{strong:corollary:009}\\label{strong:cor:choi_packets}\nThe Choi matrix of a covariant channel is block diagonal with respect to the isotypic decomposition of \\(U\\otimes\\overline U\\).\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:036}\nApply \\cref{strong:thm:isotypic_projectors} to the representation \\(U\\otimes\\overline U\\) and use \\cref{strong:thm:choi_covariance}.\n\\phantomsection\\label{strong:proof:036:end}\\end{proof}","\\begin{example}[Amplitude damping]\\label{strong:example:002}\\label{strong:ex:amplitude_damping}\nFor \\(0\\leq\\gamma\\leq1\\), let\n\\[\nK_0=\n\\begin{pmatrix}\n1&0","\\begin{example}[Amplitude damping]\\label{strong:example:002}\\label{strong:ex:amplitude_damping}\nFor \\(0\\leq\\gamma\\leq1\\), let\n\\[\nK_0=\n\\begin{pmatrix}\n1&0\\\\\n0&\\sqrt{1-\\gamma}\n\\end{pmatrix},\n\\qquad\nK_1=\n\\begin{pmatrix}\n0&\\sqrt\\gamma\\\\\n0&0\n\\end{pmatrix}.\n\\]\nThe channel\n\\[\n\\Phi_\\gamma(\\rho)=K_0\\rho K_0^*+K_1\\rho K_1^*\n\\]\nis trace preserving. It sends the off diagonal packet \\(\\ket0\\bra1\\) to \\(\\sqrt{1-\\gamma}\\ket0\\bra1\\), while the excited population transfers partly into the ground population.\n\\end{example}","0&0\n\\end{pmatrix}.\n\\]\nThe channel\n\\[\n\\Phi_\\gamma(\\rho)=K_0\\rho K_0^*+K_1\\rho K_1^*\n\\]\nis trace preserving. It sends the off diagonal packet \\(\\ket0\\bra1\\) to \\(\\sqrt{1-\\gamma}\\ket0\\bra1\\), while the excited population transfers partly into the ground population.\n\\end{example}","\\begin{standardfact}[Finite Lindblad generator]\\label{strong:standardfact:002}\\label{strong:fact:lindblad}\nA norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr\\\"odinger picture, has a generator of the form\n\\[\n\\Gcal(X)\n=\n-\\ii[H,X]\n+\n\\sum_r\n\\left(\nL_rXL_r^*\n-\n\\frac12L_r^*L_rX\n-\n\\frac12XL_r^*L_r\n\\right),\n\\]\nwhere \\(H=H^*\\). Conversely, every operator of this form generates a completely positive trace preserving semigroup \\cite{Lindblad1976}.\n\\end{standardfact}","\\begin{theorem}[Charge preservation by a Lindblad generator]\\label{strong:theorem:025}\\label{strong:thm:lindblad_charge}\nLet \\(S\\) be unitary and \\(N\\ge1\\) an integer with \\(S^N=I\\). Assume\n\\[\nSHS^*=H\n\\]\nand, for every jump operator, there is a charge \\(q_r\\) such that\n\\[\nSL_rS^*=\\omega^{q_r}L_r.\n\\]\nThen the Lindblad generator \\(\\Gcal\\) preserves every coherence order:\n\\[\n\\Gcal\\Lcal_p=\\Lcal_p\\Gcal.\n\\]\nThe quantum dynamical semigroup \\(\\e^{t\\Gcal}\\) has the same property.\n\\end{theorem}","a Lindblad generator]\\label{strong:theorem:025}\\label{strong:thm:lindblad_charge}\nLet \\(S\\) be unitary and \\(N\\ge1\\) an integer with \\(S^N=I\\). Assume\n\\[\nSHS^*=H\n\\]\nand, for every jump operator, there is a charge \\(q_r\\) such that\n\\[\nSL_rS^*=\\omega^{q_r}L_r.\n\\]\nThen the Lindblad generator \\(\\Gcal\\) preserves every coherence order:\n\\[\n\\Gcal\\Lcal_p=\\Lcal_p\\Gcal.\n\\]\nThe quantum dynamical semigroup \\(\\e^{t\\Gcal}\\) has the same property.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:037}\nLet \\(X\\) have coherence order \\(p\\). The Hamiltonian commutator has order \\(p\\) because \\(H\\) has charge zero. The jump term \\(L_rXL_r^*\\) has charge\n\\[\nq_r+p-q_r=p.\n\\]\nAlso \\(L_r^*L_r\\) has charge zero, so both anticommutator terms have charge \\(p\\). Thus \\(\\Gcal\\) preserves \\(\\Ran\\Lcal_p\\). The exponential series then shows that \\(\\e^{t\\Gcal}\\) preserves the same packets.\n\\phantomsection\\label{strong:proof:037:end}\\end{proof}","Let \\(P\\) be the orthogonal projector onto a code space\n\\[\n\\Ccal=P\\Hcal,\n\\qquad\nk=\\dim\\Ccal\\ge1.\n\\]\nThe compression of \\(A\\in\\End(\\Hcal)\\) to the code is \\(PAP\\in\\End(\\Ccal)\\). The zero code has only the zero supported operator and no normalized code state. Part III treats that branch in the mathematical completion.","\\begin{definition}[Code compression obstruction]\\label{strong:definition:005}\\label{strong:def:code_obstruction}\nLet\n\\[\nq_P:\\End(\\Ccal)\\longrightarrow\\End(\\Ccal)/(\\CC I_{\\Ccal})\n\\]\nbe the quotient map. Define\n\\[\n\\Ob_P(A)=q_P(PAP).\n\\]\nEquivalently, one may use the traceless representative\n\\[\n\\ob_P(A)\n=\nPAP-\n\\frac{\\Tr(PAP)}{k}P.\n\\]\n\\end{definition}","The class vanishes exactly when \\(A\\) acts as a scalar after compression to the code. Such an operator cannot distinguish normalized code vectors by expectation value.","Exact correction of an error span means that one completely positive trace preserving recovery $\\Rcal$ satisfies $\\Rcal(F\\rho F^*)=q_F\\rho$ for every $F$ in that span and every code-supported matrix $\\rho$, with a nonnegative scalar $q_F$ depending on $F$ but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restores the code input with total scalar one. Individual unnormalized errors are not required to have unit recovery weight.","\\begin{theorem}[Knill--Laflamme as obstruction vanishing]\\label{strong:theorem:026}\\label{strong:thm:kl_obstruction}\nLet \\(E_1,\\ldots,E_r\\) be an error family. The code \\(\\Ccal\\) corrects their linear span exactly if and only if\n\\[\n\\Ob_P(E_a^*E_b)=0\n\\]\nfor all \\(a,b\\). Equivalently, there is a matrix \\(c=(c_{ab})\\) such that\n\\[\nPE_a^*E_bP=c_{ab}P\n\\]\nfor all \\(a,b\\).\n\\end{theorem}","\\begin{theorem}[Orthogonal syndrome construction]\\label{strong:theorem:027}\\label{strong:thm:syndrome_construction}\nAssume the Knill--Laflamme identities. Then the coefficient matrix \\(c\\) is positive semidefinite. Choose a unitary matrix \\(W\\) such that\n\\[\nWcW^*=\\diag(\\lambda_1,\\ldots,\\lambda_r),\n\\qquad\n\\lambda_j\\geq0,\n\\]\nand define\n\\[\nF_j=\\sum_a\\overline{W_{ja}}E_a.\n\\]\nFor every \\(j\\) with \\(\\lambda_j>0\\), the operator\n\\[\nV_j=\\lambda_j^{-1/2}F_jP\n\\]\nis an isometry from the code into \\(\\Hcal\\). Its range projector\n\\[\nQ_j=V_jV_j^*\n\\]\nis a syndrome packet, and the nonzero \\(Q_j\\) are pairwise orthogonal.\n\\end{theorem}","syndrome construction]\\label{strong:theorem:027}\\label{strong:thm:syndrome_construction}\nAssume the Knill--Laflamme identities. Then the coefficient matrix \\(c\\) is positive semidefinite. Choose a unitary matrix \\(W\\) such that\n\\[\nWcW^*=\\diag(\\lambda_1,\\ldots,\\lambda_r),\n\\qquad\n\\lambda_j\\geq0,\n\\]\nand define\n\\[\nF_j=\\sum_a\\overline{W_{ja}}E_a.\n\\]\nFor every \\(j\\) with \\(\\lambda_j>0\\), the operator\n\\[\nV_j=\\lambda_j^{-1/2}F_jP\n\\]\nis an isometry from the code into \\(\\Hcal\\). Its range projector\n\\[\nQ_j=V_jV_j^*\n\\]\nis a syndrome packet, and the nonzero \\(Q_j\\) are pairwise orthogonal.\n\\end{theorem}",":syndrome_construction}\nAssume the Knill--Laflamme identities. Then the coefficient matrix \\(c\\) is positive semidefinite. Choose a unitary matrix \\(W\\) such that\n\\[\nWcW^*=\\diag(\\lambda_1,\\ldots,\\lambda_r),\n\\qquad\n\\lambda_j\\geq0,\n\\]\nand define\n\\[\nF_j=\\sum_a\\overline{W_{ja}}E_a.\n\\]\nFor every \\(j\\) with \\(\\lambda_j>0\\), the operator\n\\[\nV_j=\\lambda_j^{-1/2}F_jP\n\\]\nis an isometry from the code into \\(\\Hcal\\). Its range projector\n\\[\nQ_j=V_jV_j^*\n\\]\nis a syndrome packet, and the nonzero \\(Q_j\\) are pairwise orthogonal.\n\\end{theorem}","the Knill--Laflamme identities. Then the coefficient matrix \\(c\\) is positive semidefinite. Choose a unitary matrix \\(W\\) such that\n\\[\nWcW^*=\\diag(\\lambda_1,\\ldots,\\lambda_r),\n\\qquad\n\\lambda_j\\geq0,\n\\]\nand define\n\\[\nF_j=\\sum_a\\overline{W_{ja}}E_a.\n\\]\nFor every \\(j\\) with \\(\\lambda_j>0\\), the operator\n\\[\nV_j=\\lambda_j^{-1/2}F_jP\n\\]\nis an isometry from the code into \\(\\Hcal\\). Its range projector\n\\[\nQ_j=V_jV_j^*\n\\]\nis a syndrome packet, and the nonzero \\(Q_j\\) are pairwise orthogonal.\n\\end{theorem}","ix \\(c\\) is positive semidefinite. Choose a unitary matrix \\(W\\) such that\n\\[\nWcW^*=\\diag(\\lambda_1,\\ldots,\\lambda_r),\n\\qquad\n\\lambda_j\\geq0,\n\\]\nand define\n\\[\nF_j=\\sum_a\\overline{W_{ja}}E_a.\n\\]\nFor every \\(j\\) with \\(\\lambda_j>0\\), the operator\n\\[\nV_j=\\lambda_j^{-1/2}F_jP\n\\]\nis an isometry from the code into \\(\\Hcal\\). Its range projector\n\\[\nQ_j=V_jV_j^*\n\\]\nis a syndrome packet, and the nonzero \\(Q_j\\) are pairwise orthogonal.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:040}\nFor \\(z\\in\\CC^r\\), let \\(F_z=\\sum_az_aE_a\\). Then\n\\[\nPF_z^*F_zP=(z^*cz)P\\geq0,\n\\]\nso \\(c\\geq0\\). The unitary change of error coordinates gives\n\\[\nPF_j^*F_\\ell P\n=\n\\lambda_j\\delta_{j\\ell}P.\n\\]\nThus\n\\[\nV_j^*V_\\ell=\\delta_{j\\ell}P.\n\\]\nEach \\(V_j\\) is an isometry on \\(P\\Hcal\\), and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.\n\\phantomsection\\label{strong:proof:040:end}\\end{proof}","\\begin{theorem}[Exact recovery on the error span]\\label{strong:theorem:028}\\label{strong:thm:exact_recovery}\nUnder the hypotheses of \\cref{strong:thm:syndrome_construction}, define on the syndrome support\n\\[\nQ=\\sum_{\\lambda_j>0}Q_j\n\\]\nthe completely positive map\n\\[\n\\Rcal_0(X)\n=\n\\sum_{\\lambda_j>0}\nV_j^*Q_jXQ_jV_j.\n\\]\nIt extends to a quantum channel \\(\\Rcal\\) on \\(\\End(\\Hcal)\\). If\n\\[\n\\Phi(X)=\\sum_aE_aXE_a^*\n\\]\nis trace preserving on code states, then\n\\[\n\\Rcal\\Phi(\\rho)=\\rho\n\\]\nfor every density matrix \\(\\rho=P\\rho P\\).\n\\end{theorem}","strong:theorem:028}\\label{strong:thm:exact_recovery}\nUnder the hypotheses of \\cref{strong:thm:syndrome_construction}, define on the syndrome support\n\\[\nQ=\\sum_{\\lambda_j>0}Q_j\n\\]\nthe completely positive map\n\\[\n\\Rcal_0(X)\n=\n\\sum_{\\lambda_j>0}\nV_j^*Q_jXQ_jV_j.\n\\]\nIt extends to a quantum channel \\(\\Rcal\\) on \\(\\End(\\Hcal)\\). If\n\\[\n\\Phi(X)=\\sum_aE_aXE_a^*\n\\]\nis trace preserving on code states, then\n\\[\n\\Rcal\\Phi(\\rho)=\\rho\n\\]\nfor every density matrix \\(\\rho=P\\rho P\\).\n\\end{theorem}","Under the hypotheses of \\cref{strong:thm:syndrome_construction}, define on the syndrome support\n\\[\nQ=\\sum_{\\lambda_j>0}Q_j\n\\]\nthe completely positive map\n\\[\n\\Rcal_0(X)\n=\n\\sum_{\\lambda_j>0}\nV_j^*Q_jXQ_jV_j.\n\\]\nIt extends to a quantum channel \\(\\Rcal\\) on \\(\\End(\\Hcal)\\). If\n\\[\n\\Phi(X)=\\sum_aE_aXE_a^*\n\\]\nis trace preserving on code states, then\n\\[\n\\Rcal\\Phi(\\rho)=\\rho\n\\]\nfor every density matrix \\(\\rho=P\\rho P\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:041}\nThe map \\(\\Rcal_0\\) is completely positive. It is trace preserving on operators supported in \\(Q\\Hcal\\) because\n\\[\n\\Tr\\bigl(V_j^*Q_jXQ_jV_j\\bigr)\n=\n\\Tr(Q_jXQ_j).\n\\]\nFix a code state \\(\\rho_0\\) and write \\(Q^\\perp=I-Q\\). Then\n\\[\n\\Rcal(X)\n=\n\\Rcal_0(QXQ)\n+\n\\Tr(Q^\\perp X)\\rho_0\n\\]\nis completely positive and trace preserving. It extends \\(\\Rcal_0\\) on operators supported in \\(Q\\Hcal\\).","n\\label{strong:proof:041}\nThe map \\(\\Rcal_0\\) is completely positive. It is trace preserving on operators supported in \\(Q\\Hcal\\) because\n\\[\n\\Tr\\bigl(V_j^*Q_jXQ_jV_j\\bigr)\n=\n\\Tr(Q_jXQ_j).\n\\]\nFix a code state \\(\\rho_0\\) and write \\(Q^\\perp=I-Q\\). Then\n\\[\n\\Rcal(X)\n=\n\\Rcal_0(QXQ)\n+\n\\Tr(Q^\\perp X)\\rho_0\n\\]\nis completely positive and trace preserving. It extends \\(\\Rcal_0\\) on operators supported in \\(Q\\Hcal\\).","Unitary mixing of a Kraus family does not change the channel, so\n\\[\n\\Phi(\\rho)\n=\n\\sum_jF_j\\rho F_j^*\n=\n\\sum_{\\lambda_j>0}\n\\lambda_jV_j\\rho V_j^*.\n\\]\nRecovery gives\n\\[\n\\Rcal\\Phi(\\rho)\n=\n\\left(\\sum_j\\lambda_j\\right)\\rho.\n\\]\nTrace preservation on a code state implies \\(\\sum_j\\lambda_j=1\\).\n\\phantomsection\\label{strong:proof:041:end}\\end{proof}","\\begin{corollary}[Syndrome measurement preserves logical information]\\label{strong:corollary:011}\\label{strong:cor:syndrome_preserves_logic}\nFor a channel trace preserving on code states as in the preceding theorem, the completed projective measurement consisting of \\((Q_j)_{\\lambda_j>0}\\) and \\(I-\\sum_{\\lambda_j>0}Q_j\\) identifies the orthogonal error syndrome without resolving the encoded state inside the code. The complement outcome has probability zero on the corrupted code state.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:042}\nOn syndrome \\(j\\), the encoded state is carried by the isometry \\(V_j\\). The outcome probability is \\(\\lambda_j\\), independent of the normalized code state, while the postmeasurement state is \\(V_j\\rho V_j^*\\).\n\\phantomsection\\label{strong:proof:042:end}\\end{proof}","\\begin{definition}[Approximate Knill--Laflamme defect]\\label{strong:definition:006}\\label{strong:def:approximate_kl}\nFor a finite error family, define the following defect. When the family is empty, assign the maximum the value zero.\n\\[\n\\varepsilon_{\\rm KL}\n=\n\\max_{a,b}\n\\inf_{c\\in\\CC}\n\\norm{PE_a^*E_bP-cP}.\n\\]\n\\end{definition}","\\begin{proposition}[Meaning of the approximate defect]\\label{strong:proposition:003}\\label{strong:prop:approximate_kl_meaning}\nFor each pair \\((a,b)\\), choose a minimizing scalar \\(c_{ab}\\). For unit code vectors \\(\\psi,\\phi\\),\n\\[\n\\abs{\n\\ip{E_a\\psi}{E_b\\phi}\n-c_{ab}\\ip{\\psi}{\\phi}\n}\n\\leq\n\\varepsilon_{\\rm KL}.\n\\]\nThus \\(\\varepsilon_{\\rm KL}\\) controls the deviation of error overlaps from their scalar code form.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:043}\nThe left side is\n\\[\n\\abs{\n\\ip{\\psi}{(PE_a^*E_bP-c_{ab}P)\\phi}\n},\n\\]\nwhich is bounded by the operator norm.\n\\phantomsection\\label{strong:proof:043:end}\\end{proof}","\\begin{example}[Three qubit repetition code]\\label{strong:example:003}\\label{strong:ex:bit_flip_code}\nLet\n\\[\n\\ket{0_L}=\\ket{000},\n\\qquad\n\\ket{1_L}=\\ket{111},\n\\]\nand take the errors\n\\[\nI,\n\\quad\nX\\otimes I\\otimes I,\n\\quad\nI\\otimes X\\otimes I,\n\\quad\nI\\otimes I\\otimes X.\n\\]\nTheir images of the code are four pairwise orthogonal two dimensional syndrome packets whose sum is the full eight dimensional Hilbert space.\n\\end{example}","\\begin{standardfact}[Correctable observable algebras]\\label{strong:standardfact:003}\\label{strong:fact:operator_qec}\nLet \\(\\Phi\\) be a finite completely positive trace preserving channel with Kraus operators \\((E_a)\\). A finite dimensional unital star subalgebra \\(\\Mcal\\subseteq P\\End(\\Hcal)P\\) is correctable when a completely positive trace preserving recovery \\(\\Rcal\\) satisfies \\(P(\\Phi^*\\circ\\Rcal^*)(X)P=X\\) for every \\(X\\in\\Mcal\\), where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product \\(PE_a^*E_bP\\) belongs to the commutant \\(\\Mcal'\\) on the code. The scalar code criterion is the special case \\(\\Mcal=P\\End(\\Hcal)P\\) \\cite{BenyKempfKribs2007}.\n\\end{standardfact}","able observable algebras]\\label{strong:standardfact:003}\\label{strong:fact:operator_qec}\nLet \\(\\Phi\\) be a finite completely positive trace preserving channel with Kraus operators \\((E_a)\\). A finite dimensional unital star subalgebra \\(\\Mcal\\subseteq P\\End(\\Hcal)P\\) is correctable when a completely positive trace preserving recovery \\(\\Rcal\\) satisfies \\(P(\\Phi^*\\circ\\Rcal^*)(X)P=X\\) for every \\(X\\in\\Mcal\\), where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product \\(PE_a^*E_bP\\) belongs to the commutant \\(\\Mcal'\\) on the code. The scalar code criterion is the special case \\(\\Mcal=P\\End(\\Hcal)P\\) \\cite{BenyKempfKribs2007}.\n\\end{standardfact}","e completely positive trace preserving channel with Kraus operators \\((E_a)\\). A finite dimensional unital star subalgebra \\(\\Mcal\\subseteq P\\End(\\Hcal)P\\) is correctable when a completely positive trace preserving recovery \\(\\Rcal\\) satisfies \\(P(\\Phi^*\\circ\\Rcal^*)(X)P=X\\) for every \\(X\\in\\Mcal\\), where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product \\(PE_a^*E_bP\\) belongs to the commutant \\(\\Mcal'\\) on the code. The scalar code criterion is the special case \\(\\Mcal=P\\End(\\Hcal)P\\) \\cite{BenyKempfKribs2007}.\n\\end{standardfact}","hannel with Kraus operators \\((E_a)\\). A finite dimensional unital star subalgebra \\(\\Mcal\\subseteq P\\End(\\Hcal)P\\) is correctable when a completely positive trace preserving recovery \\(\\Rcal\\) satisfies \\(P(\\Phi^*\\circ\\Rcal^*)(X)P=X\\) for every \\(X\\in\\Mcal\\), where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product \\(PE_a^*E_bP\\) belongs to the commutant \\(\\Mcal'\\) on the code. The scalar code criterion is the special case \\(\\Mcal=P\\End(\\Hcal)P\\) \\cite{BenyKempfKribs2007}.\n\\end{standardfact}","r subalgebra \\(\\Mcal\\subseteq P\\End(\\Hcal)P\\) is correctable when a completely positive trace preserving recovery \\(\\Rcal\\) satisfies \\(P(\\Phi^*\\circ\\Rcal^*)(X)P=X\\) for every \\(X\\in\\Mcal\\), where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product \\(PE_a^*E_bP\\) belongs to the commutant \\(\\Mcal'\\) on the code. The scalar code criterion is the special case \\(\\Mcal=P\\End(\\Hcal)P\\) \\cite{BenyKempfKribs2007}.\n\\end{standardfact}","Let \\(P\\) be an orthogonal projector, put \\(Q=I-P\\), and write a self adjoint Hamiltonian as\n\\[\nH=\n\\begin{pmatrix}\nA&B","Let \\(P\\) be an orthogonal projector, put \\(Q=I-P\\), and write a self adjoint Hamiltonian as\n\\[\nH=\n\\begin{pmatrix}\nA&B\\\\\nB^*&D\n\\end{pmatrix}\n\\]\nrelative to \\(P\\Hcal\\oplus Q\\Hcal\\).","B^*&D\n\\end{pmatrix}\n\\]\nrelative to \\(P\\Hcal\\oplus Q\\Hcal\\).","\\begin{definition}[Feshbach map and self energy]\\label{strong:definition:007}\\label{strong:def:feshbach_map}\nFor \\(z\\notin\\Spec D\\), define\n\\[\n\\Sigma(z)=B(z-D)^{-1}B^*\n\\]\nand\n\\[\nF_P(z)=z-A-\\Sigma(z).\n\\]\n\\end{definition}","\\begin{theorem}[Exact projected resolvent]\\label{strong:theorem:029}\\label{strong:thm:feshbach_resolvent}\nIf \\(z\\notin\\Spec D\\) and \\(F_P(z)\\) is invertible, then\n\\[\nP(z-H)^{-1}P=F_P(z)^{-1}\n\\]\nas an operator on \\(P\\Hcal\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:044}\nThe lower right block of \\(z-H\\) is \\(z-D\\). Block Gaussian elimination gives\n\\[\n\\begin{pmatrix}\nI&B(z-D)^{-1}","\\begin{proof}\\phantomsection\\label{strong:proof:044}\nThe lower right block of \\(z-H\\) is \\(z-D\\). Block Gaussian elimination gives\n\\[\n\\begin{pmatrix}\nI&B(z-D)^{-1}\\\\\n0&I\n\\end{pmatrix}\n(z-H)\n\\begin{pmatrix}\nI&0\\\\\n(z-D)^{-1}B^*&I\n\\end{pmatrix}\n=\n\\begin{pmatrix}\nF_P(z)&0\\\\\n0&z-D\n\\end{pmatrix}.\n\\]\nInvert the factors. The retained resolvent is the block in the upper left corner of the resulting inverse.\n\\phantomsection\\label{strong:proof:044:end}\\end{proof}","\\begin{theorem}[Spectral equivalence]\\label{strong:theorem:030}\\label{strong:thm:feshbach_spectrum}\nLet \\(z\\notin\\Spec D\\). Then\n\\[\nz\\in\\Spec H\n\\quad\\Longleftrightarrow\\quad\n0\\in\\Spec F_P(z).\n\\]\nThe map\n\\[\nx\\longmapsto\n\\begin{pmatrix}\nx","\\begin{theorem}[Spectral equivalence]\\label{strong:theorem:030}\\label{strong:thm:feshbach_spectrum}\nLet \\(z\\notin\\Spec D\\). Then\n\\[\nz\\in\\Spec H\n\\quad\\Longleftrightarrow\\quad\n0\\in\\Spec F_P(z).\n\\]\nThe map\n\\[\nx\\longmapsto\n\\begin{pmatrix}\nx\\\\\n(z-D)^{-1}B^*x\n\\end{pmatrix}\n\\]\nis an isomorphism from \\(\\Ker F_P(z)\\) onto \\(\\Ker(z-H)\\).\n\\end{theorem}","(z-D)^{-1}B^*x\n\\end{pmatrix}\n\\]\nis an isomorphism from \\(\\Ker F_P(z)\\) onto \\(\\Ker(z-H)\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:045}\nThe equation \\((z-H)(x,y)^{\\mathsf T}=0\\) is\n\\[\n(z-A)x-By=0,\n\\qquad\n-B^*x+(z-D)y=0.\n\\]\nThe second equation gives \\(y=(z-D)^{-1}B^*x\\). Substitution into the first gives \\(F_P(z)x=0\\). The construction is reversible.\n\\phantomsection\\label{strong:proof:045:end}\\end{proof}","\\begin{theorem}[Upper half plane sign]\\label{strong:theorem:031}\\label{strong:thm:self_energy_sign}\nIf \\(\\operatorname{Im}z>0\\), then\n\\[\n\\operatorname{Im}\\Sigma(z)\n=\n-(\\operatorname{Im}z)\nB(z-D)^{-1}(\\overline z-D)^{-1}B^*\n\\leq0.\n\\]\nThus \\(-\\Sigma\\) is an operator valued Herglotz function.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:046}\nFor \\(R(z)=(z-D)^{-1}\\), the resolvent identity gives\n\\[\nR(z)-R(z)^*\n=(\\overline z-z)R(z)R(z)^*.\n\\]\nDivide by \\(2\\ii\\), then multiply the result on the left by \\(B\\) and on the right by \\(B^*\\).\n\\phantomsection\\label{strong:proof:046:end}\\end{proof}","\\begin{proposition}[Uniform self energy bounds]\\label{strong:proposition:004}\\label{strong:prop:self_energy_bounds}\nLet\n\\[\n\\delta=\\dist(z,\\Spec D)>0,\n\\qquad\ng=\\norm B.\n\\]\nThen\n\\[\n\\norm{\\Sigma(z)}\\leq\\frac{g^2}{\\delta}\n\\]\nand\n\\[\n\\norm{\\Sigma'(z)}\\leq\\frac{g^2}{\\delta^2}.\n\\]\nFor real \\(E,E'\\) in an interval at distance at least \\(\\delta\\) from \\(\\Spec D\\),\n\\[\n\\norm{\\Sigma(E)-\\Sigma(E')}\n\\leq\n\\frac{g^2}{\\delta^2}\\abs{E-E'}.\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:047}\nThe self adjoint resolvent satisfies\n\\[\n\\norm{(z-D)^{-1}}=\\frac1{\\dist(z,\\Spec D)}.\n\\]\nDifferentiate the resolvent to obtain\n\\[\n\\Sigma'(z)=-B(z-D)^{-2}B^*.\n\\]\nTaking operator norms gives the first two estimates. Applying the derivative bound along the real interval gives the final mean value estimate.\n\\phantomsection\\label{strong:proof:047:end}\\end{proof}","\\begin{proposition}[Low energy expansion]\\label{strong:proposition:005}\\label{strong:prop:low_energy_expansion}\nAssume \\(D\\) is invertible and \\(\\abs z\\norm{D^{-1}}<1\\). Then\n\\[\n\\Sigma(z)\n=-\n\\sum_{n=0}^{\\infty}\nz^nBD^{-n-1}B^*\n\\]\nwith norm convergence. In particular,\n\\[\nF_P(z)\n=\n z-A+BD^{-1}B^*+O\\bigl(\\abs z\\norm B^2\\norm{D^{-1}}^2\\bigr).\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:048}\nWrite\n\\[\n(z-D)^{-1}\n=-D^{-1}(I-zD^{-1})^{-1}\n=-\\sum_{n=0}^{\\infty}z^nD^{-n-1}\n\\]\nand multiply on the left by \\(B\\) and on the right by \\(B^*\\).\n\\phantomsection\\label{strong:proof:048:end}\\end{proof}","\\begin{proposition}[Bare truncation error identity]\\label{strong:proposition:006}\\label{strong:prop:bare_truncation}\nIf both \\(F_P(z)\\) and \\(z-A\\) are invertible, then\n\\[\nP(z-H)^{-1}P-(z-A)^{-1}\n=\nF_P(z)^{-1}\\Sigma(z)(z-A)^{-1}.\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:049}\nUse the inverse difference identity\n\\[\nX^{-1}-Y^{-1}=X^{-1}(Y-X)Y^{-1}\n\\]\nwith \\(X=F_P(z)\\) and \\(Y=z-A\\).\n\\phantomsection\\label{strong:proof:049:end}\\end{proof}","Write a Schrödinger state as \\(\\psi=x+y\\), where \\(x=P\\psi\\) and \\(y=Q\\psi\\).","\\begin{theorem}[Exact retained memory equation]\\label{strong:theorem:032}\\label{strong:thm:memory_equation}\nThe retained coordinate satisfies\n\\[\n\\begin{aligned}\n\\ii\\dot x(t)\n={}&Ax(t)+B\\e^{-\\ii tD}y(0)\\\\\n&-\\ii\\int_0^tK(t-s)x(s)\\dd s,\n\\end{aligned}\n\\]\nwhere\n\\[\nK(t)=B\\e^{-\\ii tD}B^*.\n\\]\nFor \\(\\operatorname{Re}s>0\\),\n\\[\n\\int_0^\\infty\\e^{-st}K(t)\\dd t\n=\nB(s+\\ii D)^{-1}B^*.\n\\]\n\\end{theorem}","&-\\ii\\int_0^tK(t-s)x(s)\\dd s,\n\\end{aligned}\n\\]\nwhere\n\\[\nK(t)=B\\e^{-\\ii tD}B^*.\n\\]\nFor \\(\\operatorname{Re}s>0\\),\n\\[\n\\int_0^\\infty\\e^{-st}K(t)\\dd t\n=\nB(s+\\ii D)^{-1}B^*.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:050}\nThe hidden equation is\n\\[\n\\ii\\dot y=B^*x+Dy.\n\\]\nVariation of constants gives\n\\[\ny(t)=\\e^{-\\ii tD}y(0)\n-\\ii\\int_0^t\\e^{-\\ii(t-s)D}B^*x(s)\\dd s.\n\\]\nSubstitute into \\(\\ii\\dot x=Ax+By\\). The Laplace formula follows by diagonalizing \\(D\\) or integrating the norm convergent finite dimensional exponential.\n\\phantomsection\\label{strong:proof:050:end}\\end{proof}","\\begin{theorem}[Positive definite quantum memory]\\label{strong:theorem:033}\\label{strong:thm:positive_memory}\nThe kernel \\(K(t-s)\\) is positive definite. For times \\(t_1,\\ldots,t_m\\) and retained vectors \\(v_1,\\ldots,v_m\\),\n\\[\n\\sum_{j,k=1}^{m}\n\\ip{v_j}{K(t_j-t_k)v_k}\n\\geq0.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:051}\nSet\n\\[\nw=\\sum_{k=1}^{m}\\e^{\\ii t_kD}B^*v_k.\n\\]\nThen\n\\[\n\\norm w^2\n=\n\\sum_{j,k}\n\\ip{B^*v_j}{\\e^{-\\ii(t_j-t_k)D}B^*v_k}\n=\n\\sum_{j,k}\n\\ip{v_j}{K(t_j-t_k)v_k}.\n\\]\n\\phantomsection\\label{strong:proof:051:end}\\end{proof}","\\begin{corollary}[Finite oscillatory memory expansion]\\label{strong:corollary:012}\\label{strong:cor:oscillatory_memory}\nIf\n\\[\nD=\\sum_{r}E_rQ_r\n\\]\nis the spectral decomposition of the hidden Hamiltonian, then\n\\[\nK(t)=\\sum_r\\e^{-\\ii tE_r}BQ_rB^*.\n\\]\nEvery coefficient \\(BQ_rB^*\\) is positive semidefinite.\n\\end{corollary}","\\begin{theorem}[Exact local closure criterion]\\label{strong:theorem:034}\\label{strong:thm:local_closure}\nAssume \\(y(0)=0\\). The retained equation is local and equal to\n\\[\n\\ii\\dot x=Ax\n\\]\nfor every retained initial state exactly when \\(B=0\\). Equivalently, exact local closure holds exactly when the retained and hidden packets reduce \\(H\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:053}\nIf \\(B=0\\), the retained and hidden equations decouple. Conversely, suppose that every solution with \\(y(0)=0\\) satisfies \\(x(t)=\\e^{-\\ii tA}x(0)\\). The full block equation gives\n\\[\n\\ddot x(0)\n=\n-\\bigl(A^2+BB^*\\bigr)x(0),\n\\]\nwhile the declared local equation gives\n\\[\n\\ddot x(0)=-A^2x(0).\n\\]\nSince this holds for every retained initial vector, \\(BB^*=0\\). Hence \\(B=0\\).\n\\phantomsection\\label{strong:proof:053:end}\\end{proof}","ntomsection\\label{strong:proof:053}\nIf \\(B=0\\), the retained and hidden equations decouple. Conversely, suppose that every solution with \\(y(0)=0\\) satisfies \\(x(t)=\\e^{-\\ii tA}x(0)\\). The full block equation gives\n\\[\n\\ddot x(0)\n=\n-\\bigl(A^2+BB^*\\bigr)x(0),\n\\]\nwhile the declared local equation gives\n\\[\n\\ddot x(0)=-A^2x(0).\n\\]\nSince this holds for every retained initial vector, \\(BB^*=0\\). Hence \\(B=0\\).\n\\phantomsection\\label{strong:proof:053:end}\\end{proof}","\\begin{theorem}[Stability of the self energy]\\label{strong:theorem:035}\\label{strong:thm:self_energy_stability}\nLet \\(D=D^*\\) and \\(\\widetilde D=\\widetilde D^*\\) act on one fixed hidden Hilbert space, and let \\(B,\\widetilde B\\) map that hidden space to one fixed retained Hilbert space. Define\n\\[\n\\Sigma(z)=B(z-D)^{-1}B^*,\n\\qquad\n\\widetilde\\Sigma(z)=\\widetilde B(z-\\widetilde D)^{-1}\\widetilde B^*.\n\\]\nAssume\n\\[\n\\norm B,\\norm{\\widetilde B}\\leq g\n\\]\nand\n\\[\n\\dist(z,\\Spec D),\n\\dist(z,\\Spec\\widetilde D)\n\\geq\\delta>0.\n\\]\nThen\n\\[\n\\norm{\\Sigma(z)-\\widetilde\\Sigma(z)}\n\\leq\n\\frac{2g}{\\delta}\\norm{B-\\widetilde B}\n+\n\\frac{g^2}{\\delta^2}\\norm{D-\\widetilde D}.\n\\]\n\\end{theorem}","rem}[Stability of the self energy]\\label{strong:theorem:035}\\label{strong:thm:self_energy_stability}\nLet \\(D=D^*\\) and \\(\\widetilde D=\\widetilde D^*\\) act on one fixed hidden Hilbert space, and let \\(B,\\widetilde B\\) map that hidden space to one fixed retained Hilbert space. Define\n\\[\n\\Sigma(z)=B(z-D)^{-1}B^*,\n\\qquad\n\\widetilde\\Sigma(z)=\\widetilde B(z-\\widetilde D)^{-1}\\widetilde B^*.\n\\]\nAssume\n\\[\n\\norm B,\\norm{\\widetilde B}\\leq g\n\\]\nand\n\\[\n\\dist(z,\\Spec D),\n\\dist(z,\\Spec\\widetilde D)\n\\geq\\delta>0.\n\\]\nThen\n\\[\n\\norm{\\Sigma(z)-\\widetilde\\Sigma(z)}\n\\leq\n\\frac{2g}{\\delta}\\norm{B-\\widetilde B}\n+\n\\frac{g^2}{\\delta^2}\\norm{D-\\widetilde D}.\n\\]\n\\end{theorem}","ong:theorem:035}\\label{strong:thm:self_energy_stability}\nLet \\(D=D^*\\) and \\(\\widetilde D=\\widetilde D^*\\) act on one fixed hidden Hilbert space, and let \\(B,\\widetilde B\\) map that hidden space to one fixed retained Hilbert space. Define\n\\[\n\\Sigma(z)=B(z-D)^{-1}B^*,\n\\qquad\n\\widetilde\\Sigma(z)=\\widetilde B(z-\\widetilde D)^{-1}\\widetilde B^*.\n\\]\nAssume\n\\[\n\\norm B,\\norm{\\widetilde B}\\leq g\n\\]\nand\n\\[\n\\dist(z,\\Spec D),\n\\dist(z,\\Spec\\widetilde D)\n\\geq\\delta>0.\n\\]\nThen\n\\[\n\\norm{\\Sigma(z)-\\widetilde\\Sigma(z)}\n\\leq\n\\frac{2g}{\\delta}\\norm{B-\\widetilde B}\n+\n\\frac{g^2}{\\delta^2}\\norm{D-\\widetilde D}.\n\\]\n\\end{theorem}","and \\(\\widetilde D=\\widetilde D^*\\) act on one fixed hidden Hilbert space, and let \\(B,\\widetilde B\\) map that hidden space to one fixed retained Hilbert space. Define\n\\[\n\\Sigma(z)=B(z-D)^{-1}B^*,\n\\qquad\n\\widetilde\\Sigma(z)=\\widetilde B(z-\\widetilde D)^{-1}\\widetilde B^*.\n\\]\nAssume\n\\[\n\\norm B,\\norm{\\widetilde B}\\leq g\n\\]\nand\n\\[\n\\dist(z,\\Spec D),\n\\dist(z,\\Spec\\widetilde D)\n\\geq\\delta>0.\n\\]\nThen\n\\[\n\\norm{\\Sigma(z)-\\widetilde\\Sigma(z)}\n\\leq\n\\frac{2g}{\\delta}\\norm{B-\\widetilde B}\n+\n\\frac{g^2}{\\delta^2}\\norm{D-\\widetilde D}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:054}\nAdd and subtract \\(\\widetilde B(z-D)^{-1}B^*\\) and \\(\\widetilde B(z-D)^{-1}\\widetilde B^*\\). The first two differences give the coupling term. For the last, use\n\\[\n(z-D)^{-1}-(z-\\widetilde D)^{-1}\n=(z-D)^{-1}(D-\\widetilde D)(z-\\widetilde D)^{-1}.\n\\]\nApply the resolvent bounds.\n\\phantomsection\\label{strong:proof:054:end}\\end{proof}","\\begin{corollary}[Uniform Feshbach control]\\label{strong:corollary:013}\\label{strong:cor:uniform_feshbach}\nFor a family represented on fixed retained and hidden Hilbert spaces, with self adjoint hidden Hamiltonians, a common hidden spectral separation \\(\\delta\\), and a common coupling bound \\(g\\), the self energies are uniformly bounded and uniformly Lipschitz on the declared spectral region. If the couplings and hidden Hamiltonians converge in norm, then the self energies converge uniformly there.\n\\end{corollary}","Let \\(H=H^*\\) on a nonzero finite dimensional Hilbert space and let\n\\[\nE_0=\\min\\Spec H.\n\\]\nFor a lattice time step \\(a>0\\), define\n\\[\nT=\\e^{-a(H-E_0I)}.\n\\]","\\begin{theorem}[Finite Euclidean transfer]\\label{strong:theorem:036}\\label{strong:thm:euclidean_transfer}\nThe operator \\(T\\) is positive definite, self adjoint, and contractive. Its spectrum lies in \\((0,1]\\), and\n\\[\nT^n=\\e^{-na(H-E_0I)}\n\\]\nfor every \\(n\\in\\NN\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:056}\nDiagonalize \\(H\\). Each eigenvalue of \\(T\\) is\n\\[\n\\e^{-a(E_j-E_0)}\n\\]\nand therefore lies in \\((0,1]\\). The power law follows from the functional calculus.\n\\phantomsection\\label{strong:proof:056:end}\\end{proof}","\\begin{theorem}[Hamiltonian reconstruction from transfer]\\label{strong:theorem:037}\\label{strong:thm:transfer_logarithm}\nLet \\(T\\) be positive definite, self adjoint, and contractive. Then\n\\[\nH_T=-a^{-1}\\log T\n\\]\nis a nonnegative self adjoint operator and\n\\[\nT^n=\\e^{-naH_T}.\n\\]\nIt is the unique nonnegative self adjoint operator with \\(T=\\e^{-aH_T}\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:057}\nApply the real logarithm to the positive spectrum of \\(T\\). The spectral calculus gives existence and the power identity. Injectivity of the scalar exponential on \\([0,\\infty)\\) gives uniqueness.\n\\phantomsection\\label{strong:proof:057:end}\\end{proof}","\\begin{theorem}[Finite reflection positivity]\\label{strong:theorem:038}\\label{strong:thm:finite_reflection_positivity}\nFor vectors \\(v_1,\\ldots,v_m\\in\\Hcal\\) and nonnegative integers \\(n_1,\\ldots,n_m\\), the matrix\n\\[\nG_{jk}=\\ip{v_j}{T^{n_j+n_k}v_k}\n\\]\nis positive semidefinite. Equivalently,\n\\[\n\\sum_{j,k}\\overline{c_j}c_k\n\\ip{v_j}{T^{n_j+n_k}v_k}\n\\geq0\n\\]\nfor every \\(c\\in\\CC^m\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:058}\nSince \\(T\\) is positive and self adjoint,\n\\[\n\\ip{v_j}{T^{n_j+n_k}v_k}\n=\n\\ip{T^{n_j}v_j}{T^{n_k}v_k}.\n\\]\nThe quadratic form is the squared norm of\n\\[\n\\sum_kc_kT^{n_k}v_k.\n\\]\n\\phantomsection\\label{strong:proof:058:end}\\end{proof}","\\begin{corollary}[Reflection positive correlators]\\label{strong:corollary:014}\\label{strong:cor:reflection_correlators}\nFix a vector \\(\\Omega\\) and operators \\(A_1,\\ldots,A_m\\). The matrix\n\\[\nC_{jk}\n=\n\\ip{A_j\\Omega}{T^{n_j+n_k}A_k\\Omega}\n\\]\nis positive semidefinite.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:059}\nApply \\cref{strong:thm:finite_reflection_positivity} with \\(v_j=A_j\\Omega\\).\n\\phantomsection\\label{strong:proof:059:end}\\end{proof}","\\begin{proposition}[Ground packet convergence]\\label{strong:proposition:007}\\label{strong:prop:ground_packet}\nLet \\(P_0\\) be the spectral projector of \\(H\\) at \\(E_0\\), and let\n\\[\n\\Delta=\\min\\{E_j-E_0:E_j>E_0\\}.\n\\]\nIf an excited eigenvalue exists, then\n\\[\n\\norm{T^n-P_0}=\\e^{-na\\Delta}.\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:060}\nIn the spectral decomposition, \\(T^n-P_0\\) has eigenvalues zero on the ground space and \\(\\e^{-na(E_j-E_0)}\\) on the excited eigenspaces. The largest is \\(\\e^{-na\\Delta}\\).\n\\phantomsection\\label{strong:proof:060:end}\\end{proof}","\\begin{proposition}[Symmetry resolved transfer]\\label{strong:proposition:008}\\label{strong:prop:symmetry_transfer}\nIf \\([H,P_a]=0\\) for a complete packet family, then\n\\[\nT=\\sum_aP_aTP_a\n\\]\nand each reflection positive Gram matrix decomposes as the sum of its packet Gram matrices.\n\\end{proposition}","A finite spectral datum consists of a finite dimensional Hilbert space, a nonnegative self adjoint operator \\(L\\), and any declared packet family. One may take \\(L=D^2\\) for a finite Dirac type operator or a graph Laplacian. Related geometric viewpoints appear in \\cite{Connes1994,Vassilevich2003}.","\\begin{definition}[Heat trace and spectral dimension]\\label{strong:definition:008}\\label{strong:def:spectral_dimension}\nLet the eigenvalues of \\(L\\) be \\(\\lambda_1,\\ldots,\\lambda_N\\), with at least one zero eigenvalue. Define\n\\[\nZ_L(t)=\\Tr(\\e^{-tL})\n\\]\nand\n\\[\nd_L(t)\n=-2\\frac{\\dd\\log Z_L(t)}{\\dd\\log t}\n=\n2t\n\\frac{\\Tr(L\\e^{-tL})}{\\Tr(\\e^{-tL})}.\n\\]\n\\end{definition}","\\begin{proposition}[Finite scale limits]\\label{strong:proposition:009}\\label{strong:prop:finite_spectral_limits}\nFor every finite nonnegative \\(L\\) with nontrivial kernel,\n\\[\n\\lim_{t\\downarrow0}d_L(t)=0\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}d_L(t)=0.\n\\]\nFor every \\(t>0\\),\n\\[\n0\\leq d_L(t)\\leq2t\\norm L.\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:062}\nThe formula in \\cref{strong:def:spectral_dimension} is twice \\(t\\) times a weighted average of the eigenvalues. This gives the bound and the limit at zero. At large time, the denominator converges to \\(\\dim\\Ker L\\), while every positive eigenvalue contribution in the numerator decays exponentially.\n\\phantomsection\\label{strong:proof:062:end}\\end{proof}","\\begin{proposition}[Heat packet decomposition]\\label{strong:proposition:010}\\label{strong:prop:heat_packets}\nIf \\([L,P_a]=0\\) for every packet, then\n\\[\nZ_L(t)=\\sum_aZ_{L,a}(t),\n\\qquad\nZ_{L,a}(t)=\\Tr(P_a\\e^{-tL}).\n\\]\nThe global spectral dimension is the average of the packet dimensions weighted by their heat traces:\n\\[\nd_L(t)\n=\n\\sum_{a:Z_{L,a}(t)>0}\n\\frac{Z_{L,a}(t)}{Z_L(t)}\n d_{L,a}(t),\n\\]\nwhere\n\\[\nd_{L,a}(t)\n=2t\n\\frac{\\Tr(P_aL\\e^{-tL})}{Z_{L,a}(t)}\n\\]\nfor nonempty packet heat traces.\n\\end{proposition}","begin{proposition}[Heat packet decomposition]\\label{strong:proposition:010}\\label{strong:prop:heat_packets}\nIf \\([L,P_a]=0\\) for every packet, then\n\\[\nZ_L(t)=\\sum_aZ_{L,a}(t),\n\\qquad\nZ_{L,a}(t)=\\Tr(P_a\\e^{-tL}).\n\\]\nThe global spectral dimension is the average of the packet dimensions weighted by their heat traces:\n\\[\nd_L(t)\n=\n\\sum_{a:Z_{L,a}(t)>0}\n\\frac{Z_{L,a}(t)}{Z_L(t)}\n d_{L,a}(t),\n\\]\nwhere\n\\[\nd_{L,a}(t)\n=2t\n\\frac{\\Tr(P_aL\\e^{-tL})}{Z_{L,a}(t)}\n\\]\nfor nonempty packet heat traces.\n\\end{proposition}","Let \\(N\\geq2\\) and let \\(\\Lambda_N=(\\ZZ/N\\ZZ)^d\\) be the unit flat torus grid with spacing \\(1/N\\). The standard periodic difference Laplacian has eigenvalues\n\\[\n\\lambda_N(k)\n=\n4N^2\n\\sum_{j=1}^{d}\n\\sin^2\\left(\\frac{\\pi k_j}{N}\\right),\n\\]\nwhere \\(k\\) runs through centered residue representatives. This is the standard periodic finite difference Laplacian \\cite{Strikwerda2004}.","\\begin{theorem}[Low mode dispersion in dimension \\(d\\)]\\label{strong:theorem:039}\\label{strong:thm:dispersion_bound}\nFor every centered mode \\(k\\),\n\\[\n0\n\\leq\n4\\pi^2\\abs{k}^2-\\lambda_N(k)\n\\leq\n\\frac{4\\pi^4}{3N^2}\n\\sum_{j=1}^{d}k_j^4.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:064}\nFor \\(x\\in[0,\\pi/2]\\),\n\\[\n\\sin x\\leq x\n\\]\nand\n\\[\n\\sin x\\geq x-\\frac{x^3}{6}.\n\\]\nThus\n\\[\n0\\leq x^2-\\sin^2x\\leq\\frac{x^4}{3}.\n\\]\nApply this with \\(x=\\pi\\abs{k_j}/N\\), multiply by \\(4N^2\\), and sum over \\(j\\).\n\\phantomsection\\label{strong:proof:064:end}\\end{proof}","\\begin{theorem}[Fixed time heat trace convergence]\\label{strong:theorem:040}\\label{strong:thm:heat_trace_convergence}\nFor every \\(t>0\\),\n\\[\n\\sum_{k\\in\\Lambda_N^*}\n\\e^{-t\\lambda_N(k)}\n\\longrightarrow\n\\sum_{k\\in\\ZZ^d}\n\\e^{-4\\pi^2t\\abs{k}^2}\n\\]\nas \\(N\\to\\infty\\). The first spectral moments also converge:\n\\[\n\\sum_{k\\in\\Lambda_N^*}\n\\lambda_N(k)\\e^{-t\\lambda_N(k)}\n\\longrightarrow\n\\sum_{k\\in\\ZZ^d}\n4\\pi^2\\abs{k}^2\\e^{-4\\pi^2t\\abs{k}^2}.\n\\]\nConsequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:065}\nFor a fixed integer mode, \\cref{strong:thm:dispersion_bound} gives\n\\[\n\\lambda_N(k)\\longrightarrow4\\pi^2\\abs{k}^2.\n\\]\nFor centered modes, the inequality\n\\[\n\\sin x\\geq\\frac{2x}{\\pi},\n\\qquad\n0\\leq x\\leq\\frac\\pi2,\n\\]\ngives\n\\[\n\\lambda_N(k)\\geq16\\abs{k}^2.\n\\]\nAlso \\(\\lambda_N(k)\\leq4\\pi^2\\abs{k}^2\\). Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by\n\\[\n\\e^{-16t\\abs{k}^2},\n\\]\nand the moment terms by\n\\[\n4\\pi^2\\abs{k}^2\\e^{-16t\\abs{k}^2}.\n\\]\nBoth dominating sequences are summable on \\(\\ZZ^d\\). Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.\n\\phantomsection\\label{strong:proof:065:end}\\end{proof}","ixed integer mode, \\cref{strong:thm:dispersion_bound} gives\n\\[\n\\lambda_N(k)\\longrightarrow4\\pi^2\\abs{k}^2.\n\\]\nFor centered modes, the inequality\n\\[\n\\sin x\\geq\\frac{2x}{\\pi},\n\\qquad\n0\\leq x\\leq\\frac\\pi2,\n\\]\ngives\n\\[\n\\lambda_N(k)\\geq16\\abs{k}^2.\n\\]\nAlso \\(\\lambda_N(k)\\leq4\\pi^2\\abs{k}^2\\). Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by\n\\[\n\\e^{-16t\\abs{k}^2},\n\\]\nand the moment terms by\n\\[\n4\\pi^2\\abs{k}^2\\e^{-16t\\abs{k}^2}.\n\\]\nBoth dominating sequences are summable on \\(\\ZZ^d\\). Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.\n\\phantomsection\\label{strong:proof:065:end}\\end{proof}","nd} gives\n\\[\n\\lambda_N(k)\\longrightarrow4\\pi^2\\abs{k}^2.\n\\]\nFor centered modes, the inequality\n\\[\n\\sin x\\geq\\frac{2x}{\\pi},\n\\qquad\n0\\leq x\\leq\\frac\\pi2,\n\\]\ngives\n\\[\n\\lambda_N(k)\\geq16\\abs{k}^2.\n\\]\nAlso \\(\\lambda_N(k)\\leq4\\pi^2\\abs{k}^2\\). Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by\n\\[\n\\e^{-16t\\abs{k}^2},\n\\]\nand the moment terms by\n\\[\n4\\pi^2\\abs{k}^2\\e^{-16t\\abs{k}^2}.\n\\]\nBoth dominating sequences are summable on \\(\\ZZ^d\\). Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.\n\\phantomsection\\label{strong:proof:065:end}\\end{proof}","inequality\n\\[\n\\sin x\\geq\\frac{2x}{\\pi},\n\\qquad\n0\\leq x\\leq\\frac\\pi2,\n\\]\ngives\n\\[\n\\lambda_N(k)\\geq16\\abs{k}^2.\n\\]\nAlso \\(\\lambda_N(k)\\leq4\\pi^2\\abs{k}^2\\). Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by\n\\[\n\\e^{-16t\\abs{k}^2},\n\\]\nand the moment terms by\n\\[\n4\\pi^2\\abs{k}^2\\e^{-16t\\abs{k}^2}.\n\\]\nBoth dominating sequences are summable on \\(\\ZZ^d\\). Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.\n\\phantomsection\\label{strong:proof:065:end}\\end{proof}","\\begin{standardfact}[Theta transformation]\\label{strong:standardfact:004}\\label{strong:fact:theta_transform}\nThe continuum flat torus heat trace satisfies\n\\[\nZ_d(t)\n=\n\\sum_{k\\in\\ZZ^d}\\e^{-4\\pi^2t\\abs{k}^2}\n=\n(4\\pi t)^{-d/2}\n\\sum_{n\\in\\ZZ^d}\\e^{-\\abs{n}^2/(4t)}.\n\\]\nThis is the Poisson summation formula for a Gaussian.\n\\end{standardfact}","\\begin{theorem}[Continuum torus small time dimension]\\label{strong:theorem:041}\\label{strong:thm:continuum_dimension}\nLet\n\\[\nd_{\\rm torus}(t)\n=-2\\frac{\\dd\\log Z_d(t)}{\\dd\\log t}.\n\\]\nThen\n\\[\nd_{\\rm torus}(t)\n=\n d+O\\left(t^{-1}\\e^{-1/(4t)}\\right)\n\\]\nas \\(t\\downarrow0\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:066}\nBy \\cref{strong:fact:theta_transform},\n\\[\nZ_d(t)\n=(4\\pi t)^{-d/2}(1+R(t)),\n\\]\nwhere\n\\[\nR(t)=\\sum_{n\\neq0}\\e^{-\\abs{n}^2/(4t)}\n=O\\bigl(\\e^{-1/(4t)}\\bigr).\n\\]\nDifferentiation gives\n\\[\ntR'(t)=O\\left(t^{-1}\\e^{-1/(4t)}\\right).\n\\]\nSubstituting this estimate into the logarithmic derivative gives the stated small time expansion.\n\\phantomsection\\label{strong:proof:066:end}\\end{proof}","\\begin{corollary}[Rigorous intermediate dimension window]\\label{strong:corollary:015}\\label{strong:cor:dimension_window}\nFor every \\(\\varepsilon>0\\), there is a diffusion time \\(t_\\varepsilon>0\\) and an integer \\(N_\\varepsilon\\) such that\n\\[\n\\abs{d_{L_N}(t_\\varepsilon)-d}<\\varepsilon\n\\]\nfor every \\(N\\geq N_\\varepsilon\\).\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:067}\nChoose \\(t_\\varepsilon\\) so that \\cref{strong:thm:continuum_dimension} is within \\(\\varepsilon/2\\) of \\(d\\). Then apply \\cref{strong:thm:heat_trace_convergence} to choose \\(N_\\varepsilon\\) so that the discrete spectral dimension is within \\(\\varepsilon/2\\) of the continuum value.\n\\phantomsection\\label{strong:proof:067:end}\\end{proof}","The order of limits matters. Taking \\(t\\downarrow0\\) at fixed \\(N\\) gives zero. Taking \\(N\\to\\infty\\) at fixed positive \\(t\\), then taking \\(t\\downarrow0\\), gives the continuum dimension.","Let \\(C_1,\\ldots,C_m\\) be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory \\cite{Dirac1964,HenneauxTeitelboim1992}. Write\n\\[\n\\mathfrak c=\\operatorname{span}_{\\CC}\\{C_1,\\ldots,C_m\\}\n\\subseteq\\End(\\Hcal).\n\\]","\\begin{definition}[Universal closure obstruction]\\label{strong:definition:009}\\label{strong:def:closure_obstruction}\nLet\n\\[\nq_{\\mathfrak c}:\\End(\\Hcal)\\longrightarrow\\End(\\Hcal)/\\mathfrak c\n\\]\nbe the quotient map. Define\n\\[\n\\Ob_{ab}^{\\rm cl}\n=\nq_{\\mathfrak c}([C_a,C_b]).\n\\]\n\\end{definition}","\\begin{theorem}[Finite constraint closure criterion]\\label{strong:theorem:042}\\label{strong:thm:constraint_closure}\nThe constraint span \\(\\mathfrak c\\) is closed under commutators exactly when\n\\[\n\\Ob_{ab}^{\\rm cl}=0\n\\]\nfor all \\(a,b\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:068}\nThe quotient class vanishes exactly when \\([C_a,C_b]\\in\\mathfrak c\\). This holds for every pair exactly when \\(\\mathfrak c\\) is a Lie algebra under the commutator.\n\\phantomsection\\label{strong:proof:068:end}\\end{proof}","If structure coefficients \\(f_{ab}^{\\phantom{ab}c}\\) are prescribed, define the fixed structure anomaly\n\\[\n\\mathfrak A_{ab}\n=\n[C_a,C_b]\n-\\ii\\sum_cf_{ab}^{\\phantom{ab}c}C_c.\n\\]\nExact realization of the prescribed algebra is the stronger condition \\(\\mathfrak A_{ab}=0\\).","\\begin{proposition}[Physical compression is not a closure test]\\label{strong:proposition:011}\\label{strong:prop:physical_compression_trivial}\nLet\n\\[\n\\Hcal_{\\phys}=\\bigcap_a\\Ker C_a\n\\]\nand let \\(P_{\\phys}\\) be its orthogonal projector. Then\n\\[\nP_{\\phys}[C_a,C_b]P_{\\phys}=0\n\\]\nfor every pair \\(a,b\\), whether or not the constraint span closes.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:069}\nSelf adjointness and \\(C_aP_{\\phys}=0\\) give \\(P_{\\phys}C_a=0\\). Both terms in the compressed commutator therefore vanish.\n\\phantomsection\\label{strong:proof:069:end}\\end{proof}","\\begin{theorem}[Invariance of the physical space]\\label{strong:theorem:043}\\label{strong:thm:physical_invariance}\nLet \\(H=H^*\\). Suppose\n\\[\n[H,C_a]\n=\n\\sum_cb_a^{\\phantom a c}C_c\n\\]\nfor every \\(a\\). Then \\(H\\) preserves \\(\\Hcal_{\\phys}\\), and so does \\(\\e^{-\\ii tH}\\) for every real \\(t\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:070}\nFor \\(\\psi\\in\\Hcal_{\\phys}\\),\n\\[\nC_aH\\psi\n=HC_a\\psi-[H,C_a]\\psi\n=0.\n\\]\nThus \\(H\\psi\\in\\Hcal_{\\phys}\\). The exponential preserves every invariant finite dimensional subspace.\n\\phantomsection\\label{strong:proof:070:end}\\end{proof}","Let \\((\\Hcal_\\ell,C_{\\ell,a})\\) and \\((\\Hcal_{\\ell+1},C_{\\ell+1,a})\\) be two levels, and let\n\\[\nR_\\ell:\\Hcal_\\ell\\longrightarrow\\Hcal_{\\ell+1}\n\\]\nbe an isometry.","\\begin{theorem}[Exact constraint transport]\\label{strong:theorem:044}\\label{strong:thm:constraint_transport}\nAssume\n\\[\nC_{\\ell+1,a}R_\\ell=R_\\ell C_{\\ell,a}\n\\]\nfor every \\(a\\). Then\n\\[\n[C_{\\ell+1,a},C_{\\ell+1,b}]R_\\ell\n=\nR_\\ell[C_{\\ell,a},C_{\\ell,b}].\n\\]\nIf the same structure coefficients are used at both levels, then\n\\[\n\\mathfrak A_{\\ell+1,ab}R_\\ell\n=\nR_\\ell\\mathfrak A_{\\ell,ab}.\n\\]\nThe isometry maps coarse physical states into fine physical states.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:071}\nApply the intertwining identity twice to each product in the commutator. The anomaly identity follows by subtracting the same linear combination of constraints. If \\(C_{\\ell,a}\\psi=0\\), then\n\\[\nC_{\\ell+1,a}R_\\ell\\psi=R_\\ell C_{\\ell,a}\\psi=0.\n\\]\n\\phantomsection\\label{strong:proof:071:end}\\end{proof}","\\begin{proposition}[Approximate constraint transport]\\label{strong:proposition:012}\\label{strong:prop:approx_constraint_transport}\nAssume\n\\[\n\\norm{C_{\\ell+1,a}R_\\ell-R_\\ell C_{\\ell,a}}\n\\leq\\varepsilon_a\n\\]\nand\n\\[\n\\norm{C_{\\ell,a}},\\norm{C_{\\ell+1,a}}\\leq M.\n\\]\nThen a coarse physical unit vector \\(\\psi\\) satisfies\n\\[\n\\norm{C_{\\ell+1,a}R_\\ell\\psi}\\leq\\varepsilon_a.\n\\]\nAlso,\n\\[\n\\begin{aligned}\n&\\norm{\n[C_{\\ell+1,a},C_{\\ell+1,b}]R_\\ell\n-R_\\ell[C_{\\ell,a},C_{\\ell,b}]\n}","\\begin{proposition}[Approximate constraint transport]\\label{strong:proposition:012}\\label{strong:prop:approx_constraint_transport}\nAssume\n\\[\n\\norm{C_{\\ell+1,a}R_\\ell-R_\\ell C_{\\ell,a}}\n\\leq\\varepsilon_a\n\\]\nand\n\\[\n\\norm{C_{\\ell,a}},\\norm{C_{\\ell+1,a}}\\leq M.\n\\]\nThen a coarse physical unit vector \\(\\psi\\) satisfies\n\\[\n\\norm{C_{\\ell+1,a}R_\\ell\\psi}\\leq\\varepsilon_a.\n\\]\nAlso,\n\\[\n\\begin{aligned}\n&\\norm{\n[C_{\\ell+1,a},C_{\\ell+1,b}]R_\\ell\n-R_\\ell[C_{\\ell,a},C_{\\ell,b}]\n}\\\\\n&\\hspace{4em}\\leq\n2M(\\varepsilon_a+\\varepsilon_b).\n\\end{aligned}\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{strong:proof:072}\nThe first claim follows by applying the intertwining defect to \\(\\psi\\). Write\n\\[\nE_a=C_{\\ell+1,a}R_\\ell-R_\\ell C_{\\ell,a}.\n\\]\nExpanding the two commutators gives four error terms:\n\\[\nE_aC_{\\ell,b},\n\\quad\nC_{\\ell+1,a}E_b,\n\\quad\n-E_bC_{\\ell,a},\n\\quad\n-C_{\\ell+1,b}E_a.\n\\]\nThe triangle inequality and the stated bounds on the constraints and defects give the claimed estimate.\n\\phantomsection\\label{strong:proof:072:end}\\end{proof}","\\begin{definition}[Finite quantum refinement tower]\\label{strong:definition:010}\\label{strong:def:quantum_tower}\nA finite quantum refinement tower consists of Hilbert spaces \\(\\Hcal_\\ell\\), isometries\n\\[\nR_\\ell:\\Hcal_\\ell\\longrightarrow\\Hcal_{\\ell+1},\n\\]\nself adjoint Hamiltonians \\(H_\\ell\\), packet projectors \\(P_{\\ell,a}\\), constraints \\(C_{\\ell,r}\\), and positive transfer operators \\(T_\\ell\\). Exact compatibility means that every declared object intertwines with \\(R_\\ell\\). Approximate compatibility is recorded by the corresponding operator norm defects.\n\\end{definition}","tion:010}\\label{strong:def:quantum_tower}\nA finite quantum refinement tower consists of Hilbert spaces \\(\\Hcal_\\ell\\), isometries\n\\[\nR_\\ell:\\Hcal_\\ell\\longrightarrow\\Hcal_{\\ell+1},\n\\]\nself adjoint Hamiltonians \\(H_\\ell\\), packet projectors \\(P_{\\ell,a}\\), constraints \\(C_{\\ell,r}\\), and positive transfer operators \\(T_\\ell\\). Exact compatibility means that every declared object intertwines with \\(R_\\ell\\). Approximate compatibility is recorded by the corresponding operator norm defects.\n\\end{definition}","Let \\(\\Hcal_\\infty\\) be the Hilbert direct limit of the isometric system, and let\n\\[\n\\iota_\\ell:\\Hcal_\\ell\\longrightarrow\\Hcal_\\infty\n\\]\nbe the canonical isometries.","\\begin{theorem}[Bounded Hamiltonian direct limit]\\label{strong:theorem:045}\\label{strong:thm:bounded_direct_limit}\nAssume\n\\[\nH_{\\ell+1}R_\\ell=R_\\ell H_\\ell\n\\]\nand\n\\[\n\\sup_\\ell\\norm{H_\\ell}<\\infty.\n\\]\nThen there is a unique bounded self adjoint operator \\(H_\\infty\\) on \\(\\Hcal_\\infty\\) such that\n\\[\nH_\\infty\\iota_\\ell=\\iota_\\ell H_\\ell\n\\]\nfor every \\(\\ell\\). It satisfies\n\\[\n\\e^{-\\ii tH_\\infty}\\iota_\\ell\n=\n\\iota_\\ell\\e^{-\\ii tH_\\ell}\n\\]\nand\n\\[\n\\e^{-tH_\\infty}\\iota_\\ell\n=\n\\iota_\\ell\\e^{-tH_\\ell}\n\\]\nwhen the Hamiltonians are nonnegative.\n\\end{theorem}","imit]\\label{strong:theorem:045}\\label{strong:thm:bounded_direct_limit}\nAssume\n\\[\nH_{\\ell+1}R_\\ell=R_\\ell H_\\ell\n\\]\nand\n\\[\n\\sup_\\ell\\norm{H_\\ell}<\\infty.\n\\]\nThen there is a unique bounded self adjoint operator \\(H_\\infty\\) on \\(\\Hcal_\\infty\\) such that\n\\[\nH_\\infty\\iota_\\ell=\\iota_\\ell H_\\ell\n\\]\nfor every \\(\\ell\\). It satisfies\n\\[\n\\e^{-\\ii tH_\\infty}\\iota_\\ell\n=\n\\iota_\\ell\\e^{-\\ii tH_\\ell}\n\\]\nand\n\\[\n\\e^{-tH_\\infty}\\iota_\\ell\n=\n\\iota_\\ell\\e^{-tH_\\ell}\n\\]\nwhen the Hamiltonians are nonnegative.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:073}\nDefine \\(H_\\infty\\) on the algebraic union of the embedded levels by\n\\[\nH_\\infty\\iota_\\ell x=\\iota_\\ell H_\\ell x.\n\\]\nCompatibility makes the definition independent of the chosen level. The uniform norm bound makes the operator bounded on the dense union, so it extends uniquely to the completion. It is symmetric because every \\(H_\\ell\\) is self adjoint. A bounded symmetric operator defined on the whole Hilbert space is self adjoint. The functional calculus intertwining follows first for polynomials and then for exponentials by uniform convergence.\n\\phantomsection\\label{strong:proof:073:end}\\end{proof}","\\begin{theorem}[Exact packet direct limit]\\label{strong:theorem:046}\\label{strong:thm:packet_direct_limit}\nAssume the packet families have one fixed finite address set at every level and each packet address \\(a\\) is compatible in the sense\n\\[\nP_{\\ell+1,a}R_\\ell=R_\\ell P_{\\ell,a}.\n\\]\nThen there is an orthogonal projector \\(P_{\\infty,a}\\) on \\(\\Hcal_\\infty\\) satisfying\n\\[\nP_{\\infty,a}\\iota_\\ell=\\iota_\\ell P_{\\ell,a}.\n\\]\nFinite completeness and orthogonality pass to the direct limit.\n\\end{theorem}","\\begin{theorem}[Norm summable Hamiltonian tower]\\label{strong:theorem:047}\\label{strong:thm:norm_summable_tower}\nLet \\((H_n)\\) be bounded self adjoint operators on one Hilbert space and assume\n\\[\n\\sum_{n=0}^{\\infty}\\norm{H_{n+1}-H_n}<\\infty.\n\\]\nThen \\(H_n\\) converges in norm to a bounded self adjoint operator \\(H\\). For every \\(T>0\\),\n\\[\n\\sup_{\\abs t\\leq T}\n\\norm{\\e^{-\\ii tH_n}-\\e^{-\\ii tH}}\n\\leq\nT\\norm{H_n-H}.\n\\]\nIf every \\(H_n\\geq0\\), then\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{\\e^{-tH_n}-\\e^{-tH}}\n\\leq\nT\\norm{H_n-H}.\n\\]\n\\end{theorem}","Hamiltonian tower]\\label{strong:theorem:047}\\label{strong:thm:norm_summable_tower}\nLet \\((H_n)\\) be bounded self adjoint operators on one Hilbert space and assume\n\\[\n\\sum_{n=0}^{\\infty}\\norm{H_{n+1}-H_n}<\\infty.\n\\]\nThen \\(H_n\\) converges in norm to a bounded self adjoint operator \\(H\\). For every \\(T>0\\),\n\\[\n\\sup_{\\abs t\\leq T}\n\\norm{\\e^{-\\ii tH_n}-\\e^{-\\ii tH}}\n\\leq\nT\\norm{H_n-H}.\n\\]\nIf every \\(H_n\\geq0\\), then\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{\\e^{-tH_n}-\\e^{-tH}}\n\\leq\nT\\norm{H_n-H}.\n\\]\n\\end{theorem}","\\label{strong:theorem:047}\\label{strong:thm:norm_summable_tower}\nLet \\((H_n)\\) be bounded self adjoint operators on one Hilbert space and assume\n\\[\n\\sum_{n=0}^{\\infty}\\norm{H_{n+1}-H_n}<\\infty.\n\\]\nThen \\(H_n\\) converges in norm to a bounded self adjoint operator \\(H\\). For every \\(T>0\\),\n\\[\n\\sup_{\\abs t\\leq T}\n\\norm{\\e^{-\\ii tH_n}-\\e^{-\\ii tH}}\n\\leq\nT\\norm{H_n-H}.\n\\]\nIf every \\(H_n\\geq0\\), then\n\\[\n\\sup_{0\\leq t\\leq T}\n\\norm{\\e^{-tH_n}-\\e^{-tH}}\n\\leq\nT\\norm{H_n-H}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:075}\nThe summability condition makes \\((H_n)\\) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives\n\\[\n\\e^{-\\ii tH_n}-\\e^{-\\ii tH}\n=-\\ii\\int_0^t\n\\e^{-\\ii(t-s)H_n}(H_n-H)\\e^{-\\ii sH}\\dd s.\n\\]\nBoth unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamiltonians are contractions.\n\\phantomsection\\label{strong:proof:075:end}\\end{proof}","\\begin{corollary}[Limit of packet leakage]\\label{strong:corollary:016}\\label{strong:cor:limit_leakage}\nLet \\(P_n\\) be orthogonal projectors and \\(H_n\\) bounded self adjoint operators on one Hilbert space, with \\(P_n\\to P\\) and \\(H_n\\to H\\) in norm. Then for every bounded time interval,\n\\[\n(I-P_n)\\e^{-\\ii tH_n}P_n\n\\longrightarrow\n(I-P)\\e^{-\\ii tH}P\n\\]\nuniformly in time.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{strong:proof:076}\nAdd and subtract terms so that one factor changes at a time. The projector terms converge uniformly because \\(P_n\\to P\\) in norm. Duhamel's identity gives\n\\[\n\\sup_{\\abs t\\leq T}\n\\norm{\\e^{-\\ii tH_n}-\\e^{-\\ii tH}}\n\\leq\nT\\norm{H_n-H},\n\\]\nwhich proves uniform convergence of the middle factor.\n\\phantomsection\\label{strong:proof:076:end}\\end{proof}","\\begin{theorem}[Reflection positivity is closed under limits]\\label{strong:theorem:048}\\label{strong:thm:reflection_limit}\nSuppose \\(C_n(j,k)\\) are reflection positive kernels on \\(\\NN\\), meaning every finite matrix\n\\[\n\\bigl(C_n(j_r,j_s)\\bigr)_{r,s}\n\\]\nis positive semidefinite. If\n\\[\nC_n(j,k)\\longrightarrow C(j,k)\n\\]\nfor every pair, then \\(C\\) is reflection positive.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{strong:proof:077}\nFor a fixed finite set of times and coefficients, the corresponding quadratic forms are nonnegative and converge to the quadratic form of \\(C\\). The limit is nonnegative.\n\\phantomsection\\label{strong:proof:077:end}\\end{proof}","\\begin{theorem}[Constraint closure under norm convergence]\\label{strong:theorem:049}\\label{strong:thm:constraint_limit}\nLet \\(C_{n,a}\\to C_a\\) in norm for every \\(a\\), and fix structure coefficients \\(f_{ab}^{\\phantom{ab}c}\\). If\n\\[\n\\norm{\n[C_{n,a},C_{n,b}]\n-\\ii\\sum_cf_{ab}^{\\phantom{ab}c}C_{n,c}\n}\n\\longrightarrow0,\n\\]\nthen\n\\[\n[C_a,C_b]\n=\\ii\\sum_cf_{ab}^{\\phantom{ab}c}C_c.\n\\]\n\\end{theorem}","\\begin{theorem}[Heat trace convergence from trace norm]\\label{strong:theorem:050}\\label{strong:thm:trace_norm_heat}\nLet \\(L_n,L\\) be nonnegative self adjoint operators on one Hilbert space. For a fixed \\(t>0\\), assume their heat operators are trace class and that\n\\[\n\\e^{-tL_n}\\longrightarrow\\e^{-tL}\n\\]\nin trace norm. Then\n\\[\n\\Tr(\\e^{-tL_n})\\longrightarrow\\Tr(\\e^{-tL}).\n\\]\nIf also the heat moment operators are trace class and\n\\[\nL_n\\e^{-tL_n}\\longrightarrow L\\e^{-tL}\n\\]\nin trace norm and the limiting heat trace is nonzero, then the spectral dimensions defined by the heat moment ratio converge at that time.\n\\end{theorem}","ergence from trace norm]\\label{strong:theorem:050}\\label{strong:thm:trace_norm_heat}\nLet \\(L_n,L\\) be nonnegative self adjoint operators on one Hilbert space. For a fixed \\(t>0\\), assume their heat operators are trace class and that\n\\[\n\\e^{-tL_n}\\longrightarrow\\e^{-tL}\n\\]\nin trace norm. Then\n\\[\n\\Tr(\\e^{-tL_n})\\longrightarrow\\Tr(\\e^{-tL}).\n\\]\nIf also the heat moment operators are trace class and\n\\[\nL_n\\e^{-tL_n}\\longrightarrow L\\e^{-tL}\n\\]\nin trace norm and the limiting heat trace is nonzero, then the spectral dimensions defined by the heat moment ratio converge at that time.\n\\end{theorem}","\\begin{definition}[Anomaly free continuum constraint system]\\label{strong:definition:011}\\label{strong:def:anomaly_free_continuum}\nA continuum constraint system consists of a Hilbert space, a common invariant dense domain \\(\\Dcal\\), and operators \\(C_a\\) on \\(\\Dcal\\). It is anomaly free for prescribed structure coefficients when\n\\[\n[C_a,C_b]\\psi\n=\n\\ii\\sum_cf_{ab}^{\\phantom{ab}c}C_c\\psi\n\\]\nfor every \\(\\psi\\in\\Dcal\\), and when the associated gauge evolution is defined on the physical quotient or physical subspace.\n\\end{definition}","in{definition}[Anomaly free continuum constraint system]\\label{strong:definition:011}\\label{strong:def:anomaly_free_continuum}\nA continuum constraint system consists of a Hilbert space, a common invariant dense domain \\(\\Dcal\\), and operators \\(C_a\\) on \\(\\Dcal\\). It is anomaly free for prescribed structure coefficients when\n\\[\n[C_a,C_b]\\psi\n=\n\\ii\\sum_cf_{ab}^{\\phantom{ab}c}C_c\\psi\n\\]\nfor every \\(\\psi\\in\\Dcal\\), and when the associated gauge evolution is defined on the physical quotient or physical subspace.\n\\end{definition}","\\begin{definition}[Einstein infrared criterion]\\label{strong:definition:013}\\label{strong:def:einstein_criterion}\nA quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form\n\\[\n\\Gamma_{\\rm eff}[g]\n=\n\\int\n\\sqrt{\\abs g}\n\\left(\n\\Lambda\n+\n\\frac{1}{16\\pi G}R\n+\n\\text{controlled higher order terms}\n\\right)\n\\dd^dx.\n\\]\nHere the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an estimate for the omitted terms on the declared infrared regime.\n\\end{definition}","\\section{Exact packets and arbitrary circuit length}\\label{supp1:section:001}\nLet $R$ be a unital ring, not necessarily commutative, and let $I$ be a finite address set. An exact packet family consists of elements $P_a\\in R$ satisfying\n\\[\nP_aP_b=\\begin{cases}P_a,&a=b,","\\section{Exact packets and arbitrary circuit length}\\label{supp1:section:001}\nLet $R$ be a unital ring, not necessarily commutative, and let $I$ be a finite address set. An exact packet family consists of elements $P_a\\in R$ satisfying\n\\[\nP_aP_b=\\begin{cases}P_a,&a=b,\\\\0,&a\\ne b,\\end{cases}\n\\qquad \\sum_{a\\in I}P_a=1.\n\\]\nAdjoint conditions are added when needed. For an operator $U$, define the transfer corner\n\\[\nU_{ba}=P_bUP_a.\n\\]\nThe target index comes first. Completeness and distributivity give\n\\[\n\\sum_{b,a}U_{ba}=U,\\qquad\n(UV)_{ba}=\\sum_c U_{bc}V_{ca}.\n\\]\nThe second identity uses $P_c^2=P_c$ when adjacent transfer corners are multiplied.","0,&a\\ne b,\\end{cases}\n\\qquad \\sum_{a\\in I}P_a=1.\n\\]\nAdjoint conditions are added when needed. For an operator $U$, define the transfer corner\n\\[\nU_{ba}=P_bUP_a.\n\\]\nThe target index comes first. Completeness and distributivity give\n\\[\n\\sum_{b,a}U_{ba}=U,\\qquad\n(UV)_{ba}=\\sum_c U_{bc}V_{ca}.\n\\]\nThe second identity uses $P_c^2=P_c$ when adjacent transfer corners are multiplied.","\\subsection{Circuit paths: source Theorem 4.1}\nFor $U=U_m\\cdots U_1$, the checked result is\n\\[\nP_{a_m}UP_{a_0}\n=\\sum_{a_1,\\ldots,a_{m-1}}\nP_{a_m}U_mP_{a_{m-1}}\\cdots P_{a_1}U_1P_{a_0}.\n\\]\nThe Lean implementation defines a recursive path sum and proves the identity by induction over an arbitrary finite gate list. It is not restricted to a fixed circuit depth. The empty list has circuit operator $1$ and path value $P_bP_a$. Summing the source and target indices reconstructs the full circuit.","it paths: source Theorem 4.1}\nFor $U=U_m\\cdots U_1$, the checked result is\n\\[\nP_{a_m}UP_{a_0}\n=\\sum_{a_1,\\ldots,a_{m-1}}\nP_{a_m}U_mP_{a_{m-1}}\\cdots P_{a_1}U_1P_{a_0}.\n\\]\nThe Lean implementation defines a recursive path sum and proves the identity by induction over an arbitrary finite gate list. It is not restricted to a fixed circuit depth. The empty list has circuit operator $1$ and path value $P_bP_a$. Summing the source and target indices reconstructs the full circuit.","\\subsection{Adjoint conservation: source Theorem 4.2}\nAssume a star operation with $P_a^*=P_a$. Then\n\\[\n(U_{ba})^*=P_aU^*P_b.\n\\]\nThe two checked conservation laws are\n\\begin{align*}\nU^*U=1&\\quad\\Longrightarrow\\quad\n\\sum_b(U_{ba})^*U_{bc}=\\delta_{ac}P_a,","\\subsection{Adjoint conservation: source Theorem 4.2}\nAssume a star operation with $P_a^*=P_a$. Then\n\\[\n(U_{ba})^*=P_aU^*P_b.\n\\]\nThe two checked conservation laws are\n\\begin{align*}\nU^*U=1&\\quad\\Longrightarrow\\quad\n\\sum_b(U_{ba})^*U_{bc}=\\delta_{ac}P_a,\\\\\nUU^*=1&\\quad\\Longrightarrow\\quad\n\\sum_aU_{ba}(U_{da})^*=\\delta_{bd}P_b.\n\\end{align*}\nFor the first law, expand each corner, collapse $P_b^2$ and sum:\n\\[\n\\sum_bP_aU^*P_bUP_c\n=P_aU^*\\left(\\sum_bP_b\\right)UP_c=P_aP_c.\n\\]\nThe second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The order of the operator products is preserved throughout.","UU^*=1&\\quad\\Longrightarrow\\quad\n\\sum_aU_{ba}(U_{da})^*=\\delta_{bd}P_b.\n\\end{align*}\nFor the first law, expand each corner, collapse $P_b^2$ and sum:\n\\[\n\\sum_bP_aU^*P_bUP_c\n=P_aU^*\\left(\\sum_bP_b\\right)UP_c=P_aP_c.\n\\]\nThe second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The order of the operator products is preserved throughout.","\\section{Symmetry and measurement}\\label{supp1:section:002}\n\\subsection{Exact symmetry: source Theorem 5.1}\nIf $HP_a=P_aH$ for every packet, then\n\\[\nP_bHP_a=0\\quad(b\\ne a),\\qquad H=\\sum_aP_aHP_a.\n\\]\nIn a complex Banach algebra, commutation also propagates to the actual exponential:\n\\[\ne^{zH}P_a=P_ae^{zH}\\qquad(z\\in\\C).\n\\]\nConsequently,\n\\[\ne^{zH}=\\sum_aP_ae^{zH}P_a,\n\\qquad P_be^{zH}P_a=0\\quad(b\\ne a).\n\\]\nTaking $z=-it$ gives evolution in real time. The Lean proof applies the exponential commutation theorem directly, so invariance of the packets is a proved consequence of commutation. Self-adjointness of $H$ is not needed for the algebraic reducing property, so this result applies more broadly than the Hermitian specialization in the paper.","ry and measurement}\\label{supp1:section:002}\n\\subsection{Exact symmetry: source Theorem 5.1}\nIf $HP_a=P_aH$ for every packet, then\n\\[\nP_bHP_a=0\\quad(b\\ne a),\\qquad H=\\sum_aP_aHP_a.\n\\]\nIn a complex Banach algebra, commutation also propagates to the actual exponential:\n\\[\ne^{zH}P_a=P_ae^{zH}\\qquad(z\\in\\C).\n\\]\nConsequently,\n\\[\ne^{zH}=\\sum_aP_ae^{zH}P_a,\n\\qquad P_be^{zH}P_a=0\\quad(b\\ne a).\n\\]\nTaking $z=-it$ gives evolution in real time. The Lean proof applies the exponential commutation theorem directly, so invariance of the packets is a proved consequence of commutation. Self-adjointness of $H$ is not needed for the algebraic reducing property, so this result applies more broadly than the Hermitian specialization in the paper.","Banach algebra, commutation also propagates to the actual exponential:\n\\[\ne^{zH}P_a=P_ae^{zH}\\qquad(z\\in\\C).\n\\]\nConsequently,\n\\[\ne^{zH}=\\sum_aP_ae^{zH}P_a,\n\\qquad P_be^{zH}P_a=0\\quad(b\\ne a).\n\\]\nTaking $z=-it$ gives evolution in real time. The Lean proof applies the exponential commutation theorem directly, so invariance of the packets is a proved consequence of commutation. Self-adjointness of $H$ is not needed for the algebraic reducing property, so this result applies more broadly than the Hermitian specialization in the paper.","\\subsection{Measurements: source Theorem 7.1}\nFor finite complex matrices, define packet dephasing by\n\\[\n\\deph(X)=\\sum_aP_aXP_a.\n\\]\nThe code proves idempotence and trace duality:\n\\[\n\\deph(\\deph(X))=\\deph(X),\\qquad\n\\tr\\bigl(\\deph(E)\\rho\\bigr)=\\tr\\bigl(E\\deph(\\rho)\\bigr).\n\\]\nTrace duality follows term by term from cyclicity:\n\\[\n\\tr(P_aEP_a\\rho)=\\tr(EP_a\\rho P_a).\n\\]\nTherefore an effect that is diagonal in the packets, $\\deph(E)=E$, satisfies\n\\[\n\\boxed{\\tr(E\\rho)=\\tr\\bigl(E\\deph(\\rho)\\bigr).}\n\\]\nThe proof holds for arbitrary finite complex matrices $E$ and $\\rho$. Density matrices and positive measurement effects are valid specializations rather than assumptions needed for this trace identity. Taking $E=1$ also gives trace preservation of packet dephasing.","nts: source Theorem 7.1}\nFor finite complex matrices, define packet dephasing by\n\\[\n\\deph(X)=\\sum_aP_aXP_a.\n\\]\nThe code proves idempotence and trace duality:\n\\[\n\\deph(\\deph(X))=\\deph(X),\\qquad\n\\tr\\bigl(\\deph(E)\\rho\\bigr)=\\tr\\bigl(E\\deph(\\rho)\\bigr).\n\\]\nTrace duality follows term by term from cyclicity:\n\\[\n\\tr(P_aEP_a\\rho)=\\tr(EP_a\\rho P_a).\n\\]\nTherefore an effect that is diagonal in the packets, $\\deph(E)=E$, satisfies\n\\[\n\\boxed{\\tr(E\\rho)=\\tr\\bigl(E\\deph(\\rho)\\bigr).}\n\\]\nThe proof holds for arbitrary finite complex matrices $E$ and $\\rho$. Density matrices and positive measurement effects are valid specializations rather than assumptions needed for this trace identity. Taking $E=1$ also gives trace preservation of packet dephasing.","For finite complex matrices, define packet dephasing by\n\\[\n\\deph(X)=\\sum_aP_aXP_a.\n\\]\nThe code proves idempotence and trace duality:\n\\[\n\\deph(\\deph(X))=\\deph(X),\\qquad\n\\tr\\bigl(\\deph(E)\\rho\\bigr)=\\tr\\bigl(E\\deph(\\rho)\\bigr).\n\\]\nTrace duality follows term by term from cyclicity:\n\\[\n\\tr(P_aEP_a\\rho)=\\tr(EP_a\\rho P_a).\n\\]\nTherefore an effect that is diagonal in the packets, $\\deph(E)=E$, satisfies\n\\[\n\\boxed{\\tr(E\\rho)=\\tr\\bigl(E\\deph(\\rho)\\bigr).}\n\\]\nThe proof holds for arbitrary finite complex matrices $E$ and $\\rho$. Density matrices and positive measurement effects are valid specializations rather than assumptions needed for this trace identity. Taking $E=1$ also gives trace preservation of packet dephasing.","otence and trace duality:\n\\[\n\\deph(\\deph(X))=\\deph(X),\\qquad\n\\tr\\bigl(\\deph(E)\\rho\\bigr)=\\tr\\bigl(E\\deph(\\rho)\\bigr).\n\\]\nTrace duality follows term by term from cyclicity:\n\\[\n\\tr(P_aEP_a\\rho)=\\tr(EP_a\\rho P_a).\n\\]\nTherefore an effect that is diagonal in the packets, $\\deph(E)=E$, satisfies\n\\[\n\\boxed{\\tr(E\\rho)=\\tr\\bigl(E\\deph(\\rho)\\bigr).}\n\\]\nThe proof holds for arbitrary finite complex matrices $E$ and $\\rho$. Density matrices and positive measurement effects are valid specializations rather than assumptions needed for this trace identity. Taking $E=1$ also gives trace preservation of packet dephasing.","nd trace duality:\n\\[\n\\deph(\\deph(X))=\\deph(X),\\qquad\n\\tr\\bigl(\\deph(E)\\rho\\bigr)=\\tr\\bigl(E\\deph(\\rho)\\bigr).\n\\]\nTrace duality follows term by term from cyclicity:\n\\[\n\\tr(P_aEP_a\\rho)=\\tr(EP_a\\rho P_a).\n\\]\nTherefore an effect that is diagonal in the packets, $\\deph(E)=E$, satisfies\n\\[\n\\boxed{\\tr(E\\rho)=\\tr\\bigl(E\\deph(\\rho)\\bigr).}\n\\]\nThe proof holds for arbitrary finite complex matrices $E$ and $\\rho$. Density matrices and positive measurement effects are valid specializations rather than assumptions needed for this trace identity. Taking $E=1$ also gives trace preservation of packet dephasing.","from cyclicity:\n\\[\n\\tr(P_aEP_a\\rho)=\\tr(EP_a\\rho P_a).\n\\]\nTherefore an effect that is diagonal in the packets, $\\deph(E)=E$, satisfies\n\\[\n\\boxed{\\tr(E\\rho)=\\tr\\bigl(E\\deph(\\rho)\\bigr).}\n\\]\nThe proof holds for arbitrary finite complex matrices $E$ and $\\rho$. Density matrices and positive measurement effects are valid specializations rather than assumptions needed for this trace identity. Taking $E=1$ also gives trace preservation of packet dephasing.","\\section{Corrected Feshbach resolvent}\\label{supp1:section:003}\nWrite a finite complex matrix in retained and eliminated coordinates:\n\\[\nH=\\begin{pmatrix}A&B\\\\C&D\\end{pmatrix},\\qquad\nzI-H=\\begin{pmatrix}zI_P-A&-B\\\\-C&zI_Q-D\\end{pmatrix}.\n\\]\nSet\n\\[\nD_z=zI_Q-D,\\qquad S_z=zI_P-A-BD_z^{-1}C.\n\\]\nThe effective Hamiltonian is $H_{\\mathrm{eff}}(z)=A+BD_z^{-1}C$, so\n$S_z=zI_P-H_{\\mathrm{eff}}(z)$.","-C&zI_Q-D\\end{pmatrix}.\n\\]\nSet\n\\[\nD_z=zI_Q-D,\\qquad S_z=zI_P-A-BD_z^{-1}C.\n\\]\nThe effective Hamiltonian is $H_{\\mathrm{eff}}(z)=A+BD_z^{-1}C$, so\n$S_z=zI_P-H_{\\mathrm{eff}}(z)$.","\\subsection{Correct statement: source Theorem 9.1}\nAssume both $D_z$ and $zI-H$ are invertible. Then $S_z$ is invertible and\n\\[\n\\boxed{\\bigl((zI-H)^{-1}\\bigr)_{PP}=S_z^{-1}.}\n\\]\nThe left side is the submatrix of the full inverse on the retained coordinates. This matches the paper's compressed operator when written in an orthonormal basis of the retained subspace. The proof is valid for arbitrary complex blocks of compatible sizes and does not require Hermiticity.","The checked invertibility criterion is\n\\[\nD_z\\text{ invertible}\\quad\\Longrightarrow\\quad\n\\bigl(zI-H\\text{ invertible}\\ \\Longleftrightarrow\\ S_z\\text{ invertible}\\bigr).\n\\]\nThus one may equivalently assume invertibility of $D_z$ and $S_z$.","\\textbf{Why the original assumption is insufficient.}\nTake $z=0$ and $H=\\diag(0,-1)$, retaining the first coordinate. Then\n\\[\nD_z=1,\\qquad zI-H=\\begin{pmatrix}0&0","\\textbf{Why the original assumption is insufficient.}\nTake $z=0$ and $H=\\diag(0,-1)$, retaining the first coordinate. Then\n\\[\nD_z=1,\\qquad zI-H=\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}.\n\\]\nThe eliminated block is invertible, but the full matrix is singular. Hence the original hypothesis does not ensure the existence of its stated resolvent. The Lean counterexample proves invertibility of the scalar $1$ and noninvertibility of this full $2\\times2$ matrix.","0&1\\end{pmatrix}.\n\\]\nThe eliminated block is invertible, but the full matrix is singular. Hence the original hypothesis does not ensure the existence of its stated resolvent. The Lean counterexample proves invertibility of the scalar $1$ and noninvertibility of this full $2\\times2$ matrix.","\\section{Dispersion with an explicit spacing condition}\\label{supp1:section:004}\nThe grid eigenvalue in source Theorem 11.1 is\n\\[\n\\lambda_{k,h}=\\frac{4}{h^2}\\sin^2\\left(\\frac{kh}{2}\\right).\n\\]\nThe original low-mode condition is $0\\le kh/2\\le1$. A nonzero spacing is also required for the quotient to have its intended meaning. For a physical grid, one normally takes $h>0$.","\\subsection{Checked stronger statement}\nFor every $k,h\\in\\R$ with $h\\ne0$,\n\\[\n\\boxed{0\\le k^2-\\lambda_{k,h}\\le\\frac{k^4h^2}{12}.}\n\\]\nThe code proves this without the low-mode restriction. The bound may be loose away from the low-mode regime, but it remains valid.","The proof begins with an estimate valid for every real argument:\n\\[\n0\\le x^2-\\sin^2x\\le\\frac{x^4}{3}.\n\\]\nThe lower bound follows from $|\\sin x|\\le|x|$. For the upper bound, use\n\\[\n|x-\\sin x|\\le\\frac{|x|^3}{6},\\qquad\n|x+\\sin x|\\le2|x|.\n\\]\nFactoring the difference of squares gives\n\\[\n0\\le x^2-\\sin^2x\n=|x-\\sin x|\\,|x+\\sin x|\n\\le\\frac{x^4}{3}.\n\\]\nWith $x=kh/2$ and $h\\ne0$,\n\\begin{align*}\nk^2-\\lambda_{k,h}\n&=\\frac4{h^2}\\left(x^2-\\sin^2x\\right),","The proof begins with an estimate valid for every real argument:\n\\[\n0\\le x^2-\\sin^2x\\le\\frac{x^4}{3}.\n\\]\nThe lower bound follows from $|\\sin x|\\le|x|$. For the upper bound, use\n\\[\n|x-\\sin x|\\le\\frac{|x|^3}{6},\\qquad\n|x+\\sin x|\\le2|x|.\n\\]\nFactoring the difference of squares gives\n\\[\n0\\le x^2-\\sin^2x\n=|x-\\sin x|\\,|x+\\sin x|\n\\le\\frac{x^4}{3}.\n\\]\nWith $x=kh/2$ and $h\\ne0$,\n\\begin{align*}\nk^2-\\lambda_{k,h}\n&=\\frac4{h^2}\\left(x^2-\\sin^2x\\right),\\\\\n\\frac4{h^2}\\frac{x^4}{3}\n&=\\frac{k^4h^2}{12}.\n\\end{align*}\nEvery algebraic transformation and inequality in the active source was accepted by Lean.","\\section{A constraint diagnostic that vanishes automatically}\\label{supp1:section:005}\nSection 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives $C_aP=0$ for every $a$. Absorb the factor $i$ and any structure coefficient into $f_c$, and write\n\\[\n\\mathcal A_{ab}=C_aC_b-C_bC_a-\\sum_cf_cC_c.\n\\]\n\\begin{proposition}[Exact common-kernel compression]\\label{supp1:proposition:001}\nIf $C_cP=0$ for every $c$, then\n\\[\n\\mathcal A_{ab}P=0,\\qquad P\\mathcal A_{ab}P=0.\n\\]\nNo idempotence or adjoint property of $P$ is needed for this implication.\n\\end{proposition}\n\\begin{proof}\\phantomsection\\label{supp1:proof:001}\nAssociativity gives\n\\[\n\\mathcal A_{ab}P=C_a(C_bP)-C_b(C_aP)-\\sum_cf_c(C_cP)=0.\n\\]\nLeft multiplication by $P$ gives the second identity.\n\\phantomsection\\label{supp1:proof:001:end}\\end{proof}","s automatically}\\label{supp1:section:005}\nSection 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives $C_aP=0$ for every $a$. Absorb the factor $i$ and any structure coefficient into $f_c$, and write\n\\[\n\\mathcal A_{ab}=C_aC_b-C_bC_a-\\sum_cf_cC_c.\n\\]\n\\begin{proposition}[Exact common-kernel compression]\\label{supp1:proposition:001}\nIf $C_cP=0$ for every $c$, then\n\\[\n\\mathcal A_{ab}P=0,\\qquad P\\mathcal A_{ab}P=0.\n\\]\nNo idempotence or adjoint property of $P$ is needed for this implication.\n\\end{proposition}\n\\begin{proof}\\phantomsection\\label{supp1:proof:001}\nAssociativity gives\n\\[\n\\mathcal A_{ab}P=C_a(C_bP)-C_b(C_aP)-\\sum_cf_c(C_cP)=0.\n\\]\nLeft multiplication by $P$ gives the second identity.\n\\phantomsection\\label{supp1:proof:001:end}\\end{proof}","bel{supp1:section:005}\nSection 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives $C_aP=0$ for every $a$. Absorb the factor $i$ and any structure coefficient into $f_c$, and write\n\\[\n\\mathcal A_{ab}=C_aC_b-C_bC_a-\\sum_cf_cC_c.\n\\]\n\\begin{proposition}[Exact common-kernel compression]\\label{supp1:proposition:001}\nIf $C_cP=0$ for every $c$, then\n\\[\n\\mathcal A_{ab}P=0,\\qquad P\\mathcal A_{ab}P=0.\n\\]\nNo idempotence or adjoint property of $P$ is needed for this implication.\n\\end{proposition}\n\\begin{proof}\\phantomsection\\label{supp1:proof:001}\nAssociativity gives\n\\[\n\\mathcal A_{ab}P=C_a(C_bP)-C_b(C_aP)-\\sum_cf_c(C_cP)=0.\n\\]\nLeft multiplication by $P$ gives the second identity.\n\\phantomsection\\label{supp1:proof:001:end}\\end{proof}","ection:005}\nSection 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives $C_aP=0$ for every $a$. Absorb the factor $i$ and any structure coefficient into $f_c$, and write\n\\[\n\\mathcal A_{ab}=C_aC_b-C_bC_a-\\sum_cf_cC_c.\n\\]\n\\begin{proposition}[Exact common-kernel compression]\\label{supp1:proposition:001}\nIf $C_cP=0$ for every $c$, then\n\\[\n\\mathcal A_{ab}P=0,\\qquad P\\mathcal A_{ab}P=0.\n\\]\nNo idempotence or adjoint property of $P$ is needed for this implication.\n\\end{proposition}\n\\begin{proof}\\phantomsection\\label{supp1:proof:001}\nAssociativity gives\n\\[\n\\mathcal A_{ab}P=C_a(C_bP)-C_b(C_aP)-\\sum_cf_c(C_cP)=0.\n\\]\nLeft multiplication by $P$ gives the second identity.\n\\phantomsection\\label{supp1:proof:001:end}\\end{proof}","constraints. In finite dimensions this gives $C_aP=0$ for every $a$. Absorb the factor $i$ and any structure coefficient into $f_c$, and write\n\\[\n\\mathcal A_{ab}=C_aC_b-C_bC_a-\\sum_cf_cC_c.\n\\]\n\\begin{proposition}[Exact common-kernel compression]\\label{supp1:proposition:001}\nIf $C_cP=0$ for every $c$, then\n\\[\n\\mathcal A_{ab}P=0,\\qquad P\\mathcal A_{ab}P=0.\n\\]\nNo idempotence or adjoint property of $P$ is needed for this implication.\n\\end{proposition}\n\\begin{proof}\\phantomsection\\label{supp1:proof:001}\nAssociativity gives\n\\[\n\\mathcal A_{ab}P=C_a(C_bP)-C_b(C_aP)-\\sum_cf_c(C_cP)=0.\n\\]\nLeft multiplication by $P$ gives the second identity.\n\\phantomsection\\label{supp1:proof:001:end}\\end{proof}","$f_c$, and write\n\\[\n\\mathcal A_{ab}=C_aC_b-C_bC_a-\\sum_cf_cC_c.\n\\]\n\\begin{proposition}[Exact common-kernel compression]\\label{supp1:proposition:001}\nIf $C_cP=0$ for every $c$, then\n\\[\n\\mathcal A_{ab}P=0,\\qquad P\\mathcal A_{ab}P=0.\n\\]\nNo idempotence or adjoint property of $P$ is needed for this implication.\n\\end{proposition}\n\\begin{proof}\\phantomsection\\label{supp1:proof:001}\nAssociativity gives\n\\[\n\\mathcal A_{ab}P=C_a(C_bP)-C_b(C_aP)-\\sum_cf_c(C_cP)=0.\n\\]\nLeft multiplication by $P$ gives the second identity.\n\\phantomsection\\label{supp1:proof:001:end}\\end{proof}","n}[Exact common-kernel compression]\\label{supp1:proposition:001}\nIf $C_cP=0$ for every $c$, then\n\\[\n\\mathcal A_{ab}P=0,\\qquad P\\mathcal A_{ab}P=0.\n\\]\nNo idempotence or adjoint property of $P$ is needed for this implication.\n\\end{proposition}\n\\begin{proof}\\phantomsection\\label{supp1:proof:001}\nAssociativity gives\n\\[\n\\mathcal A_{ab}P=C_a(C_bP)-C_b(C_aP)-\\sum_cf_c(C_cP)=0.\n\\]\nLeft multiplication by $P$ gives the second identity.\n\\phantomsection\\label{supp1:proof:001:end}\\end{proof}","\\subsection{Compressed vanishing without operator closure}\nOver the rationals, take\n\\[\nP=\\begin{pmatrix}1&0&0\\\\0&0&0\\\\0&0&0\\end{pmatrix},\\quad\nX=\\begin{pmatrix}0&0&0\\\\0&0&1\\\\0&1&0\\end{pmatrix},\\quad\nZ=\\begin{pmatrix}0&0&0\\\\0&1&0\\\\0&0&-1\\end{pmatrix}.\n\\]\nThese displayed matrices are symmetric. The code checks that $P^2=P$, $P\\ne0$ and $XP=ZP=0$, but\n\\[\n[X,Z]=\\begin{pmatrix}0&0&0","0&0&-1\\end{pmatrix}.\n\\]\nThese displayed matrices are symmetric. The code checks that $P^2=P$, $P\\ne0$ and $XP=ZP=0$, but\n\\[\n[X,Z]=\\begin{pmatrix}0&0&0","0&0&-1\\end{pmatrix}.\n\\]\nThese displayed matrices are symmetric. The code checks that $P^2=P$, $P\\ne0$ and $XP=ZP=0$, but\n\\[\n[X,Z]=\\begin{pmatrix}0&0&0\\\\0&0&-2\\\\0&2&0\\end{pmatrix}\\ne0,\n\\qquad P[X,Z]P=0.\n\\]\nThe commutator therefore violates the closure relation with zero structure coefficients outside the physical range, even though its physical compression vanishes exactly.","For $n\\geq 0$, let\n\\[\n A_n=(\\mathbb Z/2\\mathbb Z)^{\\{0,\\ldots,n-1\\}},\\qquad\n V_n=\\{v:A_n\\to\\C\\}.\n\\]\nFor an address $a\\in A_n$, define the standard coordinate vector\n\\[\n \\ket a(x)=\\begin{cases}1&x=a,","For $n\\geq 0$, let\n\\[\n A_n=(\\mathbb Z/2\\mathbb Z)^{\\{0,\\ldots,n-1\\}},\\qquad\n V_n=\\{v:A_n\\to\\C\\}.\n\\]\nFor an address $a\\in A_n$, define the standard coordinate vector\n\\[\n \\ket a(x)=\\begin{cases}1&x=a,\\\\0&x\\ne a.\\end{cases}\n\\]\nThe seed is $\\psi_0=\\ket0$. Define a translation operator by\n\\[\n (T_a v)(x)=v(x-a).\n\\]\nAt position $j$, the local Pauli $X$ is $X_j=T_{e_j}$, where $e_j$ has a single nonzero binary coordinate. The formal circuit uses the identity when $a_j=0$ and $X_j$ when $a_j=1$.","0&x\\ne a.\\end{cases}\n\\]\nThe seed is $\\psi_0=\\ket0$. Define a translation operator by\n\\[\n (T_a v)(x)=v(x-a).\n\\]\nAt position $j$, the local Pauli $X$ is $X_j=T_{e_j}$, where $e_j$ has a single nonzero binary coordinate. The formal circuit uses the identity when $a_j=0$ and $X_j$ when $a_j=1$.","\\begin{statement}\\label{supp2:statement:001}\nFor every binary address $a$,\n\\[\n W_a=\\prod_{j=0}^{n-1}X_j^{a_j}=T_a,\n \\qquad W_a\\psi_0=\\ket a.\n\\]\nThe family $(W_a\\psi_0)_{a\\in A_n}$ is linearly independent, spans $V_n$, and has exactly $2^n$ elements. It is the standard computational basis.\n\\end{statement}","\\begin{proof}\\phantomsection\\label{supp2:proof:001}\nComposition gives $T_aT_b=T_{a+b}$, and direct evaluation gives $T_a\\ket b=\\ket{a+b}$. Every binary address satisfies $a+a=0$, so each translation is an involution. In particular, the local operator changes precisely the addressed bit:\n\\[\n X_j\\ket a=\\ket{a\\text{ with coordinate }j\\text{ replaced by }a_j+1}.\n\\]\nThe selected factor at $j$ equals translation by the address supported at $j$ with value $a_j$. Induction on a list proves that the product of translations is translation by their sum. Each of these addresses contributes one coordinate, and their sum is $a$. Thus $W_a=T_a$ and $W_a\\psi_0=\\ket a$.","2:proof:001}\nComposition gives $T_aT_b=T_{a+b}$, and direct evaluation gives $T_a\\ket b=\\ket{a+b}$. Every binary address satisfies $a+a=0$, so each translation is an involution. In particular, the local operator changes precisely the addressed bit:\n\\[\n X_j\\ket a=\\ket{a\\text{ with coordinate }j\\text{ replaced by }a_j+1}.\n\\]\nThe selected factor at $j$ equals translation by the address supported at $j$ with value $a_j$. Induction on a list proves that the product of translations is translation by their sum. Each of these addresses contributes one coordinate, and their sum is $a$. Thus $W_a=T_a$ and $W_a\\psi_0=\\ket a$.","mposition gives $T_aT_b=T_{a+b}$, and direct evaluation gives $T_a\\ket b=\\ket{a+b}$. Every binary address satisfies $a+a=0$, so each translation is an involution. In particular, the local operator changes precisely the addressed bit:\n\\[\n X_j\\ket a=\\ket{a\\text{ with coordinate }j\\text{ replaced by }a_j+1}.\n\\]\nThe selected factor at $j$ equals translation by the address supported at $j$ with value $a_j$. Induction on a list proves that the product of translations is translation by their sum. Each of these addresses contributes one coordinate, and their sum is $a$. Thus $W_a=T_a$ and $W_a\\psi_0=\\ket a$.","particular, the local operator changes precisely the addressed bit:\n\\[\n X_j\\ket a=\\ket{a\\text{ with coordinate }j\\text{ replaced by }a_j+1}.\n\\]\nThe selected factor at $j$ equals translation by the address supported at $j$ with value $a_j$. Induction on a list proves that the product of translations is translation by their sum. Each of these addresses contributes one coordinate, and their sum is $a$. Thus $W_a=T_a$ and $W_a\\psi_0=\\ket a$.","r, the local operator changes precisely the addressed bit:\n\\[\n X_j\\ket a=\\ket{a\\text{ with coordinate }j\\text{ replaced by }a_j+1}.\n\\]\nThe selected factor at $j$ equals translation by the address supported at $j$ with value $a_j$. Induction on a list proves that the product of translations is translation by their sum. Each of these addresses contributes one coordinate, and their sum is $a$. Thus $W_a=T_a$ and $W_a\\psi_0=\\ket a$.","perator changes precisely the addressed bit:\n\\[\n X_j\\ket a=\\ket{a\\text{ with coordinate }j\\text{ replaced by }a_j+1}.\n\\]\nThe selected factor at $j$ equals translation by the address supported at $j$ with value $a_j$. Induction on a list proves that the product of translations is translation by their sum. Each of these addresses contributes one coordinate, and their sum is $a$. Thus $W_a=T_a$ and $W_a\\psi_0=\\ket a$.","Lean identifies the resulting family with \\lean{Pi.basisFun} over $\\C$. Its standard basis theorems supply linear independence and full span. The finite function-space cardinality gives $|A_n|=2^n$.\n\\phantomsection\\label{supp2:proof:001:end}\\end{proof}","\\textbf{Formal scope.} The proof constructs the sequence of gates that produces each addressed state and then identifies the resulting family with the standard coordinate basis. It includes the empty circuit for $n=0$. Orthogonality belongs to the usual computational basis interpretation. The formal endpoint records the basis identification, together with separate explicit theorems for span and linear independence.","Let $F$ be an arbitrary finite set of factors and let $D_i$ be a finite coordinate index for factor $i$. For matrix families $A_i$, define the coordinate tensor product on the product index space by\n\\[\n \\mathsf T(A)(x,y)=\\prod_{i\\in F} A_i(x_i,y_i).\n\\]\nThe proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite number of factors with the stated finite coordinate sets.","Suppose $N>0$ and each factor has a complete orthogonal self-adjoint projector family $(P^{(i)}_j)_{j\\in\\mathbb Z/N\\mathbb Z}$. For a fine address $a:F\\to\\mathbb Z/N\\mathbb Z$, set\n\\[\n Q_a=\\mathsf T\\bigl(i\\mapsto P^{(i)}_{a_i}\\bigr),\n \\qquad\n C_r=\\sum_{\\sum_i a_i=r}Q_a.\n\\]","\\begin{statement}\\label{supp2:statement:002}\nThe fine and coarse families are complete orthogonal self-adjoint projector systems. In particular,\n\\[\n C_rC_s=\\delta_{rs}C_r,\\qquad\n \\sum_r C_r=I,\\qquad C_r^*=C_r.\n\\]\nEach coarse projector is exactly the displayed sum of its fine descendants.\n\\end{statement}","\\begin{proof}\\phantomsection\\label{supp2:proof:002}\nExpanding the finite product of sums gives\n\\[\n \\mathsf T(A)\\mathsf T(B)=\\mathsf T(i\\mapsto A_iB_i).\n\\]\nConsequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so $Q_a^*=Q_a$. Summing over all address functions and using the same expansion yields\n\\[\n \\sum_a Q_a=\\mathsf T\\left(i\\mapsto\\sum_jP^{(i)}_j\\right)=I.\n\\]\nA separate algebraic regrouping theorem applies to any finite orthogonal family and any address map into a finite set. In the double sum for $C_rC_s$, orthogonality removes all terms with different fine addresses. A surviving address can lie in both classes only when $r=s$. Completeness follows by summing ea","\\begin{proof}\\phantomsection\\label{supp2:proof:002}\nExpanding the finite product of sums gives\n\\[\n \\mathsf T(A)\\mathsf T(B)=\\mathsf T(i\\mapsto A_iB_i).\n\\]\nConsequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so $Q_a^*=Q_a$. Summing over all address functions and using the same expansion yields\n\\[\n \\sum_a Q_a=\\mathsf T\\left(i\\mapsto\\sum_jP^{(i)}_j\\right)=I.\n\\]\nA separate algebraic regrouping theorem applies to any finite orthogonal family and any address map into a finite set. In the double sum for $C_rC_s$, orthogonality removes all terms with different fine addresses. A surviving address can lie in both classes only when $r=s$. Completeness follows by summing each fine projector exactly once. Adjoint preservation follows term by term. Specializing the address map to total residue proves the claim.\n\\phantomsection\\label{supp2:proof:002:end}\\end{proof}","ct of sums gives\n\\[\n \\mathsf T(A)\\mathsf T(B)=\\mathsf T(i\\mapsto A_iB_i).\n\\]\nConsequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so $Q_a^*=Q_a$. Summing over all address functions and using the same expansion yields\n\\[\n \\sum_a Q_a=\\mathsf T\\left(i\\mapsto\\sum_jP^{(i)}_j\\right)=I.\n\\]\nA separate algebraic regrouping theorem applies to any finite orthogonal family and any address map into a finite set. In the double sum for $C_rC_s$, orthogonality removes all terms with different fine addresses. A surviving address can lie in both classes only when $r=s$. Completeness follows by summing each fine projector exactly once. Adjoint preservation follows term by term. Specializing the address map to total residue proves the claim.\n\\phantomsection\\label{supp2:proof:002:end}\\end{proof}","gree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so $Q_a^*=Q_a$. Summing over all address functions and using the same expansion yields\n\\[\n \\sum_a Q_a=\\mathsf T\\left(i\\mapsto\\sum_jP^{(i)}_j\\right)=I.\n\\]\nA separate algebraic regrouping theorem applies to any finite orthogonal family and any address map into a finite set. In the double sum for $C_rC_s$, orthogonality removes all terms with different fine addresses. A surviving address can lie in both classes only when $r=s$. Completeness follows by summing each fine projector exactly once. Adjoint preservation follows term by term. Specializing the address map to total residue proves the claim.\n\\phantomsection\\label{supp2:proof:002:end}\\end{proof}","over all address functions and using the same expansion yields\n\\[\n \\sum_a Q_a=\\mathsf T\\left(i\\mapsto\\sum_jP^{(i)}_j\\right)=I.\n\\]\nA separate algebraic regrouping theorem applies to any finite orthogonal family and any address map into a finite set. In the double sum for $C_rC_s$, orthogonality removes all terms with different fine addresses. A surviving address can lie in both classes only when $r=s$. Completeness follows by summing each fine projector exactly once. Adjoint preservation follows term by term. Specializing the address map to total residue proves the claim.\n\\phantomsection\\label{supp2:proof:002:end}\\end{proof}","Fix a real time $t$ and a finite complex coordinate space. Let $U(s)$ be a unitary matrix curve, differentiable entrywise at $t$, and let $\\psi(s)$ be differentiable componentwise at $t$. Write $\\dot U$ and $\\dot\\psi$ for those derivatives. Suppose\n\\[\n i\\dot\\psi=H\\psi(t).\n\\]\nOnly the value of the Hamiltonian at the chosen time is needed. No derivative of $H$ is assumed.","\\begin{statement}\\label{supp2:statement:003}\nThe explicitly defined coordinate curve\n\\[\n y(s)=U(s)^*\\psi(s)\n\\]\nis differentiable at $t$, with\n\\[\n \\dot y=\\dot U^*\\psi(t)+U(t)^*\\dot\\psi.\n\\]\nIt satisfies\n\\[\n \\boxed{\\quad i\\dot y=H_{\\mathrm{eff}}y(t),\\qquad\n H_{\\mathrm{eff}}=U(t)^*HU(t)-iU(t)^*\\dot U.\\quad}\n\\]\nThe connection term $-iU(t)^*\\dot U$ is Hermitian. If $H^*=H$, then $H_{\\mathrm{eff}}^*=H_{\\mathrm{eff}}$.\n\\end{statement}","\\begin{proof}\\phantomsection\\label{supp2:proof:003}\nApplying the product rule entry by entry proves the differentiation formulas for matrix products and for matrices acting on vectors. Since the parameter is real, conjugation commutes with differentiation, giving the stated derivative of $U^*$ and hence of $y$.","Differentiate $U^*U=I$ using those rules and uniqueness of derivatives:\n\\[\n \\dot U^*U+U^*\\dot U=0.\n\\]\nThus $(U^*\\dot U)^*=-U^*\\dot U$, so multiplication by $-i$ produces a Hermitian matrix.","The identity $UU^*=I$ gives $\\psi=Uy$ for the constructed coordinate curve. Differentiating and inserting the original equation yields\n\\[\n i\\dot U y+iU\\dot y=HUy.\n\\]\nMultiplying by $U^*$ and using $U^*U=I$ gives the transformed equation, with the negative sign on the connection term. Finally, conjugation by $U$ preserves Hermiticity of $H$, and the sum of the two Hermitian terms is Hermitian.\n\\phantomsection\\label{supp2:proof:003:end}\\end{proof}","\\textbf{Formal scope.} The main certificate constructs $y$ and its derivative rather than assuming a separate derivative witness for the desired answer. Both unitary identities are explicit. The proofs are pointwise and apply at any time where the stated hypotheses hold. They do not assert existence of a global solution to an evolution equation.","Let $P_a(s)$ be a finite family in a normed real algebra with a continuous compatible adjoint. Assume actual derivatives $D_a=\\dot P_a(t)$ exist and, for all parameter values under consideration,\n\\[\n P_aP_b=\\delta_{ab}P_a,\\qquad \\sum_aP_a=I,\\qquad P_a^*=P_a.\n\\]\nIn the supplied Lean theorem, the curves are defined on $\\R$ and the projector laws hold at every real parameter. The algebraic formulation applies to matrices and bounded operators whenever they satisfy the stated typeclass hypotheses.","\\begin{statement}\\label{supp2:statement:004}\nDefine $C=\\sum_aD_aP_a(t)$. Then\n\\[\n \\boxed{\\quad C=\\frac12\\sum_a\\comm{D_a}{P_a(t)},\\qquad C^*=-C,\\quad}\n\\]\nand, for every $b$,\n\\[\n \\boxed{\\comm{C}{P_b(t)}=D_b.}\n\\]\n\\end{statement}","\\begin{proof}\\phantomsection\\label{supp2:proof:004}\nThe derivative product rule and uniqueness of derivatives give\n\\[\n D_aP_b+P_aD_b=\\delta_{ab}D_a.\n\\]\nDifferentiating completeness and adjoint invariance gives\n\\[\n \\sum_aD_a=0,\\qquad D_a^*=D_a.\n\\]\nThese are proved derivative consequences, not additional axioms supplied in place of differentiation.","On the diagonal, $D_aP_a+P_aD_a=D_a$. Summing shows\n\\[\n \\sum_aP_aD_a=-\\sum_aD_aP_a=-C.\n\\]\nThis proves the formula involving half the sum of commutators, together with $C^*=-C$.","Orthogonality directly yields $CP_b=D_bP_b$. For the left product, multiply the differentiated identity for $(b,a)$ on the right by $P_a$ and sum over $a$. Completeness reduces the result to\n\\[\n D_b+P_bC=D_bP_b.\n\\]\nSubtracting this equation from the identity for multiplication on the right gives $CP_b-P_bC=D_b$.\n\\phantomsection\\label{supp2:proof:004:end}\\end{proof}","For a finite family of complex matrices $K_a$, define\n\\[\n \\Phi(\\rho)=\\sum_a K_a\\rho K_a^*.\n\\]\nThe formalization constructs this as a complex-linear map and proves the following properties directly in the finite matrix representation.","\\begin{statement}\\label{supp2:statement:005}\nThe map $\\Phi$ is completely positive. If\n\\[\n \\sum_aK_a^*K_a=I,\n\\]\nthen $\\tr\\Phi(\\rho)=\\tr\\rho$ for every matrix $\\rho$. In particular, positive semidefinite matrices of trace one are mapped to positive semidefinite matrices of trace one.\n\\end{statement}","\\begin{proof}\\phantomsection\\label{supp2:proof:005}\nMatrix multiplication distributes across addition and scalar multiplication, which gives complex linearity. If $\\rho\\succeq0$, each term $K_a\\rho K_a^*$ is positive semidefinite, hence so is their finite sum.","To prove complete positivity rather than positivity alone, let $J$ be an arbitrary finite auxiliary index set. For a matrix $X$ indexed by $J\\times n$, define its blockwise amplification by\n\\[\n [\\Phi^{(J)}(X)]_{(i,u),(j,v)}\n =\\Phi\\bigl(X_{(i,\\cdot),(j,\\cdot)}\\bigr)_{u,v}.\n\\]\nThe explicit index calculation proves the identity\n\\[\n \\boxed{\\quad\n \\Phi^{(J)}(X)=\\sum_a (I_J\\otimes K_a)X(I_J\\otimes K_a)^*.\n \\quad}\n\\]\nThe displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive when $X$ is positive, and this holds for every finite $J$.","X)]_{(i,u),(j,v)}\n =\\Phi\\bigl(X_{(i,\\cdot),(j,\\cdot)}\\bigr)_{u,v}.\n\\]\nThe explicit index calculation proves the identity\n\\[\n \\boxed{\\quad\n \\Phi^{(J)}(X)=\\sum_a (I_J\\otimes K_a)X(I_J\\otimes K_a)^*.\n \\quad}\n\\]\nThe displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive when $X$ is positive, and this holds for every finite $J$.","ot)}\\bigr)_{u,v}.\n\\]\nThe explicit index calculation proves the identity\n\\[\n \\boxed{\\quad\n \\Phi^{(J)}(X)=\\sum_a (I_J\\otimes K_a)X(I_J\\otimes K_a)^*.\n \\quad}\n\\]\nThe displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive when $X$ is positive, and this holds for every finite $J$.","Finally, cyclicity and linearity of the trace give\n\\begin{align*}\n \\tr\\Phi(\\rho)\n &=\\sum_a\\tr(K_a\\rho K_a^*)\\\\\n &=\\tr\\left(\\left(\\sum_aK_a^*K_a\\right)\\rho\\right)\n =\\tr\\rho.\n\\end{align*}\nPositivity together with trace preservation gives the stated conclusion for density matrices.\n\\phantomsection\\label{supp2:proof:005:end}\\end{proof}","\\section{Actual orbits and sharp seed counts}\\label{supp3:section:001}\nLet $C\\in M_{d\\times m}(\\C)$ be the coefficient matrix of one seed. Define its orbit by actual matrix multiplication:\n\\[\n\\mathcal O(C)=\\{AC:A\\in M_d(\\C)\\}.\n\\]\nThis set is already a linear subspace. The source proves that it equals the linear span of the orbit before computing its dimension.","A matrix $X$ belongs to $\\mathcal O(C)$ exactly when each of its rows belongs to $\\operatorname{Row}(C)$. One direction follows by expanding $AC$. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of $A$. An explicit linear equivalence therefore gives\n\\[\n\\mathcal O(C)\\simeq \\operatorname{Row}(C)^d,\n\\qquad\n\\boxed{\\dim_\\C\\mathcal O(C)=d\\,\\rank C.}\n\\]\nIf $d>0$, equality with the full $d\\times m$ coefficient space is equivalent to $\\rank C=m$. Consequently a cyclic seed requires $m\\le d$. The assumption that there is at least one row is essential for canceling the factor $d$.","ngs to $\\mathcal O(C)$ exactly when each of its rows belongs to $\\operatorname{Row}(C)$. One direction follows by expanding $AC$. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of $A$. An explicit linear equivalence therefore gives\n\\[\n\\mathcal O(C)\\simeq \\operatorname{Row}(C)^d,\n\\qquad\n\\boxed{\\dim_\\C\\mathcal O(C)=d\\,\\rank C.}\n\\]\nIf $d>0$, equality with the full $d\\times m$ coefficient space is equivalent to $\\rank C=m$. Consequently a cyclic seed requires $m\\le d$. The assumption that there is at least one row is essential for canceling the factor $d$.","when each of its rows belongs to $\\operatorname{Row}(C)$. One direction follows by expanding $AC$. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of $A$. An explicit linear equivalence therefore gives\n\\[\n\\mathcal O(C)\\simeq \\operatorname{Row}(C)^d,\n\\qquad\n\\boxed{\\dim_\\C\\mathcal O(C)=d\\,\\rank C.}\n\\]\nIf $d>0$, equality with the full $d\\times m$ coefficient space is equivalent to $\\rank C=m$. Consequently a cyclic seed requires $m\\le d$. The assumption that there is at least one row is essential for canceling the factor $d$.","e{Row}(C)$. One direction follows by expanding $AC$. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of $A$. An explicit linear equivalence therefore gives\n\\[\n\\mathcal O(C)\\simeq \\operatorname{Row}(C)^d,\n\\qquad\n\\boxed{\\dim_\\C\\mathcal O(C)=d\\,\\rank C.}\n\\]\nIf $d>0$, equality with the full $d\\times m$ coefficient space is equivalent to $\\rank C=m$. Consequently a cyclic seed requires $m\\le d$. The assumption that there is at least one row is essential for canceling the factor $d$.","C$. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of $A$. An explicit linear equivalence therefore gives\n\\[\n\\mathcal O(C)\\simeq \\operatorname{Row}(C)^d,\n\\qquad\n\\boxed{\\dim_\\C\\mathcal O(C)=d\\,\\rank C.}\n\\]\nIf $d>0$, equality with the full $d\\times m$ coefficient space is equivalent to $\\rank C=m$. Consequently a cyclic seed requires $m\\le d$. The assumption that there is at least one row is essential for canceling the factor $d$.","the corresponding row of $A$. An explicit linear equivalence therefore gives\n\\[\n\\mathcal O(C)\\simeq \\operatorname{Row}(C)^d,\n\\qquad\n\\boxed{\\dim_\\C\\mathcal O(C)=d\\,\\rank C.}\n\\]\nIf $d>0$, equality with the full $d\\times m$ coefficient space is equivalent to $\\rank C=m$. Consequently a cyclic seed requires $m\\le d$. The assumption that there is at least one row is essential for canceling the factor $d$.","\\subsection{Several seeds: both necessity and construction}\nContinue under the assumption above that the number of rows is positive. For seeds $C_1,\\ldots,C_k$, generation means the concrete statement\n\\[\n\\forall X\\in M_{d\\times m}(\\C),\\quad\n\\exists A_1,\\ldots,A_k\\in M_d(\\C):\\quad\nX=\\sum_{a=1}^k A_aC_a.\n\\]\nStack the $C_a$ vertically into a $kd\\times m$ matrix $D$. The source identifies the action on the stacked matrix and proves the equivalence\n\\[\n\\text{the seeds generate}\\quad\\Longleftrightarrow\\quad\\rank D=m.\n\\]\nThus $m\\le kd$ is necessary. It is sufficient as well: embed the $m$ coordinate columns into $kd$ coordinate positions to obtain a matrix $D$ of full column rank, then split its rows back into $k$ seeds. This construction proves that generating matrices with the required dimensions exist.","truction}\nContinue under the assumption above that the number of rows is positive. For seeds $C_1,\\ldots,C_k$, generation means the concrete statement\n\\[\n\\forall X\\in M_{d\\times m}(\\C),\\quad\n\\exists A_1,\\ldots,A_k\\in M_d(\\C):\\quad\nX=\\sum_{a=1}^k A_aC_a.\n\\]\nStack the $C_a$ vertically into a $kd\\times m$ matrix $D$. The source identifies the action on the stacked matrix and proves the equivalence\n\\[\n\\text{the seeds generate}\\quad\\Longleftrightarrow\\quad\\rank D=m.\n\\]\nThus $m\\le kd$ is necessary. It is sufficient as well: embed the $m$ coordinate columns into $kd$ coordinate positions to obtain a matrix $D$ of full column rank, then split its rows back into $k$ seeds. This construction proves that generating matrices with the required dimensions exist.","sitive. For seeds $C_1,\\ldots,C_k$, generation means the concrete statement\n\\[\n\\forall X\\in M_{d\\times m}(\\C),\\quad\n\\exists A_1,\\ldots,A_k\\in M_d(\\C):\\quad\nX=\\sum_{a=1}^k A_aC_a.\n\\]\nStack the $C_a$ vertically into a $kd\\times m$ matrix $D$. The source identifies the action on the stacked matrix and proves the equivalence\n\\[\n\\text{the seeds generate}\\quad\\Longleftrightarrow\\quad\\rank D=m.\n\\]\nThus $m\\le kd$ is necessary. It is sufficient as well: embed the $m$ coordinate columns into $kd$ coordinate positions to obtain a matrix $D$ of full column rank, then split its rows back into $k$ seeds. This construction proves that generating matrices with the required dimensions exist.","crete statement\n\\[\n\\forall X\\in M_{d\\times m}(\\C),\\quad\n\\exists A_1,\\ldots,A_k\\in M_d(\\C):\\quad\nX=\\sum_{a=1}^k A_aC_a.\n\\]\nStack the $C_a$ vertically into a $kd\\times m$ matrix $D$. The source identifies the action on the stacked matrix and proves the equivalence\n\\[\n\\text{the seeds generate}\\quad\\Longleftrightarrow\\quad\\rank D=m.\n\\]\nThus $m\\le kd$ is necessary. It is sufficient as well: embed the $m$ coordinate columns into $kd$ coordinate positions to obtain a matrix $D$ of full column rank, then split its rows back into $k$ seeds. This construction proves that generating matrices with the required dimensions exist.","\\in M_{d\\times m}(\\C),\\quad\n\\exists A_1,\\ldots,A_k\\in M_d(\\C):\\quad\nX=\\sum_{a=1}^k A_aC_a.\n\\]\nStack the $C_a$ vertically into a $kd\\times m$ matrix $D$. The source identifies the action on the stacked matrix and proves the equivalence\n\\[\n\\text{the seeds generate}\\quad\\Longleftrightarrow\\quad\\rank D=m.\n\\]\nThus $m\\le kd$ is necessary. It is sufficient as well: embed the $m$ coordinate columns into $kd$ coordinate positions to obtain a matrix $D$ of full column rank, then split its rows back into $k$ seeds. This construction proves that generating matrices with the required dimensions exist.","s,A_k\\in M_d(\\C):\\quad\nX=\\sum_{a=1}^k A_aC_a.\n\\]\nStack the $C_a$ vertically into a $kd\\times m$ matrix $D$. The source identifies the action on the stacked matrix and proves the equivalence\n\\[\n\\text{the seeds generate}\\quad\\Longleftrightarrow\\quad\\rank D=m.\n\\]\nThus $m\\le kd$ is necessary. It is sufficient as well: embed the $m$ coordinate columns into $kd$ coordinate positions to obtain a matrix $D$ of full column rank, then split its rows back into $k$ seeds. This construction proves that generating matrices with the required dimensions exist.","the $C_a$ vertically into a $kd\\times m$ matrix $D$. The source identifies the action on the stacked matrix and proves the equivalence\n\\[\n\\text{the seeds generate}\\quad\\Longleftrightarrow\\quad\\rank D=m.\n\\]\nThus $m\\le kd$ is necessary. It is sufficient as well: embed the $m$ coordinate columns into $kd$ coordinate positions to obtain a matrix $D$ of full column rank, then split its rows back into $k$ seeds. This construction proves that generating matrices with the required dimensions exist.","\\subsection{Simultaneous generation in a direct sum}\nFor the stipulated block representation\n\\[\n\\mathcal A=\\bigoplus_i\\bigl(M_{d_i}(\\C)\\otimes I_{m_i}\\bigr),\\qquad d_i>0,\n\\]\nthe proof establishes that $k$ seeds generate all blocks simultaneously exactly when $m_i\\le kd_i$ for every $i$. It follows that\n\\[\n\\boxed{\\srank_{\\mathcal A}(\\mathcal H)=\\max_i\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.}\n\\]\nThe final Lean proposition is an \\code{IsLeast} statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as $0$, matching the zero module.","that $k$ seeds generate all blocks simultaneously exactly when $m_i\\le kd_i$ for every $i$. It follows that\n\\[\n\\boxed{\\srank_{\\mathcal A}(\\mathcal H)=\\max_i\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.}\n\\]\nThe final Lean proposition is an \\code{IsLeast} statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as $0$, matching the zero module.","\\section{Schmidt rank and the reduced state}\\label{supp3:section:002}\nUse coordinates $\\psi_{ij}$ in the standard product basis and let $C_{ij}=\\psi_{ij}$. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action:\n\\[\n\\operatorname{coeff}\\bigl((A\\otimes I_B)\\psi\\bigr)=AC.\n\\]\nIt also proves that the transported matrix orbit equals the span of the state vectors obtained by this Kronecker action. The dimension formula therefore applies to the physical local action on the state space.","\\subsection{An independent definition of Schmidt rank}\nThe source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length $s$ is the concrete equality\n\\[\nC_{ij}=\\sum_{a=1}^{s}u_a(i)v_a(j).\n\\]\nIt factors $C=UV$ through an $s$-dimensional intermediate space, so $\\rank C\\le s$. Conversely, choose a basis of the row space of $C$. Expanding each row in this basis constructs a decomposition with exactly $\\rank C$ product terms. Thus\n\\[\n\\boxed{r(\\psi)=\\rank C.}\n\\]\nThe definition uses a least natural number, and the equality with matrix rank is a theorem. This argument establishes the minimum number of product terms. It does not construct an orthonormal Schmidt decomposition or prove a new theorem on singular value decomposition.","nk}\nThe source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length $s$ is the concrete equality\n\\[\nC_{ij}=\\sum_{a=1}^{s}u_a(i)v_a(j).\n\\]\nIt factors $C=UV$ through an $s$-dimensional intermediate space, so $\\rank C\\le s$. Conversely, choose a basis of the row space of $C$. Expanding each row in this basis constructs a decomposition with exactly $\\rank C$ product terms. Thus\n\\[\n\\boxed{r(\\psi)=\\rank C.}\n\\]\nThe definition uses a least natural number, and the equality with matrix rank is a theorem. This argument establishes the minimum number of product terms. It does not construct an orthonormal Schmidt decomposition or prove a new theorem on singular value decomposition.","rank as the least number of summands that are product vectors. A decomposition of length $s$ is the concrete equality\n\\[\nC_{ij}=\\sum_{a=1}^{s}u_a(i)v_a(j).\n\\]\nIt factors $C=UV$ through an $s$-dimensional intermediate space, so $\\rank C\\le s$. Conversely, choose a basis of the row space of $C$. Expanding each row in this basis constructs a decomposition with exactly $\\rank C$ product terms. Thus\n\\[\n\\boxed{r(\\psi)=\\rank C.}\n\\]\nThe definition uses a least natural number, and the equality with matrix rank is a theorem. This argument establishes the minimum number of product terms. It does not construct an orthonormal Schmidt decomposition or prove a new theorem on singular value decomposition.","The matrix-orbit dimension and the coefficient equivalence now yield\n\\[\n\\boxed{\\dim_\\C\\spanof\\{(A\\otimes I_B)\\psi:A\\in M_{d_A}(\\C)\\}\n=d_A r(\\psi).}\n\\]","\\subsection{The actual partial trace and its support}\nThe reduced matrix is defined by tracing the addressed index of the density matrix of the pure state:\n\\[\n(\\rho_B)_{j\\ell}=\\sum_i\\psi_{ij}\\overline{\\psi_{i\\ell}}.\n\\]\nFor the coefficient convention used here, the exact Gram identity is\n\\[\n\\rho_B=C^{\\mathsf T}(C^{\\mathsf T})^*,\n\\qquad \\rank\\rho_B=\\rank C=r(\\psi).\n\\]\nThe transpose is important. With this convention, writing $C^*C$ instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.","al trace and its support}\nThe reduced matrix is defined by tracing the addressed index of the density matrix of the pure state:\n\\[\n(\\rho_B)_{j\\ell}=\\sum_i\\psi_{ij}\\overline{\\psi_{i\\ell}}.\n\\]\nFor the coefficient convention used here, the exact Gram identity is\n\\[\n\\rho_B=C^{\\mathsf T}(C^{\\mathsf T})^*,\n\\qquad \\rank\\rho_B=\\rank C=r(\\psi).\n\\]\nThe transpose is important. With this convention, writing $C^*C$ instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.","Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming $d_A>0$, the checked equivalence is\n\\[\n\\boxed{\\mathcal O_A(\\psi)=\\mathcal H_A\\otimes\\mathcal H_B\n\\quad\\Longleftrightarrow\\quad \\Ran\\rho_B=\\mathcal H_B.}\n\\]\nNormalization of $\\psi$ is unnecessary for these dimension and support statements. The zero state has rank $0$ because its decomposition needs no product terms.","\\section{Constructed cyclic character projectors}\\label{supp3:section:003}\nLet $N>0$ and let $S$ be a unitary complex matrix with $S^N=I$. The implementation uses the standard additive character of $\\mathbb Z/N\\mathbb Z$ and proves its expression through a root of unity:\n\\[\n\\omega=\\exp(2\\pi i/N),\\qquad\n\\chi(r)=\\omega^{\\operatorname{val}(r)}.\n\\]\nHere $\\operatorname{val}(r)$ is the canonical representative in $\\{0,\\ldots,N-1\\}$. The representation is constructed as\n\\[\n\\rho(r)=S^{\\operatorname{val}(r)}.\n\\]\nReduction of exponents modulo $N$ is proved from $S^N=I$, giving $\\rho(r+s)=\\rho(r)\\rho(s)$ and $\\rho(0)=I$. The final matrix theorem derives the representation law and character orthogonality from the stated inputs.","structed cyclic character projectors}\\label{supp3:section:003}\nLet $N>0$ and let $S$ be a unitary complex matrix with $S^N=I$. The implementation uses the standard additive character of $\\mathbb Z/N\\mathbb Z$ and proves its expression through a root of unity:\n\\[\n\\omega=\\exp(2\\pi i/N),\\qquad\n\\chi(r)=\\omega^{\\operatorname{val}(r)}.\n\\]\nHere $\\operatorname{val}(r)$ is the canonical representative in $\\{0,\\ldots,N-1\\}$. The representation is constructed as\n\\[\n\\rho(r)=S^{\\operatorname{val}(r)}.\n\\]\nReduction of exponents modulo $N$ is proved from $S^N=I$, giving $\\rho(r+s)=\\rho(r)\\rho(s)$ and $\\rho(0)=I$. The final matrix theorem derives the representation law and character orthogonality from the stated inputs.","003}\nLet $N>0$ and let $S$ be a unitary complex matrix with $S^N=I$. The implementation uses the standard additive character of $\\mathbb Z/N\\mathbb Z$ and proves its expression through a root of unity:\n\\[\n\\omega=\\exp(2\\pi i/N),\\qquad\n\\chi(r)=\\omega^{\\operatorname{val}(r)}.\n\\]\nHere $\\operatorname{val}(r)$ is the canonical representative in $\\{0,\\ldots,N-1\\}$. The representation is constructed as\n\\[\n\\rho(r)=S^{\\operatorname{val}(r)}.\n\\]\nReduction of exponents modulo $N$ is proved from $S^N=I$, giving $\\rho(r+s)=\\rho(r)\\rho(s)$ and $\\rho(0)=I$. The final matrix theorem derives the representation law and character orthogonality from the stated inputs.","he implementation uses the standard additive character of $\\mathbb Z/N\\mathbb Z$ and proves its expression through a root of unity:\n\\[\n\\omega=\\exp(2\\pi i/N),\\qquad\n\\chi(r)=\\omega^{\\operatorname{val}(r)}.\n\\]\nHere $\\operatorname{val}(r)$ is the canonical representative in $\\{0,\\ldots,N-1\\}$. The representation is constructed as\n\\[\n\\rho(r)=S^{\\operatorname{val}(r)}.\n\\]\nReduction of exponents modulo $N$ is proved from $S^N=I$, giving $\\rho(r+s)=\\rho(r)\\rho(s)$ and $\\rho(0)=I$. The final matrix theorem derives the representation law and character orthogonality from the stated inputs.","ses the standard additive character of $\\mathbb Z/N\\mathbb Z$ and proves its expression through a root of unity:\n\\[\n\\omega=\\exp(2\\pi i/N),\\qquad\n\\chi(r)=\\omega^{\\operatorname{val}(r)}.\n\\]\nHere $\\operatorname{val}(r)$ is the canonical representative in $\\{0,\\ldots,N-1\\}$. The representation is constructed as\n\\[\n\\rho(r)=S^{\\operatorname{val}(r)}.\n\\]\nReduction of exponents modulo $N$ is proved from $S^N=I$, giving $\\rho(r+s)=\\rho(r)\\rho(s)$ and $\\rho(0)=I$. The final matrix theorem derives the representation law and character orthogonality from the stated inputs.","additive character of $\\mathbb Z/N\\mathbb Z$ and proves its expression through a root of unity:\n\\[\n\\omega=\\exp(2\\pi i/N),\\qquad\n\\chi(r)=\\omega^{\\operatorname{val}(r)}.\n\\]\nHere $\\operatorname{val}(r)$ is the canonical representative in $\\{0,\\ldots,N-1\\}$. The representation is constructed as\n\\[\n\\rho(r)=S^{\\operatorname{val}(r)}.\n\\]\nReduction of exponents modulo $N$ is proved from $S^N=I$, giving $\\rho(r+s)=\\rho(r)\\rho(s)$ and $\\rho(0)=I$. The final matrix theorem derives the representation law and character orthogonality from the stated inputs.","Z/N\\mathbb Z$ and proves its expression through a root of unity:\n\\[\n\\omega=\\exp(2\\pi i/N),\\qquad\n\\chi(r)=\\omega^{\\operatorname{val}(r)}.\n\\]\nHere $\\operatorname{val}(r)$ is the canonical representative in $\\{0,\\ldots,N-1\\}$. The representation is constructed as\n\\[\n\\rho(r)=S^{\\operatorname{val}(r)}.\n\\]\nReduction of exponents modulo $N$ is proved from $S^N=I$, giving $\\rho(r+s)=\\rho(r)\\rho(s)$ and $\\rho(0)=I$. The final matrix theorem derives the representation law and character orthogonality from the stated inputs.","\\subsection{Exact coefficient formula}\nThe packet definition is a finite sum over residue classes:\n\\[\nP_j=\\frac1N\\sum_{r\\in\\mathbb Z/N\\mathbb Z}\\chi(-rj)\\rho(r).\n\\]\nThe source proves\n\\[\n\\chi(-rj)=\\bigl(\\omega^{\\operatorname{val}(r)\\operatorname{val}(j)}\\bigr)^{-1},\n\\]\nso this is precisely the manuscript's $N$-term Fourier projector formula, with canonical representatives.","\\subsection{Algebraic identities}\nPrimitivity of the standard character yields the checked finite sum\n\\[\n\\sum_r\\chi(tr)=\\begin{cases}N,&t=0,\\\\0,&t\\ne0.\\end{cases}\n\\]\nReindexing finite sums then gives\n\\[\n\\boxed{P_jP_k=\\delta_{jk}P_j,\\qquad \\sum_jP_j=I,\\qquad SP_j=\\omega^jP_j.}\n\\]\nMultiplicativity and completeness are first proved for a general representation in a complex algebra, then instantiated with the actual powers of $S$. This organization makes the algebra of finite sums reusable while preserving the final hypotheses.","0,&t\\ne0.\\end{cases}\n\\]\nReindexing finite sums then gives\n\\[\n\\boxed{P_jP_k=\\delta_{jk}P_j,\\qquad \\sum_jP_j=I,\\qquad SP_j=\\omega^jP_j.}\n\\]\nMultiplicativity and completeness are first proved for a general representation in a complex algebra, then instantiated with the actual powers of $S$. This organization makes the algebra of finite sums reusable while preserving the final hypotheses.","\\subsection{Orthogonality of the projectors}\nUnitarity supplies $\\rho(r)^*=\\rho(-r)$, while the character obeys $\\overline{\\chi(r)}=\\chi(-r)$. Reindexing $r\\mapsto-r$ proves $P_j^*=P_j$. The final concrete matrix theorem includes Hermiticity together with multiplication, completeness, and eigenaction. Together these identities establish orthogonal spectral packets.","The proof includes the degenerate modulus $N=1$: finite order then forces $S=I$ and the sole projector is $I$. It does not need an artificial assumption $N>1$.","\\section{The actual Euclidean transfer exponential}\\label{supp3:section:004}\nLet $H$ be a Hermitian matrix on a nonzero complex state space of finite dimension. The source defines\n\\[\nT_e(t)=\\exp\\bigl(-t(H-eI)\\bigr)\n\\]\nusing Lean's exponential defined by a power series. The proof then relates this defined operator to its spectral expansion.","\\subsection{Connecting power series and spectral calculus}\nThe shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus:\n\\[\n\\boxed{T_e(t)=f_t(H),\\qquad f_t(x)=e^{-t(x-e)}.}\n\\]\nThe formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually defined:\n\\[\n\\Spec_\\R T_e(t)=\\{e^{-t(x-e)}:x\\in\\Spec_\\R H\\}.\n\\]\nHermiticity also identifies the full complex spectrum with the image of these real eigenvalues. A separate checked theorem states the interval containing the complex spectrum explicitly.","tual matrix exponential with continuous functional calculus:\n\\[\n\\boxed{T_e(t)=f_t(H),\\qquad f_t(x)=e^{-t(x-e)}.}\n\\]\nThe formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually defined:\n\\[\n\\Spec_\\R T_e(t)=\\{e^{-t(x-e)}:x\\in\\Spec_\\R H\\}.\n\\]\nHermiticity also identifies the full complex spectrum with the image of these real eigenvalues. A separate checked theorem states the interval containing the complex spectrum explicitly.","\\subsection{The minimum eigenvalue}\nThe ground energy is the minimum of the finite, nonempty eigenvalue set. The proof establishes both\n\\[\nE_0\\in\\Spec_\\R H,\\qquad E_0\\le x\\quad(x\\in\\Spec_\\R H).\n\\]\nThe assumption that the state space is nonzero is recorded in the theorem because the minimum of an empty spectrum is not defined by the manuscript's formula.","\\subsection{Positivity, semigroup composition, and the interval}\nEvery eigenvalue of $T_e(t)$ is strictly positive, for arbitrary real $e,t$, since the real exponential is positive. For $e=E_0$ and $t\\ge0$ it is also at most one. Therefore\n\\[\n\\boxed{T_{E_0}(t)>0,\\qquad \\Spec T_{E_0}(t)\\subseteq(0,1].}\n\\]\nThe two scalar multiples of $H-eI$ commute, so the exponential defined by the power series obeys\n\\[\n\\boxed{T_e(t+s)=T_e(t)T_e(s),\\qquad T_e(0)=I.}\n\\]\nThe semigroup identity is proved for all real $t,s$. Nonnegative time is needed for the upper spectral bound, not for the composition identity or strict positivity.","ection{Positivity, semigroup composition, and the interval}\nEvery eigenvalue of $T_e(t)$ is strictly positive, for arbitrary real $e,t$, since the real exponential is positive. For $e=E_0$ and $t\\ge0$ it is also at most one. Therefore\n\\[\n\\boxed{T_{E_0}(t)>0,\\qquad \\Spec T_{E_0}(t)\\subseteq(0,1].}\n\\]\nThe two scalar multiples of $H-eI$ commute, so the exponential defined by the power series obeys\n\\[\n\\boxed{T_e(t+s)=T_e(t)T_e(s),\\qquad T_e(0)=I.}\n\\]\nThe semigroup identity is proved for all real $t,s$. Nonnegative time is needed for the upper spectral bound, not for the composition identity or strict positivity.","ly positive, for arbitrary real $e,t$, since the real exponential is positive. For $e=E_0$ and $t\\ge0$ it is also at most one. Therefore\n\\[\n\\boxed{T_{E_0}(t)>0,\\qquad \\Spec T_{E_0}(t)\\subseteq(0,1].}\n\\]\nThe two scalar multiples of $H-eI$ commute, so the exponential defined by the power series obeys\n\\[\n\\boxed{T_e(t+s)=T_e(t)T_e(s),\\qquad T_e(0)=I.}\n\\]\nThe semigroup identity is proved for all real $t,s$. Nonnegative time is needed for the upper spectral bound, not for the composition identity or strict positivity.","\\section{Quantitative leakage from the actual exponential}\\label{supp4:section:001}\nLet $A$ be a unital complex C-star algebra, let $H=H^*$, and let $P=P^*=P^2$. Put $Q=1-P$. The checked statement is\n\\[\n\\boxed{\\|Q\\exp(-itH)P\\|\\le |t|\\,\\|HP-PH\\|\\qquad(t\\in\\R).}\n\\]\nThe matrix specialization uses the operator norm on the Hilbert space, implemented by \\path{Matrix.Norms.L2Operator}. It is not an entrywise norm or the row-sum operator norm. The theorem includes the zero projector and empty matrix index type. The theorem also applies at time zero.","\\subsection{Deriving the estimate by differentiation}\nFor a skew-adjoint generator $G$, define the actual conjugation path\n\\[\nT_P(t)=\\exp(tG)P\\exp(-tG).\n\\]\nThe formalization derives its derivative from the exponential power series and the product rule:\n\\[\nT_P'(t)=\\exp(tG)(GP-PG)\\exp(-tG).\n\\]\nBoth exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore\n\\[\n\\|T_P'(t)\\|=\\|GP-PG\\|,\n\\qquad\n\\|T_P(t)-T_P(s)\\|\\le |t-s|\\,\\|GP-PG\\|.\n\\]\nThe second inequality follows from the mean value inequality in normed spaces and covers either ordering of $s$ and $t$. It is not a finite-difference estimate or a numerical time discretization.","g the estimate by differentiation}\nFor a skew-adjoint generator $G$, define the actual conjugation path\n\\[\nT_P(t)=\\exp(tG)P\\exp(-tG).\n\\]\nThe formalization derives its derivative from the exponential power series and the product rule:\n\\[\nT_P'(t)=\\exp(tG)(GP-PG)\\exp(-tG).\n\\]\nBoth exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore\n\\[\n\\|T_P'(t)\\|=\\|GP-PG\\|,\n\\qquad\n\\|T_P(t)-T_P(s)\\|\\le |t-s|\\,\\|GP-PG\\|.\n\\]\nThe second inequality follows from the mean value inequality in normed spaces and covers either ordering of $s$ and $t$. It is not a finite-difference estimate or a numerical time discretization.","the exponential power series and the product rule:\n\\[\nT_P'(t)=\\exp(tG)(GP-PG)\\exp(-tG).\n\\]\nBoth exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore\n\\[\n\\|T_P'(t)\\|=\\|GP-PG\\|,\n\\qquad\n\\|T_P(t)-T_P(s)\\|\\le |t-s|\\,\\|GP-PG\\|.\n\\]\nThe second inequality follows from the mean value inequality in normed spaces and covers either ordering of $s$ and $t$. It is not a finite-difference estimate or a numerical time discretization.","onential power series and the product rule:\n\\[\nT_P'(t)=\\exp(tG)(GP-PG)\\exp(-tG).\n\\]\nBoth exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore\n\\[\n\\|T_P'(t)\\|=\\|GP-PG\\|,\n\\qquad\n\\|T_P(t)-T_P(s)\\|\\le |t-s|\\,\\|GP-PG\\|.\n\\]\nThe second inequality follows from the mean value inequality in normed spaces and covers either ordering of $s$ and $t$. It is not a finite-difference estimate or a numerical time discretization.","Multiplication by $\\exp(tG)$ gives\n\\[\n(T_P(t)-P)\\exp(tG)=\\exp(tG)P-P\\exp(tG).\n\\]\nFor every orthogonal projector, $\\|P\\|\\le1$ and $\\|1-P\\|\\le1$. The exact corner identity\n\\[\n(1-P)UP=(1-P)(UP-PU)P\n\\]\nthen bounds leakage by the exponential commutator. Taking $G=-iH$ gives the stated theorem because $\\|[G,P]\\|=\\|[H,P]\\|$.","\\subsection{Checked consequences}\nUnitarity also gives the capped estimate\n\\[\n\\|Q\\exp(-itH)P\\|\\le\\min\\{1,\\ |t|\\,\\|[H,P]\\|\\}.\n\\]\nIf $H=H_0+V$, both summands are self-adjoint, and $[H_0,P]=0$, then\n\\[\n\\|Q\\exp(-it(H_0+V))P\\|\\le2|t|\\,\\|V\\|.\n\\]\nThe latter uses $\\|[V,P]\\|\\le2\\|V\\|$. Its constant is not claimed to be optimal. Exact commutation gives zero leakage. The work does not include unbounded generators or time-dependent Hamiltonians.","\\section{Actual projected differentiation and curvature}\\label{supp4:section:002}\nLet $E,F$ be real normed spaces. Let $P:E\\to\\mathcal L(F,F)$ be differentiable with $P(x)^2=P(x)$ for every $x$. Let $s:E\\to F$ be $C^2$ with $P(x)s(x)=s(x)$, and let $X,Y:E\\to E$ be differentiable vector fields. The checked connection is the actual expression\n\\[\n\\nabla_Xs(x)=P(x)\\,Ds(x)[X(x)].\n\\]\nIt is proved to preserve the range condition. It is linear in the vector field direction and obeys the scalar Leibniz rule\n\\[\n\\nabla_X(cs)=Dc[X]\\,s+c\\nabla_Xs\n\\]\nfor differentiable real scalar functions $c$. No differentiated projector identities are inserted as hypotheses in the main curvature theorem.","spaces. Let $P:E\\to\\mathcal L(F,F)$ be differentiable with $P(x)^2=P(x)$ for every $x$. Let $s:E\\to F$ be $C^2$ with $P(x)s(x)=s(x)$, and let $X,Y:E\\to E$ be differentiable vector fields. The checked connection is the actual expression\n\\[\n\\nabla_Xs(x)=P(x)\\,Ds(x)[X(x)].\n\\]\nIt is proved to preserve the range condition. It is linear in the vector field direction and obeys the scalar Leibniz rule\n\\[\n\\nabla_X(cs)=Dc[X]\\,s+c\\nabla_Xs\n\\]\nfor differentiable real scalar functions $c$. No differentiated projector identities are inserted as hypotheses in the main curvature theorem.","very $x$. Let $s:E\\to F$ be $C^2$ with $P(x)s(x)=s(x)$, and let $X,Y:E\\to E$ be differentiable vector fields. The checked connection is the actual expression\n\\[\n\\nabla_Xs(x)=P(x)\\,Ds(x)[X(x)].\n\\]\nIt is proved to preserve the range condition. It is linear in the vector field direction and obeys the scalar Leibniz rule\n\\[\n\\nabla_X(cs)=Dc[X]\\,s+c\\nabla_Xs\n\\]\nfor differentiable real scalar functions $c$. No differentiated projector identities are inserted as hypotheses in the main curvature theorem.","\\subsection{Differentiated projector and section identities}\nDifferentiating $P^2=P$ and $Ps=s$ produces\n\\begin{align*}\nDP[v]P+PDP[v]&=DP[v],","\\subsection{Differentiated projector and section identities}\nDifferentiating $P^2=P$ and $Ps=s$ produces\n\\begin{align*}\nDP[v]P+PDP[v]&=DP[v],\\\\\nPDP[v]P&=0,\\\\\nDs[v]&=DP[v]s+P Ds[v].\n\\end{align*}\nThe zero sandwich follows algebraically from the first identity. All products preserve their noncommutative order.","\\subsection{Retaining the bracket of vector fields}\nDefine the actual curvature action by\n\\[\nR_P(X,Y)s=\\nabla_X\\nabla_Ys-\\nabla_Y\\nabla_Xs-\\nabla_{[X,Y]}s.\n\\]\nThe source uses the Lie bracket defined through Fr\\'echet derivatives. Symmetry of second derivatives is supplied by the $C^2$ hypothesis, not postulated separately. Expanding the connection gives\n\\[\nR_P(X,Y)s=P\\bigl(DP[X]Ds[Y]-DP[Y]Ds[X]\\bigr).\n\\]\nSubstituting the section identity and removing the zero sandwich terms proves\n\\[\n\\boxed{R_P(X,Y)s=P\\bigl(DP[X]DP[Y]-DP[Y]DP[X]\\bigr)P s.}\n\\]\nThus the coefficient is precisely $P(\\dd P\\wedge\\dd P)P$ with the convention\n$(\\dd P\\wedge\\dd P)(v,w)=DP[v]DP[w]-DP[w]DP[v]$.","y\n\\[\nR_P(X,Y)s=\\nabla_X\\nabla_Ys-\\nabla_Y\\nabla_Xs-\\nabla_{[X,Y]}s.\n\\]\nThe source uses the Lie bracket defined through Fr\\'echet derivatives. Symmetry of second derivatives is supplied by the $C^2$ hypothesis, not postulated separately. Expanding the connection gives\n\\[\nR_P(X,Y)s=P\\bigl(DP[X]Ds[Y]-DP[Y]Ds[X]\\bigr).\n\\]\nSubstituting the section identity and removing the zero sandwich terms proves\n\\[\n\\boxed{R_P(X,Y)s=P\\bigl(DP[X]DP[Y]-DP[Y]DP[X]\\bigr)P s.}\n\\]\nThus the coefficient is precisely $P(\\dd P\\wedge\\dd P)P$ with the convention\n$(\\dd P\\wedge\\dd P)(v,w)=DP[v]DP[w]-DP[w]DP[v]$.","racket defined through Fr\\'echet derivatives. Symmetry of second derivatives is supplied by the $C^2$ hypothesis, not postulated separately. Expanding the connection gives\n\\[\nR_P(X,Y)s=P\\bigl(DP[X]Ds[Y]-DP[Y]Ds[X]\\bigr).\n\\]\nSubstituting the section identity and removing the zero sandwich terms proves\n\\[\n\\boxed{R_P(X,Y)s=P\\bigl(DP[X]DP[Y]-DP[Y]DP[X]\\bigr)P s.}\n\\]\nThus the coefficient is precisely $P(\\dd P\\wedge\\dd P)P$ with the convention\n$(\\dd P\\wedge\\dd P)(v,w)=DP[v]DP[w]-DP[w]DP[v]$.","s. Symmetry of second derivatives is supplied by the $C^2$ hypothesis, not postulated separately. Expanding the connection gives\n\\[\nR_P(X,Y)s=P\\bigl(DP[X]Ds[Y]-DP[Y]Ds[X]\\bigr).\n\\]\nSubstituting the section identity and removing the zero sandwich terms proves\n\\[\n\\boxed{R_P(X,Y)s=P\\bigl(DP[X]DP[Y]-DP[Y]DP[X]\\bigr)P s.}\n\\]\nThus the coefficient is precisely $P(\\dd P\\wedge\\dd P)P$ with the convention\n$(\\dd P\\wedge\\dd P)(v,w)=DP[v]DP[w]-DP[w]DP[v]$.","\\subsection{Tensor properties and the remaining global boundary}\nThe coefficient is proved bilinear, alternating, and supported on the range on both sides. The chain rule proves the pullback identity\n\\[\nF_{P\\circ f}(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w).\n\\]\nThe main theorem has global regularity hypotheses on normed parameter spaces. The pullback identity alone does not supply localization, smooth local trivializations, or gluing. The final supplement constructs the global range bundle and proves that the local connections glue.","\\section{A constructive and total syndrome recovery}\\label{supp4:section:003}\nLet $P$ be an orthogonal code projector. The input certificate consists of a finite family $V_a$ satisfying\n\\[\nV_aP=V_a,\n\\qquad\nV_a^*V_b=\\delta_{ab}P.\n\\]\nThese hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.","\\subsection{Constructing the recovery on the whole ambient space}\nDefine the syndrome projectors and their complement by\n\\[\nS_a=V_aV_a^*,\\qquad S=\\sum_aS_a,\\qquad Q=1-S.\n\\]\nThe source proves self-adjointness, $S_aS_b=\\delta_{ab}S_a$, $S^2=S$, and $Q^2=Q=Q^*$. It also proves $QV_a=0$ and $V_a^*Q=0$.","The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are $R_a=V_a^*$ together with $R_{\\mathrm{extra}}=Q$, giving\n\\[\n\\mathcal R(\\rho)=\\sum_aV_a^*\\rho V_a+Q\\rho Q.\n\\]\nThey satisfy exact completeness:\n\\[\n\\sum_aR_a^*R_a+R_{\\mathrm{extra}}^*R_{\\mathrm{extra}}\n=\\sum_aV_aV_a^*+Q=S+Q=I.\n\\]\nFor finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family $I_J\\otimes R_a$, including the complement term. Positivity is not merely checked on a single matrix size.","ith $R_{\\mathrm{extra}}=Q$, giving\n\\[\n\\mathcal R(\\rho)=\\sum_aV_a^*\\rho V_a+Q\\rho Q.\n\\]\nThey satisfy exact completeness:\n\\[\n\\sum_aR_a^*R_a+R_{\\mathrm{extra}}^*R_{\\mathrm{extra}}\n=\\sum_aV_aV_a^*+Q=S+Q=I.\n\\]\nFor finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family $I_J\\otimes R_a$, including the complement term. Positivity is not merely checked on a single matrix size.","\\subsection{Exact correction, including coherent errors}\nFor every operator supported on the code, $P\\rho P=\\rho$, the central identity is\n\\[\n\\boxed{\\mathcal R(V_a\\rho V_b^*)=\\delta_{ab}\\rho.}\n\\]\nConsequently normalized mixtures satisfy\n\\[\n\\mathcal R\\!\\left(\\sum_ap_aV_a\\rho V_a^*\\right)=\\rho\n\\quad\\text{when}\\quad\\sum_ap_a=1.\n\\]\nThe identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error $E_z=\\sum_a z_aV_a$, the calculation of the cross terms yields\n\\[\n\\mathcal R(E_z\\rho E_z^*)=\\left(\\sum_a z_a\\overline{z_a}\\right)\\rho.\n\\]\nThus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that follows.","very operator supported on the code, $P\\rho P=\\rho$, the central identity is\n\\[\n\\boxed{\\mathcal R(V_a\\rho V_b^*)=\\delta_{ab}\\rho.}\n\\]\nConsequently normalized mixtures satisfy\n\\[\n\\mathcal R\\!\\left(\\sum_ap_aV_a\\rho V_a^*\\right)=\\rho\n\\quad\\text{when}\\quad\\sum_ap_a=1.\n\\]\nThe identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error $E_z=\\sum_a z_aV_a$, the calculation of the cross terms yields\n\\[\n\\mathcal R(E_z\\rho E_z^*)=\\left(\\sum_a z_a\\overline{z_a}\\right)\\rho.\n\\]\nThus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that follows.","identity is\n\\[\n\\boxed{\\mathcal R(V_a\\rho V_b^*)=\\delta_{ab}\\rho.}\n\\]\nConsequently normalized mixtures satisfy\n\\[\n\\mathcal R\\!\\left(\\sum_ap_aV_a\\rho V_a^*\\right)=\\rho\n\\quad\\text{when}\\quad\\sum_ap_a=1.\n\\]\nThe identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error $E_z=\\sum_a z_aV_a$, the calculation of the cross terms yields\n\\[\n\\mathcal R(E_z\\rho E_z^*)=\\left(\\sum_a z_a\\overline{z_a}\\right)\\rho.\n\\]\nThus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that follows.","\\section{The precise finite channel statement}\\label{supp5:section:001}\nLet $\\mathcal H=\\C^n$ be a finite coordinate space and let $P=P^*=P^2$. A matrix is code supported when $P\\rho P=\\rho$. For a finite family of square matrices $E_a$, put\n\\[\n\\mathcal N_E(\\rho)=\\sum_a E_a\\rho E_a^*.\n\\]\nThe input channel normalization is the actual matrix identity\n\\[\n\\sum_a E_a^*E_a=I.\n\\]\nA finite Kraus recovery has the form $\\mathcal R(\\sigma)=\\sum_r R_r\\sigma R_r^*$, with $\\sum_rR_r^*R_r=I$. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire ambient matrix space.","recise finite channel statement}\\label{supp5:section:001}\nLet $\\mathcal H=\\C^n$ be a finite coordinate space and let $P=P^*=P^2$. A matrix is code supported when $P\\rho P=\\rho$. For a finite family of square matrices $E_a$, put\n\\[\n\\mathcal N_E(\\rho)=\\sum_a E_a\\rho E_a^*.\n\\]\nThe input channel normalization is the actual matrix identity\n\\[\n\\sum_a E_a^*E_a=I.\n\\]\nA finite Kraus recovery has the form $\\mathcal R(\\sigma)=\\sum_r R_r\\sigma R_r^*$, with $\\sum_rR_r^*R_r=I$. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire ambient matrix space.","on:001}\nLet $\\mathcal H=\\C^n$ be a finite coordinate space and let $P=P^*=P^2$. A matrix is code supported when $P\\rho P=\\rho$. For a finite family of square matrices $E_a$, put\n\\[\n\\mathcal N_E(\\rho)=\\sum_a E_a\\rho E_a^*.\n\\]\nThe input channel normalization is the actual matrix identity\n\\[\n\\sum_a E_a^*E_a=I.\n\\]\nA finite Kraus recovery has the form $\\mathcal R(\\sigma)=\\sum_r R_r\\sigma R_r^*$, with $\\sum_rR_r^*R_r=I$. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire ambient matrix space.","\\subsection{Necessary and sufficient conditions}\nUnder the stated normalization, the following equivalence is proved:\n\\[\n\\boxed{\\begin{gathered}\n\\exists\\,\\mathcal R\\text{ of the above finite Kraus form,}\\\\[-2pt]\n\\mathcal R\\mathcal N_E(\\rho)=\\rho\n\\quad\\text{for every }\\rho\\succeq0,\\ \\Tr\\rho=1,\\ P\\rho P=\\rho\n\\end{gathered}}\n\\quad\\Longleftrightarrow\\quad\n\\boxed{\\begin{gathered}\n\\exists C\\succeq0,\\\\\nPE_a^*E_bP=C_{ab}P\\quad\\text{for all }a,b.\n\\end{gathered}}\n\\]\nThe statement applies throughout the code space. The formalization separately proves equivalence with correction of \\emph{every} complex matrix supported on the code. It also proves preservation with arbitrary finite reference systems, as detailed in the discussion of density matrices and reference systems below.","\\subsection{Including degenerate cases}\nThe matrix $C$ need not be positive definite. The proof includes zero eigenvalues, redundant error operators, an empty set of positive eigenvalues, and the zero projector. If $P=0$, the only supported matrix is zero and no trace-one code density matrix exists. The proofs handle that case separately rather than dividing by the code rank or by $\\Tr P$.","For an unnormalized error family, the correct statement is generally\n\\[\n\\mathcal R\\mathcal N_E(\\rho)=\\Tr(C)\\rho,\n\\]\nnot unconditional equality to $\\rho$. Normalization forces $\\Tr C=1$ on a nonzero code. The discussion of coherent errors below treats unrestricted error families and shows when the same decoder works throughout their span.","\\section{From Gram data to an actual total recovery}\\label{supp5:section:002}\nAssume $C\\succeq0$ and $PE_a^*E_bP=C_{ab}P$. First restrict the error operators to the code, $F_a=E_aP$. The finite Hermitian spectral theorem supplies a unitary $U$ such that\n\\[\nU^*CU=\\operatorname{diag}(\\lambda_r),\\qquad \\lambda_r\\ge0.\n\\]","Define the transformed errors by an actual finite sum,\n\\[\nB_r=\\sum_a U_{ar}F_a.\n\\]\nThe formal module proves the identities\n\\[\nB_rP=B_r,\\qquad B_r^*B_s=\\delta_{rs}\\lambda_rP,\n\\qquad \\sum_rB_r\\rho B_r^*=\\sum_aF_a\\rho F_a^*.\n\\]\nFor $\\rho$ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","\\subsection{Null directions are removed by proof}\nIf $\\lambda_r=0$, then $B_r^*B_r=0$. Positivity of the ordinary complex matrix inner product gives $B_r=0$. This is the checked declaration \\path{DiagonalErrors.zero_weight_error}. The normalized syndrome operators are defined only on\n\\[\nA_+=\\{r:\\lambda_r>0\\},\\qquad V_r=\\lambda_r^{-1/2}B_r.\n\\]\nConsequently no inverse square root at a null eigenvalue is needed. Lean proves\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nThe recovery from the previous block is instantiated from these derived syndromes, rather than from an assumed orthogonal error family.","are removed by proof}\nIf $\\lambda_r=0$, then $B_r^*B_r=0$. Positivity of the ordinary complex matrix inner product gives $B_r=0$. This is the checked declaration \\path{DiagonalErrors.zero_weight_error}. The normalized syndrome operators are defined only on\n\\[\nA_+=\\{r:\\lambda_r>0\\},\\qquad V_r=\\lambda_r^{-1/2}B_r.\n\\]\nConsequently no inverse square root at a null eigenvalue is needed. Lean proves\n\\[\nV_rP=V_r,\\qquad V_r^*V_s=\\delta_{rs}P.\n\\]\nThe recovery from the previous block is instantiated from these derived syndromes, rather than from an assumed orthogonal error family.","\\subsection{A channel on the whole ambient space}\nPut $Q=I-\\sum_{r\\in A_+}V_rV_r^*$. The recovery is\n\\[\n\\boxed{\\mathcal R(\\sigma)=\\sum_{r\\in A_+}V_r^*\\sigma V_r+Q\\sigma Q.}\n\\]\nThe complement is an orthogonal projector and\n\\[\n\\sum_{r\\in A_+}V_rV_r^*+Q^*Q=I.\n\\]\nHence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.","Finally the checked calculation gives $\\mathcal R\\mathcal N_E(\\rho)=\\sum_r\\lambda_r\\rho=\\Tr(C)\\rho$. Under normalization of the input channel and $P\\ne0$, summing the diagonal Gram identities gives $\\Tr(C)P=P$, hence $\\Tr C=1$. The proof for the zero code does not use this cancellation.","\\section{Deriving the Gram matrix from recovery}\\label{supp5:section:003}\nSuppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family\n\\[\nA_{ra}=R_rE_aP.\n\\]\nFor every ambient matrix $\\rho$, correction on the code implies\n\\[\n\\sum_{r,a} A_{ra}\\rho A_{ra}^*=P\\rho P.\n\\]\nThis equality on all matrices follows by applying correction to the sandwich supported on the code, $P\\rho P$, using the projector identities.","{Deriving the Gram matrix from recovery}\\label{supp5:section:003}\nSuppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family\n\\[\nA_{ra}=R_rE_aP.\n\\]\nFor every ambient matrix $\\rho$, correction on the code implies\n\\[\n\\sum_{r,a} A_{ra}\\rho A_{ra}^*=P\\rho P.\n\\]\nThis equality on all matrices follows by applying correction to the sandwich supported on the code, $P\\rho P$, using the projector identities.","d on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family\n\\[\nA_{ra}=R_rE_aP.\n\\]\nFor every ambient matrix $\\rho$, correction on the code implies\n\\[\n\\sum_{r,a} A_{ra}\\rho A_{ra}^*=P\\rho P.\n\\]\nThis equality on all matrices follows by applying correction to the sandwich supported on the code, $P\\rho P$, using the projector identities.","\\subsection{Rank-one rigidity of the corrected channel}\nThe module tests the map equality on actual matrix units and flattens each $A_{ra}$ into a row. If $X$ is the matrix of those rows and $p$ is the vector of entries of $P$, the resulting Gram identity is\n\\[\nX^*X=\\overline p\\,p^{\\mathsf T}.\n\\]\nThe checked rank-one Gram lemma proves that every row of $X$ is proportional to $p$. For nonzero $p$, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication $X^*Xv=0\\Rightarrow Xv=0$ then gives the required proportionality. When $p=0$, the Gram identity forces $X=0$ directly.","a row. If $X$ is the matrix of those rows and $p$ is the vector of entries of $P$, the resulting Gram identity is\n\\[\nX^*X=\\overline p\\,p^{\\mathsf T}.\n\\]\nThe checked rank-one Gram lemma proves that every row of $X$ is proportional to $p$. For nonzero $p$, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication $X^*Xv=0\\Rightarrow Xv=0$ then gives the required proportionality. When $p=0$, the Gram identity forces $X=0$ directly.","es of $P$, the resulting Gram identity is\n\\[\nX^*X=\\overline p\\,p^{\\mathsf T}.\n\\]\nThe checked rank-one Gram lemma proves that every row of $X$ is proportional to $p$. For nonzero $p$, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication $X^*Xv=0\\Rightarrow Xv=0$ then gives the required proportionality. When $p=0$, the Gram identity forces $X=0$ directly.","entity is\n\\[\nX^*X=\\overline p\\,p^{\\mathsf T}.\n\\]\nThe checked rank-one Gram lemma proves that every row of $X$ is proportional to $p$. For nonzero $p$, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication $X^*Xv=0\\Rightarrow Xv=0$ then gives the required proportionality. When $p=0$, the Gram identity forces $X=0$ directly.","Applied to the corrected Kraus family, this yields scalars $z_{ra}$ with\n\\[\nR_rE_aP=z_{ra}P.\n\\]\nThe essential declarations are \\path{rank_one_gram_rigidity}, \\path{kraus_rank_one_rigidity} and \\path{corrected_kraus_family}. No rank-one Choi characterization is inserted as an axiom or as the desired conclusion in hypothesis form.","\\subsection{Trace-preserving recovery supplies the original Gram data}\nLet $Z_{ra}=z_{ra}$. The recovery normalization gives\n\\begin{align*}\nPE_a^*E_bP\n&=PE_a^*\\Bigl(\\sum_rR_r^*R_r\\Bigr)E_bP","\\subsection{Trace-preserving recovery supplies the original Gram data}\nLet $Z_{ra}=z_{ra}$. The recovery normalization gives\n\\begin{align*}\nPE_a^*E_bP\n&=PE_a^*\\Bigl(\\sum_rR_r^*R_r\\Bigr)E_bP\\\\\n&=\\sum_r(R_rE_aP)^*(R_rE_bP)\\\\\n&=\\Bigl(\\sum_r\\overline{z_{ra}}z_{rb}\\Bigr)P\n=(Z^*Z)_{ab}P.\n\\end{align*}\nThus $C=Z^*Z$ is positive semidefinite and satisfies the required compressed identities. The theorem \\path{gram_of_recovery} implements this calculation with the original error and recovery matrices.","&=\\Bigl(\\sum_r\\overline{z_{ra}}z_{rb}\\Bigr)P\n=(Z^*Z)_{ab}P.\n\\end{align*}\nThus $C=Z^*Z$ is positive semidefinite and satisfies the required compressed identities. The theorem \\path{gram_of_recovery} implements this calculation with the original error and recovery matrices.","A second rigidity argument handles a nonnegative scalar multiple of the identity code map. It uses $\\sqrt t\\,P$ as the rank-one map generator and includes $t=0$. That argument proves necessity for unnormalized error families, which need not define channels.","\\subsection{Normalization is not an extra assumption}\nA nonzero positive semidefinite matrix has strictly positive real trace. If a complex-linear map fixes every normalized code density matrix, apply that hypothesis to\n\\[\n\\frac{\\rho}{\\Tr\\rho}\n\\]\nfor each nonzero positive code-supported $\\rho$. Linearity then gives equality on $\\rho$ itself. The zero matrix is handled separately. The source proves that the trace is real and nonzero, then justifies cancellation of the scalar. These steps establish that normalization is valid.","\\subsection{Complex polarization reaches the whole operator space}\nWrite $q(x)=xx^*$. The module proves the exact identity\n\\[\n4xy^*=q(x+y)-q(x-y)\n+i\\bigl(q(x+iy)-q(x-iy)\\bigr).\n\\]\nAgreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by $P$ proves equality on every code-supported operator, including non-Hermitian ones.","\\subsection{An arbitrary finite reference system}\nFor any finite reference index set $A$, the amplified channel is defined using the actual Kronecker Kraus operators,\n\\[\n\\widetilde{\\mathcal N}_E(\\rho)\n=\\sum_a(I_A\\otimes E_a)\\rho(I_A\\otimes E_a)^*.\n\\]\nIts reference blocks are proved equal to the original channel applied blockwise. If\n\\[\n(I_A\\otimes P)\\rho(I_A\\otimes P)=\\rho,\n\\]\nthen every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives\n\\[\n\\boxed{\\widetilde{\\mathcal R}\\,\\widetilde{\\mathcal N}_E(\\rho)=\\rho.}\n\\]\nThis includes entangled density matrices supported on the combined reference and code space. In fact, the identity holds for every jointly supported matrix and every finite reference dimension.","channel is defined using the actual Kronecker Kraus operators,\n\\[\n\\widetilde{\\mathcal N}_E(\\rho)\n=\\sum_a(I_A\\otimes E_a)\\rho(I_A\\otimes E_a)^*.\n\\]\nIts reference blocks are proved equal to the original channel applied blockwise. If\n\\[\n(I_A\\otimes P)\\rho(I_A\\otimes P)=\\rho,\n\\]\nthen every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives\n\\[\n\\boxed{\\widetilde{\\mathcal R}\\,\\widetilde{\\mathcal N}_E(\\rho)=\\rho.}\n\\]\nThis includes entangled density matrices supported on the combined reference and code space. In fact, the identity holds for every jointly supported matrix and every finite reference dimension.","\\subsection{Cross terms for the original errors}\nFor arbitrary positive Gram data, the same recovery satisfies\n\\[\n\\boxed{\\mathcal R(E_a\\rho E_b^*)=C_{ba}\\rho\\qquad(P\\rho P=\\rho).}\n\\]\nThe index order $ba$ is intentional: the defining Gram relation is $PE_a^*E_bP=C_{ab}P$. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to identify the original Gram entries.","For $E(c)=\\sum_a c_aE_a$, finite-sum expansion gives\n\\[\n\\mathcal R\\bigl(E(c)\\rho E(c)^*\\bigr)\n=\\left(\\sum_{a,b}c_a\\overline{c_b}C_{ba}\\right)\\rho\n=(c^*Cc)\\rho.\n\\]\nThe formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.","\\subsection{Every normalized channel in the same span}\nSuppose $F_j=\\sum_a c_{ja}E_a$ and $\\sum_jF_j^*F_j=I$. The decoder is \\emph{not} reconstructed for the $F_j$. Summing the coherent identities gives a scalar multiple $q\\rho$. Trace preservation of both the $F$ channel and the recovery forces $q=1$ on a nonzero code. The proof uses positivity of $P$ to justify cancellation of its trace and handles $P=0$ separately. Therefore\n\\[\n\\boxed{\\mathcal R\\Bigl(\\sum_jF_j\\rho F_j^*\\Bigr)=\\rho.}\n\\]\nThis is \\path{Data.recovery_span_channel}, a quantified statement over arbitrary finite $F$ families and their complex coefficients.","ection{Every normalized channel in the same span}\nSuppose $F_j=\\sum_a c_{ja}E_a$ and $\\sum_jF_j^*F_j=I$. The decoder is \\emph{not} reconstructed for the $F_j$. Summing the coherent identities gives a scalar multiple $q\\rho$. Trace preservation of both the $F$ channel and the recovery forces $q=1$ on a nonzero code. The proof uses positivity of $P$ to justify cancellation of its trace and handles $P=0$ separately. Therefore\n\\[\n\\boxed{\\mathcal R\\Bigl(\\sum_jF_j\\rho F_j^*\\Bigr)=\\rho.}\n\\]\nThis is \\path{Data.recovery_span_channel}, a quantified statement over arbitrary finite $F$ families and their complex coefficients.","el in the same span}\nSuppose $F_j=\\sum_a c_{ja}E_a$ and $\\sum_jF_j^*F_j=I$. The decoder is \\emph{not} reconstructed for the $F_j$. Summing the coherent identities gives a scalar multiple $q\\rho$. Trace preservation of both the $F$ channel and the recovery forces $q=1$ on a nonzero code. The proof uses positivity of $P$ to justify cancellation of its trace and handles $P=0$ separately. Therefore\n\\[\n\\boxed{\\mathcal R\\Bigl(\\sum_jF_j\\rho F_j^*\\Bigr)=\\rho.}\n\\]\nThis is \\path{Data.recovery_span_channel}, a quantified statement over arbitrary finite $F$ families and their complex coefficients.","nt identities gives a scalar multiple $q\\rho$. Trace preservation of both the $F$ channel and the recovery forces $q=1$ on a nonzero code. The proof uses positivity of $P$ to justify cancellation of its trace and handles $P=0$ separately. Therefore\n\\[\n\\boxed{\\mathcal R\\Bigl(\\sum_jF_j\\rho F_j^*\\Bigr)=\\rho.}\n\\]\nThis is \\path{Data.recovery_span_channel}, a quantified statement over arbitrary finite $F$ families and their complex coefficients.","\\subsection{The equivalence for unnormalized error families}\nFor arbitrary finite $E$, define exact family recovery to mean that a finite trace-preserving Kraus recovery and a single $t\\ge0$ exist such that\n\\[\n\\mathcal R\\mathcal N_E(\\rho)=t\\rho\n\\quad\\text{for every code-supported }\\rho.\n\\]\nThen \\path{error_family_correctable_iff_gram} proves that this condition is equivalent to the existence of $C\\succeq0$ satisfying the Gram identities. Sufficiency chooses $t=\\Tr C$. Necessity constructs $C$ from the scaled rank-one rigidity argument.","r arbitrary finite $E$, define exact family recovery to mean that a finite trace-preserving Kraus recovery and a single $t\\ge0$ exist such that\n\\[\n\\mathcal R\\mathcal N_E(\\rho)=t\\rho\n\\quad\\text{for every code-supported }\\rho.\n\\]\nThen \\path{error_family_correctable_iff_gram} proves that this condition is equivalent to the existence of $C\\succeq0$ satisfying the Gram identities. Sufficiency chooses $t=\\Tr C$. Necessity constructs $C$ from the scaled rank-one rigidity argument.","xact family recovery to mean that a finite trace-preserving Kraus recovery and a single $t\\ge0$ exist such that\n\\[\n\\mathcal R\\mathcal N_E(\\rho)=t\\rho\n\\quad\\text{for every code-supported }\\rho.\n\\]\nThen \\path{error_family_correctable_iff_gram} proves that this condition is equivalent to the existence of $C\\succeq0$ satisfying the Gram identities. Sufficiency chooses $t=\\Tr C$. Necessity constructs $C$ from the scaled rank-one rigidity argument.","The scale is essential. Zero errors can give $t=0$ and cannot be inverted to a nonzero normalized state. An input with a larger total trace is not silently declared a channel. For normalized input channels, the separate channel theorem removes the scale exactly as described in the sufficiency argument above.","\\section{Exact examples, trust audit and remaining work}\\label{supp5:section:006}\n\\subsection{Singular, nonorthogonal errors are accepted}\nOn a qubit with full code $P=I$, take\n\\[\nE_0=\\frac35I,\\qquad E_1=\\frac45I,\\qquad\nC=\\begin{pmatrix}9/25&12/25","\\section{Exact examples, trust audit and remaining work}\\label{supp5:section:006}\n\\subsection{Singular, nonorthogonal errors are accepted}\nOn a qubit with full code $P=I$, take\n\\[\nE_0=\\frac35I,\\qquad E_1=\\frac45I,\\qquad\nC=\\begin{pmatrix}9/25&12/25\\\\12/25&16/25\\end{pmatrix}.\n\\]\nLean checks the channel normalization, the Gram identities, positivity of $C$, $\\det C=0$, and the nonzero off-diagonal entry. It then applies the criterion for density matrices to prove correctability. The calculation is exact and uses rational complex entries.","12/25&16/25\\end{pmatrix}.\n\\]\nLean checks the channel normalization, the Gram identities, positivity of $C$, $\\det C=0$, and the nonzero off-diagonal entry. It then applies the criterion for density matrices to prove correctability. The calculation is exact and uses rational complex entries.","\\subsection{A genuine channel can still be uncorrectable}\nFor full qubit dephasing, $E_0=\\operatorname{diag}(1,0)$ and $E_1=\\operatorname{diag}(0,1)$, the Kraus normalization is checked. But $E_0^*E_0$ cannot be a scalar multiple of $I$, as its two diagonal entries differ. Necessity therefore proves that no finite recovery channel corrects the full code space. The case of the zero code is also checked independently.","\\section{The global projected connection}\\label{supp6:section:001}\nLet $M$ be a smooth real manifold without boundary, $F$ a complete normed real vector space and $P:M\\to\\End(F)$ a smooth family of bounded idempotents. For smooth range sections $s$, define\n\\[\n\\nabla_Xs=P\\,\\dd s[X].\n\\]\nThe proof begins by constructing the range bundle from $P$. Its local connection formulas glue, and the actual covariant commutator satisfies\n\\[\n\\boxed{(\\nabla_X\\nabla_Y-\\nabla_Y\\nabla_X-\\nabla_{[X,Y]})s\n=P\\bigl(\\dd P[X]\\dd P[Y]-\\dd P[Y]\\dd P[X]\\bigr)P\\,s.}\n\\]\nThe derivative and Lie bracket are Mathlib's actual manifold operators. No global coordinate chart is assumed. The result requires idempotence, not an additional orthogonality hypothesis.","connection}\\label{supp6:section:001}\nLet $M$ be a smooth real manifold without boundary, $F$ a complete normed real vector space and $P:M\\to\\End(F)$ a smooth family of bounded idempotents. For smooth range sections $s$, define\n\\[\n\\nabla_Xs=P\\,\\dd s[X].\n\\]\nThe proof begins by constructing the range bundle from $P$. Its local connection formulas glue, and the actual covariant commutator satisfies\n\\[\n\\boxed{(\\nabla_X\\nabla_Y-\\nabla_Y\\nabla_X-\\nabla_{[X,Y]})s\n=P\\bigl(\\dd P[X]\\dd P[Y]-\\dd P[Y]\\dd P[X]\\bigr)P\\,s.}\n\\]\nThe derivative and Lie bracket are Mathlib's actual manifold operators. No global coordinate chart is assumed. The result requires idempotence, not an additional orthogonality hypothesis.","\\section{Local products constructed from the projector}\\label{supp6:section:002}\nThe fiber over $x$ is the actual fixed subspace\n\\[\n\\mathcal E_x=\\{v\\in F:P(x)v=v\\}=\\Ran P(x).\n\\]\nThe total space is the subset $\\mathcal E=\\{(x,v):P(x)v=v\\}\\subseteq M\\times F$, with the induced topology. The construction provides local homeomorphisms of this total space as well as the transition functions between them.","\\subsection{An invertible intertwiner near each anchor}\nFor idempotents $p,q$, set\n\\[\nT(p,q)=pq+(I-p)(I-q).\n\\]\nLean proves $T(p,p)=I$ and $pT(p,q)=T(p,q)q$. For an anchor $a\\in M$, define\n\\[\nU_a=\\{x:T(P(x),P(a))\\text{ is invertible}\\}.\n\\]\nThe invertible elements form an open subset of the algebra of bounded operators. Consequently $U_a$ is open and contains $a$. The family $(U_a)$ covers the whole base. Invertibility is \\emph{derived locally}, not assumed at every pair of points.","\\subsection{The mutually inverse local maps}\nWrite $T_a(x)=T(P(x),P(a))$. For $x\\in U_a$,\n\\[\nB_a(x):\\mathcal E_a\\longrightarrow\\mathcal E_x,\n\\quad v\\longmapsto T_a(x)v,\n\\]\n\\[\nA_a(x):\\mathcal E_x\\longrightarrow\\mathcal E_a,\n\\quad w\\longmapsto P(a)T_a(x)^{-1}w.\n\\]\nThe intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus\n\\[\n\\mathcal E|_{U_a}\\ \\cong\\ U_a\\times\\mathcal E_a\n\\]\nis a local product homeomorphism that preserves the base point.",")$. For $x\\in U_a$,\n\\[\nB_a(x):\\mathcal E_a\\longrightarrow\\mathcal E_x,\n\\quad v\\longmapsto T_a(x)v,\n\\]\n\\[\nA_a(x):\\mathcal E_x\\longrightarrow\\mathcal E_a,\n\\quad w\\longmapsto P(a)T_a(x)^{-1}w.\n\\]\nThe intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus\n\\[\n\\mathcal E|_{U_a}\\ \\cong\\ U_a\\times\\mathcal E_a\n\\]\nis a local product homeomorphism that preserves the base point.","\\subsection{Smooth overlap maps and rank}\nThe coordinate transition is $g_{ab}(x)=A_b(x)B_a(x)$. It is smooth on $U_a\\cap U_b$, agrees with the actual change between local homeomorphisms, and satisfies\n\\[\ng_{aa}=I,\\qquad g_{bc}g_{ab}=g_{ac}.\n\\]\nA continuous linear equivalence between $\\mathcal E_a$ and $\\mathcal E_x$ also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.","\\section{A connection on the constructed bundle}\\label{supp6:section:003}\nThe global formula is defined using the actual manifold derivative:\n\\[\n\\nabla_Xs(x)=P(x)\\,D_Ms(x)[X(x)].\n\\]\nHere $D_M$ is represented by \\path{mvfderiv}. A second definition, \\path{GlobalConnection.covariant}, has values in the actual fiber:\n\\[\n\\nabla s(x):T_xM\\longrightarrow\\mathcal E_x.\n\\]\nIt is not only an ambient expression asserted to land in the range.","\\subsection{Connection axioms and smoothness}\nThe formal proofs establish range preservation, additivity in the section, linearity over functions in the vector field argument, and the scalar Leibniz identity\n\\[\n\\nabla_X(fs)=\\dd f[X]\\,s+f\\nabla_Xs.\n\\]\nSmooth $P$, smooth sections and smooth vector fields give smooth covariant derivatives. Equality of section germs gives equality of covariant derivatives at the point, so the operation is local.","\\subsection{Smoothness of sections in bundle coordinates}\nLet $x\\in U_a$. If $P(y)s(y)=s(y)$ in a neighborhood of $x$, then\n\\[\ns\\text{ is smooth at }x\n\\quad\\Longleftrightarrow\\quad\n\\bigl(y\\mapsto A_a(y)s(y)\\bigr)\\text{ is smooth at }x.\n\\]\nThe range condition is only local. The proof uses the actual inverse local maps and equality of germs. It follows that smoothness in bundle coordinates does not depend on the chosen chart.","\\subsection{The local connection and its change of coordinates}\nFor a coordinate function $u$ valued in $\\mathcal E_a$, define\n\\[\n\\nabla^{(a)}_Xu=A_a\\bigl(P\\,D_M(B_a u)[X]\\bigr).\n\\]\nFor every overlap point, Lean proves the operator identity\n\\[\n\\boxed{\\nabla^{(b)}_X(g_{ab}u)=g_{ab}\\nabla^{(a)}_Xu.}\n\\]\nThis is proved for arbitrary local coordinate functions. It does not assume that $u$ was obtained from a globally smooth section or that a smooth extension has been supplied. Together with the local smoothness criterion, this closes the gluing step left open in the fifth block.","\\subsection{The flat ambient identity is proved in charts}\nFor a smooth vector-valued function $s$, the proof identifies directional derivatives with the pullback into each extended chart. It then proves\n\\[\nD_Ms([X,Y])=D_M(D_Ms[Y])[X]-D_M(D_Ms[X])[Y].\n\\]\nThis uses the chart pullback of the actual vector fields and the ordinary differential identity, then transfers the result back to the manifold. The auxiliary module works locally at a point, so the argument applies within arbitrary open chart domains.","\\subsection{Projector and section identities are derived}\nDifferentiating $P^2=P$ in a tangent direction $v$ yields\n\\[\n(\\dd P[v])P+P\\dd P[v]=\\dd P[v],\n\\qquad P\\dd P[v]P=0.\n\\]\nDifferentiating $Ps=s$ gives\n\\[\n\\dd s[v]=\\dd P[v]s+P\\dd s[v].\n\\]\nBoth statements are consequences of actual derivative rules. They are not additional assumptions.","\\subsection{Cancellation of the second derivatives}\nThe product rule and the identity for the ambient Lie bracket reduce curvature to\n\\[\nR^\\nabla(X,Y)s=P\\dd P[X]\\dd s[Y]-P\\dd P[Y]\\dd s[X].\n\\]\nSubstitute the identity obtained by differentiating the range condition for the section. The terms containing $P\\dd P[v]P$ vanish, leaving\n\\[\nR^\\nabla(X,Y)s=\nP\\bigl(\\dd P[X]\\dd P[Y]-\\dd P[Y]\\dd P[X]\\bigr)P s.\n\\]\nLean also proves that this depends only on $X(x)$, $Y(x)$ and $s(x)$, not on their extensions. The operator-valued form is alternating, additive and homogeneous, and lands in the actual fiber. Constant projector bundles have zero curvature form.","es}\nThe product rule and the identity for the ambient Lie bracket reduce curvature to\n\\[\nR^\\nabla(X,Y)s=P\\dd P[X]\\dd s[Y]-P\\dd P[Y]\\dd s[X].\n\\]\nSubstitute the identity obtained by differentiating the range condition for the section. The terms containing $P\\dd P[v]P$ vanish, leaving\n\\[\nR^\\nabla(X,Y)s=\nP\\bigl(\\dd P[X]\\dd P[Y]-\\dd P[Y]\\dd P[X]\\bigr)P s.\n\\]\nLean also proves that this depends only on $X(x)$, $Y(x)$ and $s(x)$, not on their extensions. The operator-valued form is alternating, additive and homogeneous, and lands in the actual fiber. Constant projector bundles have zero curvature form.","product rule and the identity for the ambient Lie bracket reduce curvature to\n\\[\nR^\\nabla(X,Y)s=P\\dd P[X]\\dd s[Y]-P\\dd P[Y]\\dd s[X].\n\\]\nSubstitute the identity obtained by differentiating the range condition for the section. The terms containing $P\\dd P[v]P$ vanish, leaving\n\\[\nR^\\nabla(X,Y)s=\nP\\bigl(\\dd P[X]\\dd P[Y]-\\dd P[Y]\\dd P[X]\\bigr)P s.\n\\]\nLean also proves that this depends only on $X(x)$, $Y(x)$ and $s(x)$, not on their extensions. The operator-valued form is alternating, additive and homogeneous, and lands in the actual fiber. Constant projector bundles have zero curvature form.","e and the identity for the ambient Lie bracket reduce curvature to\n\\[\nR^\\nabla(X,Y)s=P\\dd P[X]\\dd s[Y]-P\\dd P[Y]\\dd s[X].\n\\]\nSubstitute the identity obtained by differentiating the range condition for the section. The terms containing $P\\dd P[v]P$ vanish, leaving\n\\[\nR^\\nabla(X,Y)s=\nP\\bigl(\\dd P[X]\\dd P[Y]-\\dd P[Y]\\dd P[X]\\bigr)P s.\n\\]\nLean also proves that this depends only on $X(x)$, $Y(x)$ and $s(x)$, not on their extensions. The operator-valued form is alternating, additive and homogeneous, and lands in the actual fiber. Constant projector bundles have zero curvature form.","s retain valid mathematical arguments from the original manuscript. Their hypotheses and notation agree with the corrected finite core. The earlier abbreviated argument for error correction is replaced by the complete normalized and scaled recovery proofs above, and its substantive conclusions are included there.\n\\subsection{Duhamel proof of leakage from a fixed packet}\\label{legacy:duhamel}\nLet $H=H^*$ on a finite-dimensional complex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{","manuscript. Their hypotheses and notation agree with the corrected finite core. The earlier abbreviated argument for error correction is replaced by the complete normalized and scaled recovery proofs above, and its substantive conclusions are included there.\n\\subsection{Duhamel proof of leakage from a fixed packet}\\label{legacy:duhamel}\nLet $H=H^*$ on a finite-dimensional complex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of proje","Their hypotheses and notation agree with the corrected finite core. The earlier abbreviated argument for error correction is replaced by the complete normalized and scaled recovery proofs above, and its substantive conclusions are included there.\n\\subsection{Duhamel proof of leakage from a fixed packet}\\label{legacy:duhamel}\nLet $H=H^*$ on a finite-dimensional complex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curv","ses and notation agree with the corrected finite core. The earlier abbreviated argument for error correction is replaced by the complete normalized and scaled recovery proofs above, and its substantive conclusions are included there.\n\\subsection{Duhamel proof of leakage from a fixed packet}\\label{legacy:duhamel}\nLet $H=H^*$ on a finite-dimensional complex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior f","omplete normalized and scaled recovery proofs above, and its substantive conclusions are included there.\n\\subsection{Duhamel proof of leakage from a fixed packet}\\label{legacy:duhamel}\nLet $H=H^*$ on a finite-dimensional complex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential fo","ormalized and scaled recovery proofs above, and its substantive conclusions are included there.\n\\subsection{Duhamel proof of leakage from a fixed packet}\\label{legacy:duhamel}\nLet $H=H^*$ on a finite-dimensional complex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential forms with values in the range bund","ive conclusions are included there.\n\\subsection{Duhamel proof of leakage from a fixed packet}\\label{legacy:duhamel}\nLet $H=H^*$ on a finite-dimensional complex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square,","lex Hilbert space, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof","pace, $P=P^*=P^2$, $Q=I-P$, and $t\\in\\mathbb R$.\n\\begin{proof}[Alternative proof of the commutator leakage bound]\\phantomsection\\label{legacy:proof:001}\nBecause $QP=0$,\n\\[\nQe^{-itH}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formu","}P=Q[e^{-itH},P]P.\n\\]\nThe Duhamel identity gives\n\\[\n[e^{-itH},P]=-i\\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\\,ds.\n\\]\nBoth unitary factors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and","tors have norm one, so taking norms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\su","rms and using the integration interval's length $|t|$ yields\n$\\|Qe^{-itH}P\\|\\le |t|\\,\\|[H,P]\\|$.\n\\phantomsection\\label{legacy:proof:001:end}\\end{proof}\n\\subsection{Proof of projected curvature with exterior forms}\\label{legacy:exterior}\nLet $P$ be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restrict","ooth idempotent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1","potent field and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin","eld and extend the projected connection to differential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\s","ential forms with values in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6","in the range bundle by $\\nabla\\omega=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x","a=P\\,d\\omega$. Thus $\\nabla^2$ denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.\n\\begin{proof}[Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4","Alternative proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","ve proof of the curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","e curvature formula]\\phantomsection\\label{legacy:proof:002}\nFor a section $s$ with $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","dge ds.\n\\]\nSince $ds=d(Ps)=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","surviving term is\n\\[\nP(dP\\wedge dP)Ps.\n\\]\n\\phantomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","tomsection\\label{legacy:proof:002:end}\\end{proof}\n\\par\\Needspace{24\\baselineskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","neskip}\n\\subsection{Restricted low-mode proof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","oof}\\label{legacy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","cy:lowmode}\nLet $h\\ne0$, $x=kh/2$, and $0\\le x\\le1$.\n\\begin{proof}[Alternative proof in the original low-mode regime]\\phantomsection\\label{legacy:proof:003}\nSince $0\\le\\sin x\\le x$, the error is nonnegative. On $[0,1]$,\n\\[\n\\sin x\\ge x-\\frac{x^3}{6}\\ge0.\n\\]\nTherefore\n\\[\n\\sin^2x\\ge\\left(x-\\frac{x^3}{6}\\right)^2\\ge x^2-\\frac{x^4}{3}.\n\\]\nHence $x^2-\\sin^2x\\le x^4/3$. Multiplication by $4/h^2$ gives\n\\[\n0\\le k^2-\\frac4{h^2}\\sin^2(kh/2)\\le\\frac{k^4h^2}{12}.\n\\]\n\\phantomsection\\label{legacy:proof:003:end}\\end{proof}","\\vpage{29}{A complex range needs an adjoint}{THE INNER PRODUCT IS PART OF THE CONTRACT}{Chapter VIII: Theorem 5.6, Section F.1, Section F.2.}\nThe earlier compact example is real. Over $\\CC$, an isometry means $Q^*Q=I$, with conjugation. Take the exact Gaussian-rational factors\n\\[\nQ_L=\\mat{3/5&0","\\vpage{29}{A complex range needs an adjoint}{THE INNER PRODUCT IS PART OF THE CONTRACT}{Chapter VIII: Theorem 5.6, Section F.1, Section F.2.}\nThe earlier compact example is real. Over $\\CC$, an isometry means $Q^*Q=I$, with conjugation. Take the exact Gaussian-rational factors\n\\[\nQ_L=\\mat{3/5&0\\\\4\\mathrm i/5&0\\\\0&1},\\qquad\nQ_R=\\mat{1&0\\\\0&\\mathrm i},\\qquad G=\\diag(5,2).\n\\]\nTheir product is a genuinely complex rectangular operator:\n\\[\nA=Q_LGQ_R^*=\\mat{3&0","0&\\mathrm i},\\qquad G=\\diag(5,2).\n\\]\nTheir product is a genuinely complex rectangular operator:\n\\[\nA=Q_LGQ_R^*=\\mat{3&0\\\\4\\mathrm i&0\\\\0&-2\\mathrm i},\n\\qquad A^*A=\\diag(25,4).\n\\]\n\\vfig{complex_core}\n\\head{The small calculation exposes a wrong convention}\nThe first column of $Q_L$ has Hermitian squared length $9/25+16/25=1$. Its bilinear square without conjugation is $9/25-16/25=-7/25$. Thus $Q_L^TQ_L$ is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting isometry claim.","0&-2\\mathrm i},\n\\qquad A^*A=\\diag(25,4).\n\\]\n\\vfig{complex_core}\n\\head{The small calculation exposes a wrong convention}\nThe first column of $Q_L$ has Hermitian squared length $9/25+16/25=1$. Its bilinear square without conjugation is $9/25-16/25=-7/25$. Thus $Q_L^TQ_L$ is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting isometry claim.","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof. This two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean no","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof. This two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean norm. A transp","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof. This two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean norm. A transpose is correct in the vectorization identity on \\pagerefV{7}, but an adjoint is required for an isometr","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof. This two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean norm. A transpose is correct in the vectorization identity on \\pagerefV{7}, but an adjoint is required for an isometric range lift.}","\\vpage{30}{Inverse reachability is still an inclusion}{A FINITE INVERSE WITH A CANCELLED ROUTE}{Chapter VIII: Theorem 6.7, Theorem 6.1, Corollary 6.3.}\nReuse the signed diamond from \\pagerefV{14}, now as a complete operator:\n\\[\nA=I+K=\\mat{1&0&0&0\\\\1&1&0&0\\\\1&0&1&0\\\\0&1&-1&1}.\n\\]\nThe two routes from coordinate zero to coordinate three cancel, so $K^2=0$. Consequently\n\\[\nA^{-1}=I-K=\\mat{1&0&0&0","0&1&-1&1}.\n\\]\nThe two routes from coordinate zero to coordinate three cancel, so $K^2=0$. Consequently\n\\[\nA^{-1}=I-K=\\mat{1&0&0&0","0&1&-1&1}.\n\\]\nThe two routes from coordinate zero to coordinate three cancel, so $K^2=0$. Consequently\n\\[\nA^{-1}=I-K=\\mat{1&0&0&0\\\\-1&1&0&0\\\\-1&0&1&0\\\\0&-1&1&1}.\n\\]\n\\vfig[.52]{inverse_closure}\n\\head{What the closure permits}\nThe directed transitive closure includes $0\\to3$, because it records existence of a route rather than the signed sum of its coefficients. Nevertheless $(A^{-1})_{30}=0$. Reachability is therefore an upper envelope for inverse support, just as it is for product support.","0&-1&1&1}.\n\\]\n\\vfig[.52]{inverse_closure}\n\\head{What the closure permits}\nThe directed transitive closure includes $0\\to3$, because it records existence of a route rather than the signed sum of its coefficients. Nevertheless $(A^{-1})_{30}=0$. Reachability is therefore an upper envelope for inverse support, just as it is for product support.","The portable checker computes the inverse independently by rational row elimination, compares it with $I-K$, constructs Boolean reachability, and checks every inverse entry outside that closure. It separately rejects the assertion that every reachable transfer is nonzero.\n\\head{Why the general proof is broader}\nThis example happens to have a terminating Neumann polynomial. The general finite-dimensional theorem does not require nilpotence or a small norm. For a fixed source coordinate, its reachable-coordinate subspace is invariant under $A$. If $A$ is invertible, the restricted map is injective. Equal finite domain and codomain dimensions make it surjective, so the inverse preserves the same subspace.","checks every inverse entry outside that closure. It separately rejects the assertion that every reachable transfer is nonzero.\n\\head{Why the general proof is broader}\nThis example happens to have a terminating Neumann polynomial. The general finite-dimensional theorem does not require nilpotence or a small norm. For a fixed source coordinate, its reachable-coordinate subspace is invariant under $A$. If $A$ is invertible, the restricted map is injective. Equal finite domain and codomain dimensions make it surjective, so the inverse preserves the same subspace.","every inverse entry outside that closure. It separately rejects the assertion that every reachable transfer is nonzero.\n\\head{Why the general proof is broader}\nThis example happens to have a terminating Neumann polynomial. The general finite-dimensional theorem does not require nilpotence or a small norm. For a fixed source coordinate, its reachable-coordinate subspace is invariant under $A$. If $A$ is invertible, the restricted map is injective. Equal finite domain and codomain dimensions make it surjective, so the inverse preserves the same subspace.","bottomrule\n\\end{tabular}\n\\end{center}\n\\vfig[.45]{shared_family}\nWith reuse weights $(r_1,r_2)=(1,3)$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof app","bottomrule\n\\end{tabular}\n\\end{center}\n\\vfig[.45]{shared_family}\nWith reuse weights $(r_1,r_2)=(1,3)$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree r","$(r_1,r_2)=(1,3)$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms. It does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\take…","$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms. It does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weigh…","s are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms. It does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective bef…","\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms. It does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective before compilation. Shared opt…","6.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms. It does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective before compilation. Shared optimality is exact for the supplied common tre…","Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$. Its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms. It does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective before compilation. Shared optimality is exact for the supplied common tree and additive family costs, rather than …","\\vpage{32}{Safe integers can still be conservative}{QUANTIZATION AND THE SIZE OF THE BUDGET TABLE}{Chapter VIII: Theorem 8.7, Theorem 9.8, Corollary 9.9.}\nLet one integer unit represent $h=1/10$ of certified error. For an individual nonnegative error $e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","nit represent $h=1/10$ of certified error. For an individual nonnegative error $e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","r an individual nonnegative error $e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","eil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","Now change the errors to $(3/50,1/50)$. Their sum is $2/25\\le1/10$, but separate ceilings still give $(1,1)$. The integer model rejects an assignment that is feasible for the real-valued problem. Conservative rounding protects soundness. It need not preserve every optimum of the unquantized problem.\n\\head{Why the complexity is pseudo polynomial}\nFor a declared integer budget $B$, a table has $B+1$ possible exact-weight states. A direct merge checks at most $(B+1)^2$ state pairs. At $B=255$, this means 256 states and 65,536 candidate pairs at a merge, even though the budget fits in eight binary digits.","/10$, but separate ceilings still give $(1,1)$. The integer model rejects an assignment that is feasible for the real-valued problem. Conservative rounding protects soundness. It need not preserve every optimum of the unquantized problem.\n\\head{Why the complexity is pseudo polynomial}\nFor a declared integer budget $B$, a table has $B+1$ possible exact-weight states. A direct merge checks at most $(B+1)^2$ state pairs. At $B=255$, this means 256 states and 65,536 candidate pairs at a merge, even though the budget fits in eight binary digits.","s still give $(1,1)$. The integer model rejects an assignment that is feasible for the real-valued problem. Conservative rounding protects soundness. It need not preserve every optimum of the unquantized problem.\n\\head{Why the complexity is pseudo polynomial}\nFor a declared integer budget $B$, a table has $B+1$ possible exact-weight states. A direct merge checks at most $(B+1)^2$ state pairs. At $B=255$, this means 256 states and 65,536 candidate pairs at a merge, even though the budget fits in eight binary digits.","With a bounded number of children, the paper's direct tree implementation therefore has quadratic budget dependence at each node. Polynomial dependence on the numerical value of $B$ does not mean polynomial dependence on the binary input length. These are counts of declared states and pairs, not measured runtimes or claims that every state is reachable.\n\\takeaway{The dynamic program is exact for its integer data. Conservative conversion and the numerical size of the budget are separate obligations that determine its accuracy and practical scale.}","n 3, Appendix F.}\nA dense array, a factored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide illustrates those distinctions with calculations.\n\\vfig{three_views}\n\\head{Three models to carry through the paper}\nThe first is a six-dimensional operator $A=D+uv^T$. Every entry of $A$ is nonzero, yet three $2\\times2$ diagonal blocks and two thin vectors suffice to apply it exactly. \\pagerefV{3}, \\pagerefV{9}, \\pagerefV{10}, and \\pagerefV{27} reuse this operator or its factors.","rray, a factored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide illustrates those distinctions with calculations.\n\\vfig{three_views}\n\\head{Three models to carry through the paper}\nThe first is a six-dimensional operator $A=D+uv^T$. Every entry of $A$ is nonzero, yet three $2\\times2$ diagonal blocks and two thin vectors suffice to apply it exactly. \\pagerefV{3}, \\pagerefV{9}, \\pagerefV{10}, and \\pagerefV{27} reuse this operator or its factors.","and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide illustrates those distinctions with calculations.\n\\vfig{three_views}\n\\head{Three models to carry through the paper}\nThe first is a six-dimensional operator $A=D+uv^T$. Every entry of $A$ is nonzero, yet three $2\\times2$ diagonal blocks and two thin vectors suffice to apply it exactly. \\pagerefV{3}, \\pagerefV{9}, \\pagerefV{10}, and \\pagerefV{27} reuse this operator or its factors.","The second starts with redundant factors $L$ and $R$. Actual range coordinates reduce a $4\\times3$ product to a $2\\times3$ core. \\pagerefV{11}--\\pagerefV{12} connect rank, singular tails, and the geometric reason a small-core optimum is also an ambient optimum.","midrule\n$P_\\alpha$ & Coordinate projector & Which input or output coordinates does it select?","$A_{\\beta\\alpha}$ & A block map & What are its source and target dimensions?","$E$ & Active block addresses & Is this exact support, or only a structural upper bound?","$D+UV^*$ & Factored operator & Is the desired output a vector, factors, or a dense array?","$Q_L,G,Q_R$ & Range factors and core & Are the outer columns orthonormal, with exact reconstruction?","$E_j$ & An error operator & Is it a factor perturbation or an additive error after a product?","$s_e,\\epsilon_e$ & Sensitivity and tolerance & Is the budget input-dependent or uniform over unit inputs?","$(\\kappa_v,\\epsilon_v,\\sigma_v)$ & Candidate-tree data & Do costs, errors, and split overheads really add?","$F_v(b)$ & A budget table & Does $b$ mean exact weight, or an upper bound?","$R$ & Reuse count & Which one-time costs have been included?","bottomrule\n\\end{tabularx}\n\\head{Conventions used in the examples}\nMatrices act on column vectors. A block $A_{\\beta\\alpha}$ goes from source $\\alpha$ to target $\\beta$. Products act from right to left. Real examples use transpose, while complex examples use adjoint unless a vectorization identity explicitly requires transpose. Column-major vectorization stacks the columns of a matrix.","\\vpage{3}{A dense-looking matrix, one scalar coupling}{THE RUNNING SIX-DIMENSIONAL OPERATOR}{Chapter VIII: Theorem 2.3, Theorem 3.4.}\nLet $D$ have three identical blocks $\\mat{3&1","\\vpage{3}{A dense-looking matrix, one scalar coupling}{THE RUNNING SIX-DIMENSIONAL OPERATOR}{Chapter VIII: Theorem 2.3, Theorem 3.4.}\nLet $D$ have three identical blocks $\\mat{3&1\\\\1&3}$ and put $u=v=\\ones\\in\\QQ^6$. Then $A=D+uv^T$ has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero.\n\\[\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n\\]\n\\vfig{factored_application}\n\\head{What was avoided?}\nThe global part does not require 36 independent coefficients. It depends on the input through one scalar, $v^Tx$, which is broadcast through $u$. The diagonal part is computed by three separate local matrix products. Nothing is approximated and no channel is discarded.","1&3}$ and put $u=v=\\ones\\in\\QQ^6$. Then $A=D+uv^T$ has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero.\n\\[\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n\\]\n\\vfig{factored_application}\n\\head{What was avoided?}\nThe global part does not require 36 independent coefficients. It depends on the input through one scalar, $v^Tx$, which is broadcast through $u$. The diagonal part is computed by three separate local matrix products. Nothing is approximated and no channel is discarded.","^6$. Then $A=D+uv^T$ has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero.\n\\[\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n\\]\n\\vfig{factored_application}\n\\head{What was avoided?}\nThe global part does not require 36 independent coefficients. It depends on the input through one scalar, $v^Tx$, which is broadcast through $u$. The diagonal part is computed by three separate local matrix products. Nothing is approximated and no channel is discarded.","The sixth output is $7+3=10$, while the fifth is $-3+3=0$. A zero result can arise through cancellation even when the corresponding row contains no zero coefficient. The output vector is therefore not a support map of the input matrix.\n\\head{What has not been established?}\nThis computation proves neither that the supplied factors are the cheapest representation nor that six small tasks outperform one dense kernel. For example, the repeated diagonal blocks and all-one vectors admit further special handling. The stated BDLR model deliberately treats the blocks as dense and the factors as stored vectors so that the paper's general formulas apply literally.","The checker constructs the complete matrix independently, multiplies it by $x$, and compares all six entries with the factored route. The same input returns in the product calculation on \\pagerefV{27}.\n\\takeaway{A matrix can have full entrywise support and still admit a small exact representation. Entry sparsity and algebraic compressibility are different properties.}","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Chapter VIII: Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory i","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Chapter VIII: Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation a","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Chapter VIII: Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs,","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Chapter VIII: Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Chapter VIII: Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is n","dels dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","on with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","qual block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","$b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.",")\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","ation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","For a block band of $p$ packets and half-bandwidth $w$, the exact active-block count is\n\\[\np(2w+1)-w(w+1).\n\\]\nThe subtraction removes overcounted edge diagonals. The checker enumerates all addresses for $1\\le p\\le6$ and every admissible $w$, including $w=0$ and $w=p-1$.\n\\takeaway{An arithmetic or storage ratio explains a representation advantage. Kernel launch cost, traffic, utilization, and conversion determine whether that advantage becomes a time advantage.}","\\vpage{5}{Four coefficients, four exact frequencies}{CIRCULANT MULTIPLICATION}{Chapter VIII: Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","}{Four coefficients, four exact frequencies}{CIRCULANT MULTIPLICATION}{Chapter VIII: Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","CIRCULANT MULTIPLICATION}{Chapter VIII: Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","pter VIII: Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","athrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transfor","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transform and conv","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transform and convolution scale before interpreting a frequenc","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transform and convolution scale before interpreting a frequency-domain product","\\vpage{6}{Padding is part of the theorem}{TOEPLITZ EMBEDDING WITHOUT WRAPAROUND}{Chapter VIII: Corollary 4.4.}\nConsider\n\\[\nT=\\mat{2&-1&5\\\\3&2&-1\\\\4&3&2},\\qquad x=(1,2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as","4&3&2},\\qquad x=(1,2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and r","4&3&2},\\qquad x=(1,2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","imension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","d indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","he first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","The middle zero in $g$ is not a discarded Toeplitz coefficient: it separates the lower and reversed upper diagonals in the chosen $2n$ embedding. Other embeddings require their own index identities. It is not sufficient for an array merely to have the right length or to produce a plausible plot.\n\\takeaway{Zero padding prevents the wrong circular interactions. It is an algebraic part of the operator representation, not an optional performance tweak.}","\\vpage{7}{A transpose, not an adjoint}{KRONECKER ACTION AND ARRAY ORDER}{Chapter VIII: Theorem 4.5, Corollary 4.6.}\nWith column-major vectorization, the tensor identity is\n\\[\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n\\]\nHere the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix $A=\\mat{1&\\mathrm i","\\vpage{7}{A transpose, not an adjoint}{KRONECKER ACTION AND ARRAY ORDER}{Chapter VIII: Theorem 4.5, Corollary 4.6.}\nWith column-major vectorization, the tensor identity is\n\\[\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n\\]\nHere the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix $A=\\mat{1&\\mathrm i\\\\2&-1}$, together with $B=\\mat{2&1","2&-1}$, together with $B=\\mat{2&1\\\\0&3}$ and $X=\\mat{1&2","0&3}$ and $X=\\mat{1&2\\\\3&4}$.\n\\vfig{kronecker}\n\\head{Check the coordinates}\nThe stacked input is $(1,3,2,4)^T$. Direct tensor multiplication agrees with stacking the columns of\n\\[\nBXA^T=\\mat{5+8\\mathrm i&2","3&4}$.\n\\vfig{kronecker}\n\\head{Check the coordinates}\nThe stacked input is $(1,3,2,4)^T$. Direct tensor multiplication agrees with stacking the columns of\n\\[\nBXA^T=\\mat{5+8\\mathrm i&2","3&4}$.\n\\vfig{kronecker}\n\\head{Check the coordinates}\nThe stacked input is $(1,3,2,4)^T$. Direct tensor multiplication agrees with stacking the columns of\n\\[\nBXA^T=\\mat{5+8\\mathrm i&2\\\\9+12\\mathrm i&6}.\n\\]\nReplacing $A^T$ with $A^*$ changes the signs of the imaginary terms. A real-only test would conceal that error, which is why the worked model uses an imaginary coefficient.\n\\head{Why modewise contraction matters}\nFor $A$ of order $m$ and $B$ of order $n$, the explicit tensor matrix stores $m^2n^2$ entries. Its factors store $m^2+n^2$. Applying through $BXA^T$ costs $2n^2m+2nm^2$ modeled operations, rather than forming the full tensor matrix first.","9+12\\mathrm i&6}.\n\\]\nReplacing $A^T$ with $A^*$ changes the signs of the imaginary terms. A real-only test would conceal that error, which is why the worked model uses an imaginary coefficient.\n\\head{Why modewise contraction matters}\nFor $A$ of order $m$ and $B$ of order $n$, the explicit tensor matrix stores $m^2n^2$ entries. Its factors store $m^2+n^2$. Applying through $BXA^T$ costs $2n^2m+2nm^2$ modeled operations, rather than forming the full tensor matrix first.","\\vpage{8}{See every butterfly value}{WALSH RECURSION IN EIGHT COORDINATES}{Chapter VIII: Theorem 4.7.}\nThe unnormalized Walsh matrix satisfies $H_1=[1]$ and\n\\[\nH_{2n}=\\mat{H_n&H_n","\\vpage{8}{See every butterfly value}{WALSH RECURSION IN EIGHT COORDINATES}{Chapter VIII: Theorem 4.7.}\nThe unnormalized Walsh matrix satisfies $H_1=[1]$ and\n\\[\nH_{2n}=\\mat{H_n&H_n\\\\H_n&-H_n},\\qquad H_nH_n^T=nI.\n\\]\nApply the three add/subtract stages to $(1,2,3,4,5,6,7,8)^T$. The full intermediate ledger is displayed below.\n\\vfig{walsh}\n\\head{Arithmetic can be counted exactly}\nEach stage visits eight coordinates, producing one sum or difference per coordinate. Three stages therefore use exactly 24 additions or subtractions. No multiplication by a dense $8\\times8$ matrix is necessary.","H_n&-H_n},\\qquad H_nH_n^T=nI.\n\\]\nApply the three add/subtract stages to $(1,2,3,4,5,6,7,8)^T$. The full intermediate ledger is displayed below.\n\\vfig{walsh}\n\\head{Arithmetic can be counted exactly}\nEach stage visits eight coordinates, producing one sum or difference per coordinate. Three stages therefore use exactly 24 additions or subtractions. No multiplication by a dense $8\\times8$ matrix is necessary.","The result $(36,-4,-8,0,-16,0,0,0)^T$ still reconstructs the input: the inverse is $H_8^T/8$. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform $H_8/\\sqrt8$ would instead be orthogonal.\n\\head{What the zeros mean}\nThe zero coefficients arise from exact cancellation among the input values. They do not imply that the transform's defining matrix is sparse. The butterfly structure, not the final coefficient support, explains the arithmetic reduction.","\\vpage{9}{The cross term has somewhere to go}{CLOSED LOW-RANK PRODUCT ALGEBRA}{Chapter VIII: Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","cross term has somewhere to go}{CLOSED LOW-RANK PRODUCT ALGEBRA}{Chapter VIII: Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","Chapter VIII: Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","II: Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.",",3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","The scalar $v_a^Tu_b$ is three. Removing the quadratic term would therefore remove $3u_av_b^T$, which is nonzero. The checker compares the complete product to the factor construction and rejects a calculation that omits this contribution.\n\\head{Before and after recompression}\nA rank bound on stored factors is not always the actual matrix rank. Repeated products may introduce dependent columns. Recompression can remove those dependencies exactly, or remove small singular directions approximately under an explicit error budget.\n\\takeaway{Closure of a representation class is an algebraic statement about all product terms. It is not permission to drop an interaction merely because it is second order in the factors.","The scalar $v_a^Tu_b$ is three. Removing the quadratic term would therefore remove $3u_av_b^T$, which is nonzero. The checker compares the complete product to the factor construction and rejects a calculation that omits this contribution.\n\\head{Before and after recompression}\nA rank bound on stored factors is not always the actual matrix rank. Repeated products may introduce dependent columns. Recompression can remove those dependencies exactly, or remove small singular directions approximately under an explicit error budget.\n\\takeaway{Closure of a representation class is an algebraic statement about all product terms. It is not permission to drop an interaction merely because it is second order in the factors.}","\\vpage{10}{A large inverse, a small denominator}{WOODBURY WITH BOTH INVERTIBILITY CHECKS}{Chapter VIII: Theorem 5.3, Corollary 5.4.}\nFor the running operator, each diagonal block has inverse $\\frac18\\mat{3&-1\\\\-1&3}$. Hence $D^{-1}u=u/4$ and\n\\[\nS=1+u^TD^{-1}u=\\frac52.\n\\]\nThe Woodbury identity reduces the entire inverse to\n\\[\nA^{-1}=D^{-1}-D^{-1}uS^{-1}u^TD^{-1}\n=D^{-1}-\\frac1{40}uu^T.\n\\]\n\\vfig{woodbury}\n\\head{Two hypotheses, not one}\nInvertibility of $D$ alone does not suffice. For $D=I_2$, $u=e_1$, and $v=-e_1$, the update is $\\diag(0,1)$. The small matrix $S$ is zero, exactly where the correction formula would require an invalid division.","-1&3}$. Hence $D^{-1}u=u/4$ and\n\\[\nS=1+u^TD^{-1}u=\\frac52.\n\\]\nThe Woodbury identity reduces the entire inverse to\n\\[\nA^{-1}=D^{-1}-D^{-1}uS^{-1}u^TD^{-1}\n=D^{-1}-\\frac1{40}uu^T.\n\\]\n\\vfig{woodbury}\n\\head{Two hypotheses, not one}\nInvertibility of $D$ alone does not suffice. For $D=I_2$, $u=e_1$, and $v=-e_1$, the update is $\\diag(0,1)$. The small matrix $S$ is zero, exactly where the correction formula would require an invalid division.","For the running model, the checker multiplies the proposed inverse by $A$ in both orders and obtains the identity. It also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","he running model, the checker multiplies the proposed inverse by $A$ in both orders and obtains the identity. It also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","e identity. It also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","\\vpage{11}{Stored width is not intrinsic rank}{ACTUAL RANGE COORDINATES}{Chapter VIII: Theorem 5.6, Section F.1.}\nChoose the isometry\n\\[\nQ_L=\\mat{3/5&0\\\\4/5&0\\\\0&1\\\\0&0},\\quad\nR_L=\\mat{1&0&1\\\\0&1&0},\\quad\nR_R=\\mat{3&0&2\\\\0&2&0\\\\1&0&-1}.\n\\]\nSet $L=Q_LR_L$ and $R=R_R$. The two equal columns of $L$ do not create a third range direction. In contrast, $R$ has rank three.\n\\vfig{compact_core}\nThe actual core is\n\\[\nG=R_LR_R^T=\\mat{5&0&0","1&0&-1}.\n\\]\nSet $L=Q_LR_L$ and $R=R_R$. The two equal columns of $L$ do not create a third range direction. In contrast, $R$ has rank three.\n\\vfig{compact_core}\nThe actual core is\n\\[\nG=R_LR_R^T=\\mat{5&0&0","1&0&-1}.\n\\]\nSet $L=Q_LR_L$ and $R=R_R$. The two equal columns of $L$ do not create a third range direction. In contrast, $R$ has rank three.\n\\vfig{compact_core}\nThe actual core is\n\\[\nG=R_LR_R^T=\\mat{5&0&0\\\\0&2&0},\\qquad LR^T=Q_LG.\n\\]\nIts dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The range factor $Q_L$ has orthonormal columns and reconstructs $L$ exactly.\n\\head{The preserved formal boundary}\nThe small-core recompression construction in Appendix F constructs orthonormal bases of the actual column spaces. It then assumes exact ordered thin-SVD data for the core, as the manuscript instructs the reader to take an SVD. It does not assume an optimality or tail-error conclusion. Nor does it certify a floating-point QR or SVD implementation simply because those routines use similar notation.","0&2&0},\\qquad LR^T=Q_LG.\n\\]\nIts dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The range factor $Q_L$ has orthonormal columns and reconstructs $L$ exactly.\n\\head{The preserved formal boundary}\nThe small-core recompression construction in Appendix F constructs orthonormal bases of the actual column spaces. It then assumes exact ordered thin-SVD data for the core, as the manuscript instructs the reader to take an SVD. It does not assume an optimality or tail-error conclusion. Nor does it certify a floating-point QR or SVD implementation simply because those routines use similar notation.","\\vpage{12}{Why a core optimum is global}{COMPRESSION, KERNEL WITNESSES, AND TAILS}{Chapter VIII: Theorem 5.6, Section F.2, Section F.3, Section F.4.}\nThe product from the previous page is\n\\[\nA_0=LR^T=\\mat{3&0&0\\\\4&0&0\\\\0&2&0\\\\0&0&0}.\n\\]\nKeeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and $\\sqrt{29}$, while keeping both gives zero.\n\\vfig{optimality}\n\\head{Why an outside competitor cannot do better}\nFor isometric outer factors, compression $C(B)=Q_L^*BQ_R$ cannot increase rank or either matrix norm. Every ambient rank-$k$ competitor therefore supplies an admissible core competitor. The lifted core residual has exactly the core residual norm. These two facts transfer a core optimum to the entire ambient matrix space, not just to competitors already lying in the supplied ranges.","0&0&0}.\n\\]\nKeeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and $\\sqrt{29}$, while keeping both gives zero.\n\\vfig{optimality}\n\\head{Why an outside competitor cannot do better}\nFor isometric outer factors, compression $C(B)=Q_L^*BQ_R$ cannot increase rank or either matrix norm. Every ambient rank-$k$ competitor therefore supplies an admissible core competitor. The lifted core residual has exactly the core residual norm. These two facts transfer a core optimum to the entire ambient matrix space, not just to competitors already lying in the supplied ranges.","For a $d$-dimensional diagonal core and $0\\le k<d$, the spectral lower bound has a concrete witness. A rank-at-most-$k$ competitor has a nonzero kernel vector in the first $k+1$ coordinates. On that vector the diagonal matrix acts with norm at least $\\sigma_{k+1}$ times the vector norm. The competitor cannot cancel that direction.","The Frobenius proof uses an orthonormal basis of the competitor's kernel. Its row weights lie in $[0,1]$ and sum to the kernel dimension. Weighting the decreasing squared singular values forces at least the discarded tail. The two norm arguments are distinct. One is not a graph of the other.\n\\takeaway{Optimality is not inferred from the appearance of the singular-value plot. It follows from compression and a lower bound valid against every rank-bounded competitor.}","\\vpage{13}{Errors enter at different places}{FACTOR PERTURBATION VERSUS RECOMPRESSION}{Chapter VIII: Theorem 5.8, Theorem 5.9.}\nLet the exact scalar factors be $(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains later factors that have already been perturbed, while the additive recompression formula contains the original later factors.",".9.}\nLet the exact scalar factors be $(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains later factors that have already been perturbed, while the additive recompression formula contains the original later factors.","(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains later factors that have already been perturbed, while the additive recompression formula contains the original later factors.","art from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains later factors that have already been perturbed, while the additive recompression formula contains the original later factors.","The visual comparison uses the additive model. The first error is multiplied by both remaining factors. The last error is not amplified at all. Summing only the local error magnitudes gives $7/20$, which is smaller than the actual final error and therefore not a valid bound here.\n\\head{In a matrix chain}\nThe corresponding formulas replace scalar magnitudes with operator norms and respect multiplication order. A bound can be conservative when contributions cancel, but it must not omit possible amplification. Singular-tail errors from a recompression step become useful only after they are inserted into the correct global error model.\n\\takeaway{An error tolerance needs an entry point. A bound for perturbing factors and a bound for compressing partial products are not interchangeable.}","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Chapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Chapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on th","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Chapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpote","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Chapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Chapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an asse","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Chapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Chapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","1}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.",",\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","x of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","dicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimite","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary an","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attaina","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If o","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one e","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is a","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available,","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Chapter VIII: Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks require seven slots of",", put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks require seven slots of work. If task durations differ, the relevant critical path is weighted.","\\vpage{16}{Elimination creates new interactions}{SCHUR FILL AND ITS CANCELLATION}{Chapter VIII: Theorem 6.7, Proposition 6.8.}\nRetain the first two coordinates of\n\\[\nA=\\mat{2&0&1\\\\0&2&1\\\\1&1&2}.\n\\]\nThe third coordinate is hidden. Its invertible block is $[2]$, so eliminating it produces\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2","1&1&2}.\n\\]\nThe third coordinate is hidden. Its invertible block is $[2]$, so eliminating it produces\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2","1&1&2}.\n\\]\nThe third coordinate is hidden. Its invertible block is $[2]$, so eliminating it produces\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2\\\\-1/2&3/2}.\n\\]\n\\vfig{schur_fill}\n\\head{The hidden route has a sign}\nThe retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by $-1/2$. This is fill: a new active retained block produced by an eliminated route.","-1/2&3/2}.\n\\]\n\\vfig{schur_fill}\n\\head{The hidden route has a sign}\nThe retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by $-1/2$. This is fill: a new active retained block produced by an eliminated route.","Now change both original retained off-diagonal entries to $+1/2$. The same hidden correction cancels them, leaving the diagonal matrix $\\frac32 I$. Thus a structural route can create or remove visible coupling depending on the direct coefficient already present.\n\\head{A stronger inverse statement}\nFor an invertible finite packet matrix, inverse support lies in the directed transitive closure. A reachable-coordinate subspace is invariant under the matrix. Injectivity of its restriction implies surjectivity in finite dimension, so the inverse preserves it as well. This proof uses finite-dimensional invertibility, not a convergent geometric series.","The checker verifies both Schur complements and, in each invertible example, equality between $S^{-1}$ and the retained block of $A^{-1}$.\n\\takeaway{Removing coordinates does not remove their influence. It changes the retained operator, with possible fill, cancellation, and new conditioning constraints.}","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{Chapter VIII: Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{Chapter VIII: Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{Chapter VIII: Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM","A STRICT KRYLOV INCLUSION}{Chapter VIII: Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure fro…","es the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A …","both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can","ordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock ot…","he two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","uality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","With $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","\\vpage{18}{Measure coupling after normalization}{BLOCK JACOBI AS A GEOMETRIC CHANGE}{Chapter VIII: Theorem 7.4, Corollary 7.6, Classical input 7.5.}\nLet\n\\[\nA=\\mat{4&1\\\\1&1},\\quad D=\\diag(4,1),\\quad\nB=D^{-1/2}=\\diag(1/2,1).\n\\]\nThen $BAB=I+E=\\mat{1&1/2","1&1},\\quad D=\\diag(4,1),\\quad\nB=D^{-1/2}=\\diag(1/2,1).\n\\]\nThen $BAB=I+E=\\mat{1&1/2\\\\1/2&1}$. Its exact eigenvalues are $1/2$ and $3/2$, giving condition number three.\n\\vfig{preconditioning}\n\\head{The unit circles of the energy norms}\nThe left ellipse is $x^TAx=1$. Writing $x=By$ gives the right ellipse $y^T(BAB)y=1$. The drawing samples these explicitly defined curves. The norm and eigenvalue equalities are checked symbolically.","1/2&1}$. Its exact eigenvalues are $1/2$ and $3/2$, giving condition number three.\n\\vfig{preconditioning}\n\\head{The unit circles of the energy norms}\nThe left ellipse is $x^TAx=1$. Writing $x=By$ gives the right ellipse $y^T(BAB)y=1$. The drawing samples these explicitly defined curves. The norm and eigenvalue equalities are checked symbolically.","Here $\\|E\\|_2=1/2$. The manuscript's general inequality\n\\[\n\\kappa_2(D^{-1/2}AD^{-1/2})\n\\le\\frac{1+\\eta}{1-\\eta},\\qquad \\|E\\|_2\\le\\eta<1,\n\\]\nis attained. The raw off-diagonal entry is one. Its meaningful size is one-half only after normalization by the diagonal energy scales.\n\\head{Keep the convergence input separate}\nThe condition bound can be inserted into a conjugate-gradient convergence estimate under that method's positive-definite assumptions. The CG estimate is one of the separately tracked classical inputs. The algebraic normalization theorem does not certify every stopping rule, floating-point iteration, or arbitrary preconditioner.\n\\takeaway{A small raw coupling is not scale-invariant. Normalize aga","Here $\\|E\\|_2=1/2$. The manuscript's general inequality\n\\[\n\\kappa_2(D^{-1/2}AD^{-1/2})\n\\le\\frac{1+\\eta}{1-\\eta},\\qquad \\|E\\|_2\\le\\eta<1,\n\\]\nis attained. The raw off-diagonal entry is one. Its meaningful size is one-half only after normalization by the diagonal energy scales.\n\\head{Keep the convergence input separate}\nThe condition bound can be inserted into a conjugate-gradient convergence estimate under that method's positive-definite assumptions. The CG estimate is one of the separately tracked classical inputs. The algebraic normalization theorem does not certify every stopping rule, floating-point iteration, or arbitrary preconditioner.\n\\takeaway{A small raw coupling is not scale-invariant. Normalize against the diagonal blocks before deciding whether the coupling is weak enough for a condition bound.}","\\vpage{19}{Twelve small blocks, one coherent error}{DISCOVERY AND PERTURBATION CERTIFICATES}{Chapter VIII: Theorem 8.1, Proposition 9.2.}\nSplit $\\QQ^4$ into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","N CERTIFICATES}{Chapter VIII: Theorem 8.1, Proposition 9.2.}\nSplit $\\QQ^4$ into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","s\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","Dropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient.\n\\head{Input sensitivity is additional information}\nThe packet perturbation estimate is\n\\[\n\\|(\\widehat A-A)x\\|\\le\\sum_e\\|E_e\\|\\,\\|P_{\\alpha(e)}x\\|.\n\\]\nIt can be much tighter for an input that scarcely occupies an affected source packet. To obtain a uniform operator bound, the input dependence must be bounded uniformly over unit vectors. A certificate for one input is not automatically a certificate for every right-hand side.","\\vpage{20}{Spend error where it is cheapest}{CONTINUOUS AND CLIPPED PRECISION}{Chapter VIII: Theorem 8.4, Theorem 8.5.}\nTake sensitivities $s=(1,2,4)$, cost coefficients $c=(1,1,2)$, and budget $\\tau=1$. Minimizing $\\sum_ec_e\\log(1/\\epsilon_e)$ subject to $\\sum_es_e\\epsilon_e=1$ gives\n\\[\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n\\]\n\\vfig{precision}\n\\head{The budget is weighted}\nThe three contributions to the error allowance are $(1/4,1/4,1/2)$. Equal cost coefficients give equal weighted contributions, not equal tolerances. The third edge has twice the cost coefficient of the second, which offsets its doubled sensitivity in this example.","NUOUS AND CLIPPED PRECISION}{Chapter VIII: Theorem 8.4, Theorem 8.5.}\nTake sensitivities $s=(1,2,4)$, cost coefficients $c=(1,1,2)$, and budget $\\tau=1$. Minimizing $\\sum_ec_e\\log(1/\\epsilon_e)$ subject to $\\sum_es_e\\epsilon_e=1$ gives\n\\[\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n\\]\n\\vfig{precision}\n\\head{The budget is weighted}\nThe three contributions to the error allowance are $(1/4,1/4,1/2)$. Equal cost coefficients give equal weighted contributions, not equal tolerances. The third edge has twice the cost coefficient of the second, which offsets its doubled sensitivity in this example.","Now impose upper tolerances $m=(1/10,1/2,1/2)$. The first edge must stop at its tighter cap. With multiplier $\\lambda=10/3$, the clipped solution is\n\\[\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n\\]\nIts weighted contributions are $(1/10,3/10,3/5)$ and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly.\n\\head{From tolerance to a real data type}\nThe continuous objective is a model. Actual kernels offer discrete precisions and may use different accumulation types. A tolerance below one does not identify a particular hardware format. The finite option program on the next page is the correct next step once admissible costs and error bounds have been supplied","Now impose upper tolerances $m=(1/10,1/2,1/2)$. The first edge must stop at its tighter cap. With multiplier $\\lambda=10/3$, the clipped solution is\n\\[\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n\\]\nIts weighted contributions are $(1/10,3/10,3/5)$ and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly.\n\\head{From tolerance to a real data type}\nThe continuous objective is a model. Actual kernels offer discrete precisions and may use different accumulation types. A tolerance below one does not identify a particular hardware format. The finite option program on the next page is the correct next step once admissible costs and error bounds have been supplied.\n\\takeaway{Precision allocation is an optimization over certified sensitivit","Now impose upper tolerances $m=(1/10,1/2,1/2)$. The first edge must stop at its tighter cap. With multiplier $\\lambda=10/3$, the clipped solution is\n\\[\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n\\]\nIts weighted contributions are $(1/10,3/10,3/5)$ and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly.\n\\head{From tolerance to a real data type}\nThe continuous objective is a model. Actual kernels offer discrete precisions and may use different accumulation types. A tolerance below one does not identify a particular hardware format. The finite option program on the next page is the correct next step once admissible costs and error bounds have been supplied.\n\\takeaway{Precision allocation is an optimization over certified sensitivities and costs. It does not infer error bounds from a data-type name alone.}","\\vpage{21}{An exact budget is not a greedy choice}{THREE EDGES, TWENTY-SEVEN ASSIGNMENTS}{Chapter VIII: Theorem 8.7, Remark 8.8.}\nEach edge has three allowed $(\\text{integer weight},\\text{cost})$ options:\n\\[\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),","\\vpage{21}{An exact budget is not a greedy choice}{THREE EDGES, TWENTY-SEVEN ASSIGNMENTS}{Chapter VIII: Theorem 8.7, Remark 8.8.}\nEach edge has three allowed $(\\text{integer weight},\\text{cost})$ options:\n\\[\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),\\\\\n2 &: (0,6),(1,4),(3,1),\\\\\n3 &: (0,9),(2,5),(5,1).\n\\end{aligned}\n\\]\nThere are exactly $3^3=27$ full assignments. The table stores the minimum cost at each \\emph{exact} weight. Columns zero through eight are displayed.\n\\vfig{discrete}\n\\head{Read the final row correctly}\nAt budget four, minimize across columns zero through four. The optimum costs 14: choose weights $(2,0,2)$ with costs $(3,6,5)$. The zero-weight choice on edge two is its first, more expensive option. A locally cheap choice can consume too much of the shared budget.","3 &: (0,9),(2,5),(5,1).\n\\end{aligned}\n\\]\nThere are exactly $3^3=27$ full assignments. The table stores the minimum cost at each \\emph{exact} weight. Columns zero through eight are displayed.\n\\vfig{discrete}\n\\head{Read the final row correctly}\nAt budget four, minimize across columns zero through four. The optimum costs 14: choose weights $(2,0,2)$ with costs $(3,6,5)$. The zero-weight choice on edge two is its first, more expensive option. A locally cheap choice can consume too much of the shared budget.","The checker compares the recurrence with complete enumeration for every budget from zero through twelve. Infeasible states remain $+\\infty$. They are not zero-cost states.\n\\head{Quantization must be conservative}\nSuppose real errors are $3/50$ and $2/25$, and one integer unit represents $1/10$. Rounding down gives two zero weights, even though the true sum is $7/50>1/10$. Rounding up gives a certified upper envelope instead. The budget method is exact for its integers. It cannot repair unsafe integers supplied by a preprocessing step.\n\\takeaway{The table optimizes the declared option set exactly. Accuracy depends separately on the validity and conservativeness of the option weights.}","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Chapter VIII: Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six. Splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. T","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Chapter VIII: Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six. Splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full spl","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Chapter VIII: Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six. Splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Chapter VIII: Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six. Splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","VEX FRONTIER}{Chapter VIII: Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six. Splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six. Splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","ildren. Keeping $r$ costs one with error six. Splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","ne. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","ier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","rror})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","(7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Chapter VIII: Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If e","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Chapter VIII: Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$,","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Chapter VIII: Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Chapter VIII: Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Chapter VIII: Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Chapter VIII: Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie, so the strict inequali","A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie, so the strict inequality in the stability statement matters.","a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie, so the strict inequality in the stability statement matters.","The batching statement assumes compatible backend support and independent block tasks. If several blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","backend support and independent block tasks. If several blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","ional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","x dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","Given operation capacity $P$ and bandwidth $W$, $T\\ge\\max(F/P,B/W)$ is a lower bound, not an achieved runtime. Launch overhead, imperfect overlap, and utilization can make the actual time much larger.\n\\takeaway{Grouping can remove launches while leaving the same products and data movement. Arithmetic, traffic, and scheduling are related constraints, not one interchangeable cost.}","\\vpage{25}{The first win comes after the tie}{SETUP AND REUSE IN ONE LINEAR MODEL}{Chapter VIII: Theorem 10.4, Corollary 10.5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetim","\\vpage{25}{The first win comes after the tie}{SETUP AND REUSE IN ONE LINEAR MODEL}{Chapter VIII: Theorem 10.4, Corollary 10.5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","the tie}{SETUP AND REUSE IN ONE LINEAR MODEL}{Chapter VIII: Theorem 10.4, Corollary 10.5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","\\vpage{26}{Same output before comparing times}{A BENCHMARK ROW IS A CONTRACT}{Chapter VIII: Proposition 2.9, Section 11, Section 12.}\nConsider four synthetic timing rows for a product. Row B returns factors, while row A returns an explicit matrix. A ratio $12/2=6$ does not compare two implementations of the same returned object. Adding the decoding cost changes the candidate to row C.\n\\vfig{contract_match}\n\\head{What a fair match retains}\nThe input values, dimensions, layout, source precision, correctness reference, returned output, and timing scope must agree. Row D changes the input precision. That can be a useful accuracy tradeoff, but it is not the same case merely because the output has the same shape.","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Chapter VIII: Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Chapter VIII: Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growt","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Chapter VIII: Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Chapter VIII: Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its err…","ERENCE}{Chapter VIII: Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","perators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","\\subsection*{A low rank action and its inverse}\nSet\n\\[\nD=\\diag(2,3,4,5),\\qquad\nu=(1,0,1,0)^T,\\qquad v=(1,1,0,0)^T,\n\\]\nand define $A=D+uv^T$. Its entries are shown in the figure. The same operator can be stored by its diagonal and two thin vectors rather than by every matrix entry.\nFor $x=(1,2,3,4)^T$, first form $v^Tx=3$. Then\n\\[\nDx=(2,6,12,20)^T,\\qquad u(v^Tx)=(3,0,3,0)^T,\n\\]\nso $Ax=(5,6,15,20)^T$. Only the first and third output coordinates receive the correction.","The inverse starts with the same representation. Write $y=Ax$. Since $D$ is invertible,\n\\[\nD^{-1}u=(1/2,0,1/4,0)^T,\\qquad\n1+v^TD^{-1}u=3/2\\ne0.\n\\]\nThe scalar Woodbury core is therefore invertible. Its correction coefficient is\n\\[\n\\frac{v^TD^{-1}y}{1+v^TD^{-1}u}\n=\\frac{9/2}{3/2}=3.\n\\]\nSubtracting that correction recovers the original input:\n\\[\nD^{-1}y-3D^{-1}u\n=(5/2,2,15/4,4)^T-(3/2,0,3/4,0)^T=x.\n\\]","\\begin{interpretation}[title={What the output contract preserves}]\nThe diagonal and thin vectors use twelve scalar slots, compared with sixteen for the explicit matrix. This small storage count assumes the full thin vectors are stored. It does not predict elapsed time. A compressed action returns $Ax$. Materializing all entries of $A$ produces another output and incurs its own work. The general formulas appear in \\cref{thm:exact_evaluation,thm:woodbury}.\n\\end{interpretation}","\\clearpage\n\\subsection*{A budget can select a point that no Lagrangian selects}\nConsider a rooted candidate tree. The root $r$ has children $a,b$, and $a$ has children $a_1,a_2$. Split overheads are zero. Each retained node has the following declared cost and integer error:\n\\begin{center}\n\\begin{tabular}{lrrrrr}\\toprule\nNode & $r$ & $a$ & $b$ & $a_1$ & $a_2$","midrule\n$r$ & 1 & 6","$a,b$ & 3 & 4","$a_1,a_2,b$ & 5 & 1","bottomrule\n\\end{tabular}\n\\end{center}\nFor a Lagrangian multiplier $\\lambda\\ge0$, their objectives are $1+6\\lambda$, $3+4\\lambda$, and $5+\\lambda$. The middle pruning can be optimal only if\n\\[\n3+4\\lambda\\le1+6\\lambda\\quad\\text{and}\\quad\n3+4\\lambda\\le5+\\lambda.\n\\]\nThese inequalities require $\\lambda\\ge1$ and $\\lambda\\le2/3$ simultaneously. There is no such multiplier. The root and full split exchange optimality at $\\lambda=4/5$, where both have objective $29/5$.","abular}\n\\end{center}\nFor a Lagrangian multiplier $\\lambda\\ge0$, their objectives are $1+6\\lambda$, $3+4\\lambda$, and $5+\\lambda$. The middle pruning can be optimal only if\n\\[\n3+4\\lambda\\le1+6\\lambda\\quad\\text{and}\\quad\n3+4\\lambda\\le5+\\lambda.\n\\]\nThese inequalities require $\\lambda\\ge1$ and $\\lambda\\le2/3$ simultaneously. There is no such multiplier. The root and full split exchange optimality at $\\lambda=4/5$, where both have objective $29/5$.","tiplier $\\lambda\\ge0$, their objectives are $1+6\\lambda$, $3+4\\lambda$, and $5+\\lambda$. The middle pruning can be optimal only if\n\\[\n3+4\\lambda\\le1+6\\lambda\\quad\\text{and}\\quad\n3+4\\lambda\\le5+\\lambda.\n\\]\nThese inequalities require $\\lambda\\ge1$ and $\\lambda\\le2/3$ simultaneously. There is no such multiplier. The root and full split exchange optimality at $\\lambda=4/5$, where both have objective $29/5$.","The budget problem still selects that middle pruning. Under the constraint $\\err\\le4$, the root is infeasible. The two feasible costs are three and five, so the minimum is three. Under budget one, only the full split is feasible. Under budget six, the root is cheapest. Thus the budget program adds something that changing $\\lambda$ cannot recover.","The dynamic recurrence gives the same Lagrangian choices without listing prunings. At $a$, it compares $2+4\\lambda$ with $4+\\lambda$. At $r$, it compares $1+6\\lambda$ with $1+F_\\lambda(a)$. Its exactness follows from the finite tree and additive model in \\cref{thm:tree_compiler,thm:budget_compiler}. Overlapping decompositions or nonlocal execution costs need another argument.","bottomrule\n\\end{tabular}\n\\end{center}\nThe all-fast assignment FFF has cost three and error eight. It exceeds budget three. Changing only edge one to A gives AFF, with cost $4+1+1=6$ and error $0+2+1=3$. Changing only edges two and three gives FAA, with cost seven and error five. That choice still exceeds the budget.","The exact recurrence reaches the same result. After edge one, only $(\\err,\\cost)=(5,1)$ and $(0,4)$ are reachable. After edge two, the reachable states are $(7,2)$, $(5,4)$, $(2,5)$, and $(0,7)$. Extending these with either option on edge three gives the final eight assignments. Among states of error at most three, the lowest cost is six.","This article begins with a finite direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha\n\\]\nand a complete family of coordinate projectors\n\\[\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I.\n\\]\nAn operator has packet blocks\n\\[\nA_{\\beta\\alpha}=P_\\beta AP_\\alpha.\n\\]\nThe active addresses form a directed graph. Block dimensions, block types, approximation tolerances, and output requirements turn this graph into a computational intermediate representation.","The article separates four sources of gain:\n\\[\n\\text{representation},\n\\qquad\n\\text{device},\n\\qquad\n\\text{kernel},\n\\qquad\n\\text{amortization}.\n\\]\nA single speedup number can mix all four. The benchmark protocol therefore requires case identity, output contract, timing scope, source precision, correctness reference, discovery time, compilation time, and reuse count. The archived rows do not satisfy every field of this ideal contract: source-value hashes and explicit correctness-reference fields are absent, while setup, discovery, and reuse metadata are sparse. Recorded ratios are consequently descriptive comparisons under their stated scopes, rather than fair matches with complete provenance.","\\begin{definition}[Packet layout]\\label{main:definition:001}\\label{def:packet_layout}\nA packet layout is a finite family of finite dimensional Hilbert spaces\n\\[\n\\Pcal=(X_\\alpha)_{\\alpha\\in I}\n\\]\nwith direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nWrite\n\\[\nn_\\alpha=\\dim X_\\alpha,\n\\qquad\nn=\\sum_{\\alpha\\in I}n_\\alpha.\n\\]\nThe coordinate inclusion and projection are $\\iota_\\alpha:X_\\alpha\\to X$ and $\\pi_\\alpha:X\\to X_\\alpha$. The ambient coordinate projector is $P_\\alpha=\\iota_\\alpha\\pi_\\alpha$.\n\\end{definition}","gin{definition}[Packet layout]\\label{main:definition:001}\\label{def:packet_layout}\nA packet layout is a finite family of finite dimensional Hilbert spaces\n\\[\n\\Pcal=(X_\\alpha)_{\\alpha\\in I}\n\\]\nwith direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nWrite\n\\[\nn_\\alpha=\\dim X_\\alpha,\n\\qquad\nn=\\sum_{\\alpha\\in I}n_\\alpha.\n\\]\nThe coordinate inclusion and projection are $\\iota_\\alpha:X_\\alpha\\to X$ and $\\pi_\\alpha:X\\to X_\\alpha$. The ambient coordinate projector is $P_\\alpha=\\iota_\\alpha\\pi_\\alpha$.\n\\end{definition}","def:packet_layout}\nA packet layout is a finite family of finite dimensional Hilbert spaces\n\\[\n\\Pcal=(X_\\alpha)_{\\alpha\\in I}\n\\]\nwith direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nWrite\n\\[\nn_\\alpha=\\dim X_\\alpha,\n\\qquad\nn=\\sum_{\\alpha\\in I}n_\\alpha.\n\\]\nThe coordinate inclusion and projection are $\\iota_\\alpha:X_\\alpha\\to X$ and $\\pi_\\alpha:X\\to X_\\alpha$. The ambient coordinate projector is $P_\\alpha=\\iota_\\alpha\\pi_\\alpha$.\n\\end{definition}","\\begin{definition}[Packet operator representation]\\label{main:definition:002}\\label{def:packet_ir}\nA finite packet operator representation consists of\n\\[\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain, and $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","acket operator representation]\\label{main:definition:002}\\label{def:packet_ir}\nA finite packet operator representation consists of\n\\[\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain, and $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","et_ir}\nA finite packet operator representation consists of\n\\[\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain, and $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain, and $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","B_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain, and $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","ions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain, and $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","\\begin{theorem}[Exact evaluation]\\label{main:theorem:001}\\label{thm:exact_evaluation}\nLet \\(A\\in\\End(X)\\). Set\n\\[\nB_{\\beta\\alpha}=\\pi_\\beta A\\iota_\\alpha\n\\]\nand let \\(E\\) contain exactly the nonzero blocks. Then\n\\[\n\\boxed{\nA=\n\\sum_{(\\beta,\\alpha)\\in E}\nP_\\beta AP_\\alpha.\n}\n\\]\nFor every matrix of right hand sides \\(Z\\),\n\\[\nAZ\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha Z.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:001}\nInsert the identity on both sides of \\(A\\):\n\\[\nA\n=\n\\left(\\sum_\\beta P_\\beta\\right)\nA\n\\left(\\sum_\\alpha P_\\alpha\\right)\n=\n\\sum_{\\alpha,\\beta}P_\\beta AP_\\alpha.\n\\]\nThe zero blocks can be removed. Applying the identity to \\(Z\\) gives the second formula.\n\\phantomsection\\label{main:proof:001:end}\\end{proof}","\\begin{definition}[Support graph]\\label{main:definition:003}\\label{def:support_graph}\nThe support graph \\(G_A\\) has vertex set \\(I\\). It has a directed edge\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen \\(P_\\beta AP_\\alpha\\neq0\\).\n\\end{definition}","\\begin{theorem}[Block composition]\\label{main:theorem:002}\\label{thm:block_composition}\nLet \\(A,B\\in\\End(X)\\). Then\n\\[\n\\boxed{\n(AB)_{\\gamma\\alpha}\n=\n\\sum_{\\beta\\in I}\nA_{\\gamma\\beta}B_{\\beta\\alpha}.\n}\n\\]\nConsequently,\n\\[\n\\supp(AB)\n\\subseteq\n\\supp(A)\\circ\\supp(B),\n\\]\nwhere relational composition is defined by\n\\[\n(\\gamma,\\alpha)\\in E\\circ F\n\\]\nwhen some \\(\\beta\\) satisfies\n\\[\n(\\gamma,\\beta)\\in E,\n\\qquad\n(\\beta,\\alpha)\\in F.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:002}\nInsert the packet identity between \\(A\\) and \\(B\\):\n\\[\nP_\\gamma ABP_\\alpha\n=\n\\sum_\\beta\nP_\\gamma AP_\\beta BP_\\alpha.\n\\]\nA summand can be nonzero only when both block addresses are active.\n\\phantomsection\\label{main:proof:002:end}\\end{proof}","\\begin{corollary}[Adjoint and tensor semantics]\\label{main:corollary:001}\\label{cor:adjoint_tensor}\nThe adjoint representation has blocks\n\\[\n(A^*)_{\\alpha\\beta}=A_{\\beta\\alpha}^*.\n\\]\nIf \\(A\\) and \\(B\\) are packet resolved on \\(X\\) and \\(Y\\), then\n\\[\n(A\\otimes B)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n=\nA_{\\beta\\alpha}\\otimes B_{\\delta\\gamma}.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:003}\nBoth identities follow by inserting the corresponding coordinate projectors and using\n\\[\n(P_\\beta AP_\\alpha)^*=P_\\alpha A^*P_\\beta.\n\\]\nThe tensor identity follows from\n\\[\n(P_\\beta\\otimes Q_\\delta)\n(A\\otimes B)\n(P_\\alpha\\otimes Q_\\gamma)\n=\nA_{\\beta\\alpha}\\otimes B_{\\delta\\gamma}.\n\\]\n\\phantomsection\\label{main:proof:003:end}\\end{proof}","\\begin{proposition}[Contract conversion]\\label{main:proposition:001}\\label{prop:contract_conversion}\nSuppose one algorithm returns a compressed representation \\(R\\) and another returns an explicit object \\(Y\\). A direct timing comparison is valid only if\n\\[\n\\operatorname{decode}(R)=Y\n\\]\nand the time required by \\(\\operatorname{decode}\\) is included, or if the downstream task consumes \\(R\\) directly.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:004}\nWithout decoding, the two procedures return different mathematical objects. A timing ratio between different outputs does not compare implementations of the same map. Including the decoding cost gives two procedures with the same output. If the next operation consumes \\(R\\), decoding is not part of the declared computation.\n\\phantomsection\\label{main:proof:004:end}\\end{proof}","\\begin{remark}\\label{main:remark:001}\\label{rem:semantic_boundary}\nA compressed product and an explicit matrix product may both be exact. They still have different contracts. This distinction becomes decisive when materialization costs \\(n^2\\) entries.\n\\end{remark}","\\begin{theorem}[Dense and packet application counts]\\label{main:theorem:003}\\label{thm:packet_counts}\nA dense \\(n\\) by \\(n\\) matrix applied to \\(q\\) right hand sides has modeled cost\n\\[\n\\boxed{\n\\flop_{\\rm dense}=2n^2q.\n}\n\\]\nA packet representation with active edge set \\(E\\) has modeled cost\n\\[\n\\boxed{\n\\flop_{\\rm packet}\n=\n2q\n\\sum_{(\\beta,\\alpha)\\in E}\nn_\\beta n_\\alpha.\n}\n\\]\nIt stores\n\\[\n\\boxed{\nM_{\\rm packet}\n=\n\\sum_{(\\beta,\\alpha)\\in E}\nn_\\beta n_\\alpha\n}\n\\]\nscalar entries.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:005}\nThe active block \\(B_{\\beta\\alpha}\\) is an \\(n_\\beta\\) by \\(n_\\alpha\\) matrix. Applying it to an \\(n_\\alpha\\) by \\(q\\) input costs \\(2n_\\beta n_\\alpha q\\) modeled operations and stores \\(n_\\beta n_\\alpha\\) scalars. Summation over the active addresses gives both formulas.\n\\phantomsection\\label{main:proof:005:end}\\end{proof}","\\begin{corollary}[Equal diagonal packets]\\label{main:corollary:002}\\label{cor:equal_blocks}\nLet $p\\ge1$ and $b\\ge1$ be integers and $n=pb$. Suppose $n$ is divided into $p$ equal diagonal packets of width $b$. Then\n\\[\n\\flop_{\\rm block}=2qnb,\n\\qquad\nM_{\\rm block}=nb.\n\\]\nThe following factor describes the dense-to-block storage reduction and, when $q\\ge1$, the arithmetic reduction:\n\\[\n\\boxed{\n\\frac{n}{b}.\n}\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:006}\nThere are \\(n/b\\) active diagonal blocks. Each contains \\(b^2\\) entries.\n\\phantomsection\\label{main:proof:006:end}\\end{proof}","\\begin{definition}[Block diagonal plus low rank operator]\\label{main:definition:006}\\label{def:bdlr}\nA block diagonal plus low rank operator has the form\n\\[\nA=D+UV^*,\n\\]\nwhere\n\\[\nD=\\bigoplus_{\\alpha\\in I}D_\\alpha,\n\\qquad\nU,V\\in\\CC^{n\\times r}.\n\\]\nThe abbreviation BDLR will be used below.\n\\end{definition}","\\begin{theorem}[BDLR application and storage]\\label{main:theorem:004}\\label{thm:bdlr_counts}\nFor \\(q\\) right hand sides,\n\\[\nAZ=DZ+U(V^*Z)\n\\]\nhas modeled cost\n\\[\n\\boxed{\n2q\n\\left(\n\\sum_\\alpha n_\\alpha^2+2nr\n\\right).\n}\n\\]\nThe representation stores\n\\[\n\\boxed{\n\\sum_\\alpha n_\\alpha^2+2nr\n}\n\\]\nscalars. For $n=pb$, with integers $p,b\\ge1$, the following factor describes the ideal dense storage reduction and, when $q\\ge1$, the arithmetic reduction:\n\\[\n\\boxed{\n\\frac{n}{b+2r}.\n}\n\\]\n\\end{theorem}","rem}[BDLR application and storage]\\label{main:theorem:004}\\label{thm:bdlr_counts}\nFor \\(q\\) right hand sides,\n\\[\nAZ=DZ+U(V^*Z)\n\\]\nhas modeled cost\n\\[\n\\boxed{\n2q\n\\left(\n\\sum_\\alpha n_\\alpha^2+2nr\n\\right).\n}\n\\]\nThe representation stores\n\\[\n\\boxed{\n\\sum_\\alpha n_\\alpha^2+2nr\n}\n\\]\nscalars. For $n=pb$, with integers $p,b\\ge1$, the following factor describes the ideal dense storage reduction and, when $q\\ge1$, the arithmetic reduction:\n\\[\n\\boxed{\n\\frac{n}{b+2r}.\n}\n\\]\n\\end{theorem}","abel{main:theorem:004}\\label{thm:bdlr_counts}\nFor \\(q\\) right hand sides,\n\\[\nAZ=DZ+U(V^*Z)\n\\]\nhas modeled cost\n\\[\n\\boxed{\n2q\n\\left(\n\\sum_\\alpha n_\\alpha^2+2nr\n\\right).\n}\n\\]\nThe representation stores\n\\[\n\\boxed{\n\\sum_\\alpha n_\\alpha^2+2nr\n}\n\\]\nscalars. For $n=pb$, with integers $p,b\\ge1$, the following factor describes the ideal dense storage reduction and, when $q\\ge1$, the arithmetic reduction:\n\\[\n\\boxed{\n\\frac{n}{b+2r}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:007}\nThe block diagonal product costs\n\\[\n2q\\sum_\\alpha n_\\alpha^2.\n\\]\nThe products \\(V^*Z\\) and \\(U(V^*Z)\\) cost \\(2nrq\\) each. Storage consists of the diagonal blocks and the two low rank factors. For equal packets,\n\\[\n\\sum_\\alpha n_\\alpha^2=nb.\n\\]\n\\phantomsection\\label{main:proof:007:end}\\end{proof}","\\begin{definition}[One pass streaming model]\\label{main:definition:007}\\label{def:streaming_model}\nFix a scalar size \\(s\\) bytes. For every active block, the one pass streaming model reads the block once, reads its source packet once, and reads and writes its target packet once. It does not assume cache reuse across edges.\n\\end{definition}","\\begin{proposition}[Streaming byte count]\\label{main:proposition:002}\\label{prop:streaming_bytes}\nUnder \\cref{def:streaming_model}, applying a packet operator to \\(q\\) right hand sides moves\n\\[\n\\boxed{\nB_{\\rm stream}\n=\ns\n\\sum_{(\\beta,\\alpha)\\in E}\n\\left(\n n_\\beta n_\\alpha\n +n_\\alpha q\n +2n_\\beta q\n\\right)\n}\n\\]\nbytes.\n\\end{proposition}","\\begin{definition}[Arithmetic intensity]\\label{main:definition:008}\\label{def:intensity}\nWhen the modeled byte traffic is strictly positive, the modeled arithmetic intensity is\n\\[\n\\Ical\n=\n\\frac{\\flop}{\\bytes}.\n\\]\n\\end{definition}","\\begin{theorem}[Roofline lower bound]\\label{main:theorem:005}\\label{thm:roofline}\nSuppose a device sustains at most $P>0$ modeled operations per second and at most $W>0$ bytes per second. A computation that requires \\(F\\) operations and \\(B\\) bytes satisfies\n\\[\n\\boxed{\nT\\geq\n\\max\\left\\{\n\\frac{F}{P},\n\\frac{B}{W}\n\\right\\}.\n}\n\\]\nFor positive elapsed time and positive byte traffic, its work rate is at most\n\\[\n\\min\\{P,W\\Ical\\}.\n\\]\n\\end{theorem}","heorem}[Roofline lower bound]\\label{main:theorem:005}\\label{thm:roofline}\nSuppose a device sustains at most $P>0$ modeled operations per second and at most $W>0$ bytes per second. A computation that requires \\(F\\) operations and \\(B\\) bytes satisfies\n\\[\n\\boxed{\nT\\geq\n\\max\\left\\{\n\\frac{F}{P},\n\\frac{B}{W}\n\\right\\}.\n}\n\\]\nFor positive elapsed time and positive byte traffic, its work rate is at most\n\\[\n\\min\\{P,W\\Ical\\}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:009}\nThe operation capacity gives \\(T\\geq F/P\\). The bandwidth capacity gives \\(T\\geq B/W\\). Both restrictions hold at once.\n\\phantomsection\\label{main:proof:009:end}\\end{proof}","The roofline model is a guide rather than an exact execution law \\cite{WilliamsWatermanPatterson2009}. Packet structure changes both \\(F\\) and \\(B\\). An irregular representation may remove arithmetic while lowering intensity enough to become bandwidth or launch limited.","\\begin{corollary}[Capacity threshold]\\label{main:corollary:003}\\label{cor:capacity}\nLet a device have available memory $M\\ge0$ bytes and scalar size $s>0$, and let $n\\ge0$. The matrix representation alone can fit only when\n\\[\nn\\leq\\sqrt{M/s}.\n\\]\nThe equal packet BDLR representation alone fits when\n\\[\nn(b+2r)s\\leq M.\n\\]\n\\end{corollary}","\\begin{proposition}[Materialization cost]\\label{main:proposition:003}\\label{prop:materialization_cost}\nMaterializing \\(D+UV^*\\) as an explicit matrix requires storage for \\(n^2\\) scalars and modeled arithmetic\n\\[\n2n^2r+O(n^2)\n\\]\nwhen the low rank correction is formed by dense multiplication.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:011}\nThe product \\(UV^*\\) multiplies an \\(n\\) by \\(r\\) matrix by an \\(r\\) by \\(n\\) matrix. Adding the diagonal part changes only the lower order term.\n\\phantomsection\\label{main:proof:011:end}\\end{proof}","\\begin{proposition}[Equal block banded count]\\label{main:proposition:004}\\label{prop:block_banded_count}\nLet $p,b\\ge1$ be integers, $n=pb$, and let $p=n/b$ equal packets have width $b$. Suppose a block is active precisely when\n\\[\n\\abs{i-j}\\leq w,\n\\qquad\n0\\leq w\\leq p-1.\n\\]\nThe number of active blocks is\n\\[\n\\boxed{\nN_{p,w}\n=\np(2w+1)-w(w+1).\n}\n\\]\nFor \\(q\\) right hand sides, the modeled application cost and scalar storage are\n\\[\n\\boxed{\n2qb^2N_{p,w}\n\\quad\\text{and}\\quad\nb^2N_{p,w}.\n}\n\\]\n\\end{proposition}","ck banded count]\\label{main:proposition:004}\\label{prop:block_banded_count}\nLet $p,b\\ge1$ be integers, $n=pb$, and let $p=n/b$ equal packets have width $b$. Suppose a block is active precisely when\n\\[\n\\abs{i-j}\\leq w,\n\\qquad\n0\\leq w\\leq p-1.\n\\]\nThe number of active blocks is\n\\[\n\\boxed{\nN_{p,w}\n=\np(2w+1)-w(w+1).\n}\n\\]\nFor \\(q\\) right hand sides, the modeled application cost and scalar storage are\n\\[\n\\boxed{\n2qb^2N_{p,w}\n\\quad\\text{and}\\quad\nb^2N_{p,w}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:012}\nThe diagonal contributes \\(p\\) blocks. At offset \\(k\\), where \\(1\\leq k\\leq w\\), the upper and lower block diagonals contribute \\(2(p-k)\\) blocks. Therefore\n\\[\np+2\\sum_{k=1}^{w}(p-k)\n=\np(2w+1)-w(w+1).\n\\]\nThe cost and storage follow from \\cref{thm:packet_counts}.\n\\phantomsection\\label{main:proof:012:end}\\end{proof}","\\begin{proposition}[Sparse plus low rank application]\\label{main:proposition:005}\\label{prop:sparse_low_rank}\nLet\n\\[\nA=S+UV^*,\n\\]\nwhere \\(S\\) has \\(z\\) stored scalar nonzeros and \\(U,V\\in\\CC^{n\\times r}\\). In the model that charges two operations per stored sparse entry and right hand side, applying \\(A\\) to \\(q\\) right hand sides costs\n\\[\n\\boxed{\n2q(z+2nr)\n}\n\\]\nmodeled operations and stores\n\\[\n\\boxed{\nz+2nr\n}\n\\]\nscalar values, apart from sparse index metadata.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:013}\nThe sparse product costs \\(2zq\\). The two dense thin products cost \\(2nrq\\) each. The stored values are the nonzeros of \\(S\\) and the entries of the two factors.\n\\phantomsection\\label{main:proof:013:end}\\end{proof}","\\begin{theorem}[Circulant spectral representation]\\label{main:theorem:006}\\label{thm:circulant}\nLet $n\\ge1$ be an integer and $c\\in\\CC^n$, with indices taken modulo $n$, and define\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nLet \\(F\\) be the unitary discrete Fourier matrix\n\\[\nF_{jk}\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(-\\frac{2\\pi\\ii jk}{n}\\right).\n\\]\nThen\n\\[\n\\boxed{\nC(c)\n=\nF^*\\diag(\\sqrt n\\,Fc)F.\n}\n\\]\nThe operator is determined by \\(n\\) scalars and can be applied through two Fourier transforms and one pointwise product.\n\\end{theorem}","em}[Circulant spectral representation]\\label{main:theorem:006}\\label{thm:circulant}\nLet $n\\ge1$ be an integer and $c\\in\\CC^n$, with indices taken modulo $n$, and define\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nLet \\(F\\) be the unitary discrete Fourier matrix\n\\[\nF_{jk}\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(-\\frac{2\\pi\\ii jk}{n}\\right).\n\\]\nThen\n\\[\n\\boxed{\nC(c)\n=\nF^*\\diag(\\sqrt n\\,Fc)F.\n}\n\\]\nThe operator is determined by \\(n\\) scalars and can be applied through two Fourier transforms and one pointwise product.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:014}\nFor each frequency \\(k\\), let\n\\[\nv^{(k)}_j\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(\\frac{2\\pi\\ii jk}{n}\\right).\n\\]\nThen\n\\[\n(C(c)v^{(k)})_j\n=\n\\sum_{r=0}^{n-1}c_rv^{(k)}_{j-r}\n=\n\\left(\n\\sum_{r=0}^{n-1}\nc_r\\exp\\left(-\\frac{2\\pi\\ii rk}{n}\\right)\n\\right)v^{(k)}_j.\n\\]\nThe eigenvalue is \\(\\sqrt n(Fc)_k\\). The vectors \\(v^{(k)}\\) are the columns of \\(F^*\\), which gives the formula.\n\\phantomsection\\label{main:proof:014:end}\\end{proof}","The fast Fourier transform evaluates the two transforms in \\(O(n\\log n)\\) arithmetic for the usual composite lengths \\cite{CooleyTukey1965}. This is an algorithmic factorization of the exact identity in \\cref{thm:circulant}.","\\begin{corollary}[Toeplitz circulant embedding]\\label{main:corollary:004}\\label{cor:toeplitz_embedding}\nLet $n\\ge1$ be an integer and $T\\in\\CC^{n\\times n}$ be Toeplitz, with first column\n\\[\n(c_0,c_1,\\ldots,c_{n-1})^T\n\\]\nand first row\n\\[\n(c_0,r_1,\\ldots,r_{n-1}).\n\\]\nSet\n\\[\ng\n=\n(c_0,c_1,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1)^T\n\\in\\CC^{2n}\n\\]\nand append \\(n\\) zeros to \\(x\\in\\CC^n\\). The first \\(n\\) entries of the circulant product \\(C(g)(x,0)^T\\) equal \\(Tx\\).\n\\end{corollary}","lant embedding]\\label{main:corollary:004}\\label{cor:toeplitz_embedding}\nLet $n\\ge1$ be an integer and $T\\in\\CC^{n\\times n}$ be Toeplitz, with first column\n\\[\n(c_0,c_1,\\ldots,c_{n-1})^T\n\\]\nand first row\n\\[\n(c_0,r_1,\\ldots,r_{n-1}).\n\\]\nSet\n\\[\ng\n=\n(c_0,c_1,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1)^T\n\\in\\CC^{2n}\n\\]\nand append \\(n\\) zeros to \\(x\\in\\CC^n\\). The first \\(n\\) entries of the circulant product \\(C(g)(x,0)^T\\) equal \\(Tx\\).\n\\end{corollary}","main:corollary:004}\\label{cor:toeplitz_embedding}\nLet $n\\ge1$ be an integer and $T\\in\\CC^{n\\times n}$ be Toeplitz, with first column\n\\[\n(c_0,c_1,\\ldots,c_{n-1})^T\n\\]\nand first row\n\\[\n(c_0,r_1,\\ldots,r_{n-1}).\n\\]\nSet\n\\[\ng\n=\n(c_0,c_1,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1)^T\n\\in\\CC^{2n}\n\\]\nand append \\(n\\) zeros to \\(x\\in\\CC^n\\). The first \\(n\\) entries of the circulant product \\(C(g)(x,0)^T\\) equal \\(Tx\\).\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:015}\nFor \\(0\\leq j<n\\), the circular convolution sum separates into indices \\(k\\leq j\\), where the coefficient is \\(c_{j-k}\\), and indices \\(k>j\\), where the wrapped coefficient is \\(r_{k-j}\\). The padded coordinates do not contribute. The resulting sum is exactly the \\(j\\)th Toeplitz row applied to \\(x\\).\n\\phantomsection\\label{main:proof:015:end}\\end{proof}","\\begin{theorem}[Kronecker modewise application]\\label{main:theorem:007}\\label{thm:kronecker_apply}\nLet \\(A\\in\\CC^{m\\times m}\\), \\(B\\in\\CC^{n\\times n}\\), and \\(X\\in\\CC^{n\\times m}\\). With column major vectorization,\n\\[\n\\boxed{\n(A\\otimes B)\\operatorname{vec}(X)\n=\n\\operatorname{vec}(BXA^T).\n}\n\\]\nThe right side has modeled cost\n\\[\n\\boxed{\n2n^2m+2nm^2\n}\n\\]\nand stores \\(n^2+m^2\\) factor entries instead of \\(n^2m^2\\) entries.\n\\end{theorem}","n{theorem}[Kronecker modewise application]\\label{main:theorem:007}\\label{thm:kronecker_apply}\nLet \\(A\\in\\CC^{m\\times m}\\), \\(B\\in\\CC^{n\\times n}\\), and \\(X\\in\\CC^{n\\times m}\\). With column major vectorization,\n\\[\n\\boxed{\n(A\\otimes B)\\operatorname{vec}(X)\n=\n\\operatorname{vec}(BXA^T).\n}\n\\]\nThe right side has modeled cost\n\\[\n\\boxed{\n2n^2m+2nm^2\n}\n\\]\nand stores \\(n^2+m^2\\) factor entries instead of \\(n^2m^2\\) entries.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:016}\nThe \\((i,j)\\) entry of \\(BXA^T\\) is\n\\[\n\\sum_{k=1}^{n}\\sum_{\\ell=1}^{m}\nB_{ik}X_{k\\ell}A_{j\\ell}.\n\\]\nThis is the coordinate formula for \\((A\\otimes B)\\operatorname{vec}(X)\\). Multiplying first by \\(B\\) and then by \\(A^T\\) gives the stated count.\n\\phantomsection\\label{main:proof:016:end}\\end{proof}","\\begin{corollary}[Equal tensor factors]\\label{main:corollary:005}\\label{cor:tensor_modewise}\nFor integers $k,d\\ge0$, a product of $k$ square factors of order $d$, acting on a tensor with $d^k$ entries, can be applied by $k$ modewise contractions, reusing intermediate slices, with modeled cost\n\\[\n\\boxed{\n2k d^{k+1}\n}\n\\]\nand factor storage \\(kd^2\\).\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:017}\nFor $k=0$ there are no contractions and both modeled work and factor storage are zero. For $k\\ge1$, one mode has $d^{k-1}$ fibers of length $d$. Multiplying each fiber by a \\(d\\) by \\(d\\) matrix costs \\(2d^2\\), so one mode costs \\(2d^{k+1}\\). Sum over the \\(k\\) modes.\n\\phantomsection\\label{main:proof:017:end}\\end{proof}","\\begin{theorem}[Walsh--Hadamard recursion]\\label{main:theorem:008}\\label{thm:walsh_hadamard}\nLet \\(N=2^m\\), and define\n\\[\nH_1=[1],\n\\qquad\nH_{2N}\n=\n\\begin{bmatrix}\nH_N&H_N","\\begin{theorem}[Walsh--Hadamard recursion]\\label{main:theorem:008}\\label{thm:walsh_hadamard}\nLet \\(N=2^m\\), and define\n\\[\nH_1=[1],\n\\qquad\nH_{2N}\n=\n\\begin{bmatrix}\nH_N&H_N\\\\\nH_N&-H_N\n\\end{bmatrix}.\n\\]\nThen\n\\[\nH_NH_N^T=NI.\n\\]\nThe product \\(H_Nx\\) can be evaluated in exactly \\(N\\log_2N\\) additions or subtractions through the recursion.\n\\end{theorem}","H_N&-H_N\n\\end{bmatrix}.\n\\]\nThen\n\\[\nH_NH_N^T=NI.\n\\]\nThe product \\(H_Nx\\) can be evaluated in exactly \\(N\\log_2N\\) additions or subtractions through the recursion.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:018}\nThe orthogonality identity follows by induction from block multiplication. One recursive stage forms \\(x_0+x_1\\) and \\(x_0-x_1\\), which uses \\(N\\) additions or subtractions, and then applies two transforms of size \\(N/2\\). Thus\n\\[\nT(N)=2T(N/2)+N,\n\\qquad\nT(1)=0,\n\\]\nwhose solution is \\(T(N)=N\\log_2N\\).\n\\phantomsection\\label{main:proof:018:end}\\end{proof}","\\begin{theorem}[Exact BDLR product]\\label{main:theorem:009}\\label{thm:bdlr_product}\nLet\n\\[\nA=D_A+U_AV_A^*,\n\\qquad\nB=D_B+U_BV_B^*\n\\]\nuse the same diagonal packet layout. Then\n\\[\nAB=D_AD_B+U_CV_C^*,\n\\]\nwhere\n\\[\nU_C\n=\n\\begin{bmatrix}\nD_AU_B+U_A(V_A^*U_B)&U_A\n\\end{bmatrix}\n\\]\nand\n\\[\nV_C\n=\n\\begin{bmatrix}\nV_B&D_B^*V_A\n\\end{bmatrix}.\n\\]\nThe unrecompressed correction rank is at most \\(r_A+r_B\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:019}\nExpand the product:\n\\[\nAB\n=\nD_AD_B\n+D_AU_BV_B^*\n+U_AV_A^*D_B\n+U_A(V_A^*U_B)V_B^*.\n\\]\nThe second and fourth terms share the right factor \\(V_B^*\\). The third term equals\n\\[\nU_A(D_B^*V_A)^*.\n\\]\nConcatenation gives the stated factors.\n\\phantomsection\\label{main:proof:019:end}\\end{proof}","\\begin{corollary}[Rank growth in a product chain]\\label{main:corollary:006}\\label{cor:rank_growth}\nLet\n\\[\nA_j=D_j+U_jV_j^*,\n\\qquad\nj=1,\\ldots,m,\n\\]\nshare one diagonal packet layout. Before recompression, the product\n\\[\nA_m\\cdots A_1\n\\]\nhas a BDLR representation whose correction rank is at most\n\\[\n\\boxed{\n\\sum_{j=1}^m r_j.\n}\n\\]\n\\end{corollary}","\\begin{theorem}[Woodbury inversion]\\label{main:theorem:010}\\label{thm:woodbury}\nAssume \\(D\\) and\n\\[\nS=I+V^*D^{-1}U\n\\]\nare invertible. Then\n\\[\n\\boxed{\n(D+UV^*)^{-1}\n=\nD^{-1}\n-D^{-1}US^{-1}V^*D^{-1}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:021}\nMultiply the proposed inverse by \\(D+UV^*\\). The coefficient of \\(V^*D^{-1}\\) is\n\\[\nU-US^{-1}(I+V^*D^{-1}U)=0.\n\\]\nThe remaining term is the identity. The right inverse calculation is the same.\n\\phantomsection\\label{main:proof:021:end}\\end{proof}","The formula and its history are discussed in \\cite{Hager1989}. It reduces one large solve to packet solves with \\(D\\) and one dense solve with the small matrix \\(S\\).","\\begin{corollary}[Inverse norm bound]\\label{main:corollary:007}\\label{cor:inverse_norm}\nUnder the hypotheses of \\cref{thm:woodbury},\n\\[\n\\norm{(D+UV^*)^{-1}}\n\\leq\n\\norm{D^{-1}}\n+\n\\norm{D^{-1}U}\n\\norm{S^{-1}}\n\\norm{V^*D^{-1}}.\n\\]\n\\end{corollary}","\\begin{standardfact}[Eckart--Young--Mirsky theorem]\\label{main:standardfact:001}\\label{fact:eym}\nIf \\(M\\) has singular values\n\\[\n\\sigma_1\\geq\\sigma_2\\geq\\cdots,\n\\]\npad this finite sequence with zeros. For each integer $k\\ge0$, its truncated singular value decomposition $M_k$ is a best rank-at-most-$k$ approximation. It satisfies\n\\[\n\\norm{M-M_k}_2=\\sigma_{k+1}\n\\]\nand\n\\[\n\\norm{M-M_k}_{\\rm F}\n=\n\\left(\n\\sum_{j>k}\\sigma_j^2\n\\right)^{1/2}.\n\\]\n\\end{standardfact}","This is classical \\cite{EckartYoung1936,Mirsky1960}. The next theorem shows that an \\(n\\) dimensional singular value decomposition is unnecessary when a correction is already factored.","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equa","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduce","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero si","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular direc","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","mall_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","s\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","R=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","and let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions. Tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:023}\nThe exact factor product is\n\\[\nLR^*\n=\nQ_LGQ_R^*.\n\\]\nThe two outer maps are isometries on their column spaces. They preserve the nonzero singular values and the spectral and Frobenius norms of the core. For every ambient rank-at-most-$k$ competitor $B$, its compression $Q_L^*BQ_R$ has rank at most $k$, and compression cannot increase either norm. Thus the core best-approximation inequality also bounds the error of every ambient competitor from below. Truncating the singular value decomposition of \\(G\\) and applying \\cref{fact:eym} gives the result. The compact QR dimensions equal the ranks of the factor column spaces.\n\\phantomsection\\label{main:proof:023:end}\\end{proof}","paces. They preserve the nonzero singular values and the spectral and Frobenius norms of the core. For every ambient rank-at-most-$k$ competitor $B$, its compression $Q_L^*BQ_R$ has rank at most $k$, and compression cannot increase either norm. Thus the core best-approximation inequality also bounds the error of every ambient competitor from below. Truncating the singular value decomposition of \\(G\\) and applying \\cref{fact:eym} gives the result. The compact QR dimensions equal the ranks of the factor column spaces.\n\\phantomsection\\label{main:proof:023:end}\\end{proof}","\\begin{theorem}[Product chain perturbation]\\label{main:theorem:012}\\label{thm:product_chain_error}\nLet \\(A_1,\\ldots,A_m\\) be exact factors and let\n\\[\n\\widehat A_j=A_j+E_j.\n\\]\nThen\n\\[\n\\boxed{\n\\left\\|\n\\widehat A_m\\cdots\\widehat A_1\n-\nA_m\\cdots A_1\n\\right\\|\n\\leq\n\\sum_{j=1}^m\n\\left(\n\\prod_{i=j+1}^m\\norm{\\widehat A_i}\n\\right)\n\\norm{E_j}\n\\left(\n\\prod_{i=1}^{j-1}\\norm{A_i}\n\\right).\n}\n\\]\nEmpty products are one.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:024}\nUse the telescoping identity\n\\[\n\\widehat A_m\\cdots\\widehat A_1\n-A_m\\cdots A_1\n=\n\\sum_{j=1}^m\n\\widehat A_m\\cdots\\widehat A_{j+1}\nE_j\nA_{j-1}\\cdots A_1.\n\\]\nTake norms term by term.\n\\phantomsection\\label{main:proof:024:end}\\end{proof}","\\begin{theorem}[Certified repeated recompression]\\label{main:theorem:013}\\label{cor:recompression_chain}\nLet\n\\[\nP_0=I,\n\\qquad\nP_j=A_jP_{j-1},\n\\]\nbe exact partial products. Suppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","n{theorem}[Certified repeated recompression]\\label{main:theorem:013}\\label{cor:recompression_chain}\nLet\n\\[\nP_0=I,\n\\qquad\nP_j=A_jP_{j-1},\n\\]\nbe exact partial products. Suppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","ad\nP_j=A_jP_{j-1},\n\\]\nbe exact partial products. Suppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","uppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:025}\nSet \\(D_j=\\widehat P_j-P_j\\). Then\n\\[\nD_j=A_jD_{j-1}+E_j,\n\\qquad\nD_0=0.\n\\]\nRepeated substitution gives\n\\[\nD_m\n=\n\\sum_{j=1}^{m}\nA_mA_{m-1}\\cdots A_{j+1}E_j.\n\\]\nTaking norms yields the result.\n\\phantomsection\\label{main:proof:025:end}\\end{proof}","\\begin{theorem}[Path support of powers]\\label{main:theorem:014}\\label{thm:path_support}\nFor every positive integer \\(m\\),\n\\[\nP_\\gamma A^mP_\\alpha\n\\]\nis the sum over directed walks (vertices may repeat)\n\\[\n\\alpha=\\alpha_0\n\\longrightarrow\n\\alpha_1\n\\longrightarrow\\cdots\\longrightarrow\n\\alpha_m=\\gamma\n\\]\nin \\(G_A\\), with contribution\n\\[\nA_{\\alpha_m\\alpha_{m-1}}\n\\cdots\nA_{\\alpha_1\\alpha_0}.\n\\]\nIf no such path exists, the block vanishes.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:026}\nInsert the packet identity between every two consecutive factors of \\(A\\). Every nonzero term determines one path of length \\(m\\).\n\\phantomsection\\label{main:proof:026:end}\\end{proof}","\\begin{corollary}[Polynomial support]\\label{main:corollary:008}\\label{cor:polynomial_support}\nIf\n\\[\np(z)=\\sum_{j=0}^dc_jz^j,\n\\]\nthen\n\\[\n\\supp(p(A))\n\\]\nis contained in the union of the support relations of \\(A^j\\) for \\(0\\leq j\\leq d\\).\n\\end{corollary}","\\begin{corollary}[Acyclic transfer nilpotence]\\label{main:corollary:009}\\label{cor:acyclic_nilpotence}\nLet \\(K\\) contain only strict transfer blocks. If its support graph is acyclic and its longest directed path has \\(h\\) edges, then\n\\[\nK^{h+1}=0.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:028}\nThere is no directed path of length \\(h+1\\). Every block of \\(K^{h+1}\\) therefore vanishes by \\cref{thm:path_support}.\n\\phantomsection\\label{main:proof:028:end}\\end{proof}","\\begin{definition}[Topological level]\\label{main:definition:009}\\label{def:topological_level}\nLet \\(G\\) be a directed acyclic graph. The first level contains its source vertices. Remove them. The next level contains the new sources. Continue until every vertex has been assigned.\n\\end{definition}","\\begin{theorem}[Level schedule for a triangular solve]\\label{main:theorem:015}\\label{thm:level_schedule}\nLet \\(A=D+K\\) be block lower triangular after a topological ordering. Assume every diagonal block \\(D_\\alpha\\) is invertible. The solve\n\\[\nAx=b\n\\]\ncan be completed in one sequential stage per topological level. Blocks in the same level may be solved independently after all earlier levels are known.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:029}\nThe equation in packet \\(\\beta\\) is\n\\[\nD_\\beta x_\\beta\n=\nb_\\beta\n-\n\\sum_{\\alpha\\to\\beta}K_{\\beta\\alpha}x_\\alpha.\n\\]\nEvery predecessor of \\(\\beta\\) lies in an earlier level. Once those values are available, \\(x_\\beta\\) is obtained from the local solve with \\(D_\\beta\\). Vertices in the same level have no directed edge between them, so their solves are independent.\n\\phantomsection\\label{main:proof:029:end}\\end{proof}","\\begin{theorem}[Inverse support lies in the transitive closure]\\label{main:theorem:016}\\label{thm:inverse_support}\nLet \\(A\\) be an invertible packet operator. Reachability here includes a zero-length path from a packet to itself. If there is no directed path from packet \\(\\alpha\\) to packet \\(\\beta\\) in \\(G_A\\), then\n\\[\n\\boxed{\nP_\\beta A^{-1}P_\\alpha=0.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:031}\nLet \\(R\\) be the set of packets reachable from \\(\\alpha\\), including \\(\\alpha\\), and set\n\\[\nX_R=\\bigoplus_{\\gamma\\in R}X_\\gamma.\n\\]\nNo active edge leaves \\(R\\), so \\(AX_R\\subseteq X_R\\). The restriction of \\(A\\) to the finite dimensional space \\(X_R\\) is injective. It is therefore onto \\(X_R\\). Hence \\(A^{-1}X_R\\subseteq X_R\\). Since \\(\\beta\\notin R\\), the stated block of \\(A^{-1}\\) vanishes.\n\\phantomsection\\label{main:proof:031:end}\\end{proof}","\\label{main:proof:031}\nLet \\(R\\) be the set of packets reachable from \\(\\alpha\\), including \\(\\alpha\\), and set\n\\[\nX_R=\\bigoplus_{\\gamma\\in R}X_\\gamma.\n\\]\nNo active edge leaves \\(R\\), so \\(AX_R\\subseteq X_R\\). The restriction of \\(A\\) to the finite dimensional space \\(X_R\\) is injective. It is therefore onto \\(X_R\\). Hence \\(A^{-1}X_R\\subseteq X_R\\). Since \\(\\beta\\notin R\\), the stated block of \\(A^{-1}\\) vanishes.\n\\phantomsection\\label{main:proof:031:end}\\end{proof}","\\begin{proposition}[Schur support inclusion]\\label{main:proposition:006}\\label{prop:schur_fill}\nSplit the packets into retained indices \\(R\\) and hidden indices \\(H\\). Suppose the hidden block \\(A_{HH}\\) is invertible. The retained Schur complement is\n\\[\nS\n=\nA_{RR}\n-\nA_{RH}A_{HH}^{-1}A_{HR}.\n\\]\nA retained block $({\\beta},{\\alpha})$ can be active only if it was already active in $A_{RR}$, or if there is a directed walk from $\\alpha$ through hidden packets to $\\beta$. The internal hidden path may have length zero, so entrance and exit may use the same hidden packet.\n\\end{proposition}","pport inclusion]\\label{main:proposition:006}\\label{prop:schur_fill}\nSplit the packets into retained indices \\(R\\) and hidden indices \\(H\\). Suppose the hidden block \\(A_{HH}\\) is invertible. The retained Schur complement is\n\\[\nS\n=\nA_{RR}\n-\nA_{RH}A_{HH}^{-1}A_{HR}.\n\\]\nA retained block $({\\beta},{\\alpha})$ can be active only if it was already active in $A_{RR}$, or if there is a directed walk from $\\alpha$ through hidden packets to $\\beta$. The internal hidden path may have length zero, so entrance and exit may use the same hidden packet.\n\\end{proposition}","tion:006}\\label{prop:schur_fill}\nSplit the packets into retained indices \\(R\\) and hidden indices \\(H\\). Suppose the hidden block \\(A_{HH}\\) is invertible. The retained Schur complement is\n\\[\nS\n=\nA_{RR}\n-\nA_{RH}A_{HH}^{-1}A_{HR}.\n\\]\nA retained block $({\\beta},{\\alpha})$ can be active only if it was already active in $A_{RR}$, or if there is a directed walk from $\\alpha$ through hidden packets to $\\beta$. The internal hidden path may have length zero, so entrance and exit may use the same hidden packet.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:032}\nThe correction block is\n\\[\nP_\\beta A_{RH}A_{HH}^{-1}A_{HR}P_\\alpha.\n\\]\nIt can be nonzero only when \\(A_{HR}P_\\alpha\\) enters a hidden packet, the inverse hidden operator connects hidden packets, and \\(P_\\beta A_{RH}\\) exits to \\(\\beta\\). By \\cref{thm:inverse_support}, the hidden inverse can connect only packets joined by a hidden directed path. Cancellation may still remove an allowed block.\n\\phantomsection\\label{main:proof:032:end}\\end{proof}","\\begin{definition}[Krylov space]\\label{main:definition:010}\\label{def:krylov}\nFor a linear operator \\(A\\), a vector \\(v\\), and a positive integer \\(m\\), define\n\\[\n\\Kcal_m(A,v)\n=\n\\operatorname{span}\n\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\n\\end{definition}","\\begin{proposition}[Invariant packet Krylov decomposition]\\label{main:proposition:007}\\label{prop:krylov_invariant}\nAssume\n\\[\n[A,P_\\alpha]=0\n\\]\nfor every packet. If\n\\[\nv=\\sum_\\alpha v_\\alpha,\n\\qquad\nv_\\alpha=P_\\alpha v,\n\\]\nthen\n\\[\nA^jv\n=\n\\sum_\\alpha A_\\alpha^jv_\\alpha\n\\]\nfor every \\(j\\geq0\\), where\n\\[\nA_\\alpha=P_\\alpha AP_\\alpha.\n\\]\nHence\n\\[\n\\Kcal_m(A,v)\n\\subseteq\n\\bigoplus_\\alpha\n\\Kcal_m(A_\\alpha,v_\\alpha).\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:033}\nCommutation gives\n\\[\nA^jP_\\alpha=P_\\alpha A^j.\n\\]\nApply this identity to each packet component of \\(v\\).\n\\phantomsection\\label{main:proof:033:end}\\end{proof}","\\begin{theorem}[Approximate packet Krylov bound]\\label{main:theorem:017}\\label{thm:approx_krylov}\nWrite\n\\[\nA=D+K,\n\\]\nwhere \\(D\\) is packet diagonal. For every positive integer \\(j\\),\n\\[\n\\boxed{\n\\norm{A^j-D^j}\n\\leq\njM^{j-1}\\norm{K},\n}\n\\]\nwhere\n\\[\nM=\\max\\{\\norm{A},\\norm{D}\\}.\n\\]\nConsequently,\n\\[\n\\dist(A^jv,\\bigoplus_\\alpha\\Kcal_{j+1}(D_\\alpha,v_\\alpha))\n\\leq\njM^{j-1}\\norm{K}\\norm{v}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:034}\nUse the telescoping identity\n\\[\nA^j-D^j\n=\n\\sum_{k=0}^{j-1}\nA^{j-1-k}KD^k.\n\\]\nEach term has norm at most \\(M^{j-1}\\norm{K}\\). The vector \\(D^jv\\) lies in the direct sum of local Krylov spaces.\n\\phantomsection\\label{main:proof:034:end}\\end{proof}","\\begin{theorem}[Block Jacobi condition bound]\\label{main:theorem:018}\\label{thm:block_jacobi}\nLet\n\\[\nA=D+K\n\\]\nact on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part $D$. Here $D^{-1/2}$ is the actual inverse of the positive square root of $D$. Suppose\n\\[\nE=D^{-1/2}KD^{-1/2}\n\\]\nis Hermitian and\n\\[\n\\norm{E}_2\\leq\\eta<1.\n\\]\nThen\n\\[\n\\boxed{\n\\cond_2(D^{-1/2}AD^{-1/2})\n\\leq\n\\frac{1+\\eta}{1-\\eta}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:035}\nFor every unit vector \\(x\\),\n\\[\n\\abs{x^*Ex}\\leq\\eta.\n\\]\nThe Rayleigh quotient of \\(I+E\\) lies in \\([1-\\eta,1+\\eta]\\). The quotient of the extreme eigenvalue bounds gives the result.\n\\phantomsection\\label{main:proof:035:end}\\end{proof}","\\begin{standardfact}[Preconditioned conjugate gradient convergence]\\label{main:standardfact:002}\\label{fact:cg}\nLet $A$ and $M$ be Hermitian positive definite on a nonempty finite coordinate space, and set\n\\[\n\\kappa\n=\n\\cond_2(M^{-1/2}AM^{-1/2}).\n\\]\nIn exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies\n\\[\n\\norm{x_m-x_*}_A\n\\leq\n2\n\\left(\n\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}\n\\right)^m\n\\norm{x_0-x_*}_A.\n\\]\n\\end{standardfact}","\\begin{corollary}[Certified packet preconditioning]\\label{main:corollary:011}\\label{cor:certified_pcg}\nUnder the hypotheses of \\cref{thm:block_jacobi}, the convergence factor in \\cref{fact:cg} is bounded by the value obtained from\n\\[\n\\kappa\n=\n\\frac{1+\\eta}{1-\\eta}.\n\\]\n\\end{corollary}","\\begin{theorem}[Packet perturbation certificate]\\label{main:theorem:019}\\label{thm:packet_error}\nLet\n\\[\nA=\n\\sum_{e\\in E}A_e,\n\\qquad\n\\widehat A\n=\n\\sum_{e\\in E}(A_e+E_e),\n\\]\nwhere edge \\(e\\) starts at packet \\(\\alpha(e)\\), and assume\n\\[\nA_e=A_eP_{\\alpha(e)},\n\\qquad\nE_e=E_eP_{\\alpha(e)}.\n\\]\nThen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\norm{E_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\nIf\n\\[\n\\norm{E_e}\n\\leq\n\\eps_e\\norm{A_e},\n\\]\nthen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\eps_e\\norm{A_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\n\\end{theorem}","certificate]\\label{main:theorem:019}\\label{thm:packet_error}\nLet\n\\[\nA=\n\\sum_{e\\in E}A_e,\n\\qquad\n\\widehat A\n=\n\\sum_{e\\in E}(A_e+E_e),\n\\]\nwhere edge \\(e\\) starts at packet \\(\\alpha(e)\\), and assume\n\\[\nA_e=A_eP_{\\alpha(e)},\n\\qquad\nE_e=E_eP_{\\alpha(e)}.\n\\]\nThen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\norm{E_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\nIf\n\\[\n\\norm{E_e}\n\\leq\n\\eps_e\\norm{A_e},\n\\]\nthen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\eps_e\\norm{A_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:037}\nExact packet evaluation gives\n\\[\n(\\widehat A-A)x\n=\n\\sum_eE_eP_{\\alpha(e)}x.\n\\]\nApply the triangle inequality and the operator norm bound to each term.\n\\phantomsection\\label{main:proof:037:end}\\end{proof}","\\begin{standardfact}[Floating point matrix product model]\\label{main:standardfact:003}\\label{fact:rounding}\nFor real $A\\in\\RR^{m\\times p}$ and $B\\in\\RR^{p\\times n}$, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff $u$, no overflow or underflow, and $pu<1$. Its computed output $\\widehat C$ satisfies the componentwise forward-error bound\n\\[\n|\\widehat C-AB|\\le\\gamma_p|A||B|,\n\\qquad\\gamma_p=\\frac{pu}{1-pu}.\n\\]\nAbsolute values and the inequality are entrywise. Complex arithmetic, fused kernels, mixed accumulation, and tensor kernels require their own stated constants. This fact does not assert one common small input perturbation for every column, and an input-dependent rounded action is not automatically a fixed linear operator.\n\\end{standardfact}","\\begin{corollary}[Mixed precision product chain]\\label{main:corollary:012}\\label{cor:mixed_chain}\nIf each fixed linear factor $A_j$ is replaced by a fixed packetwise linear approximation $\\widehat A_j$, write $E_j=\\widehat A_j-A_j=\\sum_e\\Delta_{j,e}P_{\\alpha(j,e)}$. A uniform certificate is\n\\[\n\\|E_j\\|\\le\\delta_j:=\\sum_e\\|\\Delta_{j,e}\\|\\,\\|P_{\\alpha(j,e)}\\|.\n\\]\nFor orthogonal packet projectors, $\\delta_j\\le\\sum_e\\|\\Delta_{j,e}\\|$. The final operator error is bounded by \\cref{thm:product_chain_error} with these uniform $\\delta_j$. The corresponding vector error is this operator bound multiplied by the input norm.\n\\end{corollary}","r $A_j$ is replaced by a fixed packetwise linear approximation $\\widehat A_j$, write $E_j=\\widehat A_j-A_j=\\sum_e\\Delta_{j,e}P_{\\alpha(j,e)}$. A uniform certificate is\n\\[\n\\|E_j\\|\\le\\delta_j:=\\sum_e\\|\\Delta_{j,e}\\|\\,\\|P_{\\alpha(j,e)}\\|.\n\\]\nFor orthogonal packet projectors, $\\delta_j\\le\\sum_e\\|\\Delta_{j,e}\\|$. The final operator error is bounded by \\cref{thm:product_chain_error} with these uniform $\\delta_j$. The corresponding vector error is this operator bound multiplied by the input norm.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:038}\nTake the supremum over unit inputs in \\cref{thm:packet_error}, using $\\|P_{\\alpha}x\\|\\le\\|P_{\\alpha}\\|\\|x\\|$. Substitute the resulting uniform operator error bounds into \\cref{thm:product_chain_error}. The action estimate follows from submultiplicativity.\n\\phantomsection\\label{main:proof:038:end}\\end{proof}","\\begin{theorem}[Continuous precision allocation]\\label{main:theorem:020}\\label{thm:continuous_precision}\nLet $E$ be a finite nonempty edge set, let $s_e>0$ be sensitivities and $c_e>0$ be costs, and let $\\tau>0$. Consider\n\\[\n\\min_{\\eps_e>0}\n\\sum_e c_e\\log\\frac1{\\eps_e}\n\\]\nsubject to\n\\[\n\\sum_es_e\\eps_e=\\tau.\n\\]\nThe unique minimizer is\n\\[\n\\boxed{\n\\eps_e\n=\n\\frac{\\tau c_e}{s_e\\sum_fc_f}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:039}\nThe Lagrangian derivative gives\n\\[\n-\\frac{c_e}{\\eps_e}+\\lambda s_e=0.\n\\]\nThus\n\\[\n\\eps_e=\\frac{c_e}{\\lambda s_e}.\n\\]\nThe constraint gives\n\\[\n\\lambda=\\frac{\\sum_fc_f}{\\tau}.\n\\]\nThe objective is strictly convex on the positive orthant, so the stationary point is the unique minimizer.\n\\phantomsection\\label{main:proof:039:end}\\end{proof}","\\begin{theorem}[Clipped precision allocation]\\label{main:theorem:021}\\label{thm:clipped_precision}\nRetain the finite nonempty edge set, strictly positive sensitivities and costs, and add strictly positive caps $m_e>0$ with upper bounds\n\\[\n0<\\eps_e\\leq m_e.\n\\]\nAssume\n\\[\n0<\\tau\\leq\\sum_es_em_e.\n\\]\nThere is a unique \\(\\lambda>0\\), except at the maximal tolerance endpoint, such that\n\\[\n\\boxed{\n\\eps_e\n=\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n}\n\\]\nand\n\\[\n\\sum_es_e\\eps_e=\\tau.\n\\]\nThis vector minimizes the same objective over the clipped feasible set.\n\\end{theorem}","m:021}\\label{thm:clipped_precision}\nRetain the finite nonempty edge set, strictly positive sensitivities and costs, and add strictly positive caps $m_e>0$ with upper bounds\n\\[\n0<\\eps_e\\leq m_e.\n\\]\nAssume\n\\[\n0<\\tau\\leq\\sum_es_em_e.\n\\]\nThere is a unique \\(\\lambda>0\\), except at the maximal tolerance endpoint, such that\n\\[\n\\boxed{\n\\eps_e\n=\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n}\n\\]\nand\n\\[\n\\sum_es_e\\eps_e=\\tau.\n\\]\nThis vector minimizes the same objective over the clipped feasible set.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:040}\nThe Karush--Kuhn--Tucker conditions give the unconstrained form on every nonactive coordinate and equality with \\(m_e\\) on every active upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","rush--Kuhn--Tucker conditions give the unconstrained form on every nonactive coordinate and equality with \\(m_e\\) on every active upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","e\\) on every active upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","\\begin{definition}[Discrete precision option]\\label{main:definition:011}\\label{def:dtype_option}\nFor edge \\(e\\), a discrete precision option \\(k\\) has a measured or modeled cost \\(c_{e,k}\\) and a nonnegative integer error weight \\(w_{e,k}\\). Exactly one option must be selected on every edge.\n\\end{definition}","\\begin{theorem}[Exact discrete precision program]\\label{main:theorem:022}\\label{thm:discrete_precision}\nLet the total integer error budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","ct discrete precision program]\\label{main:theorem:022}\\label{thm:discrete_precision}\nLet the total integer error budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","total integer error budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","rror budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:041}\nEvery feasible assignment for the first \\(j\\) edges ends with one option \\(k\\) on edge \\(j\\). Removing it leaves an optimal or suboptimal assignment for the first \\(j-1\\) edges with residual weight \\(b-w_{j,k}\\). Replacing a suboptimal prefix by an optimal one can only lower cost. The recurrence therefore has optimal substructure. Induction on \\(j\\) proves the result.\n\\phantomsection\\label{main:proof:041:end}\\end{proof}","\\begin{proposition}[Exact block discovery certificate]\\label{main:proposition:008}\\label{prop:block_discovery_certificate}\nFor an orthogonal packet system \\((P_\\alpha)\\), define the block diagonal compression\n\\[\n\\Pi_{\\Pcal}(A)\n=\n\\sum_\\alpha P_\\alpha AP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{\n\\norm{A-\\Pi_{\\Pcal}(A)}_{\\rm F}^2\n=\n\\sum_{\\alpha\\neq\\beta}\n\\norm{P_\\beta AP_\\alpha}_{\\rm F}^2.\n}\n\\]\nIn operator norm,\n\\[\n\\boxed{\n\\norm{A-\\Pi_{\\Pcal}(A)}_2\n\\leq\n\\sum_{\\alpha\\neq\\beta}\n\\norm{P_\\beta AP_\\alpha}_2.\n}\n\\]\nThus any proposed packet discovery has an exact finite residual certificate after its projectors are known.\n\\end{proposition}","\\begin{definition}[Candidate representation tree]\\label{main:definition:012}\\label{def:candidate_tree}\nA candidate representation tree is a finite rooted tree. Every node \\(v\\) has\n\\[\n\\kappa_v\\geq0,\n\\qquad\n\\epsilon_v\\geq0,\n\\]\nrepresenting the cost and error of keeping \\(v\\) as one leaf. Every internal node also has a split overhead \\(\\sigma_v\\geq0\\). An admissible pruning contains exactly one selected node on every root to terminal path.\n\\end{definition}","The cost and error of a pruning \\(\\Lcal\\) are\n\\[\n\\cost(\\Lcal)\n=\n\\sum_{v\\in\\Lcal}\\kappa_v\n+\n\\sum_{u\\text{ split}}\\sigma_u\n\\]\nand\n\\[\n\\err(\\Lcal)=\\sum_{v\\in\\Lcal}\\epsilon_v.\n\\]\nThe model can include arithmetic, storage, launch, transfer, and precision costs inside \\(\\kappa_v\\). It remains exact only when the declared totals are additive.","\\begin{theorem}[Exact Lagrangian packet compiler]\\label{main:theorem:023}\\label{thm:tree_compiler}\nFix \\(\\lambda\\geq0\\) and define\n\\[\nJ_\\lambda(\\Lcal)\n=\n\\cost(\\Lcal)\n+\n\\lambda\\err(\\Lcal).\n\\]\nFor every node \\(v\\), set\n\\[\nK_\\lambda(v)\n=\n\\kappa_v+\\lambda\\epsilon_v\n\\]\nand\n\\[\nS_\\lambda(v)\n=\n\\sigma_v\n+\n\\sum_{w\\in\\operatorname{child}(v)}F_\\lambda(w).\n\\]\nFor a terminal node, set\n\\[\nF_\\lambda(v)=K_\\lambda(v).\n\\]\nFor an internal node, the recurrence\n\\[\n\\boxed{\nF_\\lambda(v)\n=\n\\min\\{K_\\lambda(v),S_\\lambda(v)\\}\n}\n\\]\ncomputes the exact minimum Lagrangian objective over every admissible pruning of the subtree rooted at \\(v\\).\n\\end{theorem}","er]\\label{main:theorem:023}\\label{thm:tree_compiler}\nFix \\(\\lambda\\geq0\\) and define\n\\[\nJ_\\lambda(\\Lcal)\n=\n\\cost(\\Lcal)\n+\n\\lambda\\err(\\Lcal).\n\\]\nFor every node \\(v\\), set\n\\[\nK_\\lambda(v)\n=\n\\kappa_v+\\lambda\\epsilon_v\n\\]\nand\n\\[\nS_\\lambda(v)\n=\n\\sigma_v\n+\n\\sum_{w\\in\\operatorname{child}(v)}F_\\lambda(w).\n\\]\nFor a terminal node, set\n\\[\nF_\\lambda(v)=K_\\lambda(v).\n\\]\nFor an internal node, the recurrence\n\\[\n\\boxed{\nF_\\lambda(v)\n=\n\\min\\{K_\\lambda(v),S_\\lambda(v)\\}\n}\n\\]\ncomputes the exact minimum Lagrangian objective over every admissible pruning of the subtree rooted at \\(v\\).\n\\end{theorem}","et\n\\[\nK_\\lambda(v)\n=\n\\kappa_v+\\lambda\\epsilon_v\n\\]\nand\n\\[\nS_\\lambda(v)\n=\n\\sigma_v\n+\n\\sum_{w\\in\\operatorname{child}(v)}F_\\lambda(w).\n\\]\nFor a terminal node, set\n\\[\nF_\\lambda(v)=K_\\lambda(v).\n\\]\nFor an internal node, the recurrence\n\\[\n\\boxed{\nF_\\lambda(v)\n=\n\\min\\{K_\\lambda(v),S_\\lambda(v)\\}\n}\n\\]\ncomputes the exact minimum Lagrangian objective over every admissible pruning of the subtree rooted at \\(v\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:043}\nA pruning of the subtree rooted at \\(v\\) has two possible forms. It keeps \\(v\\), in which case its objective is \\(K_\\lambda(v)\\). Otherwise it splits \\(v\\) and independently chooses an admissible pruning in every child subtree. By induction, the optimal child values are \\(F_\\lambda(w)\\). The split objective is therefore \\(S_\\lambda(v)\\). Taking the smaller value is exact.\n\\phantomsection\\label{main:proof:043:end}\\end{proof}","\\begin{corollary}[Shared compilation for an operator family]\\label{main:corollary:013}\\label{cor:shared_compiler}\nLet operators \\(A^{(1)},\\ldots,A^{(m)}\\) use the same candidate tree. Give operator \\(j\\) a nonnegative reuse weight \\(r_j\\). Replace the local keep cost and error by\n\\[\n\\kappa_v\n=\n\\kappa_v^{\\rm setup}\n+\n\\sum_{j=1}^m r_j\\kappa_v^{(j)}\n\\]\nand\n\\[\n\\epsilon_v\n=\n\\sum_{j=1}^m r_j\\epsilon_v^{(j)}.\n\\]\nThen \\cref{thm:tree_compiler} gives the exact optimal shared pruning for the additive weighted objective.\n\\end{corollary}","\\begin{proposition}[Monotone Lagrangian frontier]\\label{main:proposition:009}\\label{prop:lagrangian_frontier}\nLet \\(0\\leq\\lambda_1<\\lambda_2\\). Choose optimal prunings \\(\\Lcal_1\\) and \\(\\Lcal_2\\) for the two Lagrangian parameters. Then\n\\[\n\\boxed{\n\\err(\\Lcal_2)\n\\leq\n\\err(\\Lcal_1)\n}\n\\]\nand\n\\[\n\\boxed{\n\\cost(\\Lcal_2)\n\\geq\n\\cost(\\Lcal_1).\n}\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:045}\nOptimality gives\n\\[\n\\cost(\\Lcal_1)+\\lambda_1\\err(\\Lcal_1)\n\\leq\n\\cost(\\Lcal_2)+\\lambda_1\\err(\\Lcal_2)\n\\]\nand\n\\[\n\\cost(\\Lcal_2)+\\lambda_2\\err(\\Lcal_2)\n\\leq\n\\cost(\\Lcal_1)+\\lambda_2\\err(\\Lcal_1).\n\\]\nAdding the inequalities yields\n\\[\n(\\lambda_2-\\lambda_1)\n\\bigl(\n\\err(\\Lcal_2)-\\err(\\Lcal_1)\n\\bigr)\n\\leq0.\n\\]\nThe error ordering follows. Substitution into the first inequality gives the cost ordering.\n\\phantomsection\\label{main:proof:045:end}\\end{proof}","\\begin{corollary}[Supported Pareto points]\\label{main:corollary:014}\\label{cor:supported_pareto}\nFor \\(\\lambda>0\\), every Lagrangian optimum is Pareto optimal for the pair\n\\[\n(\\cost,\\err).\n\\]\nThe quantized budget compiler may recover Pareto points that are not optimal for any Lagrangian parameter.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:046}\nIf another pruning had no larger cost and no larger error, with one strict inequality, its Lagrangian objective would be smaller. This contradicts optimality. A finite nonconvex frontier can contain points that are not supported by a line of slope \\(-\\lambda\\), so the budget formulation is genuinely additional.\n\\phantomsection\\label{main:proof:046:end}\\end{proof}","\\begin{theorem}[Quantized budget dynamic program]\\label{main:theorem:024}\\label{thm:budget_compiler}\nAssume every keep error is a nonnegative integer \\(w_v\\). Let \\(B\\in\\NN\\). For each node \\(v\\), define \\(F_v(b)\\) as the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}",":theorem:024}\\label{thm:budget_compiler}\nAssume every keep error is a nonnegative integer \\(w_v\\). Let \\(B\\in\\NN\\). For each node \\(v\\), define \\(F_v(b)\\) as the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}","}\nAssume every keep error is a nonnegative integer \\(w_v\\). Let \\(B\\in\\NN\\). For each node \\(v\\), define \\(F_v(b)\\) as the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}","the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:047}\nThe keep case is immediate. In the split case, an admissible pruning is the union of one admissible pruning from each child. Its error is the sum of the child errors and its cost is the sum of their costs plus \\(\\sigma_v\\). Repeated min plus convolution enumerates every possible distribution of the integer budget across the child subtrees. Induction on the tree proves exactness.\n\\phantomsection\\label{main:proof:047:end}\\end{proof}","\\begin{corollary}[Pseudo polynomial complexity]\\label{main:corollary:015}\\label{cor:budget_complexity}\nIf the maximum number of children is bounded and the budget is \\(B\\), the quantized compiler has pseudo polynomial complexity in the number of tree nodes and \\(B\\). A direct pairwise convolution implementation has quadratic dependence on \\(B\\) at each merge.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:048}\nEvery table has at most \\(B+1\\) entries. Merging two tables by direct convolution examines at most \\((B+1)^2\\) pairs. A bounded number of merges occurs at every node.\n\\phantomsection\\label{main:proof:048:end}\\end{proof}","\\begin{definition}[Decision gap]\\label{main:definition:013}\\label{def:decision_gap}\nAt an internal node \\(v\\), let\n\\[\ng_v\n=\n\\abs{K_\\lambda(v)-S_\\lambda(v)}.\n\\]\nThis is the gap between the keep and split scores after the child subproblems have been solved.\n\\end{definition}","\\begin{theorem}[Compiler stability]\\label{main:theorem:025}\\label{thm:compiler_stability}\nSuppose the solved keep score and split score at every internal node change in absolute value by less than $g_v/2$. These are scores after child subproblems have been solved, rather than unconstrained changes in raw local costs. Then every strict keep or split decision in the original optimal pruning remains unchanged.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:049}\nIf the original keep score is lower by \\(g_v\\), then after perturbation it can increase by less than \\(g_v/2\\), while the split score can decrease by less than \\(g_v/2\\). The order remains strict. The split case is the same. Apply the argument from the leaves upward.\n\\phantomsection\\label{main:proof:049:end}\\end{proof}","\\begin{theorem}[Local recompilation]\\label{main:theorem:026}\\label{thm:local_recompilation}\nSuppose local costs change only inside the subtree rooted at $u$, while ancestor and sibling local models are unchanged. Use correct cached sibling values from the old solve. Then every dynamic programming value outside that subtree and outside the ancestor chain from \\(u\\) to the root remains unchanged. Recomputing the subtree at \\(u\\) and that ancestor chain gives the exact new global optimum.\n\\end{theorem}","pilation]\\label{main:theorem:026}\\label{thm:local_recompilation}\nSuppose local costs change only inside the subtree rooted at $u$, while ancestor and sibling local models are unchanged. Use correct cached sibling values from the old solve. Then every dynamic programming value outside that subtree and outside the ancestor chain from \\(u\\) to the root remains unchanged. Recomputing the subtree at \\(u\\) and that ancestor chain gives the exact new global optimum.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:050}\nA node value depends only on its own local data and the values of its children. Nodes outside the changed subtree and outside the ancestor chain have the same local data and the same child values. Their values remain unchanged. Every affected dependency lies inside the changed subtree or on the unique path from \\(u\\) to the root.\n\\phantomsection\\label{main:proof:050:end}\\end{proof}","\\begin{proposition}[Launch reduction by shape]\\label{main:proposition:010}\\label{prop:shape_launch}\nSuppose \\(B\\) independent active blocks fall into \\(g\\) shape classes, and suppose the backend accepts one batched array or pointer list for each class. Grouped execution then requires at most \\(g\\) launches instead of \\(B\\), without changing the arithmetic or the output.\n\\end{proposition}","\\begin{theorem}[Amortization threshold]\\label{main:theorem:028}\\label{thm:amortization}\nLet \\(s\\geq0\\) be a one time setup cost. Let $t_b,t_c\\ge0$ be the baseline and candidate times per use and let $R\\in\\NN$ be a reuse count. If \\(t_b>t_c\\), the candidate is faster after \\(R\\) uses exactly when\n\\[\nR>\n\\frac{s}{t_b-t_c}.\n\\]\nThe smallest integer reuse count is\n\\[\n\\boxed{\nR_{\\min}\n=\n\\left\\lfloor\n\\frac{s}{t_b-t_c}\n\\right\\rfloor+1.\n}\n\\]\nIf \\(t_b\\leq t_c\\), no positive reuse count amortizes a positive setup cost.\n\\end{theorem}","abel{main:theorem:028}\\label{thm:amortization}\nLet \\(s\\geq0\\) be a one time setup cost. Let $t_b,t_c\\ge0$ be the baseline and candidate times per use and let $R\\in\\NN$ be a reuse count. If \\(t_b>t_c\\), the candidate is faster after \\(R\\) uses exactly when\n\\[\nR>\n\\frac{s}{t_b-t_c}.\n\\]\nThe smallest integer reuse count is\n\\[\n\\boxed{\nR_{\\min}\n=\n\\left\\lfloor\n\\frac{s}{t_b-t_c}\n\\right\\rfloor+1.\n}\n\\]\nIf \\(t_b\\leq t_c\\), no positive reuse count amortizes a positive setup cost.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:053}\nCompare\n\\[\ns+Rt_c\n\\]\nwith\n\\[\nRt_b.\n\\]\nRearranging gives the stated inequality.\n\\phantomsection\\label{main:proof:053:end}\\end{proof}","\\begin{corollary}[Graph replay threshold]\\label{main:corollary:016}\\label{cor:graph_threshold}\nIf graph capture costs $c\\ge0$, ordinary execution costs $t_o\\ge0$, and replay costs $0\\le t_r<t_o$, then capture is profitable when\n\\[\nR>\n\\frac{c}{t_o-t_r}.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:054}\nApply \\cref{thm:amortization} with setup cost \\(c\\).\n\\phantomsection\\label{main:proof:054:end}\\end{proof}","\\begin{corollary}[Materialization threshold]\\label{main:corollary:017}\\label{cor:materialization_threshold}\nSuppose a compressed representation costs $t_c\\ge0$ per downstream use, an explicit representation costs $t_e\\ge0$ per use, and one time materialization costs $m\\ge0$, with reuse count $R\\in\\NN$. Materialization is profitable exactly when\n\\[\nR>\n\\frac{m}{t_c-t_e}\n\\]\nand \\(t_c>t_e\\).\n\\end{corollary}","At dimension \\(4096\\), the device resident dense rows gave the values in \\cref{tab:dense_frontier}.","\\midrule\nFP64 & 541.2803 & 253.91 & \\(2.18\\times10^{-15}\\)","FP32 & 18.1309 & 7580.35 & \\(5.76\\times10^{-7}\\)","TF32 & 8.6999 & 15797.75 & \\(2.94\\times10^{-4}\\)","FP16 & 4.0264 & 34134.72 & \\(3.60\\times10^{-4}\\)","BF16 & 4.1482 & 33132.00 & \\(2.88\\times10^{-3}\\)","For an \\(8192\\) by \\(8192\\) equal block operator with block width \\(64\\) and \\(32\\) right hand sides, the batched packet calculation took\n\\[\n0.046032\\text{ ms}.\n\\]\nIt was\n\\[\n14.8919\n\\]\ntimes faster than the recorded device resident dense GPU row with the same recorded case, output, and timing scope and\n\\[\n235.7404\n\\]\ntimes faster than the recorded CPU dense row (resident CPU scope versus device-resident GPU scope). The relative error was\n\\[\n8.17\\times10^{-8}.\n\\]","ator with block width \\(64\\) and \\(32\\) right hand sides, the batched packet calculation took\n\\[\n0.046032\\text{ ms}.\n\\]\nIt was\n\\[\n14.8919\n\\]\ntimes faster than the recorded device resident dense GPU row with the same recorded case, output, and timing scope and\n\\[\n235.7404\n\\]\ntimes faster than the recorded CPU dense row (resident CPU scope versus device-resident GPU scope). The relative error was\n\\[\n8.17\\times10^{-8}.\n\\]","For BDLR application with the same dimension and right hand side count, block width \\(64\\), and rank \\(8\\), CUDA Graph replay took\n\\[\n0.079872\\text{ ms}.\n\\]\nIt was\n\\[\n10.9355\n\\]\ntimes faster than the recorded dense GPU row (graph-replay scope versus device-resident scope) and\n\\[\n153.6527\n\\]\ntimes faster than the recorded CPU dense row (resident CPU scope versus device-resident GPU scope). Capture time was excluded from the replay scope and must be amortized separately.","For hidden permuted blocks at dimension \\(512\\), structure discovery took \\(2.1102\\) ms, dense application took \\(0.0969\\) ms, and packet application took \\(0.0151\\) ms. The exact threshold in \\cref{thm:amortization} is \\(26\\) uses. At dimension \\(1024\\), the corresponding values were \\(5.2726\\), \\(0.1896\\), and \\(0.0241\\) ms, giving \\(32\\) uses.","For one \\(2048\\) dense FP32 product, device resident graph replay took \\(1.9993\\) ms. The end to end pageable path took \\(7.3036\\) ms. The pinned path took \\(6.3084\\) ms. Two serial device resident products took \\(3.6557\\) ms, while the tested two stream path took \\(4.2368\\) ms.","\\begin{itemize}\n\\item At dimension \\(512\\), recursive Strassen multiplication with leaf size \\(128\\) took \\(7.0950\\) ms, while the recorded OpenBLAS dense multiplication took \\(1.9597\\) ms.\n\\item At dimension \\(2048\\), the tested recursive HODLR GPU path took \\(5.6079\\) ms, while dense GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh--Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","\\(512\\), recursive Strassen multiplication with leaf size \\(128\\) took \\(7.0950\\) ms, while the recorded OpenBLAS dense multiplication took \\(1.9597\\) ms.\n\\item At dimension \\(2048\\), the tested recursive HODLR GPU path took \\(5.6079\\) ms, while dense GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh--Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","PU path took \\(5.6079\\) ms, while dense GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh--Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh--Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","The accompanying verification record reports a deterministic verifier with $37392$ checks in nine groups. It covers packet evaluation, adjoints, composition, structured primitives, recompression optimality, product chain error, graph support, Schur fill, tree compilation, precision assignment, and the benchmark corpus. It also reports eleven passing unit tests and twenty-three passing original-source tests for BDLR operations, Krylov methods, mixed precision, measured data schema, and figure generation.","\\subsection{Exact packet semantics}\\label{supp1:subsection:001}\\textit{Source items 2.3, 2.5, and 2.6}\\par\nLet $R$ be a ring, $I$ a finite index type, and $(P_a)_{a\\in I}$ a family satisfying\n\\[\nP_bP_a=\\begin{cases}P_a&b=a,","\\subsection{Exact packet semantics}\\label{supp1:subsection:001}\\textit{Source items 2.3, 2.5, and 2.6}\\par\nLet $R$ be a ring, $I$ a finite index type, and $(P_a)_{a\\in I}$ a family satisfying\n\\[\nP_bP_a=\\begin{cases}P_a&b=a,\\\\0&b\\ne a,\\end{cases}\n\\qquad \\sum_aP_a=1.\n\\]\nWrite $A_{ba}=P_bAP_a$ and let $E_A=\\{(b,a):A_{ba}\\ne0\\}$. These are exact algebraic zeros, not values discarded by a floating-point threshold.","0&b\\ne a,\\end{cases}\n\\qquad \\sum_aP_a=1.\n\\]\nWrite $A_{ba}=P_bAP_a$ and let $E_A=\\{(b,a):A_{ba}\\ne0\\}$. These are exact algebraic zeros, not values discarded by a floating-point threshold.","\\textbf{Reconstruction and action.} The checked finite sum is\n\\[\n\\boxed{A=\\sum_{(b,a)\\in E_A} A_{ba}.}\n\\]\nThe proof inserts the complete projector sum on both sides and removes only zero summands. Applying this identity to an arbitrary module gives exact action. A separate finite-matrix theorem establishes\n\\[\nAZ=\\sum_{(b,a)\\in E_A}A_{ba}Z\n\\]\nfor every rectangular batch of right-hand sides $Z$.","\\textbf{Composition.} Inserting the projector identity between factors yields\n\\[\n\\boxed{(AB)_{ba}=\\sum_c A_{bc}B_{ca}.}\n\\]\nIf the left side is nonzero, at least one intermediate $c$ has both $A_{bc}\\ne0$ and $B_{ca}\\ne0$. Consequently $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$. No equality of supports is asserted.","\\textbf{Adjoint and tensor products.} With an involution and self-adjoint projectors, the formalization proves\n\\[\n(A_{ba})^*=(A^*)_{ab}.\n\\]\nFor actual finite matrices and actual Kronecker products, it proves\n\\[\n(P_b\\otimes Q_d)(A\\otimes B)(P_a\\otimes Q_c)\n=A_{ba}\\otimes B_{dc}.\n\\]\nThe self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninterpreted tensor symbol.","\\subsection{Support propagation and nilpotence}\\label{supp1:subsection:002}\\textit{Source items 6.1, 6.2, and 6.3}\\par\nA path of length $n$ in the Lean predicate \\code{TransferPath} is a directed walk: vertices may repeat. This is the notion required by matrix powers. At length zero the endpoints coincide. At length $n+1$, a nonzero last block is appended to a length-$n$ walk.","Define the active contribution sum recursively by\n\\begin{align*}\nF_0(b,a)&=\\begin{cases}P_a&b=a,\\\\0&b\\ne a,\\end{cases}\\\\\nF_{n+1}(b,a)&=\\sum_{c:\\,A_{bc}\\ne0}A_{bc}F_n(c,a).\n\\end{align*}\nThe checked identity is\n\\[\n\\boxed{P_bA^nP_a=F_n(b,a)\\qquad(n\\ge0).}\n\\]\nIts proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of $A^n$ vanishes.","F_{n+1}(b,a)&=\\sum_{c:\\,A_{bc}\\ne0}A_{bc}F_n(c,a).\n\\end{align*}\nThe checked identity is\n\\[\n\\boxed{P_bA^nP_a=F_n(b,a)\\qquad(n\\ge0).}\n\\]\nIts proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of $A^n$ vanishes.","\\textbf{Nilpotence certificate.} If no walk of length $h+1$ exists between any pair of packets, then\n\\[\n\\boxed{A^{h+1}=0.}\n\\]\nEvery block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map\n\\[\n\\ell:I\\longrightarrow\\{0,\\ldots,h\\},\\qquad\nA_{bc}\\ne0\\ \\Longrightarrow\\ \\ell(c)<\\ell(b).\n\\]\nIt proves that a length-$n$ walk increases the level by at least $n$. This certifies the acyclic case with a bounded height. This block does not implement or verify an algorithm that constructs a topological ordering or computes the longest path.","bf{Nilpotence certificate.} If no walk of length $h+1$ exists between any pair of packets, then\n\\[\n\\boxed{A^{h+1}=0.}\n\\]\nEvery block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map\n\\[\n\\ell:I\\longrightarrow\\{0,\\ldots,h\\},\\qquad\nA_{bc}\\ne0\\ \\Longrightarrow\\ \\ell(c)<\\ell(b).\n\\]\nIt proves that a length-$n$ walk increases the level by at least $n$. This certifies the acyclic case with a bounded height. This block does not implement or verify an algorithm that constructs a topological ordering or computes the longest path.","length $h+1$ exists between any pair of packets, then\n\\[\n\\boxed{A^{h+1}=0.}\n\\]\nEvery block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map\n\\[\n\\ell:I\\longrightarrow\\{0,\\ldots,h\\},\\qquad\nA_{bc}\\ne0\\ \\Longrightarrow\\ \\ell(c)<\\ell(b).\n\\]\nIt proves that a length-$n$ walk increases the level by at least $n$. This certifies the acyclic case with a bounded height. This block does not implement or verify an algorithm that constructs a topological ordering or computes the longest path.","\\textbf{Polynomial support.} For the actual finite sum\n\\[\np(A)=\\sum_{j=0}^{d}c_jA^j,\n\\]\nthe checked support lies in the union of the support relations of the powers. The formalization treats the $j=0$ contribution explicitly, so the constant term is not accidentally omitted.","\\textbf{Cancellation control.} Lean verifies the rational example\n\\[\nA=\\begin{bmatrix}1&1\\\\-1&-1\\end{bmatrix},\\qquad A^2=0.\n\\]\nThe coordinate block $A_{00}$ is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of $A^2$ is zero. This is an exact counterexample to replacing support inclusion by equality.","-1&-1\\end{bmatrix},\\qquad A^2=0.\n\\]\nThe coordinate block $A_{00}$ is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of $A^2$ is zero. This is an exact counterexample to replacing support inclusion by equality.","\\subsection{Low-rank algebra and inversion}\\label{supp1:subsection:003}\\textit{Source items 5.1, 5.3, and 5.4}\\par\nLet $A=D_A+U_AV_A^*$ and $B=D_B+U_BV_B^*$ have rectangular factor widths $r_A$ and $r_B$. The formalization constructs\n\\begin{align*}\nU_C&=\\begin{bmatrix}D_AU_B+U_A(V_A^*U_B)&U_A\\end{bmatrix},","\\subsection{Low-rank algebra and inversion}\\label{supp1:subsection:003}\\textit{Source items 5.1, 5.3, and 5.4}\\par\nLet $A=D_A+U_AV_A^*$ and $B=D_B+U_BV_B^*$ have rectangular factor widths $r_A$ and $r_B$. The formalization constructs\n\\begin{align*}\nU_C&=\\begin{bmatrix}D_AU_B+U_A(V_A^*U_B)&U_A\\end{bmatrix},\\\\\nV_C&=\\begin{bmatrix}V_B&D_B^*V_A\\end{bmatrix}\n\\end{align*}\nand proves\n\\[\n\\boxed{AB=D_AD_B+U_CV_C^*.}\n\\]\nThese are actual concatenated matrices with $r_B+r_A$ columns. Expanding the product and grouping the terms with common right factors proves the identity. The separate rank theorem establishes\n\\[\n\\rank(AB-D_AD_B)\\le r_A+r_B.\n\\]\nIt bounds actual matrix rank, not just the number of stored columns. Packet diagonality of $D_AD_B$ follows from diagonality of both factors and the checked block-composition formula.","V_C&=\\begin{bmatrix}V_B&D_B^*V_A\\end{bmatrix}\n\\end{align*}\nand proves\n\\[\n\\boxed{AB=D_AD_B+U_CV_C^*.}\n\\]\nThese are actual concatenated matrices with $r_B+r_A$ columns. Expanding the product and grouping the terms with common right factors proves the identity. The separate rank theorem establishes\n\\[\n\\rank(AB-D_AD_B)\\le r_A+r_B.\n\\]\nIt bounds actual matrix rank, not just the number of stored columns. Packet diagonality of $D_AD_B$ follows from diagonality of both factors and the checked block-composition formula.","D_B^*V_A\\end{bmatrix}\n\\end{align*}\nand proves\n\\[\n\\boxed{AB=D_AD_B+U_CV_C^*.}\n\\]\nThese are actual concatenated matrices with $r_B+r_A$ columns. Expanding the product and grouping the terms with common right factors proves the identity. The separate rank theorem establishes\n\\[\n\\rank(AB-D_AD_B)\\le r_A+r_B.\n\\]\nIt bounds actual matrix rank, not just the number of stored columns. Packet diagonality of $D_AD_B$ follows from diagonality of both factors and the checked block-composition formula.","\\textbf{Woodbury.} For $D$ and\n\\[\nS=I+V^*D^{-1}U\n\\]\nboth invertible, Lean checks the ordinary matrix inverse formula\n\\[\n\\boxed{(D+UV^*)^{-1}\n=D^{-1}-D^{-1}US^{-1}V^*D^{-1}.}\n\\]\nThe proof specializes Mathlib's classical Woodbury identity. A separate theorem derives invertibility of $D+UV^*$ from the two stated assumptions. A further theorem verifies that multiplying the proposed inverse by any batch $Z$ solves the updated system. Invertibility of the update is not inserted as an extra premise.","xtbf{Woodbury.} For $D$ and\n\\[\nS=I+V^*D^{-1}U\n\\]\nboth invertible, Lean checks the ordinary matrix inverse formula\n\\[\n\\boxed{(D+UV^*)^{-1}\n=D^{-1}-D^{-1}US^{-1}V^*D^{-1}.}\n\\]\nThe proof specializes Mathlib's classical Woodbury identity. A separate theorem derives invertibility of $D+UV^*$ from the two stated assumptions. A further theorem verifies that multiplying the proposed inverse by any batch $Z$ solves the updated system. Invertibility of the update is not inserted as an extra premise.","\\textbf{Norm bound.} Over the real or complex scalars, all matrix norms in the following theorem are Hilbert-space operator norms, including the rectangular factors:\n\\[\n\\boxed{\n\\norm{(D+UV^*)^{-1}}\n\\le\\norm{D^{-1}}+\n\\norm{D^{-1}U}\\,\\norm{S^{-1}}\\,\\norm{V^*D^{-1}}.\n}\n\\]\nAfter the checked inverse identity, the proof uses the triangle inequality and two rectangular operator-norm product bounds.","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative cer","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact b","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefor…","two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","perator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","\\textbf{Uniform operator estimates.} Two additional checked consequences are\n\\[\n\\norm{\\widehat A-A}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}}\n\\]\nand, when $\\norm{P_a}\\le1$ for every packet,\n\\[\n\\boxed{\\norm{\\widehat A-A}\\le\\sum_e\\norm{\\Delta_e}.}\n\\]\nThese estimates are uniform in the input vector and can be used as inputs to later chain analyses. The chain results themselves are not included in this block's completed inventory.","\\textbf{Implementation boundary.} The maps $\\Delta_e$ and their bounds must be supplied or justified for a numerical implementation. This formalization does not establish IEEE rounding behavior, prevent overflow or underflow, model a CUDA kernel, or certify the recorded device speedups. Those are distinct claims and require separate evidence.","\\subsection{Positive budget and feasibility}\\label{supp1:subsection:005}\\textit{Positive-budget condition for item 8.4}\\par\n\\textbf{Positive budget is necessary.} The source minimizes\n\\[\n\\sum_e c_e\\log\\frac1{\\varepsilon_e},\\qquad\n\\varepsilon_e>0,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nwith positive sensitivities and costs. The positive-budget condition is essential. For a nonempty edge set, Lean proves\n\\[\ns_e>0\\ \\text{and}\\ \\varepsilon_e>0\\ \\text{for all }e\n\\quad\\Longrightarrow\\quad\n\\sum_e s_e\\varepsilon_e>0.\n\\]\nIt follows that every nonpositive budget is infeasible. A one-edge, unit-sensitivity, zero-budget instance is checked as an exact boundary example.","ion{Positive budget and feasibility}\\label{supp1:subsection:005}\\textit{Positive-budget condition for item 8.4}\\par\n\\textbf{Positive budget is necessary.} The source minimizes\n\\[\n\\sum_e c_e\\log\\frac1{\\varepsilon_e},\\qquad\n\\varepsilon_e>0,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nwith positive sensitivities and costs. The positive-budget condition is essential. For a nonempty edge set, Lean proves\n\\[\ns_e>0\\ \\text{and}\\ \\varepsilon_e>0\\ \\text{for all }e\n\\quad\\Longrightarrow\\quad\n\\sum_e s_e\\varepsilon_e>0.\n\\]\nIt follows that every nonpositive budget is infeasible. A one-edge, unit-sensitivity, zero-budget instance is checked as an exact boundary example.","The nonempty-index statement requires $\\tau>0$. Nonemptiness also makes the sum of positive costs in the displayed optimizer's denominator positive. The empty edge set is treated separately. Existence, global optimality and uniqueness for the positive-budget problem are proved in Section~\\ref{supp2:subsection:004}.","\\subsection{Exact finite-tree compilation}\\label{supp2:subsection:001}\\textit{Source items 9.4 and 9.5}\\par\nA candidate tree is either a terminal node or an internal node with a nonempty finite family of children. Arbitrary finite branching is supported. Every node carries a label from which keep cost $\\kappa_v$, keep error $\\epsilon_v$, and split overhead $\\sigma_v$ are read.","The cost and error of a cut $L$ are computed recursively. With multiplier $\\lambda$, its objective is\n\\[\nJ_\\lambda(L)=\\cost(L)+\\lambda\\err(L).\n\\]\nLean proves that this objective equals the recursively accumulated local keep score $\\kappa_v+\\lambda\\epsilon_v$, with $\\sigma_v$ charged at every split.","\\textbf{Recurrence.} For terminal nodes,\n\\[\nF_\\lambda(v)=\\kappa_v+\\lambda\\epsilon_v.\n\\]\nFor internal nodes,\n\\[\n\\boxed{F_\\lambda(v)=\\min\\left\\{\\kappa_v+\\lambda\\epsilon_v,\n\\;\\sigma_v+\\sum_{w\\in\\mathrm{child}(v)}F_\\lambda(w)\\right\\}.}\n\\]\nThe checked result has two parts. First, $F_\\lambda(v)\\leq J_\\lambda(L)$ for every legal cut of the subtree. Second, the constructed function \\code{compile} returns a legal cut with objective exactly $F_\\lambda(v)$. Thus the minimum is attained, not merely bounded below.","des,\n\\[\nF_\\lambda(v)=\\kappa_v+\\lambda\\epsilon_v.\n\\]\nFor internal nodes,\n\\[\n\\boxed{F_\\lambda(v)=\\min\\left\\{\\kappa_v+\\lambda\\epsilon_v,\n\\;\\sigma_v+\\sum_{w\\in\\mathrm{child}(v)}F_\\lambda(w)\\right\\}.}\n\\]\nThe checked result has two parts. First, $F_\\lambda(v)\\leq J_\\lambda(L)$ for every legal cut of the subtree. Second, the constructed function \\code{compile} returns a legal cut with objective exactly $F_\\lambda(v)$. Thus the minimum is attained, not merely bounded below.","\\textbf{Shared operator families.} For reuse weights $r_j$, the local keep model can aggregate setup and per-operator costs:\n\\[\n\\kappa_v=\\kappa_v^{\\mathrm{setup}}+\\sum_jr_j\\kappa_v^{(j)},\n\\qquad \\epsilon_v=\\sum_jr_j\\epsilon_v^{(j)}.\n\\]\nLean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including $\\Q$ and $\\R$. The intended nonnegative model is a specialization.","Shared operator families.} For reuse weights $r_j$, the local keep model can aggregate setup and per-operator costs:\n\\[\n\\kappa_v=\\kappa_v^{\\mathrm{setup}}+\\sum_jr_j\\kappa_v^{(j)},\n\\qquad \\epsilon_v=\\sum_jr_j\\epsilon_v^{(j)}.\n\\]\nLean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including $\\Q$ and $\\R$. The intended nonnegative model is a specialization.","erator families.} For reuse weights $r_j$, the local keep model can aggregate setup and per-operator costs:\n\\[\n\\kappa_v=\\kappa_v^{\\mathrm{setup}}+\\sum_jr_j\\kappa_v^{(j)},\n\\qquad \\epsilon_v=\\sum_jr_j\\epsilon_v^{(j)}.\n\\]\nLean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including $\\Q$ and $\\R$. The intended nonnegative model is a specialization.","\\subsection{Frontiers and decision stability}\\label{supp2:subsection:002}\\textit{Source items 9.6, 9.7, and 9.11}\\par\nLet $L_1,L_2$ be actual optima at $0\\leq\\lambda_1<\\lambda_2$. Their optimality inequalities are\n\\begin{align*}\n\\cost(L_1)+\\lambda_1\\err(L_1)&\\leq\\cost(L_2)+\\lambda_1\\err(L_2),","\\subsection{Frontiers and decision stability}\\label{supp2:subsection:002}\\textit{Source items 9.6, 9.7, and 9.11}\\par\nLet $L_1,L_2$ be actual optima at $0\\leq\\lambda_1<\\lambda_2$. Their optimality inequalities are\n\\begin{align*}\n\\cost(L_1)+\\lambda_1\\err(L_1)&\\leq\\cost(L_2)+\\lambda_1\\err(L_2),\\\\\n\\cost(L_2)+\\lambda_2\\err(L_2)&\\leq\\cost(L_1)+\\lambda_2\\err(L_1).\n\\end{align*}\nAdding them and using the strict multiplier ordering gives\n\\[\n\\boxed{\\err(L_2)\\leq\\err(L_1),\\qquad \\cost(L_2)\\geq\\cost(L_1).}\n\\]\nThese conclusions hold for any choices among tied optima. For the cuts returned by the compiler, the required optimality premises are supplied by its constructive correctness theorem.","\\cost(L_2)+\\lambda_2\\err(L_2)&\\leq\\cost(L_1)+\\lambda_2\\err(L_1).\n\\end{align*}\nAdding them and using the strict multiplier ordering gives\n\\[\n\\boxed{\\err(L_2)\\leq\\err(L_1),\\qquad \\cost(L_2)\\geq\\cost(L_1).}\n\\]\nThese conclusions hold for any choices among tied optima. For the cuts returned by the compiler, the required optimality premises are supplied by its constructive correctness theorem.","\\textbf{Supported Pareto points.} For $\\lambda>0$, a scalarized optimum cannot be dominated in both cost and error with one strict improvement. Lean proves this directly from the two inequalities and positivity of $\\lambda$.","The converse is not asserted. For the three cost/error points\n\\[\n(0,2),\\qquad (3/2,1),\\qquad (2,0),\n\\]\nthe middle point is undominated, but it is not selected by scalarization. A checked rational inequality gives, for every $\\lambda\\in\\Q$,\n\\[\n\\min\\{2\\lambda,2\\}<3/2+\\lambda.\n\\]\nThe quantized-budget compiler can recover unsupported points. Its construction is given in Section~\\ref{supp3:subsection:005}.","\\textbf{Strict half-gap stability.} At a solved internal node write\n\\[\nK=\\kappa_v+\\lambda\\epsilon_v,\\qquad\nS=\\sigma_v+\\sum_wF_\\lambda(w),\\qquad g=|K-S|.\n\\]\nIf the updated scores satisfy\n\\[\n|K'-K|<g/2,\\qquad |S'-S|<g/2,\n\\]\nLean proves\n\\[\nK'\\leq S'\\quad\\Longleftrightarrow\\quad K\\leq S.\n\\]\nApplying this condition recursively at every internal node proves equality of the entire returned cuts before and after perturbation, not only equality of one numerical minimum.","The hypotheses concern \\emph{solved} split scores, including changes propagated from descendants. A bound on raw local costs alone is not substituted for this condition. At an exact tie $g=0$, the strict inequalities cannot hold. No robustness at ties is claimed.","Let $v$ be the new optimum of the edited subtree, and let $j$ be the edited child slot at one frame. The update performed at that frame is\n\\[\n\\boxed{\n\\operatorname{step}(v)=\n\\min\\left\\{K,\\;\\sigma+\n\\sum_i\\begin{cases}\nv,&i=j,","Let $v$ be the new optimum of the edited subtree, and let $j$ be the edited child slot at one frame. The update performed at that frame is\n\\[\n\\boxed{\n\\operatorname{step}(v)=\n\\min\\left\\{K,\\;\\sigma+\n\\sum_i\\begin{cases}\nv,&i=j,\\\\\n\\widehat F_i,&i\\ne j.\n\\end{cases}\\right\\}.\n}\n\\]\nThis result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values $\\widehat F_i$.","\\widehat F_i,&i\\ne j.\n\\end{cases}\\right\\}.\n}\n\\]\nThis result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values $\\widehat F_i$.","\\textbf{Correctness of the actual update.} For any edited subtree $T'$ and valid context $C$, Lean proves\n\\[\n\\operatorname{updateValue}\\bigl(C,F(T')\\bigr)=F\\bigl(\\operatorname{fill}(C,T')\\bigr).\n\\]\nThe proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree recurrence.","\\subsection{The continuous allocation theorem}\\label{supp2:subsection:004}\\textit{Source item 8.4, with an explicit positive budget}\\par\nLet $E$ be a finite nonempty edge set. Assume\n\\[\ns_e>0,\\qquad c_e>0,\\qquad \\tau>0,\n\\]\nand define $C=\\sum_ec_e$. The optimization problem is\n\\[\n\\min_{\\varepsilon_e>0}\\sum_ec_e\\log\\frac1{\\varepsilon_e}\n\\qquad\\text{subject to}\\qquad\n\\sum_es_e\\varepsilon_e=\\tau.\n\\]\nLean constructs the displayed candidate\n\\[\n\\boxed{\\varepsilon_e^*=\\frac{\\tau c_e}{s_eC}}\n\\]\nand proves strict positivity and the exact budget equality. It then proves global optimality and uniqueness for the actual logarithmic objective.","orem}\\label{supp2:subsection:004}\\textit{Source item 8.4, with an explicit positive budget}\\par\nLet $E$ be a finite nonempty edge set. Assume\n\\[\ns_e>0,\\qquad c_e>0,\\qquad \\tau>0,\n\\]\nand define $C=\\sum_ec_e$. The optimization problem is\n\\[\n\\min_{\\varepsilon_e>0}\\sum_ec_e\\log\\frac1{\\varepsilon_e}\n\\qquad\\text{subject to}\\qquad\n\\sum_es_e\\varepsilon_e=\\tau.\n\\]\nLean constructs the displayed candidate\n\\[\n\\boxed{\\varepsilon_e^*=\\frac{\\tau c_e}{s_eC}}\n\\]\nand proves strict positivity and the exact budget equality. It then proves global optimality and uniqueness for the actual logarithmic objective.","\\textbf{Global comparison.} For any other feasible $\\varepsilon$, set\n$r_e=\\varepsilon_e/\\varepsilon_e^*>0$. The budget identities give\n\\[\n\\sum_ec_er_e=C.\n\\]\nUsing the standard analytic inequality $\\log r\\leq r-1$ yields\n\\[\n\\sum_ec_e\\log r_e\\leq\\sum_ec_e(r_e-1)=0.\n\\]\nLean separately proves the exact objective-difference identity\n\\[\nJ(\\varepsilon)-J(\\varepsilon^*)=-\\sum_ec_e\\log r_e.\n\\]\nTherefore $J(\\varepsilon)\\geq J(\\varepsilon^*)$.","If the feasible vectors differ, at least one $r_e\\ne1$. At that coordinate the logarithm inequality is strict. Since $c_e>0$, the finite weighted sum is strict, giving\n\\[\n\\varepsilon\\ne\\varepsilon^*\\quad\\Longrightarrow\\quad\nJ(\\varepsilon)>J(\\varepsilon^*).\n\\]\nThe final theorem characterizes every feasible global minimizer as exactly the constructed formula. No unproved stationary-point, convexity, or KKT optimality assertion is supplied as a hypothesis.","\\subsection{Explicit storage and operation models}\\label{supp2:subsection:005}\\textit{Source items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2}\\par\nFor a finite collection of rectangular blocks, the scalar addresses are actual elements of\n\\[\n\\coprod_{e\\in E}\\bigl(\\operatorname{Fin}(m_e)\\times\\operatorname{Fin}(n_e)\\bigr).\n\\]\nThe corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their cardinalities rather than merely defining a formula named ``cost.''","\\begin{center}\\small\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\nRepresentation & Scalar storage & Modeled work for $q$ right sides","\\midrule\nDense & $n^2$ & $2qn^2$","Packet blocks & $\\sum_em_en_e$ & $2q\\sum_em_en_e$","Equal diagonal blocks & $nb$ & $2qnb$","BDLR & $\\sum_i n_i^2+2nr$ & $2q(\\sum_i n_i^2+2nr)$","Equal-block BDLR & $n(b+2r)$ & $2qn(b+2r)$","Sparse plus low rank & $z+2nr$ & $2q(z+2nr)$","\\bottomrule\n\\end{tabular}\n\\end{center}\nFor equal blocks, the layout condition is $n=|I|b$. Under nonzero denominator conditions, the dense-to-block reduction is $n/b$ and the BDLR reduction is $n/(b+2r)$. The BDLR output thin product accumulates into the diagonal result under the declared model of two operations per coefficient.","\\textbf{Streaming.} The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size $s$, its byte count is\n\\[\nB_{\\mathrm{stream}}=s\\sum_e(m_en_e+n_eq+2m_eq).\n\\]\nNo cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.","\\textbf{Capacity and roofline.} Positive physical capacities and the inequalities $F\\leq PT$, $B\\leq WT$ give\n\\[\nT\\geq\\max\\{F/P,B/W\\}.\n\\]\nFor positive bytes and elapsed time, the rate form is $F/T\\leq\\min\\{P,WF/B\\}$. The dense memory bound is the exact equivalence $n^2s\\leq M\\iff n\\leq\\sqrt{M/s}$ for nonnegative $n,M$ and positive $s$.","\\textbf{Materialization.} The declared dense product followed by one addition at each output entry has exactly $2n^2r+n^2$ modeled events and $n^2$ output entries. The remainder is bounded by $n^2$ for every $n,r$, giving a concrete quadratic certificate for the manuscript's $O(n^2)$ term.","\\subsection{Contracts, grouping, and amortization}\\label{supp2:subsection:006}\\textit{Source items 2.9, 10.2, 10.4--10.6, and 11.3}\\par\n\\textbf{Output conversion.} A decoded run returns the actual pair\n\\[\n\\bigl(\\operatorname{decode}(\\operatorname{compressed}(x)),\n\\;t_{\\mathrm{run}}(x)+t_{\\mathrm{decode}}(\\operatorname{compressed}(x))\\bigr).\n\\]\nUnder the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the downstream composition identity when compressed output is consumed directly. Omitting a positive decode cost strictly understates the modeled explicit-output time. This formalizes the protocol. It cannot certify the truth of measured times.","\\textbf{Exact profitability.} With setup $s$, baseline time $t_b$, candidate time $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.","ine time $t_b$, candidate time $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.",", candidate time $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.","e $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.","\\subsection{Low-rank factors through a product chain}\\label{supp3:subsection:001}\\textit{Source item 5.2}\\par\nLet\n\\[\nA_j=D_j+U_jV_j^*,\\qquad U_j,V_j\\in\\mathbb C^{n\\times r_j},\n\\]\nwith all $D_j$ diagonal in one packet layout. Set\n\\[\nP_m=A_m\\cdots A_1,\\qquad D^{(m)}=D_m\\cdots D_1.\n\\]\nThe formalization constructs thin matrices $L_m,R_m$ such that\n\\[\n\\boxed{P_m=D^{(m)}+L_mR_m^*.}\n\\]\nTheir column index type is a recursively constructed disjoint sum. Its cardinality is proved to equal $\\sum_{j=1}^m r_j$.","tit{Source item 5.2}\\par\nLet\n\\[\nA_j=D_j+U_jV_j^*,\\qquad U_j,V_j\\in\\mathbb C^{n\\times r_j},\n\\]\nwith all $D_j$ diagonal in one packet layout. Set\n\\[\nP_m=A_m\\cdots A_1,\\qquad D^{(m)}=D_m\\cdots D_1.\n\\]\nThe formalization constructs thin matrices $L_m,R_m$ such that\n\\[\n\\boxed{P_m=D^{(m)}+L_mR_m^*.}\n\\]\nTheir column index type is a recursively constructed disjoint sum. Its cardinality is proved to equal $\\sum_{j=1}^m r_j$.","\\textbf{Construction.} Suppose $P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*$. The new factors are\n\\begin{align*}\nL_m&=\\bigl[D_mL_{m-1}+U_m(V_m^*L_{m-1})\\quad U_m\\bigr],","\\textbf{Construction.} Suppose $P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*$. The new factors are\n\\begin{align*}\nL_m&=\\bigl[D_mL_{m-1}+U_m(V_m^*L_{m-1})\\quad U_m\\bigr],\\\\\nR_m&=\\bigl[R_{m-1}\\quad (D^{(m-1)})^*V_m\\bigr].\n\\end{align*}\nSubstitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","Applying the ordinary matrix-rank inequality gives\n\\[\n\\boxed{\\operatorname{rank}(P_m-D^{(m)})\\leq\\sum_{j=1}^m r_j.}\n\\]\nThe bound concerns the actual correction matrix, not just the number of allocated columns. The proof also shows that $D^{(m)}$ stays packet diagonal. The empty chain has identity base, empty column type, and zero correction. Individual $r_j=0$ are permitted.","\\subsubsection{Actual noncommutative products}\nThe error module defines the left-accumulated product recursively, with $P_0=I$ and $P_{j+1}=A_{j+1}P_j$. For a perturbed sequence $\\widehat A_j$, it derives the discrepancy recursion\n\\[\n\\widehat P_{j+1}-P_{j+1}\n=\\widehat A_{j+1}(\\widehat P_j-P_j)\n +(\\widehat A_{j+1}-A_{j+1})P_j.\n\\]\nThe identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index $j$ in source code corresponds to manuscript factor $j+1$.","I$ and $P_{j+1}=A_{j+1}P_j$. For a perturbed sequence $\\widehat A_j$, it derives the discrepancy recursion\n\\[\n\\widehat P_{j+1}-P_{j+1}\n=\\widehat A_{j+1}(\\widehat P_j-P_j)\n +(\\widehat A_{j+1}-A_{j+1})P_j.\n\\]\nThe identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index $j$ in source code corresponds to manuscript factor $j+1$.","e $\\widehat A_j$, it derives the discrepancy recursion\n\\[\n\\widehat P_{j+1}-P_{j+1}\n=\\widehat A_{j+1}(\\widehat P_j-P_j)\n +(\\widehat A_{j+1}-A_{j+1})P_j.\n\\]\nThe identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index $j$ in source code corresponds to manuscript factor $j+1$.","\\subsection{Error propagation and recompression}\\label{supp3:subsection:002}\\textit{Source items 5.8, 5.9, and 8.3}\\par\nFor $\\widehat A_j=A_j+E_j$, the checked product-chain bound is\n\\[\n\\boxed{\n\\norm{\\widehat A_m\\cdots\\widehat A_1-A_m\\cdots A_1}\n\\leq\\sum_{j=1}^m\n \\left(\\prod_{i=j+1}^m\\norm{\\widehat A_i}\\right)\n \\norm{E_j}\n \\left(\\prod_{i=1}^{j-1}\\norm{A_i}\\right).}\n\\]\nEmpty products are one. The proof first solves a scalar gain recurrence as a finite sum, then bounds the actual discrepancy recursion by it. A version with certified upper bounds $\\delta_j\\geq\\norm{E_j}$ is included. The product-norm argument uses a norm-one identity, as in operators on a nonzero normed space.","checked product-chain bound is\n\\[\n\\boxed{\n\\norm{\\widehat A_m\\cdots\\widehat A_1-A_m\\cdots A_1}\n\\leq\\sum_{j=1}^m\n \\left(\\prod_{i=j+1}^m\\norm{\\widehat A_i}\\right)\n \\norm{E_j}\n \\left(\\prod_{i=1}^{j-1}\\norm{A_i}\\right).}\n\\]\nEmpty products are one. The proof first solves a scalar gain recurrence as a finite sum, then bounds the actual discrepancy recursion by it. A version with certified upper bounds $\\delta_j\\geq\\norm{E_j}$ is included. The product-norm argument uses a norm-one identity, as in operators on a nonzero normed space.","\\textbf{Post-product recompression is different.} Define the implemented iterates by\n\\[\n\\widetilde P_0=I,\\qquad\n\\widetilde P_j=A_j\\widetilde P_{j-1}+E_j.\n\\]\nHere $E_j$ is a residual introduced after multiplication. Lean derives\n\\[\n\\widetilde P_j-P_j=A_j(\\widetilde P_{j-1}-P_{j-1})+E_j\n\\]\nand proves\n\\[\n\\boxed{\\norm{\\widetilde P_m-P_m}\n\\leq\\sum_{j=1}^m\\left(\\prod_{i=j+1}^m\\norm{A_i}\\right)\\delta_j,\n\\quad \\norm{E_j}\\leq\\delta_j.}\n\\]\nThis theorem works in a normed ring without the extra norm-one condition. It includes the empty iteration. There is no exact-prefix factor in the local residual term, because the residual is already measured at the post-product output.","\\textbf{Packetwise mixed precision.} At each stage let\n\\[\nA_j=\\sum_e A_{j,e},\\qquad\n\\widehat A_j=\\sum_e(A_{j,e}+\\Delta_{j,e}),\\qquad\n\\Delta_{j,e}=\\Delta_{j,e}P_{j,a(e)}.\n\\]\nThe previous packet certificate supplies the uniform operator bound\n\\[\n\\norm{\\widehat A_j-A_j}\\leq\n\\delta_j:=\\sum_e\\norm{\\Delta_{j,e}}\\norm{P_{j,a(e)}}.\n\\]\nThe new theorem substitutes these $\\delta_j$ into the actual product-chain estimate. The resulting vector error is at most the operator certificate times $\\norm{x}$. A vector-specific packet bound is not silently reused as an operator norm.","\\widehat A_j=\\sum_e(A_{j,e}+\\Delta_{j,e}),\\qquad\n\\Delta_{j,e}=\\Delta_{j,e}P_{j,a(e)}.\n\\]\nThe previous packet certificate supplies the uniform operator bound\n\\[\n\\norm{\\widehat A_j-A_j}\\leq\n\\delta_j:=\\sum_e\\norm{\\Delta_{j,e}}\\norm{P_{j,a(e)}}.\n\\]\nThe new theorem substitutes these $\\delta_j$ into the actual product-chain estimate. The resulting vector error is at most the operator certificate times $\\norm{x}$. A vector-specific packet bound is not silently reused as an operator norm.","\\subsection{Exact and approximate packet Krylov spaces}\\label{supp3:subsection:003}\\textit{Source items 7.2 and 7.3}\\par\nWrite\n\\[\n\\mathcal K_m(A,v)=\\operatorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","3:subsection:003}\\textit{Source items 7.2 and 7.3}\\par\nWrite\n\\[\n\\mathcal K_m(A,v)=\\operatorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","n:003}\\textit{Source items 7.2 and 7.3}\\par\nWrite\n\\[\n\\mathcal K_m(A,v)=\\operatorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","eratorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","\\textbf{The direct sum is justified.} Each local space is proved to lie in the actual range of $P_i$. Projecting a finite sum of local vectors selects its $i$th summand. Consequently a zero sum of local vectors has every summand zero. The proof does not assert equality between the global Krylov space and the full sum of local Krylov spaces.","For approximate invariance let $A=D+K$, where $D$ is packet diagonal, and set\n\\[\nM=\\max\\{\\norm{A},\\norm{D}\\}.\n\\]\nFor every positive integer $j$, the operator estimate is\n\\[\n\\boxed{\\norm{A^j-D^j}\\leq jM^{j-1}\\norm{K}.}\n\\]\nThe proof is noncommutative and uses actual powers. Its positive-index version works even in a trivial normed ring. No division by $M$ is used, so $M=0$ causes no exception.","Define the comparison space\n\\[\n\\mathcal L_{j+1}=\\bigoplus_i\\mathcal K_{j+1}(P_iDP_i,P_iv).\n\\]\nPacket diagonality is first shown to imply commutation. The exact theorem then proves $D^jv\\in\\mathcal L_{j+1}$. Using that vector as a witness in the infimum distance gives\n\\[\n\\boxed{\\operatorname{dist}(A^jv,\\mathcal L_{j+1})\n\\leq jM^{j-1}\\norm{K}\\norm{v}.}\n\\]\nThis statement uses Lean's actual metric infimum distance to the subspace. It does not assume that the comparison vector lies there. The block-Jacobi condition bound and the preconditioned conjugate-gradient corollary are not completed by these Krylov results.","For a predicate $S$ on costs, the common minimum specification means:\n\\[\n\\begin{array}{ll}\n\\code{none}: & \\text{no cost satisfies }S,","For a predicate $S$ on costs, the common minimum specification means:\n\\[\n\\begin{array}{ll}\n\\code{none}: & \\text{no cost satisfies }S,\\\\\n\\code{some}(v): & S(v)\\text{ and }v\\leq z\\text{ whenever }S(z).\n\\end{array}\n\\]\nBoth attainment and minimality are included. Optional minimum and optional addition are proved to implement union and independent sum of feasible sets.","For exact-weight tables $F,G$, min-plus convolution is\n\\[\n(F\\star G)(b)=\\min_{0\\leq k\\leq b}\\{F(k)+G(b-k)\\},\n\\]\nwith an infeasible sum whenever either operand is infeasible. Its correctness theorem identifies every finite answer with an actual feasible pair of component weights and costs. Repeated convolution handles arbitrary finite lists of components.","\\textbf{Independent precision assignments.} For every edge, a finite list contains its permitted pairs $(w,c)$ of quantized error and cost. The legal assignment type chooses exactly one listed pair per edge. It is defined independently of the dynamic program. Costs and weights are then summed over that assignment.","If $D$ is the already solved tail table, the next edge has exact-weight recurrence\n\\[\nF(b)=\\min_{(w,c)\\text{ allowed},\\;w\\leq b}\\{D(b-w)+c\\}.\n\\]\nThe final answer is $\\min_{0\\leq b\\leq B}F(b)$. Lean proves that this is the exact minimum over legal assignments with total error at most $B$, or \\code{none} exactly when no such assignment exists.","\\textbf{The subtraction guard matters.} Natural subtraction truncates at zero. Without the condition $w\\leq b$, an option of weight one could be accepted at budget zero. The implemented program checks the condition before subtraction. Dedicated regressions also cover a zero-weight option, an empty option set, and the empty collection of edges. The last case has cost and error zero.","Keeping node $v$ costs $\\kappa_v$ and incurs integer error $w_v$. Splitting costs overhead $\\sigma_v$ in addition to the selected child cuts. The exact-weight recurrence is\n\\[\nF_v(b)=\\min\\left\\{\n\\begin{array}{l}\n\\kappa_v\\quad\\text{if }b=w_v,","Keeping node $v$ costs $\\kappa_v$ and incurs integer error $w_v$. Splitting costs overhead $\\sigma_v$ in addition to the selected child cuts. The exact-weight recurrence is\n\\[\nF_v(b)=\\min\\left\\{\n\\begin{array}{l}\n\\kappa_v\\quad\\text{if }b=w_v,\\\\[2pt]\n\\displaystyle\\sigma_v+\n\\min_{\\sum_i b_i=b}\\sum_iF_{v_i}(b_i)\\quad\\text{if }v\\text{ is internal}.\n\\end{array}\\right.\n\\]\nNonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-plus convolution, not by assuming an optimal allocation.","\\textbf{Exact semantics.} For every tree and every integer weight $b$, Lean proves that $F_v(b)$ is a minimum of costs of independent legal cuts of exact weight $b$. Thus each finite answer has an attaining cut and is a lower bound for every other legal cut at that weight. Infeasibility is equivalent to the absence of such a cut.","The root scan\n\\[\n\\boxed{\\min_{0\\leq b\\leq B}F_{\\mathrm{root}}(b)}\n\\]\nis therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating \\code{none} as an ordinary real cost.","\\textbf{Why the budget program adds something.} Consider the three admissible cost/error pairs\n\\[\n(0,4),\\qquad(3,2),\\qquad(4,0).\n\\]\nAt budget two the answer is cost three. The middle pair cannot minimize cost plus $\\lambda$ times error: comparison with the first pair would require $\\lambda\\geq3/2$, while comparison with the third would require $\\lambda\\leq1/2$. The executable suite checks the budget answer on an actual unary candidate tree with these cuts. It is not obtained merely by varying the previous Lagrangian compiler's multiplier.","\\subsection{Cached arrays and a work certificate}\\label{supp3:subsection:006}\\textit{Source item 9.9 and implemented forms of 8.7 and 9.8}\\par\nA recursive value equation alone need not be an efficient program. Repeated calls can recompute the same subtree table. The new implementation therefore stores each table as an array-backed \\code{Vector} with exactly $B+1$ optional entries.","\\textbf{Entrywise implementation theorem.} For every $b\\leq B$, Lean proves\n\\[\n\\operatorname{lookup}(\\operatorname{storedTree}(B,t),b)=F_t(b).\n\\]\nThere is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at $b$ depends only on input entries at most $b$, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attainment, and infeasibility specification applies to the actual stored result.","\\textbf{Counted candidates.} One full convolution table examines indices\n\\[\n(b,k),\\qquad 0\\leq b\\leq B,\\quad0\\leq k\\leq b.\n\\]\nThese form a concrete finite dependent-sum type. Its cardinality is proved to be\n\\[\n\\sum_{b=0}^B(b+1)\\leq(B+1)^2.\n\\]\nFor a nonempty finite tree, the number of child merges plus one equals the node count $N$. This identity is proved for arbitrary finite branching.","Let $W_B(t)$ be the declared unit-cost count of table-cell constructions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves\n\\[\nW_B(t)\\leq2(B+1)N+\\bigl((B+1)+(B+1)^2\\bigr)(N-1)\n\\]\nand hence\n\\[\n\\boxed{W_B(t)\\leq N\\bigl(3(B+1)+(B+1)^2\\bigr).}\n\\]\nThe root minimum scan adds a linear pass. Thus the declared tabulation model is pseudo-polynomial in $N$ and the numeric budget $B$.","ount of table-cell constructions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves\n\\[\nW_B(t)\\leq2(B+1)N+\\bigl((B+1)+(B+1)^2\\bigr)(N-1)\n\\]\nand hence\n\\[\n\\boxed{W_B(t)\\leq N\\bigl(3(B+1)+(B+1)^2\\bigr).}\n\\]\nThe root minimum scan adds a linear pass. Thus the declared tabulation model is pseudo-polynomial in $N$ and the numeric budget $B$.","tions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves\n\\[\nW_B(t)\\leq2(B+1)N+\\bigl((B+1)+(B+1)^2\\bigr)(N-1)\n\\]\nand hence\n\\[\n\\boxed{W_B(t)\\leq N\\bigl(3(B+1)+(B+1)^2\\bigr).}\n\\]\nThe root minimum scan adds a linear pass. Thus the declared tabulation model is pseudo-polynomial in $N$ and the numeric budget $B$.","\\textbf{Scope of this certificate.} These are table and candidate counts, not a measured execution trace or a theorem about every instruction emitted by the compiler. Rational bit complexity, allocation overhead, memory traffic, and hardware timing are not included. The certificate is not polynomial in the binary input length of $B$.","\\subsection{Circulant Fourier diagonalization}\\label{supp4:subsection:001}\\textit{Source item 4.3; module CirculantFourier}\\par\nLet $n>0$, with coordinates indexed by $\\mathbb Z/n\\mathbb Z$. Define the actual circulant by\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nThe formal Fourier matrix is\n\\[\nF_{jk}=\\frac{1}{\\sqrt n}\\,\\chi(-jk),\n\\qquad\n\\chi(a)=\\exp\\!\\left(\\frac{2\\pi i a}{n}\\right).\n\\]\nThe standard character is not left abstract: a checked theorem identifies these entries with\n\\[\n\\frac{1}{\\sqrt n}\\exp\\!\\left(-\\frac{2\\pi i\\,\\operatorname{val}(j)\\operatorname{val}(k)}n\\right).\n\\]\nThe value representatives enter only this explicit exponential formula. All modular algebra uses the residue classes themselves.","n:001}\\textit{Source item 4.3; module CirculantFourier}\\par\nLet $n>0$, with coordinates indexed by $\\mathbb Z/n\\mathbb Z$. Define the actual circulant by\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nThe formal Fourier matrix is\n\\[\nF_{jk}=\\frac{1}{\\sqrt n}\\,\\chi(-jk),\n\\qquad\n\\chi(a)=\\exp\\!\\left(\\frac{2\\pi i a}{n}\\right).\n\\]\nThe standard character is not left abstract: a checked theorem identifies these entries with\n\\[\n\\frac{1}{\\sqrt n}\\exp\\!\\left(-\\frac{2\\pi i\\,\\operatorname{val}(j)\\operatorname{val}(k)}n\\right).\n\\]\nThe value representatives enter only this explicit exponential formula. All modular algebra uses the residue classes themselves.","\\textbf{Normalization and inversion.} The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform:\n\\[\nFx=n^{-1/2}\\operatorname{DFT}(x).\n\\]\nIt separately identifies $F^*x$ with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities\n\\[\n\\boxed{F^*F=I,\\qquad FF^*=I.}\n\\]\nNo unitarity assumption is inserted.","\\textbf{Convolution and reconstruction.} An explicit bijective change of index proves\n\\[\n\\operatorname{DFT}(C(c)x)_k\n=\\operatorname{DFT}(c)_k\\operatorname{DFT}(x)_k.\n\\]\nThis gives the matrix intertwining relation $FC(c)=\\operatorname{diag}(\\operatorname{DFT}c)F$. Multiplication by the proved inverse reconstructs the circulant:\n\\[\n\\boxed{C(c)=F^*\\operatorname{diag}(\\sqrt n\\,Fc)F.}\n\\]\nThe two-transform application formula is also checked for every input vector. The result includes $n=1$ and arbitrary complex coefficients. Hermiticity is not required.","\\[\n\\operatorname{DFT}(C(c)x)_k\n=\\operatorname{DFT}(c)_k\\operatorname{DFT}(x)_k.\n\\]\nThis gives the matrix intertwining relation $FC(c)=\\operatorname{diag}(\\operatorname{DFT}c)F$. Multiplication by the proved inverse reconstructs the circulant:\n\\[\n\\boxed{C(c)=F^*\\operatorname{diag}(\\sqrt n\\,Fc)F.}\n\\]\nThe two-transform application formula is also checked for every input vector. The result includes $n=1$ and arbitrary complex coefficients. Hermiticity is not required.","\\textbf{Claim boundary.} This verifies the exact Fourier representation, not an $O(n\\log n)$ implementation of the transform. The latter is an additional algorithmic factorization of the same linear map.","\\subsection{Toeplitz padding and column-major vectorization}\\label{supp4:subsection:002}\\textit{Source items 4.4 and 4.5}\\par\n\\textbf{Toeplitz embedding.} For $n>0$, define\n\\[\nT_{ij}=\n\\begin{cases}\nc_{i-j},&j\\le i,","\\subsection{Toeplitz padding and column-major vectorization}\\label{supp4:subsection:002}\\textit{Source items 4.4 and 4.5}\\par\n\\textbf{Toeplitz embedding.} For $n>0$, define\n\\[\nT_{ij}=\n\\begin{cases}\nc_{i-j},&j\\le i,\\\\\nr_{j-i},&i<j.\n\\end{cases}\n\\]\nThe diagonal is supplied by $c_0$, and the coordinate $r_0$ is unused. The constructed generator of length $2n$ is\n\\[\ng=(c_0,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1),\n\\]\nand the padding map is the actual finite-index sum eliminator that appends $n$ zeros.","r_{j-i},&i<j.\n\\end{cases}\n\\]\nThe diagonal is supplied by $c_0$, and the coordinate $r_0$ is unused. The constructed generator of length $2n$ is\n\\[\ng=(c_0,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1),\n\\]\nand the padding map is the actual finite-index sum eliminator that appends $n$ zeros.","Lean first proves the upper-left block identity\n\\[\nC(g)_{ij}=T_{ij}\\qquad(0\\le i,j<n).\n\\]\nFor $j\\le i$, finite modular subtraction has its ordinary value $i-j$. For $i<j$, it has wrapped value $2n+i-j$, whose reversed generator coordinate is $j-i$. Both branches are checked with the actual finite-index arithmetic.","Splitting the product sum into the original and padded halves then gives\n\\[\n\\boxed{\\bigl(C(g)(x,0)^T\\bigr)_{0:n}=Tx.}\n\\]\nThe generator's first half and its middle zero are checked separately. Neither symmetry nor conjugation is involved.","\\textbf{Kronecker application.} For $A\\in\\C^{m\\times m}$, $B\\in\\C^{n\\times n}$ and $X\\in\\C^{n\\times m}$, Lean proves\n\\[\n\\boxed{(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).}\n\\]\nThe formal vectorization is explicitly column-major: its pair coordinate $(j,i)$ stores $X_{ij}$. The right factor is the \\emph{ordinary transpose}, not the conjugate transpose. This distinction is essential over $\\C$.","f{Kronecker application.} For $A\\in\\C^{m\\times m}$, $B\\in\\C^{n\\times n}$ and $X\\in\\C^{n\\times m}$, Lean proves\n\\[\n\\boxed{(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).}\n\\]\nThe formal vectorization is explicitly column-major: its pair coordinate $(j,i)$ stores $X_{ij}$. The right factor is the \\emph{ordinary transpose}, not the conjugate transpose. This distinction is essential over $\\C$.","Finite types of multiplication/accumulation events and stored scalar addresses give\n\\[\n\\boxed{2n^2m+2nm^2\\ \\text{modeled operations},\\qquad n^2+m^2\\ \\text{factor entries}.}\n\\]\nThe explicit Kronecker matrix has $n^2m^2$ entries. As in the manuscript, the operation model assigns two scalar operations to each multiply-accumulate term.","\\subsection{Arbitrary tensor factors}\\label{supp4:subsection:003}\\textit{Source item 4.6; module StructuredTensor}\\par\nLet $A_0,\\ldots,A_{k-1}$ be square factors of order $d$. Tensor coordinates are represented by a nested product of $k$ copies of $\\{0,\\ldots,d-1\\}$. The empty product is a singleton. Lean proves that the coordinate type has cardinality $d^k$.","The represented matrix is defined recursively using the actual Kronecker product:\n\\[\nM_0=[1],\\qquad M_{k+1}=A_0\\otimes M_k(A_1,\\ldots,A_k).\n\\]\nThe algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies $A_0$ to the resulting first-coordinate fibers:\n\\[\ny_{i,u}=\\sum_{j=0}^{d-1}(A_0)_{ij}\\,\n\\operatorname{Apply}_{k}(A_1,\\ldots,A_k;x_{j,\\cdot})_u.\n\\]\nInduction, the Kronecker entry formula, and finite-sum distributivity prove\n\\[\n\\boxed{\\operatorname{Apply}_k(A;x)=M_k(A)x.}\n\\]\nThus correctness is not inferred merely from agreeing dimension or storage formulas.","otimes M_k(A_1,\\ldots,A_k).\n\\]\nThe algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies $A_0$ to the resulting first-coordinate fibers:\n\\[\ny_{i,u}=\\sum_{j=0}^{d-1}(A_0)_{ij}\\,\n\\operatorname{Apply}_{k}(A_1,\\ldots,A_k;x_{j,\\cdot})_u.\n\\]\nInduction, the Kronecker entry formula, and finite-sum distributivity prove\n\\[\n\\boxed{\\operatorname{Apply}_k(A;x)=M_k(A)x.}\n\\]\nThus correctness is not inferred merely from agreeing dimension or storage formulas.","\\textbf{Operation events.} The event type for $k+1$ modes is a disjoint union of $d$ copies of the trailing event type and the events for the final fiber contraction. Its cardinality obeys\n\\[\nW_0=0,\\qquad W_{k+1}=dW_k+2d^{k+2}.\n\\]\nLean proves the closed form\n\\[\n\\boxed{W_k=2kd^{k+1}.}\n\\]\nThe factor-address type has cardinality\n\\[\n\\boxed{k d^2.}\n\\]\nThese statements include $k=0$ and $d=0$. With no factors, the action is the identity on the singleton coordinate type and incurs zero modeled arithmetic. With $d=0$ and at least one factor, the coordinate type is empty.","ent type and the events for the final fiber contraction. Its cardinality obeys\n\\[\nW_0=0,\\qquad W_{k+1}=dW_k+2d^{k+2}.\n\\]\nLean proves the closed form\n\\[\n\\boxed{W_k=2kd^{k+1}.}\n\\]\nThe factor-address type has cardinality\n\\[\n\\boxed{k d^2.}\n\\]\nThese statements include $k=0$ and $d=0$. With no factors, the action is the identity on the singleton coordinate type and incurs zero modeled arithmetic. With $d=0$ and at least one factor, the coordinate type is empty.","\\subsection{Walsh butterfly and exact addition count}\\label{supp4:subsection:004}\\textit{Source item 4.7; module WalshTransform}\\par\nThe two-point matrix is constructed as\n\\[\nH_2=\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix},\n\\]\nand the full family is defined by $H_1=[1]$ and\n\\[\nH_{2N}=H_2\\otimes H_N\n=\\begin{pmatrix}H_N&H_N","1&-1\\end{pmatrix},\n\\]\nand the full family is defined by $H_1=[1]$ and\n\\[\nH_{2N}=H_2\\otimes H_N\n=\\begin{pmatrix}H_N&H_N","1&-1\\end{pmatrix},\n\\]\nand the full family is defined by $H_1=[1]$ and\n\\[\nH_{2N}=H_2\\otimes H_N\n=\\begin{pmatrix}H_N&H_N\\\\H_N&-H_N\\end{pmatrix}.\n\\]\nThe two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove\n\\[\n\\boxed{H_NH_N^T=NI,\\qquad N=2^k.}\n\\]\nThe ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is $\\C$.","H_N&-H_N\\end{pmatrix}.\n\\]\nThe two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove\n\\[\n\\boxed{H_NH_N^T=NI,\\qquad N=2^k.}\n\\]\nThe ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is $\\C$.","\\textbf{Actual butterfly.} For the two halves $x_0,x_1$ of the input, the independently defined procedure returns\n\\[\n\\operatorname{Walsh}_{2N}(x_0,x_1)\n=\\bigl(\\operatorname{Walsh}_N(x_0+x_1),\n       \\operatorname{Walsh}_N(x_0-x_1)\\bigr).\n\\]\nLean proves that this vector equals $H_{2N}x$. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already represents the matrix.","\\textbf{Exact arithmetic count.} Each pair of slices forms one sum and one difference at every trailing coordinate. The event type therefore gives\n\\[\nA_0=0,\\qquad A_{k+1}=2A_k+2^{k+1}.\n\\]\nThe closed form and its equivalent logarithmic expression are both checked:\n\\[\n\\boxed{A_k=k2^k=N\\log_2N.}\n\\]\nThe logarithm in the formal count is the natural-number binary logarithm, with its power-of-two identity proved by Mathlib. The case $N=1$ has zero arithmetic events.","} Each pair of slices forms one sum and one difference at every trailing coordinate. The event type therefore gives\n\\[\nA_0=0,\\qquad A_{k+1}=2A_k+2^{k+1}.\n\\]\nThe closed form and its equivalent logarithmic expression are both checked:\n\\[\n\\boxed{A_k=k2^k=N\\log_2N.}\n\\]\nThe logarithm in the formal count is the natural-number binary logarithm, with its power-of-two identity proved by Mathlib. The case $N=1$ has zero arithmetic events.","A further action theorem gives\n\\[\nH_N(H_N^Tx)=Nx,\n\\]\nwhich identifies the inverse after division by the nonzero scalar $N$. This result concerns exact arithmetic, not floating-point error accumulation through a butterfly implementation.","\\subsection{Clipped precision: existence and multiplier uniqueness}\\label{supp4:subsection:005}\\textit{Source item 8.5; module ClippedPrecision}\\par\nLet $E$ be a finite nonempty set and suppose\n\\[\ns_e>0,\\qquad c_e>0,\\qquad m_e>0,\n\\qquad 0<\\tau\\le\\sum_e s_em_e.\n\\]\nThe feasible set is\n\\[\n0<\\varepsilon_e\\le m_e,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nand the objective is the same logarithmic objective used in item 8.4:\n\\[\nJ(\\varepsilon)=\\sum_e c_e\\log(1/\\varepsilon_e).\n\\]\nStrictly positive caps are stated explicitly. A zero or negative cap cannot admit a positive tolerance at that coordinate.","ecision: existence and multiplier uniqueness}\\label{supp4:subsection:005}\\textit{Source item 8.5; module ClippedPrecision}\\par\nLet $E$ be a finite nonempty set and suppose\n\\[\ns_e>0,\\qquad c_e>0,\\qquad m_e>0,\n\\qquad 0<\\tau\\le\\sum_e s_em_e.\n\\]\nThe feasible set is\n\\[\n0<\\varepsilon_e\\le m_e,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nand the objective is the same logarithmic objective used in item 8.4:\n\\[\nJ(\\varepsilon)=\\sum_e c_e\\log(1/\\varepsilon_e).\n\\]\nStrictly positive caps are stated explicitly. A zero or negative cap cannot admit a positive tolerance at that coordinate.","\\textbf{Constructive parameterization.} Write $t=1/\\lambda$ and set\n\\[\nu_e(t)=\\min\\{m_e,tc_e/s_e\\},\\qquad\n\\Phi(t)=\\sum_e s_eu_e(t).\n\\]\nLean proves that $\\Phi$ is continuous and nondecreasing, with $\\Phi(0)=0$. At the finite positive scale\n\\[\nT=\\sum_e\\frac{m_es_e}{c_e},\n\\]\nevery coordinate is saturated, so $u(T)=m$ and $\\Phi(T)=\\sum_es_em_e$. The intermediate value theorem produces $t>0$ with $\\Phi(t)=\\tau$. The multiplier is not assumed to exist.","\\textbf{Strictness below saturation.} If $t<u$ and $\\Phi(t)$ is below the maximum, at least one coordinate is unsaturated at $t$. That coordinate strictly increases, and every other coordinate is nondecreasing. Hence $\\Phi(t)<\\Phi(u)$. This proves uniqueness of the scale and, by reciprocation, the exact manuscript statement\n\\[\n\\boxed{\\exists!\\lambda>0:\\quad\n\\sum_e s_e\\min\\!\\left\\{m_e,\\frac{c_e}{\\lambda s_e}\\right\\}=\\tau}\n\\]\nwhen $0<\\tau<\\sum_es_em_e$.","n.} If $t<u$ and $\\Phi(t)$ is below the maximum, at least one coordinate is unsaturated at $t$. That coordinate strictly increases, and every other coordinate is nondecreasing. Hence $\\Phi(t)<\\Phi(u)$. This proves uniqueness of the scale and, by reciprocation, the exact manuscript statement\n\\[\n\\boxed{\\exists!\\lambda>0:\\quad\n\\sum_e s_e\\min\\!\\left\\{m_e,\\frac{c_e}{\\lambda s_e}\\right\\}=\\tau}\n\\]\nwhen $0<\\tau<\\sum_es_em_e$.","At the maximal budget, the minimizing vector remains unique but the multiplier need not be. The one-coordinate case $s=c=m=1$ is checked: both $\\lambda=1$ and $\\lambda=1/2$ produce the same tolerance $1$.","Fix the feasible vector $u=u(t)$ obtained above and any feasible competitor $\\varepsilon$. For each coordinate, Lean proves\n\\[\n\\boxed{\n c_e\\left(\\frac{\\varepsilon_e}{u_e}-1\\right)\n \\le \\frac{s_e}{t}(\\varepsilon_e-u_e).\n}\n\\]\nWhen $u_e=tc_e/s_e$, this is equality. When $u_e=m_e$, the capped condition implies $s_e/t\\le c_e/m_e$, while $\\varepsilon_e-m_e\\le0$. Multiplying by this nonpositive difference gives the required inequality in the correct direction.","The equal-budget constraint then yields\n\\[\n\\sum_e c_e\\left(\\frac{\\varepsilon_e}{u_e}-1\\right)\n\\le\\frac1t\\sum_e s_e(\\varepsilon_e-u_e)=0.\n\\]\nApplying the checked inequality $\\log z\\le z-1$ for $z>0$ gives\n\\[\n\\sum_e c_e\\log(\\varepsilon_e/u_e)\\le0.\n\\]\nSince\n\\[\nJ(\\varepsilon)-J(u)\n=-\\sum_e c_e\\log(\\varepsilon_e/u_e),\n\\]\nwe obtain $J(u)\\le J(\\varepsilon)$ for every feasible competitor.","If $\\varepsilon\\ne u$, one coordinate has ratio different from $1$. The logarithmic inequality is strict there, and its positive weight preserves strictness in the sum. Consequently,\n\\[\n\\boxed{\\varepsilon\\ne u\\quad\\Longrightarrow\\quad J(u)<J(\\varepsilon).}\n\\]\nThis proves the unique global minimizing vector for every positive feasible budget, including the maximal endpoint.","\\subsection{Constructing the optimal level schedule}\\label{supp5:subsection:001}\\textit{Source items 6.6 and 10.3; module GraphSchedules}\\par\nLet $V$ be finite and let $u\\to v$ mean that task $v$ depends on task $u$. Assume there is no nonempty directed cycle. A topological order or valid level function is \\emph{not} supplied as an additional hypothesis.","For each vertex, define its strict ancestor set\n\\[\n\\operatorname{Anc}(v)=\\{u:\\text{there is a positive-length path }u\\leadsto v\\}.\n\\]\nFor every edge $u\\to v$, the proof establishes\n\\[\n\\operatorname{Anc}(u)\\subsetneq\\operatorname{Anc}(v).\n\\]\nThe inclusion follows by appending the edge. It is strict because $u$ is an ancestor of $v$ but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is defined:\n\\[\n\\boxed{\\ell(v)=1+\\max_{u\\to v}\\ell(u),\\qquad \\max\\varnothing=0.}\n\\]","r every edge $u\\to v$, the proof establishes\n\\[\n\\operatorname{Anc}(u)\\subsetneq\\operatorname{Anc}(v).\n\\]\nThe inclusion follows by appending the edge. It is strict because $u$ is an ancestor of $v$ but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is defined:\n\\[\n\\boxed{\\ell(v)=1+\\max_{u\\to v}\\ell(u),\\qquad \\max\\varnothing=0.}\n\\]","\\textbf{Validity and attainment.} Lean proves $\\ell(v)>0$ and $\\ell(u)<\\ell(v)$ on every edge. A path of $n$ edges from $a$ to $b$ satisfies\n\\[\n\\ell(a)+n\\leq\\ell(b).\n\\]\nConversely, predecessor-max attainment and induction produce an actual path with $\\ell(v)-1$ edges ending at each vertex $v$. The result is an equality with a longest path, not merely an upper bound.","A valid schedule $s:V\\to\\mathbb N$ assigns positive stages and strictly increases on each edge. Induction proves the stronger pointwise statement\n\\[\n\\ell(v)\\leq s(v)\\quad\\text{for every }v.\n\\]\nConsequently the constructed schedule achieves the least possible makespan,\n\\[\n\\boxed{d=\\max_{v\\in V}\\ell(v).}\n\\]\nFor a nonempty graph, $d$ equals the number of vertices on an attained longest path. Vertices of the same level have no dependency edge in either direction.","\\textbf{Empty graph and interpretation.} When $V=\\varnothing$, the checked depth is zero. The phrase ``one plus the longest edge-path length'' applies to nonempty graphs. These are ideal discrete scheduling statements. Weighted task costs, a finite processor budget, launch latency and communication are not part of this model. The definitions are mathematical constructions, not a certified hardware scheduler.","\\subsection{Forward substitution with an actual stage evaluator}\\label{supp5:subsection:002}\\textit{Source item 6.5; module TriangularSolve}\\par\nUse the graph and levels constructed on the previous page. A vertex $v$ carries an arbitrary vector space $X_v$ over a field. The spaces may have different dimensions. Let\n\\[\nD_v:X_v\\overset{\\sim}{\\longrightarrow}X_v,\n\\qquad T_{vu}:X_u\\longrightarrow X_v\n\\]\nbe an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action:\n\\[\n(Ax)_v=D_vx_v+\\sum_{u\\to v}T_{vu}x_u.\n\\]\nNo solution, solve equation or precomputed topological certificate is assumed.","iangularSolve}\\par\nUse the graph and levels constructed on the previous page. A vertex $v$ carries an arbitrary vector space $X_v$ over a field. The spaces may have different dimensions. Let\n\\[\nD_v:X_v\\overset{\\sim}{\\longrightarrow}X_v,\n\\qquad T_{vu}:X_u\\longrightarrow X_v\n\\]\nbe an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action:\n\\[\n(Ax)_v=D_vx_v+\\sum_{u\\to v}T_{vu}x_u.\n\\]\nNo solution, solve equation or precomputed topological certificate is assumed.","\\textbf{Constructed solution.} Well-founded recursion defines\n\\[\n\\boxed{x_v=D_v^{-1}\\!\\left(b_v-\\sum_{u\\to v}T_{vu}x_u\\right).}\n\\]\nApplying $D_v$ proves the original equation at $v$. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of $D_v$ forces agreement at $v$. Thus the construction gives existence and uniqueness, and the block operator is bijective.","\\textbf{Parallel stages.} A second, independent definition is an explicit staged evaluator. Set $x^{(0)}=0$ and\n\\[\nx^{(k+1)}_v=\n\\begin{cases}\nD_v^{-1}\\!\\left(b_v-\\displaystyle\\sum_{u\\to v}T_{vu}x^{(k)}_u\\right),\n  &\\ell(v)=k+1,","\\textbf{Parallel stages.} A second, independent definition is an explicit staged evaluator. Set $x^{(0)}=0$ and\n\\[\nx^{(k+1)}_v=\n\\begin{cases}\nD_v^{-1}\\!\\left(b_v-\\displaystyle\\sum_{u\\to v}T_{vu}x^{(k)}_u\\right),\n  &\\ell(v)=k+1,\\\\[5pt]\nx^{(k)}_v,&\\ell(v)\\ne k+1.\n\\end{cases}\n\\]\nIt reads the previous-stage vector, not partially updated coordinates. The invariant is\n\\[\n\\ell(v)\\leq k\\quad\\Longrightarrow\\quad x^{(k)}_v=x_v.\n\\]\nEvery predecessor of a level-$k+1$ vertex has level at most $k$, so the recurrence uses only coordinates already proved correct. At the canonical depth $d$ this yields\n\\[\n\\boxed{Ax^{(d)}=b.}\n\\]\nThe local independence theorem separately proves that changing coordinates outside the strict earlier levels cannot change a vertex's local solve.","[5pt]\nx^{(k)}_v,&\\ell(v)\\ne k+1.\n\\end{cases}\n\\]\nIt reads the previous-stage vector, not partially updated coordinates. The invariant is\n\\[\n\\ell(v)\\leq k\\quad\\Longrightarrow\\quad x^{(k)}_v=x_v.\n\\]\nEvery predecessor of a level-$k+1$ vertex has level at most $k$, so the recurrence uses only coordinates already proved correct. At the canonical depth $d$ this yields\n\\[\n\\boxed{Ax^{(d)}=b.}\n\\]\nThe local independence theorem separately proves that changing coordinates outside the strict earlier levels cannot change a vertex's local solve.","\\subsection{Support of inverses and Schur elimination}\\label{supp5:subsection:003}\\textit{Source items 6.7 and 6.8; modules InverseSupport and SchurSupport}\\par\nLet $(P_a)$ be a complete idempotent packet family in a finite-dimensional algebra. Define $a\\leadsto b$ by an actual finite path through nonzero blocks $P_bAP_a$. \\textbf{Zero-length paths are included}, so every packet reaches itself.","Fix the source packet $a$. In the regular representation, define the submodule\n\\[\nW_a=\\{x:P_bx=0\\ \\text{whenever }a\\not\\leadsto b\\}.\n\\]\nIt contains $P_a$. Completeness of the projectors expands each component of $Ax$ into packet transfers. If a transfer into an unreachable packet were nonzero from a reachable packet, it would extend a path, a contradiction. Thus $AW_a\\subseteq W_a$.","Suppose $BA=I$. Left multiplication by $A$ is injective, including on $W_a$. Finite dimensionality makes that restricted map surjective. Given $x\\in W_a$, choose $y\\in W_a$ with $Ay=x$. Then $Bx=BAy=y\\in W_a$. Applying this to $P_a$ proves\n\\[\n\\boxed{a\\not\\leadsto b\\quad\\Longrightarrow\\quad P_bBP_a=0.}\n\\]\nThe matrix specialization takes $B$ to Mathlib's actual nonsingular matrix inverse, with an invertible determinant. No inverse-sparsity hypothesis is inserted.","\\textbf{Hidden elimination.} For the actual Schur complement\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR},\n\\]\nthe hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices $j,k$ with nonzero entrance and exit blocks and a hidden path $j\\leadsto k$. Therefore a nonzero block $S_{ba}$ requires either a direct retained block, or\n\\[\na\\longrightarrow j\\ \\leadsto_{H}\\ k\\longrightarrow b.\n\\]\nThe entrance and exit matrices may be rectangular. Retained and hidden coordinate sizes and packet systems are independently typed. The hidden matrix must be invertible. A zero-length hidden path permits entrance and exit through the same hidden packet.","\\textbf{Inclusion is not equality.} Cancellation can remove a block permitted by a graph route. For example,\n\\[\nA=\\begin{pmatrix}1&1&1\\\\0&1&1\\\\0&0&1\\end{pmatrix}\n\\quad\\text{has}\\quad (A^{-1})_{13}=0\n\\]\ndespite a direct nonzero $(1,3)$ block. The implementation proves support inclusion and the corresponding no-route zero certificate, not a converse.","0&0&1\\end{pmatrix}\n\\quad\\text{has}\\quad (A^{-1})_{13}=0\n\\]\ndespite a direct nonzero $(1,3)$ block. The implementation proves support inclusion and the corresponding no-route zero certificate, not a converse.","\\subsection{Exact banded addresses and modeled work}\\label{supp5:subsection:004}\\textit{Source item 4.1; module BandedCount}\\par\nFor $p$ packets and half-bandwidth $w$, the actual address set is\n\\[\n\\mathcal B_{p,w}\n=\\{(i,j)\\in\\{0,\\ldots,p-1\\}^2:\n i\\leq j+w\\ \\text{and}\\ j\\leq i+w\\}.\n\\]\nThe predicate is the integer version of $|i-j|\\leq w$. The count is first defined as a double sum of indicators, then identified with the cardinality of this subtype. It is not defined by the target closed form.","Under $0\\leq w<p$, the formal proof establishes\n\\[\n\\boxed{N_{p,w}=|\\mathcal B_{p,w}|=p(2w+1)-w(w+1).}\n\\]\nThe proof uses finite prefix counting, the newly exposed boundary row and column when $p$ increases, and the saturated initial band. An intermediate subtraction-free identity avoids relying on natural-number subtraction without the required inequality. The diagonal case $w=0$ gives exactly $p$ blocks, and the saturated case $w=p-1$ gives $p^2$.","oof establishes\n\\[\n\\boxed{N_{p,w}=|\\mathcal B_{p,w}|=p(2w+1)-w(w+1).}\n\\]\nThe proof uses finite prefix counting, the newly exposed boundary row and column when $p$ increases, and the saturated initial band. An intermediate subtraction-free identity avoids relying on natural-number subtraction without the required inequality. The diagonal case $w=0$ gives exactly $p$ blocks, and the saturated case $w=p-1$ gives $p^2$.","s\n\\[\n\\boxed{N_{p,w}=|\\mathcal B_{p,w}|=p(2w+1)-w(w+1).}\n\\]\nThe proof uses finite prefix counting, the newly exposed boundary row and column when $p$ increases, and the saturated initial band. An intermediate subtraction-free identity avoids relying on natural-number subtraction without the required inequality. The diagonal case $w=0$ gives exactly $p$ blocks, and the saturated case $w=p-1$ gives $p^2$.","\\textbf{Stored scalars.} With equal block width $b$, the storage-address type is\n\\[\n\\mathcal S=\\mathcal B_{p,w}\\times\\{0,\\ldots,b-1\\}^2.\n\\]\nIts cardinality is the actual number of represented scalar entries:\n\\[\n\\boxed{|\\mathcal S|=b^2N_{p,w}.}\n\\]","\\textbf{Arithmetic events.} For $q$ right-hand sides, the declared event type adds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","$q$ right-hand sides, the declared event type adds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","ent type adds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","dds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","\\subsection{Exact residual certificates for supplied packets}\\label{supp5:subsection:005}\\textit{Source item 9.2; module DiscoveryCertificate}\\par\nLet $(P_a)$ be a complete orthogonal projector system on a finite complex coordinate space. The projectors need not be coordinate masks. For an actual matrix $A$, define\n\\[\n\\Pi(A)=\\sum_aP_aAP_a,\n\\qquad E=A-\\Pi(A).\n\\]\nReconstruction gives the exact off-diagonal residual\n\\[\nE=\\sum_{b\\ne a}P_bAP_a.\n\\]","esidual certificates for supplied packets}\\label{supp5:subsection:005}\\textit{Source item 9.2; module DiscoveryCertificate}\\par\nLet $(P_a)$ be a complete orthogonal projector system on a finite complex coordinate space. The projectors need not be coordinate masks. For an actual matrix $A$, define\n\\[\n\\Pi(A)=\\sum_aP_aAP_a,\n\\qquad E=A-\\Pi(A).\n\\]\nReconstruction gives the exact off-diagonal residual\n\\[\nE=\\sum_{b\\ne a}P_bAP_a.\n\\]","\\textbf{Hilbert--Schmidt orthogonality.} For distinct ordered packet pairs $(b,a)\\ne(d,c)$, adjoint multiplication and projector orthogonality give\n\\[\n\\operatorname{tr}\\bigl((P_bAP_a)^*P_dBP_c\\bigr)=0.\n\\]\nIf $b\\ne d$, the central projector product vanishes. Otherwise $a\\ne c$, and cyclicity of trace brings $P_cP_a$ together. This is proved for arbitrary input matrices $A,B$.","Lean identifies the actual Frobenius norm with its trace expression,\n\\[\n\\norm{M}_{\\rm F}^2=\\operatorname{Re}\\operatorname{tr}(M^*M),\n\\]\nthen expands the finite sum and removes each cross term. The result is\n\\[\n\\boxed{\\norm{A-\\Pi(A)}_{\\rm F}^2\n=\\sum_{b\\ne a}\\norm{P_bAP_a}_{\\rm F}^2.}\n\\]\nNo Pythagorean equality for the target residual is assumed.","\\textbf{Operator norm.} In a separate scoped matrix norm, the triangle inequality gives\n\\[\n\\boxed{\\norm{A-\\Pi(A)}_2\n\\leq\\sum_{b\\ne a}\\norm{P_bAP_a}_2.}\n\\]\nHere $\\norm{\\cdot}_2$ is the Euclidean induced operator norm, not the Frobenius norm. The formal declarations use distinct norm instances, preventing the two certificates from being conflated. Self-adjointness is used for Frobenius orthogonality. The sum estimate itself requires only the complete packet decomposition.","\\subsection{Block-Jacobi normalization and the CG factor}\\label{supp5:subsection:006}\\textit{Source items 7.4 and 7.6; module BlockJacobi}\\par\nWork on a nonempty finite complex coordinate space with Euclidean operator norms. Assume $D$ and $A=D+K$ are positive definite. Define\n\\[\nS=(\\sqrt D)^{-1},\\qquad B=SAS,\\qquad E=SKS.\n\\]\nThe square root is the actual positive matrix square root supplied by continuous functional calculus, and the inverse is the actual matrix inverse. Positive definiteness makes $\\sqrt D$ a unit. Its square identity and inverse identities prove\n\\[\nSDS=I,\\qquad B=I+E.\n\\]","7.4 and 7.6; module BlockJacobi}\\par\nWork on a nonempty finite complex coordinate space with Euclidean operator norms. Assume $D$ and $A=D+K$ are positive definite. Define\n\\[\nS=(\\sqrt D)^{-1},\\qquad B=SAS,\\qquad E=SKS.\n\\]\nThe square root is the actual positive matrix square root supplied by continuous functional calculus, and the inverse is the actual matrix inverse. Positive definiteness makes $\\sqrt D$ a unit. Its square identity and inverse identities prove\n\\[\nSDS=I,\\qquad B=I+E.\n\\]","lex coordinate space with Euclidean operator norms. Assume $D$ and $A=D+K$ are positive definite. Define\n\\[\nS=(\\sqrt D)^{-1},\\qquad B=SAS,\\qquad E=SKS.\n\\]\nThe square root is the actual positive matrix square root supplied by continuous functional calculus, and the inverse is the actual matrix inverse. Positive definiteness makes $\\sqrt D$ a unit. Its square identity and inverse identities prove\n\\[\nSDS=I,\\qquad B=I+E.\n\\]","\\textbf{A direct inverse bound.} If $C=(I+E)^{-1}$ and $\\norm E_2\\leq\\eta<1$, then $C=I-EC$. Consequently\n\\[\n(1-\\eta)\\norm C_2\\leq 1,\n\\qquad \\norm{I+E}_2\\leq 1+\\eta.\n\\]\nThe constructed normalization is invertible because $S$ and $A$ are. Combining the estimates gives the actual condition number bound\n\\[\n\\boxed{\\kappa_2(B):=\\norm B_2\\norm{B^{-1}}_2\n\\leq\\frac{1+\\eta}{1-\\eta}.}\n\\]\nThis norm argument does not require a separately assumed Hermitian perturbation. Nonnegative $\\eta$ follows from the norm bound. Nonemptiness is explicit because the condition-number and norm-of-identity convention uses a nonzero coordinate space.","$, then $C=I-EC$. Consequently\n\\[\n(1-\\eta)\\norm C_2\\leq 1,\n\\qquad \\norm{I+E}_2\\leq 1+\\eta.\n\\]\nThe constructed normalization is invertible because $S$ and $A$ are. Combining the estimates gives the actual condition number bound\n\\[\n\\boxed{\\kappa_2(B):=\\norm B_2\\norm{B^{-1}}_2\n\\leq\\frac{1+\\eta}{1-\\eta}.}\n\\]\nThis norm argument does not require a separately assumed Hermitian perturbation. Nonnegative $\\eta$ follows from the norm bound. Nonemptiness is explicit because the condition-number and norm-of-identity convention uses a nonzero coordinate space.","\\textbf{The preconditioning corollary.} Define\n\\[\nq(\\kappa)=\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}.\n\\]\nMonotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","onditioning corollary.} Define\n\\[\nq(\\kappa)=\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}.\n\\]\nMonotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","t\\kappa-1}{\\sqrt\\kappa+1}.\n\\]\nMonotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","Monotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","\\subsection{Compact factors from actual column spaces}\\label{supp6:subsection:001}\\textit{Modules CompactRangeFactors and SmallCoreTransfer}\\par\nLet\n\\[\nL\\in\\C^{m\\times t},\\qquad R\\in\\C^{n\\times t},\\qquad A=LR^*.\n\\]\nUse the ordinary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","tit{Modules CompactRangeFactors and SmallCoreTransfer}\\par\nLet\n\\[\nL\\in\\C^{m\\times t},\\qquad R\\in\\C^{n\\times t},\\qquad A=LR^*.\n\\]\nUse the ordinary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","s CompactRangeFactors and SmallCoreTransfer}\\par\nLet\n\\[\nL\\in\\C^{m\\times t},\\qquad R\\in\\C^{n\\times t},\\qquad A=LR^*.\n\\]\nUse the ordinary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","inary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","The checked identities are\n\\[\n\\boxed{Q_L^*Q_L=I,\\quad L=Q_LR_L,\\quad R_L=Q_L^*L,}\n\\]\nwith the analogous identities on the right. The dimension theorem is not an input:\n\\[\n\\boxed{r_L=\\rank L,\\qquad r_R=\\rank R.}\n\\]\nThe coordinate factors have full row rank. The construction therefore handles rank-deficient input factors without retaining artificial column directions.","Define the small core by actual multiplication,\n\\[\nG=R_LR_R^*\\in\\C^{r_L\\times r_R}.\n\\]\nFinite matrix associativity and adjoint identities give\n\\[\n\\boxed{A=Q_LGQ_R^*,\\qquad \\rank G=\\rank A.}\n\\]\nThe rank identity follows from both multiplication inequalities and the left inverse obtained by compression. It is not inferred from agreeing dimensions.","\\subsection{Why a small-core optimum is a global optimum}\\label{supp6:subsection:002}\\textit{Module SmallCoreNorms; arbitrary ambient competitors}\\par\nFor rectangular isometries $Q,V$ define\n\\[\n\\mathcal L(G)=QGV^*,\\qquad \\mathcal C(B)=Q^*BV,\n\\qquad Q^*Q=I,\\quad V^*V=I.\n\\]\nThe following statements are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","ubsection:002}\\textit{Module SmallCoreNorms; arbitrary ambient competitors}\\par\nFor rectangular isometries $Q,V$ define\n\\[\n\\mathcal L(G)=QGV^*,\\qquad \\mathcal C(B)=Q^*BV,\n\\qquad Q^*Q=I,\\quad V^*V=I.\n\\]\nThe following statements are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","*V=I.\n\\]\nThe following statements are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","ents are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","For the Frobenius norm, trace cyclicity proves left and right isometric invariance. The proof also establishes the projection identity\n\\[\n\\norm{B-QQ^*B}_{\\mathrm F}^{2}\n=\\norm B_{\\mathrm F}^{2}-\\norm{Q^*B}_{\\mathrm F}^{2}.\n\\]\nNonnegativity of the left side gives left compression contractivity. The right version follows by adjoints. No norm-preservation or contractivity hypothesis is inserted.","Suppose $G_k$ is a best rank-at-most-$k$ approximation to $G$ in either norm. Take an \\emph{arbitrary} ambient matrix $B$ with $\\rank B\\leq k$. Compression is an admissible small-core competitor and\n\\[\n\\begin{aligned}\n\\norm{\\mathcal L(G)-\\mathcal L(G_k)}_\\nu\n&=\\norm{G-G_k}_\\nu","Suppose $G_k$ is a best rank-at-most-$k$ approximation to $G$ in either norm. Take an \\emph{arbitrary} ambient matrix $B$ with $\\rank B\\leq k$. Compression is an admissible small-core competitor and\n\\[\n\\begin{aligned}\n\\norm{\\mathcal L(G)-\\mathcal L(G_k)}_\\nu\n&=\\norm{G-G_k}_\\nu\\\\\n&\\leq\\norm{G-\\mathcal C(B)}_\\nu\\\\\n&=\\norm{\\mathcal C(\\mathcal L(G)-B)}_\\nu\\\\\n&\\leq\\norm{\\mathcal L(G)-B}_\\nu.\n\\end{aligned}\n\\]\nThe competitor $B$ is not required to lie in the supplied row and column spaces. The next two pages discharge the intermediate small-core optimality obligation, rather than leaving it as a hypothesis in the final theorem.","&\\leq\\norm{\\mathcal L(G)-B}_\\nu.\n\\end{aligned}\n\\]\nThe competitor $B$ is not required to lie in the supplied row and column spaces. The next two pages discharge the intermediate small-core optimality obligation, rather than leaving it as a hypothesis in the final theorem.","\\subsection{Spectral optimality from a kernel witness}\\label{supp6:subsection:003}\\textit{Modules DiagonalTruncation and SpectralOptimality}\\par\nLet $\\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0$ and\n\\[\nD=\\diag(\\sigma_1,\\ldots,\\sigma_d),\\qquad\nD_k=\\diag(\\sigma_1,\\ldots,\\sigma_{\\min(k,d)},0,\\ldots,0).\n\\]\nLean uses indices beginning at zero, while the formulas on this page use the conventional one-based notation. The rank of $D_k$ is proved to be at most $k$, even when retained singular values are zero.","l optimality from a kernel witness}\\label{supp6:subsection:003}\\textit{Modules DiagonalTruncation and SpectralOptimality}\\par\nLet $\\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0$ and\n\\[\nD=\\diag(\\sigma_1,\\ldots,\\sigma_d),\\qquad\nD_k=\\diag(\\sigma_1,\\ldots,\\sigma_{\\min(k,d)},0,\\ldots,0).\n\\]\nLean uses indices beginning at zero, while the formulas on this page use the conventional one-based notation. The rank of $D_k$ is proved to be at most $k$, even when retained singular values are zero.","ness}\\label{supp6:subsection:003}\\textit{Modules DiagonalTruncation and SpectralOptimality}\\par\nLet $\\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0$ and\n\\[\nD=\\diag(\\sigma_1,\\ldots,\\sigma_d),\\qquad\nD_k=\\diag(\\sigma_1,\\ldots,\\sigma_{\\min(k,d)},0,\\ldots,0).\n\\]\nLean uses indices beginning at zero, while the formulas on this page use the conventional one-based notation. The rank of $D_k$ is proved to be at most $k$, even when retained singular values are zero.","For $k<d$, take any $d\\times d$ matrix $B$ of rank at most $k$. Restrict its domain to the first $k+1$ coordinates using the actual coordinate inclusion $J$. Since\n\\[\n\\rank(BJ)\\leq\\rank B\\leq k<k+1,\n\\]\nrank-nullity supplies a nonzero vector $z$ with $BJz=0$. The vector $y=Jz$ is nonzero, satisfies $By=0$ and is supported in the first $k+1$ coordinates. The injection and the support property are checked separately.","Monotonicity of the diagonal entries gives\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","onicity of the diagonal entries gives\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","diagonal entries gives\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","ves\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","\\subsection{Frobenius optimality from kernel weights}\\label{supp6:subsection:004}\\textit{Modules KernelFrame and FrobeniusOptimality}\\par\nFor an arbitrary competitor $B$ with $\\rank B\\leq k$, construct an orthonormal matrix $Q$ spanning its actual kernel. Then\n\\[\nBQ=0,\\qquad Q^*Q=I,\\qquad q=\\dim\\ker B=d-\\rank B.\n\\]\nDefine the row weights\n\\[\nw_i=\\sum_{j=1}^{q}|Q_{ij}|^2.\n\\]\nThe proof derives\n\\[\n\\boxed{0\\leq w_i\\leq1,\\qquad \\sum_iw_i=q\\geq\\max(d-k,0).}\n\\]\nThe upper bound uses contraction of $Q^*$ on a coordinate vector. The weight sum follows from the trace of $Q^*Q$ and rank-nullity.","s KernelFrame and FrobeniusOptimality}\\par\nFor an arbitrary competitor $B$ with $\\rank B\\leq k$, construct an orthonormal matrix $Q$ spanning its actual kernel. Then\n\\[\nBQ=0,\\qquad Q^*Q=I,\\qquad q=\\dim\\ker B=d-\\rank B.\n\\]\nDefine the row weights\n\\[\nw_i=\\sum_{j=1}^{q}|Q_{ij}|^2.\n\\]\nThe proof derives\n\\[\n\\boxed{0\\leq w_i\\leq1,\\qquad \\sum_iw_i=q\\geq\\max(d-k,0).}\n\\]\nThe upper bound uses contraction of $Q^*$ on a coordinate vector. The weight sum follows from the trace of $Q^*Q$ and rank-nullity.","$B$ with $\\rank B\\leq k$, construct an orthonormal matrix $Q$ spanning its actual kernel. Then\n\\[\nBQ=0,\\qquad Q^*Q=I,\\qquad q=\\dim\\ker B=d-\\rank B.\n\\]\nDefine the row weights\n\\[\nw_i=\\sum_{j=1}^{q}|Q_{ij}|^2.\n\\]\nThe proof derives\n\\[\n\\boxed{0\\leq w_i\\leq1,\\qquad \\sum_iw_i=q\\geq\\max(d-k,0).}\n\\]\nThe upper bound uses contraction of $Q^*$ on a coordinate vector. The weight sum follows from the trace of $Q^*Q$ and rank-nullity.","For any nonnegative decreasing sequence $a_i$, weights with these properties satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","equence $a_i$, weights with these properties satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","ights with these properties satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","s satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","Set $a_i=\\sigma_i^2$. Right compression cannot increase the Frobenius norm, and $BQ=0$, hence\n\\[\n\\begin{aligned}\n\\norm{D-B}_{\\mathrm F}^{2}\n&\\geq\\norm{(D-B)Q}_{\\mathrm F}^{2}","Set $a_i=\\sigma_i^2$. Right compression cannot increase the Frobenius norm, and $BQ=0$, hence\n\\[\n\\begin{aligned}\n\\norm{D-B}_{\\mathrm F}^{2}\n&\\geq\\norm{(D-B)Q}_{\\mathrm F}^{2}\\\\\n&=\\norm{DQ}_{\\mathrm F}^{2}\n=\\sum_i\\sigma_i^2w_i\n\\geq\\sum_{i>k}\\sigma_i^2.\n\\end{aligned}\n\\]\nDirect calculation of $D-D_k$ attains equality. Thus the best-rank Frobenius assertion is proved from kernel geometry and finite sums, independently of the spectral lower-bound proof. Repeated and zero singular values require no uniqueness claim.","&=\\norm{DQ}_{\\mathrm F}^{2}\n=\\sum_i\\sigma_i^2w_i\n\\geq\\sum_{i>k}\\sigma_i^2.\n\\end{aligned}\n\\]\nDirect calculation of $D-D_k$ attains equality. Thus the best-rank Frobenius assertion is proved from kernel geometry and finite sums, independently of the spectral lower-bound proof. Repeated and zero singular values require no uniqueness claim.","\\subsection{Thin output factors and the final theorem}\\label{supp6:subsection:005}\\textit{Modules CompactTruncation and SmallCoreRecompression}\\par\nFor the actual core $G=R_LR_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","05}\\textit{Modules CompactTruncation and SmallCoreRecompression}\\par\nFor the actual core $G=R_LR_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","ompression}\\par\nFor the actual core $G=R_LR_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,\\\\0,&i\\ne j,\\end{cases}\n\\qquad 1\\leq i\\leq d,\\quad 1\\leq j\\leq h.\n\\]\nLean proves its index cardinality and $MM^*=D_k$. Define\n\\[\n\\boxed{L_k=Q_LWM,\\qquad R_k=Q_RZM.}\n\\]\nThe product equals $Q_LWD_kZ^*Q_R^*$ and has rank at most $k$. The principal theorem proves, for every ambient $B$ with $\\rank B\\leq k$,\n\\[\n\\boxed{\\norm{LR^*-L_kR_k^*}_\\nu\\leq\\norm{LR^*-B}_\\nu,\n\\quad\\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nIt also gives the exact errors\n\\[\n\\norm{LR^*-L_kR_k^*}_2=\n\\begin{cases}\\sigma_{k+1},&k<d,","0,&i\\ne j,\\end{cases}\n\\qquad 1\\leq i\\leq d,\\quad 1\\leq j\\leq h.\n\\]\nLean proves its index cardinality and $MM^*=D_k$. Define\n\\[\n\\boxed{L_k=Q_LWM,\\qquad R_k=Q_RZM.}\n\\]\nThe product equals $Q_LWD_kZ^*Q_R^*$ and has rank at most $k$. The principal theorem proves, for every ambient $B$ with $\\rank B\\leq k$,\n\\[\n\\boxed{\\norm{LR^*-L_kR_k^*}_\\nu\\leq\\norm{LR^*-B}_\\nu,\n\\quad\\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nIt also gives the exact errors\n\\[\n\\norm{LR^*-L_kR_k^*}_2=\n\\begin{cases}\\sigma_{k+1},&k<d,","uad 1\\leq i\\leq d,\\quad 1\\leq j\\leq h.\n\\]\nLean proves its index cardinality and $MM^*=D_k$. Define\n\\[\n\\boxed{L_k=Q_LWM,\\qquad R_k=Q_RZM.}\n\\]\nThe product equals $Q_LWD_kZ^*Q_R^*$ and has rank at most $k$. The principal theorem proves, for every ambient $B$ with $\\rank B\\leq k$,\n\\[\n\\boxed{\\norm{LR^*-L_kR_k^*}_\\nu\\leq\\norm{LR^*-B}_\\nu,\n\\quad\\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nIt also gives the exact errors\n\\[\n\\norm{LR^*-L_kR_k^*}_2=\n\\begin{cases}\\sigma_{k+1},&k<d,\\\\0,&k\\geq d,\\end{cases}\n\\qquad\n\\norm{LR^*-L_kR_k^*}_{\\mathrm F}\n=\\left(\\sum_{i>k}\\sigma_i^2\\right)^{1/2}.\n\\]\nBoth the squared and square-root Frobenius formulas are checked. All these statements are combined in\\par\n\\path{KomorebiPaperVIII.SmallCore.small_core_optimal_recompression}.","\\textbf{Endpoints.} Rank zero produces the zero output, while $k\\geq d$ reconstructs the original product. An empty SVD certificate is constructed for a zero core, and diagonal SVD certificates are constructed explicitly. Zero singular values are never inverted. At repeated singular values the result asserts optimality, not uniqueness of the minimizer."]}