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complete invariants}.\n\\end{gathered}}\n\\]\nEach arrow carries a proof obligation. A semantic analogy is not a bridge. A bridge is not reversible merely because formulas exist in both directions. Local completeness is not global completeness until the local isomorphisms themselves satisfy descent.","This section expands the common formal core used by every later atlas construction.  The purpose is not to force every domain into one category.  It is to separate four layers that are often conflated:\n\\[\n\\boxed{\n\\text{typed syntax}\n\\longrightarrow\n\\text{resolved coordinates}\n\\longrightarrow\n\\text{profile-preserving realization}\n\\longrightarrow\n\\text{domain semantics}.}\n\\]\nThe first two layers are algebraic.  The third records which algebraic or universal constructions survive a realization.  The fourth supplies domain-specific hypotheses such as completeness, positivity, convergence, integrality, measurability, or analytic continuation.  A cross-domain statement is a theorem only when all four layers are present.","\\begin{definition}[Finite resolved signature]\n\\label{def:finite-resolved-signature}\nFix a commutative coefficient ring \\(\\kk\\).  A finite resolved signature is a tuple\n\\[\n\\mathbb K_{\\rm fin}\n=\n(Q,I,\\mathsf P,\\mathsf{Ref},\\mathsf{Comp},\\mathsf{Cert})\n\\]\nwith the following data.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X","\\begin{definition}[Finite resolved signature]\n\\label{def:finite-resolved-signature}\nFix a commutative coefficient ring \\(\\kk\\).  A finite resolved signature is a tuple\n\\[\n\\mathbb K_{\\rm fin}\n=\n(Q,I,\\mathsf P,\\mathsf{Ref},\\mathsf{Comp},\\mathsf{Cert})\n\\]\nwith the following data.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X","\\begin{definition}[Finite resolved signature]\n\\label{def:finite-resolved-signature}\nFix a commutative coefficient ring \\(\\kk\\).  A finite resolved signature is a tuple\n\\[\n\\mathbb K_{\\rm fin}\n=\n(Q,I,\\mathsf P,\\mathsf{Ref},\\mathsf{Comp},\\mathsf{Cert})\n\\]\nwith the following data.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}","label{def:finite-resolved-signature}\nFix a commutative coefficient ring \\(\\kk\\).  A finite resolved signature is a tuple\n\\[\n\\mathbb K_{\\rm fin}\n=\n(Q,I,\\mathsf P,\\mathsf{Ref},\\mathsf{Comp},\\mathsf{Cert})\n\\]\nwith the following data.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doct…",").  A finite resolved signature is a tuple\n\\[\n\\mathbb K_{\\rm fin}\n=\n(Q,I,\\mathsf P,\\mathsf{Ref},\\mathsf{Comp},\\mathsf{Cert})\n\\]\nwith the following data.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a mul…","P,\\mathsf{Ref},\\mathsf{Comp},\\mathsf{Cert})\n\\]\nwith the following data.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence th…","numerate}[label=\\textup{(\\roman*)}]\n\\item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types…","item \\(Q=(Q_0,Q_1,s,t)\\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies it…","nite quiver.  Its vertices are sorts and its arrows are typed primitive operations.\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, …",".\n\\item \\(I\\triangleleft \\kk Q\\) is a finitely generated two-sided ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.\n\\end{enumerate}\n\\end{d…","ideal generated by \\(\\kk\\)-linear combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.\n\\end{enumerate}\n\\end{definition}","ar combinations of paths with common source and common target.\n\\item \\(\\mathsf P\\) is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.\n\\end{enumerate}\n\\end{definition}","is a finite list of packet declarations.  A declaration on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.\n\\end{enumerate}\n\\end{definition}","on a sort \\(X\\) consists of endomorphism words \\(p_{X,r,i}\\) and the relations\n\\[\np_{X,r,i}p_{X,r,j}=\\delta_{ij}p_{X,r,i},\n\\qquad\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.\n\\end{enumerate}\n\\end{definition}","d\n\\sum_i p_{X,r,i}=1_X.\n\\]\n\\item \\(\\mathsf{Ref}\\) is a finite list of refinement declarations.  For \\(s\\succeq r\\), an address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.\n\\end{enumerate}\n\\end{definition}","address map\n\\[\n\\rho_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nis accompanied by the equations\n\\[\np_{X,r,i}\n=\n\\sum_{\\rho_{sr}(j)=i}p_{X,s,j}.\n\\]\n\\item \\(\\mathsf{Comp}\\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \\(C^*\\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.\n\\item \\(\\mathsf{Cert}\\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Realization of a finite resolved signature]\n\\label{def:resolved-realization}\nLet \\(\\mathcal C\\) be a \\(\\kk\\)-linear category.  A realization of \\(\\mathbb K_{\\rm fin}\\) in \\(\\mathcal C\\) assigns an object \\(M_X\\) to every vertex \\(X\\in Q_0\\) and a morphism\n\\[\nM_a:M_{s(a)}\\longrightarrow M_{t(a)}\n\\]\nto every arrow \\(a\\in Q_1\\), such that all relations in \\(I\\), all packet equations, and all declared refinement equations hold.  Completion and certificate tags are interpreted only when the realization profile explicitly includes them.\n\\end{definition}","\\begin{theorem}[Typed realization and module equivalence, imported]\n\\label{thm:typed-realization-module-equivalence}\nLet\n\\[\n\\Lambda_{\\mathbb K}:=\\kk Q/I.\n\\]\nIn \\(\\kk\\)-modules, the category of typed realizations of the linear grammar \\((Q,I)\\) is equivalent to the category of unital left \\(\\Lambda_{\\mathbb K}\\)-modules.  Under the equivalence, the vertex idempotent \\(e_X\\) acts as the projector onto the component of sort \\(X\\), and each arrow acts in its declared source-target block.\n\\end{theorem}","zation and module equivalence, imported]\n\\label{thm:typed-realization-module-equivalence}\nLet\n\\[\n\\Lambda_{\\mathbb K}:=\\kk Q/I.\n\\]\nIn \\(\\kk\\)-modules, the category of typed realizations of the linear grammar \\((Q,I)\\) is equivalent to the category of unital left \\(\\Lambda_{\\mathbb K}\\)-modules.  Under the equivalence, the vertex idempotent \\(e_X\\) acts as the projector onto the component of sort \\(X\\), and each arrow acts in its declared source-target block.\n\\end{theorem}","\\begin{proof}\nA typed realization defines the action of a path by composition and then extends \\(\\kk\\)-linearly.  Every element of \\(I\\) acts by zero, so the action factors through \\(\\Lambda_{\\mathbb K}\\).  Conversely, for a unital module \\(M\\), the pairwise orthogonal vertex idempotents give\n\\[\nM=\\bigoplus_{X\\in Q_0}e_XM,\n\\]\nand an arrow \\(a:X\\to Y\\) maps \\(e_XM\\) into \\(e_YM\\).  These constructions are inverse up to their canonical natural isomorphisms.  This is the typed-module theorem of Paper I \\cite{P1}.\n\\end{proof}","\\begin{example}[Two-sort transfer grammar]\n\\label{ex:two-sort-transfer}\nTake two sorts \\(X,Y\\), one arrow \\(a:X\\to Y\\), and no nontrivial relation.  A realization is a single typed map\n\\[\nM_X\\xrightarrow{M_a}M_Y.\n\\]\nAfter collapsing to \\(M_X\\oplus M_Y\\), the vertex idempotents are coordinate packets and \\(M_a\\) is one off-diagonal transfer block.  The smallest typed grammar therefore already contains the distinction between object packets and transfer channels.\n\\end{example}","\\begin{definition}[Finite packet system]\n\\label{def:finite-packet-system-expanded}\nLet \\(X\\) be an object of a preadditive category.  A finite packet system on \\(X\\) is a family\n\\[\n\\mathcal P=(p_i)_{i\\in I}\\subseteq\\operatorname{End}(X)\n\\]\nsatisfying\n\\[\np_i^2=p_i,\n\\qquad\np_ip_j=0\\quad(i\\ne j),\n\\qquad\n\\sum_{i\\in I}p_i=1_X.\n\\]\nNo self-adjointness or metric orthogonality is assumed unless stated separately.\n\\end{definition}","Assume now that \\(\\mathcal C\\) is additive and idempotent complete.  Choose a splitting\n\\[\nX_i\\xrightarrow{\\iota_i}X\\xrightarrow{\\pi_i}X_i,\n\\qquad\n\\iota_i\\pi_i=p_i,\n\\qquad\n\\pi_i\\iota_i=1_{X_i}.\n\\]\nThe packet equations imply\n\\[\n\\pi_i\\iota_j=0\\quad(i\\ne j).\n\\]","\\begin{definition}[Packet analysis and synthesis]\n\\label{def:packet-analysis-synthesis-expanded}\nThe packet analysis and synthesis maps are\n\\[\n\\Analysis_{\\mathcal P}\n:=\n(\\pi_i)_{i\\in I}:\nX\\longrightarrow\\bigoplus_{i\\in I}X_i,\n\\]\n\\[\n\\Synthesis_{\\mathcal P}\n:=\n(\\iota_i)_{i\\in I}:\n\\bigoplus_{i\\in I}X_i\\longrightarrow X.\n\\]\n\\end{definition}","\\begin{theorem}[Exact packet-coordinate isomorphism, imported]\n\\label{thm:exact-packet-coordinate-expanded}\nFor every finite packet system in an additive idempotent-complete category,\n\\[\n\\boxed{\n\\Synthesis_{\\mathcal P}\\Analysis_{\\mathcal P}=1_X,\n\\qquad\n\\Analysis_{\\mathcal P}\\Synthesis_{\\mathcal P}\n=1_{\\oplus_iX_i}.}\n\\]\nConsequently,\n\\[\nX\\cong\\bigoplus_{i\\in I}X_i.\n\\]\n\\end{theorem}","\\begin{proof}\nThe first identity is\n\\[\n\\Synthesis_{\\mathcal P}\\Analysis_{\\mathcal P}\n=\n\\sum_i\\iota_i\\pi_i\n=\n\\sum_i p_i\n=\n1_X.\n\\]\nThe \\((i,j)\\)-entry of the second composite is \\(\\pi_i\\iota_j\\), which equals \\(1_{X_i}\\) for \\(i=j\\) and zero otherwise.  This is the exact packet-coordinate theorem of Paper I \\cite{P1}.\n\\end{proof}","\\begin{corollary}[Packetwise equality]\n\\label{cor:packetwise-equality-expanded}\nFor parallel morphisms \\(f,g:Z\\to X\\),\n\\[\nf=g\n\\quad\\Longleftrightarrow\\quad\n\\pi_i f=\\pi_i g\\text{ for every }i.\n\\]\nFor parallel morphisms \\(f,g:X\\to Z\\),\n\\[\nf=g\n\\quad\\Longleftrightarrow\\quad\nf\\iota_i=g\\iota_i\\text{ for every }i.\n\\]\n\\end{corollary}","\\begin{corollary}[Universal mapping property of packet coordinates]\n\\label{cor:packet-universal-expanded}\nFor every object \\(Z\\), restriction to the packet objects gives natural bijections\n\\[\n\\operatorname{Hom}(X,Z)\n\\cong\n\\prod_i\\operatorname{Hom}(X_i,Z),\n\\qquad\n\\operatorname{Hom}(Z,X)\n\\cong\n\\prod_i\\operatorname{Hom}(Z,X_i),\n\\]\nwhere the products are finite.  Packetwise maps therefore assemble uniquely into a global map.\n\\end{corollary}","\\begin{definition}[Mixed transfer blocks]\n\\label{def:mixed-transfer-blocks-expanded}\nLet \\(\\mathcal P=(p_i)\\) be a packet system on \\(X\\), let \\(\\mathcal Q=(q_a)\\) be a packet system on \\(Y\\), and choose splittings \\((\\iota_i,\\pi_i)\\) and \\((\\jmath_a,\\varpi_a)\\).  For \\(f:X\\to Y\\), define the packet-to-packet map\n\\[\nf_{ai}:=\\varpi_a f\\iota_i:X_i\\longrightarrow Y_a.\n\\]\nIts ambient corner is\n\\[\nq_afp_i=\\jmath_a f_{ai}\\pi_i.\n\\]\n\\end{definition}","\\begin{theorem}[Exact transfer reconstruction and block composition, imported]\n\\label{thm:transfer-reconstruction-expanded}\nEvery morphism \\(f:X\\to Y\\) has the exact expansion\n\\[\n\\boxed{\nf\n=\n\\sum_{a,i}q_afp_i\n=\n\\sum_{a,i}\\jmath_a f_{ai}\\pi_i.}\n\\]\nIf \\(g:Y\\to Z\\) is resolved by a packet system \\(\\mathcal R=(r_c)\\), then\n\\[\n\\boxed{\n(gf)_{ci}\n=\n\\sum_a g_{ca}f_{ai}.}\n\\]\n\\end{theorem}","\\begin{proof}\nInsert the finite packet identities on both sides of \\(f\\):\n\\[\nf\n=\n\\left(\\sum_aq_a\\right)f\\left(\\sum_ip_i\\right).\n\\]\nFor composition, insert \\(1_Y=\\sum_aq_a\\) between \\(g\\) and \\(f\\).  This is the transfer-block reconstruction and multiplication law of Paper I \\cite{P1}.\n\\end{proof}","\\begin{corollary}[Path expansion]\n\\label{cor:packet-path-expansion}\nFor a composable string\n\\[\nX_0\\xrightarrow{f_1}X_1\\xrightarrow{f_2}\\cdots\\xrightarrow{f_m}X_m\n\\]\nwith packet systems on every \\(X_j\\), each source-target block of \\(f_m\\cdots f_1\\) is the sum over all compatible intermediate packet paths of the corresponding block products.\n\\end{corollary}","\\begin{example}[Exact oblique packets]\n\\label{ex:oblique-packets}\nOver any field, set\n\\[\np_1=\n\\begin{pmatrix}\n1&1\\\\0&0\n\\end{pmatrix},\n\\qquad\np_2=\n\\begin{pmatrix}\n0&-1\\\\0&1\n\\end{pmatrix}.\n\\]\nThen \\(p_1^2=p_1\\), \\(p_2^2=p_2\\), \\(p_1p_2=p_2p_1=0\\), and \\(p_1+p_2=I\\).  The decomposition of \\(\\kk^2\\) is exact, although the packets are not orthogonal for the standard inner product.  This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.\n\\end{example}","0&1\n\\end{pmatrix}.\n\\]\nThen \\(p_1^2=p_1\\), \\(p_2^2=p_2\\), \\(p_1p_2=p_2p_1=0\\), and \\(p_1+p_2=I\\).  The decomposition of \\(\\kk^2\\) is exact, although the packets are not orthogonal for the standard inner product.  This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.\n\\end{example}","\\begin{definition}[Finite refinement]\n\\label{def:finite-refinement-expanded}\nA packet system \\(\\mathcal P^s=(p^s_j)_{j\\in I_s}\\) refines \\(\\mathcal P^r=(p^r_i)_{i\\in I_r}\\) when there is a surjection\n\\[\n\\rho_{sr}:I_s\\twoheadrightarrow I_r\n\\]\nsuch that\n\\[\np^r_i\n=\n\\sum_{\\rho_{sr}(j)=i}p^s_j.\n\\]\nThe map \\(\\rho_{sr}\\) records the address of each fine packet inside its coarse ancestor.\n\\end{definition}","\\begin{proposition}[Aggregation and transitivity]\n\\label{prop:refinement-aggregation-transitivity}\nA finite aggregation of pairwise annihilating packets is again a packet system.  If \\(t\\succeq s\\succeq r\\), then refinement is transitive with address map\n\\[\n\\rho_{tr}=\\rho_{sr}\\circ\\rho_{ts}.\n\\]\n\\end{proposition}","\\begin{proof}\nFor an aggregated packet, idempotence follows because all cross terms vanish.  Distinct aggregates annihilate each other, and the total sum remains the identity.  Substituting the \\(t\\)-level expansion into the \\(s\\)-level expansion gives the composite address law.\n\\end{proof}","\\begin{theorem}[Exact refinement law, imported]\n\\label{thm:exact-refinement-expanded}\nLet \\(p^r_i=\\sum_{\\rho_{sr}(j)=i}p^s_j\\) and \\(q^u_a=\\sum_{\\sigma_{vu}(b)=a}q^v_b\\).  For every morphism \\(f:X\\to Y\\),\n\\[\n\\boxed{\nq^u_afp^r_i\n=\n\\sum_{\\sigma_{vu}(b)=a}\n\\sum_{\\rho_{sr}(j)=i}\nq^v_bfp^s_j.}\n\\]\nCoarsening the fine block product of two composable maps gives the coarse block product.\n\\end{theorem}","\\begin{proof}\nInsert the two refinement identities on the left and right of \\(f\\).  For composition, expand both maps at the fine level, multiply, and sum over descendant intermediate packets.  The fine intermediate projectors sum to the coarse intermediate projector.  This is the exact refinement law of Resolution Theory \\cite{RT}.\n\\end{proof}","\\begin{definition}[Support compression and residual]\n\\label{def:support-compression-expanded}\nFor packet systems \\(\\mathcal P\\) on \\(X\\) and \\(\\mathcal Q\\) on \\(Y\\), let \\(E\\subseteq I_{\\mathcal Q}\\times I_{\\mathcal P}\\).  Define\n\\[\nC_E(f):=\\sum_{(a,i)\\in E}q_afp_i,\n\\qquad\nR_E(f):=f-C_E(f).\n\\]\nThe first term is the retained transfer support and the second is its named residual.\n\\end{definition}","\\begin{proposition}[Boolean algebra of block selectors]\n\\label{prop:block-selector-algebra}\nAs endomorphisms of the \\(\\kk\\)-module \\(\\operatorname{Hom}(X,Y)\\), the selectors satisfy\n\\[\nC_E^2=C_E,\n\\qquad\nC_EC_F=C_{E\\cap F},\n\\qquad\nR_E=I-C_E.\n\\]\nIn particular,\n\\[\nC_ER_E=R_EC_E=0,\n\\qquad\nf=C_E(f)+R_E(f).\n\\]\n\\end{proposition}","\\begin{proof}\nApplying \\(C_E\\) to a block \\(q_bfp_j\\) retains it exactly when \\((b,j)\\in E\\).  Pairwise annihilation removes every mismatched source or target index.  The formulas follow block by block.\n\\end{proof}","\\begin{definition}[Relational support product]\n\\label{def:relational-support-product}\nFor \\(E\\subseteq I_Y\\times I_X\\) and \\(F\\subseteq I_Z\\times I_Y\\), define\n\\[\nF\\circ E\n:=\n\\{(c,i):\\text{there exists }a\\in I_Y\n\\text{ with }(c,a)\\in F\\text{ and }(a,i)\\in E\\}.\n\\]\n\\end{definition}","\\begin{proposition}[Support propagation under composition]\n\\label{prop:support-propagation}\nIf \\(f=C_E(f)\\) and \\(g=C_F(g)\\), then\n\\[\ngf=C_{F\\circ E}(gf).\n\\]\nThus exact zero patterns propagate by relational composition.\n\\end{proposition}","\\begin{proof}\nEvery block of \\(gf\\) is a sum \\(\\sum_a g_{ca}f_{ai}\\).  A nonzero summand can occur only along an intermediate address \\(a\\) belonging to the relational product.\n\\end{proof}","\\begin{proposition}[Exact residual product identity]\n\\label{prop:exact-residual-product}\nLet\n\\[\nf_E=C_E(f),\\quad r_f=R_E(f),\n\\qquad\ng_F=C_F(g),\\quad r_g=R_F(g).\n\\]\nThen\n\\[\n\\boxed{\ngf-g_Ff_E\n=\nr_gf_E+g_Fr_f+r_gr_f.}\n\\]\nIf the morphisms lie in a normed operator algebra with a submultiplicative norm, then\n\\[\n\\|gf-g_Ff_E\\|\n\\le\n\\|r_g\\|\\,\\|f_E\\|\n+\\|g_F\\|\\,\\|r_f\\|\n+\\|r_g\\|\\,\\|r_f\\|.\n\\]\n\\end{proposition}","\\begin{proof}\nSubstitute \\(g=g_F+r_g\\) and \\(f=f_E+r_f\\) and expand.  The norm estimate is immediate from the triangle inequality and submultiplicativity.\n\\end{proof}","\\begin{theorem}[Hilbert--Schmidt optimality, imported]\n\\label{thm:hs-optimality-expanded}\nAssume \\(X,Y\\) are Hilbert spaces, the packet idempotents are orthogonal projectors, and \\(f:X\\to Y\\) is Hilbert--Schmidt.  Then \\(C_E\\) is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and\n\\[\n\\|R_E(f)\\|_{\\mathrm{HS}}^2\n=\n\\sum_{(a,i)\\notin E}\\|q_afp_i\\|_{\\mathrm{HS}}^2.\n\\]\nConsequently, \\(C_E(f)\\) is the unique best Hilbert--Schmidt approximation to \\(f\\) among operators supported on \\(E\\).\n\\end{theorem}","orem}[Hilbert--Schmidt optimality, imported]\n\\label{thm:hs-optimality-expanded}\nAssume \\(X,Y\\) are Hilbert spaces, the packet idempotents are orthogonal projectors, and \\(f:X\\to Y\\) is Hilbert--Schmidt.  Then \\(C_E\\) is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and\n\\[\n\\|R_E(f)\\|_{\\mathrm{HS}}^2\n=\n\\sum_{(a,i)\\notin E}\\|q_afp_i\\|_{\\mathrm{HS}}^2.\n\\]\nConsequently, \\(C_E(f)\\) is the unique best Hilbert--Schmidt approximation to \\(f\\) among operators supported on \\(E\\).\n\\end{theorem}","ality-expanded}\nAssume \\(X,Y\\) are Hilbert spaces, the packet idempotents are orthogonal projectors, and \\(f:X\\to Y\\) is Hilbert--Schmidt.  Then \\(C_E\\) is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and\n\\[\n\\|R_E(f)\\|_{\\mathrm{HS}}^2\n=\n\\sum_{(a,i)\\notin E}\\|q_afp_i\\|_{\\mathrm{HS}}^2.\n\\]\nConsequently, \\(C_E(f)\\) is the unique best Hilbert--Schmidt approximation to \\(f\\) among operators supported on \\(E\\).\n\\end{theorem}","Hilbert spaces, the packet idempotents are orthogonal projectors, and \\(f:X\\to Y\\) is Hilbert--Schmidt.  Then \\(C_E\\) is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and\n\\[\n\\|R_E(f)\\|_{\\mathrm{HS}}^2\n=\n\\sum_{(a,i)\\notin E}\\|q_afp_i\\|_{\\mathrm{HS}}^2.\n\\]\nConsequently, \\(C_E(f)\\) is the unique best Hilbert--Schmidt approximation to \\(f\\) among operators supported on \\(E\\).\n\\end{theorem}","\\begin{definition}[Constructor universe]\n\\label{def:constructor-universe}\nFix a declared universe \\(\\mathfrak C\\) of constructors and universal properties.  Typical entries are\n\\[\n\\begin{gathered}\n0,\\ 1,\\ \\text{finite sums},\\ \\text{composition},\\ \\text{scalar action},\\ \\text{biproducts},","\\begin{definition}[Constructor universe]\n\\label{def:constructor-universe}\nFix a declared universe \\(\\mathfrak C\\) of constructors and universal properties.  Typical entries are\n\\[\n\\begin{gathered}\n0,\\ 1,\\ \\text{finite sums},\\ \\text{composition},\\ \\text{scalar action},\\ \\text{biproducts},\\\\\n\\text{kernels},\\ \\text{cokernels},\\ \\text{quotients},\\ \\text{tensor products},\\\\\n\\text{packet idempotents},\\ \\text{transfer blocks},\\ \\text{refinements},\\ \\text{named residuals},\\\\\n\\text{finite limits},\\ \\text{finite colimits},\\ \\text{topological completions},\\ \\text{adjoints}.\n\\end{gathered}\n\\]\nThe list may be enlarged, but every new entry must specify the exact identity or universal property to be preserved.\n\\end{definition}","\\begin{definition}[Verified preservation profile]\n\\label{def:verified-preservation-profile-expanded}\nA verified preservation profile \\(\\Pi_F\\) for a functor \\(F:\\mathcal C\\to\\mathcal D\\) consists of a subset of \\(\\mathfrak C\\) together with a proof or declared comparison isomorphism showing that \\(F\\) preserves each listed constructor.  We write \\(c\\in\\Pi_F\\) only after the corresponding preservation claim has been verified.\n\\end{definition}","\\begin{definition}[Proof-support certificate]\n\\label{def:proof-support-certificate}\nA proof-support certificate for a finite proof \\(\\mathcal P\\) is a finite subset\n\\[\n\\operatorname{csupp}(\\mathcal P)\\subseteq\\mathfrak C\n\\]\ncontaining every constructor and universal property used in the proof.  The certificate need not be minimal.  It must be sufficient and auditable.\n\\end{definition}","\\begin{proposition}[Common profile and composition]\n\\label{prop:common-profile-composition}\nLet \\(F:\\mathcal C\\to\\mathcal D\\) and \\(G:\\mathcal D\\to\\mathcal E\\).  Every constructor preserved by both functors, with compatible chosen comparison data, is preserved by \\(GF\\).  For a finite diagram of realizations and bridges, the intersection\n\\[\n\\Pi_{\\rm common}:=\\bigcap_F\\Pi_F\n\\]\nis therefore a valid common profile for every proof transported through the whole diagram.\n\\end{proposition}","\\begin{theorem}[Equational realization principle, imported]\n\\label{thm:equational-realization-expanded}\nLet \\(F:\\mathcal C\\to\\mathcal D\\) be additive between preadditive categories.  Every equality of morphism expressions built from declared generators by finite sums, additive inverses, identities, zero maps, and composition is preserved by \\(F\\).  Scalar multiplication is preserved when the categories are enriched over the same coefficient ring and \\(F\\) is linear.  If a proof additionally invokes a chosen biproduct, kernel, cokernel, quotient, tensor product, completion, or another universal construction, the proof transports only when that constructor belongs to \\(\\Pi_F\\).\n\\end{theorem}","ed]\n\\label{thm:equational-realization-expanded}\nLet \\(F:\\mathcal C\\to\\mathcal D\\) be additive between preadditive categories.  Every equality of morphism expressions built from declared generators by finite sums, additive inverses, identities, zero maps, and composition is preserved by \\(F\\).  Scalar multiplication is preserved when the categories are enriched over the same coefficient ring and \\(F\\) is linear.  If a proof additionally invokes a chosen biproduct, kernel, cokernel, quotient, tensor product, completion, or another universal construction, the proof transports only when that constructor belongs to \\(\\Pi_F\\).\n\\end{theorem}","e inverses, identities, zero maps, and composition is preserved by \\(F\\).  Scalar multiplication is preserved when the categories are enriched over the same coefficient ring and \\(F\\) is linear.  If a proof additionally invokes a chosen biproduct, kernel, cokernel, quotient, tensor product, completion, or another universal construction, the proof transports only when that constructor belongs to \\(\\Pi_F\\).\n\\end{theorem}","\\begin{proposition}[Profile-safe theorem transport]\n\\label{prop:profile-safe-theorem-transport}\nLet \\(\\mathcal P\\) prove a statement \\(T\\) in the resolved syntax, and let \\(F\\) be a realization or bridge.  Suppose\n\\[\n\\operatorname{csupp}(\\mathcal P)\\subseteq\\Pi_F.\n\\]\nAfter every additional domain hypothesis used by \\(T\\) has been verified in the target, \\(F(T)\\) is a valid target theorem.  If the inclusion fails, the source proof does not by itself establish the target statement.\n\\end{proposition}","\\begin{definition}[Linear grammar category]\n\\label{def:linear-grammar-category}\nFor a finite linear resolved presentation \\(\\mathbb K_{\\rm lin}=(Q,I,\\mathsf P,\\mathsf{Ref})\\), let\n\\[\n\\mathcal A_{\\mathbb K}:=\\kk Q/I.\n\\]\nThis is the \\(\\kk\\)-linear category whose objects are the vertices of \\(Q\\) and whose morphisms are \\(\\kk\\)-linear combinations of typed paths modulo the declared relations.\n\\end{definition}","\\begin{definition}[Finite additive hull]\n\\label{def:additive-hull-expanded}\nThe category \\(\\Add(\\mathcal A_{\\mathbb K})\\) has finite tuples\n\\[\nA=(A_1,\\dots,A_m)\n\\]\nas objects.  A morphism \\(A\\to B=(B_1,\\dots,B_n)\\) is an \\(n\\times m\\) matrix with entries in \\(\\mathcal A_{\\mathbb K}(A_j,B_i)\\).  Composition is matrix multiplication using composition and addition in \\(\\mathcal A_{\\mathbb K}\\).\n\\end{definition}","\\begin{definition}[Karoubi envelope]\n\\label{def:karoubi-envelope-expanded}\nThe Karoubi envelope \\(\\Kar(\\Add(\\mathcal A_{\\mathbb K}))\\) has pairs \\((A,e)\\), where \\(e:A\\to A\\) is idempotent, as objects.  A morphism\n\\[\nf:(A,e)\\longrightarrow(B,d)\n\\]\nis a morphism \\(f:A\\to B\\) satisfying\n\\[\nf=dfe.\n\\]\nThe identity on \\((A,e)\\) is \\(e\\).\n\\end{definition}","\\begin{theorem}[Finite additive-Karoubi resolved envelope]\n\\label{thm:additive-karoubi-envelope}\nThe category\n\\[\n\\boxed{\n\\Syn_{\\mathrm{ak}}(\\mathbb K_{\\rm lin})\n:=\n\\Kar\\!\\bigl(\\Add(\\kk Q/I)\\bigr)}\n\\]\nis small, additive, idempotent complete, and \\(\\kk\\)-linear.  For every additive idempotent-complete \\(\\kk\\)-linear category \\(\\mathcal C\\), restriction to the generating quiver induces an equivalence\n\\[\n\\Fun_{\\kk}^{\\oplus}\n\\bigl(\\Syn_{\\mathrm{ak}}(\\mathbb K_{\\rm lin}),\\mathcal C\\bigr)\n\\simeq\n\\Mod_{\\mathrm{ak}}(\\mathbb K_{\\rm lin},\\mathcal C),\n\\]\nwhere the right side is the category of typed realizations satisfying all declared relations and carrying every declared packet idempotent to a split idempotent.\n\\end{theorem}","gin{theorem}[Finite additive-Karoubi resolved envelope]\n\\label{thm:additive-karoubi-envelope}\nThe category\n\\[\n\\boxed{\n\\Syn_{\\mathrm{ak}}(\\mathbb K_{\\rm lin})\n:=\n\\Kar\\!\\bigl(\\Add(\\kk Q/I)\\bigr)}\n\\]\nis small, additive, idempotent complete, and \\(\\kk\\)-linear.  For every additive idempotent-complete \\(\\kk\\)-linear category \\(\\mathcal C\\), restriction to the generating quiver induces an equivalence\n\\[\n\\Fun_{\\kk}^{\\oplus}\n\\bigl(\\Syn_{\\mathrm{ak}}(\\mathbb K_{\\rm lin}),\\mathcal C\\bigr)\n\\simeq\n\\Mod_{\\mathrm{ak}}(\\mathbb K_{\\rm lin},\\mathcal C),\n\\]\nwhere the right side is the category of typed realizations satisfying all declared relations and carrying every declared packet idempotent to a split idempotent.\n\\end{theorem}","\\emph{Free typed stage.}  The free \\(\\kk\\)-linear category on \\(Q\\) is universal for assigning objects to vertices and morphisms to arrows.","\\emph{Relation stage.}  Quotienting by \\(I\\) imposes exactly the declared linear, packet, and refinement equations.  A functor out of \\(\\kk Q/I\\) is therefore the same as a typed realization satisfying those equations.","\\emph{Additive stage.}  Every \\(\\kk\\)-linear functor from \\(\\kk Q/I\\) into an additive category extends, uniquely up to canonical natural isomorphism, to an additive functor from the finite additive hull by sending a tuple to the corresponding biproduct and a matrix to its induced biproduct map.","\\emph{Idempotent-splitting stage.}  Every additive functor into an idempotent-complete category extends, again uniquely up to canonical natural isomorphism, across the Karoubi envelope by sending \\((A,e)\\) to the chosen image of the split idempotent \\(F(e)\\).  Morphisms satisfying \\(f=dfe\\) descend to the corresponding split summands.","\\begin{proposition}[Fully faithful grammar embedding]\n\\label{prop:grammar-embedding}\nThe canonical composite\n\\[\n\\mathcal A_{\\mathbb K}\n\\longrightarrow\n\\Add(\\mathcal A_{\\mathbb K})\n\\longrightarrow\n\\Kar(\\Add(\\mathcal A_{\\mathbb K}))\n\\]\nis fully faithful.  Thus the envelope adds finite biproducts and split summands without identifying two previously distinct grammar morphisms.\n\\end{proposition}","\\begin{proof}\nThe first functor sends an object to the one-term tuple and preserves its hom set exactly.  The second sends \\(A\\) to \\((A,1_A)\\); its morphism condition is \\(f=1_Bf1_A\\), which imposes no additional relation.\n\\end{proof}","\\begin{corollary}[Declared packets become literal objects]\n\\label{cor:packet-objects-literal}\nEvery declared idempotent \\(p:X\\to X\\) defines an object \\((X,p)\\) of \\(\\Syn_{\\mathrm{ak}}(\\mathbb K_{\\rm lin})\\).  A complete packet family \\((p_i)\\) gives the canonical isomorphism\n\\[\n(X,1_X)\\cong\\bigoplus_i(X,p_i).\n\\]\n\\end{corollary}","\\begin{corollary}[Linear resolved strata as model categories]\n\\label{cor:linear-strata-model-categories}\nEvery additive split-idempotent realization stratum of \\(\\mathbb K_{\\rm lin}\\) is a category of profile-preserving functors out of \\(\\Syn_{\\mathrm{ak}}(\\mathbb K_{\\rm lin})\\).\n\\end{corollary}","\\begin{example}[One operator and primary packets]\n\\label{ex:single-operator-envelope}\nTake one sort, one endomorphism \\(T\\), and one relation \\(\\mu(T)=0\\).  The linear grammar algebra is \\(\\kk[x]/(\\mu)\\).  When\n\\[\n\\mu=\\mu_1\\cdots\\mu_r\n\\]\nis a product of pairwise comaximal factors, the Chinese remainder idempotents define packet objects for the primary summands.  The envelope splits those idempotents formally.  Over a realization where the same factorization remains valid, the packet objects become the corresponding invariant primary sectors.\n\\end{example}","\\begin{definition}[Profile-indexed resolved envelope]\n\\label{def:profile-indexed-envelope}\nFor a preservation profile \\(\\Pi\\), a profile-indexed resolved envelope \\(\\Syn_\\Pi(\\mathbb K)\\), when it exists, is a universal source satisfying\n\\[\n\\Mod_\\Pi(\\mathbb K,\\mathcal C)\n\\simeq\n\\Fun_\\Pi(\\Syn_\\Pi(\\mathbb K),\\mathcal C)\n\\]\nfor every target category \\(\\mathcal C\\) carrying the structures named by \\(\\Pi\\).  The equivalence must preserve the chosen comparison data, not only the underlying functors.\n\\end{definition}","\\(C^*\\) / \\(W^*\\) & involution, norm, positivity, closure, multiplier or weak completion & faithful corner realization does not automatically preserve an independent tensor product","\\begin{boundary}[No single overpowered envelope]\n\\label{bnd:no-overpowered-envelope}\nThe existence of \\(\\Syn_{\\mathrm{ak}}\\) does not prove the existence of one universal category simultaneously free, complete, \\(C^*\\), \\(W^*\\), nonlinear-operadic, monoidal, and arithmetically integral in a way preserved by every target.  The mathematically safer target is a family of profile-indexed envelopes or a doctrine-valued construction with explicit forgetful and comparison maps.\n\\end{boundary}","\\begin{definition}[Profiled stratum passport]\n\\label{def:profiled-stratum-passport}\nA profiled Komorebi stratum is a record\n\\[\n\\mathbf S_\\alpha\n=\n(\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\n \\mathcal N_\\alpha,\\Sem_\\alpha,\\mathcal I_\\alpha),\n\\]\nwith the following fields.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal C_\\alpha\\) is the ambient semantic category in which the realized objects live.\n\\item \\(\\Pi_\\alpha\\) is the verified preservation profile of the realization.\n\\item \\(\\mathcal M_\\alpha\\) is a full replete category of admissible models of the declared resolved syntax.\n\\item \\(\\mathcal N_\\alpha\\subseteq\\mathcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathca","\\begin{definition}[Profiled stratum passport]\n\\label{def:profiled-stratum-passport}\nA profiled Komorebi stratum is a record\n\\[\n\\mathbf S_\\alpha\n=\n(\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\n \\mathcal N_\\alpha,\\Sem_\\alpha,\\mathcal I_\\alpha),\n\\]\nwith the following fields.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal C_\\alpha\\) is the ambient semantic category in which the realized objects live.\n\\item \\(\\Pi_\\alpha\\) is the verified preservation profile of the realization.\n\\item \\(\\mathcal M_\\alpha\\) is a full replete category of admissible models of the declared resolved syntax.\n\\item \\(\\mathcal N_\\alpha\\subseteq\\mathcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\math","ed stratum passport]\n\\label{def:profiled-stratum-passport}\nA profiled Komorebi stratum is a record\n\\[\n\\mathbf S_\\alpha\n=\n(\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\n \\mathcal N_\\alpha,\\Sem_\\alpha,\\mathcal I_\\alpha),\n\\]\nwith the following fields.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal C_\\alpha\\) is the ambient semantic category in which the realized objects live.\n\\item \\(\\Pi_\\alpha\\) is the verified preservation profile of the realization.\n\\item \\(\\mathcal M_\\alpha\\) is a full replete category of admissible models of the declared resolved syntax.\n\\item \\(\\mathcal N_\\alpha\\subseteq\\mathcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\mathcal D_\\alpha\\) and classification target are part of the record.\n\\end{enumerate}…","stratum is a record\n\\[\n\\mathbf S_\\alpha\n=\n(\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\n \\mathcal N_\\alpha,\\Sem_\\alpha,\\mathcal I_\\alpha),\n\\]\nwith the following fields.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal C_\\alpha\\) is the ambient semantic category in which the realized objects live.\n\\item \\(\\Pi_\\alpha\\) is the verified preservation profile of the realization.\n\\item \\(\\mathcal M_\\alpha\\) is a full replete category of admissible models of the declared resolved syntax.\n\\item \\(\\mathcal N_\\alpha\\subseteq\\mathcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\mathcal D_\\alpha\\) and classification target are part of the record.\n\\end{enumerate}\nThe original four-field stratum\n\\((\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\al…","a,\\Sem_\\alpha,\\mathcal I_\\alpha),\n\\]\nwith the following fields.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal C_\\alpha\\) is the ambient semantic category in which the realized objects live.\n\\item \\(\\Pi_\\alpha\\) is the verified preservation profile of the realization.\n\\item \\(\\mathcal M_\\alpha\\) is a full replete category of admissible models of the declared resolved syntax.\n\\item \\(\\mathcal N_\\alpha\\subseteq\\mathcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\mathcal D_\\alpha\\) and classification target are part of the record.\n\\end{enumerate}\nThe original four-field stratum\n\\((\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\\Sem_\\alpha)\\)\nremains the underlying realization datum.  The normalized and invariant fields are additi…","tegory in which the realized objects live.\n\\item \\(\\Pi_\\alpha\\) is the verified preservation profile of the realization.\n\\item \\(\\mathcal M_\\alpha\\) is a full replete category of admissible models of the declared resolved syntax.\n\\item \\(\\mathcal N_\\alpha\\subseteq\\mathcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\mathcal D_\\alpha\\) and classification target are part of the record.\n\\end{enumerate}\nThe original four-field stratum\n\\((\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\\Sem_\\alpha)\\)\nremains the underlying realization datum.  The normalized and invariant fields are additional assembly data rather than automatic consequences.\n\\end{definition}","realization.\n\\item \\(\\mathcal M_\\alpha\\) is a full replete category of admissible models of the declared resolved syntax.\n\\item \\(\\mathcal N_\\alpha\\subseteq\\mathcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\mathcal D_\\alpha\\) and classification target are part of the record.\n\\end{enumerate}\nThe original four-field stratum\n\\((\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\\Sem_\\alpha)\\)\nremains the underlying realization datum.  The normalized and invariant fields are additional assembly data rather than automatic consequences.\n\\end{definition}","athcal M_\\alpha\\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\mathcal D_\\alpha\\) and classification target are part of the record.\n\\end{enumerate}\nThe original four-field stratum\n\\((\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\\Sem_\\alpha)\\)\nremains the underlying realization datum.  The normalized and invariant fields are additional assembly data rather than automatic consequences.\n\\end{definition}","ated.\n\\item\n\\[\n\\Sem_\\alpha:\\mathcal M_\\alpha\\longrightarrow\\mathcal O_\\alpha\n\\]\nis the native semantics functor.\n\\item\n\\[\n\\mathcal I_\\alpha:\\Core(\\mathcal N_\\alpha)\\longrightarrow\\mathcal D_\\alpha\n\\]\nis a declared invariant functor, when one has been constructed.  Its codomain \\(\\mathcal D_\\alpha\\) and classification target are part of the record.\n\\end{enumerate}\nThe original four-field stratum\n\\((\\mathcal C_\\alpha,\\Pi_\\alpha,\\mathcal M_\\alpha,\\Sem_\\alpha)\\)\nremains the underlying realization datum.  The normalized and invariant fields are additional assembly data rather than automatic consequences.\n\\end{definition}","\\begin{definition}[Normalization witness]\n\\label{def:normalization-witness}\nA normalization witness on a full subcategory \\(\\mathcal U_\\alpha\\subseteq\\mathcal M_\\alpha\\) consists of a functor\n\\[\nN_\\alpha:\\mathcal U_\\alpha\\longrightarrow\\mathcal N_\\alpha\n\\]\nand a natural isomorphism\n\\[\n\\nu_\\alpha:\\Sem_\\alpha N_\\alpha\n\\Longrightarrow\n\\Sem_\\alpha|_{\\mathcal U_\\alpha}.\n\\]\nIt is \\emph{classification-normalizing} when every object of \\(\\mathcal U_\\alpha\\) is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in \\(\\mathcal N_\\alpha\\).  A normalization may be partial, domain-restricted, or defined only after auxiliary choices.  Those restrictions belong to the passport.\n\\end{definition}","itness]\n\\label{def:normalization-witness}\nA normalization witness on a full subcategory \\(\\mathcal U_\\alpha\\subseteq\\mathcal M_\\alpha\\) consists of a functor\n\\[\nN_\\alpha:\\mathcal U_\\alpha\\longrightarrow\\mathcal N_\\alpha\n\\]\nand a natural isomorphism\n\\[\n\\nu_\\alpha:\\Sem_\\alpha N_\\alpha\n\\Longrightarrow\n\\Sem_\\alpha|_{\\mathcal U_\\alpha}.\n\\]\nIt is \\emph{classification-normalizing} when every object of \\(\\mathcal U_\\alpha\\) is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in \\(\\mathcal N_\\alpha\\).  A normalization may be partial, domain-restricted, or defined only after auxiliary choices.  Those restrictions belong to the passport.\n\\end{definition}",") consists of a functor\n\\[\nN_\\alpha:\\mathcal U_\\alpha\\longrightarrow\\mathcal N_\\alpha\n\\]\nand a natural isomorphism\n\\[\n\\nu_\\alpha:\\Sem_\\alpha N_\\alpha\n\\Longrightarrow\n\\Sem_\\alpha|_{\\mathcal U_\\alpha}.\n\\]\nIt is \\emph{classification-normalizing} when every object of \\(\\mathcal U_\\alpha\\) is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in \\(\\mathcal N_\\alpha\\).  A normalization may be partial, domain-restricted, or defined only after auxiliary choices.  Those restrictions belong to the passport.\n\\end{definition}","\\begin{definition}[Complete invariant on a declared target]\n\\label{def:complete-invariant-target}\nAn invariant \\(\\mathcal I_\\alpha\\) is complete on \\(\\mathcal N_\\alpha\\) when\n\\[\n\\mathcal I_\\alpha(X)\\cong\\mathcal I_\\alpha(Y)\n\\quad\\Longleftrightarrow\\quad\nX\\cong Y\n\\]\nfor all \\(X,Y\\in\\mathcal N_\\alpha\\).  This statement classifies the objects of \\(\\mathcal N_\\alpha\\) up to the isomorphisms allowed there.  It does not classify arbitrary presentations outside that subcategory.\n\\end{definition}","\\begin{proposition}[Semantic conservativity of a normalized chart]\n\\label{prop:normalized-semantic-conservativity}\nIf \\(N_\\alpha\\) is classification-normalizing, then\n\\[\n\\Sem_\\alpha|_{\\mathcal N_\\alpha}\n\\]\nreflects isomorphisms.  Consequently, a complete invariant on \\(\\mathcal N_\\alpha\\) can be transported through a reversible overlap only after the overlap is shown to preserve the normalization and reflect the relevant isomorphisms.\n\\end{proposition}","\\begin{proof}\nLet \\(X,Y\\in\\mathcal N_\\alpha\\) and suppose their semantic objects are isomorphic.  The second clause of \\cref{def:normalization-witness} gives \\(X\\cong Y\\).  The final statement follows because completeness is an assertion about isomorphism classes, not merely about equality of numerical outputs.\n\\end{proof}","Digital & DFAO, finite digit kernels, regular linear representations & state maps and radix-compatible intertwiners & minimal DFAO for fixed radix and reading direction & unique minimal DFAO up to state renaming; exact finite \\(k\\)-kernel compilation & no all-moduli packet profile or general linear-state minimality for \\(k\\)-regular systems","For every stratum, the invertible part is the maximal subgroupoid\n\\[\n\\mathfrak{Rep}_\\alpha(\\bbK)\n=\n\\Core(\\mathcal M_\\alpha).\n\\]\nThis core records changes of presentation that preserve all structure required by the overlap.  Quotients, truncations, completions, specializations, differential annihilator maps, and compression maps generally remain outside it.","\\begin{definition}[Finite coordinate action groupoid]\n\\label{def:coordinate-action-groupoid}\nFix a dimension vector \\(\\mathbf d=(d_X)_{X\\in\\mathsf S}\\).  Let\n\\[\nG_{\\mathbf d}=\\prod_{X\\in\\mathsf S}GL_{d_X}(\\kk)\n\\]\nact on the finite-dimensional representation space \\(\\Rep_{\\mathbf d}(\\bbK)\\) by base change.  The action groupoid\n\\[\n[\\Rep_{\\mathbf d}(\\bbK)/G_{\\mathbf d}]\n\\]\nhas representations as objects and coordinate changes carrying one representation to another as morphisms.\n\\end{definition}","\\begin{proposition}[Coordinate groupoid realizes isomorphism]\n\\label{prop:coordinate-groupoid-isomorphism}\nTwo objects of \\(\\Rep_{\\mathbf d}(\\bbK)\\) are isomorphic as typed realizations exactly when they lie in the same \\(G_{\\mathbf d}\\)-orbit.  The stabilizer of a point is its coordinate automorphism group.\n\\end{proposition}","\\begin{proof}\nA typed isomorphism is a family of invertible linear maps on the sort spaces intertwining every generating operation.  This is precisely an element of \\(G_{\\mathbf d}\\) carrying one coordinate tuple to the other.  Elements fixing the tuple are exactly the typed automorphisms.\n\\end{proof}","\\midrule\nGauge symmetry & coordinate-change group \\(G_{\\mathbf d}\\) or a local gauge family & changes presentation coordinates while preserving the represented object & an explicitly typed action on the chart","Intrinsic symmetry & \\(\\Aut_{\\mathcal M_\\alpha}(M)\\) & automorphisms of one resolved object inside the stratum & none beyond the native category","Reconstructed symmetry & \\(\\Aut^{\\otimes}(\\omega)\\) & tensor automorphisms of a fiber functor and the corresponding affine group scheme & faithful exact tensor fiber functor","\\begin{definition}[Typed directed transformation category]\n\\label{def:directed-transformation-category}\nThe directed transformation category \\(\\mathfrak{Dir}(\\bbK)\\) has pairs \\((\\alpha,M)\\), with \\(M\\in\\mathcal M_\\alpha\\), as objects.  A morphism\n\\[\n(\\alpha,M)\\longrightarrow(\\beta,N)\n\\]\nis an admitted bridge instance equipped with a bridge passport as in \\cref{def:bridge-passport}.  Composition is defined only when the codomain profile of the first bridge satisfies the domain requirements of the second.  The union of the representation groupoids embeds as the wide invertible subcategory.\n\\end{definition}","\\begin{definition}[Bridge passport]\n\\label{def:bridge-passport}\nA certified bridge from stratum \\(\\alpha\\) to stratum \\(\\beta\\) is a record\n\\[\n\\mathfrak b\n=\n(\\mathcal U_\\mathfrak b,\\mathcal V_\\mathfrak b,\n B_\\mathfrak b,\\tau_\\mathfrak b,\n \\Pi^{\\mathrm{pres}}_\\mathfrak b,\n \\Pi^{\\mathrm{refl}}_\\mathfrak b,\n \\Lambda_\\mathfrak b,\n \\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and univer","\\begin{definition}[Bridge passport]\n\\label{def:bridge-passport}\nA certified bridge from stratum \\(\\alpha\\) to stratum \\(\\beta\\) is a record\n\\[\n\\mathfrak b\n=\n(\\mathcal U_\\mathfrak b,\\mathcal V_\\mathfrak b,\n B_\\mathfrak b,\\tau_\\mathfrak b,\n \\Pi^{\\mathrm{pres}}_\\mathfrak b,\n \\Pi^{\\mathrm{refl}}_\\mathfrak b,\n \\Lambda_\\mathfrak b,\n \\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item","\\begin{definition}[Bridge passport]\n\\label{def:bridge-passport}\nA certified bridge from stratum \\(\\alpha\\) to stratum \\(\\beta\\) is a record\n\\[\n\\mathfrak b\n=\n(\\mathcal U_\\mathfrak b,\\mathcal V_\\mathfrak b,\n B_\\mathfrak b,\\tau_\\mathfrak b,\n \\Pi^{\\mathrm{pres}}_\\mathfrak b,\n \\Pi^{\\mathrm{refl}}_\\mathfrak b,\n \\Lambda_\\mathfrak b,\n \\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfra","passport]\n\\label{def:bridge-passport}\nA certified bridge from stratum \\(\\alpha\\) to stratum \\(\\beta\\) is a record\n\\[\n\\mathfrak b\n=\n(\\mathcal U_\\mathfrak b,\\mathcal V_\\mathfrak b,\n B_\\mathfrak b,\\tau_\\mathfrak b,\n \\Pi^{\\mathrm{pres}}_\\mathfrak b,\n \\Pi^{\\mathrm{refl}}_\\mathfrak b,\n \\Lambda_\\mathfrak b,\n \\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compati…","from stratum \\(\\alpha\\) to stratum \\(\\beta\\) is a record\n\\[\n\\mathfrak b\n=\n(\\mathcal U_\\mathfrak b,\\mathcal V_\\mathfrak b,\n B_\\mathfrak b,\\tau_\\mathfrak b,\n \\Pi^{\\mathrm{pres}}_\\mathfrak b,\n \\Pi^{\\mathrm{refl}}_\\mathfrak b,\n \\Lambda_\\mathfrak b,\n \\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inve…","mathfrak b,\n B_\\mathfrak b,\\tau_\\mathfrak b,\n \\Pi^{\\mathrm{pres}}_\\mathfrak b,\n \\Pi^{\\mathrm{refl}}_\\mathfrak b,\n \\Lambda_\\mathfrak b,\n \\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy…","thrm{refl}}_\\mathfrak b,\n \\Lambda_\\mathfrak b,\n \\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy.\n\\end{definition}","\\nu_\\mathfrak b,\\chi_\\mathfrak b),\n\\]\nwhere:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\mathcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy.\n\\end{definition}","athcal U_\\mathfrak b\\subseteq\\mathcal M_\\alpha\\) and \\(\\mathcal V_\\mathfrak b\\subseteq\\mathcal M_\\beta\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy.\n\\end{definition}","\\) are the declared domain and codomain subcategories;\n\\item \\(B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy.\n\\end{definition}","B_\\mathfrak b:\\mathcal U_\\mathfrak b\\to\\mathcal V_\\mathfrak b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy.\n\\end{definition}","k b\\) is the functor;\n\\item \\(\\tau_\\mathfrak b\\) is its bridge class;\n\\item \\(\\Pi^{\\mathrm{pres}}_\\mathfrak b\\) records preserved constructors and universal properties;\n\\item \\(\\Pi^{\\mathrm{refl}}_\\mathfrak b\\) records properties or isomorphisms reflected by the bridge;\n\\item \\(\\Lambda_\\mathfrak b\\) is the information-loss profile;\n\\item \\(\\nu_\\mathfrak b\\) is the semantic comparison map;\n\\item \\(\\chi_\\mathfrak b\\) records normalization, residual, and certificate compatibility.\n\\end{enumerate}\nA passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy.\n\\end{definition}","\\begin{definition}[Information-loss profile]\n\\label{def:information-loss-profile}\nFor a functor \\(B:\\mathcal U\\to\\mathcal V\\), its information-loss profile contains at least:\n\\begin{align*}\n\\operatorname{MorKer}_B(X,Y)\n&=\n\\{(f,g)\\in\\Hom_{\\mathcal U}(X,Y)^2:Bf=Bg\\},","\\begin{definition}[Information-loss profile]\n\\label{def:information-loss-profile}\nFor a functor \\(B:\\mathcal U\\to\\mathcal V\\), its information-loss profile contains at least:\n\\begin{align*}\n\\operatorname{MorKer}_B(X,Y)\n&=\n\\{(f,g)\\in\\Hom_{\\mathcal U}(X,Y)^2:Bf=Bg\\},\\\\\n\\operatorname{Coll}_B\n&=\n\\{(X,Y)\\in\\operatorname{Ob}(\\mathcal U)^2:BX\\cong BY\\},\\\\\n\\operatorname{EssIm}(B)\n&\\subseteq\\mathcal V,\n\\end{align*}\ntogether with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by \\(B\\).  The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.\n\\end{definition}","\\operatorname{EssIm}(B)\n&\\subseteq\\mathcal V,\n\\end{align*}\ntogether with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by \\(B\\).  The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.\n\\end{definition}","\\begin{definition}[Faithfulness, conservativity, and reversibility]\n\\label{def:lossless-bridge}\nA bridge is:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\emph{morphism-faithful} when every \\(\\operatorname{MorKer}_B(X,Y)\\) is diagonal;\n\\item \\emph{conservative} when \\(Bf\\) invertible implies that \\(f\\) is invertible;\n\\item \\emph{classification-faithful} or \\emph{essentially injective on objects} when \\(BX\\cong BY\\) implies \\(X\\cong Y\\) on the declared domain;\n\\item \\emph{lossless on its image} when it is fully faithful;\n\\item \\emph{reversible} when it is an equivalence onto the declared overlap.\n\\end{enumerate}\nThese conditions answer different questions and should not be conflated.\n\\end{definition}","nition}[Faithfulness, conservativity, and reversibility]\n\\label{def:lossless-bridge}\nA bridge is:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\emph{morphism-faithful} when every \\(\\operatorname{MorKer}_B(X,Y)\\) is diagonal;\n\\item \\emph{conservative} when \\(Bf\\) invertible implies that \\(f\\) is invertible;\n\\item \\emph{classification-faithful} or \\emph{essentially injective on objects} when \\(BX\\cong BY\\) implies \\(X\\cong Y\\) on the declared domain;\n\\item \\emph{lossless on its image} when it is fully faithful;\n\\item \\emph{reversible} when it is an equivalence onto the declared overlap.\n\\end{enumerate}\nThese conditions answer different questions and should not be conflated.\n\\end{definition}","onservativity, and reversibility]\n\\label{def:lossless-bridge}\nA bridge is:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\emph{morphism-faithful} when every \\(\\operatorname{MorKer}_B(X,Y)\\) is diagonal;\n\\item \\emph{conservative} when \\(Bf\\) invertible implies that \\(f\\) is invertible;\n\\item \\emph{classification-faithful} or \\emph{essentially injective on objects} when \\(BX\\cong BY\\) implies \\(X\\cong Y\\) on the declared domain;\n\\item \\emph{lossless on its image} when it is fully faithful;\n\\item \\emph{reversible} when it is an equivalence onto the declared overlap.\n\\end{enumerate}\nThese conditions answer different questions and should not be conflated.\n\\end{definition}","\\begin{proposition}[Composition law for bridge loss]\n\\label{prop:bridge-loss-composition}\nLet\n\\[\n\\mathcal U\\xrightarrow{B}\\mathcal V\\xrightarrow{C}\\mathcal W\n\\]\nbe certified bridges.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item Every constructor or universal property preserved by both \\(B\\) and \\(C\\) is preserved by \\(CB\\).\n\\item\n\\[\n\\operatorname{MorKer}_B(X,Y)\n\\subseteq\n\\operatorname{MorKer}_{CB}(X,Y).\n\\]\n\\item\n\\[\n\\operatorname{Coll}_B\n\\subseteq\n\\operatorname{Coll}_{CB}.\n\\]\n\\item\n\\[\n\\operatorname{EssIm}(CB)\n\\subseteq\n\\operatorname{EssIm}(C).\n\\]\n\\item If both bridges are conservative, then their composite is conservative.  If both are classification-faithful, then their composite is classification-faithful.  If both are equivalences, then their composite is an equivalence.\n\\end{enumerate}\nThus information forgotten at an earlier stage cannot","\\begin{proposition}[Composition law for bridge loss]\n\\label{prop:bridge-loss-composition}\nLet\n\\[\n\\mathcal U\\xrightarrow{B}\\mathcal V\\xrightarrow{C}\\mathcal W\n\\]\nbe certified bridges.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item Every constructor or universal property preserved by both \\(B\\) and \\(C\\) is preserved by \\(CB\\).\n\\item\n\\[\n\\operatorname{MorKer}_B(X,Y)\n\\subseteq\n\\operatorname{MorKer}_{CB}(X,Y).\n\\]\n\\item\n\\[\n\\operatorname{Coll}_B\n\\subseteq\n\\operatorname{Coll}_{CB}.\n\\]\n\\item\n\\[\n\\operatorname{EssIm}(CB)\n\\subseteq\n\\operatorname{EssIm}(C).\n\\]\n\\item If both bridges are conservative, then their composite is conservative.  If both are classification-faithful, then their composite is classification-faithful.  If both are equivalences, then their composite is an equivalence.\n\\end{enumerate}\nThus information forgotten at an earlier stage cannot be recovered by merely composing more forgetful bridges, although a later bridge may attach auxiliary data through a separate construct","\\begin{proposition}[Composition law for bridge loss]\n\\label{prop:bridge-loss-composition}\nLet\n\\[\n\\mathcal U\\xrightarrow{B}\\mathcal V\\xrightarrow{C}\\mathcal W\n\\]\nbe certified bridges.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item Every constructor or universal property preserved by both \\(B\\) and \\(C\\) is preserved by \\(CB\\).\n\\item\n\\[\n\\operatorname{MorKer}_B(X,Y)\n\\subseteq\n\\operatorname{MorKer}_{CB}(X,Y).\n\\]\n\\item\n\\[\n\\operatorname{Coll}_B\n\\subseteq\n\\operatorname{Coll}_{CB}.\n\\]\n\\item\n\\[\n\\operatorname{EssIm}(CB)\n\\subseteq\n\\operatorname{EssIm}(C).\n\\]\n\\item If both bridges are conservative, then their composite is conservative.  If both are classification-faithful, then their composite is classification-faithful.  If both are equivalences, then their composite is an equivalence.\n\\end{enumerate}\nThus information forgotten at an earlier stage cannot be recovered by merely composing more forgetful bridges, although a later bridge may attach auxiliary data through a separate constructor.\n\\end{p","\\begin{proposition}[Composition law for bridge loss]\n\\label{prop:bridge-loss-composition}\nLet\n\\[\n\\mathcal U\\xrightarrow{B}\\mathcal V\\xrightarrow{C}\\mathcal W\n\\]\nbe certified bridges.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item Every constructor or universal property preserved by both \\(B\\) and \\(C\\) is preserved by \\(CB\\).\n\\item\n\\[\n\\operatorname{MorKer}_B(X,Y)\n\\subseteq\n\\operatorname{MorKer}_{CB}(X,Y).\n\\]\n\\item\n\\[\n\\operatorname{Coll}_B\n\\subseteq\n\\operatorname{Coll}_{CB}.\n\\]\n\\item\n\\[\n\\operatorname{EssIm}(CB)\n\\subseteq\n\\operatorname{EssIm}(C).\n\\]\n\\item If both bridges are conservative, then their composite is conservative.  If both are classification-faithful, then their composite is classification-faithful.  If both are equivalences, then their composite is an equivalence.\n\\end{enumerate}\nThus information forgotten at an earlier stage cannot be recovered by merely composing more forgetful bridges, although a later bridge may attach auxiliary data through a separate constructor.\n\\end{proposition}","position law for bridge loss]\n\\label{prop:bridge-loss-composition}\nLet\n\\[\n\\mathcal U\\xrightarrow{B}\\mathcal V\\xrightarrow{C}\\mathcal W\n\\]\nbe certified bridges.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item Every constructor or universal property preserved by both \\(B\\) and \\(C\\) is preserved by \\(CB\\).\n\\item\n\\[\n\\operatorname{MorKer}_B(X,Y)\n\\subseteq\n\\operatorname{MorKer}_{CB}(X,Y).\n\\]\n\\item\n\\[\n\\operatorname{Coll}_B\n\\subseteq\n\\operatorname{Coll}_{CB}.\n\\]\n\\item\n\\[\n\\operatorname{EssIm}(CB)\n\\subseteq\n\\operatorname{EssIm}(C).\n\\]\n\\item If both bridges are conservative, then their composite is conservative.  If both are classification-faithful, then their composite is classification-faithful.  If both are equivalences, then their composite is an equivalence.\n\\end{enumerate}\nThus information forgotten at an earlier stage cannot be recovered by merely composing more forgetful bridges, although a later bridge may attach auxiliary data through a separate constructor.\n\\end{proposition}","\\[\n\\mathcal U\\xrightarrow{B}\\mathcal V\\xrightarrow{C}\\mathcal W\n\\]\nbe certified bridges.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item Every constructor or universal property preserved by both \\(B\\) and \\(C\\) is preserved by \\(CB\\).\n\\item\n\\[\n\\operatorname{MorKer}_B(X,Y)\n\\subseteq\n\\operatorname{MorKer}_{CB}(X,Y).\n\\]\n\\item\n\\[\n\\operatorname{Coll}_B\n\\subseteq\n\\operatorname{Coll}_{CB}.\n\\]\n\\item\n\\[\n\\operatorname{EssIm}(CB)\n\\subseteq\n\\operatorname{EssIm}(C).\n\\]\n\\item If both bridges are conservative, then their composite is conservative.  If both are classification-faithful, then their composite is classification-faithful.  If both are equivalences, then their composite is an equivalence.\n\\end{enumerate}\nThus information forgotten at an earlier stage cannot be recovered by merely composing more forgetful bridges, although a later bridge may attach auxiliary data through a separate constructor.\n\\end{proposition}","\\begin{proof}\nFunctoriality proves the preservation claim.  If \\(Bf=Bg\\), then \\(CBf=CBg\\), proving the second inclusion.  If \\(BX\\cong BY\\), applying \\(C\\) gives \\(CBX\\cong CBY\\), proving the third.  Every object in the essential image of \\(CB\\) is visibly in the essential image of \\(C\\).  For conservativity, if \\(CBf\\) is invertible then \\(Bf\\) is invertible because \\(C\\) is conservative, and then \\(f\\) is invertible because \\(B\\) is conservative.  For classification-faithfulness, \\(CBX\\cong CBY\\) implies \\(BX\\cong BY\\) and then \\(X\\cong Y\\).  Composition of equivalences is standard.\n\\end{proof}","n{proof}\nFunctoriality proves the preservation claim.  If \\(Bf=Bg\\), then \\(CBf=CBg\\), proving the second inclusion.  If \\(BX\\cong BY\\), applying \\(C\\) gives \\(CBX\\cong CBY\\), proving the third.  Every object in the essential image of \\(CB\\) is visibly in the essential image of \\(C\\).  For conservativity, if \\(CBf\\) is invertible then \\(Bf\\) is invertible because \\(C\\) is conservative, and then \\(f\\) is invertible because \\(B\\) is conservative.  For classification-faithfulness, \\(CBX\\cong CBY\\) implies \\(BX\\cong BY\\) and then \\(X\\cong Y\\).  Composition of equivalences is standard.\n\\end{proof}","s the preservation claim.  If \\(Bf=Bg\\), then \\(CBf=CBg\\), proving the second inclusion.  If \\(BX\\cong BY\\), applying \\(C\\) gives \\(CBX\\cong CBY\\), proving the third.  Every object in the essential image of \\(CB\\) is visibly in the essential image of \\(C\\).  For conservativity, if \\(CBf\\) is invertible then \\(Bf\\) is invertible because \\(C\\) is conservative, and then \\(f\\) is invertible because \\(B\\) is conservative.  For classification-faithfulness, \\(CBX\\cong CBY\\) implies \\(BX\\cong BY\\) and then \\(X\\cong Y\\).  Composition of equivalences is standard.\n\\end{proof}",", proving the second inclusion.  If \\(BX\\cong BY\\), applying \\(C\\) gives \\(CBX\\cong CBY\\), proving the third.  Every object in the essential image of \\(CB\\) is visibly in the essential image of \\(C\\).  For conservativity, if \\(CBf\\) is invertible then \\(Bf\\) is invertible because \\(C\\) is conservative, and then \\(f\\) is invertible because \\(B\\) is conservative.  For classification-faithfulness, \\(CBX\\cong CBY\\) implies \\(BX\\cong BY\\) and then \\(X\\cong Y\\).  Composition of equivalences is standard.\n\\end{proof}","usion.  If \\(BX\\cong BY\\), applying \\(C\\) gives \\(CBX\\cong CBY\\), proving the third.  Every object in the essential image of \\(CB\\) is visibly in the essential image of \\(C\\).  For conservativity, if \\(CBf\\) is invertible then \\(Bf\\) is invertible because \\(C\\) is conservative, and then \\(f\\) is invertible because \\(B\\) is conservative.  For classification-faithfulness, \\(CBX\\cong CBY\\) implies \\(BX\\cong BY\\) and then \\(X\\cong Y\\).  Composition of equivalences is standard.\n\\end{proof}","), applying \\(C\\) gives \\(CBX\\cong CBY\\), proving the third.  Every object in the essential image of \\(CB\\) is visibly in the essential image of \\(C\\).  For conservativity, if \\(CBf\\) is invertible then \\(Bf\\) is invertible because \\(C\\) is conservative, and then \\(f\\) is invertible because \\(B\\) is conservative.  For classification-faithfulness, \\(CBX\\cong CBY\\) implies \\(BX\\cong BY\\) and then \\(X\\cong Y\\).  Composition of equivalences is standard.\n\\end{proof}","\\begin{enumerate}\n\\item A selected algebraic branch can produce a D-finite annihilator, but the operator may forget the selected branch and its minimal polynomial.  The bridge is directional unless that data are retained.\n\\item A rational functional-Mahler mask over \\(\\mathbf F_p\\) produces a Cartier representation and finite automaton under the stated hypotheses, but this does not identify it with a Newton--Mahler interpolation chart \\cite{FMH,SEQ}.\n\\item A faithful Hilbert-space corner representation realizes packet projectors and transfers, but it does not automatically preserve a tensor product strongly enough for Tannakian reconstruction \\cite{RT,GAL}.\n\\end{enumerate}","\\begin{definition}[Typed overlap]\n\\label{def:typed-overlap}\nA reversible overlap between strata \\(\\alpha\\) and \\(\\beta\\) consists of full replete subcategories\n\\[\n\\mathcal M_{\\alpha\\beta}\\subseteq\\mathcal M_\\alpha,\n\\qquad\n\\mathcal M_{\\beta\\alpha}\\subseteq\\mathcal M_\\beta,\n\\]\nan equivalence\n\\[\n\\Phi_{\\alpha\\beta}:\\mathcal M_{\\alpha\\beta}\n\\overset{\\sim}{\\longrightarrow}\n\\mathcal M_{\\beta\\alpha},\n\\]\nand a semantic natural isomorphism\n\\[\n\\sigma_{\\alpha\\beta}:\nJ_{\\alpha\\beta}\\Sem_\\alpha\n\\Longrightarrow\n\\Sem_\\beta\\Phi_{\\alpha\\beta}\n\\]\nfor an explicitly declared comparison functor \\(J_{\\alpha\\beta}\\) between the observable doctrines.  An overlap is \\emph{normalized} when the equivalence restricts to the selected normalized subcategories.\n\\end{definition}","abel{def:typed-overlap}\nA reversible overlap between strata \\(\\alpha\\) and \\(\\beta\\) consists of full replete subcategories\n\\[\n\\mathcal M_{\\alpha\\beta}\\subseteq\\mathcal M_\\alpha,\n\\qquad\n\\mathcal M_{\\beta\\alpha}\\subseteq\\mathcal M_\\beta,\n\\]\nan equivalence\n\\[\n\\Phi_{\\alpha\\beta}:\\mathcal M_{\\alpha\\beta}\n\\overset{\\sim}{\\longrightarrow}\n\\mathcal M_{\\beta\\alpha},\n\\]\nand a semantic natural isomorphism\n\\[\n\\sigma_{\\alpha\\beta}:\nJ_{\\alpha\\beta}\\Sem_\\alpha\n\\Longrightarrow\n\\Sem_\\beta\\Phi_{\\alpha\\beta}\n\\]\nfor an explicitly declared comparison functor \\(J_{\\alpha\\beta}\\) between the observable doctrines.  An overlap is \\emph{normalized} when the equivalence restricts to the selected normalized subcategories.\n\\end{definition}","beta}\\subseteq\\mathcal M_\\alpha,\n\\qquad\n\\mathcal M_{\\beta\\alpha}\\subseteq\\mathcal M_\\beta,\n\\]\nan equivalence\n\\[\n\\Phi_{\\alpha\\beta}:\\mathcal M_{\\alpha\\beta}\n\\overset{\\sim}{\\longrightarrow}\n\\mathcal M_{\\beta\\alpha},\n\\]\nand a semantic natural isomorphism\n\\[\n\\sigma_{\\alpha\\beta}:\nJ_{\\alpha\\beta}\\Sem_\\alpha\n\\Longrightarrow\n\\Sem_\\beta\\Phi_{\\alpha\\beta}\n\\]\nfor an explicitly declared comparison functor \\(J_{\\alpha\\beta}\\) between the observable doctrines.  An overlap is \\emph{normalized} when the equivalence restricts to the selected normalized subcategories.\n\\end{definition}","\\begin{definition}[Bridge-law certificate]\n\\label{def:bridge-law-certificate}\nA bridge-law certificate for \\(B:\\mathcal U\\to\\mathcal V\\) records commuting or naturally commuting diagrams for each structure used downstream:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item native semantics;\n\\item packet analysis and synthesis;\n\\item refinement and coarsening maps;\n\\item selected-support compression and named residuals;\n\\item proof or computation certificates;\n\\item normalization and invariant comparison maps.\n\\end{enumerate}\nFor example, packet naturality requires comparison maps under which\n\\[\nB(p_{Y,b}fp_{X,a})\n\\cong\np'_{B(Y),b}\\,B(f)\\,p'_{B(X),a},\n\\]\nand refinement naturality requires the image of each coarse packet to equal the declared sum of its refined descendants.\n\\end{definition}","cture used downstream:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item native semantics;\n\\item packet analysis and synthesis;\n\\item refinement and coarsening maps;\n\\item selected-support compression and named residuals;\n\\item proof or computation certificates;\n\\item normalization and invariant comparison maps.\n\\end{enumerate}\nFor example, packet naturality requires comparison maps under which\n\\[\nB(p_{Y,b}fp_{X,a})\n\\cong\np'_{B(Y),b}\\,B(f)\\,p'_{B(X),a},\n\\]\nand refinement naturality requires the image of each coarse packet to equal the declared sum of its refined descendants.\n\\end{definition}","\\begin{proposition}[Bridge-law transport]\n\\label{prop:bridge-law-transport}\nLet \\(\\mathcal P\\) be a proof whose constructor support lies in the preserved part of a bridge-law certificate.  Then the image proof is valid in the target chart.  If the proof concludes equality of normalized invariants, that equality descends across the bridge only when the normalization and invariant comparison maps are also included in the certificate.\n\\end{proposition}","On triple overlaps, pairwise equivalences are not enough.  One needs comparison cells\n\\[\n\\eta_{\\alpha\\beta\\gamma}:\n\\Phi_{\\beta\\gamma}\\Phi_{\\alpha\\beta}\n\\Longrightarrow\n\\Phi_{\\alpha\\gamma}.\n\\]\nFor four charts, there are two composites from\n\\(\n\\Phi_{\\gamma\\delta}\\Phi_{\\beta\\gamma}\\Phi_{\\alpha\\beta}\n\\)\nto \\(\\Phi_{\\alpha\\delta}\\).  Their ratio is the higher coherence defect.","\\begin{definition}[Quadruple-overlap coherence defect]\n\\label{def:quadruple-coherence-defect}\nWrite horizontal composition of natural transformations as \\(\\ast\\).  Define\n\\[\n\\Omega_{\\alpha\\beta\\gamma\\delta}\n=\n\\Bigl[\n\\eta_{\\alpha\\gamma\\delta}\n\\circ\n(\\id\\ast\\eta_{\\alpha\\beta\\gamma})\n\\Bigr]\n\\circ\n\\Bigl[\n\\eta_{\\alpha\\beta\\delta}\n\\circ\n(\\eta_{\\beta\\gamma\\delta}\\ast\\id)\n\\Bigr]^{-1}.\n\\]\nIt is a natural automorphism of \\(\\Phi_{\\alpha\\delta}\\) on the quadruple overlap.  The atlas is coherent there exactly when\n\\[\n\\Omega_{\\alpha\\beta\\gamma\\delta}=\\id.\n\\]\n\\end{definition}","}[Quadruple-overlap coherence defect]\n\\label{def:quadruple-coherence-defect}\nWrite horizontal composition of natural transformations as \\(\\ast\\).  Define\n\\[\n\\Omega_{\\alpha\\beta\\gamma\\delta}\n=\n\\Bigl[\n\\eta_{\\alpha\\gamma\\delta}\n\\circ\n(\\id\\ast\\eta_{\\alpha\\beta\\gamma})\n\\Bigr]\n\\circ\n\\Bigl[\n\\eta_{\\alpha\\beta\\delta}\n\\circ\n(\\eta_{\\beta\\gamma\\delta}\\ast\\id)\n\\Bigr]^{-1}.\n\\]\nIt is a natural automorphism of \\(\\Phi_{\\alpha\\delta}\\) on the quadruple overlap.  The atlas is coherent there exactly when\n\\[\n\\Omega_{\\alpha\\beta\\gamma\\delta}=\\id.\n\\]\n\\end{definition}","l composition of natural transformations as \\(\\ast\\).  Define\n\\[\n\\Omega_{\\alpha\\beta\\gamma\\delta}\n=\n\\Bigl[\n\\eta_{\\alpha\\gamma\\delta}\n\\circ\n(\\id\\ast\\eta_{\\alpha\\beta\\gamma})\n\\Bigr]\n\\circ\n\\Bigl[\n\\eta_{\\alpha\\beta\\delta}\n\\circ\n(\\eta_{\\beta\\gamma\\delta}\\ast\\id)\n\\Bigr]^{-1}.\n\\]\nIt is a natural automorphism of \\(\\Phi_{\\alpha\\delta}\\) on the quadruple overlap.  The atlas is coherent there exactly when\n\\[\n\\Omega_{\\alpha\\beta\\gamma\\delta}=\\id.\n\\]\n\\end{definition}","\\begin{theorem}[Split bridge rectification]\n\\label{thm:split-bridge-rectification}\nLet \\(\\mathcal A\\) be an additive idempotent-complete category and suppose\n\\[\nX\\xrightarrow{A}Y\\xrightarrow{L}X\n\\]\nsatisfies \\(LA=\\id_X\\).  Put \\(E=AL\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(E^2=E\\);\n\\item \\(Y\\cong\\im E\\oplus\\im(1-E)\\);\n\\item \\(A:X\\to\\im E\\) is an isomorphism with inverse induced by \\(L\\);\n\\item \\(L(1-E)=0\\), so \\(\\im(1-E)\\) is a named residual invisible to the decoder;\n\\item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core \\(\\im E\\).\n\\end{enumerate}\nIn a module or vector-space realization, \\(\\im(1-E)=\\ker L\\).\n\\end{theorem}","heorem}[Split bridge rectification]\n\\label{thm:split-bridge-rectification}\nLet \\(\\mathcal A\\) be an additive idempotent-complete category and suppose\n\\[\nX\\xrightarrow{A}Y\\xrightarrow{L}X\n\\]\nsatisfies \\(LA=\\id_X\\).  Put \\(E=AL\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(E^2=E\\);\n\\item \\(Y\\cong\\im E\\oplus\\im(1-E)\\);\n\\item \\(A:X\\to\\im E\\) is an isomorphism with inverse induced by \\(L\\);\n\\item \\(L(1-E)=0\\), so \\(\\im(1-E)\\) is a named residual invisible to the decoder;\n\\item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core \\(\\im E\\).\n\\end{enumerate}\nIn a module or vector-space realization, \\(\\im(1-E)=\\ker L\\).\n\\end{theorem}","bridge rectification]\n\\label{thm:split-bridge-rectification}\nLet \\(\\mathcal A\\) be an additive idempotent-complete category and suppose\n\\[\nX\\xrightarrow{A}Y\\xrightarrow{L}X\n\\]\nsatisfies \\(LA=\\id_X\\).  Put \\(E=AL\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(E^2=E\\);\n\\item \\(Y\\cong\\im E\\oplus\\im(1-E)\\);\n\\item \\(A:X\\to\\im E\\) is an isomorphism with inverse induced by \\(L\\);\n\\item \\(L(1-E)=0\\), so \\(\\im(1-E)\\) is a named residual invisible to the decoder;\n\\item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core \\(\\im E\\).\n\\end{enumerate}\nIn a module or vector-space realization, \\(\\im(1-E)=\\ker L\\).\n\\end{theorem}","tion]\n\\label{thm:split-bridge-rectification}\nLet \\(\\mathcal A\\) be an additive idempotent-complete category and suppose\n\\[\nX\\xrightarrow{A}Y\\xrightarrow{L}X\n\\]\nsatisfies \\(LA=\\id_X\\).  Put \\(E=AL\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(E^2=E\\);\n\\item \\(Y\\cong\\im E\\oplus\\im(1-E)\\);\n\\item \\(A:X\\to\\im E\\) is an isomorphism with inverse induced by \\(L\\);\n\\item \\(L(1-E)=0\\), so \\(\\im(1-E)\\) is a named residual invisible to the decoder;\n\\item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core \\(\\im E\\).\n\\end{enumerate}\nIn a module or vector-space realization, \\(\\im(1-E)=\\ker L\\).\n\\end{theorem}","\\id_X\\).  Put \\(E=AL\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(E^2=E\\);\n\\item \\(Y\\cong\\im E\\oplus\\im(1-E)\\);\n\\item \\(A:X\\to\\im E\\) is an isomorphism with inverse induced by \\(L\\);\n\\item \\(L(1-E)=0\\), so \\(\\im(1-E)\\) is a named residual invisible to the decoder;\n\\item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core \\(\\im E\\).\n\\end{enumerate}\nIn a module or vector-space realization, \\(\\im(1-E)=\\ker L\\).\n\\end{theorem}","an*)}]\n\\item \\(E^2=E\\);\n\\item \\(Y\\cong\\im E\\oplus\\im(1-E)\\);\n\\item \\(A:X\\to\\im E\\) is an isomorphism with inverse induced by \\(L\\);\n\\item \\(L(1-E)=0\\), so \\(\\im(1-E)\\) is a named residual invisible to the decoder;\n\\item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core \\(\\im E\\).\n\\end{enumerate}\nIn a module or vector-space realization, \\(\\im(1-E)=\\ker L\\).\n\\end{theorem}","\\begin{proof}\nThe identity \\(E^2=A(LA)L=AL=E\\) proves the first statement.  Idempotent completeness splits \\(E\\), producing the direct sum.  The image of \\(A\\) lies in \\(\\im E\\), and \\(L A=\\id_X\\) while \\(A L=E\\) acts as the identity on \\(\\im E\\), so the restricted maps are inverse.  Finally,\n\\[\nL(1-E)=L-LAL=L-L=0.\n\\]\nFor modules, every \\(y\\in\\ker L\\) satisfies \\((1-E)y=y\\), while every vector in \\(\\im(1-E)\\) is killed by \\(L\\).\n\\end{proof}","egin{proof}\nThe identity \\(E^2=A(LA)L=AL=E\\) proves the first statement.  Idempotent completeness splits \\(E\\), producing the direct sum.  The image of \\(A\\) lies in \\(\\im E\\), and \\(L A=\\id_X\\) while \\(A L=E\\) acts as the identity on \\(\\im E\\), so the restricted maps are inverse.  Finally,\n\\[\nL(1-E)=L-LAL=L-L=0.\n\\]\nFor modules, every \\(y\\in\\ker L\\) satisfies \\((1-E)y=y\\), while every vector in \\(\\im(1-E)\\) is killed by \\(L\\).\n\\end{proof}","(E^2=A(LA)L=AL=E\\) proves the first statement.  Idempotent completeness splits \\(E\\), producing the direct sum.  The image of \\(A\\) lies in \\(\\im E\\), and \\(L A=\\id_X\\) while \\(A L=E\\) acts as the identity on \\(\\im E\\), so the restricted maps are inverse.  Finally,\n\\[\nL(1-E)=L-LAL=L-L=0.\n\\]\nFor modules, every \\(y\\in\\ker L\\) satisfies \\((1-E)y=y\\), while every vector in \\(\\im(1-E)\\) is killed by \\(L\\).\n\\end{proof}","The integral Mahler packet theorem is a concrete instance: the analysis matrix has an integer left inverse, the consistency projector is \\(AL\\), and the residual is the complementary packet component.  The same decoder survives every base change, while rank loss of individual residue channels remains visible after reduction \\cite{CT4}.","Finite graph transport gives the sharpest reversible gluing model in the Library.  Let \\(\\Gamma=(V,E)\\) be a finite connected graph.  Attach a finite-dimensional \\(\\kk\\)-vector space \\(F_v\\) to each vertex and an isomorphism \\(T_e:F_u\\to F_v\\) to each oriented edge \\(e:u\\to v\\), with \\(T_{\\bar e}=T_e^{-1}\\).  Choose a root \\(o\\) and spanning tree \\(S\\).  If \\(G_v:F_o\\to F_v\\) is tree transport, then every non-tree edge \\(e:u\\to v\\) has fundamental holonomy\n\\[\nH_e=G_v^{-1}T_eG_u\\in GL(F_o).\n\\]\nDefine\n\\[\nC_o\n=\n\\bigcap_{e\\notin S}\\ker(H_e-I)\n=\n\\Fix\\langle H_e:e\\notin S\\rangle,\n\\qquad\nC_v=G_vC_o.\n\\]","ves the sharpest reversible gluing model in the Library.  Let \\(\\Gamma=(V,E)\\) be a finite connected graph.  Attach a finite-dimensional \\(\\kk\\)-vector space \\(F_v\\) to each vertex and an isomorphism \\(T_e:F_u\\to F_v\\) to each oriented edge \\(e:u\\to v\\), with \\(T_{\\bar e}=T_e^{-1}\\).  Choose a root \\(o\\) and spanning tree \\(S\\).  If \\(G_v:F_o\\to F_v\\) is tree transport, then every non-tree edge \\(e:u\\to v\\) has fundamental holonomy\n\\[\nH_e=G_v^{-1}T_eG_u\\in GL(F_o).\n\\]\nDefine\n\\[\nC_o\n=\n\\bigcap_{e\\notin S}\\ker(H_e-I)\n=\n\\Fix\\langle H_e:e\\notin S\\rangle,\n\\qquad\nC_v=G_vC_o.\n\\]","el in the Library.  Let \\(\\Gamma=(V,E)\\) be a finite connected graph.  Attach a finite-dimensional \\(\\kk\\)-vector space \\(F_v\\) to each vertex and an isomorphism \\(T_e:F_u\\to F_v\\) to each oriented edge \\(e:u\\to v\\), with \\(T_{\\bar e}=T_e^{-1}\\).  Choose a root \\(o\\) and spanning tree \\(S\\).  If \\(G_v:F_o\\to F_v\\) is tree transport, then every non-tree edge \\(e:u\\to v\\) has fundamental holonomy\n\\[\nH_e=G_v^{-1}T_eG_u\\in GL(F_o).\n\\]\nDefine\n\\[\nC_o\n=\n\\bigcap_{e\\notin S}\\ker(H_e-I)\n=\n\\Fix\\langle H_e:e\\notin S\\rangle,\n\\qquad\nC_v=G_vC_o.\n\\]","a finite connected graph.  Attach a finite-dimensional \\(\\kk\\)-vector space \\(F_v\\) to each vertex and an isomorphism \\(T_e:F_u\\to F_v\\) to each oriented edge \\(e:u\\to v\\), with \\(T_{\\bar e}=T_e^{-1}\\).  Choose a root \\(o\\) and spanning tree \\(S\\).  If \\(G_v:F_o\\to F_v\\) is tree transport, then every non-tree edge \\(e:u\\to v\\) has fundamental holonomy\n\\[\nH_e=G_v^{-1}T_eG_u\\in GL(F_o).\n\\]\nDefine\n\\[\nC_o\n=\n\\bigcap_{e\\notin S}\\ker(H_e-I)\n=\n\\Fix\\langle H_e:e\\notin S\\rangle,\n\\qquad\nC_v=G_vC_o.\n\\]","\\begin{theorem}[Maximal flat core and global sections]\n\\label{thm:maximal-flat-core}\nUnder the preceding hypotheses \\cite{P3,CT4}:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item evaluation at the root identifies the space of global parallel sections with \\(C_o\\);\n\\item \\((C_v)_v\\) is path-flat, meaning that all path transports with the same endpoints agree on the core;\n\\item it is the unique greatest path-flat subfamily;\n\\item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies;\n\\item for\n\\[\n\\Delta_S:F_o\\longrightarrow\\bigoplus_{e\\notin S}F_o,\n\\qquad\n\\Delta_Sx=((H_e-I)x)_{e\\notin S},\n\\]\none has\n\\[\nC_o=\\ker\\Delta_S,\n\\qquad\n\\dim C_o=\\dim F_o-\\rank\\Delta_S;\n\\]\n\\item the full fibers glue to one global frame exactly when every fundamental holonomy is the identity;\n\\item a gauge change conjugates the holonomies and transports the core, so \\(\\dim C_o\\) is gauge-invariant.\n\\end{enumerate}\n\\end{the","\\begin{theorem}[Maximal flat core and global sections]\n\\label{thm:maximal-flat-core}\nUnder the preceding hypotheses \\cite{P3,CT4}:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item evaluation at the root identifies the space of global parallel sections with \\(C_o\\);\n\\item \\((C_v)_v\\) is path-flat, meaning that all path transports with the same endpoints agree on the core;\n\\item it is the unique greatest path-flat subfamily;\n\\item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies;\n\\item for\n\\[\n\\Delta_S:F_o\\longrightarrow\\bigoplus_{e\\notin S}F_o,\n\\qquad\n\\Delta_Sx=((H_e-I)x)_{e\\notin S},\n\\]\none has\n\\[\nC_o=\\ker\\Delta_S,\n\\qquad\n\\dim C_o=\\dim F_o-\\rank\\Delta_S;\n\\]\n\\item the full fibers glue to one global frame exactly when every fundamental holonomy is the identity;\n\\item a gauge change conjugates the holonomies and transports the core, so \\(\\dim C_o\\) is gauge-invariant.\n\\end{enumerate}\n\\end{theorem}","extup{(\\roman*)}]\n\\item evaluation at the root identifies the space of global parallel sections with \\(C_o\\);\n\\item \\((C_v)_v\\) is path-flat, meaning that all path transports with the same endpoints agree on the core;\n\\item it is the unique greatest path-flat subfamily;\n\\item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies;\n\\item for\n\\[\n\\Delta_S:F_o\\longrightarrow\\bigoplus_{e\\notin S}F_o,\n\\qquad\n\\Delta_Sx=((H_e-I)x)_{e\\notin S},\n\\]\none has\n\\[\nC_o=\\ker\\Delta_S,\n\\qquad\n\\dim C_o=\\dim F_o-\\rank\\Delta_S;\n\\]\n\\item the full fibers glue to one global frame exactly when every fundamental holonomy is the identity;\n\\item a gauge change conjugates the holonomies and transports the core, so \\(\\dim C_o\\) is gauge-invariant.\n\\end{enumerate}\n\\end{theorem}",";\n\\item \\((C_v)_v\\) is path-flat, meaning that all path transports with the same endpoints agree on the core;\n\\item it is the unique greatest path-flat subfamily;\n\\item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies;\n\\item for\n\\[\n\\Delta_S:F_o\\longrightarrow\\bigoplus_{e\\notin S}F_o,\n\\qquad\n\\Delta_Sx=((H_e-I)x)_{e\\notin S},\n\\]\none has\n\\[\nC_o=\\ker\\Delta_S,\n\\qquad\n\\dim C_o=\\dim F_o-\\rank\\Delta_S;\n\\]\n\\item the full fibers glue to one global frame exactly when every fundamental holonomy is the identity;\n\\item a gauge change conjugates the holonomies and transports the core, so \\(\\dim C_o\\) is gauge-invariant.\n\\end{enumerate}\n\\end{theorem}","space of all closed-path holonomies;\n\\item for\n\\[\n\\Delta_S:F_o\\longrightarrow\\bigoplus_{e\\notin S}F_o,\n\\qquad\n\\Delta_Sx=((H_e-I)x)_{e\\notin S},\n\\]\none has\n\\[\nC_o=\\ker\\Delta_S,\n\\qquad\n\\dim C_o=\\dim F_o-\\rank\\Delta_S;\n\\]\n\\item the full fibers glue to one global frame exactly when every fundamental holonomy is the identity;\n\\item a gauge change conjugates the holonomies and transports the core, so \\(\\dim C_o\\) is gauge-invariant.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFor \\(x\\in C_o\\), set \\(x_v=G_vx\\).  Tree edges satisfy the parallel relation by construction, and a non-tree edge satisfies it because \\(H_ex=x\\).  Conversely, any global section is determined along the tree by its root value; every remaining edge forces that value to be fixed by the corresponding \\(H_e\\).  This proves the section classification.  Closed paths are words in fundamental cycles, which gives path-flatness and tree independence.  Any path-flat family must be fixed by every fundamental cycle and is therefore contained in \\((C_v)_v\\).  The rank formula is rank--nullity.  Full flatness and gauge covariance follow from spanning-tree normalization and conjugation of loop products.\n\\end{proof}","x=x\\).  Conversely, any global section is determined along the tree by its root value; every remaining edge forces that value to be fixed by the corresponding \\(H_e\\).  This proves the section classification.  Closed paths are words in fundamental cycles, which gives path-flatness and tree independence.  Any path-flat family must be fixed by every fundamental cycle and is therefore contained in \\((C_v)_v\\).  The rank formula is rank--nullity.  Full flatness and gauge covariance follow from spanning-tree normalization and conjugation of loop products.\n\\end{proof}","\\begin{corollary}[Global section versus global frame]\n\\label{cor:global-section-versus-frame}\nNonidentity holonomy can leave a nonzero maximal core.  Therefore the existence of some global compatible vectors is weaker than the existence of one global frame on the full fibers.  The former is measured by \\(\\dim C_o\\); the latter requires \\(C_o=F_o\\).\n\\end{corollary}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{flat_core_holonomy.pdf}\n\\caption{A spanning tree removes local gauge redundancy.  Each remaining edge contributes a fundamental holonomy.  Their common fixed space is the maximal path-flat core; the stacked defect map \\(\\Delta_S\\) measures the lost dimension.}\n\\label{fig:flat-core-holonomy}\n\\end{figure}","\\begin{definition}[Linear correction datum]\n\\label{def:block02-linear-correction}\nA linear correction datum is a map\n\\[\nD:U\\longrightarrow V\n\\]\nand a defect \\(r\\in V\\).  A correction is \\(u\\in U\\) satisfying \\(Du=-r\\).  The relative obstruction class is\n\\[\n\\Ob_D(r)=[r]\\in\\coker D=V/\\im D.\n\\]\n\\end{definition}","\\begin{theorem}[Cokernel criterion, universality, and naturality]\n\\label{thm:block02-cokernel-naturality}\nFor a correction datum \\((D,r)\\) \\cite{P3}:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a correction exists if and only if \\(\\Ob_D(r)=0\\);\n\\item every linear obstruction test \\(\\varphi:V\\to W\\) satisfying \\(\\varphi D=0\\) factors uniquely through \\(V\\to\\coker D\\);\n\\item for every commuting square\n\\[\n\\begin{array}{ccc}\nU & \\xrightarrow{D} & V","rem}[Cokernel criterion, universality, and naturality]\n\\label{thm:block02-cokernel-naturality}\nFor a correction datum \\((D,r)\\) \\cite{P3}:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a correction exists if and only if \\(\\Ob_D(r)=0\\);\n\\item every linear obstruction test \\(\\varphi:V\\to W\\) satisfying \\(\\varphi D=0\\) factors uniquely through \\(V\\to\\coker D\\);\n\\item for every commuting square\n\\[\n\\begin{array}{ccc}\nU & \\xrightarrow{D} & V\\\\\n\\downarrow a & & \\downarrow b\\\\\nU' & \\xrightarrow{D'} & V',\n\\end{array}\n\\qquad bD=D'a,\n\\]\nthere is a unique induced map\n\\[\n\\overline b:\\coker D\\longrightarrow\\coker D'\n\\]\nwith\n\\[\n\\overline b\\,\\Ob_D(r)=\\Ob_{D'}(br).\n\\]\n\\end{enumerate}\n\\end{theorem}","U' & \\xrightarrow{D'} & V',\n\\end{array}\n\\qquad bD=D'a,\n\\]\nthere is a unique induced map\n\\[\n\\overline b:\\coker D\\longrightarrow\\coker D'\n\\]\nwith\n\\[\n\\overline b\\,\\Ob_D(r)=\\Ob_{D'}(br).\n\\]\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nThe class vanishes exactly when \\(r\\in\\im D\\), which is equivalent to solvability of \\(Du=-r\\).  A map annihilating \\(\\im D\\) factors uniquely through the quotient.  In the square, the quotient map for \\(D'\\), composed with \\(b\\), vanishes on \\(\\im D\\), so the same universal property produces \\(\\overline b\\).\n\\end{proof}","\\begin{proposition}[Packet coordinates on the obstruction quotient]\n\\label{prop:block02-obstruction-packets}\nSuppose \\((Q_a)\\) is a complete finite family of pairwise annihilating idempotents on \\(V\\) and\n\\[\nQ_a(\\im D)\\subseteq\\im D\n\\]\nfor every \\(a\\).  Then the \\(Q_a\\) descend to quotient idempotents \\(\\overline Q_a\\) on \\(\\coker D\\),\n\\[\n\\coker D\n\\cong\n\\bigoplus_a\\overline Q_a(\\coker D),\n\\qquad\n\\overline Q_a\\Ob_D(r)=[Q_ar].\n\\]\nIf \\(U\\) and \\(V\\) already split into matching packet summands and \\(D\\) is block diagonal, this becomes\n\\[\n\\coker D\\cong\\bigoplus_a\\coker D_a.\n\\]\nOff-diagonal correction blocks can couple packet defects, so the block-diagonal hypothesis cannot be omitted.\n\\end{proposition}","s on the obstruction quotient]\n\\label{prop:block02-obstruction-packets}\nSuppose \\((Q_a)\\) is a complete finite family of pairwise annihilating idempotents on \\(V\\) and\n\\[\nQ_a(\\im D)\\subseteq\\im D\n\\]\nfor every \\(a\\).  Then the \\(Q_a\\) descend to quotient idempotents \\(\\overline Q_a\\) on \\(\\coker D\\),\n\\[\n\\coker D\n\\cong\n\\bigoplus_a\\overline Q_a(\\coker D),\n\\qquad\n\\overline Q_a\\Ob_D(r)=[Q_ar].\n\\]\nIf \\(U\\) and \\(V\\) already split into matching packet summands and \\(D\\) is block diagonal, this becomes\n\\[\n\\coker D\\cong\\bigoplus_a\\coker D_a.\n\\]\nOff-diagonal correction blocks can couple packet defects, so the block-diagonal hypothesis cannot be omitted.\n\\end{proposition}","bstruction quotient]\n\\label{prop:block02-obstruction-packets}\nSuppose \\((Q_a)\\) is a complete finite family of pairwise annihilating idempotents on \\(V\\) and\n\\[\nQ_a(\\im D)\\subseteq\\im D\n\\]\nfor every \\(a\\).  Then the \\(Q_a\\) descend to quotient idempotents \\(\\overline Q_a\\) on \\(\\coker D\\),\n\\[\n\\coker D\n\\cong\n\\bigoplus_a\\overline Q_a(\\coker D),\n\\qquad\n\\overline Q_a\\Ob_D(r)=[Q_ar].\n\\]\nIf \\(U\\) and \\(V\\) already split into matching packet summands and \\(D\\) is block diagonal, this becomes\n\\[\n\\coker D\\cong\\bigoplus_a\\coker D_a.\n\\]\nOff-diagonal correction blocks can couple packet defects, so the block-diagonal hypothesis cannot be omitted.\n\\end{proposition}","ackets}\nSuppose \\((Q_a)\\) is a complete finite family of pairwise annihilating idempotents on \\(V\\) and\n\\[\nQ_a(\\im D)\\subseteq\\im D\n\\]\nfor every \\(a\\).  Then the \\(Q_a\\) descend to quotient idempotents \\(\\overline Q_a\\) on \\(\\coker D\\),\n\\[\n\\coker D\n\\cong\n\\bigoplus_a\\overline Q_a(\\coker D),\n\\qquad\n\\overline Q_a\\Ob_D(r)=[Q_ar].\n\\]\nIf \\(U\\) and \\(V\\) already split into matching packet summands and \\(D\\) is block diagonal, this becomes\n\\[\n\\coker D\\cong\\bigoplus_a\\coker D_a.\n\\]\nOff-diagonal correction blocks can couple packet defects, so the block-diagonal hypothesis cannot be omitted.\n\\end{proposition}","e finite family of pairwise annihilating idempotents on \\(V\\) and\n\\[\nQ_a(\\im D)\\subseteq\\im D\n\\]\nfor every \\(a\\).  Then the \\(Q_a\\) descend to quotient idempotents \\(\\overline Q_a\\) on \\(\\coker D\\),\n\\[\n\\coker D\n\\cong\n\\bigoplus_a\\overline Q_a(\\coker D),\n\\qquad\n\\overline Q_a\\Ob_D(r)=[Q_ar].\n\\]\nIf \\(U\\) and \\(V\\) already split into matching packet summands and \\(D\\) is block diagonal, this becomes\n\\[\n\\coker D\\cong\\bigoplus_a\\coker D_a.\n\\]\nOff-diagonal correction blocks can couple packet defects, so the block-diagonal hypothesis cannot be omitted.\n\\end{proposition}","\\begin{proof}\nThe invariance of \\(\\im D\\) makes \\([v]\\mapsto[Q_av]\\) well defined.  Idempotence, annihilation, and reconstruction descend to the quotient.  In the block-diagonal case the image of \\(D\\) is the direct sum of the images of the restrictions.\n\\end{proof}","\\begin{theorem}[Relative obstruction exact sequence]\n\\label{thm:block02-relative-obstruction-sequence}\nSuppose the diagram\n\\[\n\\begin{array}{ccccccccc}\n0&\\to&U_0&\\to&U_f&\\to&U_c&\\to&0\\\\\n &&\\downarrow D_0&&\\downarrow D_f&&\\downarrow D_c\\\\\n0&\\to&V_0&\\to&V_f&\\to&V_c&\\to&0\n\\end{array}\n\\]\ncommutes and has exact rows.  Then there is a natural exact sequence\n\\[\n\\begin{aligned}\n0&\\longrightarrow\\ker D_0\n\\longrightarrow\\ker D_f\n\\longrightarrow\\ker D_c\n\\xrightarrow{\\delta}\\coker D_0","0&\\to&V_0&\\to&V_f&\\to&V_c&\\to&0\n\\end{array}\n\\]\ncommutes and has exact rows.  Then there is a natural exact sequence\n\\[\n\\begin{aligned}\n0&\\longrightarrow\\ker D_0\n\\longrightarrow\\ker D_f\n\\longrightarrow\\ker D_c\n\\xrightarrow{\\delta}\\coker D_0\\\\\n&\\longrightarrow\\coker D_f\n\\longrightarrow\\coker D_c\n\\longrightarrow0.\n\\end{aligned}\n\\]\nIf a fine obstruction maps to zero in \\(\\coker D_c\\), then it comes from \\(\\coker D_0\\), the obstruction sector removed by coarsening.  Coarse vanishing therefore does not imply fine vanishing.\n\\end{theorem}","&\\longrightarrow\\coker D_f\n\\longrightarrow\\coker D_c\n\\longrightarrow0.\n\\end{aligned}\n\\]\nIf a fine obstruction maps to zero in \\(\\coker D_c\\), then it comes from \\(\\coker D_0\\), the obstruction sector removed by coarsening.  Coarse vanishing therefore does not imply fine vanishing.\n\\end{theorem}","\\begin{proof}\nThis is the direct kernel--cokernel sequence proved in the strengthened Paper III.  The connecting map lifts an element of \\(\\ker D_c\\) to \\(U_f\\), applies \\(D_f\\), identifies the result in \\(V_0\\), and takes its class modulo \\(\\im D_0\\).  Exactness verifies independence of the lift and identifies each successive kernel with the preceding image \\cite{P3,MIG}.\n\\end{proof}","\\begin{example}[A fine obstruction erased by summation]\n\\label{ex:fine-obstruction-erased}\nTake all correction spaces to be zero,\n\\[\nV_f=\\kk^2,\n\\qquad\nV_c=\\kk,\n\\qquad\np(x,y)=x+y.\n\\]\nThe fine defect \\((1,-1)\\) is nonzero in \\(\\coker D_f=V_f\\), but its coarse image is zero.  It lies entirely in\n\\[\nV_0=\\ker p=\\Span\\{(1,-1)\\}.\n\\]\nThis is cancellation by coarsening, not correction of the fine defect.\n\\end{example}","\\begin{definition}[Mapping cone]\n\\label{def:block02-mapping-cone}\nFor a cochain map \\(D:C^\\bullet\\to E^\\bullet\\), define\n\\[\n\\Cone(D)^n=E^n\\oplus C^{n+1},\n\\qquad\n d_{\\Cone}(e,c)=(d_Ee+Dc,-d_Cc).\n\\]\n\\end{definition}","\\begin{proposition}[Cohomological correction criterion]\n\\label{prop:block02-cone-obstruction}\nFor a cocycle \\(e\\in E^n\\), the class of \\((e,0)\\) in \\(H^n(\\Cone(D))\\) vanishes exactly when \\([e]\\) lies in the image of \\(H^n(D)\\).  When the complexes are concentrated in degree zero,\n\\[\nH^0(\\Cone(D))\\cong\\coker D.\n\\]\n\\end{proposition}","\\begin{proof}\nThe class of \\((e,0)\\) is a boundary exactly when there are \\(e'\\in E^{n-1}\\) and \\(c\\in C^n\\) with\n\\[\ne=d_Ee'+Dc,\n\\qquad\nd_Cc=0.\n\\]\nThis is equivalent to \\([e]=H^n(D)[c]\\).  Concentration in degree zero reduces the cone quotient to \\(E^0/\\im D\\).\n\\end{proof}","\\begin{boundary}[The correction map is problem data]\n\\label{bnd:correction-map-problem-data}\nThe cokernel is canonical relative to \\(D\\).  The theory does not yet supply one choice-free rule that constructs the correct \\(D\\) for every mathematical problem and every stratum.  Calling \\([r]\\in\\coker D\\) canonical must not hide this dependence.\n\\end{boundary}","\\begin{theorem}[Effective vector descent by Hilbert 90]\n\\label{thm:block02-hilbert90}\nLet \\(L/K\\) be finite Galois with group \\(G\\).  Suppose\n\\[\na_{\\sigma\\tau}=a_\\sigma\\,\\sigma(a_\\tau),\n\\qquad\na_\\sigma\\in GL_n(L),\n\\]\ndefines a semilinear packet cocycle.  Then there is \\(B\\in GL_n(L)\\) such that\n\\[\na_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe twisted fixed packet\n\\[\nW=(L^n)^{G,\\star}\n\\]\nis an \\(n\\)-dimensional \\(K\\)-vector space and\n\\[\nW\\otimes_KL\\xrightarrow{\\sim}L^n.\n\\]\nThus every finite linear semilinear atlas of this form is effective \\cite{GAL}.\n\\end{theorem}","gin{theorem}[Effective vector descent by Hilbert 90]\n\\label{thm:block02-hilbert90}\nLet \\(L/K\\) be finite Galois with group \\(G\\).  Suppose\n\\[\na_{\\sigma\\tau}=a_\\sigma\\,\\sigma(a_\\tau),\n\\qquad\na_\\sigma\\in GL_n(L),\n\\]\ndefines a semilinear packet cocycle.  Then there is \\(B\\in GL_n(L)\\) such that\n\\[\na_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe twisted fixed packet\n\\[\nW=(L^n)^{G,\\star}\n\\]\nis an \\(n\\)-dimensional \\(K\\)-vector space and\n\\[\nW\\otimes_KL\\xrightarrow{\\sim}L^n.\n\\]\nThus every finite linear semilinear atlas of this form is effective \\cite{GAL}.\n\\end{theorem}","vector descent by Hilbert 90]\n\\label{thm:block02-hilbert90}\nLet \\(L/K\\) be finite Galois with group \\(G\\).  Suppose\n\\[\na_{\\sigma\\tau}=a_\\sigma\\,\\sigma(a_\\tau),\n\\qquad\na_\\sigma\\in GL_n(L),\n\\]\ndefines a semilinear packet cocycle.  Then there is \\(B\\in GL_n(L)\\) such that\n\\[\na_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe twisted fixed packet\n\\[\nW=(L^n)^{G,\\star}\n\\]\nis an \\(n\\)-dimensional \\(K\\)-vector space and\n\\[\nW\\otimes_KL\\xrightarrow{\\sim}L^n.\n\\]\nThus every finite linear semilinear atlas of this form is effective \\cite{GAL}.\n\\end{theorem}","\\begin{theorem}[Brauer obstruction to projective packet descent]\n\\label{thm:block02-brauer}\nProjective semilinear descent data define a class\n\\[\n\\alpha\\in H^1(G,PGL_n(L)).\n\\]\nThe connecting map for\n\\[\n1\\longrightarrow L^\\times\n\\longrightarrow GL_n(L)\n\\longrightarrow PGL_n(L)\n\\longrightarrow1\n\\]\nproduces\n\\[\n\\delta(\\alpha)\n\\in\nH^2(G,L^\\times)\n=\n\\Br(L/K).\n\\]\nThe projective atlas lifts to linear semilinear descent data exactly when \\(\\delta(\\alpha)=0\\).  When the class is nonzero, the descended object is naturally a module over a central simple \\(K\\)-algebra split by \\(L\\), rather than an ordinary \\(K\\)-vector space \\cite{GAL}.\n\\end{theorem}","to projective packet descent]\n\\label{thm:block02-brauer}\nProjective semilinear descent data define a class\n\\[\n\\alpha\\in H^1(G,PGL_n(L)).\n\\]\nThe connecting map for\n\\[\n1\\longrightarrow L^\\times\n\\longrightarrow GL_n(L)\n\\longrightarrow PGL_n(L)\n\\longrightarrow1\n\\]\nproduces\n\\[\n\\delta(\\alpha)\n\\in\nH^2(G,L^\\times)\n=\n\\Br(L/K).\n\\]\nThe projective atlas lifts to linear semilinear descent data exactly when \\(\\delta(\\alpha)=0\\).  When the class is nonzero, the descended object is naturally a module over a central simple \\(K\\)-algebra split by \\(L\\), rather than an ordinary \\(K\\)-vector space \\cite{GAL}.\n\\end{theorem}","ha\\in H^1(G,PGL_n(L)).\n\\]\nThe connecting map for\n\\[\n1\\longrightarrow L^\\times\n\\longrightarrow GL_n(L)\n\\longrightarrow PGL_n(L)\n\\longrightarrow1\n\\]\nproduces\n\\[\n\\delta(\\alpha)\n\\in\nH^2(G,L^\\times)\n=\n\\Br(L/K).\n\\]\nThe projective atlas lifts to linear semilinear descent data exactly when \\(\\delta(\\alpha)=0\\).  When the class is nonzero, the descended object is naturally a module over a central simple \\(K\\)-algebra split by \\(L\\), rather than an ordinary \\(K\\)-vector space \\cite{GAL}.\n\\end{theorem}","\\]\nThe connecting map for\n\\[\n1\\longrightarrow L^\\times\n\\longrightarrow GL_n(L)\n\\longrightarrow PGL_n(L)\n\\longrightarrow1\n\\]\nproduces\n\\[\n\\delta(\\alpha)\n\\in\nH^2(G,L^\\times)\n=\n\\Br(L/K).\n\\]\nThe projective atlas lifts to linear semilinear descent data exactly when \\(\\delta(\\alpha)=0\\).  When the class is nonzero, the descended object is naturally a module over a central simple \\(K\\)-algebra split by \\(L\\), rather than an ordinary \\(K\\)-vector space \\cite{GAL}.\n\\end{theorem}","longrightarrow L^\\times\n\\longrightarrow GL_n(L)\n\\longrightarrow PGL_n(L)\n\\longrightarrow1\n\\]\nproduces\n\\[\n\\delta(\\alpha)\n\\in\nH^2(G,L^\\times)\n=\n\\Br(L/K).\n\\]\nThe projective atlas lifts to linear semilinear descent data exactly when \\(\\delta(\\alpha)=0\\).  When the class is nonzero, the descended object is naturally a module over a central simple \\(K\\)-algebra split by \\(L\\), rather than an ordinary \\(K\\)-vector space \\cite{GAL}.\n\\end{theorem}","\\begin{proof}\nThis is the nonabelian cohomology exact sequence together with nonabelian Hilbert 90.  Vanishing of the connecting class gives a \\(GL_n(L)\\)-valued cocycle; \\cref{thm:block02-hilbert90} then trivializes it and constructs the descended vector space.  A nonzero class is the classical relative Brauer obstruction.\n\\end{proof}","\\begin{corollary}[Projective compatibility is not linear effectivity]\n\\label{cor:projective-not-linear-effectivity}\nA projective atlas may satisfy every visible overlap equation in \\(PGL_n\\) and still have no global vector-space realization.  Projective cocycle coherence and vanishing Brauer class are distinct conditions.\n\\end{corollary}","\\begin{theorem}[Tannakian boundary]\n\\label{thm:block02-tannakian-boundary}\nLet \\(\\mathcal T\\) be a neutral Tannakian category over \\(k\\) and\n\\[\n\\omega:\\mathcal T\\longrightarrow\\operatorname{Vect}_k\n\\]\na faithful exact tensor fiber functor.  Then\n\\[\n\\mathcal G=\\Aut^{\\otimes}(\\omega)\n\\]\nis an affine group scheme and\n\\[\n\\mathcal T\\simeq\\operatorname{Rep}_k(\\mathcal G)\n\\]\nas tensor categories.  A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary \\cite{GAL,RT}.\n\\end{theorem}","\\midrule\n\\endhead\nFinite transport holonomy & connected graph, invertible edge transports, chosen structure group & fundamental cycle holonomies or the conjugacy class of \\(\\pi_1(\\Gamma)\\to G\\) & one global frame on the full fibers; on a fixed subspace, path-independent transport & path dependence; possibly a smaller nonzero fixed core","Relative correction failure & correction map \\(D:U\\to V\\) and defect \\(r\\) & \\([r]\\in\\coker D\\), or a mapping-cone cohomology class & an allowed correction exists & the defect is outside the declared correction image","Projective Galois descent & projective semilinear cocycle over \\(L/K\\) & \\(\\delta(\\alpha)\\in H^2(G,L^\\times)=\\Br(L/K)\\) & lift to linear descent, then effective vector descent by Hilbert 90 & descent to a twisted central-simple-algebra object rather than an ordinary vector space","Higher atlas coherence & pairwise equivalences and comparison \\(2\\)-cells & \\(\\Omega_{\\alpha\\beta\\gamma\\delta}\\), a natural automorphism on a quadruple overlap & pseudofunctor coherence of the atlas & pairwise bridges use incompatible conventions or incomplete comparison data","\\paragraph{Relative obstruction theory.}\nOlder Paper III editions leave a broad universal packet-obstruction principle conjectural. The strengthened edition replaces the finite-projector formulation by the relative class \\([r]\\in\\coker D\\), with vanishing, naturality, packet decomposition, quotient projectors, relative exact sequences, and mapping-cone formulations proved for declared correction data \\cite{P3,MIG}. The unresolved problem is not the quotient construction itself. It is a choice-free mechanism assigning the appropriate correction datum to arbitrary problems and strata.","\\paragraph{Core-Transfer status.}\nAddendum 04 replaces the isolated \\(m=6,d=3\\) boundary by the exact composite-modulus rank law, sharp integral packet minimum, universal decoder, and all-degree stabilization \\cite{CT4}. The former example is now a sharp negative control inside a general theorem.","\\begin{itemize}\n\\item constant recurrence \\(\\leftrightarrow\\) rational ordinary generating function \\(\\leftrightarrow\\) finite linear state;\n\\item P-recursive \\(\\leftrightarrow\\) D-finite coefficient translation;\n\\item finite digit realization \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler inter","\\begin{itemize}\n\\item constant recurrence \\(\\leftrightarrow\\) rational ordinary generating function \\(\\leftrightarrow\\) finite linear state;\n\\item P-recursive \\(\\leftrightarrow\\) D-finite coefficient translation;\n\\item finite digit realization \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow","\\begin{itemize}\n\\item constant recurrence \\(\\leftrightarrow\\) rational ordinary generating function \\(\\leftrightarrow\\) finite linear state;\n\\item P-recursive \\(\\leftrightarrow\\) D-finite coefficient translation;\n\\item finite digit realization \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)","\\begin{itemize}\n\\item constant recurrence \\(\\leftrightarrow\\) rational ordinary generating function \\(\\leftrightarrow\\) finite linear state;\n\\item P-recursive \\(\\leftrightarrow\\) D-finite coefficient translation;\n\\item finite digit realization \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","m constant recurrence \\(\\leftrightarrow\\) rational ordinary generating function \\(\\leftrightarrow\\) finite linear state;\n\\item P-recursive \\(\\leftrightarrow\\) D-finite coefficient translation;\n\\item finite digit realization \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","ng function \\(\\leftrightarrow\\) finite linear state;\n\\item P-recursive \\(\\leftrightarrow\\) D-finite coefficient translation;\n\\item finite digit realization \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","recursive \\(\\leftrightarrow\\) D-finite coefficient translation;\n\\item finite digit realization \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","on \\(\\leftrightarrow\\) finite \\(k\\)-kernel;\n\\item rational functional-Mahler mask \\(\\to\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","\\) Cartier representation \\(\\to\\) finite automaton;\n\\item finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","ier transitions \\(\\leftrightarrow\\) carry matrices;\n\\item selected algebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","gebraic branch \\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","\\(\\to\\) packet polynomial system;\n\\item invertible transform \\(\\leftrightarrow\\) transported packet resolution;\n\\item Euler-product exponent coordinates \\(\\leftrightarrow\\) logarithmic-derivative recurrence data;\n\\item finite modular state \\(\\to\\) local Newton-Mahler interpolation;\n\\item local Mahler coordinates \\(\\leftrightarrow\\) affine polyphase branches;\n\\item finite resolved \\(C^*\\)-system \\(\\to\\) faithful Hilbert packet realization.\n\\end{itemize}","\\begin{definition}[Finite profiled resolved atlas]\nFix a small resolved theory \\(\\bbK\\). A finite profiled resolved atlas consists of:\n\\begin{enumerate}[label=\\textup{(A\\arabic*)}]\n\\item a finite indexing category \\(\\check{\\mathcal A}\\), retained through the overlap degrees needed for the cocycle law;\n\\item a pseudofunctor\n\\[\n\\mathfrak R:\\check{\\mathcal A}^{\\mathrm{op}}\\to\\Cat;\n\\]\n\\item a declared preservation profile \\(\\Pi_U\\) on every chart and overlap category \\(\\mathfrak R(U)\\);\n\\item coherent equivalences on the reversible overlaps and separately typed directional transformations;\n\\item a pseudonatural semantics\n\\[\n\\Sem:\\mathfrak R\\Rightarrow\\mathfrak O\n\\]\ninto a common observable doctrine;\n\\item optional pseudonatural invariant families with explicitly typed comparison","\\begin{definition}[Finite profiled resolved atlas]\nFix a small resolved theory \\(\\bbK\\). A finite profiled resolved atlas consists of:\n\\begin{enumerate}[label=\\textup{(A\\arabic*)}]\n\\item a finite indexing category \\(\\check{\\mathcal A}\\), retained through the overlap degrees needed for the cocycle law;\n\\item a pseudofunctor\n\\[\n\\mathfrak R:\\check{\\mathcal A}^{\\mathrm{op}}\\to\\Cat;\n\\]\n\\item a declared preservation profile \\(\\Pi_U\\) on every chart and overlap category \\(\\mathfrak R(U)\\);\n\\item coherent equivalences on the reversible overlaps and separately typed directional transformations;\n\\item a pseudonatural semantics\n\\[\n\\Sem:\\mathfrak R\\Rightarrow\\mathfrak O\n\\]\ninto a common observable doctrine;\n\\item optional pseudonatural invariant families with explicitly typed comparison cells.\n\\end{enumerate}\n\\end{definition}","e profiled resolved atlas]\nFix a small resolved theory \\(\\bbK\\). A finite profiled resolved atlas consists of:\n\\begin{enumerate}[label=\\textup{(A\\arabic*)}]\n\\item a finite indexing category \\(\\check{\\mathcal A}\\), retained through the overlap degrees needed for the cocycle law;\n\\item a pseudofunctor\n\\[\n\\mathfrak R:\\check{\\mathcal A}^{\\mathrm{op}}\\to\\Cat;\n\\]\n\\item a declared preservation profile \\(\\Pi_U\\) on every chart and overlap category \\(\\mathfrak R(U)\\);\n\\item coherent equivalences on the reversible overlaps and separately typed directional transformations;\n\\item a pseudonatural semantics\n\\[\n\\Sem:\\mathfrak R\\Rightarrow\\mathfrak O\n\\]\ninto a common observable doctrine;\n\\item optional pseudonatural invariant families with explicitly typed comparison cells.\n\\end{enumerate}\n\\end{definition}","d theory \\(\\bbK\\). A finite profiled resolved atlas consists of:\n\\begin{enumerate}[label=\\textup{(A\\arabic*)}]\n\\item a finite indexing category \\(\\check{\\mathcal A}\\), retained through the overlap degrees needed for the cocycle law;\n\\item a pseudofunctor\n\\[\n\\mathfrak R:\\check{\\mathcal A}^{\\mathrm{op}}\\to\\Cat;\n\\]\n\\item a declared preservation profile \\(\\Pi_U\\) on every chart and overlap category \\(\\mathfrak R(U)\\);\n\\item coherent equivalences on the reversible overlaps and separately typed directional transformations;\n\\item a pseudonatural semantics\n\\[\n\\Sem:\\mathfrak R\\Rightarrow\\mathfrak O\n\\]\ninto a common observable doctrine;\n\\item optional pseudonatural invariant families with explicitly typed comparison cells.\n\\end{enumerate}\n\\end{definition}","hcal A}\\), retained through the overlap degrees needed for the cocycle law;\n\\item a pseudofunctor\n\\[\n\\mathfrak R:\\check{\\mathcal A}^{\\mathrm{op}}\\to\\Cat;\n\\]\n\\item a declared preservation profile \\(\\Pi_U\\) on every chart and overlap category \\(\\mathfrak R(U)\\);\n\\item coherent equivalences on the reversible overlaps and separately typed directional transformations;\n\\item a pseudonatural semantics\n\\[\n\\Sem:\\mathfrak R\\Rightarrow\\mathfrak O\n\\]\ninto a common observable doctrine;\n\\item optional pseudonatural invariant families with explicitly typed comparison cells.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Descent category]\nThe category \\(\\Desc(\\mathfrak R)\\) has pseudocones over \\(\\mathfrak R\\) as objects and modifications as morphisms. Concretely, an object is a compatible family\n\\[\n(M_\\alpha,\\theta_{\\alpha\\beta})\n\\]\nwith invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.\n\\end{definition}","\\begin{theorem}[Finite profiled resolved atlas assembly]\n\\label{thm:atlas-assembly}\nLet\n\\[\n\\mathfrak R:\\check{\\mathcal A}^{\\mathrm{op}}\\to\\Cat\n\\]\nbe a finite profiled resolved atlas. Assume the chart and overlap categories are small, or locally small in one fixed universe, the reversible transitions and comparison cells satisfy the pseudofunctor coherence equations, and every construction used below belongs to the verified profile of every participating chart and bridge. Then:","\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\Desc(\\mathfrak R)\\) exists and has the universal property of the finite pseudolimit of \\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis","\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\Desc(\\mathfrak R)\\) exists and has the universal property of the finite pseudolimit of \\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item","\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\Desc(\\mathfrak R)\\) exists and has the universal property of the finite pseudolimit of \\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc","\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\Desc(\\mathfrak R)\\) exists and has the universal property of the finite pseudolimit of \\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized ch","\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(\\Desc(\\mathfrak R)\\) exists and has the universal property of the finite pseudolimit of \\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invarian…","\\(\\Desc(\\mathfrak R)\\) exists and has the universal property of the finite pseudolimit of \\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, t…","d has the universal property of the finite pseudolimit of \\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, then \\(\\Desc(I)\\) is complete on …","\\(\\mathfrak R\\).\n\\item There is a canonical equivalence\n\\[\n\\Core\\bigl(\\Desc(\\mathfrak R)\\bigr)\n\\simeq\n\\Desc\\bigl(\\mathfrak R^{\\simeq}\\bigr).\n\\]\n\\item Every pseudonatural transformation \\(T:\\mathfrak R\\Rightarrow\\mathfrak S\\) induces a functor\n\\[\n\\Desc(T):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, then \\(\\Desc(I)\\) is complete on the corresponding global normalized category.\n\\end{enumera…","thfrak S).\n\\]\nIf every \\(T_U\\) is an equivalence, then \\(\\Desc(T)\\) is an equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, then \\(\\Desc(I)\\) is complete on the corresponding global normalized category.\n\\end{enumerate}","n equivalence. A pointwise noninvertible \\(T\\) remains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, then \\(\\Desc(I)\\) is complete on the corresponding global normalized category.\n\\end{enumerate}","emains directional.\n\\item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, then \\(\\Desc(I)\\) is complete on the corresponding global normalized category.\n\\end{enumerate}","ic hypothesis is verified locally, their realizations are equal in \\(\\Desc(\\mathfrak R)\\).\n\\item A coherent pseudonatural invariant family \\(I:\\mathfrak R\\Rightarrow\\mathfrak V\\) induces\n\\[\n\\Desc(I):\\Desc(\\mathfrak R)\\to\\Desc(\\mathfrak V).\n\\]\n\\item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, then \\(\\Desc(I)\\) is complete on the corresponding global normalized category.\n\\end{enumerate}","If the semantic pseudonatural transformation is effective for objects and morphisms, then \\(\\Desc(\\Sem)\\) identifies descent data with the declared global semantic objects.\n\\end{theorem}","\\begin{proof}\nObjects and morphisms of \\(\\Desc(\\mathfrak R)\\) are finite compatible families. Identities and composition are defined componentwise. The pseudofunctor coherence equations ensure that componentwise composition remains compatible. This explicit construction has the universal property of pseudocones, proving (i).","Fix \\(d\\ge0\\). Over characteristic zero, the affine Mahler coefficient maps \\(A_{m,s}^{(d)}\\) are invertible and satisfy\n\\[\nA_{n,t}^{(d)}A_{m,s}^{(d)}=A_{mn,mt+s}^{(d)}.\n\\]\nFor coprime \\(m,n\\), the two paths around every declared Chinese-remainder diamond are equal \\cite{CT4}. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with trivial holonomy on those diamonds.","Let\n\\[\nA=P_{m,S}^{(d)}:\\mathbb Z^{d+1}\\to\\mathbb Z^{|S|(d+1)}\n\\]\nbe a stacked affine-Mahler packet analysis map. Whenever an integer left inverse \\(L\\) exists,\n\\[\nLA=I,\n\\qquad\n\\Pi=AL,\n\\qquad\n\\Pi^2=\\Pi,\n\\]\nand\n\\[\n\\mathbb Z^{|S|(d+1)}=\\im A\\oplus\\ker L.\n\\]\nThus the source chart and the normalized packet core \\(\\im A\\) are equivalent, while \\(\\ker L\\) is a named residual rather than discarded information.","Addendum 04 proves that the prefix\n\\[\nS_*=\\{0,1,\\dots,L_{\\mathbb Z}(m,d)-1\\}\n\\]\nwith\n\\[\nL_{\\mathbb Z}(m,d)=\\min(d+1,B(m)),\n\\qquad\nB(m)=\\max\\bigl(\\{1\\}\\cup\\{p^{v_p(m)}:p\\mid m\\}\\bigr),\n\\]\nalways admits an integer decoder and is cardinality-minimal \\cite{CT4}.","\\begin{proposition}[Integral split Mahler packet descent]\nThe analysis-decoder pair above gives effective object and morphism descent between the source chart and the normalized packet-core chart. The split identity survives every base change. The residual summand \\(\\ker L\\) is functorial under maps intertwining both \\(A\\) and \\(L\\).\n\\end{proposition}","\\begin{proof}\nThe identities \\(LA=I\\) and \\(\\Pi=AL\\) show that \\(A\\) identifies the source with \\(\\im A\\), with inverse \\(L|_{\\im A}\\). For any packet vector \\(y\\),\n\\[\ny=ALy+(y-ALy),\n\\]\nwhere \\(ALy\\in\\im A\\) and \\(y-ALy\\in\\ker L\\). The intersection is zero because \\(Ax\\in\\ker L\\) implies \\(x=LAx=0\\). Tensoring \\(LA=I\\) with any coefficient ring preserves the split identity. Intertwining maps preserve the image and kernel decompositions.\n\\end{proof}","Fix a field \\(K\\). A nonzero C-finite sequence \\(a=(a_n)_{n\\ge0}\\) is \\emph{persistent} when its minimal characteristic polynomial has nonzero constant term. Equivalently, its active characteristic roots are nonzero, or its minimal state transition is invertible \\cite{SEQ}.","Let \\(\\Seq_K^{\\mathrm{pers}}\\) be the discrete groupoid of persistent C-finite sequences over \\(K\\).","An object of \\(\\Rec_K^{\\mathrm{min}}\\) is a pair\n\\[\n(\\chi,\\mathbf a_0),\n\\]\nwhere\n\\[\n\\chi(x)=x^d-c_{d-1}x^{d-1}-\\cdots-c_1x-c_0,\n\\qquad c_0\\ne0,\n\\]\nis the monic minimal recurrence polynomial and \\(\\mathbf a_0=(a_0,\\dots,a_{d-1})\\) is the initial vector. It generates the sequence\n\\[\na_{n+d}=c_{d-1}a_{n+d-1}+\\cdots+c_1a_{n+1}+c_0a_n.\n\\]\nBecause the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus \\(\\Rec_K^{\\mathrm{min}}\\) is a discrete groupoid.","r\n\\[\n(\\chi,\\mathbf a_0),\n\\]\nwhere\n\\[\n\\chi(x)=x^d-c_{d-1}x^{d-1}-\\cdots-c_1x-c_0,\n\\qquad c_0\\ne0,\n\\]\nis the monic minimal recurrence polynomial and \\(\\mathbf a_0=(a_0,\\dots,a_{d-1})\\) is the initial vector. It generates the sequence\n\\[\na_{n+d}=c_{d-1}a_{n+d-1}+\\cdots+c_1a_{n+1}+c_0a_n.\n\\]\nBecause the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus \\(\\Rec_K^{\\mathrm{min}}\\) is a discrete groupoid.","An object of \\(\\Rat_K^{\\mathrm{red}}\\) is a reduced rational power series\n\\[\nA(z)=\\frac{P(z)}{Q(z)},\n\\qquad\nQ(0)=1,\n\\qquad\n\\gcd(P,Q)=1,\n\\qquad\n\\deg P<\\deg Q,\n\\]\nwhose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so \\(\\Rat_K^{\\mathrm{red}}\\) is discrete.","The recurrence-to-OGF bridge is explicit. From\n\\[\n\\chi(x)=x^d-c_{d-1}x^{d-1}-\\cdots-c_0\n\\]\nset\n\\[\nQ(z)=z^d\\chi(z^{-1})\n=1-c_{d-1}z-\\cdots-c_0z^d.\n\\]\nThe numerator is the degree-\\(<d\\) truncation of \\(Q(z)\\sum_{n\\ge0}a_nz^n\\). Conversely, clearing \\(Q\\) recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.","An object of \\(\\State_K^{\\mathrm{min}}\\) is a triple\n\\[\n(C,v,\\lambda),\n\\qquad\nC\\in M_d(K),\\quad v\\in K^d,\\quad \\lambda\\in(K^d)^*,\n\\]\nwith \\(C\\) invertible, producing\n\\[\na_n=\\lambda C^nv,\n\\]\nand satisfying reachability and observability. A morphism\n\\[\nS:(C,v,\\lambda)\\longrightarrow(C',v',\\lambda')\n\\]\nis a similarity satisfying\n\\[\nC'=SCS^{-1},\n\\qquad\nv'=Sv,\n\\qquad\n\\lambda'=\\lambda S^{-1}.\n\\]\nThe semantic functor sends a state to its coefficient sequence.","The state-to-OGF bridge is the resolvent\n\\[\nA(z)=\\lambda(I-zC)^{-1}v.\n\\]\nThe reduced denominator has degree equal to the minimal state dimension, which equals the Hankel rank. Conversely, a reduced rational OGF has a companion realization. Reducedness makes the companion realization reachable, while the companion output is observable. Any two minimal realizations of the same sequence are uniquely similar through their common shift action on the Hankel column space \\cite{SEQ}.","\\begin{theorem}[Normalized C-finite three-chart atlas]\n\\label{thm:cfinite-three-chart}\nFor every field \\(K\\), the semantic functors fit into equivalences of groupoids\n\\[\n\\Rec_K^{\\mathrm{min}}\n\\simeq\n\\Rat_K^{\\mathrm{red}}\n\\simeq\n\\State_K^{\\mathrm{min}}\n\\simeq\n\\Seq_K^{\\mathrm{pers}}.\n\\]\nThe recurrence-rational bridge is a strict inverse pair. The rational-state bridge is an equivalence with companion quasi-inverse. The comparison isomorphism on the state-side composite is the unique similarity between minimal realizations. Consequently:","\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item the three charts and their pairwise overlaps form a coherent finite profiled atlas;\n\\item the triple-overlap comparison satisfies the cocycle law by uniqueness of minimal similarity;\n\\item descent for objects and morphisms is effective;\n\\item the descent category is equivalent to \\(\\Seq_K^{\\mathrm{pers}}\\).\n\\end{enumerate}\n\\end{theorem}","ly, a reduced rational series gives a companion state. Minimal linear dimension equals Hankel rank, and the reduced denominator degree has the same value. Hence the companion state is minimal. Any two minimal states representing the same sequence identify with the forward shift on the Hankel column space and are therefore similar. The similarity is unique because a reachable state is spanned by \\(C^nv\\), so an intertwiner fixing the input vector is determined on a spanning set.","For a persistent C-finite sequence, the minimal dimension\n\\[\n\\rho(a)=\\linrank(a)=\\rank H(a)\n\\]\nis simultaneously the minimal recurrence order, the reduced denominator degree, and the dimension of a minimal reachable-observable state.","Over characteristic zero, write the unique active expansion over an algebraic closure as\n\\[\na_n=\\sum_{\\lambda\\in\\Lambda}P_\\lambda(n)\\lambda^n.\n\\]\nFor each sampling modulus \\(N\\) and address \\(j\\), the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of \\(a_{Nm+j}\\) \\cite{SEQ}.","\\midrule\nFibonacci \\(F_n\\) & \\(x^2-x-1\\) & \\(z/(1-z-z^2)\\) & \\(2\\) & The root ratio is not torsion. Every arithmetic subsequence retains rank \\(2\\).","\\(1+(-1)^n\\) & \\(x^2-1\\) & \\(2/(1-z^2)\\) & \\(2\\) & At \\(N=2\\), the even packet is constant of rank \\(1\\) and the odd packet is zero.","\\(n+1)\\) & \\((x-1)^2\\) & \\(1/(1-z)^2\\) & \\(2\\) & Every packet remains a nonzero affine polynomial in the packet index and retains rank \\(2\\).","The previous sections separated the formal assembly theorem from the nonformal work of declaring and verifying an actual atlas.  This section discharges that burden for a deliberately restricted but genuinely heterogeneous example.  The charts do not merely use different coordinates inside one linear category.  They belong to three of the principal strata:\n\\[\n\\text{finite linear and C-finite}\n\\qquad\n\\text{digital and automatic}\n\\qquad\n\\text{functional-Mahler and Cartier}.\n\\]\nThe common semantic objects are unit purely periodic sequences over a prime field.  The restriction is strong enough to make every bridge explicit, every normalized fiber rigid, every nonempty triple overlap coherent, and descent effective for both objects and morphisms.  At the same time it is broad enough to contain repeated-root recurrences in characteristic \\(p\\), nontrivial minimal automata, rational functional-Mahler masks, and redundant Cartier compiler presentations.","unctional-Mahler and Cartier}.\n\\]\nThe common semantic objects are unit purely periodic sequences over a prime field.  The restriction is strong enough to make every bridge explicit, every normalized fiber rigid, every nonempty triple overlap coherent, and descent effective for both objects and morphisms.  At the same time it is broad enough to contain repeated-root recurrences in characteristic \\(p\\), nontrivial minimal automata, rational functional-Mahler masks, and redundant Cartier compiler presentations.","Fix a prime \\(p\\).  Let \\(\\Per_p^{\\times,\\mathrm{per}}\\) denote the discrete category whose objects are sequences\n\\[\na=(a_n)_{n\\geq 0}\\in \\mathbb F_p^{\\mathbb N}\n\\]\nsatisfying\n\\[\na_0=1\n\\qquad\\text{and}\\qquad\n a_{n+T}=a_n\n\\quad(n\\geq 0)\n\\]\nfor some \\(T\\geq 1\\).  The least such positive integer is the \\emph{minimal period} \\(T(a)\\).  Write\n\\[\nw(a)=(a_0,\\ldots,a_{T(a)-1})\n\\]\nfor the primitive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than a","Fix a prime \\(p\\).  Let \\(\\Per_p^{\\times,\\mathrm{per}}\\) denote the discrete category whose objects are sequences\n\\[\na=(a_n)_{n\\geq 0}\\in \\mathbb F_p^{\\mathbb N}\n\\]\nsatisfying\n\\[\na_0=1\n\\qquad\\text{and}\\qquad\n a_{n+T}=a_n\n\\quad(n\\geq 0)\n\\]\nfor some \\(T\\geq 1\\).  The least such positive integer is the \\emph{minimal period} \\(T(a)\\).  Write\n\\[\nw(a)=(a_0,\\ldots,a_{T(a)-1})\n\\]\nfor the primitive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \\(\\mathbb F","Fix a prime \\(p\\).  Let \\(\\Per_p^{\\times,\\mathrm{per}}\\) denote the discrete category whose objects are sequences\n\\[\na=(a_n)_{n\\geq 0}\\in \\mathbb F_p^{\\mathbb N}\n\\]\nsatisfying\n\\[\na_0=1\n\\qquad\\text{and}\\qquad\n a_{n+T}=a_n\n\\quad(n\\geq 0)\n\\]\nfor some \\(T\\geq 1\\).  The least such positive integer is the \\emph{minimal period} \\(T(a)\\).  Write\n\\[\nw(a)=(a_0,\\ldots,a_{T(a)-1})\n\\]\nfor the primitive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \\(\\mathbb F_3\\), the period word \\(12\\) has\n\\[\n\\frac{1+2z}{1-z^2}=\\frac{1}{1+z}.\n\\]","\\(p\\).  Let \\(\\Per_p^{\\times,\\mathrm{per}}\\) denote the discrete category whose objects are sequences\n\\[\na=(a_n)_{n\\geq 0}\\in \\mathbb F_p^{\\mathbb N}\n\\]\nsatisfying\n\\[\na_0=1\n\\qquad\\text{and}\\qquad\n a_{n+T}=a_n\n\\quad(n\\geq 0)\n\\]\nfor some \\(T\\geq 1\\).  The least such positive integer is the \\emph{minimal period} \\(T(a)\\).  Write\n\\[\nw(a)=(a_0,\\ldots,a_{T(a)-1})\n\\]\nfor the primitive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \\(\\mathbb F_3\\), the period word \\(12\\) has\n\\[\n\\frac{1+2z}{1-z^2}=\\frac{1}{1+z}.\n\\]","screte category whose objects are sequences\n\\[\na=(a_n)_{n\\geq 0}\\in \\mathbb F_p^{\\mathbb N}\n\\]\nsatisfying\n\\[\na_0=1\n\\qquad\\text{and}\\qquad\n a_{n+T}=a_n\n\\quad(n\\geq 0)\n\\]\nfor some \\(T\\geq 1\\).  The least such positive integer is the \\emph{minimal period} \\(T(a)\\).  Write\n\\[\nw(a)=(a_0,\\ldots,a_{T(a)-1})\n\\]\nfor the primitive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \\(\\mathbb F_3\\), the period word \\(12\\) has\n\\[\n\\frac{1+2z}{1-z^2}=\\frac{1}{1+z}.\n\\]","a)=(a_0,\\ldots,a_{T(a)-1})\n\\]\nfor the primitive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \\(\\mathbb F_3\\), the period word \\(12\\) has\n\\[\n\\frac{1+2z}{1-z^2}=\\frac{1}{1+z}.\n\\]","the primitive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \\(\\mathbb F_3\\), the period word \\(12\\) has\n\\[\n\\frac{1+2z}{1-z^2}=\\frac{1}{1+z}.\n\\]","ive period word and\n\\[\nW_a(z)=\\sum_{r=0}^{T(a)-1}a_rz^r.\n\\]\nThen\n\\begin{equation}\n\\label{eq:periodic-ogf-unreduced}\nF_a(z)=\\sum_{n\\geq 0}a_nz^n\n=\n\\frac{W_a(z)}{1-z^{T(a)}}.\n\\end{equation}\nThe fraction in \\cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \\(\\mathbb F_3\\), the period word \\(12\\) has\n\\[\n\\frac{1+2z}{1-z^2}=\\frac{1}{1+z}.\n\\]","\\begin{boundary}[Why the unit condition is retained]\nThe functional-Mahler bridge uses the formal quotient\n\\[\nR_a(z)=\\frac{F_a(z)}{F_a(z^p)}.\n\\]\nIt is regular at the origin exactly when \\(F_a(0)\\neq 0\\).  The normalization \\(a_0=1\\) removes scalar ambiguity and gives \\(R_a(0)=1\\).  Purely periodic sequences with \\(a_0=0\\) remain valid C-finite and automatic objects, but they need not lie in this regular unit-mask overlap.  They are not discarded; they lie outside the selected heterogeneous atlas.\n\\end{boundary}","\\begin{proposition}[Finite-field persistence equals pure periodicity]\n\\label{prop:persistent-periodic}\nFor a sequence over \\(\\mathbb F_p\\), the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The sequence is C-finite and its minimal recurrence has nonzero constant term.\n\\item The sequence has a finite-dimensional state realization\n\\[\na_n=\\lambda C^nv\n\\]\nwith \\(C\\in GL_d(\\mathbb F_p)\\).\n\\item The sequence is purely periodic from \\(n=0\\).\n\\end{enumerate}\nIf \\(m_a(x)\\) is the monic minimal recurrence polynomial and \\(T=T(a)\\), then\n\\begin{equation}\n\\label{eq:minimal-period-annihilator}\nm_a(x)\\mid x^T-1,\n\\qquad\nT=\\min\\{N\\geq 1:m_a(x)\\mid x^N-1\\}.\n\\end{equation}\n\\end{proposition}","dicity]\n\\label{prop:persistent-periodic}\nFor a sequence over \\(\\mathbb F_p\\), the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The sequence is C-finite and its minimal recurrence has nonzero constant term.\n\\item The sequence has a finite-dimensional state realization\n\\[\na_n=\\lambda C^nv\n\\]\nwith \\(C\\in GL_d(\\mathbb F_p)\\).\n\\item The sequence is purely periodic from \\(n=0\\).\n\\end{enumerate}\nIf \\(m_a(x)\\) is the monic minimal recurrence polynomial and \\(T=T(a)\\), then\n\\begin{equation}\n\\label{eq:minimal-period-annihilator}\nm_a(x)\\mid x^T-1,\n\\qquad\nT=\\min\\{N\\geq 1:m_a(x)\\mid x^N-1\\}.\n\\end{equation}\n\\end{proposition}","nt-periodic}\nFor a sequence over \\(\\mathbb F_p\\), the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The sequence is C-finite and its minimal recurrence has nonzero constant term.\n\\item The sequence has a finite-dimensional state realization\n\\[\na_n=\\lambda C^nv\n\\]\nwith \\(C\\in GL_d(\\mathbb F_p)\\).\n\\item The sequence is purely periodic from \\(n=0\\).\n\\end{enumerate}\nIf \\(m_a(x)\\) is the monic minimal recurrence polynomial and \\(T=T(a)\\), then\n\\begin{equation}\n\\label{eq:minimal-period-annihilator}\nm_a(x)\\mid x^T-1,\n\\qquad\nT=\\min\\{N\\geq 1:m_a(x)\\mid x^N-1\\}.\n\\end{equation}\n\\end{proposition}","owing are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The sequence is C-finite and its minimal recurrence has nonzero constant term.\n\\item The sequence has a finite-dimensional state realization\n\\[\na_n=\\lambda C^nv\n\\]\nwith \\(C\\in GL_d(\\mathbb F_p)\\).\n\\item The sequence is purely periodic from \\(n=0\\).\n\\end{enumerate}\nIf \\(m_a(x)\\) is the monic minimal recurrence polynomial and \\(T=T(a)\\), then\n\\begin{equation}\n\\label{eq:minimal-period-annihilator}\nm_a(x)\\mid x^T-1,\n\\qquad\nT=\\min\\{N\\geq 1:m_a(x)\\mid x^N-1\\}.\n\\end{equation}\n\\end{proposition}","umerate}[label=\\textup{(\\roman*)}]\n\\item The sequence is C-finite and its minimal recurrence has nonzero constant term.\n\\item The sequence has a finite-dimensional state realization\n\\[\na_n=\\lambda C^nv\n\\]\nwith \\(C\\in GL_d(\\mathbb F_p)\\).\n\\item The sequence is purely periodic from \\(n=0\\).\n\\end{enumerate}\nIf \\(m_a(x)\\) is the monic minimal recurrence polynomial and \\(T=T(a)\\), then\n\\begin{equation}\n\\label{eq:minimal-period-annihilator}\nm_a(x)\\mid x^T-1,\n\\qquad\nT=\\min\\{N\\geq 1:m_a(x)\\mid x^N-1\\}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nA recurrence with nonzero constant term has an invertible companion matrix, giving (i) \\(\\Rightarrow\\) (ii).  The orbit of \\(v\\) under an invertible map of the finite set \\(\\mathbb F_p^d\\) is a cycle containing its initial state, so (ii) \\(\\Rightarrow\\) (iii).  A sequence of period \\(T\\) is annihilated by \\(E^T-I\\), where \\(E\\) is the forward shift, giving (iii) \\(\\Rightarrow\\) (i).  The minimal annihilator divides every annihilator, hence \\(m_a\\mid x^T-1\\).  Conversely, if \\(m_a\\mid x^N-1\\), then \\(E^N-I\\) annihilates the sequence.  Minimality of \\(T\\) proves \\cref{eq:minimal-period-annihilator}.\n\\end{proof}","invertible companion matrix, giving (i) \\(\\Rightarrow\\) (ii).  The orbit of \\(v\\) under an invertible map of the finite set \\(\\mathbb F_p^d\\) is a cycle containing its initial state, so (ii) \\(\\Rightarrow\\) (iii).  A sequence of period \\(T\\) is annihilated by \\(E^T-I\\), where \\(E\\) is the forward shift, giving (iii) \\(\\Rightarrow\\) (i).  The minimal annihilator divides every annihilator, hence \\(m_a\\mid x^T-1\\).  Conversely, if \\(m_a\\mid x^N-1\\), then \\(E^N-I\\) annihilates the sequence.  Minimality of \\(T\\) proves \\cref{eq:minimal-period-annihilator}.\n\\end{proof}","g (i) \\(\\Rightarrow\\) (ii).  The orbit of \\(v\\) under an invertible map of the finite set \\(\\mathbb F_p^d\\) is a cycle containing its initial state, so (ii) \\(\\Rightarrow\\) (iii).  A sequence of period \\(T\\) is annihilated by \\(E^T-I\\), where \\(E\\) is the forward shift, giving (iii) \\(\\Rightarrow\\) (i).  The minimal annihilator divides every annihilator, hence \\(m_a\\mid x^T-1\\).  Conversely, if \\(m_a\\mid x^N-1\\), then \\(E^N-I\\) annihilates the sequence.  Minimality of \\(T\\) proves \\cref{eq:minimal-period-annihilator}.\n\\end{proof}","i).  The orbit of \\(v\\) under an invertible map of the finite set \\(\\mathbb F_p^d\\) is a cycle containing its initial state, so (ii) \\(\\Rightarrow\\) (iii).  A sequence of period \\(T\\) is annihilated by \\(E^T-I\\), where \\(E\\) is the forward shift, giving (iii) \\(\\Rightarrow\\) (i).  The minimal annihilator divides every annihilator, hence \\(m_a\\mid x^T-1\\).  Conversely, if \\(m_a\\mid x^N-1\\), then \\(E^N-I\\) annihilates the sequence.  Minimality of \\(T\\) proves \\cref{eq:minimal-period-annihilator}.\n\\end{proof}","the finite set \\(\\mathbb F_p^d\\) is a cycle containing its initial state, so (ii) \\(\\Rightarrow\\) (iii).  A sequence of period \\(T\\) is annihilated by \\(E^T-I\\), where \\(E\\) is the forward shift, giving (iii) \\(\\Rightarrow\\) (i).  The minimal annihilator divides every annihilator, hence \\(m_a\\mid x^T-1\\).  Conversely, if \\(m_a\\mid x^N-1\\), then \\(E^N-I\\) annihilates the sequence.  Minimality of \\(T\\) proves \\cref{eq:minimal-period-annihilator}.\n\\end{proof}","Let\n\\[\nF_a(z)=\\frac{P_a(z)}{Q_a(z)}\n\\]\nbe reduced with\n\\[\nP_a(0)=Q_a(0)=1.\n\\]\nIf \\(d=\\deg m_a\\), then\n\\[\nQ_a(z)=z^dm_a(z^{-1}),\n\\]\nwith the evident coefficient normalization.  Taking reciprocal polynomials in \\cref{eq:minimal-period-annihilator} gives\n\\begin{equation}\n\\label{eq:minimal-period-denominator}\nQ_a(z)\\mid 1-z^T,\n\\qquad\nT=\\min\\{N\\geq 1:Q_a(z)\\mid 1-z^N\\}.\n\\end{equation}\nThus pure periodicity is visible both in the linear annihilator and in the reduced rational denominator.","An object of \\(\\Rec_p^{\\times,\\mathrm{per}}\\) is the normalized minimal recurrence polynomial\n\\[\nm(x)=x^d+c_1x^{d-1}+\\cdots+c_d,\n\\qquad c_d\\neq 0,\n\\]\ntogether with the initial vector \\((a_0,\\ldots,a_{d-1})\\), subject to \\(a_0=1\\) and the requirement that the generated sequence is purely periodic.  Minimality means that no monic recurrence of lower degree annihilates the sequence.  Morphisms are recurrence-presentation isomorphisms preserving the coefficient sequence.  Because the minimal annihilator and initial data are unique, every endomorphism is the identity.","An object of \\(\\Rat_p^{\\times,\\mathrm{per}}\\) is a reduced fraction\n\\[\nF(z)=\\frac{P(z)}{Q(z)}\\in\\mathbb F_p(z),\n\\qquad\nP(0)=Q(0)=1,\n\\]\nwhose coefficient sequence is purely periodic.  Equivalently, \\(Q\\mid 1-z^T\\) for some \\(T\\) and the reduced coefficient sequence has no transient prefix.  Morphisms preserve the rational function.  Reducedness and the constant-term normalization make this chart rigid.","The recurrence-rational bridge is the strict inverse pair already used in the C-finite atlas: multiply the generating series by the reciprocal recurrence denominator, or recover the recurrence by clearing the reduced denominator.  On the present finite-field subcategory, the additional condition \\(Q\\mid1-z^T\\) records pure periodicity.","For \\(0\\leq d<p\\), define the Cartier section of a sequence by\n\\[\n(\\Lambda_da)_n=a_{pn+d}.\n\\]\nThe \\(p\\)-kernel is\n\\begin{equation}\n\\label{eq:p-kernel}\n\\mathcal K_p(a)\n=\n\\left\\{\n\\Lambda_{d_{\\ell-1}}\\cdots\\Lambda_{d_0}a:\n\\ell\\geq 0,\\ 0\\leq d_i<p\n\\right\\}.\n\\end{equation}\nWe use least-significant-digit-first words.  Every finite digit word is allowed.  Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state \\(b\\) is \\(b_0\\) and \\((\\Lambda_0b)_0=b_0\\).","d}.\n\\]\nThe \\(p\\)-kernel is\n\\begin{equation}\n\\label{eq:p-kernel}\n\\mathcal K_p(a)\n=\n\\left\\{\n\\Lambda_{d_{\\ell-1}}\\cdots\\Lambda_{d_0}a:\n\\ell\\geq 0,\\ 0\\leq d_i<p\n\\right\\}.\n\\end{equation}\nWe use least-significant-digit-first words.  Every finite digit word is allowed.  Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state \\(b\\) is \\(b_0\\) and \\((\\Lambda_0b)_0=b_0\\).","e \\(p\\)-kernel is\n\\begin{equation}\n\\label{eq:p-kernel}\n\\mathcal K_p(a)\n=\n\\left\\{\n\\Lambda_{d_{\\ell-1}}\\cdots\\Lambda_{d_0}a:\n\\ell\\geq 0,\\ 0\\leq d_i<p\n\\right\\}.\n\\end{equation}\nWe use least-significant-digit-first words.  Every finite digit word is allowed.  Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state \\(b\\) is \\(b_0\\) and \\((\\Lambda_0b)_0=b_0\\).","el is\n\\begin{equation}\n\\label{eq:p-kernel}\n\\mathcal K_p(a)\n=\n\\left\\{\n\\Lambda_{d_{\\ell-1}}\\cdots\\Lambda_{d_0}a:\n\\ell\\geq 0,\\ 0\\leq d_i<p\n\\right\\}.\n\\end{equation}\nWe use least-significant-digit-first words.  Every finite digit word is allowed.  Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state \\(b\\) is \\(b_0\\) and \\((\\Lambda_0b)_0=b_0\\).","The canonical Moore automaton has state set \\(\\mathcal K_p(a)\\), start state \\(a\\), transition\n\\[\n\\delta_d(b)=\\Lambda_db,\n\\]\nand output\n\\[\no(b)=b_0.\n\\]\nAfter reading \\(d_0,\\ldots,d_{\\ell-1}\\), its state is\n\\[\n\\Lambda_{d_{\\ell-1}}\\cdots\\Lambda_{d_0}a\n=\n\\bigl(n\\longmapsto a_{p^\\ell n+r}\\bigr),\n\\qquad\nr=\\sum_{j=0}^{\\ell-1}d_jp^j.\n\\]","\\begin{proposition}[Finite periodic kernel and exact state model]\n\\label{prop:periodic-kernel}\nLet \\(a\\in\\Per_p^{\\times,\\mathrm{per}}\\) have minimal period \\(T\\) and period word \\(w\\).  Every kernel state has the form\n\\begin{equation}\n\\label{eq:periodic-kernel-state}\nb_{u,r}(n)=w_{un+r\\bmod T},\n\\qquad\nu\\in U_p(T),\\quad r\\in\\mathbb Z/T\\mathbb Z,\n\\end{equation}\nwhere\n\\[\nU_p(T)=\\{p^e\\bmod T:e\\geq 0\\}.\n\\]\nThe digit transition on pair representatives is\n\\begin{equation}\n\\label{eq:pair-transition}\n(u,r)\\xrightarrow{d}(pu,r+ud)\\pmod T,\n\\end{equation}\nand the output is \\(w_r\\).  Quotienting pair representatives by equality of the sequences in \\cref{eq:periodic-kernel-state} gives the canonical minimal least-significant-digit-first \\(p\\)-DFAO for \\(a\\).\n\\end{proposition}","bel{prop:periodic-kernel}\nLet \\(a\\in\\Per_p^{\\times,\\mathrm{per}}\\) have minimal period \\(T\\) and period word \\(w\\).  Every kernel state has the form\n\\begin{equation}\n\\label{eq:periodic-kernel-state}\nb_{u,r}(n)=w_{un+r\\bmod T},\n\\qquad\nu\\in U_p(T),\\quad r\\in\\mathbb Z/T\\mathbb Z,\n\\end{equation}\nwhere\n\\[\nU_p(T)=\\{p^e\\bmod T:e\\geq 0\\}.\n\\]\nThe digit transition on pair representatives is\n\\begin{equation}\n\\label{eq:pair-transition}\n(u,r)\\xrightarrow{d}(pu,r+ud)\\pmod T,\n\\end{equation}\nand the output is \\(w_r\\).  Quotienting pair representatives by equality of the sequences in \\cref{eq:periodic-kernel-state} gives the canonical minimal least-significant-digit-first \\(p\\)-DFAO for \\(a\\).\n\\end{proposition}","od word \\(w\\).  Every kernel state has the form\n\\begin{equation}\n\\label{eq:periodic-kernel-state}\nb_{u,r}(n)=w_{un+r\\bmod T},\n\\qquad\nu\\in U_p(T),\\quad r\\in\\mathbb Z/T\\mathbb Z,\n\\end{equation}\nwhere\n\\[\nU_p(T)=\\{p^e\\bmod T:e\\geq 0\\}.\n\\]\nThe digit transition on pair representatives is\n\\begin{equation}\n\\label{eq:pair-transition}\n(u,r)\\xrightarrow{d}(pu,r+ud)\\pmod T,\n\\end{equation}\nand the output is \\(w_r\\).  Quotienting pair representatives by equality of the sequences in \\cref{eq:periodic-kernel-state} gives the canonical minimal least-significant-digit-first \\(p\\)-DFAO for \\(a\\).\n\\end{proposition}","T\\mathbb Z,\n\\end{equation}\nwhere\n\\[\nU_p(T)=\\{p^e\\bmod T:e\\geq 0\\}.\n\\]\nThe digit transition on pair representatives is\n\\begin{equation}\n\\label{eq:pair-transition}\n(u,r)\\xrightarrow{d}(pu,r+ud)\\pmod T,\n\\end{equation}\nand the output is \\(w_r\\).  Quotienting pair representatives by equality of the sequences in \\cref{eq:periodic-kernel-state} gives the canonical minimal least-significant-digit-first \\(p\\)-DFAO for \\(a\\).\n\\end{proposition}","d{equation}\nwhere\n\\[\nU_p(T)=\\{p^e\\bmod T:e\\geq 0\\}.\n\\]\nThe digit transition on pair representatives is\n\\begin{equation}\n\\label{eq:pair-transition}\n(u,r)\\xrightarrow{d}(pu,r+ud)\\pmod T,\n\\end{equation}\nand the output is \\(w_r\\).  Quotienting pair representatives by equality of the sequences in \\cref{eq:periodic-kernel-state} gives the canonical minimal least-significant-digit-first \\(p\\)-DFAO for \\(a\\).\n\\end{proposition}","\\begin{proof}\nAfter \\(\\ell\\) digits, the resulting subsequence is \\(n\\mapsto a_{p^\\ell n+r}\\), which depends only on \\((p^\\ell\\bmod T,r\\bmod T)\\).  Applying \\(\\Lambda_d\\) gives\n\\[\n\\Lambda_db_{u,r}(n)\n=b_{u,r}(pn+d)\n=w_{(pu)n+(r+ud)},\n\\]\nproving \\cref{eq:pair-transition}.  The pair set is finite, so the kernel is finite.  Every kernel state is reachable by construction.  If two kernel sequences differ, they differ at some \\(n\\); reading the base-\\(p\\) digits of that \\(n\\) distinguishes the states by their outputs.  Hence the kernel automaton is minimal.  Any other reachable minimal automaton producing the same sequence is uniquely isomorphic to it by the usual behavioral quotient.\n\\end{proof}",") digits, the resulting subsequence is \\(n\\mapsto a_{p^\\ell n+r}\\), which depends only on \\((p^\\ell\\bmod T,r\\bmod T)\\).  Applying \\(\\Lambda_d\\) gives\n\\[\n\\Lambda_db_{u,r}(n)\n=b_{u,r}(pn+d)\n=w_{(pu)n+(r+ud)},\n\\]\nproving \\cref{eq:pair-transition}.  The pair set is finite, so the kernel is finite.  Every kernel state is reachable by construction.  If two kernel sequences differ, they differ at some \\(n\\); reading the base-\\(p\\) digits of that \\(n\\) distinguishes the states by their outputs.  Hence the kernel automaton is minimal.  Any other reachable minimal automaton producing the same sequence is uniquely isomorphic to it by the usual behavioral quotient.\n\\end{proof}","subsequence is \\(n\\mapsto a_{p^\\ell n+r}\\), which depends only on \\((p^\\ell\\bmod T,r\\bmod T)\\).  Applying \\(\\Lambda_d\\) gives\n\\[\n\\Lambda_db_{u,r}(n)\n=b_{u,r}(pn+d)\n=w_{(pu)n+(r+ud)},\n\\]\nproving \\cref{eq:pair-transition}.  The pair set is finite, so the kernel is finite.  Every kernel state is reachable by construction.  If two kernel sequences differ, they differ at some \\(n\\); reading the base-\\(p\\) digits of that \\(n\\) distinguishes the states by their outputs.  Hence the kernel automaton is minimal.  Any other reachable minimal automaton producing the same sequence is uniquely isomorphic to it by the usual behavioral quotient.\n\\end{proof}","psto a_{p^\\ell n+r}\\), which depends only on \\((p^\\ell\\bmod T,r\\bmod T)\\).  Applying \\(\\Lambda_d\\) gives\n\\[\n\\Lambda_db_{u,r}(n)\n=b_{u,r}(pn+d)\n=w_{(pu)n+(r+ud)},\n\\]\nproving \\cref{eq:pair-transition}.  The pair set is finite, so the kernel is finite.  Every kernel state is reachable by construction.  If two kernel sequences differ, they differ at some \\(n\\); reading the base-\\(p\\) digits of that \\(n\\) distinguishes the states by their outputs.  Hence the kernel automaton is minimal.  Any other reachable minimal automaton producing the same sequence is uniquely isomorphic to it by the usual behavioral quotient.\n\\end{proof}","The pair model gives a useful finite bound.  Write\n\\[\nT=p^\\nu M,\n\\qquad\n\\gcd(M,p)=1,\n\\]\nand set \\(\\operatorname{ord}_1(p)=1\\).","\\begin{lemma}[Exact multiplier-orbit size]\n\\label{lem:multiplier-orbit-size}\nThe set \\(U_p(T)\\) has cardinality\n\\begin{equation}\n\\label{eq:u-size}\n|U_p(T)|\n=\n\\nu+\\operatorname{ord}_M(p).\n\\end{equation}\nConsequently,\n\\begin{equation}\n\\label{eq:kernel-state-bound}\n|\\mathcal K_p(a)|\n\\leq\nT\\bigl(\\nu+\\operatorname{ord}_M(p)\\bigr).\n\\end{equation}\n\\end{lemma}","\\begin{proof}\nFor \\(0\\leq e<\\nu\\), the residues \\(p^e\\bmod p^\\nu\\) have distinct \\(p\\)-adic valuations.  For \\(e\\geq\\nu\\), the residue is zero modulo \\(p^\\nu\\), while modulo \\(M\\) it runs through one multiplicative cycle of length \\(\\operatorname{ord}_M(p)\\).  The early and eventual residues cannot coincide.  This proves \\cref{eq:u-size}.  The kernel is a quotient of the pair set \\(U_p(T)\\times\\mathbb Z/T\\mathbb Z\\), giving \\cref{eq:kernel-state-bound}.\n\\end{proof}","Let \\(\\Aut_p^{\\times,\\mathrm{per}}\\) be the groupoid of reachable minimal least-significant-digit-first \\(p\\)-DFAOs whose output sequence lies in \\(\\Per_p^{\\times,\\mathrm{per}}\\).  Morphisms are automaton isomorphisms preserving the start state, transitions, and outputs.  Reachability forces every automorphism to be the identity.","For a unit formal series \\(F(0)=1\\), define its local functional-Mahler mask by\n\\begin{equation}\n\\label{eq:unique-local-mask}\n\\mathcal M_p(F)(z)\n=\n\\frac{F(z)}{F(z^p)}.\n\\end{equation}\nThen\n\\begin{equation}\n\\label{eq:functional-mahler-equation}\nF(z)=\\mathcal M_p(F)(z)F(z^p).\n\\end{equation}\nIterating \\cref{eq:functional-mahler-equation} gives the coefficientwise product\n\\begin{equation}\n\\label{eq:mask-product}\nF(z)\n=\n\\prod_{j\\geq0}\\mathcal M_p(F)(z^{p^j}).\n\\end{equation}\nOnly finitely many factors contribute to each fixed coefficient.","If \\(F=P/Q\\) is reduced and unit normalized, then\n\\begin{equation}\n\\label{eq:rational-mask-formula}\nR_F(z)\n=\n\\frac{P(z)Q(z^p)}{Q(z)P(z^p)}.\n\\end{equation}\nThus every object of \\(\\Rat_p^{\\times,\\mathrm{per}}\\) has a rational mask.  Reduce \\cref{eq:rational-mask-formula} to\n\\[\nR_F(z)=\\frac{A(z)}{B(z)},\n\\qquad\nA(0)=B(0)=1,\n\\qquad\n\\gcd(A,B)=1.\n\\]","Define \\(\\Mask_p^{\\times,\\mathrm{per}}\\) to be the groupoid of such reduced rational masks whose normalized product solution \\cref{eq:mask-product} has purely periodic coefficients.  Morphisms preserve the reduced mask.","\\begin{theorem}[Unit rational functional-Mahler equivalence]\n\\label{thm:unit-rational-mask-equivalence}\nThe assignments\n\\[\nF\\longmapsto \\frac{F(z)}{F(z^p)}\n\\qquad\\text{and}\\qquad\nR\\longmapsto \\prod_{j\\geq0}R(z^{p^j})\n\\]\nare inverse equivalences\n\\[\n\\Rat_p^{\\times,\\mathrm{per}}\n\\simeq\n\\Mask_p^{\\times,\\mathrm{per}}.\n\\]\nThe mask is unique for the represented series.\n\\end{theorem}","\\begin{proof}\nStarting with \\(F\\), the finite product through level \\(N\\) telescopes:\n\\[\n\\prod_{j=0}^{N}\\frac{F(z^{p^j})}{F(z^{p^{j+1}})}\n=\n\\frac{F(z)}{F(z^{p^{N+1}})}.\n\\]\nAs \\(N\\to\\infty\\), the denominator converges coefficientwise to \\(F(0)=1\\), proving that the product recovers \\(F\\).  Conversely, the product solution satisfies \\(F(z)=R(z)F(z^p)\\), so \\(R=F/F(z^p)\\).  Reduced normalization makes the rational mask unique.\n\\end{proof}","\\begin{boundary}[Functional Mahler is not Newton-Mahler]\nThe mask equation in this section is\n\\[\nF(z)=R(z)F(z^p).\n\\]\nIt is a digit-section and automaton construction.  It is not the Newton-Mahler interpolation expansion\n\\[\nf(x)=\\sum_rc_r\\binom{x}{r}.\n\\]\nBoth occur elsewhere in the Library, but this atlas uses only the functional-Mahler meaning.\n\\end{boundary}","The reversible bridge core is generated by three explicit constructions:\n\\[\n\\Rec_p^{\\times,\\mathrm{per}}\n\\leftrightarrow\n\\Rat_p^{\\times,\\mathrm{per}},\n\\qquad\n\\Rat_p^{\\times,\\mathrm{per}}\n\\leftrightarrow\n\\Aut_p^{\\times,\\mathrm{per}},\n\\qquad\n\\Rat_p^{\\times,\\mathrm{per}}\n\\leftrightarrow\n\\Mask_p^{\\times,\\mathrm{per}}.\n\\]\nThe remaining reversible bridges are their normalized composites.  A separate constructor and reachable-restriction lane connects masks to the digital normal form through the Cartier compiler.","\\endfoot\n\\bottomrule\n\\endlastfoot\nRecurrence \\(\\leftrightarrow\\) reduced OGF & equivalence & minimal invertible recurrence and reduced purely periodic rational series & coefficients, minimal annihilator, initial data, period, Hankel rank & equality and isomorphism & none after reduced normalization","[4pt]\nReduced OGF \\(\\leftrightarrow\\) minimal DFAO & equivalence on the periodic essential image & unit purely periodic series and reachable minimal LSD-first \\(p\\)-DFAOs & coefficient semantics, minimal period, canonical \\(p\\)-kernel, output behavior & equality of represented sequences & a nonminimal residue-pair automaton is quotiented before entering the normalized chart\\\\[4pt]\nReduced OGF \\(\\leftrightarrow\\) rational mask & equivalence on the unit-periodic essential image & \\(F(0)=1\\), periodic rational \\(F\\), reduced regular mask \\(R\\) & exact series, functional equation, reduced rational mask, minimal period & equality of series and masks & undefined for nonunit series; unrestricted rational masks may have nonperiodic products\\\\[4pt]\nMask \\(\\to\\) Cartier numerator representation & constructor & reduced \\(A/B\\) over \\(\\mathbb F_p\\) & exact coefficient semantics, digit matrices, theorem degree bound & not generally reflective & introduces basis and denominator coordinates and may contain unreachable ambient states\\\\[4pt]\nAmbient Cartier compiler \\(\\to\\) minimal DFAO & reachable r…","Let\n\\[\nR(z)=\\frac{A(z)}{B(z)},\n\\qquad\nA(0)=B(0)=1,\n\\]\nwith\n\\[\na=\\deg A,\n\\qquad\nb=\\deg B,\n\\qquad\nc=a+(p-2)b,\n\\]\nand define\n\\begin{equation}\n\\label{eq:cartier-degree-bound-block03}\nD=\\max\\left\\{b,\\left\\lceil\\frac{c}{p-1}\\right\\rceil\\right\\}.\n\\end{equation}\nThe rational-mask Cartier theorem gives an invariant numerator space of degree at most \\(D\\) under\n\\[\nP\\longmapsto\\Lambda_d(PAB^{p-2}),\n\\qquad 0\\leq d<p.\n\\]\nThe resulting linear state vector belongs to \\(\\mathbb F_p^{D+1}\\), so the raw output automaton has at most\n\\begin{equation}\n\\label{eq:compiler-state-bound-block03}\np^{D+1}\n\\end{equation}\nstates \\cite{FMH}.  It computes the coefficient sequence exactly.","\\[\na=\\deg A,\n\\qquad\nb=\\deg B,\n\\qquad\nc=a+(p-2)b,\n\\]\nand define\n\\begin{equation}\n\\label{eq:cartier-degree-bound-block03}\nD=\\max\\left\\{b,\\left\\lceil\\frac{c}{p-1}\\right\\rceil\\right\\}.\n\\end{equation}\nThe rational-mask Cartier theorem gives an invariant numerator space of degree at most \\(D\\) under\n\\[\nP\\longmapsto\\Lambda_d(PAB^{p-2}),\n\\qquad 0\\leq d<p.\n\\]\nThe resulting linear state vector belongs to \\(\\mathbb F_p^{D+1}\\), so the raw output automaton has at most\n\\begin{equation}\n\\label{eq:compiler-state-bound-block03}\np^{D+1}\n\\end{equation}\nstates \\cite{FMH}.  It computes the coefficient sequence exactly.","n}\n\\label{eq:cartier-degree-bound-block03}\nD=\\max\\left\\{b,\\left\\lceil\\frac{c}{p-1}\\right\\rceil\\right\\}.\n\\end{equation}\nThe rational-mask Cartier theorem gives an invariant numerator space of degree at most \\(D\\) under\n\\[\nP\\longmapsto\\Lambda_d(PAB^{p-2}),\n\\qquad 0\\leq d<p.\n\\]\nThe resulting linear state vector belongs to \\(\\mathbb F_p^{D+1}\\), so the raw output automaton has at most\n\\begin{equation}\n\\label{eq:compiler-state-bound-block03}\np^{D+1}\n\\end{equation}\nstates \\cite{FMH}.  It computes the coefficient sequence exactly.","\\begin{proposition}[Compiler reachability lands in the canonical digital chart]\n\\label{prop:compiler-minimal-core}\nFor a mask in \\(\\Mask_p^{\\times,\\mathrm{per}}\\), the reachable part of the fixed-denominator Cartier compiler is uniquely isomorphic to the canonical \\(p\\)-kernel automaton of \\cref{prop:periodic-kernel}.  In particular, the subsequent quotient by equality of future output behavior is trivial.\n\\end{proposition}","\\begin{proof}\nThe fixed-denominator intertwining map sends the numerator reached by a digit word \\(w\\) to the Cartier section \\(\\Lambda_wF\\).  It is injective, as proved in the more general \\cref{thm:fixed-denominator-cartier-realization,thm:reachable-numerator-core}.  Hence distinct reachable numerator states represent distinct section series and are distinguished by some future coefficient.  The reachable compiler state set is therefore exactly the canonical \\(p\\)-kernel state set, with matching transitions and outputs.\n\\end{proof}","rator reached by a digit word \\(w\\) to the Cartier section \\(\\Lambda_wF\\).  It is injective, as proved in the more general \\cref{thm:fixed-denominator-cartier-realization,thm:reachable-numerator-core}.  Hence distinct reachable numerator states represent distinct section series and are distinguished by some future coefficient.  The reachable compiler state set is therefore exactly the canonical \\(p\\)-kernel state set, with matching transitions and outputs.\n\\end{proof}","There are four nonempty triple overlaps:\n\\[\n(\\Rec,\\Rat,\\Aut),\n\\qquad\n(\\Rec,\\Rat,\\Mask),\n\\qquad\n(\\Rec,\\Aut,\\Mask),\n\\qquad\n(\\Rat,\\Aut,\\Mask).\n\\]\nAll four charts meet on the same unit periodic semantic objects.","\\begin{lemma}[Rigidity of normalized fibers]\n\\label{lem:periodic-fiber-rigidity}\nFix \\(a\\in\\Per_p^{\\times,\\mathrm{per}}\\).  In each of the four normalized charts, the groupoid of presentations of \\(a\\) is contractible.\n\\end{lemma}","\\begin{theorem}[Coherence of the periodic heterogeneous atlas]\n\\label{thm:periodic-atlas-coherence}\nChoose every pairwise bridge by the explicit constructions above and choose every comparison \\(2\\)-cell to be the unique normalized isomorphism over the common sequence.  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item every one of the four triple-overlap comparison diagrams commutes;\n\\item the two composites of comparison cells on the four-chart overlap agree;\n\\item the quadruple-overlap coherence defect \\(\\Omega\\) is the identity.\n\\end{enumerate}\n\\end{theorem}","se bridge by the explicit constructions above and choose every comparison \\(2\\)-cell to be the unique normalized isomorphism over the common sequence.  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item every one of the four triple-overlap comparison diagrams commutes;\n\\item the two composites of comparison cells on the four-chart overlap agree;\n\\item the quadruple-overlap coherence defect \\(\\Omega\\) is the identity.\n\\end{enumerate}\n\\end{theorem}","Let \\(\\mathfrak R_p^{\\mathrm{per}}\\) denote the four-chart pseudofunctor defined by the preceding passports and comparison cells.","\\begin{theorem}[First heterogeneous effective-descent theorem]\n\\label{thm:first-heterogeneous-descent}\nThe semantic pseudonatural transformation induces an equivalence\n\\begin{equation}\n\\label{eq:first-heterogeneous-descent}\n\\Desc(\\mathfrak R_p^{\\mathrm{per}})\n\\simeq\n\\Per_p^{\\times,\\mathrm{per}}.\n\\end{equation}\nIn particular, descent is effective for both objects and morphisms.\n\\end{theorem}","\\begin{proof}\nGiven \\(a\\in\\Per_p^{\\times,\\mathrm{per}}\\), form its minimal recurrence, reduced rational OGF, canonical minimal \\(p\\)-kernel automaton, and reduced rational mask.  The explicit bridges supply a descent object.  This defines a functor\n\\[\nN:\\Per_p^{\\times,\\mathrm{per}}\n\\longrightarrow\n\\Desc(\\mathfrak R_p^{\\mathrm{per}}).\n\\]\nConversely, any descent object has local presentations whose semantic images agree on every overlap.  Since the semantic target is discrete, they determine one sequence \\(a\\).  Normalization and \\cref{lem:periodic-fiber-rigidity} identify the local presentations with those constructed by \\(N(a)\\).  Thus \\(N\\) is essentially surjective and fully faithful.","anonical minimal \\(p\\)-kernel automaton, and reduced rational mask.  The explicit bridges supply a descent object.  This defines a functor\n\\[\nN:\\Per_p^{\\times,\\mathrm{per}}\n\\longrightarrow\n\\Desc(\\mathfrak R_p^{\\mathrm{per}}).\n\\]\nConversely, any descent object has local presentations whose semantic images agree on every overlap.  Since the semantic target is discrete, they determine one sequence \\(a\\).  Normalization and \\cref{lem:periodic-fiber-rigidity} identify the local presentations with those constructed by \\(N(a)\\).  Thus \\(N\\) is essentially surjective and fully faithful.","defines a functor\n\\[\nN:\\Per_p^{\\times,\\mathrm{per}}\n\\longrightarrow\n\\Desc(\\mathfrak R_p^{\\mathrm{per}}).\n\\]\nConversely, any descent object has local presentations whose semantic images agree on every overlap.  Since the semantic target is discrete, they determine one sequence \\(a\\).  Normalization and \\cref{lem:periodic-fiber-rigidity} identify the local presentations with those constructed by \\(N(a)\\).  Thus \\(N\\) is essentially surjective and fully faithful.","or\n\\[\nN:\\Per_p^{\\times,\\mathrm{per}}\n\\longrightarrow\n\\Desc(\\mathfrak R_p^{\\mathrm{per}}).\n\\]\nConversely, any descent object has local presentations whose semantic images agree on every overlap.  Since the semantic target is discrete, they determine one sequence \\(a\\).  Normalization and \\cref{lem:periodic-fiber-rigidity} identify the local presentations with those constructed by \\(N(a)\\).  Thus \\(N\\) is essentially surjective and fully faithful.","\\begin{corollary}[A verified instance of the assembly theorem]\nThe atlas \\(\\mathfrak R_p^{\\mathrm{per}}\\) satisfies realization, semantic adequacy, typed overlaps, higher coherence, effective descent, and preservation for the finite algebraic constructions used in this section.  Therefore \\cref{thm:atlas-assembly} applies without a remaining categorical hypothesis on this restricted domain.\n\\end{corollary}","For \\(a\\in\\Per_p^{\\times,\\mathrm{per}}\\), define the periodic resolved invariant package\n\\begin{equation}\n\\label{eq:periodic-invariant-package}\n\\mathcal J_p(a)\n=\n\\bigl(\nT(a),\n w(a),\n m_a(x),\n P_a(z)/Q_a(z),\n \\mathcal K_p(a),\n \\mathcal A_p(a),\n R_a(z)\n\\bigr),\n\\end{equation}\nwhere \\(\\mathcal A_p(a)\\) is the canonical minimal digital automaton.  Its comparison identities include\n\\begin{equation}\n\\label{eq:periodic-invariant-identities}\n\\begin{aligned}\n\\deg m_a\n&=\\deg Q_a\n=\\rank H(a)\n=\\dim \\State_{\\min}(a),","For \\(a\\in\\Per_p^{\\times,\\mathrm{per}}\\), define the periodic resolved invariant package\n\\begin{equation}\n\\label{eq:periodic-invariant-package}\n\\mathcal J_p(a)\n=\n\\bigl(\nT(a),\n w(a),\n m_a(x),\n P_a(z)/Q_a(z),\n \\mathcal K_p(a),\n \\mathcal A_p(a),\n R_a(z)\n\\bigr),\n\\end{equation}\nwhere \\(\\mathcal A_p(a)\\) is the canonical minimal digital automaton.  Its comparison identities include\n\\begin{equation}\n\\label{eq:periodic-invariant-identities}\n\\begin{aligned}\n\\deg m_a\n&=\\deg Q_a\n=\\rank H(a)\n=\\dim \\State_{\\min}(a),\\\\\nT(a)\n&=\\min\\{N:m_a\\mid x^N-1\\}\n=\\min\\{N:Q_a\\mid 1-z^N\\},\\\\\n|\\mathcal A_p(a)|\n&=|\\mathcal K_p(a)|\n\\leq T(a)\\bigl(\\nu_p(T(a))+\\operatorname{ord}_{M}(p)\\bigr),\\\\\nR_a(z)\n&=\\frac{F_a(z)}{F_a(z^p)},\n\\qquad T(a)=p^{\\nu_p(T(a))}M,\n\\quad \\gcd(M,p)=1.\n\\end{aligned}\n\\end{equation}","\\begin{theorem}[Invariant descent and global completeness on the periodic atlas]\n\\label{thm:periodic-invariant-completeness}\nThe native invariants in the recurrence, rational, digital, and mask charts admit coherent comparison maps to \\(\\mathcal J_p\\).  These maps descend to\n\\[\n\\mathcal J_p:\n\\Desc(\\mathfrak R_p^{\\mathrm{per}})\n\\longrightarrow\n\\mathbf{Inv}_p^{\\mathrm{per}}.\n\\]\nThe descended invariant is complete on the declared semantic category.\n\\end{theorem}","\\begin{proof}\nEvery component of \\(\\mathcal J_p(a)\\) is reconstructed from the coefficient sequence by the explicit normalization procedures above.  Hence every pairwise bridge preserves the package and the comparison cells are identities after semantic identification.  Triple coherence follows from \\cref{thm:periodic-atlas-coherence}.  If \\(\\mathcal J_p(a)=\\mathcal J_p(b)\\), then the primitive period words agree, so \\(a=b\\).  Under \\cref{eq:first-heterogeneous-descent}, equality of semantic objects is exactly isomorphism in the descent category.\n\\end{proof}","ponent of \\(\\mathcal J_p(a)\\) is reconstructed from the coefficient sequence by the explicit normalization procedures above.  Hence every pairwise bridge preserves the package and the comparison cells are identities after semantic identification.  Triple coherence follows from \\cref{thm:periodic-atlas-coherence}.  If \\(\\mathcal J_p(a)=\\mathcal J_p(b)\\), then the primitive period words agree, so \\(a=b\\).  Under \\cref{eq:first-heterogeneous-descent}, equality of semantic objects is exactly isomorphism in the descent category.\n\\end{proof}","The completeness claim has a narrow classification target: unit purely periodic sequences over \\(\\mathbb F_p\\).  It is not a complete invariant for all C-finite, all automatic, or all rational functional-Mahler sequences.","\\paragraph{Even indicator over \\(\\mathbb F_2\\).}\nFor the word \\(10\\),\n\\[\nT=2,\n\\qquad\nm(x)=x^2+1=(x+1)^2,\n\\qquad\nF(z)=\\frac{1}{1+z^2},\n\\qquad\nR(z)=1+z^2.\n\\]\nThe repeated recurrence root is compatible with pure periodicity because the field has characteristic two.  The canonical \\(2\\)-kernel has three states: the original sequence, the constant-one sequence, and the constant-zero sequence.  The rational-mask compiler gives the safe ambient bound \\(2^3=8\\) before reachability restriction.","or the word \\(10\\),\n\\[\nT=2,\n\\qquad\nm(x)=x^2+1=(x+1)^2,\n\\qquad\nF(z)=\\frac{1}{1+z^2},\n\\qquad\nR(z)=1+z^2.\n\\]\nThe repeated recurrence root is compatible with pure periodicity because the field has characteristic two.  The canonical \\(2\\)-kernel has three states: the original sequence, the constant-one sequence, and the constant-zero sequence.  The rational-mask compiler gives the safe ambient bound \\(2^3=8\\) before reachability restriction.","\\paragraph{Alternating unit over \\(\\mathbb F_3\\).}\nFor the word \\(12\\),\n\\[\nT=2,\n\\qquad\nm(x)=x+1,\n\\qquad\nF(z)=\\frac{1}{1+z},\n\\qquad\nR(z)=\\frac{1+z^3}{1+z}=1+2z+z^2.\n\\]\nThe recurrence order is one although the minimal period is two.  The \\(3\\)-kernel consists of the sequence and its negative, giving two minimal digital states.  The compiler bound is nine.","\\paragraph{Period-three unit word over \\(\\mathbb F_2\\).}\nFor the word \\(110\\),\n\\[\nT=3,\n\\qquad\nm(x)=x^2+x+1,\n\\qquad\nF(z)=\\frac{1}{1+z+z^2},\n\\qquad\nR(z)=1+z+z^2.\n\\]\nThe reachable pair model has six representatives, while equality of subsequence behavior collapses them to three kernel states.  This is a concrete instance of the directional quotient in \\cref{prop:compiler-minimal-core}.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.99\\textwidth]{periodic_example_automata.pdf}\n\\caption{Three exact examples.  A state label \\(s\\mid o\\) gives the state identifier and output.  The compiler bounds are ambient theorem bounds before reachability restriction, not claims of minimal state count.}\n\\label{fig:periodic-example-automata}\n\\end{figure}","\\paragraph{Negative control: a nonunit periodic sequence.}\nOver \\(\\mathbb F_2\\), the word \\(01\\) gives\n\\[\nF(z)=\\frac{z}{1+z^2}.\n\\]\nThen\n\\[\n\\frac{F(z)}{F(z^2)}\n=\n\\frac{1+z^2}{z},\n\\]\nwhich is Laurent rather than regular at the origin.  The sequence remains purely periodic, C-finite, and automatic, but it is not an object of \\(\\Mask_2^{\\times,\\mathrm{per}}\\).  This verifies that the unit overlap condition is structural rather than cosmetic.","The supplied verifier enumerates every primitive unit-periodic word over \\(\\mathbb F_2\\) through period eight and over \\(\\mathbb F_3\\) through period six.  For each object it checks, using exact prime-field and polynomial arithmetic:\n\\begin{enumerate}\n\\item Berlekamp-Massey recovery of the minimal recurrence;\n\\item divisibility by \\(x^T-1\\) and \\(1-z^T\\);\n\\item recovery of the least period from both normalized linear charts;\n\\item the reduced rational functional-Mahler identity;\n\\item the exact reachable pair model and its behavioral quotient;\n\\item the multiplier-orbit formula \\cref{eq:u-size};\n\\item coefficient output of the minimal digit automaton;\n\\item the Cartier compiler state bound \\cref{eq:compiler-state-bound-block03}.\n\\end{enumerate}\nThe finite computation supports","The supplied verifier enumerates every primitive unit-periodic word over \\(\\mathbb F_2\\) through period eight and over \\(\\mathbb F_3\\) through period six.  For each object it checks, using exact prime-field and polynomial arithmetic:\n\\begin{enumerate}\n\\item Berlekamp-Massey recovery of the minimal recurrence;\n\\item divisibility by \\(x^T-1\\) and \\(1-z^T\\);\n\\item recovery of the least period from both normalized linear charts;\n\\item the reduced rational functional-Mahler identity;\n\\item the exact reachable pair model and its behavioral quotient;\n\\item the multiplier-orbit formula \\cref{eq:u-size};\n\\item coefficient output of the minimal digit automaton;\n\\item the Cartier compiler state bound \\cref{eq:compiler-state-bound-block03}.\n\\end{enumerate}\nThe finite computation supports the implementation and examples.  The symboli","The supplied verifier enumerates every primitive unit-periodic word over \\(\\mathbb F_2\\) through period eight and over \\(\\mathbb F_3\\) through period six.  For each object it checks, using exact prime-field and polynomial arithmetic:\n\\begin{enumerate}\n\\item Berlekamp-Massey recovery of the minimal recurrence;\n\\item divisibility by \\(x^T-1\\) and \\(1-z^T\\);\n\\item recovery of the least period from both normalized linear charts;\n\\item the reduced rational functional-Mahler identity;\n\\item the exact reachable pair model and its behavioral quotient;\n\\item the multiplier-orbit formula \\cref{eq:u-size};\n\\item coefficient output of the minimal digit automaton;\n\\item the Cartier compiler state bound \\cref{eq:compiler-state-bound-block03}.\n\\end{enumerate}\nThe finite computation supports the implementation and examples.  The symbolic theorems above do not depend on the enumeration.","Fix a prime \\(p\\).  Let\n\\[\n\\RMah_p^{\\times}\n=\n\\left\\{\nF(z)\\in 1+z\\mathbb F_p[[z]]:\n\\frac{F(z)}{F(z^p)}\\in\\mathbb F_p(z)\n\\right\\}\n\\]\nbe the discrete category of unit formal series with rational canonical functional-Mahler mask.  This category contains the periodic objects of \\cref{sec:periodic-heterogeneous-atlas}, but for odd \\(p\\) it also contains genuinely nonrational and non-C-finite automatic series.","For a unit formal series, define its canonical local mask by\n\\begin{equation}\n\\label{eq:block04-canonical-mask}\nR_F(z)\n=\n\\frac{F(z)}{F(z^p)}.\n\\end{equation}\nThe coefficientwise infinite product\n\\begin{equation}\n\\label{eq:block04-mask-product}\n\\mathcal M_p[R](z)\n=\n\\prod_{j\\geq 0}R\\!\\left(z^{p^j}\\right)\n\\end{equation}\nis well defined for every \\(R\\in1+z\\mathbb F_p[[z]]\\), because only finitely many factors contribute to a fixed coefficient.","\\begin{theorem}[Frobenius-radical characterization]\n\\label{thm:frobenius-radical-characterization}\nLet \\(F\\in1+z\\mathbb F_p[[z]]\\).  Then\n\\begin{equation}\n\\label{eq:block04-frobenius-mask}\nR_F(z)=F(z)^{1-p}.\n\\end{equation}\nConsequently, the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto","\\begin{theorem}[Frobenius-radical characterization]\n\\label{thm:frobenius-radical-characterization}\nLet \\(F\\in1+z\\mathbb F_p[[z]]\\).  Then\n\\begin{equation}\n\\label{eq:block04-frobenius-mask}\nR_F(z)=F(z)^{1-p}.\n\\end{equation}\nConsequently, the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RM","\\begin{theorem}[Frobenius-radical characterization]\n\\label{thm:frobenius-radical-characterization}\nLet \\(F\\in1+z\\mathbb F_p[[z]]\\).  Then\n\\begin{equation}\n\\label{eq:block04-frobenius-mask}\nR_F(z)=F(z)^{1-p}.\n\\end{equation}\nConsequently, the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","abel{thm:frobenius-radical-characterization}\nLet \\(F\\in1+z\\mathbb F_p[[z]]\\).  Then\n\\begin{equation}\n\\label{eq:block04-frobenius-mask}\nR_F(z)=F(z)^{1-p}.\n\\end{equation}\nConsequently, the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","racterization}\nLet \\(F\\in1+z\\mathbb F_p[[z]]\\).  Then\n\\begin{equation}\n\\label{eq:block04-frobenius-mask}\nR_F(z)=F(z)^{1-p}.\n\\end{equation}\nConsequently, the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","F_p[[z]]\\).  Then\n\\begin{equation}\n\\label{eq:block04-frobenius-mask}\nR_F(z)=F(z)^{1-p}.\n\\end{equation}\nConsequently, the following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","he following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","{enumerate}[label=\\textup{(\\roman*)}]\n\\item The canonical mask \\(R_F\\) is rational.\n\\item The power \\(F^{p-1}\\) belongs to \\(\\mathbb F_p(z)\\).\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}",".\n\\item There are coprime polynomials \\(A,B\\in\\mathbb F_p[z]\\) with\n\\[\nA(0)=B(0)=1\n\\]\nand\n\\begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","begin{equation}\n\\label{eq:block04-rational-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","nal-mask-equation}\nB(z)F(z)=A(z)F(z^p).\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}",").\n\\end{equation}\n\\end{enumerate}\nFor the reduced mask \\(R_F=A/B\\), one has\n\\begin{equation}\n\\label{eq:block04-radical-equation}\nF(z)^{p-1}=\\frac{B(z)}{A(z)}.\n\\end{equation}\nThe ratio and product constructions are strict inverses:\n\\[\nF\\longmapsto R_F,\n\\qquad\nR\\longmapsto\\mathcal M_p[R].\n\\]\nIn particular, every object of \\(\\RMah_p^{\\times}\\) is algebraic over \\(\\mathbb F_p(z)\\) of degree at most \\(p-1\\).\n\\end{theorem}","\\begin{proof}\nIn characteristic \\(p\\), Frobenius gives\n\\[\nF(z^p)=F(z)^p.\n\\]\nSubstitution into \\cref{eq:block04-canonical-mask} gives \\cref{eq:block04-frobenius-mask}.  Thus \\(R_F\\) is rational exactly when \\(F^{p-1}=R_F^{-1}\\) is rational.  Writing the reduced rational mask as \\(A/B\\) gives \\cref{eq:block04-rational-mask-equation} and \\cref{eq:block04-radical-equation}.","Conversely, a reduced rational mask \\(A/B\\) with unit constant terms defines a unique formal solution of \\cref{eq:block04-rational-mask-equation}: the product in \\cref{eq:block04-mask-product}.  Telescoping gives\n\\[\n\\frac{\\mathcal M_p[R](z)}{\\mathcal M_p[R](z^p)}=R(z),\n\\]\nwhile uniqueness follows by comparing coefficients recursively.  Hence the ratio and product constructions are strict inverses.  Finally, \\cref{eq:block04-radical-equation} is an algebraic equation of degree \\(p-1\\).\n\\end{proof}","ue formal solution of \\cref{eq:block04-rational-mask-equation}: the product in \\cref{eq:block04-mask-product}.  Telescoping gives\n\\[\n\\frac{\\mathcal M_p[R](z)}{\\mathcal M_p[R](z^p)}=R(z),\n\\]\nwhile uniqueness follows by comparing coefficients recursively.  Hence the ratio and product constructions are strict inverses.  Finally, \\cref{eq:block04-radical-equation} is an algebraic equation of degree \\(p-1\\).\n\\end{proof}","\\begin{boundary}[Functional Mahler versus Newton-Mahler]\nThe word ``Mahler'' continues to name two distinct constructions.  This section concerns the functional equation\n\\[\nF(z)=R(z)F(z^p).\n\\]\nIt does not identify rational masks with Newton-Mahler expansions\n\\[\nf(x)=\\sum_{r\\geq0}c_r\\binom xr\n\\]\nor with \\(p\\)-adic analytic interpolation.  Any bridge to that local analytic chart still requires its own passport and proof.\n\\end{boundary}","Let \\(R=A/B\\) be reduced and normalized.  For each monic irreducible polynomial \\(\\pi\\in\\mathbb F_p[z]\\), define\n\\[\n\\omega_{\\pi}(R)\n=\n\\bigl(v_{\\pi}(B)-v_{\\pi}(A)\\bigr)\\bmod(p-1).\n\\]\nInclude the valuation at infinity\n\\[\n\\omega_{\\infty}(R)\n=\n\\bigl(\\deg A-\\deg B\\bigr)\\bmod(p-1).\n\\]\nThe finite-support family\n\\begin{equation}\n\\label{eq:block04-collapse-obstruction}\n\\omega_{p-1}(R)\n=\n\\bigl(\\omega_v(R)\\bigr)_v\n\\end{equation}\nis the \\emph{rational-collapse obstruction}.","\\begin{theorem}[Exact rational-collapse criterion]\n\\label{thm:rational-collapse-criterion}\nLet \\(F=\\mathcal M_p[A/B]\\).  The following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(F\\in\\mathbb F_p(z)\\).\n\\item \\(B/A\\) is a \\((p-1)\\)-st power in \\(\\mathbb F_p(z)\\).\n\\item \\(\\omega_{p-1}(A/B)=0\\).\n\\end{enumerate}\nWhen these conditions hold, there is a unique rational function \\(G\\in\\mathbb F_p(z)\\) with \\(G(0)=1\\) and\n\\[\nG^{p-1}=B/A,\n\\]\nand \\(F=G\\).","nal-collapse criterion]\n\\label{thm:rational-collapse-criterion}\nLet \\(F=\\mathcal M_p[A/B]\\).  The following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(F\\in\\mathbb F_p(z)\\).\n\\item \\(B/A\\) is a \\((p-1)\\)-st power in \\(\\mathbb F_p(z)\\).\n\\item \\(\\omega_{p-1}(A/B)=0\\).\n\\end{enumerate}\nWhen these conditions hold, there is a unique rational function \\(G\\in\\mathbb F_p(z)\\) with \\(G(0)=1\\) and\n\\[\nG^{p-1}=B/A,\n\\]\nand \\(F=G\\).","iterion]\n\\label{thm:rational-collapse-criterion}\nLet \\(F=\\mathcal M_p[A/B]\\).  The following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(F\\in\\mathbb F_p(z)\\).\n\\item \\(B/A\\) is a \\((p-1)\\)-st power in \\(\\mathbb F_p(z)\\).\n\\item \\(\\omega_{p-1}(A/B)=0\\).\n\\end{enumerate}\nWhen these conditions hold, there is a unique rational function \\(G\\in\\mathbb F_p(z)\\) with \\(G(0)=1\\) and\n\\[\nG^{p-1}=B/A,\n\\]\nand \\(F=G\\).","ional-collapse-criterion}\nLet \\(F=\\mathcal M_p[A/B]\\).  The following are equivalent.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item \\(F\\in\\mathbb F_p(z)\\).\n\\item \\(B/A\\) is a \\((p-1)\\)-st power in \\(\\mathbb F_p(z)\\).\n\\item \\(\\omega_{p-1}(A/B)=0\\).\n\\end{enumerate}\nWhen these conditions hold, there is a unique rational function \\(G\\in\\mathbb F_p(z)\\) with \\(G(0)=1\\) and\n\\[\nG^{p-1}=B/A,\n\\]\nand \\(F=G\\).","If \\(p=2\\), every rational mask collapses to a rational series.  If \\(p>2\\) and \\(\\omega_{p-1}(A/B)\\neq0\\), then \\(F\\) is nonrational, non-C-finite, and not eventually periodic.\n\\end{theorem}","\\begin{proof}\nThe equivalence of (i) and (ii) follows from \\cref{eq:block04-radical-equation}.  If \\(B/A=G^{p-1}\\), every valuation of \\(B/A\\) is divisible by \\(p-1\\), so (ii) implies (iii).","Conversely, suppose all valuation residues vanish.  Since \\(A\\) and \\(B\\) are coprime, every irreducible exponent in each polynomial is divisible by \\(p-1\\).  Factoring into monic irreducibles therefore gives\n\\[\nA=C^{p-1},\n\\qquad\nB=D^{p-1}\n\\]\nfor polynomials \\(C,D\\) after a constant normalization.  The unit conditions \\(A(0)=B(0)=1\\) remove the remaining scalar obstruction.  Thus \\(B/A=(D/C)^{p-1}\\).  Multiplying \\(D/C\\) by its inverse constant term does not change its \\((p-1)\\)-st power, and gives the unique root with constant term one.  The formal solution is unique, so \\(F=G\\).","all valuation residues vanish.  Since \\(A\\) and \\(B\\) are coprime, every irreducible exponent in each polynomial is divisible by \\(p-1\\).  Factoring into monic irreducibles therefore gives\n\\[\nA=C^{p-1},\n\\qquad\nB=D^{p-1}\n\\]\nfor polynomials \\(C,D\\) after a constant normalization.  The unit conditions \\(A(0)=B(0)=1\\) remove the remaining scalar obstruction.  Thus \\(B/A=(D/C)^{p-1}\\).  Multiplying \\(D/C\\) by its inverse constant term does not change its \\((p-1)\\)-st power, and gives the unique root with constant term one.  The formal solution is unique, so \\(F=G\\).","e coprime, every irreducible exponent in each polynomial is divisible by \\(p-1\\).  Factoring into monic irreducibles therefore gives\n\\[\nA=C^{p-1},\n\\qquad\nB=D^{p-1}\n\\]\nfor polynomials \\(C,D\\) after a constant normalization.  The unit conditions \\(A(0)=B(0)=1\\) remove the remaining scalar obstruction.  Thus \\(B/A=(D/C)^{p-1}\\).  Multiplying \\(D/C\\) by its inverse constant term does not change its \\((p-1)\\)-st power, and gives the unique root with constant term one.  The formal solution is unique, so \\(F=G\\).","rreducibles therefore gives\n\\[\nA=C^{p-1},\n\\qquad\nB=D^{p-1}\n\\]\nfor polynomials \\(C,D\\) after a constant normalization.  The unit conditions \\(A(0)=B(0)=1\\) remove the remaining scalar obstruction.  Thus \\(B/A=(D/C)^{p-1}\\).  Multiplying \\(D/C\\) by its inverse constant term does not change its \\((p-1)\\)-st power, and gives the unique root with constant term one.  The formal solution is unique, so \\(F=G\\).","For \\(p=2\\), the exponent \\(p-1\\) is one, hence the obstruction group is trivial and \\(F=B/A\\).  Over a finite field, a rational coefficient series is C-finite and eventually periodic, while an eventually periodic sequence has a rational generating function.  Therefore a nonzero obstruction forces all three failures stated in the theorem.\n\\end{proof}","Write\n\\[\na=\\deg A,\n\\qquad\nb=\\deg B,\n\\qquad\nH(z)=A(z)B(z)^{p-2},\n\\]\nand define\n\\begin{equation}\n\\label{eq:block04-degree-bound}\nD(A,B)\n=\n\\max\\left\\{\nb,\n\\left\\lceil\\frac{a+(p-2)b}{p-1}\\right\\rceil\n\\right\\}.\n\\end{equation}\nLet \\(\\mathcal P_D=\\mathbb F_p[z]_{\\leq D}\\).  For \\(0\\leq d<p\\), define\n\\begin{equation}\n\\label{eq:block04-numerator-transition}\nT_d(P)=\\Lambda_d(PH),\n\\end{equation}\nwhere \\(\\Lambda_d\\) selects the coefficients in degrees congruent to \\(d\\) modulo \\(p\\).","p-2},\n\\]\nand define\n\\begin{equation}\n\\label{eq:block04-degree-bound}\nD(A,B)\n=\n\\max\\left\\{\nb,\n\\left\\lceil\\frac{a+(p-2)b}{p-1}\\right\\rceil\n\\right\\}.\n\\end{equation}\nLet \\(\\mathcal P_D=\\mathbb F_p[z]_{\\leq D}\\).  For \\(0\\leq d<p\\), define\n\\begin{equation}\n\\label{eq:block04-numerator-transition}\nT_d(P)=\\Lambda_d(PH),\n\\end{equation}\nwhere \\(\\Lambda_d\\) selects the coefficients in degrees congruent to \\(d\\) modulo \\(p\\).","define\n\\begin{equation}\n\\label{eq:block04-degree-bound}\nD(A,B)\n=\n\\max\\left\\{\nb,\n\\left\\lceil\\frac{a+(p-2)b}{p-1}\\right\\rceil\n\\right\\}.\n\\end{equation}\nLet \\(\\mathcal P_D=\\mathbb F_p[z]_{\\leq D}\\).  For \\(0\\leq d<p\\), define\n\\begin{equation}\n\\label{eq:block04-numerator-transition}\nT_d(P)=\\Lambda_d(PH),\n\\end{equation}\nwhere \\(\\Lambda_d\\) selects the coefficients in degrees congruent to \\(d\\) modulo \\(p\\).","\\begin{theorem}[Fixed-denominator Cartier realization]\n\\label{thm:fixed-denominator-cartier-realization}\nLet \\(F=\\mathcal M_p[A/B]\\) and \\(D=D(A,B)\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item every \\(T_d\\) preserves \\(\\mathcal P_D\\);\n\\item the linear map\n\\begin{equation}\n\\label{eq:block04-theta-map}\n\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\b","\\begin{theorem}[Fixed-denominator Cartier realization]\n\\label{thm:fixed-denominator-cartier-realization}\nLet \\(F=\\mathcal M_p[A/B]\\) and \\(D=D(A,B)\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item every \\(T_d\\) preserves \\(\\mathcal P_D\\);\n\\item the linear map\n\\begin{equation}\n\\label{eq:block04-theta-map}\n\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\en","\\begin{theorem}[Fixed-denominator Cartier realization]\n\\label{thm:fixed-denominator-cartier-realization}\nLet \\(F=\\mathcal M_p[A/B]\\) and \\(D=D(A,B)\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item every \\(T_d\\) preserves \\(\\mathcal P_D\\);\n\\item the linear map\n\\begin{equation}\n\\label{eq:block04-theta-map}\n\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\end{equation}\nwhere \\(\\epsilon_0(G)=[z^0]G\\).\n\\end{enumerate}\n\\end{theorem","\\begin{theorem}[Fixed-denominator Cartier realization]\n\\label{thm:fixed-denominator-cartier-realization}\nLet \\(F=\\mathcal M_p[A/B]\\) and \\(D=D(A,B)\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item every \\(T_d\\) preserves \\(\\mathcal P_D\\);\n\\item the linear map\n\\begin{equation}\n\\label{eq:block04-theta-map}\n\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\end{equation}\nwhere \\(\\epsilon_0(G)=[z^0]G\\).\n\\end{enumerate}\n\\end{theorem}","tion}\nLet \\(F=\\mathcal M_p[A/B]\\) and \\(D=D(A,B)\\).  Then:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item every \\(T_d\\) preserves \\(\\mathcal P_D\\);\n\\item the linear map\n\\begin{equation}\n\\label{eq:block04-theta-map}\n\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\end{equation}\nwhere \\(\\epsilon_0(G)=[z^0]G\\).\n\\end{enumerate}\n\\end{theorem}","d\\) preserves \\(\\mathcal P_D\\);\n\\item the linear map\n\\begin{equation}\n\\label{eq:block04-theta-map}\n\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\end{equation}\nwhere \\(\\epsilon_0(G)=[z^0]G\\).\n\\end{enumerate}\n\\end{theorem}","m the linear map\n\\begin{equation}\n\\label{eq:block04-theta-map}\n\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\end{equation}\nwhere \\(\\epsilon_0(G)=[z^0]G\\).\n\\end{enumerate}\n\\end{theorem}","\\Theta_{A,B}:\\mathcal P_D\\longrightarrow\\mathbb F_p[[z]],\n\\qquad\n\\Theta_{A,B}(P)=\\frac{P}{B}F,\n\\end{equation}\nis injective;\n\\item the transitions intertwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\end{equation}\nwhere \\(\\epsilon_0(G)=[z^0]G\\).\n\\end{enumerate}\n\\end{theorem}","rtwine exactly:\n\\begin{equation}\n\\label{eq:block04-intertwining}\n\\Lambda_d\\Theta_{A,B}(P)\n=\n\\Theta_{A,B}(T_d(P));\n\\end{equation}\n\\item starting from \\(P_0=B\\), the coefficient of index\n\\[\nn=d_0+d_1p+\\cdots+d_{\\ell-1}p^{\\ell-1}\n\\]\nis\n\\begin{equation}\n\\label{eq:block04-compiler-output}\n[z^n]F\n=\n\\epsilon_0\\!\\left(\n\\Theta_{A,B}\n\\bigl(T_{d_{\\ell-1}}\\cdots T_{d_0}B\\bigr)\n\\right),\n\\end{equation}\nwhere \\(\\epsilon_0(G)=[z^0]G\\).\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nIf \\(\\deg P\\leq D\\), then\n\\[\n\\deg(PH)\n\\leq\nD+a+(p-2)b.\n\\]\nCartier selection divides the retained degrees by \\(p\\).  The definition of \\(D\\) gives\n\\[\n\\deg T_d(P)\n\\leq\n\\left\\lfloor\\frac{D+a+(p-2)b}{p}\\right\\rfloor\n\\leq D.\n\\]\nThis proves (i).","The series \\(F/B\\) is a unit in \\(\\mathbb F_p[[z]]\\).  Multiplication by a unit is injective, so \\(\\Theta_{A,B}\\) is injective.  The rational-mask Cartier identity imported from \\cite{FMH} is precisely \\cref{eq:block04-intertwining}.  Iterating it from \\(\\Theta(B)=F\\) and taking constant terms proves \\cref{eq:block04-compiler-output}.\n\\end{proof}","For a digit word \\(w=d_0\\cdots d_{\\ell-1}\\), read least significant digit first and write\n\\[\n\\Lambda_w\n=\n\\Lambda_{d_{\\ell-1}}\\cdots\\Lambda_{d_0}.\n\\]\nDefine the Cartier kernel and its linear span by\n\\begin{equation}\n\\label{eq:block04-kernel-span}\n\\mathcal K_p(F)\n=\n\\{\\Lambda_wF:w\\in\\{0,\\ldots,p-1\\}^{*}\\},\n\\qquad\n\\mathcal V_p(F)\n=\n\\operatorname{span}_{\\mathbb F_p}\\mathcal K_p(F).\n\\end{equation}","\\begin{definition}[Canonical Cartier module and Cartier rank]\nThe canonical Cartier module of \\(F\\) is\n\\begin{equation}\n\\label{eq:block04-cartier-module}\n\\mathsf C_p(F)\n=\n\\left(\n\\mathcal V_p(F),\n(\\Lambda_d|_{\\mathcal V_p(F)})_{0\\leq d<p},\nF,\n\\epsilon_0\n\\right).\n\\end{equation}\nIts dimension\n\\begin{equation}\n\\label{eq:block04-cartier-rank}\nr_p(F)=\\dim_{\\mathbb F_p}\\mathcal V_p(F)\n\\end{equation}\nis the \\emph{Cartier rank}.\n\\end{definition}","\\begin{theorem}[Minimality of the canonical Cartier module]\n\\label{thm:canonical-cartier-minimality}\nFor every \\(F\\in\\RMah_p^{\\times}\\), the module \\(\\mathsf C_p(F)\\) is finite dimensional, reachable, and observable.  It is therefore a minimal digit-linear realization of the coefficient function.  Every other reachable-observable digit-linear realization of the same coefficient function is uniquely similar to \\(\\mathsf C_p(F)\\).\n\\end{theorem}","[Minimality of the canonical Cartier module]\n\\label{thm:canonical-cartier-minimality}\nFor every \\(F\\in\\RMah_p^{\\times}\\), the module \\(\\mathsf C_p(F)\\) is finite dimensional, reachable, and observable.  It is therefore a minimal digit-linear realization of the coefficient function.  Every other reachable-observable digit-linear realization of the same coefficient function is uniquely similar to \\(\\mathsf C_p(F)\\).\n\\end{theorem}","\\begin{proof}\nBy \\cref{thm:fixed-denominator-cartier-realization}, every section \\(\\Lambda_wF\\) lies in the \\((D+1)\\)-dimensional image of \\(\\Theta_{A,B}\\).  Hence \\(\\mathcal V_p(F)\\) is finite dimensional.  Reachability is built into its definition as the span of the orbit of \\(F\\).","For observability, let \\(G\\in\\mathcal V_p(F)\\) satisfy\n\\[\n\\epsilon_0(\\Lambda_wG)=0\n\\]\nfor every digit word \\(w\\).  For any \\(n\\geq0\\), choose the base-\\(p\\) digits of \\(n\\).  The section identity gives\n\\[\n[z^n]G=\\epsilon_0(\\Lambda_wG)=0.\n\\]\nThus \\(G=0\\), so the observation family separates points.","For uniqueness, let \\((V,(L_d),v,\\lambda)\\) be another reachable-observable realization of the same digit function.  Send the reachable vector \\(L_wv\\) to \\(\\Lambda_wF\\).  If a linear combination of reachable vectors vanishes, applying every continuation and then \\(\\lambda\\) shows that the corresponding combination of Cartier sections has every coefficient zero.  Observability makes the assignment well defined.  Reachability makes it surjective, and the symmetric construction gives an inverse.  The intertwining, initial-vector, and output identities are immediate.  Any such similarity is determined on the reachable spanning set, so it is unique.\n\\end{proof}","Let \\(\\mathcal R(A,B)\\subseteq\\mathcal P_D\\) be the finite set of numerator states reachable from \\(B\\) under the transitions \\(T_d\\), and let\n\\[\nW(A,B)=\\operatorname{span}_{\\mathbb F_p}\\mathcal R(A,B).\n\\]","\\begin{theorem}[Reachable numerator core]\n\\label{thm:reachable-numerator-core}\nThe map \\(\\Theta_{A,B}\\) restricts to\n\\begin{equation}\n\\label{eq:block04-reachable-bijections}\n\\mathcal R(A,B)\n\\overset{\\sim}{\\longrightarrow}\n\\mathcal K_p(F)\n\\end{equation}\nand to a linear isomorphism\n\\begin{equation}\n\\label{eq:block04-reachable-span-isomorphism}\nW(A,B)\n\\overset{\\sim}{\\longrightarrow}\n\\mathcal V_p(F).\n\\end{equation}\nThe reachable deterministic output automaton of the fixed-denominator compiler is already minimal.  No nontrivial output-equivalence quotient remains after reachability restriction.\n\\end{theorem}","\\begin{proof}\nFor every word \\(w\\), repeated use of \\cref{eq:block04-intertwining} gives\n\\[\n\\Theta_{A,B}(T_wB)=\\Lambda_wF.\n\\]\nThus \\(\\Theta_{A,B}\\) maps the reachable numerator set onto the Cartier kernel.  Injectivity of \\(\\Theta_{A,B}\\) makes the map bijective, and taking linear spans gives \\cref{eq:block04-reachable-span-isomorphism}.","If two reachable numerators \\(P\\neq Q\\) had identical future output behavior, the two series \\(\\Theta(P)\\) and \\(\\Theta(Q)\\) would have identical coefficients and would therefore be equal.  Injectivity would force \\(P=Q\\), a contradiction.  Hence all reachable states are behaviorally distinct.  Reachability and distinguishability prove minimality.\n\\end{proof}","Let\n\\begin{equation}\n\\label{eq:block04-kernel-size}\n\\kappa_p(F)=|\\mathcal K_p(F)|.\n\\end{equation}","\\begin{corollary}[Exact state-complexity hierarchy]\n\\label{cor:cartier-state-hierarchy}\nFor a reduced rational mask \\(A/B\\),\n\\begin{equation}\n\\label{eq:block04-state-hierarchy}\nr_p(F)\n\\leq\n\\kappa_p(F)\n\\leq\np^{r_p(F)}\n\\leq\np^{D(A,B)+1}.\n\\end{equation}\nMoreover,\n\\[\nr_p(F)=\\dim W(A,B),\n\\qquad\n\\kappa_p(F)=|\\mathcal R(A,B)|.\n\\]\n\\end{corollary}","\\begin{proof}\nThe Cartier kernel spans \\(\\mathcal V_p(F)\\), so its cardinality is at least the dimension.  It is a subset of the \\(p^{r_p(F)}\\) vectors in \\(\\mathcal V_p(F)\\).  Finally, \\(\\mathcal V_p(F)\\) embeds into the \\((D+1)\\)-dimensional compiler space by \\cref{thm:reachable-numerator-core}.\n\\end{proof}","\\paragraph{Reduced mask chart.}\nAn object of \\(\\Mask_p^{\\mathrm{rat}}\\) is a reduced rational function\n\\[\nR(z)=\\frac{A(z)}{B(z)},\n\\qquad\nA(0)=B(0)=1.\n\\]\nMorphisms preserve the rational function.  The semantics is \\(R\\mapsto\\mathcal M_p[R]\\).  The inverse chart map is the canonical ratio \\(F\\mapsto F/F(z^p)\\).","\\paragraph{Minimal Cartier-module chart.}\nAn object of \\(\\mathbf{Cart}_p^{\\mathrm{min,rat}}\\) is a finite-dimensional tuple\n\\[\n(V,(L_d)_{d=0}^{p-1},v,\\lambda)\n\\]\nthat is reachable and observable, whose digit coefficient function belongs to \\(\\RMah_p^{\\times}\\).  Morphisms are similarities preserving every \\(L_d\\), the initial vector, and the output functional.  Its normalized representative is \\(\\mathsf C_p(F)\\).","\\paragraph{Minimal digital chart.}\nAn object of \\(\\mathbf{Aut}_p^{\\mathrm{min,rat}}\\) is a reachable minimal least-significant-digit-first \\(p\\)-DFAO whose coefficient series belongs to \\(\\RMah_p^{\\times}\\).  Its canonical representative has state set \\(\\mathcal K_p(F)\\), transition \\(G\\mapsto\\Lambda_dG\\), start state \\(F\\), and output \\(G\\mapsto G(0)\\).","\\midrule\nMask \\(\\leftrightarrow\\) series & strict inverse pair & every coefficient and the canonical mask & the mask must be rational and unit normalized","Series \\(\\leftrightarrow\\) Cartier module & normalized equivalence & all Cartier sections, linear relations, initial vector, output & the module is restricted to reachable-observable realizations","Series \\(\\leftrightarrow\\) minimal DFAO & normalized equivalence & the complete \\(p\\)-kernel and coefficient function & the automaton uses the fixed radix and LSD-first convention","Mask \\(\\to\\) ambient compiler & directional constructor & exact coefficients and fixed-denominator transitions & adds unreachable ambient numerator states","Ambient compiler \\(\\to\\) normalized core & reachable restriction & all reachable series and all future outputs & discards only states outside the orbit of \\(B\\)","\\begin{theorem}[Rational-mask three-chart equivalence]\n\\label{thm:rational-mask-three-chart-equivalence}\nThe semantic functors induce equivalences of groupoids\n\\begin{equation}\n\\label{eq:block04-three-chart-equivalence}\n\\Mask_p^{\\mathrm{rat}}\n\\simeq\n\\mathbf{Cart}_p^{\\mathrm{min,rat}}\n\\simeq\n\\mathbf{Aut}_p^{\\mathrm{min,rat}}\n\\simeq\n\\RMah_p^{\\times}.\n\\end{equation}\nThe mask-series bridge is a strict inverse pair.  The Cartier and digital bridges are normalized equivalences with unique comparison isomorphisms.\n\\end{theorem}","\\begin{proof}\nThe strict mask-series equivalence is \\cref{thm:frobenius-radical-characterization}.  The canonical Cartier construction gives an object of the second chart, and \\cref{thm:canonical-cartier-minimality} gives a unique similarity from every other minimal realization of the same series.  The canonical \\(p\\)-kernel automaton is finite by \\cref{thm:fixed-denominator-cartier-realization}.  It is reachable by construction.  Distinct kernel series differ in some coefficient, and the digit word for that coefficient distinguishes their future outputs, so the automaton is minimal.  The usual behavioral map gives the unique isomorphism from any other reachable minimal DFAO for the same series.\n\\end{proof}","\\begin{lemma}[Rigidity of rational-mask normalized fibers]\n\\label{lem:rational-mask-fiber-rigidity}\nFix \\(F\\in\\RMah_p^{\\times}\\).  The groupoid of normalized presentations of \\(F\\) in each of the three charts is contractible.\n\\end{lemma}","Let \\(\\mathfrak R_p^{\\mathrm{rm}}\\) denote the three-chart pseudofunctor defined by the passports above.","\\begin{theorem}[Rational-mask coherence and effective descent]\n\\label{thm:rational-mask-effective-descent}\nChoose every pairwise bridge by the canonical constructions and every comparison cell by the unique normalized isomorphism over the common series.  Then the triple-overlap coherence law holds, and\n\\begin{equation}\n\\label{eq:block04-effective-descent}\n\\Desc(\\mathfrak R_p^{\\mathrm{rm}})\n\\simeq\n\\RMah_p^{\\times}.\n\\end{equation}\nDescent is effective for objects and morphisms.\n\\end{theorem}","Given \\(F\\in\\RMah_p^{\\times}\\), its reduced mask, canonical Cartier module, and minimal \\(p\\)-kernel automaton form descent data.  Conversely, compatible local objects have one common semantic series.  Fiber rigidity identifies them uniquely with its normalized presentations.  The same argument applies to morphisms.  Hence the normalization functor is fully faithful and essentially surjective.\n\\end{proof}","For \\(F\\in\\RMah_p^{\\times}\\), define\n\\begin{equation}\n\\label{eq:block04-invariant-package}\n\\mathcal J_p^{\\mathrm{rm}}(F)\n=\n\\left(\nR_F,\n\\omega_{p-1}(R_F),\nD(R_F),\n\\mathcal V_p(F),\nr_p(F),\n\\mathcal K_p(F),\n\\kappa_p(F),\n\\mathcal A_p(F)\n\\right).\n\\end{equation}","\\begin{theorem}[Invariant descent on the rational-mask atlas]\n\\label{thm:rational-mask-invariant-descent}\nThe native mask, Cartier-module, and automaton invariants admit coherent comparison maps to \\(\\mathcal J_p^{\\mathrm{rm}}\\).  They descend to the category in \\cref{eq:block04-effective-descent}.  The descended invariant is complete on \\(\\RMah_p^{\\times}\\).\n\\end{theorem}","\\begin{proof}\nThe fixed-denominator intertwining theorem identifies the reachable numerator span with \\(\\mathcal V_p(F)\\) and the reachable numerator set with \\(\\mathcal K_p(F)\\).  Hence all components of the package agree under the bridges.  Coherence follows from \\cref{thm:rational-mask-effective-descent}.  If two packages agree, their reduced canonical masks agree.  The coefficientwise mask product then reconstructs the same formal series, so the semantic objects are equal.\n\\end{proof}","The obstruction \\(\\omega_{p-1}\\) divides the semantic category into two exact loci:\n\\[\n\\RMah_p^{\\times}\n=\n\\RMah_{p,\\mathrm{col}}^{\\times}\n\\sqcup\n\\RMah_{p,\\mathrm{ncol}}^{\\times}.\n\\]\nThe collapse locus consists of rational, hence ultimately periodic, series.  The noncollapse locus consists of non-C-finite automatic series.  The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside \\(\\RMah_{p,\\mathrm{col}}^{\\times}\\).","omega_{p-1}\\) divides the semantic category into two exact loci:\n\\[\n\\RMah_p^{\\times}\n=\n\\RMah_{p,\\mathrm{col}}^{\\times}\n\\sqcup\n\\RMah_{p,\\mathrm{ncol}}^{\\times}.\n\\]\nThe collapse locus consists of rational, hence ultimately periodic, series.  The noncollapse locus consists of non-C-finite automatic series.  The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside \\(\\RMah_{p,\\mathrm{col}}^{\\times}\\).","\\paragraph{Cantor-digit indicator.}\nOver \\(\\mathbb F_3\\), let\n\\[\nR(z)=1+z.\n\\]\nThen\n\\[\nF(z)=\\prod_{j\\geq0}(1+z^{3^j})\n\\]\nhas coefficient one exactly when the ternary expansion of the index uses only the digits zero and one.  The functional equation is\n\\[\nF(z)=(1+z)F(z^3),\n\\qquad\nF(z)^2=\\frac1{1+z}.\n\\]\nThe factor \\(1+z\\) has odd divisor multiplicity, so the collapse obstruction is nonzero.  Thus the sequence is automatic but not C-finite or eventually periodic.  The canonical automaton has two states, while the Cartier rank is one.","\\paragraph{A genuine denominator example.}\nOver \\(\\mathbb F_3\\), take\n\\[\nR(z)=\\frac1{1+z}.\n\\]\nThen\n\\[\nF(z)^2=1+z.\n\\]\nThe reduced denominator remains fixed under every Cartier section.  The ambient compiler has dimension two, its reachable numerator set is\n\\[\n\\{0,1,2,1+z\\},\n\\]\nand its reachable span has dimension two.  Hence\n\\[\nr_3(F)=2,\n\\qquad\n\\kappa_3(F)=4.\n\\]\nThe nonzero collapse obstruction again proves that the automatic coefficient sequence is not eventually periodic.","\\paragraph{Collapse back to the periodic atlas.}\nFor\n\\[\nR(z)=(1+z)^2=1+2z+z^2\n\\]\nover \\(\\mathbb F_3\\), one has\n\\[\nF(z)^2=\\frac1{(1+z)^2},\n\\qquad\nF(z)=\\frac1{1+z}.\n\\]\nThe coefficients alternate between one and two.  This object lies in the rational-collapse locus and in the unit-periodic atlas of Periodic atlas.","\\paragraph{Characteristic-two transient.}\nOver \\(\\mathbb F_2\\), choose\n\\[\nR(z)=\\frac1{1+z+z^2}.\n\\]\nSince \\(p-1=1\\),\n\\[\nF(z)=1+z+z^2.\n\\]\nIts coefficient sequence is \\(1,1,1,0,0,\\ldots\\).  It is ultimately periodic but not purely periodic.  This is the simplest exact witness that Rational-mask atlas extends the semantic domain even in characteristic two.","\\paragraph{Negative control: automatic does not imply rational mask.}\nLet \\(t_n\\in\\mathbb F_2\\) be the parity of the binary digit sum of \\(n\\), and put\n\\[\nu_n=1+t_n.\n\\]\nThe sequence \\(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","ative control: automatic does not imply rational mask.}\nLet \\(t_n\\in\\mathbb F_2\\) be the parity of the binary digit sum of \\(n\\), and put\n\\[\nu_n=1+t_n.\n\\]\nThe sequence \\(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","atic does not imply rational mask.}\nLet \\(t_n\\in\\mathbb F_2\\) be the parity of the binary digit sum of \\(n\\), and put\n\\[\nu_n=1+t_n.\n\\]\nThe sequence \\(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","et \\(t_n\\in\\mathbb F_2\\) be the parity of the binary digit sum of \\(n\\), and put\n\\[\nu_n=1+t_n.\n\\]\nThe sequence \\(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","2\\) be the parity of the binary digit sum of \\(n\\), and put\n\\[\nu_n=1+t_n.\n\\]\nThe sequence \\(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","\\), and put\n\\[\nu_n=1+t_n.\n\\]\nThe sequence \\(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","e sequence \\(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","(u\\) is unit and \\(2\\)-automatic.  It is not eventually periodic.  Indeed, eventual period \\(T\\) would propagate backwards through the identities\n\\[\nt_{2^k+n}=1+t_n\n\\qquad\n(0\\leq n<2^k)\n\\]\nand force a pure period \\(T\\).  Writing \\(T=2^{\\nu}q\\) with \\(q\\) odd, the binary digit count of \\(2^m-T\\) has parity\n\\[\nm-\\nu-s_2(q-1)\n\\]\nfor all sufficiently large \\(m\\).  Choosing the parity of \\(m\\) makes\n\\[\nt_{2^m-T}\\neq t_{2^m},\n\\]\ncontradicting pure periodicity.","Let \\(U(z)=\\sum_{n\\geq0}u_nz^n\\).  If its canonical mask were rational, then \\cref{thm:rational-collapse-criterion} with \\(p=2\\) would make \\(U\\) rational, hence eventually periodic.  Therefore \\(U/U(z^2)\\) is not rational.  The rational-mask atlas is a proper normalized subatlas of the unit automatic stratum.","The Rational-mask atlas verifier enumerates every reduced unit mask with\n\\[\n\\deg A,\\deg B\\leq3\\quad(p=2),\n\\qquad\n\\deg A,\\deg B\\leq2\\quad(p=3),\n\\qquad\n\\deg A,\\deg B\\leq1\\quad(p=5).\n\\]\nFor every mask it checks, using exact finite-field and polynomial arithmetic:\n\\begin{enumerate}\n\\item the invariant degree bound on the full ambient numerator space;\n\\item the functional-equation coefficient recursion against compiler output;\n\\item transition closure and fixed-denominator intertwining;\n\\item distinguishability of every pair of reachable numerator states;\n\\item the identities\n\\[\nW(A,B)\\cong\\mathcal V_p(F),\n\\qquad\n\\mathcal R(A,B)\\cong\\mathcal K_p(F);\n\\]\n\\item the state hierarchy in \\cref{eq:block04-state-hierarchy};\n\\item characteristic-two collapse and the named collapse or noncollapse examples.\n\\end{enumerate}\nThe finite calculation validates the implementation on the rec","\\]\nFor every mask it checks, using exact finite-field and polynomial arithmetic:\n\\begin{enumerate}\n\\item the invariant degree bound on the full ambient numerator space;\n\\item the functional-equation coefficient recursion against compiler output;\n\\item transition closure and fixed-denominator intertwining;\n\\item distinguishability of every pair of reachable numerator states;\n\\item the identities\n\\[\nW(A,B)\\cong\\mathcal V_p(F),\n\\qquad\n\\mathcal R(A,B)\\cong\\mathcal K_p(F);\n\\]\n\\item the state hierarchy in \\cref{eq:block04-state-hierarchy};\n\\item characteristic-two collapse and the named collapse or noncollapse examples.\n\\end{enumerate}\nThe finite calculation validates the implementation on the recorded ranges.  It is not used as a substitute for the unbounded proofs.","\\begin{resultbox}\n\\textbf{Section result.}\nThe verified heterogeneous atlas now extends beyond pure periodicity.  Every unit rational functional-Mahler mask over \\(\\mathbb F_p\\) has a finite canonical Cartier module and a finite canonical \\(p\\)-kernel automaton.  The three normalized charts satisfy coherence and effective descent.  The fixed-denominator compiler has an exact reachable core: reachable numerator states are already the minimal digital states, and their span is the minimal Cartier module.  The rational-collapse obstruction separates the ultimately periodic sublocus from genuinely non-C-finite automatic series.  The resulting invariant package is globally complete on the declared rational-mask semantic category, but no claim is made for automatic sequences whose canonical mask is nonrational.\n\\end{resultbox}","This section answers that question over a field.  It separates two invariants that coincide only in special cases:\n\\[\n\\text{linear section rank}\n\\qquad\\text{and}\\qquad\n\\text{cardinality of the section orbit}.\n\\]\nThe first controls regularity and minimal digit-linear dimension.  The second controls automaticity and minimal deterministic-state cardinality.  Their exact relationship is the automatic--regular boundary developed below.","Fix an integer radix \\(k\\geq2\\) and a field \\(K\\).  Let\n\\[\n\\Sigma_k=\\{0,1,\\ldots,k-1\\}.\n\\]\nWords are read from least significant digit to most significant digit.  For\n\\[\nw=d_0d_1\\cdots d_{\\ell-1}\\in\\Sigma_k^*,\n\\]\ndefine\n\\[\n|w|=\\ell,\n\\qquad\n[w]_k=\\sum_{j=0}^{\\ell-1}d_jk^j.\n\\]\nThe empty word \\(\\varepsilon\\) has length zero and value zero.","For a sequence \\(a=(a_n)_{n\\geq0}\\in K^{\\mathbb N}\\), define the digit-section operators\n\\[\n(\\Lambda_d a)(n)=a(kn+d),\n\\qquad 0\\leq d<k.\n\\]\nFor a word \\(w=d_0\\cdots d_{\\ell-1}\\), put\n\\[\n\\Lambda_w\n=\n\\Lambda_{d_{\\ell-1}}\\cdots\\Lambda_{d_1}\\Lambda_{d_0}.\n\\]\nThen\n\\begin{equation}\n(\\Lambda_w a)(n)\n=\n a\\bigl(k^{|w|}n+[w]_k\\bigr).\n\\label{eq:block05-word-section}\n\\end{equation}","\\begin{definition}[Fixed-radix kernel and section module]\nThe \\(k\\)-kernel and its linear span are\n\\[\n\\mathcal K_k(a)\n=\n\\{\\Lambda_w a:w\\in\\Sigma_k^*\\},\n\\qquad\n\\mathcal V_k(a)\n=\n\\Span_K\\mathcal K_k(a).\n\\]\nThe sequence is \\emph{\\(k\\)-regular over \\(K\\)} when \\(\\mathcal V_k(a)\\) is finite dimensional.  Its regular rank is\n\\[\nr_k(a)=\\dim_K\\mathcal V_k(a).\n\\]\nIt is \\emph{\\(k\\)-automatic in the present reading convention} when \\(\\mathcal K_k(a)\\) is finite.  In that case set\n\\[\n\\kappa_k(a)=|\\mathcal K_k(a)|.\n\\]\n\\end{definition}","dix kernel and section module]\nThe \\(k\\)-kernel and its linear span are\n\\[\n\\mathcal K_k(a)\n=\n\\{\\Lambda_w a:w\\in\\Sigma_k^*\\},\n\\qquad\n\\mathcal V_k(a)\n=\n\\Span_K\\mathcal K_k(a).\n\\]\nThe sequence is \\emph{\\(k\\)-regular over \\(K\\)} when \\(\\mathcal V_k(a)\\) is finite dimensional.  Its regular rank is\n\\[\nr_k(a)=\\dim_K\\mathcal V_k(a).\n\\]\nIt is \\emph{\\(k\\)-automatic in the present reading convention} when \\(\\mathcal K_k(a)\\) is finite.  In that case set\n\\[\n\\kappa_k(a)=|\\mathcal K_k(a)|.\n\\]\n\\end{definition}","nd its linear span are\n\\[\n\\mathcal K_k(a)\n=\n\\{\\Lambda_w a:w\\in\\Sigma_k^*\\},\n\\qquad\n\\mathcal V_k(a)\n=\n\\Span_K\\mathcal K_k(a).\n\\]\nThe sequence is \\emph{\\(k\\)-regular over \\(K\\)} when \\(\\mathcal V_k(a)\\) is finite dimensional.  Its regular rank is\n\\[\nr_k(a)=\\dim_K\\mathcal V_k(a).\n\\]\nIt is \\emph{\\(k\\)-automatic in the present reading convention} when \\(\\mathcal K_k(a)\\) is finite.  In that case set\n\\[\n\\kappa_k(a)=|\\mathcal K_k(a)|.\n\\]\n\\end{definition}","\\begin{definition}[Digit-linear realization]\nA digit-linear realization of \\(a\\) is a tuple\n\\[\n\\mathsf L=(W,A_0,\\ldots,A_{k-1},v,\\lambda),\n\\]\nwhere \\(W\\) is a finite-dimensional \\(K\\)-vector space, \\(A_d\\in\\End_K(W)\\), \\(v\\in W\\), and \\(\\lambda\\in W^*\\), such that for every word \\(w=d_0\\cdots d_{\\ell-1}\\),\n\\begin{equation}\na_{[w]_k}\n=\n\\lambda A_wv,\n\\qquad\nA_w=A_{d_{\\ell-1}}\\cdots A_{d_0}.\n\\label{eq:block05-digit-linear-semantics}\n\\end{equation}\nBecause different words can represent the same integer after terminal high zeros are appended, equation \\eqref{eq:block05-digit-linear-semantics} includes the required leading-zero consistency.\n\\end{definition}","\\begin{theorem}[Section-module realization theorem]\n\\label{thm:block05-section-module-realization}\nFor a sequence \\(a\\in K^{\\mathbb N}\\), the following are equivalent:\n\\begin{enumerate}\n\\item \\(a\\) is \\(k\\)-regular over \\(K\\);\n\\item \\(a\\) admits a finite-dimensional digit-linear realization.\n\\end{enumerate}\nWhen these conditions hold, the tuple\n\\begin{equation}\n\\mathsf C_k(a)\n=\n\\bigl(\n\\mathcal V_k(a),\n\\Lambda_0,\\ldots,\\Lambda_{k-1},\n a,\n\\epsilon_0\n\\bigr),\n\\qquad\n\\epsilon_0(f)=f(0),\n\\label{eq:block05-canonical-section-module}\n\\end{equation}\nis a canonical realization of dimension \\(r_k(a)\\).\n\\end{theorem}","tem \\(a\\) is \\(k\\)-regular over \\(K\\);\n\\item \\(a\\) admits a finite-dimensional digit-linear realization.\n\\end{enumerate}\nWhen these conditions hold, the tuple\n\\begin{equation}\n\\mathsf C_k(a)\n=\n\\bigl(\n\\mathcal V_k(a),\n\\Lambda_0,\\ldots,\\Lambda_{k-1},\n a,\n\\epsilon_0\n\\bigr),\n\\qquad\n\\epsilon_0(f)=f(0),\n\\label{eq:block05-canonical-section-module}\n\\end{equation}\nis a canonical realization of dimension \\(r_k(a)\\).\n\\end{theorem}","\\begin{proof}\nAssume first that \\(\\mathcal V_k(a)\\) is finite dimensional.  It is invariant under every \\(\\Lambda_d\\) by construction.  Choose any basis, represent the section operators by matrices, take the initial vector to be \\(a\\), and take output to be evaluation at zero.  For every word \\(w\\),\n\\[\n\\epsilon_0(\\Lambda_wa)\n=(\\Lambda_wa)(0)\n=a_{[w]_k},\n\\]\nso the resulting realization satisfies \\eqref{eq:block05-digit-linear-semantics}.","Conversely, suppose \\(a\\) has a realization \\((W,A_d,v,\\lambda)\\).  For \\(x\\in W\\), define its represented behavior by\n\\[\n\\mathcal B(x)(n)\n=\n\\lambda A_{\\operatorname{dig}_k(n)}x,\n\\]\nwhere \\(A_{\\operatorname{dig}_k(n)}\\) is the product associated with the canonical least-significant-digit-first expansion of \\(n\\).  Equation \\eqref{eq:block05-digit-linear-semantics} and concatenation of digit words give\n\\[\n\\Lambda_w a\n=\n\\mathcal B(A_wv).\n\\]\nHence every kernel section belongs to \\(\\mathcal B(W)\\), so\n\\[\n\\dim_K\\mathcal V_k(a)\n\\leq\n\\dim_K W.\n\\]\nThus \\(a\\) is regular.  The realization in \\eqref{eq:block05-canonical-section-module} has dimension exactly \\(r_k(a)\\).\n\\end{proof}","ely, suppose \\(a\\) has a realization \\((W,A_d,v,\\lambda)\\).  For \\(x\\in W\\), define its represented behavior by\n\\[\n\\mathcal B(x)(n)\n=\n\\lambda A_{\\operatorname{dig}_k(n)}x,\n\\]\nwhere \\(A_{\\operatorname{dig}_k(n)}\\) is the product associated with the canonical least-significant-digit-first expansion of \\(n\\).  Equation \\eqref{eq:block05-digit-linear-semantics} and concatenation of digit words give\n\\[\n\\Lambda_w a\n=\n\\mathcal B(A_wv).\n\\]\nHence every kernel section belongs to \\(\\mathcal B(W)\\), so\n\\[\n\\dim_K\\mathcal V_k(a)\n\\leq\n\\dim_K W.\n\\]\nThus \\(a\\) is regular.  The realization in \\eqref{eq:block05-canonical-section-module} has dimension exactly \\(r_k(a)\\).\n\\end{proof}","define its represented behavior by\n\\[\n\\mathcal B(x)(n)\n=\n\\lambda A_{\\operatorname{dig}_k(n)}x,\n\\]\nwhere \\(A_{\\operatorname{dig}_k(n)}\\) is the product associated with the canonical least-significant-digit-first expansion of \\(n\\).  Equation \\eqref{eq:block05-digit-linear-semantics} and concatenation of digit words give\n\\[\n\\Lambda_w a\n=\n\\mathcal B(A_wv).\n\\]\nHence every kernel section belongs to \\(\\mathcal B(W)\\), so\n\\[\n\\dim_K\\mathcal V_k(a)\n\\leq\n\\dim_K W.\n\\]\nThus \\(a\\) is regular.  The realization in \\eqref{eq:block05-canonical-section-module} has dimension exactly \\(r_k(a)\\).\n\\end{proof}","d behavior by\n\\[\n\\mathcal B(x)(n)\n=\n\\lambda A_{\\operatorname{dig}_k(n)}x,\n\\]\nwhere \\(A_{\\operatorname{dig}_k(n)}\\) is the product associated with the canonical least-significant-digit-first expansion of \\(n\\).  Equation \\eqref{eq:block05-digit-linear-semantics} and concatenation of digit words give\n\\[\n\\Lambda_w a\n=\n\\mathcal B(A_wv).\n\\]\nHence every kernel section belongs to \\(\\mathcal B(W)\\), so\n\\[\n\\dim_K\\mathcal V_k(a)\n\\leq\n\\dim_K W.\n\\]\nThus \\(a\\) is regular.  The realization in \\eqref{eq:block05-canonical-section-module} has dimension exactly \\(r_k(a)\\).\n\\end{proof}","operatorname{dig}_k(n)}x,\n\\]\nwhere \\(A_{\\operatorname{dig}_k(n)}\\) is the product associated with the canonical least-significant-digit-first expansion of \\(n\\).  Equation \\eqref{eq:block05-digit-linear-semantics} and concatenation of digit words give\n\\[\n\\Lambda_w a\n=\n\\mathcal B(A_wv).\n\\]\nHence every kernel section belongs to \\(\\mathcal B(W)\\), so\n\\[\n\\dim_K\\mathcal V_k(a)\n\\leq\n\\dim_K W.\n\\]\nThus \\(a\\) is regular.  The realization in \\eqref{eq:block05-canonical-section-module} has dimension exactly \\(r_k(a)\\).\n\\end{proof}","nonical least-significant-digit-first expansion of \\(n\\).  Equation \\eqref{eq:block05-digit-linear-semantics} and concatenation of digit words give\n\\[\n\\Lambda_w a\n=\n\\mathcal B(A_wv).\n\\]\nHence every kernel section belongs to \\(\\mathcal B(W)\\), so\n\\[\n\\dim_K\\mathcal V_k(a)\n\\leq\n\\dim_K W.\n\\]\nThus \\(a\\) is regular.  The realization in \\eqref{eq:block05-canonical-section-module} has dimension exactly \\(r_k(a)\\).\n\\end{proof}","\\begin{proposition}[Reachability and observability]\n\\label{prop:block05-reach-observe}\nThe canonical section realization \\(\\mathsf C_k(a)\\) is linearly reachable and observable:\n\\[\n\\Span_K\\{\\Lambda_wa:w\\in\\Sigma_k^*\\}\n=\n\\mathcal V_k(a),\n\\]\nand\n\\[\n\\bigcap_{w\\in\\Sigma_k^*}\n\\ker(\\epsilon_0\\Lambda_w)\n=\n\\{0\\}.\n\\]\n\\end{proposition}","\\begin{proof}\nReachability is the definition of \\(\\mathcal V_k(a)\\).  If \\(f\\in\\mathcal V_k(a)\\) satisfies \\((\\epsilon_0\\Lambda_w)(f)=0\\) for every word \\(w\\), then\n\\[\nf([w]_k)=0\n\\]\nfor every word.  Every nonnegative integer has such an expansion, so \\(f=0\\).\n\\end{proof}","\\begin{definition}[Word-Hankel matrix]\nThe fixed-radix word-Hankel matrix of \\(a\\) is the infinite matrix\n\\begin{equation}\nH_a(u,v)\n=\n a_{[u]_k+k^{|u|}[v]_k}\n=\n(\\Lambda_u a)([v]_k),\n\\qquad\nu,v\\in\\Sigma_k^*.\n\\label{eq:block05-word-hankel}\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Regular rank theorem]\n\\label{thm:block05-regular-rank}\nLet \\(a\\) be \\(k\\)-regular over \\(K\\).  Then\n\\begin{equation}\n\\boxed{\n r_k(a)\n =\n \\rank_K H_a\n =\n \\min\\{\\dim_K W:W\\text{ supports a digit-linear realization of }a\\}.\n}\n\\label{eq:block05-rank-identity}\n\\end{equation}\nMoreover:\n\\begin{enumerate}\n\\item there exist finite word sets \\(U,V\\subset\\Sigma_k^*\\), each of cardinality \\(r_k(a)\\), for which the minor \\(H_a[U,V]\\) is nonsingular;\n\\item every reachable-observable realization of dimension \\(r_k(a)\\) is isomorphic to \\(\\mathsf C_k(a)\\);\n\\item any two minimal reachable-observable realizations are related by a unique similarity intertwining all digit operators, the initial vector, and the output functional.\n\\end{enumerate}\n\\end{theorem}","heorem]\n\\label{thm:block05-regular-rank}\nLet \\(a\\) be \\(k\\)-regular over \\(K\\).  Then\n\\begin{equation}\n\\boxed{\n r_k(a)\n =\n \\rank_K H_a\n =\n \\min\\{\\dim_K W:W\\text{ supports a digit-linear realization of }a\\}.\n}\n\\label{eq:block05-rank-identity}\n\\end{equation}\nMoreover:\n\\begin{enumerate}\n\\item there exist finite word sets \\(U,V\\subset\\Sigma_k^*\\), each of cardinality \\(r_k(a)\\), for which the minor \\(H_a[U,V]\\) is nonsingular;\n\\item every reachable-observable realization of dimension \\(r_k(a)\\) is isomorphic to \\(\\mathsf C_k(a)\\);\n\\item any two minimal reachable-observable realizations are related by a unique similarity intertwining all digit operators, the initial vector, and the output functional.\n\\end{enumerate}\n\\end{theorem}","begin{equation}\n\\boxed{\n r_k(a)\n =\n \\rank_K H_a\n =\n \\min\\{\\dim_K W:W\\text{ supports a digit-linear realization of }a\\}.\n}\n\\label{eq:block05-rank-identity}\n\\end{equation}\nMoreover:\n\\begin{enumerate}\n\\item there exist finite word sets \\(U,V\\subset\\Sigma_k^*\\), each of cardinality \\(r_k(a)\\), for which the minor \\(H_a[U,V]\\) is nonsingular;\n\\item every reachable-observable realization of dimension \\(r_k(a)\\) is isomorphic to \\(\\mathsf C_k(a)\\);\n\\item any two minimal reachable-observable realizations are related by a unique similarity intertwining all digit operators, the initial vector, and the output functional.\n\\end{enumerate}\n\\end{theorem}","(a)\n =\n \\rank_K H_a\n =\n \\min\\{\\dim_K W:W\\text{ supports a digit-linear realization of }a\\}.\n}\n\\label{eq:block05-rank-identity}\n\\end{equation}\nMoreover:\n\\begin{enumerate}\n\\item there exist finite word sets \\(U,V\\subset\\Sigma_k^*\\), each of cardinality \\(r_k(a)\\), for which the minor \\(H_a[U,V]\\) is nonsingular;\n\\item every reachable-observable realization of dimension \\(r_k(a)\\) is isomorphic to \\(\\mathsf C_k(a)\\);\n\\item any two minimal reachable-observable realizations are related by a unique similarity intertwining all digit operators, the initial vector, and the output functional.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nThe row indexed by \\(u\\) in \\(H_a\\) is the behavior of the section \\(\\Lambda_u a\\) evaluated on all continuation words.  By observability in \\cref{prop:block05-reach-observe}, evaluation on all words is injective on \\(\\mathcal V_k(a)\\).  Therefore the row span of \\(H_a\\) is naturally isomorphic to \\(\\mathcal V_k(a)\\), and\n\\[\n\\rank_K H_a=r_k(a).\n\\]","For an arbitrary realization \\((W,A_d,v_{\\mathrm{in}},\\lambda)\\), concatenation gives the factorization\n\\[\nH_a(u,v)\n=\n\\lambda A_vA_uv_{\\mathrm{in}}.\n\\]\nThus \\(H_a\\) factors through \\(W\\), so \\(\\rank H_a\\leq\\dim W\\).  The canonical realization has dimension \\(r_k(a)\\), proving the minimality formula.","An infinite matrix of finite rank \\(r_k(a)\\) has a nonsingular finite \\(r_k(a)\\times r_k(a)\\) minor, which gives the first assertion.  For the remaining assertions, define the behavior map of any realization by\n\\[\n\\mathcal B_W(x)(n)=\\lambda A_{\\operatorname{dig}_k(n)}x.\n\\]\nObservability makes this map injective.  Reachability and equality of the represented sequence make its image equal to \\(\\mathcal V_k(a)\\).  In minimal dimension it is therefore an isomorphism intertwining the digit operators, initial vector, and output.  Composing the two behavior isomorphisms gives the unique similarity between any two minimal reachable-observable realizations.\n\\end{proof}","\\begin{remark}[Finite certification]\nThe theorem is infinite in statement but finite in certificate.  A proposed rank \\(r\\) can be certified by:\n\\begin{enumerate}\n\\item an explicit \\(r\\)-dimensional section-stable span containing \\(a\\), which proves \\(r_k(a)\\leq r\\);\n\\item an explicit nonsingular \\(r\\times r\\) word-Hankel minor, which proves \\(r_k(a)\\geq r\\).\n\\end{enumerate}\nWhen both are supplied, the regular rank is exact.\n\\end{remark}","Order \\(\\Sigma_k^*\\) by shortlex order: shorter words precede longer words, and words of equal length are ordered lexicographically.  Starting from the empty list, scan the words in this order and retain a word precisely when its section is not in the span of the previously retained sections.","\\begin{definition}[Greedy shortlex kernel basis]\nFor a regular sequence \\(a\\), let\n\\[\nU_a=(u_1,\\ldots,u_{r_k(a)})\n\\]\nbe the unique ordered list produced by the greedy shortlex rule, and put\n\\[\nb_i=\\Lambda_{u_i}a.\n\\]\nFor the zero sequence, the list is empty.\n\\end{definition}","\\begin{construction}[Strict shortlex realization]\n\\label{con:block05-shortlex}\nWrite\n\\[\n\\Lambda_db_j\n=\n\\sum_{i=1}^{r_k(a)}(M_d(a))_{ij}b_i,\n\\qquad\n a=\\sum_i v_i(a)b_i,\n\\qquad\n\\ell_i(a)=b_i(0).\n\\]\nThe strict shortlex record is\n\\begin{equation}\n\\mathsf{SL}_{k,K}(a)\n=\n\\bigl(\nU_a,\nM_0(a),\\ldots,M_{k-1}(a),\n v(a),\n\\ell(a)\n\\bigr).\n\\label{eq:block05-shortlex-record}\n\\end{equation}\n\\end{construction}","\\begin{theorem}[Strict fixed-radix normal form]\n\\label{thm:block05-shortlex-normal-form}\nThe record \\(\\mathsf{SL}_{k,K}(a)\\) is uniquely determined by \\(a\\) and reconstructs \\(a\\) exactly through\n\\[\na_{[w]_k}\n=\n\\ell(a)M_w(a)v(a).\n\\]\nIt has dimension \\(r_k(a)\\).  Two regular sequences have the same strict shortlex record if and only if they are equal.\n\\end{theorem}","\\begin{theorem}[Fixed-radix regular normalized atlas]\n\\label{thm:block05-regular-atlas}\nLet \\(\\mathbf{Reg}_k(K)\\) be the discrete semantic category of \\(K\\)-valued \\(k\\)-regular sequences.  Let \\(\\mathbf{Sec}_k^{\\mathrm{can}}(K)\\) be the category of canonical section-module records \\eqref{eq:block05-canonical-section-module}, and let \\(\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}\\) be the category of strict shortlex records \\eqref{eq:block05-shortlex-record}, with morphisms required to preserve the represented sequence.  Then\n\\begin{equation}\n\\boxed{\n\\mathbf{Reg}_k(K)\n\\simeq\n\\mathbf{Sec}_k^{\\mathrm{can}}(K)\n\\simeq\n\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}.\n}\n\\label{eq:block05-regular-atlas-equivalence}\n\\end{equation}\nEvery normalized fiber over a fixed sequence is contractible.  Consequently the associated finite present","\\begin{theorem}[Fixed-radix regular normalized atlas]\n\\label{thm:block05-regular-atlas}\nLet \\(\\mathbf{Reg}_k(K)\\) be the discrete semantic category of \\(K\\)-valued \\(k\\)-regular sequences.  Let \\(\\mathbf{Sec}_k^{\\mathrm{can}}(K)\\) be the category of canonical section-module records \\eqref{eq:block05-canonical-section-module}, and let \\(\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}\\) be the category of strict shortlex records \\eqref{eq:block05-shortlex-record}, with morphisms required to preserve the represented sequence.  Then\n\\begin{equation}\n\\boxed{\n\\mathbf{Reg}_k(K)\n\\simeq\n\\mathbf{Sec}_k^{\\mathrm{can}}(K)\n\\simeq\n\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}.\n}\n\\label{eq:block05-regular-atlas-equivalence}\n\\end{equation}\nEvery normalized fiber over a fixed sequence is contractible.  Consequently the associated finite presentation atlas has effective object and morphi","\\begin{theorem}[Fixed-radix regular normalized atlas]\n\\label{thm:block05-regular-atlas}\nLet \\(\\mathbf{Reg}_k(K)\\) be the discrete semantic category of \\(K\\)-valued \\(k\\)-regular sequences.  Let \\(\\mathbf{Sec}_k^{\\mathrm{can}}(K)\\) be the category of canonical section-module records \\eqref{eq:block05-canonical-section-module}, and let \\(\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}\\) be the category of strict shortlex records \\eqref{eq:block05-shortlex-record}, with morphisms required to preserve the represented sequence.  Then\n\\begin{equation}\n\\boxed{\n\\mathbf{Reg}_k(K)\n\\simeq\n\\mathbf{Sec}_k^{\\mathrm{can}}(K)\n\\simeq\n\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}.\n}\n\\label{eq:block05-regular-atlas-equivalence}\n\\end{equation}\nEvery normalized fiber over a fixed sequence is contractible.  Consequently the associated finite presentation atlas has effective object and morphism descent.\n\\","\\begin{theorem}[Fixed-radix regular normalized atlas]\n\\label{thm:block05-regular-atlas}\nLet \\(\\mathbf{Reg}_k(K)\\) be the discrete semantic category of \\(K\\)-valued \\(k\\)-regular sequences.  Let \\(\\mathbf{Sec}_k^{\\mathrm{can}}(K)\\) be the category of canonical section-module records \\eqref{eq:block05-canonical-section-module}, and let \\(\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}\\) be the category of strict shortlex records \\eqref{eq:block05-shortlex-record}, with morphisms required to preserve the represented sequence.  Then\n\\begin{equation}\n\\boxed{\n\\mathbf{Reg}_k(K)\n\\simeq\n\\mathbf{Sec}_k^{\\mathrm{can}}(K)\n\\simeq\n\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}.\n}\n\\label{eq:block05-regular-atlas-equivalence}\n\\end{equation}\nEvery normalized fiber over a fixed sequence is contractible.  Consequently the associated finite presentation atlas has effective object and morphism descent.\n\\end{theorem}","ic category of \\(K\\)-valued \\(k\\)-regular sequences.  Let \\(\\mathbf{Sec}_k^{\\mathrm{can}}(K)\\) be the category of canonical section-module records \\eqref{eq:block05-canonical-section-module}, and let \\(\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}\\) be the category of strict shortlex records \\eqref{eq:block05-shortlex-record}, with morphisms required to preserve the represented sequence.  Then\n\\begin{equation}\n\\boxed{\n\\mathbf{Reg}_k(K)\n\\simeq\n\\mathbf{Sec}_k^{\\mathrm{can}}(K)\n\\simeq\n\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}.\n}\n\\label{eq:block05-regular-atlas-equivalence}\n\\end{equation}\nEvery normalized fiber over a fixed sequence is contractible.  Consequently the associated finite presentation atlas has effective object and morphism descent.\n\\end{theorem}","\\begin{definition}[Fixed-radix regular invariant]\nThe normalized fixed-radix invariant is\n\\begin{equation}\n\\mathcal J_{k,K}^{\\mathrm{reg}}(a)\n=\n\\bigl(\n r_k(a),\n \\mathsf C_k(a),\n \\mathsf{SL}_{k,K}(a),\n [H_a[U,V]]\n\\bigr),\n\\label{eq:block05-regular-invariant}\n\\end{equation}\nwhere \\([H_a[U,V]]\\) denotes any recorded nonsingular rank certificate together with its word addresses.  The strict normal form makes this package complete on \\(\\mathbf{Reg}_k(K)\\).\n\\end{definition}","ular invariant]\nThe normalized fixed-radix invariant is\n\\begin{equation}\n\\mathcal J_{k,K}^{\\mathrm{reg}}(a)\n=\n\\bigl(\n r_k(a),\n \\mathsf C_k(a),\n \\mathsf{SL}_{k,K}(a),\n [H_a[U,V]]\n\\bigr),\n\\label{eq:block05-regular-invariant}\n\\end{equation}\nwhere \\([H_a[U,V]]\\) denotes any recorded nonsingular rank certificate together with its word addresses.  The strict normal form makes this package complete on \\(\\mathbf{Reg}_k(K)\\).\n\\end{definition}","\\begin{theorem}[Kernel automaton theorem]\n\\label{thm:block05-kernel-automaton}\nAssume \\(\\mathcal K_k(a)\\) is finite.  Define a deterministic automaton with output by\n\\[\n\\text{state set }\\mathcal K_k(a),\n\\qquad\nf\\xrightarrow{d}\\Lambda_df,\n\\qquad\n\\text{initial state }a,\n\\qquad\n\\text{output }f(0).\n\\]\nThen:\n\\begin{enumerate}\n\\item the automaton represents \\(a\\) under least-significant-digit-first reading;\n\\item every state is reachable;\n\\item distinct states are behaviorally distinguishable;\n\\item it is the unique minimal DFAO up to state renaming.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nAfter reading a word \\(w\\), the state is \\(\\Lambda_wa\\), whose output is \\(a_{[w]_k}\\).  Reachability is immediate from the definition of the kernel.  If \\(f\\neq g\\), then there is an integer \\(n\\) with \\(f(n)\\neq g(n)\\).  Reading a digit word for \\(n\\) from either state distinguishes them.  Thus no two states can be merged, and the reachable automaton is minimal.  The usual residual-behavior argument then gives uniqueness up to renaming.\n\\end{proof}","\\begin{theorem}[Automatic--regular finite-orbit equivalence]\n\\label{thm:block05-automatic-regular-equivalence}\nFor a \\(k\\)-regular sequence \\(a\\), the following are equivalent:\n\\begin{enumerate}\n\\item \\(a\\) is \\(k\\)-automatic;\n\\item the orbit \\(\\{\\Lambda_wa:w\\in\\Sigma_k^*\\}\\) of the initial vector in \\(\\mathsf C_k(a)\\) is finite;\n\\item the orbit of \\(v(a)\\) under the strict shortlex digit matrices is finite.\n\\end{enumerate}\nOn this locus, the minimal DFAO is recovered as the orbit itself.  Hence\n\\begin{equation}\n\\mathbf{Aut}_k(K)\n\\simeq\n\\mathbf{FinOrbSec}_k(K)\n\\simeq\n\\mathbf{FinOrbLin}_{k,K}^{\\mathrm{sl}}\n\\simeq\n\\mathbf{DFAO}_{k,K}^{\\min}.\n\\label{eq:block05-automatic-finite-orbit-atlas}\n\\end{equation}\nIf \\(K=\\mathbb F_q\\) is finite, every \\(k\\)-regular \\(K\\)-valued sequence is automatic.\n\\end{theorem}","\\(k\\)-regular sequence \\(a\\), the following are equivalent:\n\\begin{enumerate}\n\\item \\(a\\) is \\(k\\)-automatic;\n\\item the orbit \\(\\{\\Lambda_wa:w\\in\\Sigma_k^*\\}\\) of the initial vector in \\(\\mathsf C_k(a)\\) is finite;\n\\item the orbit of \\(v(a)\\) under the strict shortlex digit matrices is finite.\n\\end{enumerate}\nOn this locus, the minimal DFAO is recovered as the orbit itself.  Hence\n\\begin{equation}\n\\mathbf{Aut}_k(K)\n\\simeq\n\\mathbf{FinOrbSec}_k(K)\n\\simeq\n\\mathbf{FinOrbLin}_{k,K}^{\\mathrm{sl}}\n\\simeq\n\\mathbf{DFAO}_{k,K}^{\\min}.\n\\label{eq:block05-automatic-finite-orbit-atlas}\n\\end{equation}\nIf \\(K=\\mathbb F_q\\) is finite, every \\(k\\)-regular \\(K\\)-valued sequence is automatic.\n\\end{theorem}","a)\\) is finite;\n\\item the orbit of \\(v(a)\\) under the strict shortlex digit matrices is finite.\n\\end{enumerate}\nOn this locus, the minimal DFAO is recovered as the orbit itself.  Hence\n\\begin{equation}\n\\mathbf{Aut}_k(K)\n\\simeq\n\\mathbf{FinOrbSec}_k(K)\n\\simeq\n\\mathbf{FinOrbLin}_{k,K}^{\\mathrm{sl}}\n\\simeq\n\\mathbf{DFAO}_{k,K}^{\\min}.\n\\label{eq:block05-automatic-finite-orbit-atlas}\n\\end{equation}\nIf \\(K=\\mathbb F_q\\) is finite, every \\(k\\)-regular \\(K\\)-valued sequence is automatic.\n\\end{theorem}","\\(v(a)\\) under the strict shortlex digit matrices is finite.\n\\end{enumerate}\nOn this locus, the minimal DFAO is recovered as the orbit itself.  Hence\n\\begin{equation}\n\\mathbf{Aut}_k(K)\n\\simeq\n\\mathbf{FinOrbSec}_k(K)\n\\simeq\n\\mathbf{FinOrbLin}_{k,K}^{\\mathrm{sl}}\n\\simeq\n\\mathbf{DFAO}_{k,K}^{\\min}.\n\\label{eq:block05-automatic-finite-orbit-atlas}\n\\end{equation}\nIf \\(K=\\mathbb F_q\\) is finite, every \\(k\\)-regular \\(K\\)-valued sequence is automatic.\n\\end{theorem}","the strict shortlex digit matrices is finite.\n\\end{enumerate}\nOn this locus, the minimal DFAO is recovered as the orbit itself.  Hence\n\\begin{equation}\n\\mathbf{Aut}_k(K)\n\\simeq\n\\mathbf{FinOrbSec}_k(K)\n\\simeq\n\\mathbf{FinOrbLin}_{k,K}^{\\mathrm{sl}}\n\\simeq\n\\mathbf{DFAO}_{k,K}^{\\min}.\n\\label{eq:block05-automatic-finite-orbit-atlas}\n\\end{equation}\nIf \\(K=\\mathbb F_q\\) is finite, every \\(k\\)-regular \\(K\\)-valued sequence is automatic.\n\\end{theorem}","\\begin{proof}\nThe first three conditions are the same orbit statement written in semantic, intrinsic-module, and coordinate form.  The kernel automaton in \\cref{thm:block05-kernel-automaton} recovers the deterministic chart, while its state-to-section map recovers the finite orbit inside the canonical module.  These constructions are inverse on normalized objects.  If \\(K=\\mathbb F_q\\) and \\(r_k(a)<\\infty\\), the vector space \\(\\mathcal V_k(a)\\) has only \\(q^{r_k(a)}\\) elements, so the section orbit is automatically finite.\n\\end{proof}","onditions are the same orbit statement written in semantic, intrinsic-module, and coordinate form.  The kernel automaton in \\cref{thm:block05-kernel-automaton} recovers the deterministic chart, while its state-to-section map recovers the finite orbit inside the canonical module.  These constructions are inverse on normalized objects.  If \\(K=\\mathbb F_q\\) and \\(r_k(a)<\\infty\\), the vector space \\(\\mathcal V_k(a)\\) has only \\(q^{r_k(a)}\\) elements, so the section orbit is automatically finite.\n\\end{proof}","statement written in semantic, intrinsic-module, and coordinate form.  The kernel automaton in \\cref{thm:block05-kernel-automaton} recovers the deterministic chart, while its state-to-section map recovers the finite orbit inside the canonical module.  These constructions are inverse on normalized objects.  If \\(K=\\mathbb F_q\\) and \\(r_k(a)<\\infty\\), the vector space \\(\\mathcal V_k(a)\\) has only \\(q^{r_k(a)}\\) elements, so the section orbit is automatically finite.\n\\end{proof}","\\begin{corollary}[State-rank hierarchy]\n\\label{cor:block05-state-rank}\nLet \\(a\\) be automatic with values in \\(\\mathbb F_q\\).  Then\n\\begin{equation}\n\\boxed{\n r_k(a)\n \\leq\n \\kappa_k(a)\n \\leq\n q^{r_k(a)}.\n}\n\\label{eq:block05-state-rank-bound}\n\\end{equation}\nThe first inequality can be strict: behavior-distinct kernel states may be linearly dependent.\n\\end{corollary}","\\begin{proof}\nThe kernel states span \\(\\mathcal V_k(a)\\), so their number is at least its dimension.  They form a subset of the \\(q^{r_k(a)}\\)-element vector space \\(\\mathcal V_k(a)\\), proving the upper bound.\n\\end{proof}","\\begin{theorem}[Polynomial regular rank in characteristic zero]\n\\label{thm:block05-polynomial-rank}\nLet \\(K\\) have characteristic zero and let \\(P\\in K[x]\\) be nonzero of degree \\(m\\).  For the sequence \\(a_n=P(n)\\),\n\\[\nr_k(a)=m+1.\n\\]\nIf \\(m\\geq1\\), its \\(k\\)-kernel is infinite.\n\\end{theorem}","\\begin{proof}\nEvery section has the form\n\\[\n(\\Lambda_wa)(n)\n=\nP(k^{|w|}n+[w]_k),\n\\]\nwhich is a polynomial in \\(n\\) of degree at most \\(m\\).  Hence the section span has dimension at most \\(m+1\\).","Choose \\(\\ell\\) with \\(k^\\ell>m\\) and consider the \\(m+1\\) sections\n\\[\nP(k^\\ell n+r),\n\\qquad 0\\leq r\\leq m.\n\\]\nWhen expanded in the basis \\(1,n,\\ldots,n^m\\), their coefficient functions as functions of \\(r\\) have successive degrees \\(m,m-1,\\ldots,0\\), with nonzero leading coefficients.  Evaluation at the distinct points \\(0,1,\\ldots,m\\) therefore gives an invertible generalized Vandermonde matrix.  These sections are linearly independent, proving rank \\(m+1\\).","sections\n\\[\nP(k^\\ell n+r),\n\\qquad 0\\leq r\\leq m.\n\\]\nWhen expanded in the basis \\(1,n,\\ldots,n^m\\), their coefficient functions as functions of \\(r\\) have successive degrees \\(m,m-1,\\ldots,0\\), with nonzero leading coefficients.  Evaluation at the distinct points \\(0,1,\\ldots,m\\) therefore gives an invertible generalized Vandermonde matrix.  These sections are linearly independent, proving rank \\(m+1\\).","For \\(m\\geq1\\), the sections \\(P(k^\\ell n+r)\\) are distinct as \\(r\\) varies through arbitrarily large complete residue ranges, so the kernel orbit is infinite.\n\\end{proof}","\\begin{corollary}\nFor \\(a_n=n\\), one has \\(r_k(a)=2\\) and exactly \\(k^\\ell\\) distinct sections at depth \\(\\ell\\).  For \\(a_n=n^2\\), one has \\(r_k(a)=3\\) and again exactly \\(k^\\ell\\) distinct sections at depth \\(\\ell\\).\n\\end{corollary}","\\begin{theorem}[Digit-sum regularity]\n\\label{thm:block05-digit-sum}\nLet \\(s_k(n)\\) be the sum of the base-\\(k\\) digits of \\(n\\), regarded as a sequence over a characteristic-zero field.  Then\n\\[\nr_k(s_k)=2,\n\\]\nand\n\\begin{equation}\n\\mathcal K_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\}.\n\\label{eq:block05-digit-sum-kernel}\n\\end{equation}\nAt exact depth \\(\\ell\\), the number of distinct sections is\n\\[\n(k-1)\\ell+1.\n\\]\nThus \\(s_k\\) is \\(k\\)-regular but not \\(k\\)-automatic over a characteristic-zero field.\n\\end{theorem}","-sum regularity]\n\\label{thm:block05-digit-sum}\nLet \\(s_k(n)\\) be the sum of the base-\\(k\\) digits of \\(n\\), regarded as a sequence over a characteristic-zero field.  Then\n\\[\nr_k(s_k)=2,\n\\]\nand\n\\begin{equation}\n\\mathcal K_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\}.\n\\label{eq:block05-digit-sum-kernel}\n\\end{equation}\nAt exact depth \\(\\ell\\), the number of distinct sections is\n\\[\n(k-1)\\ell+1.\n\\]\nThus \\(s_k\\) is \\(k\\)-regular but not \\(k\\)-automatic over a characteristic-zero field.\n\\end{theorem}","rity]\n\\label{thm:block05-digit-sum}\nLet \\(s_k(n)\\) be the sum of the base-\\(k\\) digits of \\(n\\), regarded as a sequence over a characteristic-zero field.  Then\n\\[\nr_k(s_k)=2,\n\\]\nand\n\\begin{equation}\n\\mathcal K_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\}.\n\\label{eq:block05-digit-sum-kernel}\n\\end{equation}\nAt exact depth \\(\\ell\\), the number of distinct sections is\n\\[\n(k-1)\\ell+1.\n\\]\nThus \\(s_k\\) is \\(k\\)-regular but not \\(k\\)-automatic over a characteristic-zero field.\n\\end{theorem}","05-digit-sum}\nLet \\(s_k(n)\\) be the sum of the base-\\(k\\) digits of \\(n\\), regarded as a sequence over a characteristic-zero field.  Then\n\\[\nr_k(s_k)=2,\n\\]\nand\n\\begin{equation}\n\\mathcal K_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\}.\n\\label{eq:block05-digit-sum-kernel}\n\\end{equation}\nAt exact depth \\(\\ell\\), the number of distinct sections is\n\\[\n(k-1)\\ell+1.\n\\]\nThus \\(s_k\\) is \\(k\\)-regular but not \\(k\\)-automatic over a characteristic-zero field.\n\\end{theorem}","\\begin{proof}\nThe digit identity\n\\[\ns_k(kn+d)=s_k(n)+d\n\\]\ngives\n\\[\n\\Lambda_d s_k=s_k+d.\n\\]\nIterating shows that every section is \\(s_k+c\\), so the span is contained in \\(\\Span\\{1,s_k\\}\\).  The two sequences are independent, giving rank two.  At depth \\(\\ell\\), the additive constant is the digit sum of an \\(\\ell\\)-digit residue.  Every integer from \\(0\\) through \\((k-1)\\ell\\) occurs, giving exactly \\((k-1)\\ell+1\\) states.\n\\end{proof}","gin{proof}\nThe digit identity\n\\[\ns_k(kn+d)=s_k(n)+d\n\\]\ngives\n\\[\n\\Lambda_d s_k=s_k+d.\n\\]\nIterating shows that every section is \\(s_k+c\\), so the span is contained in \\(\\Span\\{1,s_k\\}\\).  The two sequences are independent, giving rank two.  At depth \\(\\ell\\), the additive constant is the digit sum of an \\(\\ell\\)-digit residue.  Every integer from \\(0\\) through \\((k-1)\\ell\\) occurs, giving exactly \\((k-1)\\ell+1\\) states.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.99\\textwidth]{regular_infinite_orbit_examples.pdf}\n\\caption{Fixed section rank does not bound orbit cardinality over an infinite field.  The Thue--Morse orbit is finite, while \\(n\\), \\(n^2\\), and the digit sum have finite linear rank but unboundedly many distinct sections.}\n\\label{fig:regular-infinite-orbit}\n\\end{figure}","First, it exhaustively enumerates bounded deterministic automata in three configurations:\n\\[\n(k,q,s_{\\max})=(2,2,3),\\qquad(2,3,3),\\qquad(3,2,2).\n\\]\nEvery candidate is restricted to reachable states, minimized by behavioral equivalence, and filtered by the terminal-high-zero consistency required by integer semantics.  For every resulting normalized automaton the verifier checks:\n\\begin{enumerate}\n\\item reachability and pairwise behavioral distinguishability;\n\\item construction of the canonical linear quotient from continuation-output rows;\n\\item transition and output equivariance of the state embedding;\n\\item the inequalities in \\cref{eq:block05-state-rank-bound};\n\\item equality of word-Hankel rank and regular rank;\n\\item exact equality between the finite linear orbit and the minimized automaton state set;\n\\item coefficient agreement","Second, it checks unbounded symbolic families using exact integer and rational matrices.  Polynomial sequences \\(n^m\\) are checked for bases two and three through degree six, and digit sums are checked in bases two, three, and four, including their exact depth-orbit growth formulas.","Fixed-radix regular atlas proves a complete fixed-radix normalization theorem over a field.  The next question is not obtained by replacing the coefficient field with \\(\\mathbb Z\\) symbol by symbol.  Over \\(\\mathbb Z\\), two canonical lattices appear inside the same rational section space.  Their finite-index difference is invisible over \\(\\mathbb Q\\), but it controls reduction modulo every integer, the appearance of ghost coordinates, and the distinction between an effective \\(p\\)-adic lift and an effective integral lift.","a field.  The next question is not obtained by replacing the coefficient field with \\(\\mathbb Z\\) symbol by symbol.  Over \\(\\mathbb Z\\), two canonical lattices appear inside the same rational section space.  Their finite-index difference is invisible over \\(\\mathbb Q\\), but it controls reduction modulo every integer, the appearance of ghost coordinates, and the distinction between an effective \\(p\\)-adic lift and an effective integral lift.","Throughout this section, \\(k\\geq2\\) is fixed and\n\\[\na:\\mathbb N\\longrightarrow\\mathbb Z\n\\]\nis an integer-valued sequence whose rational section space\n\\[\nV_{\\mathbb Q}(a)\n=\n\\operatorname{span}_{\\mathbb Q}\\mathcal K_k(a)\n\\subseteq\n\\mathbb Q^{\\mathbb N}\n\\]\nhas finite dimension\n\\[\n\\rho_k(a)=\\dim_{\\mathbb Q}V_{\\mathbb Q}(a).\n\\]\nThus \\(a\\) is \\(k\\)-regular after extension of scalars to \\(\\mathbb Q\\).","\\begin{definition}[Raw and saturated section lattices]\n\\label{def:block06-lattices}\nThe \\emph{raw section lattice} and the \\emph{saturated section lattice} are\n\\[\nM_k(a)\n=\n\\operatorname{span}_{\\mathbb Z}\\mathcal K_k(a),\n\\qquad\nL_k(a)\n=\nV_{\\mathbb Q}(a)\\cap\\mathbb Z^{\\mathbb N}.\n\\]\nThe intersection defining \\(L_k(a)\\) is taken inside \\(\\mathbb Q^{\\mathbb N}\\).\n\\end{definition}","The raw lattice is generated by the actual radix sections.  The saturated lattice contains every integer-valued sequence lying in their rational span.  These lattices agree over \\(\\mathbb Q\\), but need not agree over \\(\\mathbb Z\\).","\\begin{lemma}[Finite coordinate detection]\n\\label{lem:block06-coordinate-detection}\nLet \\(V\\subseteq\\mathbb Q^{\\mathbb N}\\) be a vector space of finite dimension \\(\\rho\\).  There exist indices\n\\[\n0\\leq n_1<\\cdots<n_\\rho\n\\]\nsuch that the evaluation map\n\\[\nE_{\\mathbf n}:V\\longrightarrow\\mathbb Q^\\rho,\n\\qquad\nf\\longmapsto\n\\bigl(f(n_1),\\ldots,f(n_\\rho)\\bigr)\n\\]\nis an isomorphism.\n\\end{lemma}","\\begin{proof}\nThe coordinate functionals on \\(\\mathbb Q^{\\mathbb N}\\) separate points of \\(V\\).  Choose \\(n_1\\) for which some element of \\(V\\) has nonzero value.  If \\(n_1,\\ldots,n_j\\) have been chosen and the joint evaluation map has rank \\(j<\\rho\\), its kernel contains a nonzero \\(f\\).  Choose \\(n_{j+1}\\) with \\(f(n_{j+1})\\neq0\\).  The rank increases.  After \\(\\rho\\) steps, the evaluation map has full rank.\n\\end{proof}","\\begin{theorem}[Canonical finite saturated section lattice]\n\\label{thm:block06-saturation}\nThe groups \\(M_k(a)\\) and \\(L_k(a)\\) are free abelian of rank \\(\\rho_k(a)\\).  They are stable under every digit section \\(\\Lambda_d\\), and\n\\[\nM_k(a)\\subseteq L_k(a)\n\\]\nhas finite index.  Moreover, \\(L_k(a)\\) is the unique maximal subgroup of \\(V_{\\mathbb Q}(a)\\) whose elements are integer-valued.\n\\end{theorem}","\\begin{proof}\nChoose a coordinate-detection set \\(\\mathbf n\\) by \\cref{lem:block06-coordinate-detection}.  Evaluation embeds \\(L_k(a)\\) into \\(\\mathbb Z^\\rho\\), so \\(L_k(a)\\) is a finitely generated free abelian group of rank at most \\(\\rho\\).  The raw lattice \\(M_k(a)\\) is contained in \\(L_k(a)\\), and its rational span is all of \\(V_{\\mathbb Q}(a)\\).  Hence \\(M_k(a)\\) has rank \\(\\rho\\), and therefore \\(L_k(a)\\) also has rank \\(\\rho\\).  A subgroup of a finitely generated free abelian group is finitely generated and free, so \\(M_k(a)\\) is free of rank \\(\\rho\\).  Equal-rank lattice inclusion has finite index.","proof}\nChoose a coordinate-detection set \\(\\mathbf n\\) by \\cref{lem:block06-coordinate-detection}.  Evaluation embeds \\(L_k(a)\\) into \\(\\mathbb Z^\\rho\\), so \\(L_k(a)\\) is a finitely generated free abelian group of rank at most \\(\\rho\\).  The raw lattice \\(M_k(a)\\) is contained in \\(L_k(a)\\), and its rational span is all of \\(V_{\\mathbb Q}(a)\\).  Hence \\(M_k(a)\\) has rank \\(\\rho\\), and therefore \\(L_k(a)\\) also has rank \\(\\rho\\).  A subgroup of a finitely generated free abelian group is finitely generated and free, so \\(M_k(a)\\) is free of rank \\(\\rho\\).  Equal-rank lattice inclusion has finite index.","e-detection set \\(\\mathbf n\\) by \\cref{lem:block06-coordinate-detection}.  Evaluation embeds \\(L_k(a)\\) into \\(\\mathbb Z^\\rho\\), so \\(L_k(a)\\) is a finitely generated free abelian group of rank at most \\(\\rho\\).  The raw lattice \\(M_k(a)\\) is contained in \\(L_k(a)\\), and its rational span is all of \\(V_{\\mathbb Q}(a)\\).  Hence \\(M_k(a)\\) has rank \\(\\rho\\), and therefore \\(L_k(a)\\) also has rank \\(\\rho\\).  A subgroup of a finitely generated free abelian group is finitely generated and free, so \\(M_k(a)\\) is free of rank \\(\\rho\\).  Equal-rank lattice inclusion has finite index.",")\\) into \\(\\mathbb Z^\\rho\\), so \\(L_k(a)\\) is a finitely generated free abelian group of rank at most \\(\\rho\\).  The raw lattice \\(M_k(a)\\) is contained in \\(L_k(a)\\), and its rational span is all of \\(V_{\\mathbb Q}(a)\\).  Hence \\(M_k(a)\\) has rank \\(\\rho\\), and therefore \\(L_k(a)\\) also has rank \\(\\rho\\).  A subgroup of a finitely generated free abelian group is finitely generated and free, so \\(M_k(a)\\) is free of rank \\(\\rho\\).  Equal-rank lattice inclusion has finite index.",", so \\(L_k(a)\\) is a finitely generated free abelian group of rank at most \\(\\rho\\).  The raw lattice \\(M_k(a)\\) is contained in \\(L_k(a)\\), and its rational span is all of \\(V_{\\mathbb Q}(a)\\).  Hence \\(M_k(a)\\) has rank \\(\\rho\\), and therefore \\(L_k(a)\\) also has rank \\(\\rho\\).  A subgroup of a finitely generated free abelian group is finitely generated and free, so \\(M_k(a)\\) is free of rank \\(\\rho\\).  Equal-rank lattice inclusion has finite index.","Every \\(\\Lambda_d\\) preserves \\(V_{\\mathbb Q}(a)\\) and preserves integer-valued sequences.  Hence it preserves \\(L_k(a)\\).  It also maps each radix section to another radix section, so it preserves \\(M_k(a)\\).  Finally, any integer-valued subgroup of \\(V_{\\mathbb Q}(a)\\) lies in \\(V_{\\mathbb Q}(a)\\cap\\mathbb Z^{\\mathbb N}=L_k(a)\\).\n\\end{proof}","\\begin{definition}[Integral saturation defect]\n\\label{def:block06-defect}\nThe finite abelian group\n\\[\nC_k(a)\n=\nL_k(a)/M_k(a)\n\\]\nis the \\emph{integral saturation defect}.  Its order\n\\[\n\\delta_k(a)\n=\n[L_k(a):M_k(a)]\n\\]\nis the \\emph{saturation index}.\n\\end{definition}","By the Smith normal form, there are uniquely determined integers\n\\[\n1\\leq d_1(a)\\mid d_2(a)\\mid\\cdots\\mid d_{\\rho}(a)\n\\]\nsuch that\n\\[\nC_k(a)\n\\cong\n\\bigoplus_{i=1}^{\\rho}\\mathbb Z/d_i(a)\\mathbb Z.\n\\]\nFactors with \\(d_i=1\\) are harmless but are retained so that the list has length \\(\\rho\\).","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.93\\textwidth]{saturation_reduction_defect.pdf}\n\\caption{The rational section space contains the raw section lattice and its canonical integer-valued saturation.  The finite quotient is invisible over \\(\\mathbb Q\\), but it controls every modular reduction.}\n\\label{fig:block06-saturation-defect}\n\\end{figure}","\\begin{example}[A rank-two nontrivial defect]\n\\label{ex:block06-double-digitsum}\nLet \\(s_k(n)\\) be the base-\\(k\\) digit-sum sequence and let \\(a=2s_k\\).  Since\n\\[\n\\Lambda_d s_k=s_k+d,\n\\]\nthe rational section space is spanned by \\(s_k\\) and the constant sequence \\(1\\).  The actual sections of \\(2s_k\\) generate\n\\[\nM_k(2s_k)\n=\n2\\mathbb Z\\,s_k\\oplus2\\mathbb Z\\,1,\n\\]\nwhereas\n\\[\nL_k(2s_k)\n=\n\\mathbb Z\\,s_k\\oplus\\mathbb Z\\,1.\n\\]\nThus\n\\[\nC_k(2s_k)\n\\cong\n(\\mathbb Z/2\\mathbb Z)^2.\n\\]\nThe rational regular rank is two, but reduction modulo two loses the entire semantic lattice.\n\\end{example}","ivial defect]\n\\label{ex:block06-double-digitsum}\nLet \\(s_k(n)\\) be the base-\\(k\\) digit-sum sequence and let \\(a=2s_k\\).  Since\n\\[\n\\Lambda_d s_k=s_k+d,\n\\]\nthe rational section space is spanned by \\(s_k\\) and the constant sequence \\(1\\).  The actual sections of \\(2s_k\\) generate\n\\[\nM_k(2s_k)\n=\n2\\mathbb Z\\,s_k\\oplus2\\mathbb Z\\,1,\n\\]\nwhereas\n\\[\nL_k(2s_k)\n=\n\\mathbb Z\\,s_k\\oplus\\mathbb Z\\,1.\n\\]\nThus\n\\[\nC_k(2s_k)\n\\cong\n(\\mathbb Z/2\\mathbb Z)^2.\n\\]\nThe rational regular rank is two, but reduction modulo two loses the entire semantic lattice.\n\\end{example}","For an integer \\(m\\geq2\\), the raw coefficient reduction is\n\\[\nM_k(a)/mM_k(a),\n\\]\nwhile the semantic reduction inside the saturated lattice is\n\\[\n\\overline M_{k,m}(a)\n=\n\\operatorname{im}\n\\left(\nM_k(a)/mM_k(a)\n\\longrightarrow\nL_k(a)/mL_k(a)\n\\right).\n\\]\nThese need not agree.","\\begin{definition}[Ghost and missing reduction modules]\n\\label{def:block06-ghost-missing}\nDefine\n\\[\nG_{k,m}(a)\n=\n\\ker\n\\left(\nM_k(a)/mM_k(a)\n\\longrightarrow\nL_k(a)/mL_k(a)\n\\right)\n\\]\nand\n\\[\nQ_{k,m}(a)\n=\n\\operatorname{coker}\n\\left(\nM_k(a)/mM_k(a)\n\\longrightarrow\nL_k(a)/mL_k(a)\n\\right).\n\\]\nThe first records raw residue coordinates that represent the zero semantic sequence.  The second records saturated residue classes not reached by raw sections.\n\\end{definition}","\\begin{theorem}[Smith--Tor reduction sequence]\n\\label{thm:block06-tor-sequence}\nFor every \\(m\\geq2\\), tensoring\n\\[\n0\\longrightarrow M_k(a)\\longrightarrow L_k(a)\\longrightarrow C_k(a)\\longrightarrow0\n\\]\nwith \\(\\mathbb Z/m\\mathbb Z\\) gives a canonical digit-equivariant exact sequence\n\\[\n0\n\\longrightarrow\nC_k(a)[m]\n\\longrightarrow\nM_k(a)/mM_k(a)\n\\longrightarrow\nL_k(a)/mL_k(a)\n\\longrightarrow\nC_k(a)/mC_k(a)\n\\longrightarrow\n0.\n\\]\nConsequently,\n\\[\nG_{k,m}(a)\\cong C_k(a)[m],\n\\qquad\nQ_{k,m}(a)\\cong C_k(a)/mC_k(a).\n\\]\n\\end{theorem}","]\n\\label{thm:block06-tor-sequence}\nFor every \\(m\\geq2\\), tensoring\n\\[\n0\\longrightarrow M_k(a)\\longrightarrow L_k(a)\\longrightarrow C_k(a)\\longrightarrow0\n\\]\nwith \\(\\mathbb Z/m\\mathbb Z\\) gives a canonical digit-equivariant exact sequence\n\\[\n0\n\\longrightarrow\nC_k(a)[m]\n\\longrightarrow\nM_k(a)/mM_k(a)\n\\longrightarrow\nL_k(a)/mL_k(a)\n\\longrightarrow\nC_k(a)/mC_k(a)\n\\longrightarrow\n0.\n\\]\nConsequently,\n\\[\nG_{k,m}(a)\\cong C_k(a)[m],\n\\qquad\nQ_{k,m}(a)\\cong C_k(a)/mC_k(a).\n\\]\n\\end{theorem}","\\begin{proof}\nBoth \\(M_k(a)\\) and \\(L_k(a)\\) are free abelian.  The long exact Tor sequence therefore begins\n\\[\n0\\longrightarrow\n\\operatorname{Tor}_1^{\\mathbb Z}\n(C_k(a),\\mathbb Z/m\\mathbb Z)\n\\longrightarrow\nM_k(a)\\otimes\\mathbb Z/m\\mathbb Z\n\\longrightarrow\nL_k(a)\\otimes\\mathbb Z/m\\mathbb Z.\n\\]\nFor any abelian group \\(C\\),\n\\[\n\\operatorname{Tor}_1^{\\mathbb Z}(C,\\mathbb Z/m\\mathbb Z)\n\\cong C[m],\n\\qquad\nC\\otimes\\mathbb Z/m\\mathbb Z\n\\cong C/mC.\n\\]\nThe digit-section maps preserve the original short exact sequence, so naturality of Tor makes the five-term sequence digit-equivariant.\n\\end{proof}","\\begin{corollary}[Exact Smith formulas]\n\\label{cor:block06-smith-formulas}\nLet \\(g_i(m)=\\gcd(m,d_i(a))\\).  Then\n\\[\nG_{k,m}(a)\n\\cong\nQ_{k,m}(a)\n\\cong\n\\bigoplus_{i=1}^{\\rho}\n\\mathbb Z/g_i(m)\\mathbb Z,\n\\]\nand\n\\[\n\\left|\\overline M_{k,m}(a)\\right|\n=\n\\prod_{i=1}^{\\rho}\n\\frac{m}{g_i(m)}.\n\\]\nThe reduction map is an isomorphism if and only if\n\\[\n\\gcd\\bigl(m,d_\\rho(a)\\bigr)=1.\n\\]\nEquivalently, every prime dividing a reduction defect divides the saturation index.\n\\end{corollary}","\\begin{proof}\nChoose Smith bases in which the inclusion \\(M_k(a)\\hookrightarrow L_k(a)\\) is the diagonal matrix\n\\[\nD=\\operatorname{diag}(d_1,\\ldots,d_\\rho).\n\\]\nModulo \\(m\\), the kernel and cokernel of multiplication by \\(d_i\\) are both cyclic of order \\(\\gcd(m,d_i)\\), and the image has order \\(m/\\gcd(m,d_i)\\).  The map is invertible in every coordinate exactly when every \\(d_i\\) is a unit modulo \\(m\\), which is equivalent to \\(\\gcd(m,d_\\rho)=1\\).\n\\end{proof}","begin{proof}\nChoose Smith bases in which the inclusion \\(M_k(a)\\hookrightarrow L_k(a)\\) is the diagonal matrix\n\\[\nD=\\operatorname{diag}(d_1,\\ldots,d_\\rho).\n\\]\nModulo \\(m\\), the kernel and cokernel of multiplication by \\(d_i\\) are both cyclic of order \\(\\gcd(m,d_i)\\), and the image has order \\(m/\\gcd(m,d_i)\\).  The map is invertible in every coordinate exactly when every \\(d_i\\) is a unit modulo \\(m\\), which is equivalent to \\(\\gcd(m,d_\\rho)=1\\).\n\\end{proof}","s in which the inclusion \\(M_k(a)\\hookrightarrow L_k(a)\\) is the diagonal matrix\n\\[\nD=\\operatorname{diag}(d_1,\\ldots,d_\\rho).\n\\]\nModulo \\(m\\), the kernel and cokernel of multiplication by \\(d_i\\) are both cyclic of order \\(\\gcd(m,d_i)\\), and the image has order \\(m/\\gcd(m,d_i)\\).  The map is invertible in every coordinate exactly when every \\(d_i\\) is a unit modulo \\(m\\), which is equivalent to \\(\\gcd(m,d_\\rho)=1\\).\n\\end{proof}","\\begin{proposition}[Fitting-ideal certificate]\n\\label{prop:block06-fitting}\nFor \\(0\\leq j<\\rho\\),\n\\[\n\\operatorname{Fitt}_j(C_k(a))\n=\n\\left(\nd_1(a)d_2(a)\\cdots d_{\\rho-j}(a)\n\\right),\n\\]\nand\n\\[\n\\operatorname{Fitt}_j(C_k(a))=\\mathbb Z\n\\qquad\n(j\\geq\\rho).\n\\]\nThus the Smith profile and the Fitting filtration contain the same reduction-defect data at different resolutions.\n\\end{proposition}","\\begin{proof}\nUse the diagonal presentation \\(D:\\mathbb Z^\\rho\\to\\mathbb Z^\\rho\\).  The \\(j\\)-th Fitting ideal is generated by the minors of size \\(\\rho-j\\).  Since \\(d_1\\mid\\cdots\\mid d_\\rho\\), the product of the first \\(\\rho-j\\) diagonal entries divides every other such minor and is itself one of the minors.\n\\end{proof}","\\begin{boundary}[What the defect classifies]\n\\label{bnd:block06-defect-boundary}\nThe group \\(C_k(a)\\) classifies the finite-index failure of the raw section lattice to be saturated.  It does not classify the whole integral digit module, the sequence, or its section orbit.  Two nonisomorphic digit modules can have the same Smith profile.\n\\end{boundary}","Choose Smith bases for \\(M_k(a)\\) and \\(L_k(a)\\), and let\n\\[\nD=\\operatorname{diag}(d_1,\\ldots,d_\\rho)\n\\]\nbe the inclusion matrix.  Let \\(A_d\\) and \\(B_d\\) be the integer matrices of \\(\\Lambda_d\\) on the raw and saturated lattices.","\\begin{proposition}[Intertwining across saturation]\n\\label{prop:block06-intertwining}\nFor every digit \\(d\\),\n\\[\nB_dD=DA_d.\n\\]\nConsequently, the raw modules, semantic images, ghost modules, and missing modules at every modulus form compatible digit modules.\n\\end{proposition}","\\begin{proof}\nThe inclusion \\(M_k(a)\\hookrightarrow L_k(a)\\) commutes with \\(\\Lambda_d\\).  The displayed matrix identity is that commutative square in the chosen bases.  Reduction modulo \\(m\\), passage to kernel, image, and cokernel preserve the induced digit action.\n\\end{proof}","Fix a prime \\(p\\) and write\n\\[\n\\nu_i=\\nu_{p,i}(a)=v_p(d_i(a)).\n\\]","\\begin{theorem}[Prime-power defect profile]\n\\label{thm:block06-prime-power-profile}\nFor \\(r\\geq1\\),\n\\[\nG_{k,p^r}(a)\n\\cong\nQ_{k,p^r}(a)\n\\cong\n\\bigoplus_{i=1}^{\\rho}\n\\mathbb Z/p^{\\min(r,\\nu_i)}\\mathbb Z,\n\\]\nand\n\\[\n\\left|\\overline M_{k,p^r}(a)\\right|\n=\np^{\\sum_{i=1}^{\\rho}(r-\\min(r,\\nu_i))}.\n\\]\nThe minimal number of generators of \\(\\overline M_{k,p^r}(a)\\) as a \\(\\mathbb Z/p^r\\mathbb Z\\)-module is\n\\[\n\\mu_{p^r}(a)\n=\n\\#\\{i:\\nu_i<r\\}.\n\\]\nIn particular, \\(p\\) is a good reduction prime exactly when every \\(\\nu_i=0\\).\n\\end{theorem}","m:block06-prime-power-profile}\nFor \\(r\\geq1\\),\n\\[\nG_{k,p^r}(a)\n\\cong\nQ_{k,p^r}(a)\n\\cong\n\\bigoplus_{i=1}^{\\rho}\n\\mathbb Z/p^{\\min(r,\\nu_i)}\\mathbb Z,\n\\]\nand\n\\[\n\\left|\\overline M_{k,p^r}(a)\\right|\n=\np^{\\sum_{i=1}^{\\rho}(r-\\min(r,\\nu_i))}.\n\\]\nThe minimal number of generators of \\(\\overline M_{k,p^r}(a)\\) as a \\(\\mathbb Z/p^r\\mathbb Z\\)-module is\n\\[\n\\mu_{p^r}(a)\n=\n\\#\\{i:\\nu_i<r\\}.\n\\]\nIn particular, \\(p\\) is a good reduction prime exactly when every \\(\\nu_i=0\\).\n\\end{theorem}","_{k,p^r}(a)\n\\cong\nQ_{k,p^r}(a)\n\\cong\n\\bigoplus_{i=1}^{\\rho}\n\\mathbb Z/p^{\\min(r,\\nu_i)}\\mathbb Z,\n\\]\nand\n\\[\n\\left|\\overline M_{k,p^r}(a)\\right|\n=\np^{\\sum_{i=1}^{\\rho}(r-\\min(r,\\nu_i))}.\n\\]\nThe minimal number of generators of \\(\\overline M_{k,p^r}(a)\\) as a \\(\\mathbb Z/p^r\\mathbb Z\\)-module is\n\\[\n\\mu_{p^r}(a)\n=\n\\#\\{i:\\nu_i<r\\}.\n\\]\nIn particular, \\(p\\) is a good reduction prime exactly when every \\(\\nu_i=0\\).\n\\end{theorem}","\\begin{proof}\nApply \\cref{cor:block06-smith-formulas} to \\(m=p^r\\).  The \\(i\\)-th semantic image is the ideal\n\\[\np^{\\min(r,\\nu_i)}\\mathbb Z/p^r\\mathbb Z.\n\\]\nIt is nonzero and cyclic exactly when \\(\\nu_i<r\\).  Over the local ring \\(\\mathbb Z/p^r\\mathbb Z\\), the minimal number of generators of a direct sum of cyclic ideals is the number of nonzero summands.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.94\\textwidth]{prime_power_regular_tower.pdf}\n\\caption{The raw residue tower maps onto the semantic prime-power tower.  Ghost coordinates occur at finite depth when the raw lattice is not \\(p\\)-saturated.  The compatible semantic tower converges to the embedded \\(p\\)-adic raw lattice.}\n\\label{fig:block06-prime-power-tower}\n\\end{figure}","\\begin{definition}[Prime-power section automaton]\n\\label{def:block06-prime-power-automaton}\nThe semantic \\(p^r\\)-kernel is\n\\[\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n=\n\\left\\{\n\\Lambda_w a\\bmod p^rL_k(a):\nw\\in\\Sigma_k^*\n\\right\\}\n\\subseteq\n\\overline M_{k,p^r}(a).\n\\]\nIts cardinality is denoted\n\\[\n\\kappa_{k,p^r}^{\\mathrm{sem}}(a).\n\\]\n\\end{definition}","\\begin{theorem}[Finite prime-power automata]\n\\label{thm:block06-prime-power-automata}\nFor every \\(r\\geq1\\), the semantic \\(p^r\\)-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence \\(a\\bmod p^r\\).  Moreover,\n\\[\n\\kappa_{k,p^r}^{\\mathrm{sem}}(a)\n\\leq\n\\left|\\overline M_{k,p^r}(a)\\right|\n=\np^{\\sum_i(r-\\min(r,\\nu_i))}.\n\\]\nReduction induces surjective digit-equivariant maps\n\\[\n\\mathcal K_{k,p^{r+1}}^{\\mathrm{sem}}(a)\n\\twoheadrightarrow\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a).\n\\]\n\\end{theorem}","ower-automata}\nFor every \\(r\\geq1\\), the semantic \\(p^r\\)-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence \\(a\\bmod p^r\\).  Moreover,\n\\[\n\\kappa_{k,p^r}^{\\mathrm{sem}}(a)\n\\leq\n\\left|\\overline M_{k,p^r}(a)\\right|\n=\np^{\\sum_i(r-\\min(r,\\nu_i))}.\n\\]\nReduction induces surjective digit-equivariant maps\n\\[\n\\mathcal K_{k,p^{r+1}}^{\\mathrm{sem}}(a)\n\\twoheadrightarrow\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a).\n\\]\n\\end{theorem}","nite, so its orbit under the digit maps is finite.  As in the field-valued proof of \\cref{thm:block05-kernel-automaton}, every orbit state is reachable.  Distinct section sequences have a continuation on which they differ, so they are behaviorally distinguishable.  This gives the minimal deterministic automaton.  The cardinality bound is the ambient module bound.  The same section word at level \\(r+1\\) reduces to the corresponding section word at level \\(r\\), so the transition map on orbit states is surjective.\n\\end{proof}","he field-valued proof of \\cref{thm:block05-kernel-automaton}, every orbit state is reachable.  Distinct section sequences have a continuation on which they differ, so they are behaviorally distinguishable.  This gives the minimal deterministic automaton.  The cardinality bound is the ambient module bound.  The same section word at level \\(r+1\\) reduces to the corresponding section word at level \\(r\\), so the transition map on orbit states is surjective.\n\\end{proof}","\\begin{corollary}[Good-prime transfer]\n\\label{cor:block06-good-prime}\nIf \\(p\\nmid\\delta_k(a)\\), then for every \\(r\\geq1\\),\n\\[\nM_k(a)/p^rM_k(a)\n\\overset{\\sim}{\\longrightarrow}\n\\overline M_{k,p^r}(a)\n=\nL_k(a)/p^rL_k(a).\n\\]\nAt a good prime, the raw and saturated prime-power towers agree exactly.\n\\end{corollary}","Define\n\\[\nM_{k,p}(a)=M_k(a)\\otimes_{\\mathbb Z}\\mathbb Z_p,\n\\qquad\nL_{k,p}(a)=L_k(a)\\otimes_{\\mathbb Z}\\mathbb Z_p.\n\\]","\\begin{theorem}[\\(p\\)-adic saturation sequence]\n\\label{thm:block06-padic-saturation}\nThere is a digit-equivariant short exact sequence\n\\[\n0\n\\longrightarrow\nM_{k,p}(a)\n\\longrightarrow\nL_{k,p}(a)\n\\longrightarrow\nC_k(a)_{(p)}\n\\longrightarrow\n0,\n\\]\nwhere \\(C_k(a)_{(p)}\\) is the \\(p\\)-primary subgroup of \\(C_k(a)\\).  In Smith coordinates,\n\\[\nM_{k,p}(a)\n=\n\\bigoplus_{i=1}^{\\rho}p^{\\nu_i}\\mathbb Z_p e_i\n\\subseteq\n\\bigoplus_{i=1}^{\\rho}\\mathbb Z_p e_i\n=\nL_{k,p}(a).\n\\]\n\\end{theorem}","\\begin{proof}\nThe \\(\\mathbb Z\\)-module \\(\\mathbb Z_p\\) is torsion free and hence flat.  Tensor the integral short exact sequence with \\(\\mathbb Z_p\\).  For a finite abelian group \\(C\\),\n\\[\nC\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong C_{(p)}.\n\\]\nThe Smith-coordinate formula follows by tensoring the diagonal inclusion.\n\\end{proof}","\\begin{theorem}[Raw and semantic inverse limits]\n\\label{thm:block06-inverse-limits}\nThere are canonical digit-equivariant identifications\n\\[\n\\varprojlim_r M_k(a)/p^rM_k(a)\n\\cong\nM_{k,p}(a)\n\\]\nand\n\\[\n\\varprojlim_r \\overline M_{k,p^r}(a)\n\\cong\nM_{k,p}(a)\n\\subseteq\nL_{k,p}(a).\n\\]\nFurthermore,\n\\[\n\\varprojlim_r G_{k,p^r}(a)=0,\n\\qquad\n\\varprojlim_r Q_{k,p^r}(a)\n\\cong\nC_k(a)_{(p)}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe first identification is the definition of the \\(p\\)-adic completion of a finite free lattice.  In Smith coordinate \\(i\\), the semantic image at level \\(r\\) is\n\\[\np^{\\min(r,\\nu_i)}\\mathbb Z/p^r\\mathbb Z.\n\\]\nIts inverse limit is \\(p^{\\nu_i}\\mathbb Z_p\\), giving the second identification.","A compatible ghost tower represents an element \\(x\\in\\mathbb Z_p\\) satisfying \\(p^{\\nu_i}x=0\\).  Since \\(\\mathbb Z_p\\) is torsion free, \\(x=0\\).  Hence the inverse limit of the ghost modules vanishes.  The missing quotient stabilizes coordinatewise at \\(\\mathbb Z/p^{\\nu_i}\\mathbb Z\\), whose product is the \\(p\\)-primary Smith quotient.\n\\end{proof}","The finite ghosts are therefore real finite-level defects, but they do not survive as nonzero points of the completed raw lattice.  By contrast, the missing saturated quotient survives exactly as the \\(p\\)-primary saturation defect.","Let\n\\[\n\\mathcal O_k(a)=\\{\\Lambda_w a:w\\in\\Sigma_k^*\\}\\subseteq M_k(a)\n\\]\nbe the actual section orbit, and let\n\\[\n\\mathcal O^{\\mathrm{raw}}_{k,p^r}(a)\n\\]\nbe its image in \\(M_k(a)/p^rM_k(a)\\).","\\begin{theorem}[Raw orbit-closure descent]\n\\label{thm:block06-orbit-closure}\nThe reduction maps\n\\[\n\\mathcal O^{\\mathrm{raw}}_{k,p^{r+1}}(a)\n\\twoheadrightarrow\n\\mathcal O^{\\mathrm{raw}}_{k,p^r}(a)\n\\]\nare surjective, and\n\\[\n\\overline{\\mathcal O_k(a)}^{\\,p}\n\\overset{\\sim}{\\longrightarrow}\n\\varprojlim_r\n\\mathcal O^{\\mathrm{raw}}_{k,p^r}(a),\n\\]\nwhere the closure is taken in \\(M_{k,p}(a)\\).\n\\end{theorem}","\\begin{proof}\nA section word at level \\(r\\) is lifted by the same word at level \\(r+1\\), proving surjectivity.  Every \\(p\\)-adic limit of orbit points gives a compatible tower.  Conversely, choose for each compatible residue state an orbit representative \\(x_r\\).  Compatibility gives\n\\[\nx_{r+1}-x_r\\in p^rM_k(a),\n\\]\nso \\((x_r)\\) is Cauchy in \\(M_{k,p}(a)\\).  Its limit lies in the orbit closure and maps to the prescribed tower.\n\\end{proof}","\\begin{definition}[Three effectivity obstructions]\n\\label{def:block06-effectivity}\nLet \\(x\\in M_{k,p}(a)\\).\n\\begin{enumerate}\n\\item The \\emph{\\(p\\)-adic effectivity condition} is simply \\(x\\in M_{k,p}(a)\\); every compatible full semantic module tower satisfies it by \\cref{thm:block06-inverse-limits}.\n\\item The \\emph{integral lattice obstruction} is\n\\[\n\\epsilon_{\\mathrm{int}}(x)\n=\n[x]\\in M_{k,p}(a)/M_k(a).\n\\]\nIt vanishes exactly when \\(x\\) is represented by one integral lattice element.\n\\item The \\emph{section-orbit obstruction} is membership in\n\\[\nM_k(a)\\setminus\\mathcal O_k(a).\n\\]\nEven after integral effectivity holds, the lift need not be an actual radix section.\n\\end{enumerate}\n\\end{definition}","structions]\n\\label{def:block06-effectivity}\nLet \\(x\\in M_{k,p}(a)\\).\n\\begin{enumerate}\n\\item The \\emph{\\(p\\)-adic effectivity condition} is simply \\(x\\in M_{k,p}(a)\\); every compatible full semantic module tower satisfies it by \\cref{thm:block06-inverse-limits}.\n\\item The \\emph{integral lattice obstruction} is\n\\[\n\\epsilon_{\\mathrm{int}}(x)\n=\n[x]\\in M_{k,p}(a)/M_k(a).\n\\]\nIt vanishes exactly when \\(x\\) is represented by one integral lattice element.\n\\item The \\emph{section-orbit obstruction} is membership in\n\\[\nM_k(a)\\setminus\\mathcal O_k(a).\n\\]\nEven after integral effectivity holds, the lift need not be an actual radix section.\n\\end{enumerate}\n\\end{definition}","effectivity condition} is simply \\(x\\in M_{k,p}(a)\\); every compatible full semantic module tower satisfies it by \\cref{thm:block06-inverse-limits}.\n\\item The \\emph{integral lattice obstruction} is\n\\[\n\\epsilon_{\\mathrm{int}}(x)\n=\n[x]\\in M_{k,p}(a)/M_k(a).\n\\]\nIt vanishes exactly when \\(x\\) is represented by one integral lattice element.\n\\item The \\emph{section-orbit obstruction} is membership in\n\\[\nM_k(a)\\setminus\\mathcal O_k(a).\n\\]\nEven after integral effectivity holds, the lift need not be an actual radix section.\n\\end{enumerate}\n\\end{definition}","\\begin{boundary}[Semantic orbit towers]\n\\label{bnd:block06-semantic-orbit}\nThe inverse limit of the full semantic modules is always \\(M_{k,p}(a)\\).  For the smaller semantic orbit sets, surjectivity of reduction remains automatic, but lifting to an actual orbit closure can be obscured by finite ghost fibers.  The unconditional orbit-closure theorem is therefore stated for the raw orbit tower.  A semantic-orbit effectivity theorem requires a compatible choice of raw lifts or an equivalent fiberwise Mittag--Leffler condition.\n\\end{boundary}","\\begin{example}[Digit sum separates all three levels]\n\\label{ex:block06-digitsum-effectivity}\nFor the base-\\(k\\) digit sum \\(s_k\\),\n\\[\n\\mathcal O_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\},\n\\]\nand\n\\[\nM_k(s_k)=L_k(s_k)=\\mathbb Zs_k\\oplus\\mathbb Z1.\n\\]\nHence\n\\[\n\\overline{\\mathcal O_k(s_k)}^{\\,p}\n=\n\\{s_k+c:c\\in\\mathbb Z_p\\}.\n\\]\nThe partial sums\n\\[\nc_N=\\sum_{j=0}^{N}p^{2j}\n\\]\nare nonnegative integers and therefore produce actual section states \\(s_k+c_N\\).  They converge in \\(\\mathbb Z_p\\) to\n\\[\nc_\\infty\n=\n\\frac{1}{1-p^2},\n\\]\nwhich is not an ordinary integer.  Thus \\(s_k+c_\\infty\\) is \\(p\\)-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.","e levels]\n\\label{ex:block06-digitsum-effectivity}\nFor the base-\\(k\\) digit sum \\(s_k\\),\n\\[\n\\mathcal O_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\},\n\\]\nand\n\\[\nM_k(s_k)=L_k(s_k)=\\mathbb Zs_k\\oplus\\mathbb Z1.\n\\]\nHence\n\\[\n\\overline{\\mathcal O_k(s_k)}^{\\,p}\n=\n\\{s_k+c:c\\in\\mathbb Z_p\\}.\n\\]\nThe partial sums\n\\[\nc_N=\\sum_{j=0}^{N}p^{2j}\n\\]\nare nonnegative integers and therefore produce actual section states \\(s_k+c_N\\).  They converge in \\(\\mathbb Z_p\\) to\n\\[\nc_\\infty\n=\n\\frac{1}{1-p^2},\n\\]\nwhich is not an ordinary integer.  Thus \\(s_k+c_\\infty\\) is \\(p\\)-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.","itsum-effectivity}\nFor the base-\\(k\\) digit sum \\(s_k\\),\n\\[\n\\mathcal O_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\},\n\\]\nand\n\\[\nM_k(s_k)=L_k(s_k)=\\mathbb Zs_k\\oplus\\mathbb Z1.\n\\]\nHence\n\\[\n\\overline{\\mathcal O_k(s_k)}^{\\,p}\n=\n\\{s_k+c:c\\in\\mathbb Z_p\\}.\n\\]\nThe partial sums\n\\[\nc_N=\\sum_{j=0}^{N}p^{2j}\n\\]\nare nonnegative integers and therefore produce actual section states \\(s_k+c_N\\).  They converge in \\(\\mathbb Z_p\\) to\n\\[\nc_\\infty\n=\n\\frac{1}{1-p^2},\n\\]\nwhich is not an ordinary integer.  Thus \\(s_k+c_\\infty\\) is \\(p\\)-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.","For the base-\\(k\\) digit sum \\(s_k\\),\n\\[\n\\mathcal O_k(s_k)\n=\n\\{s_k+c:c\\in\\mathbb Z_{\\geq0}\\},\n\\]\nand\n\\[\nM_k(s_k)=L_k(s_k)=\\mathbb Zs_k\\oplus\\mathbb Z1.\n\\]\nHence\n\\[\n\\overline{\\mathcal O_k(s_k)}^{\\,p}\n=\n\\{s_k+c:c\\in\\mathbb Z_p\\}.\n\\]\nThe partial sums\n\\[\nc_N=\\sum_{j=0}^{N}p^{2j}\n\\]\nare nonnegative integers and therefore produce actual section states \\(s_k+c_N\\).  They converge in \\(\\mathbb Z_p\\) to\n\\[\nc_\\infty\n=\n\\frac{1}{1-p^2},\n\\]\nwhich is not an ordinary integer.  Thus \\(s_k+c_\\infty\\) is \\(p\\)-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.","mathbb Z_{\\geq0}\\},\n\\]\nand\n\\[\nM_k(s_k)=L_k(s_k)=\\mathbb Zs_k\\oplus\\mathbb Z1.\n\\]\nHence\n\\[\n\\overline{\\mathcal O_k(s_k)}^{\\,p}\n=\n\\{s_k+c:c\\in\\mathbb Z_p\\}.\n\\]\nThe partial sums\n\\[\nc_N=\\sum_{j=0}^{N}p^{2j}\n\\]\nare nonnegative integers and therefore produce actual section states \\(s_k+c_N\\).  They converge in \\(\\mathbb Z_p\\) to\n\\[\nc_\\infty\n=\n\\frac{1}{1-p^2},\n\\]\nwhich is not an ordinary integer.  Thus \\(s_k+c_\\infty\\) is \\(p\\)-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.","\\]\nand\n\\[\nM_k(s_k)=L_k(s_k)=\\mathbb Zs_k\\oplus\\mathbb Z1.\n\\]\nHence\n\\[\n\\overline{\\mathcal O_k(s_k)}^{\\,p}\n=\n\\{s_k+c:c\\in\\mathbb Z_p\\}.\n\\]\nThe partial sums\n\\[\nc_N=\\sum_{j=0}^{N}p^{2j}\n\\]\nare nonnegative integers and therefore produce actual section states \\(s_k+c_N\\).  They converge in \\(\\mathbb Z_p\\) to\n\\[\nc_\\infty\n=\n\\frac{1}{1-p^2},\n\\]\nwhich is not an ordinary integer.  Thus \\(s_k+c_\\infty\\) is \\(p\\)-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.","By contrast, \\(s_k-1\\) is an integral lattice point and lies in the \\(p\\)-adic orbit closure because \\(-1\\in\\mathbb Z_p\\) is a limit of nonnegative integers.  It is not an actual section, since every actual offset is nonnegative.  Integral effectivity therefore does not imply section-orbit effectivity.\n\\end{example}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.93\\textwidth]{digit_sum_padic_effectivity.pdf}\n\\caption{For the digit-sum sequence, actual section states form a nonnegative integral ray.  Its \\(p\\)-adic closure contains all \\(p\\)-adic offsets, including nonintegral limits and integral points that are not actual sections.}\n\\label{fig:block06-digitsum-effectivity}\n\\end{figure}","Let\n\\[\n\\widehat{M_k(a)}\n=\n\\varprojlim_m M_k(a)/mM_k(a)\n\\]\nbe the profinite completion over divisibility of moduli.","\\begin{theorem}[Profinite saturation sequence]\n\\label{thm:block06-profinite}\nThere are canonical digit-equivariant identifications\n\\[\n\\widehat{M_k(a)}\n\\cong\n\\prod_p M_{k,p}(a),\n\\qquad\n\\widehat{L_k(a)}\n\\cong\n\\prod_p L_{k,p}(a),\n\\]\nand a short exact sequence\n\\[\n0\n\\longrightarrow\n\\widehat{M_k(a)}\n\\longrightarrow\n\\widehat{L_k(a)}\n\\longrightarrow\nC_k(a)\n\\longrightarrow\n0.\n\\]\nA compatible all-moduli raw tower has an ordinary integral lift exactly when its class in\n\\[\n\\widehat{M_k(a)}/M_k(a)\n\\]\nvanishes.\n\\end{theorem}","-profinite}\nThere are canonical digit-equivariant identifications\n\\[\n\\widehat{M_k(a)}\n\\cong\n\\prod_p M_{k,p}(a),\n\\qquad\n\\widehat{L_k(a)}\n\\cong\n\\prod_p L_{k,p}(a),\n\\]\nand a short exact sequence\n\\[\n0\n\\longrightarrow\n\\widehat{M_k(a)}\n\\longrightarrow\n\\widehat{L_k(a)}\n\\longrightarrow\nC_k(a)\n\\longrightarrow\n0.\n\\]\nA compatible all-moduli raw tower has an ordinary integral lift exactly when its class in\n\\[\n\\widehat{M_k(a)}/M_k(a)\n\\]\nvanishes.\n\\end{theorem}","\\begin{proof}\nFinite free abelian groups have profinite completion equal to the product of their \\(p\\)-adic completions.  In Smith coordinates, the inclusion is multiplication by \\(d_i\\) in each profinite coordinate.  Its cokernel is\n\\[\n\\widehat{\\mathbb Z}/d_i\\widehat{\\mathbb Z}\n\\cong\n\\mathbb Z/d_i\\mathbb Z.\n\\]\nTaking the direct sum over \\(i\\) gives \\(C_k(a)\\).  The final assertion is the definition of belonging to the diagonal image of the integral lattice inside its profinite completion.\n\\end{proof}","\\begin{corollary}[Local-global separation]\n\\label{cor:block06-local-global}\nEvery compatible prime-power tower is effective in a \\(p\\)-adic completion.  A compatible family over all primes is effective in the profinite completion.  Neither statement alone implies an ordinary integral lift.  Integral effectivity is the separate diagonal condition\n\\[\n\\widehat{x}\\in M_k(a)\\subseteq\\widehat{M_k(a)}.\n\\]\n\\end{corollary}","\\begin{definition}[Canonical integral regular record]\n\\label{def:block06-integral-record}\nThe canonical integral record of \\(a\\) is\n\\[\n\\mathcal I_k(a)\n=\n\\left(\nV_{\\mathbb Q}(a),\nM_k(a),\nL_k(a),\nC_k(a),\n(\\Lambda_d)_{0\\leq d<k},\na,\n\\operatorname{ev}_0\n\\right).\n\\]\nIts Smith--Fitting profile is\n\\[\n\\mathsf{SF}_k(a)\n=\n\\left(\nd_1,\\ldots,d_\\rho;\n\\operatorname{Fitt}_0(C_k(a)),\\ldots,\n\\operatorname{Fitt}_{\\rho-1}(C_k(a))\n\\right).\n\\]\n\\end{definition}","\\begin{theorem}[Integral normalization and directional local reduction]\n\\label{thm:block06-integral-normalization}\nFor fixed \\(k\\), the assignment\n\\[\na\\longmapsto\\mathcal I_k(a)\n\\]\nis functorial for digit-module morphisms preserving the distinguished sequence and evaluation.  It is uniquely determined by the semantic sequence.  For every prime \\(p\\), reduction defines a directional bridge\n\\[\n\\mathcal I_k(a)\n\\longmapsto\n\\left(\n\\overline M_{k,p^r}(a),\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n\\right)_{r\\geq1}\n\\]\nwhose exact information-loss passport is:\n\\begin{enumerate}\n\\item finite-depth ghost kernel \\(C_k(a)[p^r]\\);\n\\item finite-depth missing quotient \\(C_k(a)/p^rC_k(a)\\);\n\\item completed essential image \\(M_{k,p}(a)\\subseteq L_{k,p}(a)\\);\n\\item integral effectivity obstruction \\(M_{k,p}(a)/M_k(a)\\);\n\\item actual-section","\\begin{theorem}[Integral normalization and directional local reduction]\n\\label{thm:block06-integral-normalization}\nFor fixed \\(k\\), the assignment\n\\[\na\\longmapsto\\mathcal I_k(a)\n\\]\nis functorial for digit-module morphisms preserving the distinguished sequence and evaluation.  It is uniquely determined by the semantic sequence.  For every prime \\(p\\), reduction defines a directional bridge\n\\[\n\\mathcal I_k(a)\n\\longmapsto\n\\left(\n\\overline M_{k,p^r}(a),\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n\\right)_{r\\geq1}\n\\]\nwhose exact information-loss passport is:\n\\begin{enumerate}\n\\item finite-depth ghost kernel \\(C_k(a)[p^r]\\);\n\\item finite-depth missing quotient \\(C_k(a)/p^rC_k(a)\\);\n\\item completed essential image \\(M_{k,p}(a)\\subseteq L_{k,p}(a)\\);\n\\item integral effectivity obstruction \\(M_{k,p}(a)/M_k(a)\\);\n\\item actual-section obstruction inside the integral lattice.\n\\end{enu","\\begin{theorem}[Integral normalization and directional local reduction]\n\\label{thm:block06-integral-normalization}\nFor fixed \\(k\\), the assignment\n\\[\na\\longmapsto\\mathcal I_k(a)\n\\]\nis functorial for digit-module morphisms preserving the distinguished sequence and evaluation.  It is uniquely determined by the semantic sequence.  For every prime \\(p\\), reduction defines a directional bridge\n\\[\n\\mathcal I_k(a)\n\\longmapsto\n\\left(\n\\overline M_{k,p^r}(a),\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n\\right)_{r\\geq1}\n\\]\nwhose exact information-loss passport is:\n\\begin{enumerate}\n\\item finite-depth ghost kernel \\(C_k(a)[p^r]\\);\n\\item finite-depth missing quotient \\(C_k(a)/p^rC_k(a)\\);\n\\item completed essential image \\(M_{k,p}(a)\\subseteq L_{k,p}(a)\\);\n\\item integral effectivity obstruction \\(M_{k,p}(a)/M_k(a)\\);\n\\item actual-section obstruction inside the integral lattice.\n\\end{enumerate}\n\\end{theorem}","module morphisms preserving the distinguished sequence and evaluation.  It is uniquely determined by the semantic sequence.  For every prime \\(p\\), reduction defines a directional bridge\n\\[\n\\mathcal I_k(a)\n\\longmapsto\n\\left(\n\\overline M_{k,p^r}(a),\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n\\right)_{r\\geq1}\n\\]\nwhose exact information-loss passport is:\n\\begin{enumerate}\n\\item finite-depth ghost kernel \\(C_k(a)[p^r]\\);\n\\item finite-depth missing quotient \\(C_k(a)/p^rC_k(a)\\);\n\\item completed essential image \\(M_{k,p}(a)\\subseteq L_{k,p}(a)\\);\n\\item integral effectivity obstruction \\(M_{k,p}(a)/M_k(a)\\);\n\\item actual-section obstruction inside the integral lattice.\n\\end{enumerate}\n\\end{theorem}","e and evaluation.  It is uniquely determined by the semantic sequence.  For every prime \\(p\\), reduction defines a directional bridge\n\\[\n\\mathcal I_k(a)\n\\longmapsto\n\\left(\n\\overline M_{k,p^r}(a),\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n\\right)_{r\\geq1}\n\\]\nwhose exact information-loss passport is:\n\\begin{enumerate}\n\\item finite-depth ghost kernel \\(C_k(a)[p^r]\\);\n\\item finite-depth missing quotient \\(C_k(a)/p^rC_k(a)\\);\n\\item completed essential image \\(M_{k,p}(a)\\subseteq L_{k,p}(a)\\);\n\\item integral effectivity obstruction \\(M_{k,p}(a)/M_k(a)\\);\n\\item actual-section obstruction inside the integral lattice.\n\\end{enumerate}\n\\end{theorem}","mantic sequence.  For every prime \\(p\\), reduction defines a directional bridge\n\\[\n\\mathcal I_k(a)\n\\longmapsto\n\\left(\n\\overline M_{k,p^r}(a),\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n\\right)_{r\\geq1}\n\\]\nwhose exact information-loss passport is:\n\\begin{enumerate}\n\\item finite-depth ghost kernel \\(C_k(a)[p^r]\\);\n\\item finite-depth missing quotient \\(C_k(a)/p^rC_k(a)\\);\n\\item completed essential image \\(M_{k,p}(a)\\subseteq L_{k,p}(a)\\);\n\\item integral effectivity obstruction \\(M_{k,p}(a)/M_k(a)\\);\n\\item actual-section obstruction inside the integral lattice.\n\\end{enumerate}\n\\end{theorem}","idge\n\\[\n\\mathcal I_k(a)\n\\longmapsto\n\\left(\n\\overline M_{k,p^r}(a),\n\\mathcal K_{k,p^r}^{\\mathrm{sem}}(a)\n\\right)_{r\\geq1}\n\\]\nwhose exact information-loss passport is:\n\\begin{enumerate}\n\\item finite-depth ghost kernel \\(C_k(a)[p^r]\\);\n\\item finite-depth missing quotient \\(C_k(a)/p^rC_k(a)\\);\n\\item completed essential image \\(M_{k,p}(a)\\subseteq L_{k,p}(a)\\);\n\\item integral effectivity obstruction \\(M_{k,p}(a)/M_k(a)\\);\n\\item actual-section obstruction inside the integral lattice.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nEvery component of \\(\\mathcal I_k(a)\\) is defined from the semantic sequence and its digit sections.  A compatible morphism carries actual sections to actual sections and integer-valued rational combinations to integer-valued rational combinations, so it induces maps on \\(M_k(a)\\), \\(L_k(a)\\), and the quotient.  The local reduction statements are \\cref{thm:block06-tor-sequence,thm:block06-inverse-limits,def:block06-effectivity}.\n\\end{proof}","\\begin{resultbox}\n\\textbf{Section result.}\nFor every integer-valued fixed-radix sequence that is regular over \\(\\mathbb Q\\), the actual sections generate a canonical finite free lattice \\(M_k(a)\\), and the integer-valued part of the rational section space is a canonical finite free saturation \\(L_k(a)\\).  Their Smith--Fitting quotient controls every modular reduction exactly.  At each prime power, the coefficient sequence has a finite minimal semantic section automaton.  The compatible tower reconstructs the \\(p\\)-adic raw lattice, while finite ghost coordinates vanish in the inverse limit and the missing saturated quotient survives as the \\(p\\)-primary defect.  \\(p\\)-adic, profinite, integral-lattice, and actual-section effectivity are distinct and are separated by explicit obstruction quotients.\n\\end{resultb","\\begin{resultbox}\n\\textbf{Section result.}\nFor every integer-valued fixed-radix sequence that is regular over \\(\\mathbb Q\\), the actual sections generate a canonical finite free lattice \\(M_k(a)\\), and the integer-valued part of the rational section space is a canonical finite free saturation \\(L_k(a)\\).  Their Smith--Fitting quotient controls every modular reduction exactly.  At each prime power, the coefficient sequence has a finite minimal semantic section automaton.  The compatible tower reconstructs the \\(p\\)-adic raw lattice, while finite ghost coordinates vanish in the inverse limit and the missing saturated quotient survives as the \\(p\\)-primary defect.  \\(p\\)-adic, profinite, integral-lattice, and actual-section effectivity are distinct and are separated by explicit obstruction quotients.\n\\end{resultbox}","hbb Q\\), the actual sections generate a canonical finite free lattice \\(M_k(a)\\), and the integer-valued part of the rational section space is a canonical finite free saturation \\(L_k(a)\\).  Their Smith--Fitting quotient controls every modular reduction exactly.  At each prime power, the coefficient sequence has a finite minimal semantic section automaton.  The compatible tower reconstructs the \\(p\\)-adic raw lattice, while finite ghost coordinates vanish in the inverse limit and the missing saturated quotient survives as the \\(p\\)-primary defect.  \\(p\\)-adic, profinite, integral-lattice, and actual-section effectivity are distinct and are separated by explicit obstruction quotients.\n\\end{resultbox}","space is a canonical finite free saturation \\(L_k(a)\\).  Their Smith--Fitting quotient controls every modular reduction exactly.  At each prime power, the coefficient sequence has a finite minimal semantic section automaton.  The compatible tower reconstructs the \\(p\\)-adic raw lattice, while finite ghost coordinates vanish in the inverse limit and the missing saturated quotient survives as the \\(p\\)-primary defect.  \\(p\\)-adic, profinite, integral-lattice, and actual-section effectivity are distinct and are separated by explicit obstruction quotients.\n\\end{resultbox}","nite free saturation \\(L_k(a)\\).  Their Smith--Fitting quotient controls every modular reduction exactly.  At each prime power, the coefficient sequence has a finite minimal semantic section automaton.  The compatible tower reconstructs the \\(p\\)-adic raw lattice, while finite ghost coordinates vanish in the inverse limit and the missing saturated quotient survives as the \\(p\\)-primary defect.  \\(p\\)-adic, profinite, integral-lattice, and actual-section effectivity are distinct and are separated by explicit obstruction quotients.\n\\end{resultbox}","\\begin{definition}[Coefficientwise filtered product]\nLet $R$ be a commutative ring.  A filtered factor family is a sequence\n\\[\nF_m(z)\\in 1+z^{w(m)}R[[z]],\n\\qquad\nw(m)\\longrightarrow\\infty.\n\\]\nIts coefficientwise product is\n\\[\n\\prod_{m\\geq1}F_m(z),\n\\]\nwhere the coefficient of $z^N$ is computed from the finitely many factors satisfying $w(m)\\leq N$.\n\\end{definition}","\\begin{proposition}[Filtered product completion]\nEvery filtered factor family has a unique coefficientwise product in $1+zR[[z]]$.  For every $N$, the truncation modulo $z^{N+1}$ is equal to the product of the finitely many factors with $w(m)\\leq N$.\n\\end{proposition}","\\begin{proof}\nFix $N$.  Every factor with $w(m)>N$ is congruent to $1$ modulo $z^{N+1}$, so only finitely many factors can affect the coefficient prefix through degree $N$.  These finite products are compatible as $N$ increases.  Their inverse limit in the $z$-adic filtration is the required series.  Uniqueness follows because every coefficient is fixed at a finite stage.\n\\end{proof}","Let\n\\[\n\\mathcal U_{\\mathbb Z}=1+z\\mathbb Z[[z]]\n\\]\nbe the multiplicative group of integral unit series, and let\n\\[\n\\mathcal E_{\\mathbb Z}=\\mathbb Z^{\\mathbb N_{\\geq1}}\n\\]\nbe the additive group of integer exponent sequences.","\\begin{definition}[Euler-coordinate map]\nFor $b=(b_m)_{m\\geq1}\\in\\mathcal E_{\\mathbb Z}$, define\n\\[\n\\mathcal E(b)(z)\n=\n\\prod_{m\\geq1}(1-z^m)^{-b_m}.\n\\]\nThe product is coefficientwise filtered because the $m$th factor is congruent to $1$ modulo $z^m$.\n\\end{definition}","\\begin{theorem}[Integral Euler-coordinate classification]\n\\label{thm:block07-euler-classification}\nThe map\n\\[\n\\mathcal E:\n(\\mathcal E_{\\mathbb Z},+)\n\\longrightarrow\n(\\mathcal U_{\\mathbb Z},\\cdot)\n\\]\nis a group isomorphism.  Consequently, every $A(z)\\in1+z\\mathbb Z[[z]]$ has a unique representation\n\\[\n\\boxed{\nA(z)=\\prod_{m\\geq1}(1-z^m)^{-b_m},\n\\qquad b_m\\in\\mathbb Z.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFiltered multiplication gives\n\\[\n\\mathcal E(b+b')=\\mathcal E(b)\\mathcal E(b').\n\\]\nTo prove surjectivity and uniqueness, construct the exponents recursively.  Suppose $b_1,\\ldots,b_{n-1}$ have been chosen so that\n\\[\nA(z)\\prod_{m<n}(1-z^m)^{b_m}\n\\equiv 1+r_nz^n\\pmod{z^{n+1}}.\n\\]\nBecause\n\\[\n(1-z^n)^{-r_n}\\equiv1+r_nz^n\\pmod{z^{n+1}},\n\\]\nthe unique choice $b_n=r_n$ cancels the degree-$n$ residual.  Every $r_n$ is integral because all preceding operations occur in $\\mathbb Z[[z]]$.  Induction produces one exponent sequence with the required product.  If two exponent sequences differed first at $n$, their quotient would have nonzero coefficient $(b_n-b_n')z^n$ modulo $z^{n+1}$, contradicting equality.  Hence the representation is unique.\n\\end{proof}","red multiplication gives\n\\[\n\\mathcal E(b+b')=\\mathcal E(b)\\mathcal E(b').\n\\]\nTo prove surjectivity and uniqueness, construct the exponents recursively.  Suppose $b_1,\\ldots,b_{n-1}$ have been chosen so that\n\\[\nA(z)\\prod_{m<n}(1-z^m)^{b_m}\n\\equiv 1+r_nz^n\\pmod{z^{n+1}}.\n\\]\nBecause\n\\[\n(1-z^n)^{-r_n}\\equiv1+r_nz^n\\pmod{z^{n+1}},\n\\]\nthe unique choice $b_n=r_n$ cancels the degree-$n$ residual.  Every $r_n$ is integral because all preceding operations occur in $\\mathbb Z[[z]]$.  Induction produces one exponent sequence with the required product.  If two exponent sequences differed first at $n$, their quotient would have nonzero coefficient $(b_n-b_n')z^n$ modulo $z^{n+1}$, contradicting equality.  Hence the representation is unique.\n\\end{proof}","athcal E(b').\n\\]\nTo prove surjectivity and uniqueness, construct the exponents recursively.  Suppose $b_1,\\ldots,b_{n-1}$ have been chosen so that\n\\[\nA(z)\\prod_{m<n}(1-z^m)^{b_m}\n\\equiv 1+r_nz^n\\pmod{z^{n+1}}.\n\\]\nBecause\n\\[\n(1-z^n)^{-r_n}\\equiv1+r_nz^n\\pmod{z^{n+1}},\n\\]\nthe unique choice $b_n=r_n$ cancels the degree-$n$ residual.  Every $r_n$ is integral because all preceding operations occur in $\\mathbb Z[[z]]$.  Induction produces one exponent sequence with the required product.  If two exponent sequences differed first at $n$, their quotient would have nonzero coefficient $(b_n-b_n')z^n$ modulo $z^{n+1}$, contradicting equality.  Hence the representation is unique.\n\\end{proof}","chosen so that\n\\[\nA(z)\\prod_{m<n}(1-z^m)^{b_m}\n\\equiv 1+r_nz^n\\pmod{z^{n+1}}.\n\\]\nBecause\n\\[\n(1-z^n)^{-r_n}\\equiv1+r_nz^n\\pmod{z^{n+1}},\n\\]\nthe unique choice $b_n=r_n$ cancels the degree-$n$ residual.  Every $r_n$ is integral because all preceding operations occur in $\\mathbb Z[[z]]$.  Induction produces one exponent sequence with the required product.  If two exponent sequences differed first at $n$, their quotient would have nonzero coefficient $(b_n-b_n')z^n$ modulo $z^{n+1}$, contradicting equality.  Hence the representation is unique.\n\\end{proof}","iv 1+r_nz^n\\pmod{z^{n+1}}.\n\\]\nBecause\n\\[\n(1-z^n)^{-r_n}\\equiv1+r_nz^n\\pmod{z^{n+1}},\n\\]\nthe unique choice $b_n=r_n$ cancels the degree-$n$ residual.  Every $r_n$ is integral because all preceding operations occur in $\\mathbb Z[[z]]$.  Induction produces one exponent sequence with the required product.  If two exponent sequences differed first at $n$, their quotient would have nonzero coefficient $(b_n-b_n')z^n$ modulo $z^{n+1}$, contradicting equality.  Hence the representation is unique.\n\\end{proof}","}}.\n\\]\nBecause\n\\[\n(1-z^n)^{-r_n}\\equiv1+r_nz^n\\pmod{z^{n+1}},\n\\]\nthe unique choice $b_n=r_n$ cancels the degree-$n$ residual.  Every $r_n$ is integral because all preceding operations occur in $\\mathbb Z[[z]]$.  Induction produces one exponent sequence with the required product.  If two exponent sequences differed first at $n$, their quotient would have nonzero coefficient $(b_n-b_n')z^n$ modulo $z^{n+1}$, contradicting equality.  Hence the representation is unique.\n\\end{proof}","\\begin{corollary}[Finite-prefix triangularity]\n\\label{cor:block07-euler-prefix}\nFor each $N$, the coefficient prefix\n\\[\n(a_1,\\ldots,a_N)\n\\quad\\text{of}\\quad\nA(z)=1+\\sum_{n\\geq1}a_nz^n\n\\]\nand the exponent prefix $(b_1,\\ldots,b_N)$ determine one another by integral triangular recursions.  No exponent $b_m$ with $m>N$ affects the coefficient prefix through degree $N$.\n\\end{corollary}","\\begin{definition}[Euler ghost coordinates]\nFor $b\\in\\mathcal E_{\\mathbb Z}$, define\n\\[\nc_n(b)=\\sum_{d\\mid n}d\\,b_d,\n\\qquad n\\geq1.\n\\]\nLet $\\mathcal G_{\\mathbb Z}$ be the set of integer sequences $c=(c_n)$ satisfying\n\\[\n\\frac1n\\sum_{d\\mid n}\\mu\\!\\left(\\frac nd\\right)c_d\\in\\mathbb Z\n\\qquad(n\\geq1).\n\\]\n\\end{definition}","\\begin{theorem}[Ghost-coordinate equivalence]\n\\label{thm:block07-ghost-equivalence}\nThe divisor ghost map\n\\[\n\\mathcal W:\n\\mathcal E_{\\mathbb Z}\n\\longrightarrow\n\\mathcal G_{\\mathbb Z},\n\\qquad\nb\\longmapsto c(b),\n\\]\nis an additive group isomorphism.  Its inverse is\n\\[\n\\boxed{\nb_n\n=\n\\frac1n\\sum_{d\\mid n}\n\\mu\\!\\left(\\frac nd\\right)c_d.\n}\n\\]\nIf $A(z)=\\mathcal E(b)(z)$, then\n\\[\n\\boxed{\n\\frac{zA'(z)}{A(z)}\n=\n\\sum_{n\\geq1}c_nz^n.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFormal logarithmic differentiation gives\n\\[\n\\frac{zA'}A\n=\n\\sum_{m\\geq1}b_m\\frac{mz^m}{1-z^m}\n=\n\\sum_{m\\geq1}\\sum_{r\\geq1}m b_m z^{mr}.\n\\]\nThe coefficient at $z^n$ is $\\sum_{m\\mid n}m b_m=c_n$.  M\\\"obius inversion gives the displayed inverse formula.  The definition of $\\mathcal G_{\\mathbb Z}$ is exactly the integrality condition for that inverse.\n\\end{proof}","\\begin{theorem}[Exact coefficient recurrence]\n\\label{thm:block07-euler-recurrence}\nLet\n\\[\nA(z)=1+\\sum_{n\\geq1}a_nz^n\n\\]\nand let $c_n$ be its Euler ghost coordinates.  Then\n\\[\n\\boxed{\nna_n\n=\n\\sum_{j=1}^{n}c_j a_{n-j}\n\\qquad(n\\geq1),\n}\n\\]\nwith $a_0=1$.  Conversely, any $c\\in\\mathcal G_{\\mathbb Z}$ and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.\n\\end{theorem}","\\begin{proof}\nThe logarithmic-derivative identity is equivalent to\n\\[\nzA'(z)=\\left(\\sum_{j\\geq1}c_jz^j\\right)A(z).\n\\]\nComparing coefficients gives the recurrence.  Conversely, the recurrence uniquely determines the coefficient sequence over $\\mathbb Q$.  Since $c\\in\\mathcal G_{\\mathbb Z}$ reconstructs an integral exponent sequence, the resulting series is the Euler product from \\cref{thm:block07-euler-classification} and therefore has integral coefficients.\n\\end{proof}","\\begin{example}[Partition and power families]\nThe following fixed-basis exponent functions give familiar filtered products:\n\\[\n\\begin{array}{c|c|c}\n b_n & A_b(z) & c_n\\\\\n\\hline\n1 & \\displaystyle\\prod_{n\\geq1}(1-z^n)^{-1} & \\sigma_1(n)\\\\[4pt]\nn & \\displaystyle\\prod_{n\\geq1}(1-z^n)^{-n} & \\sigma_2(n)\\\\[4pt]\nn^r & \\displaystyle\\prod_{n\\geq1}(1-z^n)^{-n^r} & \\sigma_{r+1}(n).\n\\end{array}\n\\]\nThe first row is the ordinary partition series, and the second is the plane-partition series.  More generally, the power family converts the exponent rule $n^r$ into the divisor-sum ghost $\\sigma_{r+1}$.\n\\end{example}","[4pt]\nn^r & \\displaystyle\\prod_{n\\geq1}(1-z^n)^{-n^r} & \\sigma_{r+1}(n).\n\\end{array}\n\\]\nThe first row is the ordinary partition series, and the second is the plane-partition series.  More generally, the power family converts the exponent rule $n^r$ into the divisor-sum ghost $\\sigma_{r+1}$.\n\\end{example}","\\begin{example}[Distinct parts and normalization]\nThe product\n\\[\n\\prod_{m\\geq1}(1+z^m)\n=\n\\prod_{m\\geq1}\\frac{1-z^{2m}}{1-z^m}\n\\]\nhas normalized Euler exponent\n\\[\nb_n=\n\\begin{cases}\n1,&n\\text{ odd},","\\begin{example}[Distinct parts and normalization]\nThe product\n\\[\n\\prod_{m\\geq1}(1+z^m)\n=\n\\prod_{m\\geq1}\\frac{1-z^{2m}}{1-z^m}\n\\]\nhas normalized Euler exponent\n\\[\nb_n=\n\\begin{cases}\n1,&n\\text{ odd},\\\\\n0,&n\\text{ even}.\n\\end{cases}\n\\]\nThus the distinct-part and odd-part products become the same object after normalization in the fixed Euler basis.  This is an exact equality inside the declared product grammar, not a claim that arbitrary product grammars possess one preferred factorization.\n\\end{example}","For arithmetic functions $f,g:\\mathbb N_{\\geq1}\\to\\mathbb Z$, write\n\\[\n(f\\mathbin{\\ast_D}g)(n)\n=\n\\sum_{d\\mid n}f(d)g(n/d)\n\\]\nfor Dirichlet convolution.  Its identity is $\\varepsilon(1)=1$ and $\\varepsilon(n)=0$ for $n>1$.","\\begin{definition}[Integral multiplicative category]\nLet $\\mathbf{Mult}_{\\mathbb Z}$ be the discrete category whose objects are multiplicative arithmetic functions $f:\\mathbb N_{\\geq1}\\to\\mathbb Z$ satisfying\n\\[\nf(1)=1,\n\\qquad\nf(mn)=f(m)f(n)\n\\quad\\text{when }(m,n)=1.\n\\]\nThe monoidal operation is Dirichlet convolution.\n\\end{definition}","\\begin{proposition}[Dirichlet inverse and multiplicative closure]\nIf $f(1)=1$, then $f$ has a unique integral Dirichlet inverse.  If $f$ and $g$ are multiplicative, then $f\\mathbin{\\ast_D}g$ and $f^{-1_D}$ are multiplicative.\n\\end{proposition}","\\begin{proof}\nThe coefficient equation at $n$ is triangular:\n\\[\ng(1)=1,\n\\qquad\ng(n)=-\\sum_{\\substack{d\\mid n","\\begin{proof}\nThe coefficient equation at $n$ is triangular:\n\\[\ng(1)=1,\n\\qquad\ng(n)=-\\sum_{\\substack{d\\mid n\\\\d<n}}f(n/d)g(d).\n\\]\nHence the inverse exists integrally and uniquely.  The standard coprime-divisor factorization of a divisor of $mn$ proves closure of multiplicative functions under convolution; uniqueness of the inverse then proves multiplicativity of $f^{-1_D}$.\n\\end{proof}","d<n}}f(n/d)g(d).\n\\]\nHence the inverse exists integrally and uniquely.  The standard coprime-divisor factorization of a divisor of $mn$ proves closure of multiplicative functions under convolution; uniqueness of the inverse then proves multiplicativity of $f^{-1_D}$.\n\\end{proof}","\\begin{definition}[Prime-power seed series]\nFor a multiplicative function $f$ and a prime $p$, define\n\\[\nF_{f,p}(T)\n=\n\\sum_{e\\geq0}f(p^e)T^e\n\\in1+T\\mathbb Z[[T]].\n\\]\n\\end{definition}","\\begin{theorem}[Prime-power descent equivalence]\n\\label{thm:block07-prime-power-descent}\nThe assignment\n\\[\nf\\longmapsto\\bigl(F_{f,p}\\bigr)_p\n\\]\nis a group isomorphism\n\\[\n\\boxed{\n(\\mathbf{Mult}_{\\mathbb Z},\\mathbin{\\ast_D})\n\\simeq\n\\prod_{p\\text{ prime}}\n(1+T\\mathbb Z[[T]],\\cdot).\n}\n\\]\nThe inverse reconstructs\n\\[\n\\boxed{\nf(n)=\\prod_{p^e\\parallel n}f(p^e).}\n\\]\nUnder this correspondence,\n\\[\nF_{f\\ast_D g,p}(T)\n=\nF_{f,p}(T)F_{g,p}(T).\n\\]\n\\end{theorem}","\\begin{proof}\nUnique factorization writes every positive integer as a finite product of pairwise coprime prime powers, so multiplicativity gives the reconstruction formula.  Conversely, an arbitrary family of local unit series defines a multiplicative function by the same finite product.  For $n=p^e$, Dirichlet convolution becomes\n\\[\n(f\\mathbin{\\ast_D}g)(p^e)\n=\n\\sum_{r=0}^{e}f(p^r)g(p^{e-r}),\n\\]\nwhich is exactly the coefficient of $T^e$ in the local Cauchy product.\n\\end{proof}","rites every positive integer as a finite product of pairwise coprime prime powers, so multiplicativity gives the reconstruction formula.  Conversely, an arbitrary family of local unit series defines a multiplicative function by the same finite product.  For $n=p^e$, Dirichlet convolution becomes\n\\[\n(f\\mathbin{\\ast_D}g)(p^e)\n=\n\\sum_{r=0}^{e}f(p^r)g(p^{e-r}),\n\\]\nwhich is exactly the coefficient of $T^e$ in the local Cauchy product.\n\\end{proof}","Apply the Euler-coordinate theorem independently at each prime:\n\\[\nF_{f,p}(T)\n=\n\\prod_{m\\geq1}(1-T^m)^{-\\beta_{p,m}(f)}.\n\\]\nDefine local ghosts\n\\[\n\\gamma_{p,e}(f)\n=\n\\sum_{d\\mid e}d\\,\\beta_{p,d}(f).\n\\]","\\begin{theorem}[Complete prime-Euler coordinates]\n\\label{thm:block07-prime-euler}\nFor integral multiplicative functions, the following normalized data are equivalent and complete:\n\\[\n f,\n\\qquad\n\\bigl(F_{f,p}\\bigr)_p,\n\\qquad\n\\bigl(\\beta_{p,m}(f)\\bigr)_{p,m},\n\\qquad\n\\bigl(\\gamma_{p,e}(f)\\bigr)_{p,e}.\n\\]\nDirichlet convolution is linearized by the last two charts:\n\\[\n\\beta_{p,m}(f\\mathbin{\\ast_D}g)\n=\n\\beta_{p,m}(f)+\\beta_{p,m}(g),\n\\]\n\\[\n\\gamma_{p,e}(f\\mathbin{\\ast_D}g)\n=\n\\gamma_{p,e}(f)+\\gamma_{p,e}(g).\n\\]\nThe ghost chart consists exactly of the prime-indexed sequences satisfying the local M\\\"obius divisibility conditions\n\\[\n\\frac1e\\sum_{d\\mid e}\\mu\\!\\left(\\frac ed\\right)\\gamma_{p,d}\\in\\mathbb Z.\n\\]\n\\end{theorem}","coordinates]\n\\label{thm:block07-prime-euler}\nFor integral multiplicative functions, the following normalized data are equivalent and complete:\n\\[\n f,\n\\qquad\n\\bigl(F_{f,p}\\bigr)_p,\n\\qquad\n\\bigl(\\beta_{p,m}(f)\\bigr)_{p,m},\n\\qquad\n\\bigl(\\gamma_{p,e}(f)\\bigr)_{p,e}.\n\\]\nDirichlet convolution is linearized by the last two charts:\n\\[\n\\beta_{p,m}(f\\mathbin{\\ast_D}g)\n=\n\\beta_{p,m}(f)+\\beta_{p,m}(g),\n\\]\n\\[\n\\gamma_{p,e}(f\\mathbin{\\ast_D}g)\n=\n\\gamma_{p,e}(f)+\\gamma_{p,e}(g).\n\\]\nThe ghost chart consists exactly of the prime-indexed sequences satisfying the local M\\\"obius divisibility conditions\n\\[\n\\frac1e\\sum_{d\\mid e}\\mu\\!\\left(\\frac ed\\right)\\gamma_{p,d}\\in\\mathbb Z.\n\\]\n\\end{theorem}","l(\\beta_{p,m}(f)\\bigr)_{p,m},\n\\qquad\n\\bigl(\\gamma_{p,e}(f)\\bigr)_{p,e}.\n\\]\nDirichlet convolution is linearized by the last two charts:\n\\[\n\\beta_{p,m}(f\\mathbin{\\ast_D}g)\n=\n\\beta_{p,m}(f)+\\beta_{p,m}(g),\n\\]\n\\[\n\\gamma_{p,e}(f\\mathbin{\\ast_D}g)\n=\n\\gamma_{p,e}(f)+\\gamma_{p,e}(g).\n\\]\nThe ghost chart consists exactly of the prime-indexed sequences satisfying the local M\\\"obius divisibility conditions\n\\[\n\\frac1e\\sum_{d\\mid e}\\mu\\!\\left(\\frac ed\\right)\\gamma_{p,d}\\in\\mathbb Z.\n\\]\n\\end{theorem}","\\begin{proof}\nTheorem~\\ref{thm:block07-euler-classification} and Theorem~\\ref{thm:block07-ghost-equivalence} apply to each local unit series $F_{f,p}$.  The prime-power descent theorem then reconstructs the global function.  Local multiplication corresponds to Dirichlet convolution, while Euler exponents and ghosts add under multiplication.\n\\end{proof}","\\begin{theorem}[Valuation logarithmic derivative]\n\\label{thm:block07-valuation-derivative}\nFix a prime $p$.  Let $\\Gamma_{f,p}$ be the arithmetic function supported on powers of $p$ with\n\\[\n\\Gamma_{f,p}(1)=0,\n\\qquad\n\\Gamma_{f,p}(p^e)=\\gamma_{p,e}(f)\n\\quad(e\\geq1).\n\\]\nThen\n\\[\n\\boxed{\n(v_p\\cdot f)(n)\n=\n(\\Gamma_{f,p}\\mathbin{\\ast_D}f)(n)\n\\qquad(n\\geq1).\n}\n\\]\nEquivalently,\n\\[\n\\frac{T F_{f,p}'(T)}{F_{f,p}(T)}\n=\n\\sum_{e\\geq1}\\gamma_{p,e}(f)T^e.\n\\]\n\\end{theorem}","\\begin{proof}\nThe local logarithmic-derivative recurrence gives\n\\[\ne f(p^e)\n=\n\\sum_{j=1}^{e}\\gamma_{p,j}(f)f(p^{e-j}).\n\\]\nWrite $n=p^e u$ with $(u,p)=1$ and multiply by $f(u)$.  Since $\\Gamma_{f,p}$ is supported on $p$-powers, the resulting sum is exactly the Dirichlet convolution at $n$.\n\\end{proof}","\\begin{proposition}[Multiplicative residue-packet law]\nFor $N\\geq1$, let $P_{N,r}f$ be the arithmetic function equal to $f(n)$ when $n\\equiv r\\pmod N$ and zero otherwise.  Then\n\\[\n\\boxed{\nP_{N,k}(f\\mathbin{\\ast_D}g)\n=\n\\sum_{rs\\equiv k\\, (N)}\n(P_{N,r}f)\\mathbin{\\ast_D}(P_{N,s}g).\n}\n\\]\nThus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid $\\mathbb Z/N\\mathbb Z$, including its zero divisors.\n\\end{proposition}","\\begin{proof}\nIn each divisor pair $n=d(n/d)$, the residue of $n$ modulo $N$ is the product of the two divisor residues.  Group the divisor sum by those two residues.\n\\end{proof}","The Euler ghost transform can itself be written in the Dirichlet-convolution language.  Let $\\mathbf{1}(n)=1$, let $\\operatorname{id}(n)=n$, and let pointwise multiplication be denoted by a centered dot.","\\begin{definition}[Divisor--ghost transform]\nFor an arithmetic exponent function $b:\\mathbb N_{\\geq1}\\to\\mathbb Z$, define\n\\[\n\\mathfrak G(b)\n=\n\\mathbf{1}\\mathbin{\\ast_D}(\\operatorname{id}\\cdot b),\n\\]\nso that\n\\[\n\\mathfrak G(b)(n)\n=\n\\sum_{d\\mid n}d\\,b(d).\n\\]\n\\end{definition}","\\begin{theorem}[Exact divisor--ghost bridge]\n\\label{thm:block07-divisor-ghost}\nThe map $b\\mapsto c=\\mathfrak G(b)$ is injective, with inverse\n\\[\n\\boxed{\nb(n)\n=\n\\frac1n\n(\\mu\\mathbin{\\ast_D}c)(n).\n}\n\\]\nIts integral image consists exactly of the arithmetic functions $c$ satisfying\n\\[\nn\\mid(\\mu\\mathbin{\\ast_D}c)(n)\n\\qquad(n\\geq1).\n\\]\nFor the Euler product\n\\[\nA_b(z)=\\prod_{n\\geq1}(1-z^n)^{-b(n)},\n\\]\none has\n\\[\n\\frac{zA_b'(z)}{A_b(z)}\n=\n\\sum_{n\\geq1}c(n)z^n.\n\\]\n\\end{theorem}","heorem}[Exact divisor--ghost bridge]\n\\label{thm:block07-divisor-ghost}\nThe map $b\\mapsto c=\\mathfrak G(b)$ is injective, with inverse\n\\[\n\\boxed{\nb(n)\n=\n\\frac1n\n(\\mu\\mathbin{\\ast_D}c)(n).\n}\n\\]\nIts integral image consists exactly of the arithmetic functions $c$ satisfying\n\\[\nn\\mid(\\mu\\mathbin{\\ast_D}c)(n)\n\\qquad(n\\geq1).\n\\]\nFor the Euler product\n\\[\nA_b(z)=\\prod_{n\\geq1}(1-z^n)^{-b(n)},\n\\]\none has\n\\[\n\\frac{zA_b'(z)}{A_b(z)}\n=\n\\sum_{n\\geq1}c(n)z^n.\n\\]\n\\end{theorem}","\\begin{proof}\nThe first identity is the convolution form of the divisor sum.  Since $\\mu\\mathbin{\\ast_D}\\mathbf{1}=\\varepsilon$,\n\\[\n\\mu\\mathbin{\\ast_D}c\n=\n\\operatorname{id}\\cdot b.\n\\]\nDivision by $n$ gives the inverse and its exact integrality condition.  The logarithmic-derivative identity is Theorem~\\ref{thm:block07-ghost-equivalence} with the exponent sequence regarded as an arithmetic function.\n\\end{proof}","\\begin{theorem}[Multiplicativity is preserved and reflected]\n\\label{thm:block07-multiplicative-ghost}\nAssume $b(1)=1$ and $c=\\mathfrak G(b)$.  Then\n\\[\n\\boxed{\nb\\text{ is multiplicative}\n\\quad\\Longleftrightarrow\\quad\nc\\text{ is multiplicative}.}\n\\]\nOn this locus, both $b$ and $c$ are reconstructed from prime-power seeds.  For every prime $p$ and $e\\geq1$,\n\\[\n\\boxed{\nc(p^e)=\\sum_{j=0}^{e}p^j b(p^j),}\n\\]\n\\[\n\\boxed{\nb(p^e)=\\frac{c(p^e)-c(p^{e-1})}{p^e}.}\n\\]\n\\end{theorem}","theorem}[Multiplicativity is preserved and reflected]\n\\label{thm:block07-multiplicative-ghost}\nAssume $b(1)=1$ and $c=\\mathfrak G(b)$.  Then\n\\[\n\\boxed{\nb\\text{ is multiplicative}\n\\quad\\Longleftrightarrow\\quad\nc\\text{ is multiplicative}.}\n\\]\nOn this locus, both $b$ and $c$ are reconstructed from prime-power seeds.  For every prime $p$ and $e\\geq1$,\n\\[\n\\boxed{\nc(p^e)=\\sum_{j=0}^{e}p^j b(p^j),}\n\\]\n\\[\n\\boxed{\nb(p^e)=\\frac{c(p^e)-c(p^{e-1})}{p^e}.}\n\\]\n\\end{theorem}","\\begin{proof}\nIf $b$ is multiplicative, then $\\operatorname{id}\\cdot b$ is multiplicative, and Dirichlet convolution with $\\mathbf{1}$ preserves multiplicativity.  Conversely, if $c$ is multiplicative, then $\\mu\\mathbin{\\ast_D}c$ is multiplicative.  Dividing pointwise by the completely multiplicative function $\\operatorname{id}$ preserves multiplicativity over $\\mathbb Q$; the integral image condition ensures that the resulting values are integers.  The prime-power formulas follow by listing the divisors of $p^e$ and subtracting consecutive sums.\n\\end{proof}","on with $\\mathbf{1}$ preserves multiplicativity.  Conversely, if $c$ is multiplicative, then $\\mu\\mathbin{\\ast_D}c$ is multiplicative.  Dividing pointwise by the completely multiplicative function $\\operatorname{id}$ preserves multiplicativity over $\\mathbb Q$; the integral image condition ensures that the resulting values are integers.  The prime-power formulas follow by listing the divisors of $p^e$ and subtracting consecutive sums.\n\\end{proof}","\\begin{definition}[Multiplicative Euler-overlap category]\nLet $\\mathbf{EulMult}_{\\mathbb Z}$ be the discrete category of multiplicative exponent functions\n\\[\nb:\\mathbb N_{\\geq1}\\to\\mathbb Z,\n\\qquad b(1)=1.\n\\]\nAssociate to $b$ the four normalized charts\n\\[\n\\mathsf B(b)=b,\n\\qquad\n\\mathsf A(b)=A_b(z),\n\\qquad\n\\mathsf C(b)=\\mathfrak G(b),\n\\qquad\n\\mathsf P(b)=\\bigl(b(p^e),c(p^e)\\bigr)_{p,e}.\n\\]\n\\end{definition}","\\begin{theorem}[Effective descent on the divisor--ghost overlap]\n\\label{thm:block07-product-multiplicative-descent}\nThe exponent, fixed Euler-product, multiplicative ghost, and prime-power seed charts form a finite normalized atlas\n\\[\n\\mathfrak R_{\\mathrm{DG}}.\n\\]\nEvery bridge is reversible on the declared overlap, every multiple-overlap comparison is the unique identity over the represented exponent function, and\n\\[\n\\boxed{\n\\Desc(\\mathfrak R_{\\mathrm{DG}})\n\\simeq\n\\mathbf{EulMult}_{\\mathbb Z}.\n}\n\\]\nDescent is effective for objects and morphisms.\n\\end{theorem}","ctive descent on the divisor--ghost overlap]\n\\label{thm:block07-product-multiplicative-descent}\nThe exponent, fixed Euler-product, multiplicative ghost, and prime-power seed charts form a finite normalized atlas\n\\[\n\\mathfrak R_{\\mathrm{DG}}.\n\\]\nEvery bridge is reversible on the declared overlap, every multiple-overlap comparison is the unique identity over the represented exponent function, and\n\\[\n\\boxed{\n\\Desc(\\mathfrak R_{\\mathrm{DG}})\n\\simeq\n\\mathbf{EulMult}_{\\mathbb Z}.\n}\n\\]\nDescent is effective for objects and morphisms.\n\\end{theorem}","\\begin{proof}\nThe Euler-product chart is uniquely inverted by Theorem~\\ref{thm:block07-euler-classification}.  The ghost chart is uniquely inverted by Theorem~\\ref{thm:block07-divisor-ghost}.  The prime-power chart is uniquely inverted by multiplicativity and unique factorization.  Hence every normalized fiber over $b$ is a singleton.  All comparison cells and all multiple-overlap coherence laws are therefore forced to be identities.  A compatible local family glues to its common exponent function, and morphism effectivity is automatic because the chart categories are discrete.\n\\end{proof}","\\begin{theorem}[Complete descended prime-power invariant]\n\\label{thm:block07-complete-product-invariant}\nThe prime-power package\n\\[\n\\boxed{\n\\mathcal J_{\\mathrm{DG}}(b)\n=\n\\left(\n b(p^e),\n c(p^e)\n\\right)_{p\\text{ prime},\\ e\\geq0}\n}\n\\]\nsubject to\n\\[\np^e b(p^e)=c(p^e)-c(p^{e-1})\n\\quad(e\\geq1)\n\\]\nis complete on $\\mathbf{EulMult}_{\\mathbb Z}$.  It reconstructs $b$, $c$, the Euler product $A_b$, and the coefficient recurrence of $A_b$.\n\\end{theorem}","scended prime-power invariant]\n\\label{thm:block07-complete-product-invariant}\nThe prime-power package\n\\[\n\\boxed{\n\\mathcal J_{\\mathrm{DG}}(b)\n=\n\\left(\n b(p^e),\n c(p^e)\n\\right)_{p\\text{ prime},\\ e\\geq0}\n}\n\\]\nsubject to\n\\[\np^e b(p^e)=c(p^e)-c(p^{e-1})\n\\quad(e\\geq1)\n\\]\nis complete on $\\mathbf{EulMult}_{\\mathbb Z}$.  It reconstructs $b$, $c$, the Euler product $A_b$, and the coefficient recurrence of $A_b$.\n\\end{theorem}","\\begin{proof}\nPrime-power values reconstruct the multiplicative functions $b$ and $c$.  Theorem~\\ref{thm:block07-euler-classification} reconstructs $A_b$, and Theorem~\\ref{thm:block07-euler-recurrence} reconstructs its coefficient sequence from $c$.  Therefore equal invariant packages imply equality in every chart.\n\\end{proof}","\\begin{example}[Power bridge]\nFor $b(n)=n^r$, which is completely multiplicative, the divisor ghost is\n\\[\nc(n)=\\sum_{d\\mid n}d^{r+1}=\\sigma_{r+1}(n).\n\\]\nAt a prime power,\n\\[\n\\sigma_{r+1}(p^e)\n=\n1+p^{r+1}+\\cdots+p^{e(r+1)}.\n\\]\nThus one simple multiplicative exponent rule simultaneously produces a filtered ordinary product, a multiplicative divisor-sum ghost, and a one-dimensional geometric local rule at every prime.\n\\end{example}","\\begin{proposition}[The multiplicative exponent locus is not a product subgroup]\nThe set of multiplicative Euler exponent functions is not closed under the addition of exponent coordinates.  Consequently, multiplication of the corresponding ordinary Euler products does not define an internal group law on $\\mathbf{EulMult}_{\\mathbb Z}$.\n\\end{proposition}","\\begin{proof}\nThe constant function $b(n)=1$ is multiplicative.  But $b+b$ has value $2$ at $1$ and therefore is not multiplicative under the normalization $f(1)=1$.  Since ordinary Euler-product multiplication adds exponents, the multiplicative exponent locus is not closed under that operation.\n\\end{proof}","\\begin{proposition}[Euler coefficients are not multiplicative semantics]\nThe coefficient sequence of $A_b(z)$ need not be a multiplicative arithmetic function even when $b$ is multiplicative.  For example, $b(n)=1$ gives the partition numbers, and\n\\[\np(6)=11\\neq p(2)p(3)=6.\n\\]\n\\end{proposition}","The Product-multiplicative atlas verifier performs four exact audits.  First, it exhaustively checks the coefficient, exponent, and ghost triangular maps for every exponent prefix in $\\{-2,-1,0,1,2\\}^6$ and every unit coefficient prefix in $\\{-2,-1,0,1,2\\}^5$.  Second, it constructs deterministic prime-power systems, reconstructs global multiplicative functions, and verifies local Euler inversion, local logarithmic derivatives, and Dirichlet convolution.  Third, it checks the divisor--ghost bridge and the power family $b(n)=n^r$.  Fourth, it verifies the named local Euler coordinates printed in \\cref{tab:block07-named-local}.  The recorded 595,964 exact checks have no failures.","t, and ghost triangular maps for every exponent prefix in $\\{-2,-1,0,1,2\\}^6$ and every unit coefficient prefix in $\\{-2,-1,0,1,2\\}^5$.  Second, it constructs deterministic prime-power systems, reconstructs global multiplicative functions, and verifies local Euler inversion, local logarithmic derivatives, and Dirichlet convolution.  Third, it checks the divisor--ghost bridge and the power family $b(n)=n^r$.  Fourth, it verifies the named local Euler coordinates printed in \\cref{tab:block07-named-local}.  The recorded 595,964 exact checks have no failures.","\\begin{enumerate}\n\\item A \\emph{functional-Mahler equation}\n\\[\n F(z)=R(z)F(z^p)\n\\]\nis an identity for a formal coefficient series in the variable \\(z\\).  Cartier operators act on the coefficient index.  the periodic and rational-mask atlases use this channel over finite fields.\n\\item A \\emph{Newton--Mahler expansion}\n\\[\n f(x)=\\sum_{r\\geq0}c_r\\binom xr\n\\]\nis a coordinate expansion for a function of a \\(p\\)-adic input variable \\(x\\).  Affine substitutions \\(x=Mq+s\\) produce polyphase branches of the input domain.\n\\end{enumerate}","n{enumerate}\n\\item A \\emph{functional-Mahler equation}\n\\[\n F(z)=R(z)F(z^p)\n\\]\nis an identity for a formal coefficient series in the variable \\(z\\).  Cartier operators act on the coefficient index.  the periodic and rational-mask atlases use this channel over finite fields.\n\\item A \\emph{Newton--Mahler expansion}\n\\[\n f(x)=\\sum_{r\\geq0}c_r\\binom xr\n\\]\nis a coordinate expansion for a function of a \\(p\\)-adic input variable \\(x\\).  Affine substitutions \\(x=Mq+s\\) produce polyphase branches of the input domain.\n\\end{enumerate}","{functional-Mahler equation}\n\\[\n F(z)=R(z)F(z^p)\n\\]\nis an identity for a formal coefficient series in the variable \\(z\\).  Cartier operators act on the coefficient index.  the periodic and rational-mask atlases use this channel over finite fields.\n\\item A \\emph{Newton--Mahler expansion}\n\\[\n f(x)=\\sum_{r\\geq0}c_r\\binom xr\n\\]\nis a coordinate expansion for a function of a \\(p\\)-adic input variable \\(x\\).  Affine substitutions \\(x=Mq+s\\) produce polyphase branches of the input domain.\n\\end{enumerate}","\\[\n F(z)=R(z)F(z^p)\n\\]\nis an identity for a formal coefficient series in the variable \\(z\\).  Cartier operators act on the coefficient index.  the periodic and rational-mask atlases use this channel over finite fields.\n\\item A \\emph{Newton--Mahler expansion}\n\\[\n f(x)=\\sum_{r\\geq0}c_r\\binom xr\n\\]\nis a coordinate expansion for a function of a \\(p\\)-adic input variable \\(x\\).  Affine substitutions \\(x=Mq+s\\) produce polyphase branches of the input domain.\n\\end{enumerate}","This section treats the second channel.  Fix a prime \\(p\\).  Let\n\\[\n \\mathcal C_p=C(\\mathbb Z_p,\\mathbb Z_p)\n\\]\nwith the uniform norm, and let\n\\[\n c_0(\\mathbb Z_p)\n =\n \\left\\{c=(c_r)_{r\\geq0}:c_r\\in\\mathbb Z_p,\\ c_r\\longrightarrow0\\right\\}\n\\]\nwith the coefficient sup norm.","\\begin{theorem}[Mahler coordinate theorem]\n\\label{thm:block08-mahler-coordinate}\nThe synthesis map\n\\[\n \\mathcal M_p:c_0(\\mathbb Z_p)\\longrightarrow\\mathcal C_p,\n \\qquad\n c\\longmapsto\n \\left(x\\longmapsto\\sum_{r\\geq0}c_r\\binom xr\\right),\n\\]\nis an isometric isomorphism of \\(\\mathbb Z_p\\)-modules.  Its inverse is\n\\[\n \\boxed{c_r=\\Delta^r f(0).}\n\\]\nIn particular, the Newton--Mahler coefficient sequence is a canonical complete coordinate for a continuous \\(\\mathbb Z_p\\)-valued function.\n\\end{theorem}","block08-mahler-coordinate}\nThe synthesis map\n\\[\n \\mathcal M_p:c_0(\\mathbb Z_p)\\longrightarrow\\mathcal C_p,\n \\qquad\n c\\longmapsto\n \\left(x\\longmapsto\\sum_{r\\geq0}c_r\\binom xr\\right),\n\\]\nis an isometric isomorphism of \\(\\mathbb Z_p\\)-modules.  Its inverse is\n\\[\n \\boxed{c_r=\\Delta^r f(0).}\n\\]\nIn particular, the Newton--Mahler coefficient sequence is a canonical complete coordinate for a continuous \\(\\mathbb Z_p\\)-valued function.\n\\end{theorem}","\\begin{proof}\nThis is the classical Mahler coordinate theorem \\cite{Mahler1958}.  Finite Newton interpolation gives the coefficient identity on nonnegative integers.  Continuity implies that the forward-difference coefficients tend to zero, so the binomial series converges uniformly on \\(\\mathbb Z_p\\).  Conversely, every coefficient sequence tending to zero gives a uniformly convergent continuous function.  Uniqueness follows from the finite-difference formula.\n\\end{proof}","\\begin{definition}[Exact valuation profile and certified class]\n\\label{def:block08-weight-class}\nFor \\(f=\\mathcal M_p(c)\\), define\n\\[\n \\nu_f(r)=v_p(c_r)\\in\\mathbb Z_{\\geq0}\\cup\\{\\infty\\}.\n\\]\nIf \\(w:\\mathbb N\\to\\mathbb Z_{\\geq0}\\) is nondecreasing and tends to infinity, put\n\\[\n \\mathcal M_w\n =\n \\{c\\in c_0(\\mathbb Z_p):v_p(c_r)\\geq w(r)\\},\n \\qquad\n \\mathcal C_w=\\mathcal M_p(\\mathcal M_w).\n\\]\nThe function \\(w\\) is a declared precision certificate.  The class \\(\\mathcal C_w\\) is not asserted to be a characterization of every possible notion of local analyticity.\n\\end{definition}","efinition}[Exact valuation profile and certified class]\n\\label{def:block08-weight-class}\nFor \\(f=\\mathcal M_p(c)\\), define\n\\[\n \\nu_f(r)=v_p(c_r)\\in\\mathbb Z_{\\geq0}\\cup\\{\\infty\\}.\n\\]\nIf \\(w:\\mathbb N\\to\\mathbb Z_{\\geq0}\\) is nondecreasing and tends to infinity, put\n\\[\n \\mathcal M_w\n =\n \\{c\\in c_0(\\mathbb Z_p):v_p(c_r)\\geq w(r)\\},\n \\qquad\n \\mathcal C_w=\\mathcal M_p(\\mathcal M_w).\n\\]\nThe function \\(w\\) is a declared precision certificate.  The class \\(\\mathcal C_w\\) is not asserted to be a characterization of every possible notion of local analyticity.\n\\end{definition}","]\n\\label{def:block08-weight-class}\nFor \\(f=\\mathcal M_p(c)\\), define\n\\[\n \\nu_f(r)=v_p(c_r)\\in\\mathbb Z_{\\geq0}\\cup\\{\\infty\\}.\n\\]\nIf \\(w:\\mathbb N\\to\\mathbb Z_{\\geq0}\\) is nondecreasing and tends to infinity, put\n\\[\n \\mathcal M_w\n =\n \\{c\\in c_0(\\mathbb Z_p):v_p(c_r)\\geq w(r)\\},\n \\qquad\n \\mathcal C_w=\\mathcal M_p(\\mathcal M_w).\n\\]\nThe function \\(w\\) is a declared precision certificate.  The class \\(\\mathcal C_w\\) is not asserted to be a characterization of every possible notion of local analyticity.\n\\end{definition}","For \\(a\\geq1\\) and \\(0\\leq s<p^a\\), define the branch\n\\[\n \\mathcal B_{a,s}f(q)=f(p^a q+s),\n \\qquad q\\in\\mathbb Z_p,\n\\]\nand the complete packet\n\\[\n \\mathcal B_a f=\\bigl(\\mathcal B_{a,s}f\\bigr)_{0\\leq s<p^a}.\n\\]","\\begin{theorem}[Complete clopen packet equivalence]\n\\label{thm:block08-clopen-equivalence}\nThe analysis map\n\\[\n \\mathcal B_a:\\mathcal C_p\n \\longrightarrow\n \\prod_{s=0}^{p^a-1}\\mathcal C_p\n\\]\nis an isometric isomorphism.  Its inverse is the piecewise synthesis\n\\[\n \\boxed{\n (\\mathcal S_a g)(x)\n =g_s\\!\\left(\\frac{x-s}{p^a}\\right)\n \\quad\\text{for }x\\in s+p^a\\mathbb Z_p.}\n\\]\nThus the clopen residue cover of \\(\\mathbb Z_p\\) has effective descent for both objects and equality morphisms.\n\\end{theorem}","heorem}[Complete clopen packet equivalence]\n\\label{thm:block08-clopen-equivalence}\nThe analysis map\n\\[\n \\mathcal B_a:\\mathcal C_p\n \\longrightarrow\n \\prod_{s=0}^{p^a-1}\\mathcal C_p\n\\]\nis an isometric isomorphism.  Its inverse is the piecewise synthesis\n\\[\n \\boxed{\n (\\mathcal S_a g)(x)\n =g_s\\!\\left(\\frac{x-s}{p^a}\\right)\n \\quad\\text{for }x\\in s+p^a\\mathbb Z_p.}\n\\]\nThus the clopen residue cover of \\(\\mathbb Z_p\\) has effective descent for both objects and equality morphisms.\n\\end{theorem}","\\begin{proof}\nThe cosets \\(s+p^a\\mathbb Z_p\\) are pairwise disjoint, clopen, and cover \\(\\mathbb Z_p\\).  Restriction followed by the displayed synthesis is the identity.  Conversely, synthesis followed by restriction recovers every component.  Continuity is checked separately on each clopen component, and the uniform norm is the maximum of the branch norms.\n\\end{proof}","\\begin{theorem}[Strict nested-branch law]\n\\label{thm:block08-nested-branches}\nFor \\(a,b\\geq1\\), \\(0\\leq s<p^a\\), and \\(0\\leq t<p^b\\),\n\\[\n \\boxed{(f_{a,s})_{b,t}=f_{a+b,s+p^a t}.}\n\\]\nAll multiple-overlap comparison cells for the nested \\(p\\)-power branch atlas are identities.  Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.\n\\end{theorem}","\\begin{proof}\nDirect substitution gives\n\\[\n f_{a,s}(p^bq+t)\n =f(p^a(p^bq+t)+s)\n =f(p^{a+b}q+s+p^a t).\n\\]\nThe address is uniquely determined in the interval \\(0\\leq s+p^a t<p^{a+b}\\).\n\\end{proof}","A single branch \\(\\mathcal B_{a,s}\\) is not injective on \\(\\mathcal C_p\\) when \\(a\\geq1\\): any nonzero continuous function supported on another clopen coset lies in its kernel.  The complete packet is therefore the reversible object.  This observation will be contrasted below with the finite-degree characteristic-zero polynomial chart, where one affine branch is invertible.","For \\(M\\geq1\\), \\(s\\geq0\\), and \\(r,k\\geq0\\), define\n\\[\n \\Phi_M(z)=(1+z)^M-1\n\\]\nand\n\\[\n \\boxed{\n \\Theta_{M,s}(r,k)\n =[z^r](1+z)^s\\Phi_M(z)^k.}\n\\]\nEquivalently,\n\\[\n \\boxed{\n \\Theta_{M,s}(r,k)\n =\\sum_{j=0}^{k}(-1)^{k-j}\\binom kj\\binom{Mj+s}{r}.}\n\\]\nThe support is finite:\n\\[\n \\Theta_{M,s}(r,k)=0\n \\quad\\text{unless}\\quad\n k\\leq r\\leq Mk+s.\n\\]","\\begin{theorem}[Affine Newton--Mahler transfer]\n\\label{thm:block08-affine-transfer}\nLet\n\\[\n f(x)=\\sum_{r\\geq0}c_r\\binom xr\n\\]\nbe a convergent Mahler expansion and put \\(g_s(q)=f(Mq+s)\\).  Then\n\\[\n g_s(q)=\\sum_{k\\geq0}d_{s,k}\\binom qk,\n\\]\nwhere\n\\[\n \\boxed{\n d_{s,k}\n =\\sum_{r=k}^{Mk+s}c_r\\Theta_{M,s}(r,k).}\n\\]\nEvery output coefficient depends on a finite source window.\n\\end{theorem}","\\begin{proof}\nExpand\n\\[\n (1+z)^{Mq+s}\n =(1+z)^s\\bigl(1+\\Phi_M(z)\\bigr)^q\n =\\sum_{k\\geq0}\\binom qk(1+z)^s\\Phi_M(z)^k.\n\\]\nTaking \\([z^r]\\) gives the affine expansion of \\(\\binom{Mq+s}{r}\\).  Substitution into the Mahler series gives the formula.  The finite support of \\(\\Theta_{M,s}\\) makes each coefficient sum finite.\n\\end{proof}","Let \\(\\mathcal T_{M,s}\\) denote this coefficient operator.  Conjugating the clopen packet theorem by \\(\\mathcal M_p\\) gives the following normalized coefficient atlas.","\\begin{corollary}[Complete coefficient-packet equivalence]\n\\label{cor:block08-coefficient-packet}\nFor \\(M=p^a\\), the map\n\\[\n \\mathcal T_a:c_0(\\mathbb Z_p)\n \\longrightarrow\n \\prod_{s=0}^{p^a-1}c_0(\\mathbb Z_p),\n \\qquad\n c\\longmapsto(\\mathcal T_{p^a,s}c)_s,\n\\]\nis a topological isomorphism.  It is the coefficient conjugate of complete clopen analysis.\n\\end{corollary}","\\begin{theorem}[Mixed-radix kernel composition]\n\\label{thm:block08-kernel-composition}\nFor \\(M,N\\geq1\\),\n\\[\n \\boxed{\n \\sum_{k\\geq0}\n \\Theta_{M,s}(r,k)\\Theta_{N,t}(k,\\ell)\n =\n \\Theta_{MN,Mt+s}(r,\\ell).}\n\\]\nEquivalently,\n\\[\n \\boxed{\n \\mathcal T_{N,t}\\mathcal T_{M,s}\n =\n \\mathcal T_{MN,Mt+s}.}\n\\]\nWhen \\(\\gcd(M,N)=1\\), the two Chinese-remainder refinement paths to a common address agree exactly.  Their holonomy is the identity.\n\\end{theorem}","\\begin{proof}\nApply the affine substitution \\(x=Mq+s\\), then \\(q=Ny+t\\).  The composite is \\(x=MNy+Mt+s\\).  Uniqueness of Mahler coefficients yields the kernel identity.  The Chinese-remainder diamond is the same equality written along two factorizations of one affine address.\n\\end{proof}","\\begin{resultbox}\n\\textbf{Normalized local atlas.}\nFor every fixed prime \\(p\\) and depth \\(a\\), the following normalized charts are equivalent:\n\\[\n \\boxed{\n \\mathbf{Loc}^{\\mathrm{cts}}_p\n \\simeq\n \\mathbf{Mah}^{c_0}_p\n \\simeq\n \\mathbf{Branch}^{\\mathrm{cts}}_{p,a}\n \\simeq\n \\mathbf{BranchMah}^{c_0}_{p,a}.}\n\\]\nThe first chart contains continuous functions on \\(\\mathbb Z_p\\), the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet.  Every normalized fiber is contractible.  The descent category is therefore equivalent to the discrete semantic category \\(\\mathbf{Loc}^{\\mathrm{cts}}_p\\).\n\\end{resultbox}","cts}}_{p,a}\n \\simeq\n \\mathbf{BranchMah}^{c_0}_{p,a}.}\n\\]\nThe first chart contains continuous functions on \\(\\mathbb Z_p\\), the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet.  Every normalized fiber is contractible.  The descent category is therefore equivalent to the discrete semantic category \\(\\mathbf{Loc}^{\\mathrm{cts}}_p\\).\n\\end{resultbox}","The same integer kernel matrix has different categorical status after changing the coefficient doctrine.  Fix \\(d\\geq0\\) and write\n\\[\n \\mathcal M_d(R)=R^{d+1}\n\\]\nfor the formal truncated Mahler coefficient module.  Let\n\\[\n A_{M,s}^{(d)}\n =\\bigl[\\Theta_{M,s}(r,k)\\bigr]_{0\\leq k,r\\leq d},\n\\]\nwhere the target degree \\(k\\) is the row index.","\\begin{theorem}[Typed affine-channel ladder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly a","\\begin{theorem}[Typed affine-channel ladder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes div","\\begin{theorem}[Typed affine-channel ladder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\it","\\begin{theorem}[Typed affine-channel ladder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the sourc","\\begin{theorem}[Typed affine-channel ladder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standa","\\begin{theorem}[Typed affine-channel ladder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after bas","\\begin{theorem}[Typed affine-channel ladder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","dder]\n\\label{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","{thm:block08-channel-ladder}\nFor every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","every \\(M\\geq1\\), \\(s\\geq0\\), and \\(d\\geq0\\):\n\\begin{enumerate}\n\\item The matrix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","ix \\(A_{M,s}^{(d)}\\) is upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","upper triangular with diagonal\n\\[\n 1,M,M^2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","2,\\ldots,M^d.\n\\]\nHence\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","e\n\\[\n \\boxed{\\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}\n\\]\nEvery characteristic-zero polynomial chart is invertible.\n\\item On the integral lattice \\(\\mathbb Z^{d+1}\\), one channel is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","is injective.  If \\(M>1\\) and \\(d\\geq1\\), its image has index \\(M^{d(d+1)/2}\\), so the channel has no integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","integer linear left inverse.\n\\item Let \\(p\\) be prime and put \\(\\beta_p=p^{v_p(M)}\\).  Reduction modulo \\(p\\) has exact rank\n\\[\n \\boxed{\n \\rank_{\\mathbb F_p}\\overline A_{M,s}^{(d)}\n =\\left\\lfloor\\frac d{\\beta_p}\\right\\rfloor+1.}\n\\]\nA local channel loses residual rank exactly at primes dividing \\(M\\).\n\\item A complete neighboring packet can recover the source integrally.  The canonical prefix of \\(\\min(M,d+1)\\) standard shifts has an integer left inverse, valid after base change to every commutative ring.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nThe diagonal follows from the leading term of \\(\\binom{Mq+s}{r}\\), or from the generating identity for \\(\\Theta_{M,s}\\).  The determinant gives the integral index statement.  Modulo \\(p\\), the first nonzero source degree in target row \\(k\\) is \\(\\beta_pk\\), which gives the rank formula.  For the prefix packet, choose from channel \\(s\\) and target row \\(k\\) the equation indexed by \\(n=Mk+s\\).  The selected square coefficient matrix is unit lower triangular, so its inverse is integral and remains a left inverse after every base change.\n\\end{proof}","\\begin{theorem}[Sharp standard packet size]\n\\label{thm:block08-sharp-packet}\nDefine\n\\[\n B(M)\n =\n \\max\\left(\\{1\\}\\cup\\{p^{v_p(M)}:p\\mid M\\}\\right).\n\\]\nThe least number of standard affine channels whose stacked map has an integer left inverse is\n\\[\n \\boxed{L_{\\mathbb Z}(M,d)=\\min(d+1,B(M)).}\n\\]\nThe prefix \\(\\{0,1,\\ldots,L_{\\mathbb Z}(M,d)-1\\}\\) attains the minimum.","For a fixed modulus \\(N\\geq2\\), set\n\\[\n B_N(M)\n =\n \\max\\left(\\{1\\}\\cup\n \\{p^{v_p(M)}:p\\mid\\gcd(M,N)\\}\\right).\n\\]\nThe least packet size for a left inverse over \\(\\mathbb Z/N\\mathbb Z\\) is\n\\[\n \\boxed{L_N(M,d)=\\min(d+1,B_N(M)).}\n\\]\nThe exponents \\(v_p(N)\\) do not change the minimum; the first residue layer already detects the obstruction.\n\\end{theorem}","\\begin{proof}\nThis is the primary-rank and Smith/Hermite criterion of Core-Transfer Addendum 04.  At a prime \\(p\\mid M\\), selected shifts are visible only through their residues modulo \\(p^{v_p(M)}\\).  A prefix of the stated length has full rank at every obstructing prime, hence a unimodular-row left inverse.  Any smaller packet fails at a prime attaining the largest relevant primary component.  A nonzero first-layer kernel vector lifts to a kernel vector at every higher prime-power depth.\n\\end{proof}","\\begin{theorem}[Exact branchwise valuation ledger]\n\\label{thm:block08-valuation-ledger}\nFor every refined coefficient,\n\\[\n \\boxed{\n v_p(d_{s,k})\n \\geq\n \\min_{k\\leq r\\leq Mk+s}\n \\left(v_p(c_r)+v_p(\\Theta_{M,s}(r,k))\\right).}\n\\]\nIf \\(v_p(c_r)\\geq w(r)\\) for a nondecreasing weight \\(w\\), then\n\\[\n \\boxed{v_p(d_{s,k})\\geq w(k).}\n\\]\nThus every declared monotone Mahler precision ladder survives affine polyphase refinement.\n\\end{theorem}","\\begin{proof}\nThe transfer sum is finite.  Apply the non-Archimedean triangle inequality.  Since \\(r\\geq k\\), monotonicity gives \\(w(r)\\geq w(k)\\), and every kernel entry is integral.\n\\end{proof}","Define the min-plus certificate transform\n\\[\n \\boxed{\n (\\mathfrak W_{M,s}w)(k)\n =\n \\min_{k\\leq r\\leq Mk+s}\n \\left(w(r)+v_p(\\Theta_{M,s}(r,k))\\right).}\n\\]\nThis transform is a certified lower bound.  It need not be the exact output valuation, because cancellation can increase the valuation beyond the termwise minimum.","\\begin{theorem}[Prime-power Frobenius sparsification]\n\\label{thm:block08-frobenius-sparsity}\nFor \\(M=p^a\\),\n\\[\n \\boxed{\n \\Theta_{p^a,s}(r,k)\n \\equiv\n \\binom{s}{r-p^ak}\n \\pmod p,}\n\\]\nwhere the binomial coefficient is zero outside \\(0\\leq r-p^ak\\leq s\\).  Therefore\n\\[\n \\boxed{\n d_{s,k}\n \\equiv\n \\sum_{u=0}^{s}\\binom su c_{p^ak+u}\n \\pmod p.}\n\\]\nIn particular,\n\\[\n \\boxed{d_{0,k}\\equiv c_{p^ak}\\pmod p.}\n\\]\n\\end{theorem}","\\begin{proof}\nIn characteristic \\(p\\),\n\\[\n (1+z)^{p^a}-1\\equiv z^{p^a}.\n\\]\nSubstitute this into the generating definition of \\(\\Theta_{p^a,s}\\) and compare coefficients.\n\\end{proof}","Finite modular state dynamics provide a directional constructor into the local Mahler chart.  Let\n\\[\n v_{n+1}=Cv_n,\n \\qquad\n a_n=w^{\\mathsf T}C^nv_0,\n\\]\nwith \\(C\\in M_d(\\mathbb Z_p)\\) invertible modulo \\(p\\).","\\begin{theorem}[Matrix Carmichael--Mahler constructor]\n\\label{thm:block08-matrix-carmichael}\nAssume\n\\[\n C^H=I+p^LB,\n \\qquad L\\geq1.\n\\]\nFor every address \\(j\\),\n\\[\n \\boxed{\n a_{Hx+j}\n =\n \\sum_{r\\geq0}\n p^{Lr}\\binom xr\n w^{\\mathsf T}C^jB^rv_0,}\n\\]\nwhere the series converges for \\(x\\in\\mathbb Z_p\\).  Its Mahler coefficients satisfy\n\\[\n \\boxed{v_p(c_{j,r})\\geq Lr,}\n\\]\nand\n\\[\n \\boxed{\\Delta_H^r a_n\\equiv0\\pmod{p^{Lr}}.}\n\\]\n\\end{theorem}","\\begin{proof}\nUse\n\\[\n C^{Hx+j}=C^j(I+p^LB)^x\n\\]\nand expand the principal-unit matrix power by the convergent binomial series.  One forward difference in \\(x\\) contributes the factor \\(p^LB\\).\n\\end{proof}","\\begin{theorem}[Principal-unit precision amplification]\n\\label{thm:block08-precision-amplification}\nIf \\(X=I+p^LB\\), then for every \\(a\\geq0\\) there exists \\(B_a\\in M_d(\\mathbb Z_p)\\) such that\n\\[\n \\boxed{X^{p^a}=I+p^{L+a}B_a.}\n\\]\nConsequently, for \\(H_a=Hp^a\\), every branch \\(x\\mapsto a_{H_ax+j}\\) has Mahler coefficients satisfying\n\\[\n \\boxed{v_p(c_{a,j,r})\\geq(L+a)r.}\n\\]\nEach additional \\(p\\)-power refinement contributes at least one further digit of precision per Mahler depth.\n\\end{theorem}","\\begin{proof}\nInduct on \\(a\\).  If \\(X_a=I+p^{L+a}B_a\\), the binomial expansion of \\(X_a^p\\) has linear term divisible by \\(p^{L+a+1}\\), and every higher term has at least the same valuation.  Apply \\cref{thm:block08-matrix-carmichael} at the refined jump.\n\\end{proof}","\\begin{example}[Refined dyadic Fibonacci packets]\nFor the Fibonacci companion matrix\n\\[\n C=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix},\n\\]\none has\n\\[\n C^{12}\n =I+8\\begin{pmatrix}29&18","1&0\\end{pmatrix},\n\\]\none has\n\\[\n C^{12}\n =I+8\\begin{pmatrix}29&18\\\\18&11\\end{pmatrix}.\n\\]\nHence\n\\[\n \\boxed{\n \\Delta_{12\\cdot2^a}^{\\,r}F_n\n \\equiv0\\pmod{2^{(a+3)r}}.}\n\\]\n\\end{example}","18&11\\end{pmatrix}.\n\\]\nHence\n\\[\n \\boxed{\n \\Delta_{12\\cdot2^a}^{\\,r}F_n\n \\equiv0\\pmod{2^{(a+3)r}}.}\n\\]\n\\end{example}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\textwidth]{local_precision_tree.pdf}\n\\caption{A principal-unit state packet acquires one additional guaranteed \\(p\\)-adic digit per Mahler depth at each \\(p\\)-power refinement.}\n\\label{fig:block08-precision-tree}\n\\end{figure}","\\begin{proposition}[No canonical unrestricted bridge between the two Mahler channels]\n\\label{prop:block08-no-unrestricted-mahler-bridge}\nFunctional-Mahler mask data over \\(\\mathbb F_p\\) does not determine a unique Newton--Mahler chart over \\(\\mathbb Z_p\\), and a continuous Newton--Mahler function does not in general determine a rational functional-Mahler mask.\n\\end{proposition}","\\begin{proof}\nA residue sequence \\((\\bar a_n)\\) admits many integral lifts.  If \\((a_n)\\) is one lift, then \\((a_n+pn)\\) is another with the same reduction modulo \\(p\\).  Their Newton--Mahler coefficients differ by the nonzero coefficient sequence of \\(pn\\).  Thus the residue-series or functional-mask presentation cannot select one local interpolation without a declared lifting rule.","Conversely, a continuous function \\(f:\\mathbb Z_p\\to\\mathbb Z_p\\) canonically determines its Newton--Mahler coefficients, but rationality of an ordinary generating series or of the mask \\(F(z)/F(z^p)\\) is an additional property of a separate coefficient-series presentation.  Continuity alone does not supply it.\n\\end{proof}","\\begin{boundarybox}\n\\textbf{Exact local boundary.}\nLocal Mahler-polyphase atlas proves a complete normalized atlas for continuous \\(\\mathbb Z_p\\)-valued functions and their complete \\(p^a\\)-branch packets.  It proves exact affine coefficient transport, mixed-radix coherence, finite-degree characteristic-zero invertibility, integral split recovery, modular rank loss, valuation preservation, and principal-unit precision amplification under the printed hypotheses.  It does not identify every continuous function as locally analytic, compute exact higher valuations of every \\(\\Theta_{p^a,s}(r,k)\\), choose an optimal mixed-radix refinement tree, or construct the combined spectral--local invariant requested by the broader program.\n\\end{boundarybox}","ete \\(p^a\\)-branch packets.  It proves exact affine coefficient transport, mixed-radix coherence, finite-degree characteristic-zero invertibility, integral split recovery, modular rank loss, valuation preservation, and principal-unit precision amplification under the printed hypotheses.  It does not identify every continuous function as locally analytic, compute exact higher valuations of every \\(\\Theta_{p^a,s}(r,k)\\), choose an optimal mixed-radix refinement tree, or construct the combined spectral--local invariant requested by the broader program.\n\\end{boundarybox}","For \\(f(x)=x^2\\), the source Mahler vector is\n\\[\n (c_0,c_1,c_2)=(0,1,2).\n\\]\nThe two dyadic branches are\n\\[\n f(2q)=4\\binom q1+8\\binom q2,\n\\]\nand\n\\[\n f(2q+1)=1+8\\binom q1+8\\binom q2.\n\\]\nThis example exhibits exact finite-window coefficient transport.","For a principal unit \\(u=1+p\\),\n\\[\n u^x=\\sum_{r\\geq0}p^r\\binom xr.\n\\]\nThe branch \\(x=pq\\) has coefficients\n\\[\n \\bigl(u^p-1\\bigr)^k,\n\\]\nwhose valuation is at least \\(2k\\).  The branch therefore improves the original slope-one certificate.","At \\(p=2\\), \\(M=2\\), and \\(d=3\\), each single reduced channel has rank two, while the two-channel packet has rank four.  This is the smallest visible instance of collective recovery.","e strictly and transport declared valuation ladders.  The finite-degree restriction exposes a four-level bridge ladder: reversible characteristic-zero charts, nonsplit integral channels, rank-deficient residue channels, and split neighboring packets.  Carmichael--Mahler transfer embeds finite principal-unit state dynamics into the local atlas with a certified precision slope that improves under \\(p\\)-power refinement.  Functional-Mahler masks remain a distinct typed channel unless coefficient lifts and interpolation data are supplied.\n\\end{resultbox}","or fixed-radix regular systems, 697,958 for integral saturation and prime-power towers, 595,964 for Euler-product and multiplicative coordinates, 229,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute \\(\\BlockThirteenBridgeChecks\\) bridge assertions and \\(\\BlockThirteenInvariantChecks\\) weighted Fuss and tagged-invariant assertions. They include \\(\\BlockThirteenRegressionChecks\\) authenticated replay assertions already counted in the preceding records and contribute \\(\\BlockThirteenNewChecks\\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \\(\\BlockThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-compo…","saturation and prime-power towers, 595,964 for Euler-product and multiplicative coordinates, 229,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute \\(\\BlockThirteenBridgeChecks\\) bridge assertions and \\(\\BlockThirteenInvariantChecks\\) weighted Fuss and tagged-invariant assertions. They include \\(\\BlockThirteenRegressionChecks\\) authenticated replay assertions already counted in the preceding records and contribute \\(\\BlockThirteenNewChecks\\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \\(\\BlockThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial impl…","29,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute \\(\\BlockThirteenBridgeChecks\\) bridge assertions and \\(\\BlockThirteenInvariantChecks\\) weighted Fuss and tagged-invariant assertions. They include \\(\\BlockThirteenRegressionChecks\\) authenticated replay assertions already counted in the preceding records and contribute \\(\\BlockThirteenNewChecks\\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \\(\\BlockThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \\(\\BlockSixteenValidationAssertions\\) exact…","ts, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute \\(\\BlockThirteenBridgeChecks\\) bridge assertions and \\(\\BlockThirteenInvariantChecks\\) weighted Fuss and tagged-invariant assertions. They include \\(\\BlockThirteenRegressionChecks\\) authenticated replay assertions already counted in the preceding records and contribute \\(\\BlockThirteenNewChecks\\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \\(\\BlockThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \\(\\BlockSixteenValidationAssertions\\) exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible sele…","BlockThirteenBridgeChecks\\) bridge assertions and \\(\\BlockThirteenInvariantChecks\\) weighted Fuss and tagged-invariant assertions. They include \\(\\BlockThirteenRegressionChecks\\) authenticated replay assertions already counted in the preceding records and contribute \\(\\BlockThirteenNewChecks\\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \\(\\BlockThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \\(\\BlockSixteenValidationAssertions\\) exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit projectors, and theorem-source contracts, with \\(\\BlockSixteenValidationFa…","invariant assertions. They include \\(\\BlockThirteenRegressionChecks\\) authenticated replay assertions already counted in the preceding records and contribute \\(\\BlockThirteenNewChecks\\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \\(\\BlockThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \\(\\BlockSixteenValidationAssertions\\) exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit projectors, and theorem-source contracts, with \\(\\BlockSixteenValidationFailures\\) recorded failures. Detailed tables and machine-readable JSON records are included in the source pack…","ted replay assertions already counted in the preceding records and contribute \\(\\BlockThirteenNewChecks\\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \\(\\BlockThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \\(\\BlockSixteenValidationAssertions\\) exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit projectors, and theorem-source contracts, with \\(\\BlockSixteenValidationFailures\\) recorded failures. Detailed tables and machine-readable JSON records are included in the source package. The Part IV semantic-diamond audit additionally executes exact rational lin…","kThirteenCumulative\\) exact assertions with no failures; the raw executed total, including replay, is \\(\\BlockThirteenCumulativeExecuted\\). The Block 15 verifier separately executes \\(\\BlockFifteenValidationAssertions\\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \\(\\BlockSixteenValidationAssertions\\) exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit projectors, and theorem-source contracts, with \\(\\BlockSixteenValidationFailures\\) recorded failures. Detailed tables and machine-readable JSON records are included in the source package. The Part IV semantic-diamond audit additionally executes exact rational linear-algebra trials for quotient dimensions, factorization, relative residuals, and compatibility sequences. Its role is implementation validation; the universal statements are proved in the text.","ne-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \\(\\BlockSixteenValidationAssertions\\) exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit projectors, and theorem-source contracts, with \\(\\BlockSixteenValidationFailures\\) recorded failures. Detailed tables and machine-readable JSON records are included in the source package. The Part IV semantic-diamond audit additionally executes exact rational linear-algebra trials for quotient dimensions, factorization, relative residuals, and compatibility sequences. Its role is implementation validation; the universal statements are proved in the text.","\\bibitem{BeukersHeckman1989}\nF. Beukers and G. Heckman, ``Monodromy for the hypergeometric function \\({}_nF_{n-1}\\),'' \\emph{Inventiones Mathematicae} \\textbf{95} (1989), 325--354.","\\bibitem{Mahler1958}\nK. Mahler, ``An interpolation series for continuous functions of a \\(p\\)-adic variable,'' \\emph{Journal f\\\"ur die reine und angewandte Mathematik} \\textbf{199} (1958), 23--34.","g can erase a nonzero fine obstruction.\n\\item Under the stated classical hypotheses, Hilbert 90 gives effective vector descent and the Brauer class is the exact projective-to-linear lifting obstruction.\n\\item Restricted complete invariants exist in several strata. The persistent C-finite slice has a complete global-to-packet minimality transfer theorem.\n\\item The unit purely periodic atlas over \\(\\mathbb F_p\\) is a verified heterogeneous atlas across linear, digital, and functional-Mahler strata. Its objects and morphisms descend effectively, and its periodic resolved invariant package is complete on the declared normalized semantic category.\n\\item The unit rational-mask atlas over \\(\\mathbb F_p\\) is a verified heterogeneous atlas across reduced functional-Mahler masks, minimal Cartier modules, and minimal digital automata. Its exact collapse obstruction separates the ultimately periodic locus from the genuinely non-C-finite automatic locus, and its descended invariant package is complete on the declared semantic category.\n\\item For a fixed radix and field, every regular sequence has…","e has a complete global-to-packet minimality transfer theorem.\n\\item The unit purely periodic atlas over \\(\\mathbb F_p\\) is a verified heterogeneous atlas across linear, digital, and functional-Mahler strata. Its objects and morphisms descend effectively, and its periodic resolved invariant package is complete on the declared normalized semantic category.\n\\item The unit rational-mask atlas over \\(\\mathbb F_p\\) is a verified heterogeneous atlas across reduced functional-Mahler masks, minimal Cartier modules, and minimal digital automata. Its exact collapse obstruction separates the ultimately periodic locus from the genuinely non-C-finite automatic locus, and its descended invariant package is complete on the declared semantic category.\n\\item For a fixed radix and field, every regular sequence has a canonical minimal section module, a strict shortlex coordinate record, and a word-Hankel rank certificate. These charts satisfy effective descent, and the pointed section module is complete on the declared regular category.\n\\item Over a finite field, regularity is equivalent to automaticit…","-valued saturation of the same rank.\n\\item The Smith invariants and Fitting ideals of the finite saturation quotient determine the ghost kernel, semantic image size, missing quotient, good moduli, and prime-power generator profile exactly.\n\\item Every prime-power coefficient reduction has a finite minimal semantic section automaton, and the compatible full module tower reconstructs the embedded \\(p\\)-adic raw lattice.\n\\item \\(p\\)-adic completion, profinite completion, ordinary integral lifting, and actual section realization are distinct effectivity levels with explicit obstruction quotients.\n\\item Every integral unit series has unique fixed Euler exponents and unique divisor ghosts, with exact M\\\"obius integrality conditions and coefficient recurrence.\n\\item Every integral multiplicative arithmetic function is reconstructed exactly from prime-power local series, and local Euler exponents and ghosts form complete additive coordinates for Dirichlet convolution.\n\\item On the multiplicative exponent locus, the divisor--ghost transform gives an effective four-chart atlas and a complete p…","rank.\n\\item The Smith invariants and Fitting ideals of the finite saturation quotient determine the ghost kernel, semantic image size, missing quotient, good moduli, and prime-power generator profile exactly.\n\\item Every prime-power coefficient reduction has a finite minimal semantic section automaton, and the compatible full module tower reconstructs the embedded \\(p\\)-adic raw lattice.\n\\item \\(p\\)-adic completion, profinite completion, ordinary integral lifting, and actual section realization are distinct effectivity levels with explicit obstruction quotients.\n\\item Every integral unit series has unique fixed Euler exponents and unique divisor ghosts, with exact M\\\"obius integrality conditions and coefficient recurrence.\n\\item Every integral multiplicative arithmetic function is reconstructed exactly from prime-power local series, and local Euler exponents and ghosts form complete additive coordinates for Dirichlet convolution.\n\\item On the multiplicative exponent locus, the divisor--ghost transform gives an effective four-chart atlas and a complete prime-power descended invariant.","recurrence.\n\\item Every integral multiplicative arithmetic function is reconstructed exactly from prime-power local series, and local Euler exponents and ghosts form complete additive coordinates for Dirichlet convolution.\n\\item On the multiplicative exponent locus, the divisor--ghost transform gives an effective four-chart atlas and a complete prime-power descended invariant.\n\\item Continuous \\(\\mathbb Z_p\\)-valued functions, their canonical Mahler coefficients, complete clopen branch packets, and branchwise Mahler coefficients form an effective normalized local atlas.\n\\item Affine Mahler kernels have finite triangular support, compose by exact mixed-radix substitution, and satisfy strict Chinese-remainder coherence.\n\\item Finite-degree affine channels have distinct characteristic-zero, integral, modular, and collective-packet bridge passports, with sharp minimal packet sizes under the printed hypotheses.\n\\item Declared monotone Mahler valuation profiles survive affine polyphase refinement, and principal-unit state packets acquire a certified linearly improving precision slope under…","ese-remainder coherence.\n\\item Finite-degree affine channels have distinct characteristic-zero, integral, modular, and collective-packet bridge passports, with sharp minimal packet sizes under the printed hypotheses.\n\\item Declared monotone Mahler valuation profiles survive affine polyphase refinement, and principal-unit state packets acquire a certified linearly improving precision slope under \\(p\\)-power refinement.\n\\item Every nonzero D-finite series and non-eventually-zero P-recursive sequence over a characteristic-zero field has a canonical monic minimal Ore annihilator; a finite transient ledger upgrades the tail operator to an exact semantic chart.\n\\item Arithmetic multisection preserves holonomicity, and every fixed closure pipeline has an exact excess-order certificate relative to the unique minimal packet operator.\n\\item In the marked binomial--Fuchsian family, nonintegral exponent gives minimal packet differential order \\(N\\), positive integral exponent gives rational order one, and nonpositive integral exponent gives the exact finite-support alias/cancellation phase diagr…","ical monic minimal Ore annihilator; a finite transient ledger upgrades the tail operator to an exact semantic chart.\n\\item Arithmetic multisection preserves holonomicity, and every fixed closure pipeline has an exact excess-order certificate relative to the unique minimal packet operator.\n\\item In the marked binomial--Fuchsian family, nonintegral exponent gives minimal packet differential order \\(N\\), positive integral exponent gives rational order one, and nonpositive integral exponent gives the exact finite-support alias/cancellation phase diagram.\n\\item For a finite Galois extension in the semisimple setting, subgroups, fixed fields, and faithful Reynolds projectors form an order-reversing normalized atlas.\n\\item Nonnormal intermediate fields form conjugate projector atlases with canonical core and normal-closure envelopes, while Mackey double cosets give exact induction--restriction transition packets.\n\\item Ramification groups form a nested projector resolution, and the Artin and Swan conductors are weighted unresolved-rank sums.\n\\item Finite Galois vector descent is effective o…","ctional loss, partial constructors, and absent bridges.\n\\item On the five declared connected components, local complete invariants have coherent comparison maps and descend to a complete tagged invariant that reflects isomorphisms.\n\\item The marked weighted Fuss family gives a genuine reversible overlap among tree, selected algebraic, and P-recursive presentations, classified by the finite pair \\((q,\\alpha)\\).\n\\item At every declared complete finite resolution in an additive idempotent-complete category, analysis and synthesis are inverse, morphisms have unique lane-transfer matrices, composition is block contraction, and compatible refinement aggregates those matrices functorially.\n\\item Finite lane selectors split retained transport from an exact residual. Under the separately printed abelian or strict-fold hypotheses, the discarded or collided part is measured by the corresponding kernel-exact sequence.\n\\item Factorial residue lanes realize the formal theorem exactly, and the stated polyphase, Cartier, holonomic, local, nonlinear-tree, algebraic, and Galois cases are valid only wi…","canonical factorization across arbitrary product grammars;\n\\item ordinary Cauchy convolution and Dirichlet convolution are globally isomorphic, or prime-power seeds classify arithmetic functions without the multiplicativity axiom;\n\\item the multiplicative Euler-exponent locus is closed under ordinary Euler-product multiplication;\n\\item functional-Mahler coefficient equations and Newton--Mahler \\(p\\)-adic interpolation are canonically or unrestrictedly equivalent;\n\\item every continuous \\(p\\)-adic function is locally analytic, every affine-kernel valuation lower bound is always sharp, or one optimal mixed-radix refinement tree has been constructed;\n\\item the local atlas already carries a complete combined spectral--local invariant or a universal state-recognition theorem;\n\\item arbitrary D-finite multisections have a proved finite torsion signature, a closed minimal-order formula, or a canonical apparent-singularity removal algorithm;\n\\item local exponent eigenvalues alone determine the full monodromy representation, logarithmic extension data, or global differential Galois group;\n\\it","and Dirichlet convolution are globally isomorphic, or prime-power seeds classify arithmetic functions without the multiplicativity axiom;\n\\item the multiplicative Euler-exponent locus is closed under ordinary Euler-product multiplication;\n\\item functional-Mahler coefficient equations and Newton--Mahler \\(p\\)-adic interpolation are canonically or unrestrictedly equivalent;\n\\item every continuous \\(p\\)-adic function is locally analytic, every affine-kernel valuation lower bound is always sharp, or one optimal mixed-radix refinement tree has been constructed;\n\\item the local atlas already carries a complete combined spectral--local invariant or a universal state-recognition theorem;\n\\item arbitrary D-finite multisections have a proved finite torsion signature, a closed minimal-order formula, or a canonical apparent-singularity removal algorithm;\n\\item local exponent eigenvalues alone determine the full monodromy representation, logarithmic extension data, or global differential Galois group;\n\\item the marked binomial--Fuchsian invariant classifies unmarked holonomic series or recovers a…","rsion signature, a closed minimal-order formula, or a canonical apparent-singularity removal algorithm;\n\\item local exponent eigenvalues alone determine the full monodromy representation, logarithmic extension data, or global differential Galois group;\n\\item the marked binomial--Fuchsian invariant classifies unmarked holonomic series or recovers an algebraic branch from an annihilator;\n\\item the $d^N$ algebraic packet bound is always minimal, or every resultant automatically selects the intended branch factor;\n\\item selected algebraic branches admit one canonical minimal multivariate system dimension across arbitrary polynomial grammars;\n\\item algebraic and D-finite presentations are equivalent without retained branch data;\n\\item Reynolds averaging remains exact when the group order is not invertible, or finite semisimple Galois charts automatically extend to modular and wild integral representations;\n\\item every projective cocycle descends to a vector space, or the Brauer class alone classifies the full central simple algebra without its remaining algebra data;\n\\item every tensor ca…","effective heterogeneous digital/Mahler atlases. The fixed-radix regular stage adds the full fixed-radix field-valued regular atlas, a canonical word-Hankel and shortlex normal form, and the automatic finite-orbit theorem. The integral prime-power stage adds the canonical integral raw/saturated lattice pair, the exact Smith--Fitting reduction defect, finite prime-power semantic automata, and the \\(p\\)-adic/profinite effectivity hierarchy. The product-multiplicative stage adds canonical fixed-basis Euler and ghost coordinates, prime-power effective descent for multiplicative functions, and the complete restricted divisor--ghost atlas. The local Mahler-polyphase stage adds the normalized continuous Newton--Mahler atlas, strict complete polyphase descent, sharp finite-degree channel packets, valuation-profile transport, and principal-unit precision amplification. The holonomic stage adds cano","degree channel packets, valuation-profile transport, and principal-unit precision amplification. The holonomic stage adds canonical minimal Ore charts with finite transients, the general multisection alias/cancellation ledger, and an exact marked binomial--Fuchsian packet-minimality theorem. The algebraic stage adds finite selected-branch normalization, exact packet polynomial systems, a generic $d^N$ packet-degree theorem, and collision signatures with sharp Catalan and radical model cases. The nonlinear tree stage adds a marked polynomial fixed-point atlas, exact diagonal packet Fa\\`a di Bruno compression, an address-preserving cyclic-core dimension, and finite simply-generated-tree support and singularity signatures. The Galois-Tannakian stage adds the normalized finite subgroup--fixed-field--projector atlas, conjugate and Mackey transition packets, ramification details, effective Hilber","The remaining heterogeneous work has a strict topological order:\n\\[\n\\boxed{\n\\begin{gathered}\n\\text{profiled envelope}\n\\longrightarrow\n\\text{declared atlas}\n\\longrightarrow\n\\text{typed bridge laws}\\\\[-1pt]\n\\longrightarrow\n\\text{higher coherence}\n\\longrightarrow\n\\text{effective descent}\n\\longrightarrow\n\\text{compatible invariants}\n\\end{gathered}}\n\\]\nThe first three arrows cannot be skipped. Pairwise formulas do not themselves define a pseudofunctor. Local completeness does not itself construct compatible global isomorphisms.","ce equation, effective object and morphism descent, and a globally complete compatible invariant package;\n\\item the Frobenius-radical characterization of rational canonical masks and the exact divisor-residue obstruction to rational collapse;\n\\item the canonical finite Cartier module, its reachable-observable minimality theorem, and the exact identification of reachable compiler states with the \\(p\\)-kernel;\n\\item effective descent across reduced mask, minimal Cartier-module, and minimal DFAO charts on the full unit rational-mask domain;\n\\item a globally complete rational-mask invariant package and a strict negative control showing that the rational-mask atlas is a proper subatlas of the automatic stratum;\n\\item the canonical section-module, strict shortlex, minimal digit-linear, and word-Hankel normalization theorem for every fixed-radix field-valued regular sequence;\n\\item the automatic finite-orbit theorem over finite fields, including exact state-to-rank bounds and a strict compression corpus;\n\\item the canonical raw and saturated integral section lattices for every integer-value…","ion corpus;\n\\item the canonical raw and saturated integral section lattices for every integer-valued sequence of finite rational section rank;\n\\item the digit-equivariant Smith--Tor reduction sequence, exact Smith and Fitting formulas, and good-prime criterion;\n\\item finite minimal prime-power semantic section automata with explicit ambient module bounds and compatible reduction maps;\n\\item the \\(p\\)-adic and profinite completion sequences, disappearance of finite ghost coordinates, persistence of the missing saturation quotient, and the separate integral and actual-section effectivity obstructions;\n\\item the integral fixed-Euler-basis classification, divisor-ghost integrality locus, exact logarithmic-derivative recurrence, and finite-prefix triangular normalization;\n\\item prime-power effective descent for integral multiplicative arithmetic functions, complete local Euler and ghost coordinates, valuation logarithmic derivatives, and multiplicative residue packets;\n\\item the restricted divisor--ghost product--multiplicative atlas with effective object and morphism descent and a comple…","efect decomposition into orbit aliasing and address cancellation, with singularity-orbit and local-exponent passports;\n\\item the exact hypergeometric packet equation, minimal differential-order phase diagram, orbit rank, and complete marked invariant for the binomial--Fuchsian model family;\n\\item finite selected-branch algebraic normalization, exact residue-packet polynomial systems, the generic $d^N$ packet-degree theorem, and explicit tuple/stabilizer collision signatures;\n\\item the marked polynomial fixed-point atlas across complete packet jets, colored operation trees, and exact packet equations;\n\\item exact repeated-direction packet Fa\\`a di Bruno compression by colored integer-partition profiles;\n\\item exact SCC elimination to the address-preserving cyclic nonlinear core;\n\\item the finite arity-gcd tree signature controlling arithmetic packet support and the complete dominant-singularity orbit, together with the full $q$-ary phase theorem;\n\\item the normalized finite subgroup--fixed-field--Reynolds atlas, exact normality and quotient-compression criterion, conjugate nonnormal a…","ete packet jets, colored operation trees, and exact packet equations;\n\\item exact repeated-direction packet Fa\\`a di Bruno compression by colored integer-partition profiles;\n\\item exact SCC elimination to the address-preserving cyclic nonlinear core;\n\\item the finite arity-gcd tree signature controlling arithmetic packet support and the complete dominant-singularity orbit, together with the full $q$-ary phase theorem;\n\\item the normalized finite subgroup--fixed-field--Reynolds atlas, exact normality and quotient-compression criterion, conjugate nonnormal atlas envelopes, and Mackey double-coset transition packets;\n\\item nested ramification projectors and the weighted unresolved-rank formulas for Artin and Swan conductors;\n\\item effective finite Galois vector descent, the framed Hilbert-90 chart, the exact relative Brauer obstruction, profinite finite-level reconstruction, and the neutral Tannakian tensor-fiber gate;\n\\item an authoritative bridge matrix with named restricted equivalence loci, directional-loss entries, partial constructors, and explicit absent bridges;\n\\item normalizer…","Throughout this section, $K$ is a characteristic-zero field.  When local monodromy is discussed, $K$ is embedded in $\\mathbb C$ and all local exponents are taken over an algebraic closure.","Let\n\\[\n \\mathcal D_K=K(z)\\langle D\\rangle,\n \\qquad\n Dq=qD+q',\n\\]\nbe the differential Ore algebra, and let\n\\[\n \\mathcal S_K=K(n)\\langle S\\rangle,\n \\qquad\n Sq(n)=q(n+1)S,\n\\]\nbe the forward-shift Ore algebra.  Both are Euclidean skew-polynomial rings in their operator variable, so their nonzero left ideals have unique monic generators \\cite{Ore1933}.","\\begin{definition}[Minimal differential annihilator]\nFor a nonzero D-finite series $F\\in K[[z]]$, define\n\\[\n \\operatorname{Ann}_{\\mathcal D}(F)\n =\n \\{L\\in\\mathcal D_K:LF=0\\}.\n\\]\nIts unique monic generator is denoted\n\\[\n \\boxed{L_F.}\n\\]\nThe differential order\n\\[\n r_D(F)=\\operatorname{ord}_D L_F\n\\]\nis the minimal scalar differential order of $F$ over $K(z)$.\n\\end{definition}","\\begin{definition}[Minimal tail recurrence]\nFor a sequence $a=(a_n)_{n\\geq0}$, define its tail annihilator by\n\\[\n \\operatorname{Ann}_{\\rm tail}(a)\n =\n \\{R\\in\\mathcal S_K:(Ra)_n=0\\text{ for all sufficiently large }n\\}.\n\\]\nFor a non-eventually-zero P-recursive sequence, its unique monic generator is denoted\n\\[\n \\boxed{R_a.}\n\\]\nIf the sequence is eventually zero, the tail generator is $1$ and has order zero.\n\\end{definition}","Write\n\\[\n R_a\n =\n S^s+\\sum_{j=0}^{s-1}r_j(n)S^j.\n\\]\nLet $\\nu_a$ be the least index from which $R_a$ annihilates the sequence, and define $N_a$ to be the least integer $N\\geq\\nu_a$ larger than every nonnegative pole of the rational functions $r_j(n)$.  Then\n\\[\n \\mathfrak R(a)\n =\n \\bigl(R_a,N_a,a_0,\\ldots,a_{N_a+s-1}\\bigr)\n\\]\nis a canonical complete recurrence chart.  The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.","\\begin{proposition}[Derivative-module rank]\n\\label{prop:block09-derivative-rank}\nFor a nonzero D-finite series $F$,\n\\[\n \\boxed{\n r_D(F)\n =\n \\dim_{K(z)}\\operatorname{span}_{K(z)}\n \\{F,DF,D^2F,\\ldots\\}.}\n\\]\n\\end{proposition}","\\begin{proof}\nA differential operator of order $r$ gives a linear dependence among $F,DF,\\ldots,D^rF$.  Minimality of $L_F$ says that no dependence exists among the first $r$ derivative vectors.  Therefore they form a basis of the derivative span.\n\\end{proof}","Let\n\\[\n F(z)=\\sum_{n\\geq0}a_nz^n\n\\]\nand\n\\[\n L\n =\n \\sum_{j=0}^{r}q_j(z)D^j,\n \\qquad\n q_j(z)=\\sum_{m=0}^{d_j}q_{j,m}z^m.\n\\]","\\begin{theorem}[Differential-to-recurrence coefficient formula]\n\\label{thm:block09-diff-to-rec}\nFor every $n\\geq0$,\n\\[\n \\boxed{\n [z^n]LF\n =\n \\sum_{j=0}^{r}\n \\sum_{m=0}^{\\min(d_j,n)}\n q_{j,m}\n (n-m+j)^{\\underline{j}}\n a_{n-m+j},}\n\\]\nwhere\n\\[\n x^{\\underline j}=x(x-1)\\cdots(x-j+1).\n\\]\nConsequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.\n\\end{theorem}","\\begin{proof}\nThe coefficient of $z^{n-m}$ in $D^jF$ is\n\\[\n (n-m+j)^{\\underline j}a_{n-m+j}.\n\\]\nMultiplication by $q_{j,m}z^m$ shifts that contribution to degree $n$.  Summing over $j$ and $m$ gives the formula.\n\\end{proof}","Conversely, suppose\n\\[\n \\sum_{s=0}^{q}p_s(n)a_{n+s}=0\n \\qquad(n\\geq N),\n\\]\nwhere $p_s\\in K[n]$, and put\n\\[\n \\theta=zD.\n\\]","\\begin{theorem}[Recurrence residual as an Euler-operator polynomial]\n\\label{thm:block09-rec-to-diff}\nDefine\n\\[\n \\boxed{\n Q\n =\n \\sum_{s=0}^{q}\n z^{q-s}p_s(\\theta-s).}\n\\]\nThen\n\\[\n \\boxed{\n [z^{n+q}]QF\n =\n \\sum_{s=0}^{q}p_s(n)a_{n+s}.}\n\\]\nHence $QF$ is a polynomial of degree less than $N+q$, and\n\\[\n \\boxed{D^{N+q}QF=0.}\n\\]\n\\end{theorem}","\\begin{proof}\nThe operator $p_s(\\theta-s)$ multiplies the coefficient of $z^k$ by $p_s(k-s)$.  After multiplication by $z^{q-s}$, the coefficient at $z^{n+q}$ is $p_s(n)a_{n+s}$.  The recurrence kills every coefficient from degree $N+q$ onward.  Differentiating more times than the residual degree gives a homogeneous annihilator.\n\\end{proof}","\\begin{definition}[Ordinary-point locus]\nA nonzero D-finite series $F\\in K[[z]]$ belongs to\n\\[\n \\mathbf{Hol}^{\\rm ord}_K\n\\]\nwhen its monic minimal operator can be written\n\\[\n L_F\n =\n D^r+\\sum_{j=0}^{r-1}\\ell_j(z)D^j\n\\]\nwith every $\\ell_j$ regular at $z=0$.\n\\end{definition}","The normalized jet is\n\\[\n J_0(F)\n =\n \\bigl(F(0),F'(0),\\ldots,F^{(r-1)}(0)\\bigr).\n\\]","\\begin{theorem}[Ordinary-point reconstruction]\n\\label{thm:block09-ordinary-reconstruction}\nLet\n\\[\n L=D^r+\\sum_{j=0}^{r-1}\\ell_j(z)D^j\n\\]\nwith all $\\ell_j$ regular at zero.  For every jet\n\\[\n (y_0,\\ldots,y_{r-1})\\in K^r\n\\]\nthere is a unique formal series $F\\in K[[z]]$ satisfying\n\\[\n LF=0,\n \\qquad\n F^{(j)}(0)=y_j.\n\\]\n\\end{theorem}","\\begin{proof}\nWrite $F(z)=\\sum c_nz^n$.  At coefficient degree $n$, the monic term $D^rF$ contributes\n\\[\n \\frac{(n+r)!}{n!}c_{n+r},\n\\]\nwhile every lower derivative involves only $c_0,\\ldots,c_{n+r-1}$.  Characteristic zero makes the displayed coefficient invertible.  The equation therefore determines $c_{n+r}$ recursively from the jet.\n\\end{proof}","\\begin{theorem}[Normalized ordinary holonomic atlas]\n\\label{thm:block09-ordinary-atlas}\nOn the ordinary-point locus, the following normalized presentations determine one another exactly:\n\\[\n \\boxed{\n \\begin{aligned}\n &F\\in K[[z]],\\\\\n &(L_F,J_0(F)),\\\\\n &\\operatorname{span}_{K(z)}\\{D^jF:j\\geq0\\}\n   \\text{ with cyclic vector }F,\\\\\n &(R_a,N_a,a_0,\\ldots,a_{N_a+s-1}),\n \\qquad a_n=[z^n]F.\n \\end{aligned}}\n\\]\nWith equality morphisms, these charts form equivalent normalized groupoids, and every fiber over a fixed semantic series is contractible.\n\\end{theorem}","\\begin{proof}\nThe series gives its unique minimal Ore operators and finite data.  The derivative module recovers $L_F$ as the first monic dependence among consecutive derivatives.  The operator and jet recover $F$ by \\cref{thm:block09-ordinary-reconstruction}.  The recurrence chart reconstructs every coefficient after its singular prefix.  The two explicit coefficient formulas transport between the differential and recurrence channels.  Minimal monic normalization removes coordinate automorphisms, so each normalized fiber has only the identity.\n\\end{proof}","For $N\\geq1$ and $0\\leq j<N$, define\n\\[\n \\mathfrak M_{N,j}F(x)\n =\n \\sum_{m\\geq0}a_{Nm+j}x^m.\n\\]\nIf $\\omega_N$ is a primitive $N$th root of unity and $x=t^N$, then\n\\[\n \\boxed{\n t^j\\mathfrak M_{N,j}F(t^N)\n =\n \\frac1N\n \\sum_{q=0}^{N-1}\n \\omega_N^{-jq}F(\\omega_N^qt).}\n\\]\nThis is the character projector for the cyclic rotation action.","\\begin{theorem}[Holonomic multisection closure]\n\\label{thm:block09-multisection-closure}\nIf $F$ is D-finite, then every $\\mathfrak M_{N,j}F$ is D-finite.  Equivalently, arithmetic sections of P-recursive sequences are P-recursive.\n\\end{theorem}","\\begin{theorem}[Nonautonomous state jump]\n\\label{thm:block09-state-jump}\nSuppose a P-recursive sequence is represented, beyond its singular prefix, by\n\\[\n v_{n+1}=C(n)v_n,\n \\qquad\n a_n=\\lambda(n)v_n,\n\\]\nwhere $v_n\\in K^s$ and $C(n)$ is rational.  Then the $j$th $N$-section has the jump system\n\\[\n \\boxed{\n w_{m+1}=J_{N,j}(m)w_m,}\n\\]\nwhere\n\\[\n \\boxed{\n J_{N,j}(m)\n =\n C(Nm+j+N-1)\\cdots C(Nm+j).}\n\\]\nAfter denominator clearing, every scalar output has a polynomial recurrence of order at most $s$.\n\\end{theorem}","e sequence is represented, beyond its singular prefix, by\n\\[\n v_{n+1}=C(n)v_n,\n \\qquad\n a_n=\\lambda(n)v_n,\n\\]\nwhere $v_n\\in K^s$ and $C(n)$ is rational.  Then the $j$th $N$-section has the jump system\n\\[\n \\boxed{\n w_{m+1}=J_{N,j}(m)w_m,}\n\\]\nwhere\n\\[\n \\boxed{\n J_{N,j}(m)\n =\n C(Nm+j+N-1)\\cdots C(Nm+j).}\n\\]\nAfter denominator clearing, every scalar output has a polynomial recurrence of order at most $s$.\n\\end{theorem}","\\begin{proof}\nCompose the $N$ consecutive one-step transfers.  The $s+1$ consecutive scalar output rows are linearly dependent over $K(m)$, producing a recurrence of order at most $s$.\n\\end{proof}","The root filter also gives a construction-independent order ledger before the algebraic descent from $t$ to $x=t^N$.  Work over a finite extension $E/K$ containing $\\omega_N$.  Put\n\\[\n F_q(t)=F(\\omega_N^qt),\n \\qquad\n G_{N,j}(t)=\\frac1N\\sum_{q=0}^{N-1}\\omega_N^{-jq}F_q(t).\n\\]\nLet $L_q\\in E(t)\\langle D_t\\rangle$ be the monic minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","t filter also gives a construction-independent order ledger before the algebraic descent from $t$ to $x=t^N$.  Work over a finite extension $E/K$ containing $\\omega_N$.  Put\n\\[\n F_q(t)=F(\\omega_N^qt),\n \\qquad\n G_{N,j}(t)=\\frac1N\\sum_{q=0}^{N-1}\\omega_N^{-jq}F_q(t).\n\\]\nLet $L_q\\in E(t)\\langle D_t\\rangle$ be the monic minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","ves a construction-independent order ledger before the algebraic descent from $t$ to $x=t^N$.  Work over a finite extension $E/K$ containing $\\omega_N$.  Put\n\\[\n F_q(t)=F(\\omega_N^qt),\n \\qquad\n G_{N,j}(t)=\\frac1N\\sum_{q=0}^{N-1}\\omega_N^{-jq}F_q(t).\n\\]\nLet $L_q\\in E(t)\\langle D_t\\rangle$ be the monic minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","der ledger before the algebraic descent from $t$ to $x=t^N$.  Work over a finite extension $E/K$ containing $\\omega_N$.  Put\n\\[\n F_q(t)=F(\\omega_N^qt),\n \\qquad\n G_{N,j}(t)=\\frac1N\\sum_{q=0}^{N-1}\\omega_N^{-jq}F_q(t).\n\\]\nLet $L_q\\in E(t)\\langle D_t\\rangle$ be the monic minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","F_q(t)=F(\\omega_N^qt),\n \\qquad\n G_{N,j}(t)=\\frac1N\\sum_{q=0}^{N-1}\\omega_N^{-jq}F_q(t).\n\\]\nLet $L_q\\in E(t)\\langle D_t\\rangle$ be the monic minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","qt),\n \\qquad\n G_{N,j}(t)=\\frac1N\\sum_{q=0}^{N-1}\\omega_N^{-jq}F_q(t).\n\\]\nLet $L_q\\in E(t)\\langle D_t\\rangle$ be the monic minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","(t).\n\\]\nLet $L_q\\in E(t)\\langle D_t\\rangle$ be the monic minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","c minimal annihilator of $F_q$.  Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\\operatorname{ord}L_F$.  Let\n\\[\n L_N^{\\rm orb}=\\operatorname{LCLM}(L_0,\\ldots,L_{N-1}),\n \\qquad\n r_N^{\\rm orb}=\\operatorname{ord}L_N^{\\rm orb},\n\\]\nand, when $G_{N,j}\\neq0$, let $L_{N,j}^{\\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\\rm char}$.  For a zero character packet put $r_{N,j}^{\\rm char}=0$.","\\begin{definition}[Root-filter alias and address-cancellation defects]\n\\label{def:block09-general-defects}\nDefine\n\\[\n \\mathfrak a_N(F)=Nr-r_N^{\\rm orb},\n \\qquad\n \\mathfrak c_{N,j}(F)=r_N^{\\rm orb}-r_{N,j}^{\\rm char}.\n\\]\nThe first quantity measures dependence already present among the rotated holonomic modules.  The second measures the additional loss caused by the $j$th character projector after the rotated orbit has been assembled.\n\\end{definition}","\\begin{theorem}[Exact two-stage root-filter defect identity]\n\\label{thm:block09-general-defect-identity}\nThe defects in \\cref{def:block09-general-defects} are nonnegative and satisfy\n\\[\n \\boxed{\n Nr-r_{N,j}^{\\rm char}\n =\n \\mathfrak a_N(F)+\\mathfrak c_{N,j}(F).}\n\\]\nThus orbit aliasing occurs before address selection, while address cancellation occurs after the Fourier character has been applied.\n\\end{theorem}","\\begin{proof}\nThe least common left multiple has order at most the sum of the orders of its inputs, hence $r_N^{\\rm orb}\\leq Nr$.  It annihilates every rotated series and therefore annihilates their character sum $G_{N,j}$.  By minimality, $L_N^{\\rm orb}$ belongs to the principal left annihilator ideal generated by $L_{N,j}^{\\rm char}$ whenever the character sum is nonzero, so $r_{N,j}^{\\rm char}\\leq r_N^{\\rm orb}$.  The zero case is immediate.  The displayed identity is the telescoping decomposition\n\\[\n Nr-r_{N,j}^{\\rm char}\n =\n (Nr-r_N^{\\rm orb})+(r_N^{\\rm orb}-r_{N,j}^{\\rm char}).\n\\]\n\\end{proof}","_N^{\\rm orb}\\leq Nr$.  It annihilates every rotated series and therefore annihilates their character sum $G_{N,j}$.  By minimality, $L_N^{\\rm orb}$ belongs to the principal left annihilator ideal generated by $L_{N,j}^{\\rm char}$ whenever the character sum is nonzero, so $r_{N,j}^{\\rm char}\\leq r_N^{\\rm orb}$.  The zero case is immediate.  The displayed identity is the telescoping decomposition\n\\[\n Nr-r_{N,j}^{\\rm char}\n =\n (Nr-r_N^{\\rm orb})+(r_N^{\\rm orb}-r_{N,j}^{\\rm char}).\n\\]\n\\end{proof}","\\begin{boundarybox}\n\\textbf{Root-filter order versus packet order.}\nThe preceding theorem is exact in the rotated $t$-coordinate over $E(t)$.  The arithmetic packet is the series $\\mathfrak M_{N,j}F(x)$ over the base field, related by\n\\[\n G_{N,j}(t)=t^j\\mathfrak M_{N,j}F(t^N).\n\\]\nAlgebraic pullback, base extension, gauge multiplication, and descent may change the minimal scalar operator visible in the $x$-chart.  That comparison is a separately typed descent problem.  The exact binomial--Fuchsian family below computes both sides and therefore resolves this additional step in a nontrivial model case.\n\\end{boundarybox}","{boundarybox}\n\\textbf{Root-filter order versus packet order.}\nThe preceding theorem is exact in the rotated $t$-coordinate over $E(t)$.  The arithmetic packet is the series $\\mathfrak M_{N,j}F(x)$ over the base field, related by\n\\[\n G_{N,j}(t)=t^j\\mathfrak M_{N,j}F(t^N).\n\\]\nAlgebraic pullback, base extension, gauge multiplication, and descent may change the minimal scalar operator visible in the $x$-chart.  That comparison is a separately typed descent problem.  The exact binomial--Fuchsian family below computes both sides and therefore resolves this additional step in a nontrivial model case.\n\\end{boundarybox}","Let $Q^{\\rm cl}_{N,j}$ be an annihilator produced by a declared closure or elimination pipeline, and let $L_{N,j}$ be the monic minimal operator of the section.  Since every annihilator belongs to the principal left ideal generated by $L_{N,j}$,\n\\[\n Q^{\\rm cl}_{N,j}=A_{N,j}L_{N,j}\n\\]\nfor some Ore operator $A_{N,j}$.  The nonnegative quantity\n\\[\n \\delta^{\\rm cl}_{N,j}\n =\n \\operatorname{ord}Q^{\\rm cl}_{N,j}\n -\n \\operatorname{ord}L_{N,j}\n\\]\nmeasures cancellation relative to that construction.  It is not intrinsic until the construction has been fixed.","Multiply $L_F$ by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic.  This gives a unique primitive polynomial representative\n\\[\n \\widetilde L_F\n =\n \\sum_{j=0}^{r}q_j(z)D^j.\n\\]\nIts operator singular support is\n\\[\n \\Sigma_{\\rm op}(F)\n =\n \\{q_r(z)=0\\}\\cup\\{\\infty\\}.\n\\]\nApparent singularities remain part of this operator coordinate until an explicit desingularization has been performed.","Let $\\alpha$ be a finite point and put\n\\[\n v_j=v_\\alpha(q_j),\n \\qquad\n \\mu=\\min_j(v_j-j).\n\\]\nWrite the first nonzero local coefficient of $q_j$ as $q_{j,v_j}$.  The indicial polynomial is\n\\[\n \\boxed{\n I_{\\alpha,L}(\\rho)\n =\n \\sum_{v_j-j=\\mu}\n q_{j,v_j}\\rho^{\\underline j}.}\n\\]\nAt a regular singular point, its roots form the local exponent packet.  Over $\\mathbb C$, the corresponding local monodromy eigenvalues are\n\\[\n e^{2\\pi i\\rho}.\n\\]\nInteger differences between exponents signal resonance.  Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.","finite point and put\n\\[\n v_j=v_\\alpha(q_j),\n \\qquad\n \\mu=\\min_j(v_j-j).\n\\]\nWrite the first nonzero local coefficient of $q_j$ as $q_{j,v_j}$.  The indicial polynomial is\n\\[\n \\boxed{\n I_{\\alpha,L}(\\rho)\n =\n \\sum_{v_j-j=\\mu}\n q_{j,v_j}\\rho^{\\underline j}.}\n\\]\nAt a regular singular point, its roots form the local exponent packet.  Over $\\mathbb C$, the corresponding local monodromy eigenvalues are\n\\[\n e^{2\\pi i\\rho}.\n\\]\nInteger differences between exponents signal resonance.  Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.","\\begin{proposition}[Singularity orbit under multisection]\n\\label{prop:block09-singularity-orbit}\nAssume $F$ is an analytic germ at zero with analytic continuation away from a finite set $\\Sigma$.  Then every finite nonzero analytic singularity of $\\mathfrak M_{N,j}F$ lies in\n\\[\n \\boxed{\n \\Sigma^{[N]}\n =\n \\{\\alpha^N:\\alpha\\in\\Sigma\\}.}\n\\]\nDistinct singularities in one fiber of $\\alpha\\mapsto\\alpha^N$ are aliased by the multisection map.  A character projection may remove some aliased local contributions.\n\\end{proposition}","\\begin{proof}\nAway from $\\Sigma^{[N]}$ and zero, choose a local analytic branch $t=x^{1/N}$.  Every rotated argument $\\omega_N^qt$ avoids $\\Sigma$, so the root-filter expression is analytic.  A singularity can therefore occur only when one rotated preimage meets $\\Sigma$.\n\\end{proof}","At nonzero regular singular points, the map $t\\mapsto t^N$ is unramified.  It preserves local exponents before character projection.  At zero and infinity it is ramified, so exponent scaling must be recorded separately.","The general minimal multisection problem remains open.  The family\n\\[\n \\boxed{\n F_{\\lambda,\\alpha}(z)\n =\n (1-\\alpha z)^{-\\lambda}}\n\\]\nprovides a nontrivial exact model that contains irreducible Fuchsian packets, rational collapse, finite-support aliasing, and zero-address cancellation in one parameter space.","Its coefficients are\n\\[\n a_n\n =\n \\frac{(\\lambda)_n}{n!}\\alpha^n.\n\\]\nFor $0\\leq j<N$, put\n\\[\n G_{N,j}^{\\lambda,\\alpha}(x)\n =\n \\sum_{m\\geq0}\n \\frac{(\\lambda)_{Nm+j}}{(Nm+j)!}\n \\alpha^{Nm+j}x^m.\n\\]\nDefine the upper and lower parameter packets\n\\[\n A_{N,j}(\\lambda)\n =\n \\left\\{\n \\frac{\\lambda+j+r}{N}:0\\leq r<N\n \\right\\},\n\\]\n\\[\n B_{N,j}\n =\n \\left\\{\n \\frac{j+1+r}{N}:0\\leq r<N\n \\right\\}\\setminus\\{1\\}.\n\\]","\\begin{theorem}[Exact hypergeometric packet form]\n\\label{thm:block09-hypergeom-packet}\nIf $(\\lambda)_j\\neq0$, then\n\\[\n \\boxed{\n G_{N,j}^{\\lambda,\\alpha}(x)\n =\n \\frac{(\\lambda)_j\\alpha^j}{j!}\n {}_NF_{N-1}\n \\!\\left(\n \\begin{matrix}A_{N,j}(\\lambda)","\\begin{theorem}[Exact hypergeometric packet form]\n\\label{thm:block09-hypergeom-packet}\nIf $(\\lambda)_j\\neq0$, then\n\\[\n \\boxed{\n G_{N,j}^{\\lambda,\\alpha}(x)\n =\n \\frac{(\\lambda)_j\\alpha^j}{j!}\n {}_NF_{N-1}\n \\!\\left(\n \\begin{matrix}A_{N,j}(\\lambda)\\\\ B_{N,j}\\end{matrix}\n ;\\alpha^Nx\n \\right).}\n\\]\nIts coefficients satisfy the first-order polynomial recurrence\n\\[\n \\boxed{\n \\prod_{r=0}^{N-1}(Nm+j+1+r)u_{m+1}\n =\n \\alpha^N\n \\prod_{r=0}^{N-1}(Nm+j+\\lambda+r)u_m.}\n\\]\nThe series is annihilated by\n\\[\n \\boxed{\n \\mathcal H_{N,j}^{\\lambda,\\alpha}\n =\n \\theta\n \\prod_{b\\in B_{N,j}}(\\theta+b-1)\n -\n \\alpha^Nx\n \\prod_{a\\in A_{N,j}(\\lambda)}(\\theta+a),}\n\\]\nwhere $\\theta=xD_x$.\n\\end{theorem}","B_{N,j}\\end{matrix}\n ;\\alpha^Nx\n \\right).}\n\\]\nIts coefficients satisfy the first-order polynomial recurrence\n\\[\n \\boxed{\n \\prod_{r=0}^{N-1}(Nm+j+1+r)u_{m+1}\n =\n \\alpha^N\n \\prod_{r=0}^{N-1}(Nm+j+\\lambda+r)u_m.}\n\\]\nThe series is annihilated by\n\\[\n \\boxed{\n \\mathcal H_{N,j}^{\\lambda,\\alpha}\n =\n \\theta\n \\prod_{b\\in B_{N,j}}(\\theta+b-1)\n -\n \\alpha^Nx\n \\prod_{a\\in A_{N,j}(\\lambda)}(\\theta+a),}\n\\]\nwhere $\\theta=xD_x$.\n\\end{theorem}","_{N,j}\\end{matrix}\n ;\\alpha^Nx\n \\right).}\n\\]\nIts coefficients satisfy the first-order polynomial recurrence\n\\[\n \\boxed{\n \\prod_{r=0}^{N-1}(Nm+j+1+r)u_{m+1}\n =\n \\alpha^N\n \\prod_{r=0}^{N-1}(Nm+j+\\lambda+r)u_m.}\n\\]\nThe series is annihilated by\n\\[\n \\boxed{\n \\mathcal H_{N,j}^{\\lambda,\\alpha}\n =\n \\theta\n \\prod_{b\\in B_{N,j}}(\\theta+b-1)\n -\n \\alpha^Nx\n \\prod_{a\\in A_{N,j}(\\lambda)}(\\theta+a),}\n\\]\nwhere $\\theta=xD_x$.\n\\end{theorem}","\\begin{proof}\nSplit the rising factorials and ordinary factorials into their $N$ residue factors.  The multiplication formula gives the displayed hypergeometric coefficients.  Taking the ratio of consecutive packet coefficients gives the recurrence.  Coefficient comparison with the standard ${}_NF_{N-1}$ operator gives the differential equation.\n\\end{proof}","\\begin{theorem}[Minimal differential order phase diagram]\n\\label{thm:block09-order-phase}\nLet\n\\[\n r_D(N,j;\\lambda)\n =\n r_D\\bigl(G_{N,j}^{\\lambda,\\alpha}\\bigr),\n \\qquad \\alpha\\neq0.\n\\]\nThen:","\\begin{enumerate}\n\\item If $\\lambda\\notin\\mathbb Z$, then\n\\[\n \\boxed{r_D(N,j;\\lambda)=N.}\n\\]\n\\item If $\\lambda=L\\in\\mathbb Z_{>0}$, then every packet is a nonzero rational function with exact denominator\n\\[\n (1-\\alpha^Nx)^L,\n\\]\nand\n\\[\n \\boxed{r_D(N,j;L)=1.}\n\\]\n\\item If $\\lambda=-M\\in\\mathbb Z_{\\leq0}$, then $G_{N,j}^{-M,\\alpha}$ is zero when $j>M$.  Otherwise it is a nonzero polynomial of degree\n\\[\n \\left\\lfloor\\frac{M-j}{N}\\right\\rfloor\n\\]\nand\n\\[\n \\boxed{\n r_D(N,j;-M)\n =\n \\begin{cases}\n 1,&j\\leq M,","begin{enumerate}\n\\item If $\\lambda\\notin\\mathbb Z$, then\n\\[\n \\boxed{r_D(N,j;\\lambda)=N.}\n\\]\n\\item If $\\lambda=L\\in\\mathbb Z_{>0}$, then every packet is a nonzero rational function with exact denominator\n\\[\n (1-\\alpha^Nx)^L,\n\\]\nand\n\\[\n \\boxed{r_D(N,j;L)=1.}\n\\]\n\\item If $\\lambda=-M\\in\\mathbb Z_{\\leq0}$, then $G_{N,j}^{-M,\\alpha}$ is zero when $j>M$.  Otherwise it is a nonzero polynomial of degree\n\\[\n \\left\\lfloor\\frac{M-j}{N}\\right\\rfloor\n\\]\nand\n\\[\n \\boxed{\n r_D(N,j;-M)\n =\n \\begin{cases}\n 1,&j\\leq M,",", then\n\\[\n \\boxed{r_D(N,j;\\lambda)=N.}\n\\]\n\\item If $\\lambda=L\\in\\mathbb Z_{>0}$, then every packet is a nonzero rational function with exact denominator\n\\[\n (1-\\alpha^Nx)^L,\n\\]\nand\n\\[\n \\boxed{r_D(N,j;L)=1.}\n\\]\n\\item If $\\lambda=-M\\in\\mathbb Z_{\\leq0}$, then $G_{N,j}^{-M,\\alpha}$ is zero when $j>M$.  Otherwise it is a nonzero polynomial of degree\n\\[\n \\left\\lfloor\\frac{M-j}{N}\\right\\rfloor\n\\]\nand\n\\[\n \\boxed{\n r_D(N,j;-M)\n =\n \\begin{cases}\n 1,&j\\leq M,\\\\\n 0,&j>M.\n \\end{cases}}\n\\]\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFor nonintegral $\\lambda$, no difference between an upper parameter and a completed lower parameter is an integer.  The classical irreducibility criterion for generalized hypergeometric operators therefore makes $\\mathcal H_{N,j}^{\\lambda,\\alpha}$ irreducible of order $N$ \\cite{BeukersHeckman1989}.  Since the packet is nonzero, its minimal operator has order $N$.","For $L>0$, the coefficient is a polynomial of degree $L-1$ in $m$ times $\\alpha^{Nm+j}$.  Its generating function is rational with denominator $(1-\\alpha^Nx)^L$, and the leading polynomial term prevents cancellation of the highest denominator power.  Every nonconstant rational function has a first-order logarithmic-derivative annihilator.","For $\\lambda=-M$, the source series is the polynomial $(1-\\alpha z)^M$.  Selecting one residue class leaves exactly the displayed finite support.  A nonzero polynomial has differential order one, while the zero function has order zero.\n\\end{proof}","Let $\\omega_N$ be primitive and consider the rotated orbit\n\\[\n \\mathcal O_N(F_{\\lambda,\\alpha})\n =\n \\left\n \\{(1-\\alpha\\omega_N^rt)^{-\\lambda}:0\\leq r<N\\right\\}.\n\\]\nLet $r_{\\rm orb}$ be its dimension over the constant field.","\\begin{theorem}[Exact orbit rank]\n\\label{thm:block09-orbit-rank}\nFor the binomial family,\n\\[\n \\boxed{\n r_{\\rm orb}(N;\\lambda)\n =\n \\begin{cases}\n N,&\\lambda\\notin\\mathbb Z_{\\leq0},\\\\\n \\min(N,M+1),&\\lambda=-M\\in\\mathbb Z_{\\leq0}.\n \\end{cases}}\n\\]\n\\end{theorem}","\\begin{proof}\nIf $\\lambda$ is nonintegral, each rotated branch has a branch singularity at a different point.  If $\\lambda$ is a positive integer, each has a pole at a different point.  A constant linear relation would therefore have zero coefficient at every branch.","If $\\lambda=-M$, expand\n\\[\n (1-\\alpha\\omega_N^rt)^M\n =\n \\sum_{n=0}^{M}\n \\binom Mn(-\\alpha)^n\\omega_N^{rn}t^n.\n\\]\nThe rotation matrix is a truncated discrete Fourier matrix.  Its rank is the number of available residue classes, namely $\\min(N,M+1)$.\n\\end{proof}","Define the alias and cancellation deficits by\n\\[\n \\boxed{\n A_N(\\lambda)=N-r_{\\rm orb}(N;\\lambda),}\n\\]\n\\[\n \\boxed{\n C_{N,j}(\\lambda)\n =\n r_{\\rm orb}(N;\\lambda)\n -r_D(N,j;\\lambda).}\n\\]\nThen\n\\[\n \\boxed{\n N-r_D(N,j;\\lambda)\n =\n A_N(\\lambda)+C_{N,j}(\\lambda).}\n\\]","\\begin{enumerate}\n\\item If $\\lambda\\notin\\mathbb Z$, then\n\\[\n A_N(\\lambda)=0,\n \\qquad\n C_{N,j}(\\lambda)=0.\n\\]\n\\item If $\\lambda=L>0$, then\n\\[\n A_N(L)=0,\n \\qquad\n C_{N,j}(L)=N-1.\n\\]\n\\item If $\\lambda=-M\\leq0$, then\n\\[\n A_N(-M)=\\max(0,N-M-1),\n\\]\nand\n\\[\n C_{N,j}(-M)\n =\n \\begin{cases}\n \\min(N,M+1)-1,&j\\leq M,","\\begin{enumerate}\n\\item If $\\lambda\\notin\\mathbb Z$, then\n\\[\n A_N(\\lambda)=0,\n \\qquad\n C_{N,j}(\\lambda)=0.\n\\]\n\\item If $\\lambda=L>0$, then\n\\[\n A_N(L)=0,\n \\qquad\n C_{N,j}(L)=N-1.\n\\]\n\\item If $\\lambda=-M\\leq0$, then\n\\[\n A_N(-M)=\\max(0,N-M-1),\n\\]\nand\n\\[\n C_{N,j}(-M)\n =\n \\begin{cases}\n \\min(N,M+1)-1,&j\\leq M,\\\\\n \\min(N,M+1),&j>M.\n \\end{cases}\n\\]\n\\end{enumerate}\n\\end{corollary}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\textwidth]{holonomic_order_phase.pdf}\n\\caption{Minimal differential order in the binomial packet family.  Nonintegral exponents retain order $N$, positive integers collapse to rational order one, and nonpositive integers terminate with explicit aliasing and zero-address packets.}\n\\label{fig:block09-order-phase}\n\\end{figure}","For $\\lambda\\notin\\mathbb Z$, the hypergeometric operator in \\cref{thm:block09-hypergeom-packet} is Fuchsian with singularities\n\\[\n \\boxed{\n 0,\n \\qquad\n \\alpha^{-N},\n \\qquad\n \\infty.}\n\\]\nIts exponent packets are\n\\[\n \\boxed{\n E_0\n =\n \\{0\\}\\cup\\{1-b:b\\in B_{N,j}\\},}\n\\]\n\\[\n \\boxed{\n E_{\\alpha^{-N}}\n =\n \\{0,1,\\ldots,N-2,-\\lambda\\},}\n\\]\nand\n\\[\n \\boxed{\n E_\\infty\n =\n A_{N,j}(\\lambda).}\n\\]\nModulo integers, $E_0$ is the complete $N$-torsion set\n\\[\n \\left\\{0,\\frac1N,\\ldots,\\frac{N-1}{N}\\right\\},\n\\]\nand the pairwise differences in $E_\\infty$ are also $N$-torsion.","$\\lambda\\notin\\mathbb Z$, the hypergeometric operator in \\cref{thm:block09-hypergeom-packet} is Fuchsian with singularities\n\\[\n \\boxed{\n 0,\n \\qquad\n \\alpha^{-N},\n \\qquad\n \\infty.}\n\\]\nIts exponent packets are\n\\[\n \\boxed{\n E_0\n =\n \\{0\\}\\cup\\{1-b:b\\in B_{N,j}\\},}\n\\]\n\\[\n \\boxed{\n E_{\\alpha^{-N}}\n =\n \\{0,1,\\ldots,N-2,-\\lambda\\},}\n\\]\nand\n\\[\n \\boxed{\n E_\\infty\n =\n A_{N,j}(\\lambda).}\n\\]\nModulo integers, $E_0$ is the complete $N$-torsion set\n\\[\n \\left\\{0,\\frac1N,\\ldots,\\frac{N-1}{N}\\right\\},\n\\]\nand the pairwise differences in $E_\\infty$ are also $N$-torsion.","Z$, the hypergeometric operator in \\cref{thm:block09-hypergeom-packet} is Fuchsian with singularities\n\\[\n \\boxed{\n 0,\n \\qquad\n \\alpha^{-N},\n \\qquad\n \\infty.}\n\\]\nIts exponent packets are\n\\[\n \\boxed{\n E_0\n =\n \\{0\\}\\cup\\{1-b:b\\in B_{N,j}\\},}\n\\]\n\\[\n \\boxed{\n E_{\\alpha^{-N}}\n =\n \\{0,1,\\ldots,N-2,-\\lambda\\},}\n\\]\nand\n\\[\n \\boxed{\n E_\\infty\n =\n A_{N,j}(\\lambda).}\n\\]\nModulo integers, $E_0$ is the complete $N$-torsion set\n\\[\n \\left\\{0,\\frac1N,\\ldots,\\frac{N-1}{N}\\right\\},\n\\]\nand the pairwise differences in $E_\\infty$ are also $N$-torsion.","etric operator in \\cref{thm:block09-hypergeom-packet} is Fuchsian with singularities\n\\[\n \\boxed{\n 0,\n \\qquad\n \\alpha^{-N},\n \\qquad\n \\infty.}\n\\]\nIts exponent packets are\n\\[\n \\boxed{\n E_0\n =\n \\{0\\}\\cup\\{1-b:b\\in B_{N,j}\\},}\n\\]\n\\[\n \\boxed{\n E_{\\alpha^{-N}}\n =\n \\{0,1,\\ldots,N-2,-\\lambda\\},}\n\\]\nand\n\\[\n \\boxed{\n E_\\infty\n =\n A_{N,j}(\\lambda).}\n\\]\nModulo integers, $E_0$ is the complete $N$-torsion set\n\\[\n \\left\\{0,\\frac1N,\\ldots,\\frac{N-1}{N}\\right\\},\n\\]\nand the pairwise differences in $E_\\infty$ are also $N$-torsion.","\\[\n \\boxed{\n 0,\n \\qquad\n \\alpha^{-N},\n \\qquad\n \\infty.}\n\\]\nIts exponent packets are\n\\[\n \\boxed{\n E_0\n =\n \\{0\\}\\cup\\{1-b:b\\in B_{N,j}\\},}\n\\]\n\\[\n \\boxed{\n E_{\\alpha^{-N}}\n =\n \\{0,1,\\ldots,N-2,-\\lambda\\},}\n\\]\nand\n\\[\n \\boxed{\n E_\\infty\n =\n A_{N,j}(\\lambda).}\n\\]\nModulo integers, $E_0$ is the complete $N$-torsion set\n\\[\n \\left\\{0,\\frac1N,\\ldots,\\frac{N-1}{N}\\right\\},\n\\]\nand the pairwise differences in $E_\\infty$ are also $N$-torsion.","uad\n \\alpha^{-N},\n \\qquad\n \\infty.}\n\\]\nIts exponent packets are\n\\[\n \\boxed{\n E_0\n =\n \\{0\\}\\cup\\{1-b:b\\in B_{N,j}\\},}\n\\]\n\\[\n \\boxed{\n E_{\\alpha^{-N}}\n =\n \\{0,1,\\ldots,N-2,-\\lambda\\},}\n\\]\nand\n\\[\n \\boxed{\n E_\\infty\n =\n A_{N,j}(\\lambda).}\n\\]\nModulo integers, $E_0$ is the complete $N$-torsion set\n\\[\n \\left\\{0,\\frac1N,\\ldots,\\frac{N-1}{N}\\right\\},\n\\]\nand the pairwise differences in $E_\\infty$ are also $N$-torsion.","\\begin{proposition}[Local monodromy ledger]\n\\label{prop:block09-local-monodromy}\nFor nonintegral rational $\\lambda$:","\\begin{enumerate}\n\\item the monodromy eigenvalues at zero form the complete group of $N$th roots of unity;\n\\item the ratios of monodromy eigenvalues at infinity are $N$th roots of unity;\n\\item the distinguished nontrivial eigenvalue at $\\alpha^{-N}$ is\n\\[\n e^{-2\\pi i\\lambda}.\n\\]\n\\end{enumerate}\n\\end{proposition}","The local eigenvalues are finite-order when $\\lambda\\in\\mathbb Q$.  This does not imply that the global monodromy group is finite.  Connection matrices and logarithmic extension data remain additional invariants.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.94\\textwidth]{holonomic_exponent_packet.pdf}\n\\caption{The exact Fuchsian exponent packet of a nonintegral binomial multisection.  Root-orbit aliasing moves $N$ rotated source singularities into one finite singularity while the complete $N$-torsion exponent sets remain visible at zero and infinity.}\n\\label{fig:block09-exponent-packet}\n\\end{figure}","\\begin{definition}[Marked binomial--Fuchsian packet]\n\\label{def:block09-marked-packet}\nA marked packet is a tuple\n\\[\n \\mathfrak b=(\\alpha,\\lambda,N,j;G),\n\\]\nwhere $\\alpha\\neq0$, $N\\geq1$, $0\\leq j<N$, and\n\\[\n G(x)=G_{N,j}^{\\lambda,\\alpha}(x).\n\\]\nThe parameters $(\\alpha,\\lambda,N,j)$ are part of the semantic object rather than metadata to be forgotten.\n\\end{definition}","For a nonzero marked packet, consider the following normalized charts:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item the marked coefficient series $G$;\n\\item the primitive first-order tail recurrence obtained from the exact coefficient ratio, together with its finite initial record;\n\\item the monic minimal differential operator from \\cref{thm:block09-order-phase}, together with the distinguished power-series solution selected by the marked coefficient record;\n\\item the hypergeometric parameter packet $\\bigl(A_{N,j}(\\lambda),B_{N,j};\\alpha^N\\bigr)$ and leading coefficient;\n\\item the primitive minimal-operator singular and indicial record\n\\[\n \\mathscr E(G)\n =\n \\left(\n \\Sigma_{\\rm op}(G),\n \\bigl(I_{\\xi,L_G}\\bigr)_{\\xi\\in\\Sigma_{\\rm op}(G)}\n \\right),\n\\]\ntogether with the orbit-rank and defect data\n\\[\n \\left(r_{\\rm orb},r_D,A_N,C_{N,j}\\right)",")}]\n\\item the marked coefficient series $G$;\n\\item the primitive first-order tail recurrence obtained from the exact coefficient ratio, together with its finite initial record;\n\\item the monic minimal differential operator from \\cref{thm:block09-order-phase}, together with the distinguished power-series solution selected by the marked coefficient record;\n\\item the hypergeometric parameter packet $\\bigl(A_{N,j}(\\lambda),B_{N,j};\\alpha^N\\bigr)$ and leading coefficient;\n\\item the primitive minimal-operator singular and indicial record\n\\[\n \\mathscr E(G)\n =\n \\left(\n \\Sigma_{\\rm op}(G),\n \\bigl(I_{\\xi,L_G}\\bigr)_{\\xi\\in\\Sigma_{\\rm op}(G)}\n \\right),\n\\]\ntogether with the orbit-rank and defect data\n\\[\n \\left(r_{\\rm orb},r_D,A_N,C_{N,j}\\right).\n\\]\nIn the nonintegral Fuchsian regime, \\(\\mathscr E(G)\\) is the explicit three-point packet\n\\[\n \\left(E_0,E_{\\alpha^{-N}},E_\\infty\\right)\n\\]\ncomputed above.  In the integral regimes it is computed from the corresponding minimal first-order operator.  The zero packets in the terminated regime are retained as separately marked zero objects with the zero-op…","ther with its finite initial record;\n\\item the monic minimal differential operator from \\cref{thm:block09-order-phase}, together with the distinguished power-series solution selected by the marked coefficient record;\n\\item the hypergeometric parameter packet $\\bigl(A_{N,j}(\\lambda),B_{N,j};\\alpha^N\\bigr)$ and leading coefficient;\n\\item the primitive minimal-operator singular and indicial record\n\\[\n \\mathscr E(G)\n =\n \\left(\n \\Sigma_{\\rm op}(G),\n \\bigl(I_{\\xi,L_G}\\bigr)_{\\xi\\in\\Sigma_{\\rm op}(G)}\n \\right),\n\\]\ntogether with the orbit-rank and defect data\n\\[\n \\left(r_{\\rm orb},r_D,A_N,C_{N,j}\\right).\n\\]\nIn the nonintegral Fuchsian regime, \\(\\mathscr E(G)\\) is the explicit three-point packet\n\\[\n \\left(E_0,E_{\\alpha^{-N}},E_\\infty\\right)\n\\]\ncomputed above.  In the integral regimes it is computed from the corresponding minimal first-order operator.  The zero packets in the terminated regime are retained as separately marked zero objects with the zero-operator convention.\n\\end{enumerate}","ries solution selected by the marked coefficient record;\n\\item the hypergeometric parameter packet $\\bigl(A_{N,j}(\\lambda),B_{N,j};\\alpha^N\\bigr)$ and leading coefficient;\n\\item the primitive minimal-operator singular and indicial record\n\\[\n \\mathscr E(G)\n =\n \\left(\n \\Sigma_{\\rm op}(G),\n \\bigl(I_{\\xi,L_G}\\bigr)_{\\xi\\in\\Sigma_{\\rm op}(G)}\n \\right),\n\\]\ntogether with the orbit-rank and defect data\n\\[\n \\left(r_{\\rm orb},r_D,A_N,C_{N,j}\\right).\n\\]\nIn the nonintegral Fuchsian regime, \\(\\mathscr E(G)\\) is the explicit three-point packet\n\\[\n \\left(E_0,E_{\\alpha^{-N}},E_\\infty\\right)\n\\]\ncomputed above.  In the integral regimes it is computed from the corresponding minimal first-order operator.  The zero packets in the terminated regime are retained as separately marked zero objects with the zero-operator convention.\n\\end{enumerate}","ameter packet $\\bigl(A_{N,j}(\\lambda),B_{N,j};\\alpha^N\\bigr)$ and leading coefficient;\n\\item the primitive minimal-operator singular and indicial record\n\\[\n \\mathscr E(G)\n =\n \\left(\n \\Sigma_{\\rm op}(G),\n \\bigl(I_{\\xi,L_G}\\bigr)_{\\xi\\in\\Sigma_{\\rm op}(G)}\n \\right),\n\\]\ntogether with the orbit-rank and defect data\n\\[\n \\left(r_{\\rm orb},r_D,A_N,C_{N,j}\\right).\n\\]\nIn the nonintegral Fuchsian regime, \\(\\mathscr E(G)\\) is the explicit three-point packet\n\\[\n \\left(E_0,E_{\\alpha^{-N}},E_\\infty\\right)\n\\]\ncomputed above.  In the integral regimes it is computed from the corresponding minimal first-order operator.  The zero packets in the terminated regime are retained as separately marked zero objects with the zero-operator convention.\n\\end{enumerate}","N\\bigr)$ and leading coefficient;\n\\item the primitive minimal-operator singular and indicial record\n\\[\n \\mathscr E(G)\n =\n \\left(\n \\Sigma_{\\rm op}(G),\n \\bigl(I_{\\xi,L_G}\\bigr)_{\\xi\\in\\Sigma_{\\rm op}(G)}\n \\right),\n\\]\ntogether with the orbit-rank and defect data\n\\[\n \\left(r_{\\rm orb},r_D,A_N,C_{N,j}\\right).\n\\]\nIn the nonintegral Fuchsian regime, \\(\\mathscr E(G)\\) is the explicit three-point packet\n\\[\n \\left(E_0,E_{\\alpha^{-N}},E_\\infty\\right)\n\\]\ncomputed above.  In the integral regimes it is computed from the corresponding minimal first-order operator.  The zero packets in the terminated regime are retained as separately marked zero objects with the zero-operator convention.\n\\end{enumerate}","\\begin{theorem}[Effective marked binomial--Fuchsian atlas]\n\\label{thm:block09-marked-descent}\nThe five normalized presentations above determine one another on the declared marked family.  With equality morphisms they form equivalent discrete groupoids, every normalized fiber is contractible, and objects and morphisms descend effectively.  The package\n\\[\n \\boxed{\n \\mathcal J_{\\rm BF}\n =\n \\left(\n \\alpha,\\lambda,N,j,\n R_{N,j},L_{N,j},\n A_{N,j}(\\lambda),B_{N,j},\n \\mathscr E(G),\n r_{\\rm orb},r_D,A_N,C_{N,j}\n \\right)}\n\\]\nis complete on this marked category.\n\\end{theorem}","\\begin{proof}\nThe mark constructs the coefficient packet directly.  Consecutive coefficient ratios give the normalized recurrence chart, and the multiplication formula gives the hypergeometric chart.  \\Cref{thm:block09-order-phase} identifies the monic minimal differential operator in every parameter regime.  The primitive polynomial representative of that operator determines \\(\\mathscr E(G)\\) by the indicial construction in \\cref{subsec:block09-local-exponents}.  In the nonintegral regime, \\cref{prop:block09-local-monodromy} gives its explicit Fuchsian eigenvalue ledger.  \\Cref{thm:block09-orbit-rank,cor:block09-defect-formulas} compute the orbit and cancellation data in every regime.  Conversely, the displayed invariant retains the complete mark and therefore reconstructs the source packet and every normalized chart.  Uniqueness of monic Ore normalization and of the marked coefficient record leaves no nontrivial automorphism in a fiber.\n\\end{proof}","The parity sections of\n\\[\n \\frac{2}{1-z^2}\n\\]\ngive the smallest singularity-alias example.  The source singularities $1$ and $-1$ both map to $1$ under squaring.  The even address is $2/(1-x)$, while the odd address vanishes.","For\n\\[\n 2\\cosh z,\n\\]\nthe even packet is nonzero and the odd packet is zero.  There is no finite singularity, so the cancellation occurs at the level of the character-projected solution rather than by finite singularity collision.","nt locus, with minimal Ore annihilators, finite prefix and jet certificates, and exact coefficient transport.  General arithmetic multisection has a typed root-filter and state-jump construction, together with an explicit construction-relative minimality defect.  The binomial--Fuchsian family supplies the first exact holonomic packet phase diagram: nonintegral exponents retain differential order $N$, positive integers collapse to rational order one, nonpositive integers terminate with a closed orbit-alias and address-cancellation formula, and the nonintegral local exponent packet exposes the complete finite torsion at zero and infinity.\n\\end{resultbox}","Throughout this section, $K$ is a characteristic-zero field and $\\overline K$ is an algebraic closure.  Algebraic degree always means degree over the displayed rational-function field.  All assertions about formal branches are made inside $\\overline K[[z]]$ unless a different local field is named.","Let $F\\in K[[z]]$ be algebraic over $K(z)$.  Its monic irreducible polynomial is\n\\[\n P_F(z,Y)\\in K(z)[Y],\n \\qquad\n P_F(z,F(z))=0.\n\\]\nWrite\n\\[\n d(F)=\\deg_Y P_F.\n\\]\nAfter multiplying by a common denominator and dividing by the greatest common divisor of the polynomial coefficients, one obtains a primitive representative in $K[z,Y]$, unique up to a nonzero scalar.  We normalize that scalar by fixing the first nonzero coefficient in graded lexicographic order to be one.","A polynomial alone does not select a branch.  Let\n\\[\n \\operatorname{Br}_0(P_F)\n =\n \\{G\\in\\overline K[[z]]:P_F(z,G(z))=0\\}.\n\\]\nThis set has cardinality at most $d(F)$.","\\begin{definition}[Branch-separation index]\nFor $F\\in\\operatorname{Br}_0(P_F)$, define\n\\[\n \\sigma(P_F,F)\n =\n \\begin{cases}\n 0,&\\operatorname{Br}_0(P_F)=\\{F\\},","\\begin{definition}[Branch-separation index]\nFor $F\\in\\operatorname{Br}_0(P_F)$, define\n\\[\n \\sigma(P_F,F)\n =\n \\begin{cases}\n 0,&\\operatorname{Br}_0(P_F)=\\{F\\},\\\\[3pt]\n 1+\\displaystyle\\max_{G\\in\\operatorname{Br}_0(P_F),\\,G\\neq F}\n \\operatorname{ord}_z(F-G),&\\text{otherwise}.\n \\end{cases}\n\\]\nThe corresponding branch selector is the finite coefficient jet\n\\[\n \\boxed{\n J_{\\rm alg}(F)\n =\n (a_0,a_1,\\ldots,a_{\\sigma-1}),\n \\qquad\n F(z)=\\sum_{n\\geq0}a_nz^n.}\n\\]\nWhen $\\sigma=0$, the selector is empty.\n\\end{definition}","[3pt]\n 1+\\displaystyle\\max_{G\\in\\operatorname{Br}_0(P_F),\\,G\\neq F}\n \\operatorname{ord}_z(F-G),&\\text{otherwise}.\n \\end{cases}\n\\]\nThe corresponding branch selector is the finite coefficient jet\n\\[\n \\boxed{\n J_{\\rm alg}(F)\n =\n (a_0,a_1,\\ldots,a_{\\sigma-1}),\n \\qquad\n F(z)=\\sum_{n\\geq0}a_nz^n.}\n\\]\nWhen $\\sigma=0$, the selector is empty.\n\\end{definition}","\\begin{theorem}[Finite branch separation]\n\\label{thm:block10-finite-branch-separation}\nThe pair\n\\[\n \\boxed{(P_F,J_{\\rm alg}(F))}\n\\]\nselects $F$ uniquely among the formal power-series branches of $P_F$ at the origin.\n\\end{theorem}","\\begin{proof}\nThere are at most $d(F)$ formal branches.  The difference of two distinct formal series has finite $z$-adic order.  By definition of $\\sigma$, every other branch differs from $F$ in one of the coefficients recorded by $J_{\\rm alg}(F)$.\n\\end{proof}","\\begin{corollary}[Simple-branch normalization]\n\\label{cor:block10-simple-branch}\nIf\n\\[\n P_F(0,a_0)=0,\n \\qquad\n \\partial_Y P_F(0,a_0)\\neq0,\n\\]\nthen $a_0$ alone selects a unique formal branch.  Equivalently, $\\sigma(P_F,F)\\leq1$.\n\\end{corollary}","\\begin{proof}\nCoefficient comparison in $P_F(z,F(z))=0$ determines each next coefficient linearly, with diagonal coefficient $\\partial_Y P_F(0,a_0)$.\n\\end{proof}","The finite selector is semantic data.  It cannot be dropped merely because the polynomial is irreducible over $K(z)$.  Irreducibility controls the generic function-field extension, while the jet chooses one local embedding of that extension into formal series.","Let\n\\[\n A_F=K(z)[Y]/(P_F),\n\\]\nand let $\\bar Y$ denote the residue class of $Y$.  In the basis\n\\[\n 1,\\bar Y,\\ldots,\\bar Y^{d-1},\n\\]\nmultiplication by $\\bar Y$ is represented by the companion matrix $M_F$.  Its characteristic and minimal polynomials are both $P_F$.","\\begin{theorem}[Normalized selected-branch algebraic atlas]\n\\label{thm:block10-branch-atlas}\nThe following normalized presentations determine one another exactly:\n\\[\n \\boxed{\n \\begin{aligned}\n &F\\in K[[z]],\\\\\n &(P_F,J_{\\rm alg}(F)),\\\\\n &(A_F,\\bar Y,J_{\\rm alg}(F)),\\\\\n &(M_F,J_{\\rm alg}(F)).\n \\end{aligned}}\n\\]\nWith equality morphisms, these presentations form equivalent normalized groupoids.  Every fiber over a fixed semantic branch is contractible.\n\\end{theorem}","\\begin{proof}\nThe semantic branch gives its unique monic minimal polynomial and its finite branch selector.  The polynomial gives the finite field algebra and its companion multiplication matrix.  Conversely, the companion matrix recovers $P_F$ as its minimal polynomial.  The branch selector then recovers the selected formal root by \\cref{thm:block10-finite-branch-separation}.  Every normalized presentation is determined literally, so a normalized automorphism fixing the semantic branch is the identity.\n\\end{proof}","Fix $N\\geq1$.  For\n\\[\n F(z)=\\sum_{n\\geq0}a_nz^n,\n\\]\ndefine the arithmetic packets\n\\[\n U_j(x)\n =\n \\sum_{m\\geq0}a_{Nm+j}x^m,\n \\qquad\n 0\\leq j<N.\n\\]\nThey satisfy the exact synthesis identity\n\\[\n \\boxed{\n F(z)\n =\n \\sum_{j=0}^{N-1}z^jU_j(z^N).}\n\\]","Let $P(z,Y)\\in K[z,Y]$ be the normalized primitive representative of $P_F$.  Introduce indeterminates\n\\[\n \\mathbf U=(U_0,\\ldots,U_{N-1})\n\\]\nand substitute\n\\[\n Y=\\sum_{j=0}^{N-1}z^jU_j.\n\\]\nEvery polynomial in $z$ has a unique decomposition modulo the relation $z^N=x$:\n\\[\n P\\!\\left(z,\\sum_{j=0}^{N-1}z^jU_j\\right)\n =\n \\sum_{r=0}^{N-1}z^r\\Phi_{N,r}(x,\\mathbf U).\n\\]\nThe polynomials\n\\[\n \\boxed{\n \\Phi_{N,r}(x,\\mathbf U)\\in K[x,\\mathbf U]}\n\\]\nare the algebraic packet equations.","\\begin{theorem}[Exact algebraic packet system]\n\\label{thm:block10-exact-packet-system}\nLet $d=\\deg_Y P_F$.  Then:\n\\begin{enumerate}\n\\item the semantic packet tuple $(U_0,\\ldots,U_{N-1})$ satisfies\n\\[\n \\Phi_{N,r}(x,\\mathbf U)=0\n \\qquad(0\\leq r<N);\n\\]\n\\item every $\\Phi_{N,r}$ has total degree at most $d$ in the packet variables;\n\\item a formal packet tuple satisfying all $N$ equations synthesizes a formal branch of $P_F$;\n\\item the packet tuple selected by the transported coefficient jet reconstructs $F$ uniquely.\n\\end{enumerate}\n\\end{theorem}","act algebraic packet system]\n\\label{thm:block10-exact-packet-system}\nLet $d=\\deg_Y P_F$.  Then:\n\\begin{enumerate}\n\\item the semantic packet tuple $(U_0,\\ldots,U_{N-1})$ satisfies\n\\[\n \\Phi_{N,r}(x,\\mathbf U)=0\n \\qquad(0\\leq r<N);\n\\]\n\\item every $\\Phi_{N,r}$ has total degree at most $d$ in the packet variables;\n\\item a formal packet tuple satisfying all $N$ equations synthesizes a formal branch of $P_F$;\n\\item the packet tuple selected by the transported coefficient jet reconstructs $F$ uniquely.\n\\end{enumerate}\n\\end{theorem}",".  Then:\n\\begin{enumerate}\n\\item the semantic packet tuple $(U_0,\\ldots,U_{N-1})$ satisfies\n\\[\n \\Phi_{N,r}(x,\\mathbf U)=0\n \\qquad(0\\leq r<N);\n\\]\n\\item every $\\Phi_{N,r}$ has total degree at most $d$ in the packet variables;\n\\item a formal packet tuple satisfying all $N$ equations synthesizes a formal branch of $P_F$;\n\\item the packet tuple selected by the transported coefficient jet reconstructs $F$ uniquely.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nSubstituting the semantic synthesis formula into $P_F(z,F(z))=0$ gives the first statement.  Substitution does not increase degree in the $U_j$, so the second follows.  If all residue polynomials vanish, then the synthesized series satisfies $P_F=0$.  The source branch selector is distributed among the packets by residue class.  The resulting packet jet synthesizes the original jet, so \\cref{thm:block10-finite-branch-separation} identifies the selected branch.\n\\end{proof}","If the source selector records $a_0,\\ldots,a_{\\sigma-1}$, the transported packet selector records precisely\n\\[\n [x^m]U_j=a_{Nm+j}\n \\qquad\n \\text{for }Nm+j<\\sigma.\n\\]\nNo information is added or lost.  The coefficient prefix is merely rewritten in packet addresses.","Put\n\\[\n R=K(x),\n \\qquad\n E_N=R[z]/(z^N-x).\n\\]\nCharacteristic zero makes $z^N-x$ separable over $R$.  Interpret the monic polynomial $P_F(z,Y)$ as a polynomial in $E_N[Y]$, and define\n\\[\n B_{N,F}=E_N[Y]/(P_F).\n\\]\nThe packet polynomial system is the coordinate presentation of the finite Weil restriction\n\\[\n \\operatorname{Res}_{E_N/R}\\operatorname{Spec}B_{N,F}.\n\\]","\\begin{theorem}[Algebraic packet degree bound]\n\\label{thm:block10-packet-degree-bound}\nLet $F$ have algebraic degree $d$ over $K(z)$.  For every packet width $N$:\n\\begin{enumerate}\n\\item the generic geometric degree of the complete packet scheme is\n\\[\n \\boxed{d^N;}\n\\]\n\\item the joint field generated by a selected complete packet satisfies\n\\[\n \\boxed{\n [K(x)(U_0,\\ldots,U_{N-1}):K(x)]\\leq d^N;}\n\\]\n\\item every individual packet satisfies\n\\[\n \\boxed{\n [K(x)(U_j):K(x)]\\leq d^N.}\n\\]\n\\end{enumerate}\n\\end{theorem}","ound]\n\\label{thm:block10-packet-degree-bound}\nLet $F$ have algebraic degree $d$ over $K(z)$.  For every packet width $N$:\n\\begin{enumerate}\n\\item the generic geometric degree of the complete packet scheme is\n\\[\n \\boxed{d^N;}\n\\]\n\\item the joint field generated by a selected complete packet satisfies\n\\[\n \\boxed{\n [K(x)(U_0,\\ldots,U_{N-1}):K(x)]\\leq d^N;}\n\\]\n\\item every individual packet satisfies\n\\[\n \\boxed{\n [K(x)(U_j):K(x)]\\leq d^N.}\n\\]\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nOver an algebraic closure of $K(x)$, the equation $z^N=x$ has $N$ distinct roots.  The algebra $E_N$ therefore splits into a product of $N$ copies of the algebraic closure.  At each address, the separable degree-$d$ polynomial $P_F$ has $d$ geometric roots.  The Weil-restricted fiber is their Cartesian product and has $d^N$ points.  A selected packet tuple is one point of this finite scheme, so its residue field and every coordinate subfield have degree at most the scheme degree.\n\\end{proof}","This bound is constructive.  The quotient of\n\\[\n K(x)[U_0,\\ldots,U_{N-1}]\n\\]\nby the packet equations is a finite algebra of generic rank $d^N$.  Multiplication by $U_j$ is therefore a matrix of size $d^N$.  Its characteristic polynomial annihilates $U_j$ by Cayley--Hamilton.","\\begin{corollary}[Constructive packet eliminant]\n\\label{cor:block10-constructive-eliminant}\nFor each $j$, one can compute an annihilating polynomial\n\\[\n Q_{N,j}(x,T)\\in K(x)[T]\n\\]\nwith\n\\[\n \\deg_TQ_{N,j}\\leq d^N\n\\]\nby multiplication matrices, resultants, or a Gr\\\"obner elimination of the exact packet system.\n\\end{corollary}","The bound is a universal degree ceiling for the minimal-polynomial grammar.  It does not assert that every coordinate attains degree $d^N$ or that one elimination algorithm returns the minimal polynomial without factor selection.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.98\\textwidth]{algebraic_packet_weil_restriction.pdf}\n\\caption{After the address extension $x=z^N$ splits, each of the $N$ addresses carries $d$ algebraic branches.  The complete packet scheme is their Cartesian product and has generic degree $d^N$.}\n\\label{fig:block10-weil-restriction}\n\\end{figure}","Let $\\Omega$ be a normal splitting field of the generic packet scheme over $K(x)$, and let\n\\[\n \\Gamma=\\operatorname{Gal}(\\Omega/K(x)).\n\\]\nAfter splitting $E_N$, the geometric packet branches form a set\n\\[\n \\mathcal B_N\n\\]\nof cardinality $d^N$.  Let\n\\[\n \\pi_j:\\mathcal B_N\\to\\Omega\n\\]\nbe the $j$th packet coordinate, and set\n\\[\n \\mathcal C_{N,j}=\\pi_j(\\mathcal B_N).\n\\]\nFor the selected semantic packet $U_j$, separability gives\n\\[\n [K(x)(U_j):K(x)]\n =\n |\\Gamma\\cdot U_j|.\n\\]","$\\Omega$ be a normal splitting field of the generic packet scheme over $K(x)$, and let\n\\[\n \\Gamma=\\operatorname{Gal}(\\Omega/K(x)).\n\\]\nAfter splitting $E_N$, the geometric packet branches form a set\n\\[\n \\mathcal B_N\n\\]\nof cardinality $d^N$.  Let\n\\[\n \\pi_j:\\mathcal B_N\\to\\Omega\n\\]\nbe the $j$th packet coordinate, and set\n\\[\n \\mathcal C_{N,j}=\\pi_j(\\mathcal B_N).\n\\]\nFor the selected semantic packet $U_j$, separability gives\n\\[\n [K(x)(U_j):K(x)]\n =\n |\\Gamma\\cdot U_j|.\n\\]","ting field of the generic packet scheme over $K(x)$, and let\n\\[\n \\Gamma=\\operatorname{Gal}(\\Omega/K(x)).\n\\]\nAfter splitting $E_N$, the geometric packet branches form a set\n\\[\n \\mathcal B_N\n\\]\nof cardinality $d^N$.  Let\n\\[\n \\pi_j:\\mathcal B_N\\to\\Omega\n\\]\nbe the $j$th packet coordinate, and set\n\\[\n \\mathcal C_{N,j}=\\pi_j(\\mathcal B_N).\n\\]\nFor the selected semantic packet $U_j$, separability gives\n\\[\n [K(x)(U_j):K(x)]\n =\n |\\Gamma\\cdot U_j|.\n\\]","\\begin{definition}[Algebraic packet collision defects]\nDefine\n\\[\n \\mathfrak c^{\\rm tuple}_{N,j}\n =\n d^N-|\\mathcal C_{N,j}|,\n\\]\nand\n\\[\n \\mathfrak s^{\\rm Gal}_{N,j}\n =\n |\\mathcal C_{N,j}|-|\\Gamma\\cdot U_j|.\n\\]\nThe first records collisions caused by projection from complete branch tuples to one packet coordinate.  The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.\n\\end{definition}","\\begin{theorem}[Exact algebraic degree-defect identity]\n\\label{thm:block10-degree-defect}\nThe two defects are nonnegative and satisfy\n\\[\n \\boxed{\n d^N-[K(x)(U_j):K(x)]\n =\n \\mathfrak c^{\\rm tuple}_{N,j}\n +\n \\mathfrak s^{\\rm Gal}_{N,j}.}\n\\]\n\\end{theorem}","\\begin{proof}\nThe coordinate image $\\mathcal C_{N,j}$ has at most $d^N$ values.  The Galois orbit of the selected value is contained in that image.  The displayed identity is the telescoping decomposition\n\\[\n d^N-|\\Gamma\\cdot U_j|\n =\n (d^N-|\\mathcal C_{N,j}|)\n +\n (|\\mathcal C_{N,j}|-|\\Gamma\\cdot U_j|).\n\\]\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\textwidth]{algebraic_collision_defects.pdf}\n\\caption{The universal degree $d^N$ can drop first because several branch tuples have the same packet coordinate, then because the selected coordinate has a nontrivial Galois stabilizer.}\n\\label{fig:block10-collision-ledger}\n\\end{figure}","Let\n\\[\n C(z)=1+zC(z)^2\n\\]\nbe the Catalan branch with $C(0)=1$, and write\n\\[\n C(z)=E(z^2)+zO(z^2).\n\\]\nPut $x=z^2$.","\\begin{theorem}[Exact Catalan packet system]\n\\label{thm:block10-catalan-system}\nThe parity packets satisfy\n\\[\n \\boxed{\n E=1+2xEO,\n \\qquad\n O=E^2+xO^2.}\n\\]\nElimination gives\n\\[\n \\boxed{4xE^4-E^2+1=0}\n\\]\nand\n\\[\n \\boxed{4x^3O^4-8x^2O^3+5xO^2-O+1=0.}\n\\]\nBoth packets have algebraic degree four over $K(x)$.  Hence the universal bound\n\\[\n d^N=2^2=4\n\\]\nis sharp.\n\\end{theorem}","\\begin{proof}\nSubstitute $C=E+zO$ into $C=1+zC^2$ and separate even and odd powers.  The two resultants give the displayed quartics.","For the even packet, put $T=E^2$.  Then\n\\[\n 4xT^2-T+1=0.\n\\]\nIts discriminant $1-16x$ is not a square in $K(x)$, so $[K(x)(T):K(x)]=2$.  The norm of $T$ from this quadratic field is $1/(4x)$, which is not a square in $K(x)$.  Therefore $T$ is not a square in its quadratic field and $E$ has degree four.","The first packet equation gives\n\\[\n O=\\frac{E-1}{2xE},\n \\qquad\n E=\\frac{1}{1-2xO}.\n\\]\nThus $K(x)(E)=K(x)(O)$, and the odd packet also has degree four.\n\\end{proof}","Let\n\\[\n F_{\\alpha,\\beta}(z)\n =\n \\sqrt{(1-\\alpha z)(1-\\beta z)},\n \\qquad\n F_{\\alpha,\\beta}(0)=1,\n\\]\nand assume first that $\\alpha\\neq\\beta$, so the normalized source degree is two.  Write\n\\[\n F_{\\alpha,\\beta}(z)=E(x)+zO(x),\n \\qquad\n x=z^2,\n\\]\nand put\n\\[\n s=\\alpha+\\beta,\n \\qquad\n p=\\alpha\\beta.\n\\]","\\begin{theorem}[Quadratic-radical parity packets]\n\\label{thm:block10-radical-packets}\nThe packets satisfy\n\\[\n \\boxed{\n E^2+xO^2=1+px,\n \\qquad\n 2EO=-s.}\n\\]\nTheir eliminants are\n\\[\n \\boxed{\n E^4-(1+px)E^2+\\frac{s^2x}{4}=0}\n\\]\nand\n\\[\n \\boxed{\n xO^4-(1+px)O^2+\\frac{s^2}{4}=0.}\n\\]\nIf\n\\[\n s\\neq0,\n \\qquad\n \\alpha^2\\neq\\beta^2,\n\\]\nthen both selected packets have degree four over $K(x)$.\n\\end{theorem}","\\begin{proof}\nThe two equations follow by adding and subtracting the squares of\n\\[\n F(z)=E+zO,\n \\qquad\n F(-z)=E-zO.\n\\]\nElimination gives the quartics.  Their quadratic discriminant is\n\\[\n \\Delta(x)\n =\n (1-\\alpha^2x)(1-\\beta^2x),\n\\]\nwhich is not a square under the stated hypotheses.  Hence $E^2$ generates a quadratic extension.  Its norm is $s^2x/4$, which is not a square in $K(x)$, so $E$ has degree four.  Since $EO=-s/2$, the fields generated by $E$ and $O$ coincide.\n\\end{proof}","two equations follow by adding and subtracting the squares of\n\\[\n F(z)=E+zO,\n \\qquad\n F(-z)=E-zO.\n\\]\nElimination gives the quartics.  Their quadratic discriminant is\n\\[\n \\Delta(x)\n =\n (1-\\alpha^2x)(1-\\beta^2x),\n\\]\nwhich is not a square under the stated hypotheses.  Hence $E^2$ generates a quadratic extension.  Its norm is $s^2x/4$, which is not a square in $K(x)$, so $E$ has degree four.  Since $EO=-s/2$, the fields generated by $E$ and $O$ coincide.\n\\end{proof}","subtracting the squares of\n\\[\n F(z)=E+zO,\n \\qquad\n F(-z)=E-zO.\n\\]\nElimination gives the quartics.  Their quadratic discriminant is\n\\[\n \\Delta(x)\n =\n (1-\\alpha^2x)(1-\\beta^2x),\n\\]\nwhich is not a square under the stated hypotheses.  Hence $E^2$ generates a quadratic extension.  Its norm is $s^2x/4$, which is not a square in $K(x)$, so $E$ has degree four.  Since $EO=-s/2$, the fields generated by $E$ and $O$ coincide.\n\\end{proof}","ting the squares of\n\\[\n F(z)=E+zO,\n \\qquad\n F(-z)=E-zO.\n\\]\nElimination gives the quartics.  Their quadratic discriminant is\n\\[\n \\Delta(x)\n =\n (1-\\alpha^2x)(1-\\beta^2x),\n\\]\nwhich is not a square under the stated hypotheses.  Hence $E^2$ generates a quadratic extension.  Its norm is $s^2x/4$, which is not a square in $K(x)$, so $E$ has degree four.  Since $EO=-s/2$, the fields generated by $E$ and $O$ coincide.\n\\end{proof}","\\begin{corollary}[Antisymmetric branch collision]\n\\label{cor:block10-antisymmetric-collision}\nIf $\\beta=-\\alpha\\neq0$, then\n\\[\n \\boxed{O=0,\n \\qquad\n E^2=1-\\alpha^2x.}\n\\]\nFor the even coordinate,\n\\[\n |\\mathcal B_2|=4,\n \\qquad\n |\\mathcal C_{2,0}|=3,\n \\qquad\n |\\Gamma\\cdot E|=2.\n\\]\nThus\n\\[\n \\mathfrak c^{\\rm tuple}_{2,0}=1,\n \\qquad\n \\mathfrak s^{\\rm Gal}_{2,0}=1.\n\\]\nFor the selected odd coordinate,\n\\[\n |\\mathcal C_{2,1}|=3,\n \\qquad\n |\\Gamma\\cdot O|=1,\n\\]\nso\n\\[\n \\mathfrak c^{\\rm tuple}_{2,1}=1,\n \\qquad\n \\mathfrak s^{\\rm Gal}_{2,1}=2.\n\\]\n\\end{corollary}","r:block10-antisymmetric-collision}\nIf $\\beta=-\\alpha\\neq0$, then\n\\[\n \\boxed{O=0,\n \\qquad\n E^2=1-\\alpha^2x.}\n\\]\nFor the even coordinate,\n\\[\n |\\mathcal B_2|=4,\n \\qquad\n |\\mathcal C_{2,0}|=3,\n \\qquad\n |\\Gamma\\cdot E|=2.\n\\]\nThus\n\\[\n \\mathfrak c^{\\rm tuple}_{2,0}=1,\n \\qquad\n \\mathfrak s^{\\rm Gal}_{2,0}=1.\n\\]\nFor the selected odd coordinate,\n\\[\n |\\mathcal C_{2,1}|=3,\n \\qquad\n |\\Gamma\\cdot O|=1,\n\\]\nso\n\\[\n \\mathfrak c^{\\rm tuple}_{2,1}=1,\n \\qquad\n \\mathfrak s^{\\rm Gal}_{2,1}=2.\n\\]\n\\end{corollary}","\\begin{proof}\nNow $F(z)=F(-z)=\\sqrt{1-\\alpha^2z^2}$ for the selected branch, so the odd packet vanishes.  Over the split branch tuple set, the two signs are chosen independently.  The even projection has values $E,0,-E$, while the odd projection has values $0,E/z,-E/z$.  The selected Galois orbits have sizes two and one, respectively.\n\\end{proof}","If $\\alpha=\\beta$, the quadratic equation\n\\[\n Y^2-(1-\\alpha z)^2=0\n\\]\nis not the normalized source polynomial.  The selected branch is rational:\n\\[\n F_{\\alpha,\\alpha}(z)=1-\\alpha z,\n\\]\nso the correct source degree is one.  This is a negative control against using a nonminimal defining equation in the degree ledger.","\\begin{proposition}[Algebraic derivative closure]\n\\label{prop:block10-algebraic-dfinite}\nLet $F$ have separable minimal polynomial $P_F$ of degree $d$.  Then\n\\[\n F'\n =\n -\\frac{\\partial_zP_F(z,F)}{\\partial_YP_F(z,F)}\n \\in\n K(z,F).\n\\]\nEvery higher derivative lies in the same degree-$d$ function field.  Consequently, $F$ has a differential annihilator of order at most $d$.\n\\end{proposition}","\\begin{proof}\nDifferentiate $P_F(z,F(z))=0$.  Separability makes $\\partial_YP_F(z,F)$ nonzero in the function field.  The field is closed under the induced derivation.  The $d+1$ elements $F,F',\\ldots,F^{(d)}$ are linearly dependent over $K(z)$.\n\\end{proof}","The annihilator can forget both the polynomial relation and the selected branch.  For example,\n\\[\n \\sqrt{1+z}\n \\quad\\text{and}\\quad\n -\\sqrt{1+z}\n\\]\nsatisfy the same first-order differential operator.  Their initial values distinguish them.  More generally, a D-finite equation may have transcendental solutions, so an Ore chart does not reconstruct an algebraic minimal polynomial without additional recognition data.","\\begin{boundarybox}\n\\textbf{Exact algebraic boundary.}\nAlgebraic atlas proves finite selected-branch normalization, an exact packet polynomial compiler, the generic $d^N$ Weil-restriction degree, a constructive coordinate eliminant, and a two-stage collision invariant.  It proves sharp packet degrees for Catalan parity and a complete collision phase for a marked quadratic-radical family.  It does not prove that every elimination output is minimal without factor selection, one canonical minimal polynomial-system dimension across arbitrary multivariate grammars, an optimal packet-degree bound using all source symmetries, or unrestricted equivalence between algebraic and D-finite presentations.\n\\end{boundarybox}","The verifier independently constructs packet equations by substitution and reduction modulo $z^N=x$.  It checks Catalan, Motzkin, and marked radical branches at several packet widths.  It verifies the sharp Catalan quartics, the quadratic-radical trace and norm systems, the generic and antisymmetric collision phases, the rejection of nonminimal quadratic equations, and finite branch-selector certificates.","\\begin{resultbox}\n\\textbf{Section result.}\nA selected algebraic branch has a canonical minimal polynomial and a finite branch-separating jet.  Complete arithmetic packets are the solutions of an explicit $N$-equation polynomial system obtained by residue reduction.  The generic packet scheme is a finite Weil restriction of degree $d^N$, giving a constructive degree bound for every packet coordinate.  Degree loss splits exactly into tuple projection collisions and Galois stabilizer loss.  Catalan parity attains the universal quadratic bound, while the marked radical family exhibits generic sharpness, antisymmetric cancellation, and rational collapse.  The algebraic-to-holonomic bridge remains directional unless the minimal polynomial and branch selector are retained.\n\\end{resultbox}","Throughout the analytic part, $X$ is a Banach space over $\\mathbb R$ or $\\mathbb C$.  A finite packet system on $X$ is a family of bounded projections\n\\[\n P_iP_j=\\delta_{ij}P_i,\n \\qquad\n \\sum_{i\\in I}P_i=I_X.\n\\]\nAll sums over packet addresses are finite.  The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.","Let\n\\[\n A_m:X^m\\longrightarrow X,\n \\qquad 2\\le m\\le D,\n\\]\nbe bounded symmetric multilinear maps, and let $s\\in X$.  Consider the marked equation\n\\[\n \\boxed{\n u=s+\\Phi(u),\n \\qquad\n \\Phi(u)=\\sum_{m=2}^{D}A_m[u,\\ldots,u].}\n \\tag{11.1}\n\\]\nThe word \\emph{marked} means that the source term, every multilinear operation, the packet system, and the selected convergence or filtration certificate are retained as part of the presentation.  The fixed point $u$ alone is not assumed to reconstruct those data.","le D,\n\\]\nbe bounded symmetric multilinear maps, and let $s\\in X$.  Consider the marked equation\n\\[\n \\boxed{\n u=s+\\Phi(u),\n \\qquad\n \\Phi(u)=\\sum_{m=2}^{D}A_m[u,\\ldots,u].}\n \\tag{11.1}\n\\]\nThe word \\emph{marked} means that the source term, every multilinear operation, the packet system, and the selected convergence or filtration certificate are retained as part of the presentation.  The fixed point $u$ alone is not assumed to reconstruct those data.","Put\n\\[\n a=\\lVert s\\rVert,\n \\qquad\n b_m=\\lVert A_m\\rVert,\n \\qquad\n \\varphi(t)=\\sum_{m=2}^{D}b_mt^m.\n\\]\nFor a radius $R>0$, define\n\\[\n \\kappa_R=\\varphi'(R)\n =\\sum_{m=2}^{D}m b_mR^{m-1}.\n\\]","\\begin{theorem}[Selected polynomial fixed point and tree expansion]\n\\label{thm:block11-polynomial-fixed-point}\nAssume\n\\[\n a+\\varphi(R)\\le R,\n \\qquad\n \\kappa_R<1.\n\\]\nThen equation \\textup{(11.1)} has a unique solution in the closed ball $\\overline B_R$.  Moreover, that solution is the absolutely convergent sum\n\\[\n \\boxed{\n u=\\sum_{T\\in\\mathcal T_{\\mathcal A}}\\operatorname{ev}(T),}\n \\tag{11.2}\n\\]\nwhere $\\mathcal A=\\{m:A_m\\neq0\\}$, the leaves of $T$ evaluate to $s$, and an internal vertex of arity $m$ evaluates by $A_m$ applied to the ordered values of its children.\n\\end{theorem}","n{theorem}[Selected polynomial fixed point and tree expansion]\n\\label{thm:block11-polynomial-fixed-point}\nAssume\n\\[\n a+\\varphi(R)\\le R,\n \\qquad\n \\kappa_R<1.\n\\]\nThen equation \\textup{(11.1)} has a unique solution in the closed ball $\\overline B_R$.  Moreover, that solution is the absolutely convergent sum\n\\[\n \\boxed{\n u=\\sum_{T\\in\\mathcal T_{\\mathcal A}}\\operatorname{ev}(T),}\n \\tag{11.2}\n\\]\nwhere $\\mathcal A=\\{m:A_m\\neq0\\}$, the leaves of $T$ evaluate to $s$, and an internal vertex of arity $m$ evaluates by $A_m$ applied to the ordered values of its children.\n\\end{theorem}","and tree expansion]\n\\label{thm:block11-polynomial-fixed-point}\nAssume\n\\[\n a+\\varphi(R)\\le R,\n \\qquad\n \\kappa_R<1.\n\\]\nThen equation \\textup{(11.1)} has a unique solution in the closed ball $\\overline B_R$.  Moreover, that solution is the absolutely convergent sum\n\\[\n \\boxed{\n u=\\sum_{T\\in\\mathcal T_{\\mathcal A}}\\operatorname{ev}(T),}\n \\tag{11.2}\n\\]\nwhere $\\mathcal A=\\{m:A_m\\neq0\\}$, the leaves of $T$ evaluate to $s$, and an internal vertex of arity $m$ evaluates by $A_m$ applied to the ordered values of its children.\n\\end{theorem}","on]\n\\label{thm:block11-polynomial-fixed-point}\nAssume\n\\[\n a+\\varphi(R)\\le R,\n \\qquad\n \\kappa_R<1.\n\\]\nThen equation \\textup{(11.1)} has a unique solution in the closed ball $\\overline B_R$.  Moreover, that solution is the absolutely convergent sum\n\\[\n \\boxed{\n u=\\sum_{T\\in\\mathcal T_{\\mathcal A}}\\operatorname{ev}(T),}\n \\tag{11.2}\n\\]\nwhere $\\mathcal A=\\{m:A_m\\neq0\\}$, the leaves of $T$ evaluate to $s$, and an internal vertex of arity $m$ evaluates by $A_m$ applied to the ordered values of its children.\n\\end{theorem}","ed-point}\nAssume\n\\[\n a+\\varphi(R)\\le R,\n \\qquad\n \\kappa_R<1.\n\\]\nThen equation \\textup{(11.1)} has a unique solution in the closed ball $\\overline B_R$.  Moreover, that solution is the absolutely convergent sum\n\\[\n \\boxed{\n u=\\sum_{T\\in\\mathcal T_{\\mathcal A}}\\operatorname{ev}(T),}\n \\tag{11.2}\n\\]\nwhere $\\mathcal A=\\{m:A_m\\neq0\\}$, the leaves of $T$ evaluate to $s$, and an internal vertex of arity $m$ evaluates by $A_m$ applied to the ordered values of its children.\n\\end{theorem}","\\[\n a+\\varphi(R)\\le R,\n \\qquad\n \\kappa_R<1.\n\\]\nThen equation \\textup{(11.1)} has a unique solution in the closed ball $\\overline B_R$.  Moreover, that solution is the absolutely convergent sum\n\\[\n \\boxed{\n u=\\sum_{T\\in\\mathcal T_{\\mathcal A}}\\operatorname{ev}(T),}\n \\tag{11.2}\n\\]\nwhere $\\mathcal A=\\{m:A_m\\neq0\\}$, the leaves of $T$ evaluate to $s$, and an internal vertex of arity $m$ evaluates by $A_m$ applied to the ordered values of its children.\n\\end{theorem}","\\begin{proof}\nThe map\n\\[\n \\Psi(x)=s+\\Phi(x)\n\\]\npreserves $\\overline B_R$ because\n\\[\n \\lVert\\Psi(x)\\rVert\n \\le a+\\sum_{m=2}^{D}b_mR^m\n \\le R.\n\\]\nFor $x,y\\in\\overline B_R$, multilinearity gives\n\\[\n A_m[x^m]-A_m[y^m]\n =\n \\sum_{r=1}^{m}\n A_m[y^{r-1},x-y,x^{m-r}],\n\\]\nso\n\\[\n \\lVert\\Psi(x)-\\Psi(y)\\rVert\n \\le\n \\kappa_R\\lVert x-y\\rVert.\n\\]\nThe contraction theorem gives the unique fixed point.","Let $u^{(0)}=0$ and $u^{(h+1)}=\\Psi(u^{(h)})$.  Expanding recursively shows that $u^{(h)}$ is the sum of the evaluations of all allowed plane rooted trees of height at most $h-1$.  If $r_0=0$ and\n\\[\n r_{h+1}=a+\\varphi(r_h),\n\\]\nthen the sum of the norms of those tree evaluations is at most $r_h$.  The scalar iterates increase and remain bounded by $R$, hence converge.  Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum.  Regrouping trees by the arity of the root gives equation \\textup{(11.1)}.\n\\end{proof}","\\begin{remark}[Formal filtered version]\nIf $X$ is replaced by a complete filtered module, $s$ begins in positive filtration, and every $A_m$ raises total filtration, then the same tree expansion is coefficientwise finite at each filtration depth.  It therefore defines the unique formal solution without a Banach norm.  The analytic and formal doctrines are related but are not silently identified.\n\\end{remark}","\\begin{boundary}[The selected object is a marked system]\nTwo different polynomial systems can have the same fixed point.  Conversely, a single polynomial system can have several fixed points outside the contraction ball.  The normalized object in this section is therefore\n\\[\n (s,A_2,\\ldots,A_D;\\mathcal P;\\text{certificate}),\n\\]\nnot an unmarked semantic vector $u$.\n\\end{boundary}","For $i,j_1,\\ldots,j_m\\in I$, define the resolved multilinear block\n\\[\n A_{m;i\\mid j_1,\\ldots,j_m}\n =\n P_iA_m(P_{j_1}\\,\\cdot,\\ldots,P_{j_m}\\,\\cdot).\n\\]\nCompleteness of the packet family gives the exact reconstruction\n\\[\n \\boxed{\n A_m\n =\n \\sum_{i,j_1,\\ldots,j_m}\n A_{m;i\\mid j_1,\\ldots,j_m}.}\n \\tag{11.3}\n\\]\nWrite\n\\[\n u_i=P_iu,\n \\qquad\n s_i=P_is.\n\\]\nThen equation \\textup{(11.1)} is equivalent to the complete packet system\n\\[\n \\boxed{\n u_i\n =\n s_i+\n \\sum_{m=2}^{D}\n \\sum_{j_1,\\ldots,j_m}\n A_{m;i\\mid j_1,\\ldots,j_m}\n [u_{j_1},\\ldots,u_{j_m}].}\n \\tag{11.4}\n\\]","I$, define the resolved multilinear block\n\\[\n A_{m;i\\mid j_1,\\ldots,j_m}\n =\n P_iA_m(P_{j_1}\\,\\cdot,\\ldots,P_{j_m}\\,\\cdot).\n\\]\nCompleteness of the packet family gives the exact reconstruction\n\\[\n \\boxed{\n A_m\n =\n \\sum_{i,j_1,\\ldots,j_m}\n A_{m;i\\mid j_1,\\ldots,j_m}.}\n \\tag{11.3}\n\\]\nWrite\n\\[\n u_i=P_iu,\n \\qquad\n s_i=P_is.\n\\]\nThen equation \\textup{(11.1)} is equivalent to the complete packet system\n\\[\n \\boxed{\n u_i\n =\n s_i+\n \\sum_{m=2}^{D}\n \\sum_{j_1,\\ldots,j_m}\n A_{m;i\\mid j_1,\\ldots,j_m}\n [u_{j_1},\\ldots,u_{j_m}].}\n \\tag{11.4}\n\\]","The active nonlinear transfer hypergraph has packet addresses as colors and one typed hyperedge\n\\[\n (j_1,\\ldots,j_m)\\longrightarrow i\n\\]\nfor every nonzero block $A_{m;i\\mid j_1,\\ldots,j_m}$.  Rooted substitution of these hyperedges is precisely the colored symmetric operad already present in the nonlinear core sources \\cite{RT,NLO}.","\\begin{theorem}[Normalized atlas of a marked polynomial system]\n\\label{thm:block11-marked-atlas}\nFix a finite packet system and a certificate satisfying the hypotheses of \\cref{thm:block11-polynomial-fixed-point}, or a declared complete filtration satisfying the formal version.  The following presentations determine one another exactly:\n\\[\n \\boxed{\n \\begin{aligned}\n &\\text{the marked polynomial system }(s,A_2,\\ldots,A_D),\\\\\n &\\text{the complete family of packet blocks }A_{m;i\\mid j_1,\\ldots,j_m},\\\\\n &\\text{the colored operadic tree grammar with its actual vertex operations},\\\\\n &\\text{the exact packet fixed-point system \\textup{(11.4)}}.\n \\end{aligned}}\n\\]\nEach chart reconstructs the same selected fixed point.  With literal normalized data and equality morphisms, every semantic fiber is contractible.\n\\end{theorem}","The packet Fa\\`a di Bruno theorem from the nonlinear source expands an $n$th derivative over all set partitions and all intermediate packet labels \\cite{P3,RT,NLO}.  When all directions are equal, the symmetric derivative tensors make many labeled terms identical.  They can be grouped exactly by block-size and packet-color multiplicities.","Let $F:Y\\to Z$ and $G:X\\to Y$ be $n$ times differentiable.  Fix an input packet $\\beta$, an output packet $\\delta$, a point $x$, and a direction $h\\in P_\\beta X$.  For every intermediate packet $\\gamma$ and every $r\\ge1$, put\n\\[\n g_{r,\\gamma}\n =\n Q_\\gamma D^rG(x)[h,\\ldots,h].\n\\]\nA colored partition profile is a finite family\n\\[\n \\mu=(\\mu_{r,\\gamma})_{r\\ge1,\\,\\gamma\\in\\Gamma}\n\\]\nof nonnegative integers satisfying\n\\[\n \\sum_{r,\\gamma}r\\mu_{r,\\gamma}=n.\n\\]\nLet\n\\[\n k(\\mu)=\\sum_{r,\\gamma}\\mu_{r,\\gamma}.\n\\]","$G:X\\to Y$ be $n$ times differentiable.  Fix an input packet $\\beta$, an output packet $\\delta$, a point $x$, and a direction $h\\in P_\\beta X$.  For every intermediate packet $\\gamma$ and every $r\\ge1$, put\n\\[\n g_{r,\\gamma}\n =\n Q_\\gamma D^rG(x)[h,\\ldots,h].\n\\]\nA colored partition profile is a finite family\n\\[\n \\mu=(\\mu_{r,\\gamma})_{r\\ge1,\\,\\gamma\\in\\Gamma}\n\\]\nof nonnegative integers satisfying\n\\[\n \\sum_{r,\\gamma}r\\mu_{r,\\gamma}=n.\n\\]\nLet\n\\[\n k(\\mu)=\\sum_{r,\\gamma}\\mu_{r,\\gamma}.\n\\]","an input packet $\\beta$, an output packet $\\delta$, a point $x$, and a direction $h\\in P_\\beta X$.  For every intermediate packet $\\gamma$ and every $r\\ge1$, put\n\\[\n g_{r,\\gamma}\n =\n Q_\\gamma D^rG(x)[h,\\ldots,h].\n\\]\nA colored partition profile is a finite family\n\\[\n \\mu=(\\mu_{r,\\gamma})_{r\\ge1,\\,\\gamma\\in\\Gamma}\n\\]\nof nonnegative integers satisfying\n\\[\n \\sum_{r,\\gamma}r\\mu_{r,\\gamma}=n.\n\\]\nLet\n\\[\n k(\\mu)=\\sum_{r,\\gamma}\\mu_{r,\\gamma}.\n\\]","\\begin{theorem}[Compressed diagonal packet Fa\\`a di Bruno formula]\n\\label{thm:block11-compressed-faa}\nWith the notation above,\n\\[\n\\boxed{\n\\begin{aligned}\n&P_\\delta D^n(F\\circ G)(x)[h,\\ldots,h]\\\\\n&=\\sum_{\\mu}\n\\frac{n!}\n{\\displaystyle\\prod_{r,\\gamma}\n (r!)^{\\mu_{r,\\gamma}}\\mu_{r,\\gamma}!}\n\\,P_\\delta D^{k(\\mu)}F(Gx)\n\\bigl[\n (g_{r,\\gamma})^{\\mu_{r,\\gamma}}\n\\bigr]_{r,\\gamma}.\n\\end{aligned}}\n\\tag{11.5}\n\\]\nHere the last line means that $g_{r,\\gamma}$ is inserted $\\mu_{r,\\gamma}$ times; symmetry makes the insertion order irrelevant.\n\\end{theorem}","&=\\sum_{\\mu}\n\\frac{n!}\n{\\displaystyle\\prod_{r,\\gamma}\n (r!)^{\\mu_{r,\\gamma}}\\mu_{r,\\gamma}!}\n\\,P_\\delta D^{k(\\mu)}F(Gx)\n\\bigl[\n (g_{r,\\gamma})^{\\mu_{r,\\gamma}}\n\\bigr]_{r,\\gamma}.\n\\end{aligned}}\n\\tag{11.5}\n\\]\nHere the last line means that $g_{r,\\gamma}$ is inserted $\\mu_{r,\\gamma}$ times; symmetry makes the insertion order irrelevant.\n\\end{theorem}","\\begin{proof}\nStart with the packet Fa\\`a di Bruno sum over a set partition $\\pi$ of $\\{1,\\ldots,n\\}$ and a packet color attached to every block.  Because every input direction is $h$, the value of one term depends only on the number $\\mu_{r,\\gamma}$ of blocks of size $r$ carrying color $\\gamma$.  The symmetric group acts transitively on all colored set partitions with one fixed profile.  The stabilizer permutes the $r$ entries inside each block and permutes the indistinguishable blocks of a fixed type $(r,\\gamma)$.  Its order is\n\\[\n \\prod_{r,\\gamma}(r!)^{\\mu_{r,\\gamma}}\\mu_{r,\\gamma}!,\n\\]\nso the orbit has the coefficient printed in \\textup{(11.5)}.  Summing one representative per orbit gives the result.\n\\end{proof}","ith the packet Fa\\`a di Bruno sum over a set partition $\\pi$ of $\\{1,\\ldots,n\\}$ and a packet color attached to every block.  Because every input direction is $h$, the value of one term depends only on the number $\\mu_{r,\\gamma}$ of blocks of size $r$ carrying color $\\gamma$.  The symmetric group acts transitively on all colored set partitions with one fixed profile.  The stabilizer permutes the $r$ entries inside each block and permutes the indistinguishable blocks of a fixed type $(r,\\gamma)$.  Its order is\n\\[\n \\prod_{r,\\gamma}(r!)^{\\mu_{r,\\gamma}}\\mu_{r,\\gamma}!,\n\\]\nso the orbit has the coefficient printed in \\textup{(11.5)}.  Summing one representative per orbit gives the result.\n\\end{proof}","acket color attached to every block.  Because every input direction is $h$, the value of one term depends only on the number $\\mu_{r,\\gamma}$ of blocks of size $r$ carrying color $\\gamma$.  The symmetric group acts transitively on all colored set partitions with one fixed profile.  The stabilizer permutes the $r$ entries inside each block and permutes the indistinguishable blocks of a fixed type $(r,\\gamma)$.  Its order is\n\\[\n \\prod_{r,\\gamma}(r!)^{\\mu_{r,\\gamma}}\\mu_{r,\\gamma}!,\n\\]\nso the orbit has the coefficient printed in \\textup{(11.5)}.  Summing one representative per orbit gives the result.\n\\end{proof}","block.  Because every input direction is $h$, the value of one term depends only on the number $\\mu_{r,\\gamma}$ of blocks of size $r$ carrying color $\\gamma$.  The symmetric group acts transitively on all colored set partitions with one fixed profile.  The stabilizer permutes the $r$ entries inside each block and permutes the indistinguishable blocks of a fixed type $(r,\\gamma)$.  Its order is\n\\[\n \\prod_{r,\\gamma}(r!)^{\\mu_{r,\\gamma}}\\mu_{r,\\gamma}!,\n\\]\nso the orbit has the coefficient printed in \\textup{(11.5)}.  Summing one representative per orbit gives the result.\n\\end{proof}","\\begin{corollary}[Colored integer-partition term count]\n\\label{cor:block11-compressed-count}\nIf $Y$ has $M$ intermediate packets, the number of compressed diagonal profiles at order $n$ is\n\\[\n \\boxed{\n C_{n,M}\n =[x^n]\\prod_{r\\ge1}(1-x^r)^{-M}.}\n \\tag{11.6}\n\\]\nBefore compression, the number of packet-labeled set-partition terms is\n\\[\n N_{n,M}=\\sum_{k=1}^{n}S(n,k)M^k.\n\\]\n\\end{corollary}","\\begin{proof}\nFor each block size $r$, the multiplicities among $M$ colors contribute\n\\[\n (1-x^r)^{-M}.\n\\]\nMultiplying over all $r$ and taking the coefficient of $x^n$ counts exactly the profiles satisfying the weighted-size constraint.  The raw count is the existing packet term-count formula, grouped by the number of blocks.\n\\end{proof}","For the packet system \\textup{(11.4)}, define the directed dependency graph $G_\\Phi$ on $I$ by placing an edge\n\\[\n j\\longrightarrow i\n\\]\nwhen some active block contributing to the equation for $u_i$ contains $u_j$ as an input.  Let\n\\[\n \\operatorname{SCC}(G_\\Phi)\n\\]\nbe its strongly connected components.  A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.","\\begin{definition}[Address-preserving cyclic dimension]\nThe cyclic packet dimension of the marked system is\n\\[\n \\boxed{\n \\operatorname{cdim}_{\\mathcal P}(\\Phi)\n =\n \\sum_{C\\text{ cyclic SCC}}|C|.}\n \\tag{11.7}\n\\]\nIt measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.\n\\end{definition}","ted exactly once its predecessors have been determined.  Repeating this elimination gives a solution-preserving bijection between:\n\\begin{enumerate}\n\\item solutions of the full packet system, and\n\\item solutions of the subsystem supported on the cyclic SCCs, followed by deterministic reconstruction along the condensation graph.\n\\end{enumerate}\nConsequently, the coupled nonlinear core has exactly $\\operatorname{cdim}_{\\mathcal P}(\\Phi)$ packet variables.\n\\end{theorem}","\\begin{boundary}[Not a universal nonlinear state minimum]\nThe quantity \\(\\operatorname{cdim}_{\\mathcal P}\\) is invariant under relabeling of the declared packet system and exact elimination within its dependency grammar.  A nonlinear change of variables, hidden-variable introduction, algebraic elimination, or another packet family can change it.  It is therefore a certified address-preserving system dimension, not a universal minimum over all nonlinear presentations.\n\\end{boundary}","Consider the scalar marked equation\n\\[\n \\boxed{\n u(z)=z+\\Phi(u(z)),\n \\qquad\n \\Phi(y)=\\sum_{m\\in\\mathcal A}w_my^m,}\n \\tag{11.8}\n\\]\nwhere $\\mathcal A\\subseteq\\{2,3,\\ldots\\}$ is finite and every $w_m$ is a positive integer.  Its unique formal solution belongs to $z\\mathbb Z[[z]]$ and counts plane rooted trees whose internal arities lie in $\\mathcal A$, weighted by the $w_m$.","Define the arity period\n\\[\n \\boxed{\n d_{\\mathcal A}\n =\n \\gcd\\{m-1:m\\in\\mathcal A\\}.}\n \\tag{11.9}\n\\]\nEvery tree with internal vertices $v$ has leaf count\n\\[\n \\ell(T)=1+\\sum_{v}(\\deg(v)-1).\n\\]","\\begin{theorem}[Finite tree-address signature]\n\\label{thm:block11-tree-address}\nFix a packet width $N$ and put\n\\[\n g_N=\\gcd(N,d_{\\mathcal A}).\n\\]\nFor the arithmetic packets\n\\[\n U_j(x)=\\sum_{r\\ge0}[z^{Nr+j}]u(z)\\,x^r,\n\\]\nthe following are equivalent:\n\\begin{enumerate}\n\\item $U_j\\neq0$;\n\\item $j-1$ belongs to the subgroup of $\\mathbb Z/N\\mathbb Z$ generated by $\\{m-1:m\\in\\mathcal A\\}$;\n\\item\n\\[\n \\boxed{j\\equiv1\\pmod{g_N}.}\n\\]\n\\end{enumerate}\nHence the exact number of active packet addresses is\n\\[\n \\boxed{\n \\#\\operatorname{Supp}_N(u)=\\frac{N}{g_N}.}\n \\tag{11.10}\n\\]\n\\end{theorem}","$N$ and put\n\\[\n g_N=\\gcd(N,d_{\\mathcal A}).\n\\]\nFor the arithmetic packets\n\\[\n U_j(x)=\\sum_{r\\ge0}[z^{Nr+j}]u(z)\\,x^r,\n\\]\nthe following are equivalent:\n\\begin{enumerate}\n\\item $U_j\\neq0$;\n\\item $j-1$ belongs to the subgroup of $\\mathbb Z/N\\mathbb Z$ generated by $\\{m-1:m\\in\\mathcal A\\}$;\n\\item\n\\[\n \\boxed{j\\equiv1\\pmod{g_N}.}\n\\]\n\\end{enumerate}\nHence the exact number of active packet addresses is\n\\[\n \\boxed{\n \\#\\operatorname{Supp}_N(u)=\\frac{N}{g_N}.}\n \\tag{11.10}\n\\]\n\\end{theorem}","\\begin{proof}\nReplacing one leaf by an $m$-ary vertex increases the leaf count by $m-1$.  Starting from the one-leaf tree, every nonnegative sum of allowed increments is realized by a sequence of leaf replacements.  Positivity of the weights prevents cancellation, so the coefficient support is exactly\n\\[\n 1+\\left\\langle m-1:m\\in\\mathcal A\\right\\rangle_{\\mathbb N}.\n\\]\nModulo $N$, the monoid generated by the increments equals the subgroup they generate because every element of a finite cyclic subgroup has a nonnegative representative.  That subgroup is $g_N\\mathbb Z/N\\mathbb Z$, which proves the congruence and the count.\n\\end{proof}","crements is realized by a sequence of leaf replacements.  Positivity of the weights prevents cancellation, so the coefficient support is exactly\n\\[\n 1+\\left\\langle m-1:m\\in\\mathcal A\\right\\rangle_{\\mathbb N}.\n\\]\nModulo $N$, the monoid generated by the increments equals the subgroup they generate because every element of a finite cyclic subgroup has a nonnegative representative.  That subgroup is $g_N\\mathbb Z/N\\mathbb Z$, which proves the congruence and the count.\n\\end{proof}","\\begin{corollary}[Finite bottom-up residue automaton]\nLabel every leaf by $1\\in\\mathbb Z/N\\mathbb Z$ and every $m$-ary vertex by the sum of its child labels.  The reachable state set is exactly\n\\[\n \\{j:j\\equiv1\\pmod{g_N}\\}.\n\\]\nThus the finite pair\n\\[\n \\boxed{\n \\mathfrak T_N(\\mathcal A)\n =\n (d_{\\mathcal A},g_N,N/g_N)}\n \\tag{11.11}\n\\]\nclassifies packet support for the marked simply generated family.\n\\end{corollary}","Assume now that the coefficients $w_m$ are positive real numbers.  The classical smooth implicit-function schema for simply generated trees applies when there is a positive solution $\\tau$ of\n\\[\n \\Phi'(\\tau)=1\n\\]\nand\n\\[\n \\rho=\\tau-\\Phi(\\tau)>0.\n\\]\nUnder its standard nondegeneracy hypotheses, the reduced aperiodic series has a square-root singularity at its positive radius \\cite{FlajoletSedgewick2009}.  The arity period restores the complete source orbit.","\\begin{theorem}[Complete dominant-orbit signature]\n\\label{thm:block11-dominant-orbit}\nUnder the smooth implicit schema, the dominant singularities of $u(z)$ form\n\\[\n \\boxed{\n \\rho\\,\\mu_{d_{\\mathcal A}},}\n \\tag{11.12}\n\\]\nwhere $\\mu_d$ is the group of $d$th roots of unity.  After packetization with $x=z^N$, the union of dominant singularities of the complete packet tuple is\n\\[\n \\boxed{\n \\rho^N\\,\\mu_{h_N},\n \\qquad\n h_N=\\frac{d_{\\mathcal A}}{\\gcd(N,d_{\\mathcal A})}.}\n \\tag{11.13}\n\\]\nTherefore the same finite arity signature determines both the number of active packet addresses and the size of the complete dominant-singularity orbit.\n\\end{theorem}","\\begin{proof}\nBy \\cref{thm:block11-tree-address}, the coefficient support is contained in $1+d_{\\mathcal A}\\mathbb N$ and the reduced exponent set is aperiodic by maximality of the gcd.  Hence\n\\[\n u(z)=zV(z^{d_{\\mathcal A}})\n\\]\nwith $V$ aperiodic.  The smooth implicit schema gives one positive dominant singularity for $V$, and pulling back through $z^{d_{\\mathcal A}}$ gives \\textup{(11.12)}.  The map $z\\mapsto z^N$ sends $\\mu_{d_{\\mathcal A}}$ onto a root group of cardinality\n\\[\n d_{\\mathcal A}/\\gcd(N,d_{\\mathcal A}).\n\\]\nNo packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N)\n\\]\nwould make $u$ analytic at every source point above that image, contradicting \\textup{(11.12)}.  Thus the uni","\\begin{proof}\nBy \\cref{thm:block11-tree-address}, the coefficient support is contained in $1+d_{\\mathcal A}\\mathbb N$ and the reduced exponent set is aperiodic by maximality of the gcd.  Hence\n\\[\n u(z)=zV(z^{d_{\\mathcal A}})\n\\]\nwith $V$ aperiodic.  The smooth implicit schema gives one positive dominant singularity for $V$, and pulling back through $z^{d_{\\mathcal A}}$ gives \\textup{(11.12)}.  The map $z\\mapsto z^N$ sends $\\mu_{d_{\\mathcal A}}$ onto a root group of cardinality\n\\[\n d_{\\mathcal A}/\\gcd(N,d_{\\mathcal A}).\n\\]\nNo packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N)\n\\]\nwould make $u$ analytic at every source point above that image, contradicting \\textup{(11.12)}.  Thus the union of packet singularities is exactly \\textup{(11.13)}.\n\\end{proof}","n{proof}\nBy \\cref{thm:block11-tree-address}, the coefficient support is contained in $1+d_{\\mathcal A}\\mathbb N$ and the reduced exponent set is aperiodic by maximality of the gcd.  Hence\n\\[\n u(z)=zV(z^{d_{\\mathcal A}})\n\\]\nwith $V$ aperiodic.  The smooth implicit schema gives one positive dominant singularity for $V$, and pulling back through $z^{d_{\\mathcal A}}$ gives \\textup{(11.12)}.  The map $z\\mapsto z^N$ sends $\\mu_{d_{\\mathcal A}}$ onto a root group of cardinality\n\\[\n d_{\\mathcal A}/\\gcd(N,d_{\\mathcal A}).\n\\]\nNo packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N)\n\\]\nwould make $u$ analytic at every source point above that image, contradicting \\textup{(11.12)}.  Thus the union of packet singularities is exactly \\textup{(11.13)}.\n\\end{proof}",":block11-tree-address}, the coefficient support is contained in $1+d_{\\mathcal A}\\mathbb N$ and the reduced exponent set is aperiodic by maximality of the gcd.  Hence\n\\[\n u(z)=zV(z^{d_{\\mathcal A}})\n\\]\nwith $V$ aperiodic.  The smooth implicit schema gives one positive dominant singularity for $V$, and pulling back through $z^{d_{\\mathcal A}}$ gives \\textup{(11.12)}.  The map $z\\mapsto z^N$ sends $\\mu_{d_{\\mathcal A}}$ onto a root group of cardinality\n\\[\n d_{\\mathcal A}/\\gcd(N,d_{\\mathcal A}).\n\\]\nNo packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N)\n\\]\nwould make $u$ analytic at every source point above that image, contradicting \\textup{(11.12)}.  Thus the union of packet singularities is exactly \\textup{(11.13)}.\n\\end{proof}","ned in $1+d_{\\mathcal A}\\mathbb N$ and the reduced exponent set is aperiodic by maximality of the gcd.  Hence\n\\[\n u(z)=zV(z^{d_{\\mathcal A}})\n\\]\nwith $V$ aperiodic.  The smooth implicit schema gives one positive dominant singularity for $V$, and pulling back through $z^{d_{\\mathcal A}}$ gives \\textup{(11.12)}.  The map $z\\mapsto z^N$ sends $\\mu_{d_{\\mathcal A}}$ onto a root group of cardinality\n\\[\n d_{\\mathcal A}/\\gcd(N,d_{\\mathcal A}).\n\\]\nNo packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N)\n\\]\nwould make $u$ analytic at every source point above that image, contradicting \\textup{(11.12)}.  Thus the union of packet singularities is exactly \\textup{(11.13)}.\n\\end{proof}","schema gives one positive dominant singularity for $V$, and pulling back through $z^{d_{\\mathcal A}}$ gives \\textup{(11.12)}.  The map $z\\mapsto z^N$ sends $\\mu_{d_{\\mathcal A}}$ onto a root group of cardinality\n\\[\n d_{\\mathcal A}/\\gcd(N,d_{\\mathcal A}).\n\\]\nNo packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N)\n\\]\nwould make $u$ analytic at every source point above that image, contradicting \\textup{(11.12)}.  Thus the union of packet singularities is exactly \\textup{(11.13)}.\n\\end{proof}","$V$, and pulling back through $z^{d_{\\mathcal A}}$ gives \\textup{(11.12)}.  The map $z\\mapsto z^N$ sends $\\mu_{d_{\\mathcal A}}$ onto a root group of cardinality\n\\[\n d_{\\mathcal A}/\\gcd(N,d_{\\mathcal A}).\n\\]\nNo packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N)\n\\]\nwould make $u$ analytic at every source point above that image, contradicting \\textup{(11.12)}.  Thus the union of packet singularities is exactly \\textup{(11.13)}.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.94\\textwidth]{tree_singularity_orbit.pdf}\n\\caption{Arithmetic packetization maps the source root-of-unity singularity orbit through $z\\mapsto z^N$.  Distinct source singularities can alias to one packet-variable singularity.}\n\\label{fig:block11-tree-singularity}\n\\end{figure}","\\subsection{\\texorpdfstring{The sharp full \\(q\\)-ary family}{The sharp full q-ary family}}\n\\label{subsec:block11-qary}","Take\n\\[\n \\boxed{u=z+u^q,\\qquad q\\ge2.}\n \\tag{11.14}\n\\]\nLagrange inversion gives the exact coefficient formula\n\\[\n \\boxed{\n [z^n]u\n =\n \\begin{cases}\n \\displaystyle\n \\frac{1}{(q-1)m+1}\\binom{qm}{m},\n &n=1+(q-1)m,","Take\n\\[\n \\boxed{u=z+u^q,\\qquad q\\ge2.}\n \\tag{11.14}\n\\]\nLagrange inversion gives the exact coefficient formula\n\\[\n \\boxed{\n [z^n]u\n =\n \\begin{cases}\n \\displaystyle\n \\frac{1}{(q-1)m+1}\\binom{qm}{m},\n &n=1+(q-1)m,\\\\[9pt]\n 0,&n\\not\\equiv1\\pmod{q-1}.\n \\end{cases}}\n \\tag{11.15}\n\\]\nThe nonzero coefficients are the Fuss--Catalan numbers of full plane $q$-ary trees.","[9pt]\n 0,&n\\not\\equiv1\\pmod{q-1}.\n \\end{cases}}\n \\tag{11.15}\n\\]\nThe nonzero coefficients are the Fuss--Catalan numbers of full plane $q$-ary trees.","Write\n\\[\n u(z)=\\sum_{j=0}^{N-1}z^jU_j(z^N),\n \\qquad x=z^N.\n\\]","\\begin{theorem}[Exact $q$-ary packet phase theorem]\n\\label{thm:block11-qary-phase}\nPut\n\\[\n g=\\gcd(N,q-1),\n \\qquad\n h=\\frac{q-1}{g}.\n\\]\nThen:\n\\begin{enumerate}\n\\item the active packet addresses are exactly\n\\[\n \\boxed{j\\equiv1\\pmod g;}\n\\]\n\\item the complete packet system is\n\\[\n\\boxed{\n U_j(x)\n =\n \\delta_{j,1}\n +\n \\sum_{\\substack{0\\le r_1,\\ldots,r_q<N","\\begin{theorem}[Exact $q$-ary packet phase theorem]\n\\label{thm:block11-qary-phase}\nPut\n\\[\n g=\\gcd(N,q-1),\n \\qquad\n h=\\frac{q-1}{g}.\n\\]\nThen:\n\\begin{enumerate}\n\\item the active packet addresses are exactly\n\\[\n \\boxed{j\\equiv1\\pmod g;}\n\\]\n\\item the complete packet system is\n\\[\n\\boxed{\n U_j(x)\n =\n \\delta_{j,1}\n +\n \\sum_{\\substack{0\\le r_1,\\ldots,r_q<N\\\\\n r_1+\\cdots+r_q\\equiv j\\ (\\mathrm{mod}\\ N)}}\n x^{(r_1+\\cdots+r_q-j)/N}\n \\prod_{a=1}^{q}U_{r_a}(x);}\n \\tag{11.16}\n\\]\n\\item the smooth critical values are\n\\[\n \\tau=q^{-1/(q-1)},\n \\qquad\n \\rho=(q-1)q^{-q/(q-1)};\n\\]\n\\item the source dominant orbit has $q-1$ points, while the complete packet orbit has exactly $h$ points.\n\\end{enumerate}\nThe phase is therefore determined by the finite tuple\n\\[\n \\boxed{(q,N,g,N/g,h).}\n \\tag{11.17}\n\\]\n\\end{theorem}","r_1+\\cdots+r_q\\equiv j\\ (\\mathrm{mod}\\ N)}}\n x^{(r_1+\\cdots+r_q-j)/N}\n \\prod_{a=1}^{q}U_{r_a}(x);}\n \\tag{11.16}\n\\]\n\\item the smooth critical values are\n\\[\n \\tau=q^{-1/(q-1)},\n \\qquad\n \\rho=(q-1)q^{-q/(q-1)};\n\\]\n\\item the source dominant orbit has $q-1$ points, while the complete packet orbit has exactly $h$ points.\n\\end{enumerate}\nThe phase is therefore determined by the finite tuple\n\\[\n \\boxed{(q,N,g,N/g,h).}\n \\tag{11.17}\n\\]\n\\end{theorem}","\\begin{proof}\nFormula \\textup{(11.15)} follows from Lagrange inversion applied to\n\\[\n z=u(1-u^{q-1}).\n\\]\nThe support and active-address statements are \\cref{thm:block11-tree-address} with $d_{\\mathcal A}=q-1$.  Expanding $u^q$ in complete packets and collecting the exponent carry\n\\[\n r_1+\\cdots+r_q=j+Nc\n\\]\ngives \\textup{(11.16)}.  The critical equations\n\\[\n q\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\tau^q\n\\]\ngive the printed values, and \\cref{thm:block11-dominant-orbit} gives the orbit count.\n\\end{proof}","audits.  It enumerates colored integer-partition profiles and compares their orbit-weighted Fa\\`a di Bruno values with the full set-partition expansion.  It constructs exact fixed-point prefixes for marked polynomial tree families and checks the finite arity-gcd support theorem over many packet widths.  It verifies Fuss--Catalan formulas and the complete carry-form packet equations for the full $q$-ary family.  Finally, it checks dependency-graph reductions against exhaustive bounded graph instances.","on compresses exactly from colored set partitions to colored integer-partition profiles.  The packet dependency graph has an exact SCC reduction whose cyclic vertices form the address-preserving nonlinear core.  For positive simply generated tree systems, the arity gcd is a finite signature determining every arithmetic packet-support pattern and the complete dominant-singularity orbit.  The full $q$-ary family supplies a closed Fuss--Catalan phase theorem and exact packet equations.\n\\end{resultbox}","cal descent, profinite refinement, and the tensor-fiber boundary at which a symmetry group can be reconstructed.  The strongest Galois source already contains Reynolds projectors, the subgroup--fixed-field correspondence, normality and quotient compression, conjugate projector atlases, Mackey double-coset packets, ramification details, Haar projectors, Hilbert 90, the Brauer obstruction, finite \\(\\acute{e}\\)tale permutation resolutions, and neutral Tannakian reconstruction \\cite{GAL}.  The purpose here is to place those results into the same passport, bridge, obstruction, and effective-descent language used by the preceding sections.","Let \\(L/K\\) be a finite Galois extension with group\n\\[\n G=\\operatorname{Gal}(L/K).\n\\]\nLet \\(V\\) be a finite-dimensional representation of \\(G\\) over a characteristic-zero field, or a unitary complex realization obtained after scalar extension.  For every subgroup \\(H\\leq G\\), define\n\\[\n \\boxed{\n P_H=\\frac1{|H|}\\sum_{h\\in H}U_h.}\n \\tag{12.1}\n\\]","\\begin{theorem}[Reynolds fixed packet and reverse order]\n\\label{thm:block12-reynolds}\nThe operator \\(P_H\\) is the projector onto the fixed packet\n\\[\n V^H=\\{v:U_hv=v\\text{ for all }h\\in H\\}.\n\\]\nIf \\(H\\leq J\\), then\n\\[\n \\boxed{P_HP_J=P_JP_H=P_J.}\n \\tag{12.2}\n\\]\nConsequently, subgroup inclusion reverses the fixed-projector order.\n\\end{theorem}","\\begin{proof}\nThe average is invariant under left and right multiplication by elements of \\(H\\).  Therefore \\(P_H^2=P_H\\), and its range is exactly the fixed subspace.  If \\(H\\leq J\\), then \\(V^J\\subseteq V^H\\).  The two projections are nested and multiply to the projector with smaller range.\n\\end{proof}","In the faithful regular realization, or in the field realization supplied by a normal basis, the projector determines the subgroup.  Indeed, the support of the group-algebra idempotent\n\\[\n e_H=\\frac1{|H|}\\sum_{h\\in H}h\n\\]\nis exactly \\(H\\).","\\begin{theorem}[Normalized finite Galois chart]\n\\label{thm:block12-normalized-galois-chart}\nFix the finite Galois extension \\(L/K\\).  The following data determine one another exactly:\n\\[\n \\boxed{\n H\n \\quad\\longleftrightarrow\\quad\n L^H\n \\quad\\longleftrightarrow\\quad\n P_H\n }\n \\tag{12.3}\n\\]\nwhere the projector is taken in a faithful field or regular realization.  The correspondence is order reversing:\n\\[\n H\\leq J\n \\quad\\Longleftrightarrow\\quad\n L^H\\supseteq L^J\n \\quad\\Longleftrightarrow\\quad\n P_H\\geq P_J.\n\\]\nThus the subgroup poset, the intermediate-field poset, and the faithful fixed-projector poset form three normalized charts of one finite semisimple Galois object.\n\\end{theorem}","in{theorem}[Normalized finite Galois chart]\n\\label{thm:block12-normalized-galois-chart}\nFix the finite Galois extension \\(L/K\\).  The following data determine one another exactly:\n\\[\n \\boxed{\n H\n \\quad\\longleftrightarrow\\quad\n L^H\n \\quad\\longleftrightarrow\\quad\n P_H\n }\n \\tag{12.3}\n\\]\nwhere the projector is taken in a faithful field or regular realization.  The correspondence is order reversing:\n\\[\n H\\leq J\n \\quad\\Longleftrightarrow\\quad\n L^H\\supseteq L^J\n \\quad\\Longleftrightarrow\\quad\n P_H\\geq P_J.\n\\]\nThus the subgroup poset, the intermediate-field poset, and the faithful fixed-projector poset form three normalized charts of one finite semisimple Galois object.\n\\end{theorem}","\\begin{proof}\nThe finite Galois correspondence gives \\(H=\\operatorname{Gal}(L/L^H)\\) and \\(E=L^{\\operatorname{Gal}(L/E)}\\).  The range of \\(P_H\\) is \\(L^H\\) after faithful scalar extension, and the support of \\(e_H\\) recovers \\(H\\).  All three assignments are therefore mutually inverse on the declared charts.\n\\end{proof}","For two subgroups \\(H,J\\leq G\\), the common fixed packet is always\n\\[\n V^H\\cap V^J=V^{\\langle H,J\\rangle}.\n\\]\nThe two averaging projectors need not commute.","\\begin{theorem}[Permutable subgroups and exact common refinement]\n\\label{thm:block12-permutable}\nIn a faithful regular realization, the following are equivalent:\n\\begin{enumerate}\n\\item \\(HJ=JH\\);\n\\item \\(HJ\\) is a subgroup of \\(G\\);\n\\item \\(P_HP_J=P_JP_H\\).\n\\end{enumerate}\nWhen these conditions hold,\n\\[\n \\boxed{P_HP_J=P_{HJ}.}\n \\tag{12.4}\n\\]\nIf they fail, \\(P_HP_J\\) and \\(P_JP_H\\) are mixed atlas corners rather than projectors.\n\\end{theorem}","\\begin{proof}\nIf \\(HJ=JH\\), every element of \\(HJ\\) has \\(|H\\cap J|\\) representations as \\(hj\\).  Multiplying the two group averages therefore gives the average over \\(HJ\\).  Conversely, the support of \\(e_He_J\\) is \\(HJ\\), while the support of \\(e_Je_H\\) is \\(JH\\).  Equality in the regular group algebra forces equality of the product sets.\n\\end{proof}","\\begin{theorem}[Normality and quotient compression]\n\\label{thm:block12-normality}\nFor a subgroup \\(H\\leq G\\), the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\(H\\triangleleft G\\);\n\\item \\(e_H\\) is central in the group algebra;\n\\item \\(P_HU_g=U_gP_H\\) for every \\(g\\in G\\);\n\\item the fixed packet \\(V^H\\) is \\(G\\)-invariant;\n\\item the fixed field \\(L^H\\) is Galois over \\(K\\).\n\\end{enumerate}\nWhen they hold, the compressed action on \\(V^H\\) factors through \\(G/H\\), and\n\\[\n \\boxed{\\operatorname{Gal}(L^H/K)\\cong G/H.}\n \\tag{12.5}\n\\]\n\\end{theorem}","\\label{thm:block12-normality}\nFor a subgroup \\(H\\leq G\\), the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\(H\\triangleleft G\\);\n\\item \\(e_H\\) is central in the group algebra;\n\\item \\(P_HU_g=U_gP_H\\) for every \\(g\\in G\\);\n\\item the fixed packet \\(V^H\\) is \\(G\\)-invariant;\n\\item the fixed field \\(L^H\\) is Galois over \\(K\\).\n\\end{enumerate}\nWhen they hold, the compressed action on \\(V^H\\) factors through \\(G/H\\), and\n\\[\n \\boxed{\\operatorname{Gal}(L^H/K)\\cong G/H.}\n \\tag{12.5}\n\\]\n\\end{theorem}","lity}\nFor a subgroup \\(H\\leq G\\), the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\(H\\triangleleft G\\);\n\\item \\(e_H\\) is central in the group algebra;\n\\item \\(P_HU_g=U_gP_H\\) for every \\(g\\in G\\);\n\\item the fixed packet \\(V^H\\) is \\(G\\)-invariant;\n\\item the fixed field \\(L^H\\) is Galois over \\(K\\).\n\\end{enumerate}\nWhen they hold, the compressed action on \\(V^H\\) factors through \\(G/H\\), and\n\\[\n \\boxed{\\operatorname{Gal}(L^H/K)\\cong G/H.}\n \\tag{12.5}\n\\]\n\\end{theorem}","ubgroup \\(H\\leq G\\), the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\(H\\triangleleft G\\);\n\\item \\(e_H\\) is central in the group algebra;\n\\item \\(P_HU_g=U_gP_H\\) for every \\(g\\in G\\);\n\\item the fixed packet \\(V^H\\) is \\(G\\)-invariant;\n\\item the fixed field \\(L^H\\) is Galois over \\(K\\).\n\\end{enumerate}\nWhen they hold, the compressed action on \\(V^H\\) factors through \\(G/H\\), and\n\\[\n \\boxed{\\operatorname{Gal}(L^H/K)\\cong G/H.}\n \\tag{12.5}\n\\]\n\\end{theorem}","\\begin{proof}\nConjugation sends \\(e_H\\) to \\(e_{gHg^{-1}}\\).  Centrality is therefore equivalent to normality.  The represented identity\n\\[\n U_gP_HU_g^{-1}=P_{gHg^{-1}}\n\\]\ngives the operator conditions.  The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.\n\\end{proof}","If \\(H\\) is not normal, its conjugates form a genuine atlas rather than one quotient chart.  Put\n\\[\n \\operatorname{Core}_G(H)=\\bigcap_{g\\in G}gHg^{-1},\n \\qquad\n \\operatorname{Ncl}_G(H)=\\left\\langle gHg^{-1}:g\\in G\\right\\rangle.\n\\]","\\begin{theorem}[Canonical envelopes of a nonnormal Galois atlas]\n\\label{thm:block12-core-normal-closure}\nLet \\(M=L^H\\).  Then\n\\[\n \\boxed{\n L^{\\operatorname{Core}_G(H)}\n =\\prod_{g\\in G}g(M)}\n \\tag{12.6}\n\\]\nis the smallest Galois extension inside \\(L\\) containing every conjugate chart, while\n\\[\n \\boxed{\n L^{\\operatorname{Ncl}_G(H)}\n =\\bigcap_{g\\in G}g(M)}\n \\tag{12.7}\n\\]\nis the largest Galois subfield contained in every chart.  The core and normal-closure projectors are the two canonical global envelopes of the conjugate atlas.\n\\end{theorem}","\\begin{proof}\nComposita of intermediate fields correspond to intersections of subgroups, while intersections of fields correspond to generated subgroups.  The core is the largest normal subgroup inside \\(H\\), and the normal closure is the smallest normal subgroup containing \\(H\\).\n\\end{proof}","\\begin{example}[The \\(S_3\\) nonnormal atlas]\nLet \\(G=S_3\\) and let \\(H\\) be generated by one transposition.  There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts.  Their subgroup core is trivial, while their normal closure is all of \\(S_3\\).  By contrast, the order-three subgroup \\(A_3\\) is normal and its fixed chart carries the quotient symmetry \\(S_3/A_3\\cong C_2\\).\n\\end{example}","Restriction and induction connect subgroup charts that need not commute.  Their exact address set is the double-coset space\n\\[\n K\\backslash G/H.\n\\]","\\begin{theorem}[Mackey transition-packet decomposition]\n\\label{thm:block12-mackey}\nLet \\(H,K\\leq G\\), let \\(W\\) be a finite-dimensional representation of \\(H\\), and choose representatives \\(x\\) for \\(K\\backslash G/H\\).  Then\n\\[\n \\boxed{\n \\operatorname{Res}_K^G\\operatorname{Ind}_H^GW\n \\cong\n \\bigoplus_{x\\in K\\backslash G/H}\n \\operatorname{Ind}_{K\\cap xHx^{-1}}^K\n \\left({}^x\\operatorname{Res}_{H\\cap x^{-1}Kx}^HW\\right).}\n \\tag{12.8}\n\\]\nEach double coset is one exact transfer packet between the \\(H\\)- and \\(K\\)-charts.\n\\end{theorem}","), let \\(W\\) be a finite-dimensional representation of \\(H\\), and choose representatives \\(x\\) for \\(K\\backslash G/H\\).  Then\n\\[\n \\boxed{\n \\operatorname{Res}_K^G\\operatorname{Ind}_H^GW\n \\cong\n \\bigoplus_{x\\in K\\backslash G/H}\n \\operatorname{Ind}_{K\\cap xHx^{-1}}^K\n \\left({}^x\\operatorname{Res}_{H\\cap x^{-1}Kx}^HW\\right).}\n \\tag{12.8}\n\\]\nEach double coset is one exact transfer packet between the \\(H\\)- and \\(K\\)-charts.\n\\end{theorem}",") be a finite-dimensional representation of \\(H\\), and choose representatives \\(x\\) for \\(K\\backslash G/H\\).  Then\n\\[\n \\boxed{\n \\operatorname{Res}_K^G\\operatorname{Ind}_H^GW\n \\cong\n \\bigoplus_{x\\in K\\backslash G/H}\n \\operatorname{Ind}_{K\\cap xHx^{-1}}^K\n \\left({}^x\\operatorname{Res}_{H\\cap x^{-1}Kx}^HW\\right).}\n \\tag{12.8}\n\\]\nEach double coset is one exact transfer packet between the \\(H\\)- and \\(K\\)-charts.\n\\end{theorem}","\\begin{proof}\nThe induced representation is a function space on \\(G\\) with \\(H\\)-covariance.  Functions supported on different double cosets are invariant direct summands under \\(K\\).  Identifying the summand on \\(KxH\\) with an induced representation from \\(K\\cap xHx^{-1}\\) gives the decomposition.\n\\end{proof}","The dimension identity for the trivial \\(H\\)-representation is\n\\[\n \\boxed{\n [G:H]\n =\n \\sum_{x\\in K\\backslash G/H}\n [K:K\\cap xHx^{-1}].}\n \\tag{12.9}\n\\]\nThis is the finite packet-count law audited by the Galois-Tannakian atlas verifier.","Let \\(L/K\\) be a finite Galois extension of local fields.  Write\n\\[\n G_0\\supseteq G_1\\supseteq G_2\\supseteq\\cdots\n\\]\nfor the lower ramification filtration after inertia, and let \\(V\\) be a finite-dimensional complex representation of \\(G\\).  Define\n\\[\n P_i=\\frac1{|G_i|}\\sum_{g\\in G_i}U_g.\n\\]\nThe ramification groups are nested and normal, so\n\\[\n P_0\\leq P_1\\leq P_2\\leq\\cdots\\leq I.\n\\]\nPut\n\\[\n D_i=P_{i+1}-P_i.\n\\]","\\begin{theorem}[Orthogonal ramification details]\n\\label{thm:block12-ramification-details}\nThe operators \\(D_i\\) are mutually orthogonal projectors, and\n\\[\n \\boxed{\n V=V^{G_0}\\oplus\\bigoplus_i D_iV.}\n \\tag{12.10}\n\\]\nEvery detail packet is stable under the quotient action permitted by the corresponding normal subgroup.\n\\end{theorem}","\\begin{theorem}[Artin conductor as weighted unresolved rank]\n\\label{thm:block12-artin-conductor}\nThe Artin and Swan conductors satisfy\n\\[\n \\boxed{\n a(V)=\\sum_{i\\geq0}\\frac{|G_i|}{|G_0|}\\operatorname{Tr}(I-P_i),}\n \\tag{12.11}\n\\]\n\\[\n \\boxed{\n \\operatorname{Sw}(V)=\\sum_{i\\geq1}\\frac{|G_i|}{|G_0|}\\operatorname{Tr}(I-P_i).}\n \\tag{12.12}\n\\]\nIf \\(d_j=\\operatorname{rank}D_j\\), then\n\\[\n \\boxed{\n a(V)=\\sum_jd_j\\sum_{i=0}^{j}\\frac{|G_i|}{|G_0|}.}\n \\tag{12.13}\n\\]\nThus the conductor is the weighted total rank that remains unresolved at each ramification scale.\n\\end{theorem}","\\begin{proof}\nThe classical conductor formula uses \\(\\operatorname{codim}V^{G_i}\\).  Since \\(P_i\\) projects onto \\(V^{G_i}\\), this codimension is \\(\\operatorname{Tr}(I-P_i)\\).  Also\n\\[\n \\operatorname{codim}V^{G_i}=\\sum_{j\\geq i}d_j.\n\\]\nReversing the finite sums gives the detail formula.\n\\end{proof}","Let \\(G\\) be a compact group and let \\(U:G\\to\\mathcal U(\\mathcal H)\\) be strongly continuous.  For a closed subgroup \\(H\\), normalized Haar measure defines\n\\[\n \\boxed{\n P_H=\\int_HU_h\\,d\\mu_H(h).}\n \\tag{12.14}\n\\]\nThe same averaging proof as in the finite case shows that \\(P_H\\) is the orthogonal projector onto \\(\\mathcal H^H\\).","\\begin{theorem}[Shrinking-subgroup reconstruction]\n\\label{thm:block12-shrinking-subgroups}\nLet \\(G\\) be profinite and let\n\\[\n U_1\\supseteq U_2\\supseteq\\cdots\n\\]\nbe a neighborhood basis of the identity consisting of open normal subgroups.  Then\n\\[\n \\boxed{P_{U_n}\\longrightarrow I}\n \\tag{12.15}\n\\]\nstrongly in every strongly continuous unitary representation.  More generally, if \\(N\\triangleleft G\\) is closed and \\(U_n/N\\) shrinks to the identity in \\(G/N\\), then \\(P_{U_n}\\to P_N\\) strongly.\n\\end{theorem}","Shrinking-subgroup reconstruction]\n\\label{thm:block12-shrinking-subgroups}\nLet \\(G\\) be profinite and let\n\\[\n U_1\\supseteq U_2\\supseteq\\cdots\n\\]\nbe a neighborhood basis of the identity consisting of open normal subgroups.  Then\n\\[\n \\boxed{P_{U_n}\\longrightarrow I}\n \\tag{12.15}\n\\]\nstrongly in every strongly continuous unitary representation.  More generally, if \\(N\\triangleleft G\\) is closed and \\(U_n/N\\) shrinks to the identity in \\(G/N\\), then \\(P_{U_n}\\to P_N\\) strongly.\n\\end{theorem}","abel{thm:block12-shrinking-subgroups}\nLet \\(G\\) be profinite and let\n\\[\n U_1\\supseteq U_2\\supseteq\\cdots\n\\]\nbe a neighborhood basis of the identity consisting of open normal subgroups.  Then\n\\[\n \\boxed{P_{U_n}\\longrightarrow I}\n \\tag{12.15}\n\\]\nstrongly in every strongly continuous unitary representation.  More generally, if \\(N\\triangleleft G\\) is closed and \\(U_n/N\\) shrinks to the identity in \\(G/N\\), then \\(P_{U_n}\\to P_N\\) strongly.\n\\end{theorem}","12-shrinking-subgroups}\nLet \\(G\\) be profinite and let\n\\[\n U_1\\supseteq U_2\\supseteq\\cdots\n\\]\nbe a neighborhood basis of the identity consisting of open normal subgroups.  Then\n\\[\n \\boxed{P_{U_n}\\longrightarrow I}\n \\tag{12.15}\n\\]\nstrongly in every strongly continuous unitary representation.  More generally, if \\(N\\triangleleft G\\) is closed and \\(U_n/N\\) shrinks to the identity in \\(G/N\\), then \\(P_{U_n}\\to P_N\\) strongly.\n\\end{theorem}","\\begin{proof}\nFor each \\(v\\in\\mathcal H\\), strong continuity gives\n\\[\n \\sup_{u\\in U_n}\\|U_uv-v\\|\\longrightarrow0.\n\\]\nAveraging this estimate over \\(U_n\\) proves the first statement.  Apply it to the quotient action on the \\(N\\)-fixed packet for the second.\n\\end{proof}","\\begin{proof}\nThe image is a compact totally disconnected subgroup of \\(\\operatorname{GL}_n(\\mathbb C)\\).  After conjugation it lies in a compact Lie group.  A compact totally disconnected Lie group is finite.\n\\end{proof}","Let \\(L/K\\) be finite Galois with group \\(G\\).  A semilinear \\(G\\)-object is an \\(L\\)-vector space \\(V\\) with maps\n\\[\n \\rho_\\sigma:V\\to V,\n \\qquad\n \\rho_\\sigma(\\lambda v)=\\sigma(\\lambda)\\rho_\\sigma(v),\n\\]\nsatisfying the group law.  Scalar extension and fixed vectors define functors\n\\[\n L\\otimes_K-:\\operatorname{Vect}_K\\longrightarrow\\operatorname{Semilin}_G(L),\n\\]\n\\[\n (-)^G:\\operatorname{Semilin}_G(L)\\longrightarrow\\operatorname{Vect}_K.\n\\]","\\begin{theorem}[Effective finite Galois vector descent]\n\\label{thm:block12-vector-descent}\nThe two functors above are quasi-inverse equivalences:\n\\[\n \\boxed{\n \\operatorname{Vect}_K\n \\simeq\n \\operatorname{Semilin}_G(L).}\n \\tag{12.16}\n\\]\nFor every semilinear object \\(V\\), the fixed packet \\(W=V^G\\) satisfies\n\\[\n \\boxed{W\\otimes_KL\\cong V.}\n \\tag{12.17}\n\\]\nThe equivalence is effective on both objects and morphisms.\n\\end{theorem}","\\begin{proof}\nThis is finite Galois descent.  After an \\(L\\)-basis is chosen, the semilinear action is represented by a cocycle\n\\[\n a_{\\sigma\\tau}=a_\\sigma\\sigma(a_\\tau),\n \\qquad\n a_\\sigma\\in\\operatorname{GL}_n(L).\n\\]\nNonabelian Hilbert 90 gives \\(B\\in\\operatorname{GL}_n(L)\\) with\n\\[\n a_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe columns of \\(B^{-1}\\) applied to a \\(K\\)-basis are fixed by the twisted action.  They form an \\(n\\)-dimensional fixed packet, and scalar extension reconstructs \\(V\\).  An equivariant \\(L\\)-linear map restricts to a \\(K\\)-linear map on fixed packets, and scalar extension recovers it.\n\\end{proof}","his is finite Galois descent.  After an \\(L\\)-basis is chosen, the semilinear action is represented by a cocycle\n\\[\n a_{\\sigma\\tau}=a_\\sigma\\sigma(a_\\tau),\n \\qquad\n a_\\sigma\\in\\operatorname{GL}_n(L).\n\\]\nNonabelian Hilbert 90 gives \\(B\\in\\operatorname{GL}_n(L)\\) with\n\\[\n a_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe columns of \\(B^{-1}\\) applied to a \\(K\\)-basis are fixed by the twisted action.  They form an \\(n\\)-dimensional fixed packet, and scalar extension reconstructs \\(V\\).  An equivariant \\(L\\)-linear map restricts to a \\(K\\)-linear map on fixed packets, and scalar extension recovers it.\n\\end{proof}","semilinear action is represented by a cocycle\n\\[\n a_{\\sigma\\tau}=a_\\sigma\\sigma(a_\\tau),\n \\qquad\n a_\\sigma\\in\\operatorname{GL}_n(L).\n\\]\nNonabelian Hilbert 90 gives \\(B\\in\\operatorname{GL}_n(L)\\) with\n\\[\n a_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe columns of \\(B^{-1}\\) applied to a \\(K\\)-basis are fixed by the twisted action.  They form an \\(n\\)-dimensional fixed packet, and scalar extension reconstructs \\(V\\).  An equivariant \\(L\\)-linear map restricts to a \\(K\\)-linear map on fixed packets, and scalar extension recovers it.\n\\end{proof}","epresented by a cocycle\n\\[\n a_{\\sigma\\tau}=a_\\sigma\\sigma(a_\\tau),\n \\qquad\n a_\\sigma\\in\\operatorname{GL}_n(L).\n\\]\nNonabelian Hilbert 90 gives \\(B\\in\\operatorname{GL}_n(L)\\) with\n\\[\n a_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe columns of \\(B^{-1}\\) applied to a \\(K\\)-basis are fixed by the twisted action.  They form an \\(n\\)-dimensional fixed packet, and scalar extension reconstructs \\(V\\).  An equivariant \\(L\\)-linear map restricts to a \\(K\\)-linear map on fixed packets, and scalar extension recovers it.\n\\end{proof}","sigma\\tau}=a_\\sigma\\sigma(a_\\tau),\n \\qquad\n a_\\sigma\\in\\operatorname{GL}_n(L).\n\\]\nNonabelian Hilbert 90 gives \\(B\\in\\operatorname{GL}_n(L)\\) with\n\\[\n a_\\sigma=B^{-1}\\sigma(B).\n\\]\nThe columns of \\(B^{-1}\\) applied to a \\(K\\)-basis are fixed by the twisted action.  They form an \\(n\\)-dimensional fixed packet, and scalar extension reconstructs \\(V\\).  An equivariant \\(L\\)-linear map restricts to a \\(K\\)-linear map on fixed packets, and scalar extension recovers it.\n\\end{proof}","A useful constructive form is obtained by choosing any matrix \\(C\\in M_n(L)\\) and setting\n\\[\n T(C)=\\sum_{\\sigma\\in G}a_\\sigma\\sigma(C).\n\\]\nThen\n\\[\n a_\\tau\\tau(T(C))=T(C).\n\\]\nWhenever \\(T(C)\\) is invertible,\n\\[\n B=T(C)^{-1}\n\\]\nis a Hilbert 90 trivialization.  Standard descent guarantees that such a \\(C\\) exists.","\\begin{definition}[Framed semilinear descent chart]\nA framed descent chart retains an ordered \\(K\\)-basis of the fixed packet \\(W\\).  Its scalar extension gives a distinguished \\(L\\)-basis of \\(V\\), and hence a literal cocycle matrix family.\n\\end{definition}","\\begin{proposition}[Gauge and intrinsic automorphisms remain distinct]\n\\label{prop:block12-framed-unframed}\nFor fixed rank \\(n\\), nonabelian Hilbert 90 implies that every \\(\\operatorname{GL}_n(L)\\)-cocycle is gauge equivalent to the trivial cocycle.  The unframed descent groupoid is not contractible: its automorphism group is \\(\\operatorname{GL}_n(K)\\).  After a global \\(K\\)-frame is retained, the normalized fiber becomes contractible.\n\\end{proposition}","\\begin{proof}\nHilbert 90 gives one isomorphism class.  Automorphisms of the descended \\(K\\)-space are exactly \\(\\operatorname{GL}_n(K)\\).  A chosen ordered basis kills those automorphisms.\n\\end{proof}","A projective cocycle takes values in \\(\\operatorname{PGL}_n(L)\\).  Choose lifts\n\\[\n A_\\sigma\\in\\operatorname{GL}_n(L).\n\\]\nTheir cocycle failure is scalar:\n\\[\n \\boxed{\n c(\\sigma,\\tau)\n =A_\\sigma\\sigma(A_\\tau)A_{\\sigma\\tau}^{-1}\n \\in L^\\times.}\n \\tag{12.18}\n\\]\nChanging the lifts alters \\(c\\) by a scalar coboundary.","\\begin{theorem}[Exact projective descent obstruction]\n\\label{thm:block12-brauer}\nThe scalar class\n\\[\n \\boxed{\n [c]=\\delta([\\alpha])\n \\in H^2(G,L^\\times)\n =\\operatorname{Br}(L/K)}\n \\tag{12.19}\n\\]\nis independent of the chosen lifts.  It vanishes exactly when the projective cocycle lifts to a linear semilinear cocycle.  In that case \\cref{thm:block12-vector-descent} produces a descended vector space.  If it does not vanish, the descended object is a module over a central simple \\(K\\)-algebra split by \\(L\\), not an ordinary global vector packet.\n\\end{theorem}","he scalar class\n\\[\n \\boxed{\n [c]=\\delta([\\alpha])\n \\in H^2(G,L^\\times)\n =\\operatorname{Br}(L/K)}\n \\tag{12.19}\n\\]\nis independent of the chosen lifts.  It vanishes exactly when the projective cocycle lifts to a linear semilinear cocycle.  In that case \\cref{thm:block12-vector-descent} produces a descended vector space.  If it does not vanish, the descended object is a module over a central simple \\(K\\)-algebra split by \\(L\\), not an ordinary global vector packet.\n\\end{theorem}","xed{\n [c]=\\delta([\\alpha])\n \\in H^2(G,L^\\times)\n =\\operatorname{Br}(L/K)}\n \\tag{12.19}\n\\]\nis independent of the chosen lifts.  It vanishes exactly when the projective cocycle lifts to a linear semilinear cocycle.  In that case \\cref{thm:block12-vector-descent} produces a descended vector space.  If it does not vanish, the descended object is a module over a central simple \\(K\\)-algebra split by \\(L\\), not an ordinary global vector packet.\n\\end{theorem}","\\begin{proof}\nApply the connecting map in nonabelian cohomology to\n\\[\n 1\\longrightarrow L^\\times\n \\longrightarrow\\operatorname{GL}_n(L)\n \\longrightarrow\\operatorname{PGL}_n(L)\n \\longrightarrow1.\n\\]\nVanishing is equivalent to rescaling the lifts so that the scalar defect is removed.  The lifted cocycle then descends by Hilbert 90.  The nonzero classes are the relative Brauer classes of central simple algebras split by \\(L\\).\n\\end{proof}","connecting map in nonabelian cohomology to\n\\[\n 1\\longrightarrow L^\\times\n \\longrightarrow\\operatorname{GL}_n(L)\n \\longrightarrow\\operatorname{PGL}_n(L)\n \\longrightarrow1.\n\\]\nVanishing is equivalent to rescaling the lifts so that the scalar defect is removed.  The lifted cocycle then descends by Hilbert 90.  The nonzero classes are the relative Brauer classes of central simple algebras split by \\(L\\).\n\\end{proof}","\\begin{example}[The quaternionic obstruction over \\(\\mathbb C/\\mathbb R\\)]\nLet \\(G=\\{1,\\sigma\\}\\), where \\(\\sigma\\) is complex conjugation, and put\n\\[\n A_\\sigma=\n \\begin{pmatrix}\n 0&1","\\begin{example}[The quaternionic obstruction over \\(\\mathbb C/\\mathbb R\\)]\nLet \\(G=\\{1,\\sigma\\}\\), where \\(\\sigma\\) is complex conjugation, and put\n\\[\n A_\\sigma=\n \\begin{pmatrix}\n 0&1\\\\\n -1&0\n \\end{pmatrix}.\n\\]\nThen\n\\[\n A_\\sigma\\sigma(A_\\sigma)=-I.\n\\]\nThe projective cocycle condition holds because \\(-I\\) is scalar.  Its boundary is represented by \\(-1\\) in\n\\[\n \\mathbb R^\\times/N_{\\mathbb C/\\mathbb R}(\\mathbb C^\\times).\n\\]\nEvery complex norm is positive, so \\(-1\\) is nontrivial.  The descended central simple algebra is the Hamilton quaternion algebra.  Replacing \\(A_\\sigma\\) by\n\\[\n \\begin{pmatrix}0&1","-1&0\n \\end{pmatrix}.\n\\]\nThen\n\\[\n A_\\sigma\\sigma(A_\\sigma)=-I.\n\\]\nThe projective cocycle condition holds because \\(-I\\) is scalar.  Its boundary is represented by \\(-1\\) in\n\\[\n \\mathbb R^\\times/N_{\\mathbb C/\\mathbb R}(\\mathbb C^\\times).\n\\]\nEvery complex norm is positive, so \\(-1\\) is nontrivial.  The descended central simple algebra is the Hamilton quaternion algebra.  Replacing \\(A_\\sigma\\) by\n\\[\n \\begin{pmatrix}0&1","-1&0\n \\end{pmatrix}.\n\\]\nThen\n\\[\n A_\\sigma\\sigma(A_\\sigma)=-I.\n\\]\nThe projective cocycle condition holds because \\(-I\\) is scalar.  Its boundary is represented by \\(-1\\) in\n\\[\n \\mathbb R^\\times/N_{\\mathbb C/\\mathbb R}(\\mathbb C^\\times).\n\\]\nEvery complex norm is positive, so \\(-1\\) is nontrivial.  The descended central simple algebra is the Hamilton quaternion algebra.  Replacing \\(A_\\sigma\\) by\n\\[\n \\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}\n\\]\ngives scalar defect \\(+1\\) and hence split vector descent.\n\\end{example}","1&0\\end{pmatrix}\n\\]\ngives scalar defect \\(+1\\) and hence split vector descent.\n\\end{example}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.97\\textwidth]{galois_descent_obstruction.pdf}\n\\caption{Linear cocycles are removed by Hilbert 90.  Projective cocycles can satisfy every visible \\(\\operatorname{PGL}\\) overlap law while retaining a nonzero degree-two Brauer obstruction.}\n\\label{fig:block12-descent-obstruction}\n\\end{figure}","\\subsection{Finite \\texorpdfstring{\\(\\acute{e}\\)}{e}tale and neutral Tannakian symmetry}\n\\label{subsec:block12-tannakian}","For a connected scheme \\(X\\) with geometric point \\(\\bar x\\), the fiber functor on finite \\(\\acute{e}\\)tale covers identifies that category with finite continuous \\(\\pi_1^{\\acute{e}t}(X,\\bar x)\\)-sets.  Every finite fiber \\(S\\) produces a permutation resolution on \\(\\ell^2(S)\\): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms give equivariant packet transfers.  The profinite fundamental group is the automorphism group of this fiber functor.","\\begin{theorem}[Neutral Tannakian reconstruction]\n\\label{thm:block12-tannakian}\nLet \\(\\mathcal C\\) be a neutral Tannakian category over \\(k\\), and let\n\\[\n \\omega:\\mathcal C\\longrightarrow\\operatorname{Vect}_k\n\\]\nbe a faithful exact tensor fiber functor.  Then\n\\[\n \\boxed{\n \\mathcal G=\\operatorname{Aut}^{\\otimes}(\\omega)}\n \\tag{12.20}\n\\]\nis an affine group scheme, and\n\\[\n \\boxed{\n \\mathcal C\\simeq\\operatorname{Rep}_k(\\mathcal G)}\n \\tag{12.21}\n\\]\nas tensor categories.  Declared packet idempotents and refinement maps preserved by \\(\\omega\\) become tensor-compatible concrete projector resolutions in every fiber.\n\\end{theorem}","r \\(k\\), and let\n\\[\n \\omega:\\mathcal C\\longrightarrow\\operatorname{Vect}_k\n\\]\nbe a faithful exact tensor fiber functor.  Then\n\\[\n \\boxed{\n \\mathcal G=\\operatorname{Aut}^{\\otimes}(\\omega)}\n \\tag{12.20}\n\\]\nis an affine group scheme, and\n\\[\n \\boxed{\n \\mathcal C\\simeq\\operatorname{Rep}_k(\\mathcal G)}\n \\tag{12.21}\n\\]\nas tensor categories.  Declared packet idempotents and refinement maps preserved by \\(\\omega\\) become tensor-compatible concrete projector resolutions in every fiber.\n\\end{theorem}","\\begin{boundary}[Faithful corner realization is not a fiber functor]\nA faithful additive or Hilbert-space realization can preserve packet idempotents, transfer blocks, and operator equations without preserving tensor products, duals, exact sequences, or the tensor unit.  Such a realization does not by itself determine \\(\\operatorname{Aut}^{\\otimes}(\\omega)\\).  Tannakian symmetry begins only after the faithful exact tensor gate has been verified.\n\\end{boundary}","\\begin{proposition}[Finite group model of the tensor boundary]\n\\label{prop:block12-finite-tensor-model}\nLet \\(G\\) be finite and let \\(\\omega:\\operatorname{Rep}_k(G)\\to\\operatorname{Vect}_k\\) be the forgetful functor.  The regular representation is faithful, the action on tensor products is diagonal, the unit is fixed, and the dual action is inverse transpose.  These data distinguish every \\(g\\in G\\) and realize the finite constant-group case of \\cref{thm:block12-tannakian}.\n\\end{proposition}","The preceding results can be collected into one typed finite atlas.  The reversible charts are:\n\\begin{enumerate}\n\\item subgroups \\(H\\leq G\\), fixed fields \\(L^H\\), and faithful Reynolds projectors \\(P_H\\);\n\\item finite-dimensional \\(K\\)-vector spaces and semilinear \\(L\\)-spaces with effective descent data;\n\\item finite \\(\\acute{e}\\)tale fibers and their continuous permutation actions;\n\\item neutral Tannakian categories equipped with a specified faithful exact tensor fiber functor.\n\\end{enumerate}\nDirectional or obstructed bridges include:\n\\begin{enumerate}\n\\item restriction and induction between subgroup charts, decomposed by Mackey packets;\n\\item passage from a nonnormal chart to its core or normal-closure envelope;\n\\item projectivization from linear to projective cocycles;\n\\item forgetting tensor structure from a Tannakian f","The preceding results can be collected into one typed finite atlas.  The reversible charts are:\n\\begin{enumerate}\n\\item subgroups \\(H\\leq G\\), fixed fields \\(L^H\\), and faithful Reynolds projectors \\(P_H\\);\n\\item finite-dimensional \\(K\\)-vector spaces and semilinear \\(L\\)-spaces with effective descent data;\n\\item finite \\(\\acute{e}\\)tale fibers and their continuous permutation actions;\n\\item neutral Tannakian categories equipped with a specified faithful exact tensor fiber functor.\n\\end{enumerate}\nDirectional or obstructed bridges include:\n\\begin{enumerate}\n\\item restriction and induction between subgroup charts, decomposed by Mackey packets;\n\\item passage from a nonnormal chart to its core or normal-closure envelope;\n\\item projectivization from linear to projective cocycles;\n\\item forgetting tensor structure from a Tannakian fiber functor to an add","The preceding results can be collected into one typed finite atlas.  The reversible charts are:\n\\begin{enumerate}\n\\item subgroups \\(H\\leq G\\), fixed fields \\(L^H\\), and faithful Reynolds projectors \\(P_H\\);\n\\item finite-dimensional \\(K\\)-vector spaces and semilinear \\(L\\)-spaces with effective descent data;\n\\item finite \\(\\acute{e}\\)tale fibers and their continuous permutation actions;\n\\item neutral Tannakian categories equipped with a specified faithful exact tensor fiber functor.\n\\end{enumerate}\nDirectional or obstructed bridges include:\n\\begin{enumerate}\n\\item restriction and induction between subgroup charts, decomposed by Mackey packets;\n\\item passage from a nonnormal chart to its core or normal-closure envelope;\n\\item projectivization from linear to projective cocycles;\n\\item forgetting tensor structure from a Tannakian fiber functor to an additive or corner realization;\n\\item passage","The preceding results can be collected into one typed finite atlas.  The reversible charts are:\n\\begin{enumerate}\n\\item subgroups \\(H\\leq G\\), fixed fields \\(L^H\\), and faithful Reynolds projectors \\(P_H\\);\n\\item finite-dimensional \\(K\\)-vector spaces and semilinear \\(L\\)-spaces with effective descent data;\n\\item finite \\(\\acute{e}\\)tale fibers and their continuous permutation actions;\n\\item neutral Tannakian categories equipped with a specified faithful exact tensor fiber functor.\n\\end{enumerate}\nDirectional or obstructed bridges include:\n\\begin{enumerate}\n\\item restriction and induction between subgroup charts, decomposed by Mackey packets;\n\\item passage from a nonnormal chart to its core or normal-closure envelope;\n\\item projectivization from linear to projective cocycles;\n\\item forgetting tensor structure from a Tannakian fiber functor to an additive or corner realization;\n\\item passage from finite quotients to a prof","The preceding results can be collected into one typed finite atlas.  The reversible charts are:\n\\begin{enumerate}\n\\item subgroups \\(H\\leq G\\), fixed fields \\(L^H\\), and faithful Reynolds projectors \\(P_H\\);\n\\item finite-dimensional \\(K\\)-vector spaces and semilinear \\(L\\)-spaces with effective descent data;\n\\item finite \\(\\acute{e}\\)tale fibers and their continuous permutation actions;\n\\item neutral Tannakian categories equipped with a specified faithful exact tensor fiber functor.\n\\end{enumerate}\nDirectional or obstructed bridges include:\n\\begin{enumerate}\n\\item restriction and induction between subgroup charts, decomposed by Mackey packets;\n\\item passage from a nonnormal chart to its core or normal-closure envelope;\n\\item projectivization from linear to projective cocycles;\n\\item forgetting tensor structure from a Tannakian fiber functor to an additive or corner realization;\n\\item passage from finite quotients to a profinite inverse system.\n\\end{enumerat","The preceding results can be collected into one typed finite atlas.  The reversible charts are:\n\\begin{enumerate}\n\\item subgroups \\(H\\leq G\\), fixed fields \\(L^H\\), and faithful Reynolds projectors \\(P_H\\);\n\\item finite-dimensional \\(K\\)-vector spaces and semilinear \\(L\\)-spaces with effective descent data;\n\\item finite \\(\\acute{e}\\)tale fibers and their continuous permutation actions;\n\\item neutral Tannakian categories equipped with a specified faithful exact tensor fiber functor.\n\\end{enumerate}\nDirectional or obstructed bridges include:\n\\begin{enumerate}\n\\item restriction and induction between subgroup charts, decomposed by Mackey packets;\n\\item passage from a nonnormal chart to its core or normal-closure envelope;\n\\item projectivization from linear to projective cocycles;\n\\item forgetting tensor structure from a Tannakian fiber functor to an additive or corner realization;\n\\item passage from finite quotients to a profinite inverse system.\n\\end{enumerate}","A complete chart record on the finite subgroup locus is\n\\[\n \\mathcal J_{L/K}(H)\n =\n \\left(\n H,\n L^H,\n P_H,\n \\operatorname{Core}_G(H),\n \\operatorname{Ncl}_G(H),\n \\{K\\backslash G/H\\}_{K\\leq G}\n \\right).\n\\]\nThe first three entries already classify the normalized object, while the remaining entries record its atlas position and transition structure.  For a framed semilinear vector object, the descended \\(K\\)-space with its frame is complete.  For a projective object, the Brauer class is an obstruction coordinate, not a complete invariant of all central simple algebras without the remaining algebra data.","mplete chart record on the finite subgroup locus is\n\\[\n \\mathcal J_{L/K}(H)\n =\n \\left(\n H,\n L^H,\n P_H,\n \\operatorname{Core}_G(H),\n \\operatorname{Ncl}_G(H),\n \\{K\\backslash G/H\\}_{K\\leq G}\n \\right).\n\\]\nThe first three entries already classify the normalized object, while the remaining entries record its atlas position and transition structure.  For a framed semilinear vector object, the descended \\(K\\)-space with its frame is complete.  For a projective object, the Brauer class is an obstruction coordinate, not a complete invariant of all central simple algebras without the remaining algebra data.","The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra.  It audits cyclic groups, \\(V_4\\), \\(S_3\\), \\(D_4\\), and \\(A_4\\); every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commuting and noncommuting subgroup pairs; double-coset partitions and the Mackey dimension law; cyclic ramification-style filtrations; explicit Hilbert 90 coboundaries over \\(\\mathbb C/\\mathbb R\\); the split and quaternionic projective examples; and faithful regular tensor actions.","ssian-rational matrix algebra.  It audits cyclic groups, \\(V_4\\), \\(S_3\\), \\(D_4\\), and \\(A_4\\); every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commuting and noncommuting subgroup pairs; double-coset partitions and the Mackey dimension law; cyclic ramification-style filtrations; explicit Hilbert 90 coboundaries over \\(\\mathbb C/\\mathbb R\\); the split and quaternionic projective examples; and faithful regular tensor actions.","The verifier records \\(11{,}195\\) exact checks and no failures.  Together with the preceding certified atlas program, the cumulative exact record is\n\\[\n \\boxed{2{,}038{,}860}\n\\]\nchecks with no recorded failures.","Write\n\\[\n \\mathsf E,\n \\quad\n \\mathsf{FF},\n \\quad\n \\mathsf D,\n \\quad\n \\mathsf P,\n \\quad\n \\varnothing\n\\]\nfor an equivalence on a named normalized overlap, a fully faithful inclusion, an information-losing directional functor, a partial or hypothesis-dependent bridge, and the absence of a certified broad bridge, respectively.  The matrix below records only relations already established in the preceding sections or in the audited source concordance.  A symbol carrying a subscript refers to the named overlap rather than to the whole broad stratum.","The entries \\(E_{\\mathrm{per}}\\), \\(E_{\\mathrm{rm}}\\), and \\(E_{\\mathrm{mark}}\\) refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap.  In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate from neutral Tannakian reconstruction.","\\begin{definition}[Certified Global synthesis atlas atlas]\n\\label{def:block13-certified-atlas}\nFix a prime \\(p\\), a characteristic-zero field \\(K\\), and a finite Galois extension \\(L/K\\) for the Galois component.  The certified atlas \\(\\mathfrak A_{13}\\) is the finite coproduct of the following five connected profiled atlases.","\\begin{enumerate}[label=\\textup{(C\\arabic*)}]\n\\item \\(\\mathfrak A_{\\mathrm{DRLM},p}\\), the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts;\n\\item \\(\\mathfrak A_{\\mathrm{Loc},p}\\), the continuous Newton--Mahler and complete clopen-branch component;\n\\item \\(\\mathfrak A_{\\mathrm{Fuss}}\\), the marked Fuss--Catalan nonlinear--algebraic--holonomic component;\n\\item \\(\\mathfrak A_{\\mathrm{EM}}\\), the fixed-Euler-basis and prime-uniform multiplicative component;\n\\item \\(\\mathfrak A_{\\mathrm{Gal},L/K}\\), the finite semisimple Galois component for the fixed extension \\(L/K\\).\n\\end{enumerate}","}[label=\\textup{(C\\arabic*)}]\n\\item \\(\\mathfrak A_{\\mathrm{DRLM},p}\\), the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts;\n\\item \\(\\mathfrak A_{\\mathrm{Loc},p}\\), the continuous Newton--Mahler and complete clopen-branch component;\n\\item \\(\\mathfrak A_{\\mathrm{Fuss}}\\), the marked Fuss--Catalan nonlinear--algebraic--holonomic component;\n\\item \\(\\mathfrak A_{\\mathrm{EM}}\\), the fixed-Euler-basis and prime-uniform multiplicative component;\n\\item \\(\\mathfrak A_{\\mathrm{Gal},L/K}\\), the finite semisimple Galois component for the fixed extension \\(L/K\\).\n\\end{enumerate}","component with rational functional-Mahler and periodic linear subcharts;\n\\item \\(\\mathfrak A_{\\mathrm{Loc},p}\\), the continuous Newton--Mahler and complete clopen-branch component;\n\\item \\(\\mathfrak A_{\\mathrm{Fuss}}\\), the marked Fuss--Catalan nonlinear--algebraic--holonomic component;\n\\item \\(\\mathfrak A_{\\mathrm{EM}}\\), the fixed-Euler-basis and prime-uniform multiplicative component;\n\\item \\(\\mathfrak A_{\\mathrm{Gal},L/K}\\), the finite semisimple Galois component for the fixed extension \\(L/K\\).\n\\end{enumerate}","hrm{Loc},p}\\), the continuous Newton--Mahler and complete clopen-branch component;\n\\item \\(\\mathfrak A_{\\mathrm{Fuss}}\\), the marked Fuss--Catalan nonlinear--algebraic--holonomic component;\n\\item \\(\\mathfrak A_{\\mathrm{EM}}\\), the fixed-Euler-basis and prime-uniform multiplicative component;\n\\item \\(\\mathfrak A_{\\mathrm{Gal},L/K}\\), the finite semisimple Galois component for the fixed extension \\(L/K\\).\n\\end{enumerate}","The semantic category of the total atlas is\n\\[\n \\mathcal O_{13}\n =\n \\mathcal O_{\\mathrm{DRLM}}\n \\sqcup\n \\mathcal O_{\\mathrm{Loc}}\n \\sqcup\n \\mathcal O_{\\mathrm{Fuss}}\n \\sqcup\n \\mathcal O_{\\mathrm{EM}}\n \\sqcup\n \\mathcal O_{\\mathrm{Gal}}.\n\\]\nThe coproduct symbol is literal.  It does not assert that a finite-field regular sequence, a continuous \\(p\\)-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category.  The tag records which semantic doctrine is in force.","Every connected component in \\cref{def:block13-certified-atlas} has a canonical semantic normalizer.  For a chart \\(\\mathcal C_{\\lambda,a}\\) in component \\(\\lambda\\), write\n\\[\n N_{\\lambda,a}:\\mathcal C_{\\lambda,a}\\longrightarrow\\mathcal O_\\lambda\n\\]\nfor the semantic normalization functor and choose its canonical inverse chart\n\\[\n Q_{\\lambda,a}:\\mathcal O_\\lambda\\longrightarrow\\mathcal C_{\\lambda,a}.\n\\]\nThe certified bridge from chart \\(a\\) to chart \\(b\\) is\n\\[\n \\Phi_{ab}=Q_{\\lambda,b}N_{\\lambda,a}.\n\\]","lock13-certified-atlas} has a canonical semantic normalizer.  For a chart \\(\\mathcal C_{\\lambda,a}\\) in component \\(\\lambda\\), write\n\\[\n N_{\\lambda,a}:\\mathcal C_{\\lambda,a}\\longrightarrow\\mathcal O_\\lambda\n\\]\nfor the semantic normalization functor and choose its canonical inverse chart\n\\[\n Q_{\\lambda,a}:\\mathcal O_\\lambda\\longrightarrow\\mathcal C_{\\lambda,a}.\n\\]\nThe certified bridge from chart \\(a\\) to chart \\(b\\) is\n\\[\n \\Phi_{ab}=Q_{\\lambda,b}N_{\\lambda,a}.\n\\]","d-atlas} has a canonical semantic normalizer.  For a chart \\(\\mathcal C_{\\lambda,a}\\) in component \\(\\lambda\\), write\n\\[\n N_{\\lambda,a}:\\mathcal C_{\\lambda,a}\\longrightarrow\\mathcal O_\\lambda\n\\]\nfor the semantic normalization functor and choose its canonical inverse chart\n\\[\n Q_{\\lambda,a}:\\mathcal O_\\lambda\\longrightarrow\\mathcal C_{\\lambda,a}.\n\\]\nThe certified bridge from chart \\(a\\) to chart \\(b\\) is\n\\[\n \\Phi_{ab}=Q_{\\lambda,b}N_{\\lambda,a}.\n\\]","has a canonical semantic normalizer.  For a chart \\(\\mathcal C_{\\lambda,a}\\) in component \\(\\lambda\\), write\n\\[\n N_{\\lambda,a}:\\mathcal C_{\\lambda,a}\\longrightarrow\\mathcal O_\\lambda\n\\]\nfor the semantic normalization functor and choose its canonical inverse chart\n\\[\n Q_{\\lambda,a}:\\mathcal O_\\lambda\\longrightarrow\\mathcal C_{\\lambda,a}.\n\\]\nThe certified bridge from chart \\(a\\) to chart \\(b\\) is\n\\[\n \\Phi_{ab}=Q_{\\lambda,b}N_{\\lambda,a}.\n\\]","\\begin{theorem}[Normalizer-induced coherence]\n\\label{thm:block13-normalizer-coherence}\nAssume that every normalized fiber of \\(N_{\\lambda,a}\\) is contractible.  Then the comparison isomorphism\n\\[\n \\eta_{abc}:\n \\Phi_{bc}\\Phi_{ab}\n \\Longrightarrow\n \\Phi_{ac}\n\\]\nis unique.  Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the normalized chart.\n\\end{theorem}","\\begin{proof}\nBoth \\(\\Phi_{bc}\\Phi_{ab}(X)\\) and \\(\\Phi_{ac}(X)\\) are normalized presentations of the same semantic object \\(N_{\\lambda,a}(X)\\).  Contractibility of the normalized fiber gives a unique isomorphism between them.  Any two composites occurring in a coherence diagram are isomorphisms between the same two normalized presentations, so uniqueness makes them equal.  The cycle statement follows by applying the same argument to the starting chart and itself.\n\\end{proof}","For every chart \\(\\mathcal C_{\\lambda,a}\\), let\n\\[\n I_{\\lambda,a}:\\mathcal C_{\\lambda,a}\\longrightarrow\\mathcal V_{\\lambda,a}\n\\]\nbe its native normalized invariant.  A bridge comparison is a natural isomorphism\n\\[\n \\kappa_{ab}:\n J_{ab}I_{\\lambda,a}\n \\Longrightarrow\n I_{\\lambda,b}\\Phi_{ab},\n\\]\nwhere \\(J_{ab}\\) transports invariant coordinates between their chart-specific target categories.","\\begin{definition}[Coherent invariant family]\nThe family \\((I_{\\lambda,a},\\kappa_{ab})\\) is coherent when the two induced comparison maps from \\(a\\) to \\(c\\) agree on every triple overlap and the corresponding quadruple-overlap equation holds.\n\\end{definition}","\\begin{theorem}[Certified finite invariant descent]\n\\label{thm:block13-certified-invariant-descent}\nLet \\(\\mathfrak A_{13}\\) be the atlas of \\cref{def:block13-certified-atlas}.  Suppose that, in every connected component:\n\\begin{enumerate}\n\\item the local invariant is complete on every normalized chart;\n\\item the bridge comparisons are coherent;\n\\item invariant-compatible local isomorphisms satisfy effective descent.\n\\end{enumerate}\nThen the local invariants induce a functor\n\\[\n \\mathcal I_{13}:\n \\Desc(\\mathfrak A_{13})\n \\longrightarrow\n \\Desc(\\mathfrak V_{13})\n\\]\nthat reflects isomorphisms.  If each semantic component is discrete, equality of descended invariants is equivalent to equality of the represented semantic objects.\n\\end{theorem}","nvariant-descent}\nLet \\(\\mathfrak A_{13}\\) be the atlas of \\cref{def:block13-certified-atlas}.  Suppose that, in every connected component:\n\\begin{enumerate}\n\\item the local invariant is complete on every normalized chart;\n\\item the bridge comparisons are coherent;\n\\item invariant-compatible local isomorphisms satisfy effective descent.\n\\end{enumerate}\nThen the local invariants induce a functor\n\\[\n \\mathcal I_{13}:\n \\Desc(\\mathfrak A_{13})\n \\longrightarrow\n \\Desc(\\mathfrak V_{13})\n\\]\nthat reflects isomorphisms.  If each semantic component is discrete, equality of descended invariants is equivalent to equality of the represented semantic objects.\n\\end{theorem}","Fix the prime \\(p\\) and let\n\\[\n \\mathcal O_{\\mathrm{DRLM}}\n =\\mathbf{Reg}_p(\\mathbb F_p)\n\\]\nbe the discrete semantic category of \\(p\\)-regular sequences over \\(\\mathbb F_p\\).  By the finite-field fixed-radix theorem, every object is also \\(p\\)-automatic.  Its globally defined normalized charts are the canonical section module, the strict shortlex pointed digit-linear record, and the minimal least-significant-digit-first \\(p\\)-DFAO.","t\n\\[\n \\mathcal O_{\\mathrm{DRLM}}\n =\\mathbf{Reg}_p(\\mathbb F_p)\n\\]\nbe the discrete semantic category of \\(p\\)-regular sequences over \\(\\mathbb F_p\\).  By the finite-field fixed-radix theorem, every object is also \\(p\\)-automatic.  Its globally defined normalized charts are the canonical section module, the strict shortlex pointed digit-linear record, and the minimal least-significant-digit-first \\(p\\)-DFAO.","Two full replete subcategories carry additional charts.\n\\begin{enumerate}\n\\item On the rational-mask locus \\(\\RMah_p^{\\times}\\), the reduced functional-Mahler mask and the minimal reachable-observable Cartier module are equivalent to the digital charts.\n\\item On the unit purely periodic locus, the minimal recurrence, reduced rational OGF, minimal finite linear state, primitive period word, and rational-mask chart are all equivalent.\n\\end{enumerate}","For \\(a\\in\\mathcal O_{\\mathrm{DRLM}}\\), define\n\\[\n \\mathcal J_{\\mathrm{DRLM},p}(a)\n =\n \\left(\n \\mathsf{SL}_{p,\\mathbb F_p}(a),\n \\mathcal P_{\\mathrm{rm}}(a),\n \\mathcal P_{\\mathrm{per}}(a)\n \\right),\n\\]\nwhere \\(\\mathsf{SL}_{p,\\mathbb F_p}(a)\\) is the strict shortlex record, \\(\\mathcal P_{\\mathrm{rm}}(a)\\) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and \\(\\mathcal P_{\\mathrm{per}}(a)\\) is empty off the purely periodic locus and otherwise contains the primitive period word, exact period, minimal recurrence, reduced OGF, and periodic packet profile.","_{\\mathrm{DRLM},p}(a)\n =\n \\left(\n \\mathsf{SL}_{p,\\mathbb F_p}(a),\n \\mathcal P_{\\mathrm{rm}}(a),\n \\mathcal P_{\\mathrm{per}}(a)\n \\right),\n\\]\nwhere \\(\\mathsf{SL}_{p,\\mathbb F_p}(a)\\) is the strict shortlex record, \\(\\mathcal P_{\\mathrm{rm}}(a)\\) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and \\(\\mathcal P_{\\mathrm{per}}(a)\\) is empty off the purely periodic locus and otherwise contains the primitive period word, exact period, minimal recurrence, reduced OGF, and periodic packet profile.","\\begin{theorem}[Digital--regular--linear--Mahler comparison theorem]\n\\label{thm:block13-drlm-comparison}\nThe invariant \\(\\mathcal J_{\\mathrm{DRLM},p}\\) is preserved by every reversible bridge in \\(\\mathfrak A_{\\mathrm{DRLM},p}\\).  Its comparison cells satisfy all higher coherence equations.  The strict shortlex coordinate reconstructs the represented regular sequence, so the descended invariant is complete on \\(\\mathbf{Reg}_p(\\mathbb F_p)\\).  The mask and periodic packages are compatible exact refinements on their respective subloci.\n\\end{theorem}","[Digital--regular--linear--Mahler comparison theorem]\n\\label{thm:block13-drlm-comparison}\nThe invariant \\(\\mathcal J_{\\mathrm{DRLM},p}\\) is preserved by every reversible bridge in \\(\\mathfrak A_{\\mathrm{DRLM},p}\\).  Its comparison cells satisfy all higher coherence equations.  The strict shortlex coordinate reconstructs the represented regular sequence, so the descended invariant is complete on \\(\\mathbf{Reg}_p(\\mathbb F_p)\\).  The mask and periodic packages are compatible exact refinements on their respective subloci.\n\\end{theorem}","The inclusions of the rational-mask and periodic loci are not essentially surjective.  Over a finite field every regular sequence is automatic, but many automatic sequences have no rational canonical mask, and many rational-mask sequences are not periodic.  The optional fields in \\(\\mathcal J_{\\mathrm{DRLM},p}\\) record these exact domain boundaries rather than forcing one chart to exist everywhere.","Let\n\\[\n \\mathcal O_{\\mathrm{Loc}}\n =C(\\mathbb Z_p,\\mathbb Z_p)\n\\]\nwith equality morphisms.  Its canonical coordinate is the Mahler sequence\n\\[\n \\mathcal M_p(f)\n =\n (c_r(f))_{r\\ge0},\n \\qquad\n c_r(f)=\\Delta^r f(0),\n \\qquad\n c_r(f)\\longrightarrow0.\n\\]\nAt every declared depth \\(p^a\\), the complete clopen branch packet\n\\[\n \\bigl(f(p^aq+s)\\bigr)_{0\\le s<p^a}\n\\]\nand the corresponding branchwise Mahler coefficients are reversible by explicit piecewise synthesis.  Affine branch kernels and mixed-radix refinements are comparison charts of the same semantic function.","\\begin{theorem}[Continuous local invariant comparison]\n\\label{thm:block13-local-comparison}\nThe Mahler coefficient sequence\n\\[\n \\mathcal J_{\\mathrm{Loc},p}(f)\n =\n (c_r(f))_{r\\ge0}\n\\]\nis a complete invariant of \\(\\mathfrak A_{\\mathrm{Loc},p}\\).  Complete branch packets, branchwise Mahler coordinates, affine coefficient kernels, and nested refinements admit coherent comparison maps to \\(\\mathcal J_{\\mathrm{Loc},p}\\).\n\\end{theorem}","\\begin{proof}\nMahler analysis and synthesis are mutually inverse on \\(C(\\mathbb Z_p,\\mathbb Z_p)\\).  Complete clopen analysis and piecewise synthesis are mutually inverse at each depth, and the nested branch law is strict.  Conjugating those maps by Mahler analysis gives the affine coefficient comparisons, whose mixed-radix composition was proved in Local Mahler-polyphase atlas.  All normalized charts therefore return the same function and the same Mahler sequence, so \\cref{thm:block13-normalizer-coherence} gives higher coherence.\n\\end{proof}","Let \\(K\\) be a characteristic-zero field.  For an integer \\(q\\geq2\\) and \\(\\alpha\\in K^\\times\\), let\n\\[\n U_{q,\\alpha}(z)\\in zK[[z]]\n\\]\nbe the unique selected branch satisfying\n\\begin{equation}\n U=z+\\alpha U^q.\n \\label{eq:block13-fuss-tree}\n\\end{equation}\nThe derivative of\n\\[\n P_{q,\\alpha}(z,Y)=Y-z-\\alpha Y^q\n\\]\nwith respect to \\(Y\\) is one at \\((0,0)\\), so the branch is simple and selected by \\(U(0)=0\\).","Put\n\\[\n x=z^{q-1},\n \\qquad\n U_{q,\\alpha}(z)=zC_{q,\\alpha}(x).\n\\]\nThen\n\\begin{equation}\n C=1+\\alpha xC^q.\n \\label{eq:block13-fuss-compressed}\n\\end{equation}\nLagrange inversion gives\n\\begin{equation}\n [x^n]C_{q,\\alpha}(x)\n =\n c_n(q,\\alpha)\n =\n \\frac{\\alpha^n}{(q-1)n+1}\n \\binom{qn}{n}.\n \\label{eq:block13-fuss-coefficients}\n\\end{equation}\nDefine\n\\[\n A_q(n)\n =(n+1)\\prod_{r=2}^{q}\\bigl((q-1)n+r\\bigr),\n\\]\n\\[\n B_q(n)\n =\\prod_{r=1}^{q}(qn+r).\n\\]\nThen the coefficient sequence satisfies the first-order polynomial recurrence\n\\begin{equation}\n \\boxed{\n A_q(n)c_{n+1}\n =\n \\alpha B_q(n)c_n.}\n \\label{eq:block13-fuss-recurrence}\n\\end{equation}\nAfter dividing the polynomial gcd of \\(A_q\\) and \\(B_q\\), this is the normalized first-order tail recurrence with initial value","Put\n\\[\n x=z^{q-1},\n \\qquad\n U_{q,\\alpha}(z)=zC_{q,\\alpha}(x).\n\\]\nThen\n\\begin{equation}\n C=1+\\alpha xC^q.\n \\label{eq:block13-fuss-compressed}\n\\end{equation}\nLagrange inversion gives\n\\begin{equation}\n [x^n]C_{q,\\alpha}(x)\n =\n c_n(q,\\alpha)\n =\n \\frac{\\alpha^n}{(q-1)n+1}\n \\binom{qn}{n}.\n \\label{eq:block13-fuss-coefficients}\n\\end{equation}\nDefine\n\\[\n A_q(n)\n =(n+1)\\prod_{r=2}^{q}\\bigl((q-1)n+r\\bigr),\n\\]\n\\[\n B_q(n)\n =\\prod_{r=1}^{q}(qn+r).\n\\]\nThen the coefficient sequence satisfies the first-order polynomial recurrence\n\\begin{equation}\n \\boxed{\n A_q(n)c_{n+1}\n =\n \\alpha B_q(n)c_n.}\n \\label{eq:block13-fuss-recurrence}\n\\end{equation}\nAfter dividing the polynomial gcd of \\(A_q\\) and \\(B_q\\), this is the normalized first-order tail recurrence with initial value \\(c_0=1\\).","\\[\n x=z^{q-1},\n \\qquad\n U_{q,\\alpha}(z)=zC_{q,\\alpha}(x).\n\\]\nThen\n\\begin{equation}\n C=1+\\alpha xC^q.\n \\label{eq:block13-fuss-compressed}\n\\end{equation}\nLagrange inversion gives\n\\begin{equation}\n [x^n]C_{q,\\alpha}(x)\n =\n c_n(q,\\alpha)\n =\n \\frac{\\alpha^n}{(q-1)n+1}\n \\binom{qn}{n}.\n \\label{eq:block13-fuss-coefficients}\n\\end{equation}\nDefine\n\\[\n A_q(n)\n =(n+1)\\prod_{r=2}^{q}\\bigl((q-1)n+r\\bigr),\n\\]\n\\[\n B_q(n)\n =\\prod_{r=1}^{q}(qn+r).\n\\]\nThen the coefficient sequence satisfies the first-order polynomial recurrence\n\\begin{equation}\n \\boxed{\n A_q(n)c_{n+1}\n =\n \\alpha B_q(n)c_n.}\n \\label{eq:block13-fuss-recurrence}\n\\end{equation}\nAfter dividing the polynomial gcd of \\(A_q\\) and \\(B_q\\), this is the normalized first-order tail recurrence with initial value \\(c_0=1\\).","ssed}\n\\end{equation}\nLagrange inversion gives\n\\begin{equation}\n [x^n]C_{q,\\alpha}(x)\n =\n c_n(q,\\alpha)\n =\n \\frac{\\alpha^n}{(q-1)n+1}\n \\binom{qn}{n}.\n \\label{eq:block13-fuss-coefficients}\n\\end{equation}\nDefine\n\\[\n A_q(n)\n =(n+1)\\prod_{r=2}^{q}\\bigl((q-1)n+r\\bigr),\n\\]\n\\[\n B_q(n)\n =\\prod_{r=1}^{q}(qn+r).\n\\]\nThen the coefficient sequence satisfies the first-order polynomial recurrence\n\\begin{equation}\n \\boxed{\n A_q(n)c_{n+1}\n =\n \\alpha B_q(n)c_n.}\n \\label{eq:block13-fuss-recurrence}\n\\end{equation}\nAfter dividing the polynomial gcd of \\(A_q\\) and \\(B_q\\), this is the normalized first-order tail recurrence with initial value \\(c_0=1\\).","om{qn}{n}.\n \\label{eq:block13-fuss-coefficients}\n\\end{equation}\nDefine\n\\[\n A_q(n)\n =(n+1)\\prod_{r=2}^{q}\\bigl((q-1)n+r\\bigr),\n\\]\n\\[\n B_q(n)\n =\\prod_{r=1}^{q}(qn+r).\n\\]\nThen the coefficient sequence satisfies the first-order polynomial recurrence\n\\begin{equation}\n \\boxed{\n A_q(n)c_{n+1}\n =\n \\alpha B_q(n)c_n.}\n \\label{eq:block13-fuss-recurrence}\n\\end{equation}\nAfter dividing the polynomial gcd of \\(A_q\\) and \\(B_q\\), this is the normalized first-order tail recurrence with initial value \\(c_0=1\\).","\\label{eq:block13-fuss-coefficients}\n\\end{equation}\nDefine\n\\[\n A_q(n)\n =(n+1)\\prod_{r=2}^{q}\\bigl((q-1)n+r\\bigr),\n\\]\n\\[\n B_q(n)\n =\\prod_{r=1}^{q}(qn+r).\n\\]\nThen the coefficient sequence satisfies the first-order polynomial recurrence\n\\begin{equation}\n \\boxed{\n A_q(n)c_{n+1}\n =\n \\alpha B_q(n)c_n.}\n \\label{eq:block13-fuss-recurrence}\n\\end{equation}\nAfter dividing the polynomial gcd of \\(A_q\\) and \\(B_q\\), this is the normalized first-order tail recurrence with initial value \\(c_0=1\\).","rod_{r=2}^{q}\\bigl((q-1)n+r\\bigr),\n\\]\n\\[\n B_q(n)\n =\\prod_{r=1}^{q}(qn+r).\n\\]\nThen the coefficient sequence satisfies the first-order polynomial recurrence\n\\begin{equation}\n \\boxed{\n A_q(n)c_{n+1}\n =\n \\alpha B_q(n)c_n.}\n \\label{eq:block13-fuss-recurrence}\n\\end{equation}\nAfter dividing the polynomial gcd of \\(A_q\\) and \\(B_q\\), this is the normalized first-order tail recurrence with initial value \\(c_0=1\\).","\\begin{theorem}[Fuss--Catalan three-stratum equivalence]\n\\label{thm:block13-fuss-equivalence}\nThe following normalized marked presentations determine one another exactly:\n\\begin{enumerate}\n\\item the polynomial tree system \\(U=z+\\alpha U^q\\);\n\\item the selected algebraic branch \\((P_{q,\\alpha},U(0)=0)\\);\n\\item the compressed P-recursive coefficient chart \\((\\cref{eq:block13-fuss-recurrence},c_0=1)\\) together with the support embedding \\(U(z)=zC(z^{q-1})\\).\n\\end{enumerate}\nEvery normalized fiber is contractible.  All triangle comparisons return the same parameter pair \\((q,\\alpha)\\).\n\\end{theorem}","tum equivalence]\n\\label{thm:block13-fuss-equivalence}\nThe following normalized marked presentations determine one another exactly:\n\\begin{enumerate}\n\\item the polynomial tree system \\(U=z+\\alpha U^q\\);\n\\item the selected algebraic branch \\((P_{q,\\alpha},U(0)=0)\\);\n\\item the compressed P-recursive coefficient chart \\((\\cref{eq:block13-fuss-recurrence},c_0=1)\\) together with the support embedding \\(U(z)=zC(z^{q-1})\\).\n\\end{enumerate}\nEvery normalized fiber is contractible.  All triangle comparisons return the same parameter pair \\((q,\\alpha)\\).\n\\end{theorem}","in{enumerate}\n\\item the polynomial tree system \\(U=z+\\alpha U^q\\);\n\\item the selected algebraic branch \\((P_{q,\\alpha},U(0)=0)\\);\n\\item the compressed P-recursive coefficient chart \\((\\cref{eq:block13-fuss-recurrence},c_0=1)\\) together with the support embedding \\(U(z)=zC(z^{q-1})\\).\n\\end{enumerate}\nEvery normalized fiber is contractible.  All triangle comparisons return the same parameter pair \\((q,\\alpha)\\).\n\\end{theorem}","\\begin{proof}\nThe marked tree equation is the selected algebraic equation.  Simplicity at the origin gives its unique formal branch.  Substitution \\(U=zC(z^{q-1})\\) gives \\cref{eq:block13-fuss-compressed}.  Lagrange inversion gives \\cref{eq:block13-fuss-coefficients}, and taking the ratio of consecutive coefficients gives \\cref{eq:block13-fuss-recurrence}.  Conversely, the recurrence and initial value reconstruct \\(C\\), the support embedding reconstructs \\(U\\), and the reconstructed series satisfies the selected algebraic equation.  A normalized automorphism fixes \\(q\\), \\(\\alpha\\), and the unique branch, hence is the identity.\n\\end{proof}","arked tree equation is the selected algebraic equation.  Simplicity at the origin gives its unique formal branch.  Substitution \\(U=zC(z^{q-1})\\) gives \\cref{eq:block13-fuss-compressed}.  Lagrange inversion gives \\cref{eq:block13-fuss-coefficients}, and taking the ratio of consecutive coefficients gives \\cref{eq:block13-fuss-recurrence}.  Conversely, the recurrence and initial value reconstruct \\(C\\), the support embedding reconstructs \\(U\\), and the reconstructed series satisfies the selected algebraic equation.  A normalized automorphism fixes \\(q\\), \\(\\alpha\\), and the unique branch, hence is the identity.\n\\end{proof}","aic equation.  Simplicity at the origin gives its unique formal branch.  Substitution \\(U=zC(z^{q-1})\\) gives \\cref{eq:block13-fuss-compressed}.  Lagrange inversion gives \\cref{eq:block13-fuss-coefficients}, and taking the ratio of consecutive coefficients gives \\cref{eq:block13-fuss-recurrence}.  Conversely, the recurrence and initial value reconstruct \\(C\\), the support embedding reconstructs \\(U\\), and the reconstructed series satisfies the selected algebraic equation.  A normalized automorphism fixes \\(q\\), \\(\\alpha\\), and the unique branch, hence is the identity.\n\\end{proof}","ref{eq:block13-fuss-compressed}.  Lagrange inversion gives \\cref{eq:block13-fuss-coefficients}, and taking the ratio of consecutive coefficients gives \\cref{eq:block13-fuss-recurrence}.  Conversely, the recurrence and initial value reconstruct \\(C\\), the support embedding reconstructs \\(U\\), and the reconstructed series satisfies the selected algebraic equation.  A normalized automorphism fixes \\(q\\), \\(\\alpha\\), and the unique branch, hence is the identity.\n\\end{proof}","block13-fuss-compressed}.  Lagrange inversion gives \\cref{eq:block13-fuss-coefficients}, and taking the ratio of consecutive coefficients gives \\cref{eq:block13-fuss-recurrence}.  Conversely, the recurrence and initial value reconstruct \\(C\\), the support embedding reconstructs \\(U\\), and the reconstructed series satisfies the selected algebraic equation.  A normalized automorphism fixes \\(q\\), \\(\\alpha\\), and the unique branch, hence is the identity.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.87\\textwidth]{fuss_catalan_three_stratum.pdf}\n\\caption{The marked Fuss--Catalan family gives a genuine overlap among the nonlinear tree, selected algebraic, and holonomic recurrence strata.  Every path returns the same marked pair \\((q,\\alpha)\\).}\n\\label{fig:block13-fuss-triangle}\n\\end{figure}","The tree arity period is \\(q-1\\).  For packet modulus \\(N\\), set\n\\[\n g_N=\\gcd(N,q-1).\n\\]\nThe nonzero coefficient addresses of \\(U_{q,\\alpha}\\) are exactly\n\\[\n j\\equiv1\\pmod{g_N},\n\\]\nand the complete dominant singularity orbit has size\n\\[\n \\frac{q-1}{g_N}\n\\]\nunder the positive analytic hypotheses of \\cref{thm:block11-qary-phase}.  These packet signatures agree in the tree and algebraic charts because both are computed from the same selected branch.","Define\n\\[\n \\mathcal J_{\\mathrm{Fuss}}(q,\\alpha)\n =\n \\left(\n q,\n \\alpha,\n P_{q,\\alpha},\n \\{c_n(q,\\alpha)\\}_{n\\geq0},\n \\{g_N\\}_{N\\geq1},\n \\left\\{\\frac{q-1}{g_N}\\right\\}_{N\\geq1}\n \\right).\n\\]","\\begin{corollary}[Complete cross-stratum invariant on the marked Fuss family]\n\\label{cor:block13-fuss-invariant}\nThe nonlinear, algebraic, holonomic, packet-support, and dominant-orbit invariants admit coherent comparison maps to \\(\\mathcal J_{\\mathrm{Fuss}}\\).  The package is complete on the declared marked family.\n\\end{corollary}","The examples show that the finite invariant retains the weight parameter as well as the arity.  Support period alone recovers only \\(q-1\\); the coefficient \\([z^q]U=\\alpha\\) supplies the second coordinate.","Let\n\\[\n A_b(z)=\\prod_{m\\geq1}(1-z^m)^{-b_m}\n\\]\nbe an integral unit series in the fixed Euler basis, and let\n\\[\n c_n=\\sum_{d\\mid n}d\\,b_d\n\\]\nbe its divisor-ghost sequence.  Define the prime-uniform multiplicative function\n\\[\n f_b(n)=\\prod_{p^e\\parallel n}[z^e]A_b(z).\n\\]\nEvery prime has the same local factor \\(A_b(T)\\).  By \\cref{thm:block07-euler-classification,thm:block07-product-multiplicative-descent}, the coefficient, exponent, ghost, and prime-uniform multiplicative charts are reversible on this declared locus.","\\begin{theorem}[Euler--multiplicative invariant comparison]\n\\label{thm:block13-em-comparison}\nThe exponent sequence\n\\[\n \\mathcal J_{\\mathrm{EM}}(A_b)=b\n\\]\nis a complete invariant of the certified Euler--multiplicative component.  In every chart it is recovered by the same M\\\"obius formula\n\\[\n b_n\n =\n \\frac1n\n \\sum_{d\\mid n}\\mu(n/d)c_d,\n\\]\nand Dirichlet convolution corresponds to exponent addition.\n\\end{theorem}","\\begin{proof}\nUnique Euler factorization reconstructs \\(A_b\\) from \\(b\\).  Prime-power reconstruction then reconstructs \\(f_b\\).  The logarithmic-derivative identity gives \\(c\\), and M\\\"obius inversion recovers \\(b\\).  Every bridge therefore factors through the same exponent normalizer, so higher coherence follows from \\cref{thm:block13-normalizer-coherence}.\n\\end{proof}","Fix a finite Galois extension \\(L/K\\) in the semisimple setting.  For each subgroup \\(H\\leq G\\), define\n\\[\n \\mathcal J_{\\mathrm{Gal}}(H)\n =\n \\left(\n H,\n L^H,\n P_H,\n \\operatorname{Core}_G(H),\n \\operatorname{Ncl}_G(H),\n \\{J\\backslash G/H\\}_{J\\leq G}\n \\right).\n\\]","\\begin{theorem}[Finite Galois invariant comparison]\n\\label{thm:block13-gal-comparison}\nThe subgroup, fixed-field, and faithful Reynolds-projector charts have coherent comparison maps to \\(\\mathcal J_{\\mathrm{Gal}}\\).  The invariant is complete on the normalized subgroup atlas for the fixed extension \\(L/K\\).\n\\end{theorem}","Let\n\\[\n \\Lambda_{13}\n =\n \\{\\mathrm{DRLM},\\mathrm{Loc},\\mathrm{Fuss},\\mathrm{EM},\\mathrm{Gal}\\}\n\\]\nbe the component-tag set.  Define\n\\[\n \\mathcal J_{13}(X)\n =\n \\bigl(\\lambda(X),\\mathcal J_{\\lambda(X)}(X)\\bigr).\n\\]","\\begin{theorem}[Certified global completeness theorem]\n\\label{thm:block13-global-completeness}\nThe invariant \\(\\mathcal J_{13}\\) descends across every reversible bridge in \\(\\mathfrak A_{13}\\) and is complete on\n\\[\n \\mathcal O_{13}\n =\n \\mathcal O_{\\mathrm{DRLM}}\n \\sqcup\n \\mathcal O_{\\mathrm{Loc}}\n \\sqcup\n \\mathcal O_{\\mathrm{Fuss}}\n \\sqcup\n \\mathcal O_{\\mathrm{EM}}\n \\sqcup\n \\mathcal O_{\\mathrm{Gal}}.\n\\]\nMore precisely,\n\\[\n \\boxed{\n \\mathcal J_{13}(X)=\\mathcal J_{13}(Y)\n \\quad\\Longleftrightarrow\\quad\n X=Y\n }\n\\]\ninside the declared discrete semantic category.  The associated invariant functor reflects isomorphisms on the descent category.\n\\end{theorem}","global completeness theorem]\n\\label{thm:block13-global-completeness}\nThe invariant \\(\\mathcal J_{13}\\) descends across every reversible bridge in \\(\\mathfrak A_{13}\\) and is complete on\n\\[\n \\mathcal O_{13}\n =\n \\mathcal O_{\\mathrm{DRLM}}\n \\sqcup\n \\mathcal O_{\\mathrm{Loc}}\n \\sqcup\n \\mathcal O_{\\mathrm{Fuss}}\n \\sqcup\n \\mathcal O_{\\mathrm{EM}}\n \\sqcup\n \\mathcal O_{\\mathrm{Gal}}.\n\\]\nMore precisely,\n\\[\n \\boxed{\n \\mathcal J_{13}(X)=\\mathcal J_{13}(Y)\n \\quad\\Longleftrightarrow\\quad\n X=Y\n }\n\\]\ninside the declared discrete semantic category.  The associated invariant functor reflects isomorphisms on the descent category.\n\\end{theorem}","The certified reversible core is surrounded by directional bridges that remain useful for theorem transport and computation.  Their failure to enter \\(\\mathcal J_{13}\\) is controlled by explicit loss data.","\\midrule\nIntegral regular lattice \\(\\to\\) residue section module & digit transitions and residue semantics & Smith--Tor ghost kernel and missing quotient","General regular sequence \\(\\dashrightarrow\\) rational mask & coefficient semantics on the essential image & rational-mask existence and collapse obstruction","Algebraic branch \\(\\to\\) holonomic annihilator & D-finite coefficient semantics & selected branch and minimal polynomial may be forgotten","Marked nonlinear system \\(\\to\\) semantic fixed point & the represented fixed point & generator labels, arities, and tree grammar may be forgotten","Finite-state dynamics \\(\\to\\) Newton--Mahler chart & congruence branch values and precision ladder & depends on a chosen lift and principal-unit hypothesis","Projective Galois cocycle \\(\\dashrightarrow\\) linear descent & projective action & relative Brauer class","Faithful corner realization \\(\\dashrightarrow\\) Tannakian symmetry & additive packet and operator equations & exact tensor, dual, unit, and fiber-functor data are absent","\\begin{proposition}[No completeness transport through an unreflected directional bridge]\n\\label{prop:block13-no-directional-completeness}\nLet \\(T:\\mathcal C\\to\\mathcal D\\) be a directional bridge.  A complete invariant on \\(\\mathcal D\\) pulls back to a complete invariant on \\(\\mathcal C\\) only if \\(T\\) reflects isomorphisms on the declared source subcategory.  In particular, any nontrivial object-collapse relation or obstruction fiber prevents automatic completeness transport.\n\\end{proposition}","\\begin{proof}\nIf \\(T\\) does not reflect isomorphisms, there exist nonisomorphic \\(X,Y\\) with \\(T(X)\\cong T(Y)\\).  Every invariant depending only on the target object then takes the same value on \\(X\\) and \\(Y\\), so its pullback cannot be complete.\n\\end{proof}","Two complementary exact suites audit Global synthesis atlas.  The first authenticates the frozen preceding verification records and checks the broad bridge matrix, periodic and product--multiplicative overlap replays, directional-loss controls, finite \\v{C}ech holonomy models, and the five completeness gates.  The second checks the weighted \\(q\\)-ary tree--algebraic--holonomic overlap, its coefficient and packet equations, parameter recovery, detailed bridge ledger, and tagged component separation.","The two suites execute \\(\\BlockThirteenChecks\\) exact assertions with no failures.  Of these, \\(\\BlockThirteenRegressionChecks\\) authenticate or replay assertions already represented in the preceding cumulative record.  The global synthesis therefore contributes \\(\\BlockThirteenNewChecks\\) nonduplicative comparison, coherence, packet, and loss-control assertions.  The blockwise cumulative record is\n\\[\n \\boxed{\\BlockThirteenCumulative}\n\\]\nexact assertions with no recorded failures.  Counting every regression assertion executed during the Global synthesis atlas replay gives \\(\\BlockThirteenCumulativeExecuted\\) total executed assertions.  Neither number is a count of independent theorems.","two suites execute \\(\\BlockThirteenChecks\\) exact assertions with no failures.  Of these, \\(\\BlockThirteenRegressionChecks\\) authenticate or replay assertions already represented in the preceding cumulative record.  The global synthesis therefore contributes \\(\\BlockThirteenNewChecks\\) nonduplicative comparison, coherence, packet, and loss-control assertions.  The blockwise cumulative record is\n\\[\n \\boxed{\\BlockThirteenCumulative}\n\\]\nexact assertions with no recorded failures.  Counting every regression assertion executed during the Global synthesis atlas replay gives \\(\\BlockThirteenCumulativeExecuted\\) total executed assertions.  Neither number is a count of independent theorems.","d in the preceding cumulative record.  The global synthesis therefore contributes \\(\\BlockThirteenNewChecks\\) nonduplicative comparison, coherence, packet, and loss-control assertions.  The blockwise cumulative record is\n\\[\n \\boxed{\\BlockThirteenCumulative}\n\\]\nexact assertions with no recorded failures.  Counting every regression assertion executed during the Global synthesis atlas replay gives \\(\\BlockThirteenCumulativeExecuted\\) total executed assertions.  Neither number is a count of independent theorems.","The bridge-synthesis suite replays primitive unit-periodic words over finite prime fields, compares their recurrence, OGF, automaton, section-module, and functional-Mahler coordinates, verifies the prime-uniform Euler--multiplicative overlap, and tests finite nontrivial \\v{C}ech cycle models.  The weighted \\(q\\)-ary suite checks the fixed-point identity, Lagrange coefficients, first-order P-recursive recurrence, support phases, exact packet synthesis, address-carry equations, and all triangle normalizers over a declared finite parameter grid.  Both machine-readable records are included in the source package.","Let\n\\[\n \\mathfrak A_{\\mathrm{cert}}:=\\mathfrak A_{13}\n\\]\ndenote the finite certified atlas constructed in \\cref{sec:block13-global-invariant}.  Its reversible core is the disjoint union of five connected semantic components, while its directional satellites retain explicit loss or obstruction passports.","\\begin{theorem}[Stratified resolved representation, descent, and complete invariant theorem]\n\\label{thm:block14-main}\nLet \\(\\mathbb K\\) be the resolved theory used in this manuscript and let \\(\\mathfrak A_{\\mathrm{cert}}\\) be the declared finite atlas.  Then the following statements hold.\n\\begin{enumerate}\n\\item Every chart is a profile-preserving realization of the common resolved syntax, and every reversible edge is an equivalence on its named overlap category.\n\\item The descent category exists as the finite pseudolimit of the atlas and decomposes as the coproduct of its five connected semantic components.\n\\item Every proof whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object.\n\\item The semantic normalizer","\\begin{theorem}[Stratified resolved representation, descent, and complete invariant theorem]\n\\label{thm:block14-main}\nLet \\(\\mathbb K\\) be the resolved theory used in this manuscript and let \\(\\mathfrak A_{\\mathrm{cert}}\\) be the declared finite atlas.  Then the following statements hold.\n\\begin{enumerate}\n\\item Every chart is a profile-preserving realization of the common resolved syntax, and every reversible edge is an equivalence on its named overlap category.\n\\item The descent category exists as the finite pseudolimit of the atlas and decomposes as the coproduct of its five connected semantic components.\n\\item Every proof whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object.\n\\item The semantic normalizers induce the comparison 2-cells on all nonempty triple overlaps and force the quadruple","whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object.\n\\item The semantic normalizers induce the comparison 2-cells on all nonempty triple overlaps and force the quadruple-overlap coherence equation.\n\\item The compatible native invariant family descends to a tagged global invariant\n\\[\n \\mathcal J_{\\mathrm{cert}}:\n \\Desc(\\mathfrak A_{\\mathrm{cert}})\n \\longrightarrow\n \\Desc(\\mathcal V_{\\mathrm{cert}}).\n\\]\n\\item The descended invariant is complete on the declared normalized global category:\n\\[\n \\mathcal J_{\\mathrm{cert}}(X)\\cong\n \\mathcal J_{\\mathrm{cert}}(Y)\n \\quad\\Longleftrightarrow\\quad\n X\\cong Y.\n\\]\n\\item Every noninvertible construction remains outside the reversible core and carries a typed record of its preserved structure, forgotten information, named residual, or obstruction class.\n\\end{enumerate}\n\\end{theorem}","e comparison 2-cells on all nonempty triple overlaps and force the quadruple-overlap coherence equation.\n\\item The compatible native invariant family descends to a tagged global invariant\n\\[\n \\mathcal J_{\\mathrm{cert}}:\n \\Desc(\\mathfrak A_{\\mathrm{cert}})\n \\longrightarrow\n \\Desc(\\mathcal V_{\\mathrm{cert}}).\n\\]\n\\item The descended invariant is complete on the declared normalized global category:\n\\[\n \\mathcal J_{\\mathrm{cert}}(X)\\cong\n \\mathcal J_{\\mathrm{cert}}(Y)\n \\quad\\Longleftrightarrow\\quad\n X\\cong Y.\n\\]\n\\item Every noninvertible construction remains outside the reversible core and carries a typed record of its preserved structure, forgotten information, named residual, or obstruction class.\n\\end{enumerate}\n\\end{theorem}","\\begin{definition}[Finite resolution shape]\n\\label{def:block15-resolution-diagram}\nA finite resolution shape is a finite partially ordered set\n\\(\\mathsf R\\).  We write \\(s\\succeq r\\) when \\(s\\) is finer than \\(r\\).\nThe shape contains no address sets: object-specific addresses and their\nrefinement maps are declared in the lane system below.\n\\end{definition}","\\begin{definition}[Finite profiled lane system]\n\\label{def:block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also cont","\\begin{definition}[Finite profiled lane system]\n\\label{def:block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}","\\begin{definition}[Finite profiled lane system]\n\\label{def:block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i","\\begin{definition}[Finite profiled lane system]\n\\label{def:block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho","\\begin{definition}[Finite profiled lane system]\n\\label{def:block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X","\\begin{definition}[Finite profiled lane system]\n\\label{def:block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinem","\\begin{definition}[Finite profiled lane system]\n\\label{def:block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinement maps are\ndeclared data; they are not inf…","block15-finite-profiled-lane-system}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category, let\n\\(X\\in\\mathcal C\\), and let \\(\\mathsf R\\) be a finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinement maps are\ndeclared data; they are not inferred from their cardinalities.\nIts verified profile contai…","finite resolution shape.  A\nfinite profiled lane system \\(\\mathcal L_X\\) consists, for every\n\\(r\\in\\mathsf R\\), of a finite address set \\(I_{X,r}\\) and a family\n\\[\n \\mathcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinement maps are\ndeclared data; they are not inferred from their cardinalities.\nIts verified profile contains finite biproducts, addition of morphisms,\ncomposition, the displayed splittings, and these projector-sum refinement\nidentities.  Any further construc…","athcal P_X^r=(p_{X,i}^r)_{i\\in I_{X,r}}\n \\subseteq\\End_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinement maps are\ndeclared data; they are not inferred from their cardinalities.\nIts verified profile contains finite biproducts, addition of morphisms,\ncomposition, the displayed splittings, and these projector-sum refinement\nidentities.  Any further constructor is included only after its preservation\nhas been verified.\n\\end{definition}","_{\\mathcal C}(X)\n\\]\nwith\n\\[\n (p_{X,i}^r)^2=p_{X,i}^r,\n \\qquad\n p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinement maps are\ndeclared data; they are not inferred from their cardinalities.\nIts verified profile contains finite biproducts, addition of morphisms,\ncomposition, the displayed splittings, and these projector-sum refinement\nidentities.  Any further constructor is included only after its preservation\nhas been verified.\n\\end{definition}","p_{X,i}^rp_{X,j}^r=0\\quad(i\\ne j),\n \\qquad\n \\sum_{i\\in I_{X,r}}p_{X,i}^r=1_X,\n\\]\ntogether with chosen splittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinement maps are\ndeclared data; they are not inferred from their cardinalities.\nIts verified profile contains finite biproducts, addition of morphisms,\ncomposition, the displayed splittings, and these projector-sum refinement\nidentities.  Any further constructor is included only after its preservation\nhas been verified.\n\\end{definition}","ittings\n\\[\n X_i^r\\xrightarrow{\\iota_{X,i}^r}X\n \\xrightarrow{\\pi_{X,i}^r}X_i^r,\n \\qquad\n \\iota_{X,i}^r\\pi_{X,i}^r=p_{X,i}^r,\n \\qquad\n \\pi_{X,i}^r\\iota_{X,i}^r=1_{X_i^r}.\n\\]\nFor \\(s\\succeq r\\), the system also contains a surjection\n\\[\n \\rho^X_{sr}:I_{X,s}\\twoheadrightarrow I_{X,r}\n\\]\nsuch that\n\\[\n p_{X,i}^r\n =\n \\sum_{\\rho^X_{sr}(j)=i}p_{X,j}^s,\n \\qquad\n \\rho^X_{tr}=\\rho^X_{sr}\\rho^X_{ts}.\n\\]\nHere \\(\\rho^X_{rr}=1_{I_{X,r}}\\).  The address sets and refinement maps are\ndeclared data; they are not inferred from their cardinalities.\nIts verified profile contains finite biproducts, addition of morphisms,\ncomposition, the displayed splittings, and these projector-sum refinement\nidentities.  Any further constructor is included only after its preservation\nhas been verified.\n\\end{definition}","At resolution \\(r\\), write\n\\[\n \\Analysis_X^r\n :=(\\pi_{X,i}^r)_i:\n X\\longrightarrow\\bigoplus_iX_i^r,\n \\qquad\n \\Synthesis_X^r\n :=(\\iota_{X,i}^r)_i:\n \\bigoplus_iX_i^r\\longrightarrow X.\n\\]\nFor profiled lane systems on \\(X\\) and \\(Y\\), define the coordinate block of\n\\(f:X\\to Y\\) at resolutions \\(r,u\\) by\n\\[\n [f]^{u,r}_{a i}\n :=\n \\pi_{Y,a}^u f\\iota_{X,i}^r:\n X_i^r\\longrightarrow Y_a^u.\n\\]\nIts ambient corner is\n\\[\n p_{Y,a}^u f p_{X,i}^r\n =\n \\iota_{Y,a}^u[f]^{u,r}_{a i}\\pi_{X,i}^r.\n\\]","ion \\(r\\), write\n\\[\n \\Analysis_X^r\n :=(\\pi_{X,i}^r)_i:\n X\\longrightarrow\\bigoplus_iX_i^r,\n \\qquad\n \\Synthesis_X^r\n :=(\\iota_{X,i}^r)_i:\n \\bigoplus_iX_i^r\\longrightarrow X.\n\\]\nFor profiled lane systems on \\(X\\) and \\(Y\\), define the coordinate block of\n\\(f:X\\to Y\\) at resolutions \\(r,u\\) by\n\\[\n [f]^{u,r}_{a i}\n :=\n \\pi_{Y,a}^u f\\iota_{X,i}^r:\n X_i^r\\longrightarrow Y_a^u.\n\\]\nIts ambient corner is\n\\[\n p_{Y,a}^u f p_{X,i}^r\n =\n \\iota_{Y,a}^u[f]^{u,r}_{a i}\\pi_{X,i}^r.\n\\]","\\begin{theorem}[Universal Lane Transport Theorem, finite profiled form]\n\\label{thm:block15-universal-lane-transport}\nLet \\(\\mathcal C\\) be an additive idempotent-complete category.  Give each of\nthe objects \\(X_0,\\ldots,X_m\\) a finite profiled lane system.  Then all of the\nfollowing statements hold.","\\begin{enumerate}[label=\\textup{(L\\arabic*)}]\n\\item \\textbf{Exact analysis and synthesis.}  At every declared resolution,\n\\[\n \\Synthesis_X^r\\Analysis_X^r=1_X,\n \\qquad\n \\Analysis_X^r\\Synthesis_X^r\n =1_{\\oplus_iX_i^r}.\n\\]\nHence \\(X\\cong\\bigoplus_iX_i^r\\).","\\item \\textbf{Unique lane matrix.}  Every morphism \\(f:X\\to Y\\) is recovered\nuniquely from its lane blocks:\n\\[\n f\n =\n \\sum_{a,i}p_{Y,a}^u f p_{X,i}^r\n =\n \\sum_{a,i}\\iota_{Y,a}^u\n [f]^{u,r}_{a i}\\pi_{X,i}^r.\n\\]\nEquivalently, the assignment\n\\[\n f\\longmapsto\n \\bigl([f]^{u,r}_{a i}\\bigr)_{a,i}\n\\]\nis a bijection from \\(\\Hom_{\\mathcal C}(X,Y)\\) to the group of rectangular\nmatrices with entries in\n\\(\\Hom_{\\mathcal C}(X_i^r,Y_a^u)\\).","\\item \\textbf{Composition and path law.}  If \\(f:X\\to Y\\) and \\(g:Y\\to Z\\),\nwith the same resolution \\(u\\) used on the intermediate object, then\n\\[\n [gf]^{v,r}_{c i}\n =\n \\sum_{a\\in I_{Y,u}}[g]^{v,u}_{c a}[f]^{u,r}_{a i}.\n\\]\nMore generally, a block of a composable string is the finite sum of the\nproducts along all compatible intermediate address paths.","\\item \\textbf{Projector-sum refinement.}  If \\(s\\succeq r\\) on \\(X\\) and\n\\(v\\succeq u\\) on \\(Y\\), then\n\\[\n p_{Y,a}^u f p_{X,i}^r\n =\n \\sum_{\\rho^Y_{vu}(b)=a}\n \\sum_{\\rho^X_{sr}(j)=i}\n p_{Y,b}^v f p_{X,j}^s.\n\\]\nCoarsening a fine composite gives the same ambient morphism as composing at\nthe coarse level.","\\item \\textbf{Selectors and named residuals.}  For\n\\(E\\subseteq I_{Y,u}\\times I_{X,r}\\), put\n\\[\n C_E^{u,r}(f)\n :=\\sum_{(a,i)\\in E}p_{Y,a}^ufp_{X,i}^r,\n \\qquad\n R_E^{u,r}(f):=f-C_E^{u,r}(f).\n\\]\nAs additive endomorphisms of \\(\\Hom_{\\mathcal C}(X,Y)\\),\n\\[\n C_E^2=C_E,\n \\qquad\n C_EC_F=C_{E\\cap F},\n \\qquad\n R_E=1-C_E,\n \\qquad\n f=C_E(f)+R_E(f).\n\\]\nThus deletion of a lane block is never silent.  Its complement is an exact,\nnamed residual.","\\item \\textbf{Support transport.}  If \\(f=C_E(f)\\) and \\(g=C_F(g)\\), then\n\\[\n gf=C_{F\\circ E}(gf),\n\\]\nwhere \\(F\\circ E\\) is relational composition through the intermediate\naddresses.  For arbitrary \\(f,g\\), writing\n\\(f_E=C_E(f)\\), \\(r_f=R_E(f)\\), \\(g_F=C_F(g)\\), and\n\\(r_g=R_F(g)\\), one has the exact identity\n\\[\n gf-g_Ff_E\n =r_gf_E+g_Fr_f+r_gr_f.\n\\]","\\item \\textbf{Additive realization.}  Let\n\\(F:\\mathcal C\\to\\mathcal D\\) be an additive functor to another additive\nidempotent-complete category.  If the verified profile of \\(F\\) contains the\nchosen splittings and refinement identities, then\n\\[\n F([f]^{u,r}_{a i})\n =\n [F(f)]^{u,r}_{a i}\n\\]\nafter the declared splitting comparisons.  The functor carries analysis,\nsynthesis, matrix multiplication, projector-sum refinement, selectors, and\nnamed residuals to their corresponding constructions in \\(\\mathcal D\\).","another additive\nidempotent-complete category.  If the verified profile of \\(F\\) contains the\nchosen splittings and refinement identities, then\n\\[\n F([f]^{u,r}_{a i})\n =\n [F(f)]^{u,r}_{a i}\n\\]\nafter the declared splitting comparisons.  The functor carries analysis,\nsynthesis, matrix multiplication, projector-sum refinement, selectors, and\nnamed residuals to their corresponding constructions in \\(\\mathcal D\\).","\\item \\textbf{Exact fold loss.}  Let \\(E\\) be a finite set of retained lane\npositions and set\n\\[\n B_e:=\\Hom_{\\mathcal C}(X_i^r,Y_a^u)\n \\quad\\text{for }e=(a,i)\\in E.\n\\]\nLet \\(T\\) be a finite set, let \\(\\kappa:E\\to T\\), let \\(C_t\\) be abelian\ngroups, and let \\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) be homomorphisms.  With\n\\(C=\\bigoplus_{t\\in T}C_t\\) and coordinate inclusions\n\\(j_t:C_t\\to C\\), the fold\n\\[\n \\Phi_{\\kappa,\\tau}:\n \\bigoplus_{e\\in E}B_e\\longrightarrow C,\n \\qquad\n (b_e)_e\\longmapsto\n \\sum_{e\\in E}j_{\\kappa(e)}\\tau_e(b_e)\n\\]\nhas exact information-loss group \\(\\ker\\Phi_{\\kappa,\\tau}\\).  In particular,\ntwo retained lane matrices have the same folded record exactly when their\ndifference belongs to this kernel.","\\]\nLet \\(T\\) be a finite set, let \\(\\kappa:E\\to T\\), let \\(C_t\\) be abelian\ngroups, and let \\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) be homomorphisms.  With\n\\(C=\\bigoplus_{t\\in T}C_t\\) and coordinate inclusions\n\\(j_t:C_t\\to C\\), the fold\n\\[\n \\Phi_{\\kappa,\\tau}:\n \\bigoplus_{e\\in E}B_e\\longrightarrow C,\n \\qquad\n (b_e)_e\\longmapsto\n \\sum_{e\\in E}j_{\\kappa(e)}\\tau_e(b_e)\n\\]\nhas exact information-loss group \\(\\ker\\Phi_{\\kappa,\\tau}\\).  In particular,\ntwo retained lane matrices have the same folded record exactly when their\ndifference belongs to this kernel.","Suppose, more strongly, that every \\(\\tau_e\\) is injective and that there are\nglobally typed selectors \\(Q_e\\in\\End(C)\\) satisfying, for every\n\\(e,e'\\in E\\) and every \\(b_{e'}\\in B_{e'}\\),\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}(b_{e'})\n =\n \\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}(b_{e'}).\n\\]\nThen \\(\\Phi_{\\kappa,\\tau}\\) is injective.  If there are also homomorphisms\n\\[\n r_e:C\\longrightarrow B_e,\n \\qquad\n r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen\n\\[\n L:=\\bigl(r_eQ_e\\bigr)_{e\\in E}:C\\longrightarrow\\bigoplus_{e\\in E}B_e\n\\]\nis a left inverse of \\(\\Phi_{\\kappa,\\tau}\\), so the fold is split lossless.\n\\end{enumerate}\n\\end{theorem}","\\tau_e\\) is injective and that there are\nglobally typed selectors \\(Q_e\\in\\End(C)\\) satisfying, for every\n\\(e,e'\\in E\\) and every \\(b_{e'}\\in B_{e'}\\),\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}(b_{e'})\n =\n \\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}(b_{e'}).\n\\]\nThen \\(\\Phi_{\\kappa,\\tau}\\) is injective.  If there are also homomorphisms\n\\[\n r_e:C\\longrightarrow B_e,\n \\qquad\n r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen\n\\[\n L:=\\bigl(r_eQ_e\\bigr)_{e\\in E}:C\\longrightarrow\\bigoplus_{e\\in E}B_e\n\\]\nis a left inverse of \\(\\Phi_{\\kappa,\\tau}\\), so the fold is split lossless.\n\\end{enumerate}\n\\end{theorem}","or every\n\\(e,e'\\in E\\) and every \\(b_{e'}\\in B_{e'}\\),\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}(b_{e'})\n =\n \\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}(b_{e'}).\n\\]\nThen \\(\\Phi_{\\kappa,\\tau}\\) is injective.  If there are also homomorphisms\n\\[\n r_e:C\\longrightarrow B_e,\n \\qquad\n r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen\n\\[\n L:=\\bigl(r_eQ_e\\bigr)_{e\\in E}:C\\longrightarrow\\bigoplus_{e\\in E}B_e\n\\]\nis a left inverse of \\(\\Phi_{\\kappa,\\tau}\\), so the fold is split lossless.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFor (L1), the first composite is\n\\[\n \\sum_i\\iota_{X,i}^r\\pi_{X,i}^r\n =\\sum_i p_{X,i}^r=1_X.\n\\]\nThe \\((i,j)\\)-entry of the other composite is\n\\(\\pi_{X,i}^r\\iota_{X,j}^r\\).  It is the identity when \\(i=j\\), and it is\nzero otherwise because the corresponding idempotents annihilate one another.","For (L2), insert the two reconstruction identities on the two sides of\n\\(f\\).  Applying \\(\\pi_{Y,a}^u\\) on the left and\n\\(\\iota_{X,i}^r\\) on the right proves uniqueness of every entry.  For (L3),\ninsert\n\\(1_Y=\\sum_a\\iota_{Y,a}^u\\pi_{Y,a}^u\\) between \\(g\\) and \\(f\\).\nIteration gives the path formula.","For (L5), each selector either fixes an ambient block or kills it.  Applying\ntwo selectors therefore retains precisely the intersection.  The residual\nidentities follow by subtraction.  For (L6), expand each block of \\(gf\\)\nthrough the intermediate resolution.  A surviving summand supplies an\nintermediate address in the relational product.  The residual formula is the\nfinite expansion of\n\\((g_F+r_g)(f_E+r_f)-g_Ff_E\\).","For (L8), equality of two fold values is equivalent to the difference of the\ntwo input families lying in the kernel.  Under the selector hypothesis, if\n\\(\\Phi_{\\kappa,\\tau}((b_e)_e)=0\\), applying \\(Q_e\\) gives\n\\(j_{\\kappa(e)}\\tau_e(b_e)=0\\).  Both \\(j_{\\kappa(e)}\\) and \\(\\tau_e\\) are\ninjective, hence \\(b_e=0\\) for every \\(e\\).  The chosen retractions compose\nwith the \\(Q_e\\) to give the displayed left inverse: its \\(e\\)-coordinate\non the fold is \\(r_eQ_ej_{\\kappa(e)}\\tau_e=1_{B_e}\\), while strict\nseparation kills every other coordinate.\n\\end{proof}","f two fold values is equivalent to the difference of the\ntwo input families lying in the kernel.  Under the selector hypothesis, if\n\\(\\Phi_{\\kappa,\\tau}((b_e)_e)=0\\), applying \\(Q_e\\) gives\n\\(j_{\\kappa(e)}\\tau_e(b_e)=0\\).  Both \\(j_{\\kappa(e)}\\) and \\(\\tau_e\\) are\ninjective, hence \\(b_e=0\\) for every \\(e\\).  The chosen retractions compose\nwith the \\(Q_e\\) to give the displayed left inverse: its \\(e\\)-coordinate\non the fold is \\(r_eQ_ej_{\\kappa(e)}\\tau_e=1_{B_e}\\), while strict\nseparation kills every other coordinate.\n\\end{proof}","to the difference of the\ntwo input families lying in the kernel.  Under the selector hypothesis, if\n\\(\\Phi_{\\kappa,\\tau}((b_e)_e)=0\\), applying \\(Q_e\\) gives\n\\(j_{\\kappa(e)}\\tau_e(b_e)=0\\).  Both \\(j_{\\kappa(e)}\\) and \\(\\tau_e\\) are\ninjective, hence \\(b_e=0\\) for every \\(e\\).  The chosen retractions compose\nwith the \\(Q_e\\) to give the displayed left inverse: its \\(e\\)-coordinate\non the fold is \\(r_eQ_ej_{\\kappa(e)}\\tau_e=1_{B_e}\\), while strict\nseparation kills every other coordinate.\n\\end{proof}","\\begin{remark}[Why the fold selectors live in one group]\n\\label{rem:block15-global-fold-typing}\nThe endomorphisms \\(Q_e\\) in (L8) all act on the single group\n\\(C=\\bigoplus_tC_t\\).  Writing separate selectors on unrelated codomains\nwould make the Kronecker relation ill-typed and would not prove injectivity of\nthe combined fold.\n\\end{remark}","\\begin{proposition}[Exact visible-support loss ledger]\n\\label{prop:block15-visible-support-ledger}\nUse the finite fold data of (L8) and fix an input family\n\\(b=(b_e)_{e\\in E}\\).  Define\n\\[\n S_0:=\\{e\\in E:b_e\\ne0\\},\n \\qquad\n A_0:=\\{e\\in S_0:\\tau_e(b_e)=0\\},\n \\qquad\n S_1:=S_0\\setminus A_0,\n\\]\nand let\n\\[\n \\Gamma:=\\kappa(S_1)\\subseteq T.\n\\]\nFor \\(t\\in T\\), write the folded coordinate as\n\\[\n \\overline b_t\n :=\n \\sum_{\\substack{e\\in S_1","\\begin{proposition}[Exact visible-support loss ledger]\n\\label{prop:block15-visible-support-ledger}\nUse the finite fold data of (L8) and fix an input family\n\\(b=(b_e)_{e\\in E}\\).  Define\n\\[\n S_0:=\\{e\\in E:b_e\\ne0\\},\n \\qquad\n A_0:=\\{e\\in S_0:\\tau_e(b_e)=0\\},\n \\qquad\n S_1:=S_0\\setminus A_0,\n\\]\nand let\n\\[\n \\Gamma:=\\kappa(S_1)\\subseteq T.\n\\]\nFor \\(t\\in T\\), write the folded coordinate as\n\\[\n \\overline b_t\n :=\n \\sum_{\\substack{e\\in S_1\\\\\\kappa(e)=t}}\\tau_e(b_e)\n \\in C_t,\n\\]\nand put\n\\[\n \\operatorname{supp}(\\overline b)\n :=\\{t\\in T:\\overline b_t\\ne0\\},\n \\qquad\n K_0:=\\{t\\in\\Gamma:\\overline b_t=0\\}.\n\\]\nThen\n\\[\n \\boxed{\n |S_0|-|\\operatorname{supp}(\\overline b)|\n =\n |A_0|\n +\\bigl(|S_1|-|\\Gamma|\\bigr)\n +|K_0|.}\n\\]\nThe three terms record, respectively, annihilation by the local maps\n\\(\\tau_e\\), collision of surviving lane addresses under \\(\\kappa\\), and\ncancellation among folded values at a common address.  If every \\(\\tau_e\\) is\ninjective, then \\(A_0=\\varnothing\\).  If globally typed selectors satisfy the\nstrict separation law of (L8), then \\(K_0=\\varnothing\\).\n\\end{proposition}","kappa(e)=t}}\\tau_e(b_e)\n \\in C_t,\n\\]\nand put\n\\[\n \\operatorname{supp}(\\overline b)\n :=\\{t\\in T:\\overline b_t\\ne0\\},\n \\qquad\n K_0:=\\{t\\in\\Gamma:\\overline b_t=0\\}.\n\\]\nThen\n\\[\n \\boxed{\n |S_0|-|\\operatorname{supp}(\\overline b)|\n =\n |A_0|\n +\\bigl(|S_1|-|\\Gamma|\\bigr)\n +|K_0|.}\n\\]\nThe three terms record, respectively, annihilation by the local maps\n\\(\\tau_e\\), collision of surviving lane addresses under \\(\\kappa\\), and\ncancellation among folded values at a common address.  If every \\(\\tau_e\\) is\ninjective, then \\(A_0=\\varnothing\\).  If globally typed selectors satisfy the\nstrict separation law of (L8), then \\(K_0=\\varnothing\\).\n\\end{proposition}","ppa(e)=t}}\\tau_e(b_e)\n \\in C_t,\n\\]\nand put\n\\[\n \\operatorname{supp}(\\overline b)\n :=\\{t\\in T:\\overline b_t\\ne0\\},\n \\qquad\n K_0:=\\{t\\in\\Gamma:\\overline b_t=0\\}.\n\\]\nThen\n\\[\n \\boxed{\n |S_0|-|\\operatorname{supp}(\\overline b)|\n =\n |A_0|\n +\\bigl(|S_1|-|\\Gamma|\\bigr)\n +|K_0|.}\n\\]\nThe three terms record, respectively, annihilation by the local maps\n\\(\\tau_e\\), collision of surviving lane addresses under \\(\\kappa\\), and\ncancellation among folded values at a common address.  If every \\(\\tau_e\\) is\ninjective, then \\(A_0=\\varnothing\\).  If globally typed selectors satisfy the\nstrict separation law of (L8), then \\(K_0=\\varnothing\\).\n\\end{proposition}",":=\\{t\\in T:\\overline b_t\\ne0\\},\n \\qquad\n K_0:=\\{t\\in\\Gamma:\\overline b_t=0\\}.\n\\]\nThen\n\\[\n \\boxed{\n |S_0|-|\\operatorname{supp}(\\overline b)|\n =\n |A_0|\n +\\bigl(|S_1|-|\\Gamma|\\bigr)\n +|K_0|.}\n\\]\nThe three terms record, respectively, annihilation by the local maps\n\\(\\tau_e\\), collision of surviving lane addresses under \\(\\kappa\\), and\ncancellation among folded values at a common address.  If every \\(\\tau_e\\) is\ninjective, then \\(A_0=\\varnothing\\).  If globally typed selectors satisfy the\nstrict separation law of (L8), then \\(K_0=\\varnothing\\).\n\\end{proposition}","},\n \\qquad\n K_0:=\\{t\\in\\Gamma:\\overline b_t=0\\}.\n\\]\nThen\n\\[\n \\boxed{\n |S_0|-|\\operatorname{supp}(\\overline b)|\n =\n |A_0|\n +\\bigl(|S_1|-|\\Gamma|\\bigr)\n +|K_0|.}\n\\]\nThe three terms record, respectively, annihilation by the local maps\n\\(\\tau_e\\), collision of surviving lane addresses under \\(\\kappa\\), and\ncancellation among folded values at a common address.  If every \\(\\tau_e\\) is\ninjective, then \\(A_0=\\varnothing\\).  If globally typed selectors satisfy the\nstrict separation law of (L8), then \\(K_0=\\varnothing\\).\n\\end{proposition}","overline b)|\n =\n |A_0|\n +\\bigl(|S_1|-|\\Gamma|\\bigr)\n +|K_0|.}\n\\]\nThe three terms record, respectively, annihilation by the local maps\n\\(\\tau_e\\), collision of surviving lane addresses under \\(\\kappa\\), and\ncancellation among folded values at a common address.  If every \\(\\tau_e\\) is\ninjective, then \\(A_0=\\varnothing\\).  If globally typed selectors satisfy the\nstrict separation law of (L8), then \\(K_0=\\varnothing\\).\n\\end{proposition}","\\begin{proof}\nThe first partition gives \\(|S_0|=|A_0|+|S_1|\\).  The definition of\n\\(K_0\\) gives the disjoint partition\n\\[\n \\Gamma\n =\\operatorname{supp}(\\overline b)\\sqcup K_0,\n\\]\nso \\(|\\Gamma|=|\\operatorname{supp}(\\overline b)|+|K_0|\\).  Subtracting the\nsecond equality from the first proves the formula.","Injectivity of \\(\\tau_e\\) makes \\(\\tau_e(b_e)\\ne0\\) for every \\(e\\in S_0\\),\nso there is no annihilation.  For strict selectors, suppose\n\\(t\\in K_0\\) and choose \\(e\\in S_1\\) with \\(\\kappa(e)=t\\).  Applying \\(Q_e\\)\nto\n\\[\n j_t\\overline b_t\n =\\sum_{\\kappa(e')=t}j_t\\tau_{e'}(b_{e'})=0\n\\]\nleaves \\(j_t\\tau_e(b_e)=0\\), contrary to \\(e\\in S_1\\).  Hence\n\\(K_0=\\varnothing\\).\n\\end{proof}","For resolutions \\(s\\succeq r\\) on \\(X\\) and \\(v\\succeq u\\) on \\(Y\\), let\n\\[\n \\mathsf B^{v,s}(X,Y)\n :=\n \\bigoplus_{b\\in I_{Y,v}}\n \\bigoplus_{j\\in I_{X,s}}\n \\Hom_{\\mathcal C}(X_j^s,Y_b^v),\n\\]\nand define \\(\\mathsf B^{u,r}(X,Y)\\) similarly.  Analysis of a morphism gives\nan isomorphism\n\\[\n \\mathfrak a^{v,s}_{X,Y}:\n \\Hom_{\\mathcal C}(X,Y)\\xrightarrow{\\sim}\n \\mathsf B^{v,s}(X,Y),\n\\]\nwhose inverse is lane synthesis.","\\begin{proposition}[Certified coarsening is an exact coordinate change]\n\\label{prop:block15-certified-coarsening}\nThe change from fine to coarse block coordinates, canonical relative to the\nchosen splittings, is\n\\[\n \\Gamma^{(u,r)}_{(v,s)}\n :=\n \\mathfrak a^{u,r}_{X,Y}\n \\bigl(\\mathfrak a^{v,s}_{X,Y}\\bigr)^{-1}.\n\\]\nFor a fine matrix \\(h=(h_{bj})\\), its \\((a,i)\\)-entry is\n\\[\n \\bigl(\\Gamma^{(u,r)}_{(v,s)}h\\bigr)_{a i}\n =\n \\sum_{b,j}\n \\bigl(\\pi_{Y,a}^u\\iota_{Y,b}^v\\bigr)\n h_{bj}\n \\bigl(\\pi_{X,j}^s\\iota_{X,i}^r\\bigr).\n\\]\nOnly descendants of \\(a\\) and \\(i\\) can contribute.  The inverse is the\ncorresponding fine analysis after coarse synthesis.  Consequently, changing a\ncomplete lane resolution loses no information.  Loss occurs only after a\nsubsequent selector, quotient, summary, or fold is applied.\n\\end{proposition}","se block coordinates, canonical relative to the\nchosen splittings, is\n\\[\n \\Gamma^{(u,r)}_{(v,s)}\n :=\n \\mathfrak a^{u,r}_{X,Y}\n \\bigl(\\mathfrak a^{v,s}_{X,Y}\\bigr)^{-1}.\n\\]\nFor a fine matrix \\(h=(h_{bj})\\), its \\((a,i)\\)-entry is\n\\[\n \\bigl(\\Gamma^{(u,r)}_{(v,s)}h\\bigr)_{a i}\n =\n \\sum_{b,j}\n \\bigl(\\pi_{Y,a}^u\\iota_{Y,b}^v\\bigr)\n h_{bj}\n \\bigl(\\pi_{X,j}^s\\iota_{X,i}^r\\bigr).\n\\]\nOnly descendants of \\(a\\) and \\(i\\) can contribute.  The inverse is the\ncorresponding fine analysis after coarse synthesis.  Consequently, changing a\ncomplete lane resolution loses no information.  Loss occurs only after a\nsubsequent selector, quotient, summary, or fold is applied.\n\\end{proposition}","oordinates, canonical relative to the\nchosen splittings, is\n\\[\n \\Gamma^{(u,r)}_{(v,s)}\n :=\n \\mathfrak a^{u,r}_{X,Y}\n \\bigl(\\mathfrak a^{v,s}_{X,Y}\\bigr)^{-1}.\n\\]\nFor a fine matrix \\(h=(h_{bj})\\), its \\((a,i)\\)-entry is\n\\[\n \\bigl(\\Gamma^{(u,r)}_{(v,s)}h\\bigr)_{a i}\n =\n \\sum_{b,j}\n \\bigl(\\pi_{Y,a}^u\\iota_{Y,b}^v\\bigr)\n h_{bj}\n \\bigl(\\pi_{X,j}^s\\iota_{X,i}^r\\bigr).\n\\]\nOnly descendants of \\(a\\) and \\(i\\) can contribute.  The inverse is the\ncorresponding fine analysis after coarse synthesis.  Consequently, changing a\ncomplete lane resolution loses no information.  Loss occurs only after a\nsubsequent selector, quotient, summary, or fold is applied.\n\\end{proposition}","\\begin{proof}\nThe displayed definition is a composite of two isomorphisms from\n\\cref{thm:block15-universal-lane-transport}.  Expanding coarse analysis after\nfine synthesis gives the entry formula.  If \\(b\\) is not a descendant of\n\\(a\\), then\n\\(p_{Y,a}^up_{Y,b}^v=0\\), hence\n\\(\\pi_{Y,a}^u\\iota_{Y,b}^v=0\\).  The source factor vanishes by the same\nargument when \\(j\\) is not a descendant of \\(i\\).  Reversing analysis and\nsynthesis produces the inverse.\n\\end{proof}","\\begin{corollary}[Functorial coarsening]\n\\label{cor:block15-functorial-coarsening}\nFor \\(t\\succeq s\\succeq r\\) and compatible target resolutions,\nthe certified coordinate changes compose strictly.  They also intertwine\nmatrix multiplication.  Thus a finite refinement diagram gives a genuine\ndiagram of equivalent matrix-coordinate categories, not merely a family of\nnumerically similar arrays.\n\\end{corollary}","\\begin{theorem}[Abelian fold-loss exact sequence]\n\\label{thm:block15-abelian-fold-exact-sequence}\nLet \\(\\mathcal A\\) be an abelian category.  Suppose a finite family of lane\nobjects \\((B_e)_{e\\in E}\\), target objects \\((C_t)_{t\\in T}\\), and morphisms\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Sup","\\begin{theorem}[Abelian fold-loss exact sequence]\n\\label{thm:block15-abelian-fold-exact-sequence}\nLet \\(\\mathcal A\\) be an abelian category.  Suppose a finite family of lane\nobjects \\((B_e)_{e\\in E}\\), target objects \\((C_t)_{t\\in T}\\), and morphisms\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC","\\begin{theorem}[Abelian fold-loss exact sequence]\n\\label{thm:block15-abelian-fold-exact-sequence}\nLet \\(\\mathcal A\\) be an abelian category.  Suppose a finite family of lane\nobjects \\((B_e)_{e\\in E}\\), target objects \\((C_t)_{t\\in T}\\), and morphisms\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e","\\begin{theorem}[Abelian fold-loss exact sequence]\n\\label{thm:block15-abelian-fold-exact-sequence}\nLet \\(\\mathcal A\\) be an abelian category.  Suppose a finite family of lane\nobjects \\((B_e)_{e\\in E}\\), target objects \\((C_t)_{t\\in T}\\), and morphisms\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_","\\begin{theorem}[Abelian fold-loss exact sequence]\n\\label{thm:block15-abelian-fold-exact-sequence}\nLet \\(\\mathcal A\\) be an abelian category.  Suppose a finite family of lane\nobjects \\((B_e)_{e\\in E}\\), target objects \\((C_t)_{t\\in T}\\), and morphisms\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\big","\\begin{theorem}[Abelian fold-loss exact sequence]\n\\label{thm:block15-abelian-fold-exact-sequence}\nLet \\(\\mathcal A\\) be an abelian category.  Suppose a finite family of lane\nobjects \\((B_e)_{e\\in E}\\), target objects \\((C_t)_{t\\in T}\\), and morphisms\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there a…","Suppose a finite family of lane\nobjects \\((B_e)_{e\\in E}\\), target objects \\((C_t)_{t\\in T}\\), and morphisms\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of …","s\n\\(\\tau_e:B_e\\to C_{\\kappa(e)}\\) is given.  Let\n\\(q_e:\\bigoplus_{e'\\in E}B_{e'}\\to B_e\\) be the coordinate projection and\ndefine\n\\[\n \\Phi:=\\sum_{e\\in E}j_{\\kappa(e)}\\tau_eq_e:\n \\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exac…","\\bigoplus_{e\\in E}B_e\\longrightarrow\n \\bigoplus_{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","{t\\in T}C_t.\n\\]\nThen there are canonical short exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","hort exact sequences\n\\[\n 0\\longrightarrow\\ker\\Phi\n \\longrightarrow\\bigoplus_eB_e\n \\longrightarrow\\im\\Phi\n \\longrightarrow0\n\\]\nand\n\\[\n 0\\longrightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","ightarrow\\im\\Phi\n \\longrightarrow\\bigoplus_tC_t\n \\longrightarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","htarrow\\coker\\Phi\n \\longrightarrow0.\n\\]\nThe first sequence is the exact fold-loss certificate.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","te.  Suppose every\n\\(\\tau_e\\) is monic and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","and there are endomorphisms\n\\(Q_e\\in\\End(\\bigoplus_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","us_tC_t)\\) satisfying\n\\[\n Q_ej_{\\kappa(e')}\\tau_{e'}\n =\\delta_{e,e'}j_{\\kappa(e')}\\tau_{e'}\n \\quad\\text{as morphisms }B_{e'}\\longrightarrow\\bigoplus_tC_t.\n\\]\nThen \\(\\Phi\\) is monic.  If, in addition, there are morphisms\n\\[\n r_e:\\bigoplus_tC_t\\longrightarrow B_e,\n \\qquad r_ej_{\\kappa(e)}\\tau_e=1_{B_e},\n\\]\nthen \\(L=(r_eQ_e)_e\\) is a left inverse of \\(\\Phi\\).  Thus the first exact\nsequence splits, \\(\\ker\\Phi=0\\), and the image inclusion in the second exact\nsequence also splits.\n\\end{theorem}","\\begin{proof}\nThe two sequences are the kernel--image and image--cokernel factorizations in\nan abelian category.  The selector argument in (L8), interpreted internally\nin \\(\\mathcal A\\), makes \\(\\Phi\\) monic.  The displayed morphism \\(L\\) is a\nleft inverse by strict separation.  Hence the first sequence splits;\nmoreover, \\(\\Phi L\\) is an idempotent with image \\(\\im\\Phi\\), so the image\ninclusion in the second sequence splits as well.\n\\end{proof}","\\begin{definition}[Lane matrix category at a fixed resolution]\n\\label{def:block15-lane-matrix-category}\nLet \\(\\mathcal L_{\\mathsf R}\\) be the category whose objects are objects of\n\\(\\mathcal C\\) equipped with full finite profiled lane systems over\n\\(\\mathsf R\\), with all underlying \\(\\mathcal C\\)-morphisms allowed.  For\n\\(r\\in\\mathsf R\\), define\n\\(\\operatorname{Mat}_r(\\mathcal L_{\\mathsf R})\\) to have the tagged finite\nfamilies\n\\([X;(X_i^r)_{i\\in I_{X,r}}]\\) as objects and rectangular matrices\n\\[\n (h_{a i}),\n \\qquad h_{a i}:X_i^r\\to Y_a^r,\n\\]\nas morphisms.  Composition is matrix contraction and identities are diagonal\nidentity matrices.","egory at a fixed resolution]\n\\label{def:block15-lane-matrix-category}\nLet \\(\\mathcal L_{\\mathsf R}\\) be the category whose objects are objects of\n\\(\\mathcal C\\) equipped with full finite profiled lane systems over\n\\(\\mathsf R\\), with all underlying \\(\\mathcal C\\)-morphisms allowed.  For\n\\(r\\in\\mathsf R\\), define\n\\(\\operatorname{Mat}_r(\\mathcal L_{\\mathsf R})\\) to have the tagged finite\nfamilies\n\\([X;(X_i^r)_{i\\in I_{X,r}}]\\) as objects and rectangular matrices\n\\[\n (h_{a i}),\n \\qquad h_{a i}:X_i^r\\to Y_a^r,\n\\]\nas morphisms.  Composition is matrix contraction and identities are diagonal\nidentity matrices.","y}\nLet \\(\\mathcal L_{\\mathsf R}\\) be the category whose objects are objects of\n\\(\\mathcal C\\) equipped with full finite profiled lane systems over\n\\(\\mathsf R\\), with all underlying \\(\\mathcal C\\)-morphisms allowed.  For\n\\(r\\in\\mathsf R\\), define\n\\(\\operatorname{Mat}_r(\\mathcal L_{\\mathsf R})\\) to have the tagged finite\nfamilies\n\\([X;(X_i^r)_{i\\in I_{X,r}}]\\) as objects and rectangular matrices\n\\[\n (h_{a i}),\n \\qquad h_{a i}:X_i^r\\to Y_a^r,\n\\]\nas morphisms.  Composition is matrix contraction and identities are diagonal\nidentity matrices.","For \\(s\\succeq r\\), define\n\\[\n \\Gamma_{sr}:\\operatorname{Mat}_s(\\mathcal L_{\\mathsf R})\n \\longrightarrow\\operatorname{Mat}_r(\\mathcal L_{\\mathsf R})\n\\]\nby replacing each object tag with its \\(r\\)-tag and sending a fine matrix to\nthe certified source-and-target coordinate change of\n\\cref{prop:block15-certified-coarsening}.\n\\end{definition}","\\begin{theorem}[Fixed-resolution matrix-category equivalence]\n\\label{thm:block15-matrix-category-equivalence}\nThe functor\n\\[\n \\mathfrak M_r:\\mathcal L_{\\mathsf R}\\longrightarrow\n \\operatorname{Mat}_r(\\mathcal L_{\\mathsf R}),\n \\qquad\n X\\longmapsto[X;(X_i^r)_i],\n \\qquad\n f\\longmapsto([f]_{a i}^{r,r}),\n\\]\nis an equivalence.  A quasi-inverse returns the lane-decorated object named by\nthe tag and sends a matrix to\n\\[\n \\sum_{a,i}\\iota_{Y,a}^r h_{a i}\\pi_{X,i}^r.\n\\]\nFor every \\(s\\succeq r\\), the coordinate changes are functors and satisfy\n\\[\n \\Gamma_{sr}\\mathfrak M_s=\\mathfrak M_r.\n\\]\nFor \\(t\\succeq s\\succeq r\\), they also satisfy\n\\[\n \\Gamma_{tr}=\\Gamma_{sr}\\Gamma_{ts}.\n\\]\nThus the equivalences are compatible with every certified change of\nresolution.\n\\end{theorem}","[Fixed-resolution matrix-category equivalence]\n\\label{thm:block15-matrix-category-equivalence}\nThe functor\n\\[\n \\mathfrak M_r:\\mathcal L_{\\mathsf R}\\longrightarrow\n \\operatorname{Mat}_r(\\mathcal L_{\\mathsf R}),\n \\qquad\n X\\longmapsto[X;(X_i^r)_i],\n \\qquad\n f\\longmapsto([f]_{a i}^{r,r}),\n\\]\nis an equivalence.  A quasi-inverse returns the lane-decorated object named by\nthe tag and sends a matrix to\n\\[\n \\sum_{a,i}\\iota_{Y,a}^r h_{a i}\\pi_{X,i}^r.\n\\]\nFor every \\(s\\succeq r\\), the coordinate changes are functors and satisfy\n\\[\n \\Gamma_{sr}\\mathfrak M_s=\\mathfrak M_r.\n\\]\nFor \\(t\\succeq s\\succeq r\\), they also satisfy\n\\[\n \\Gamma_{tr}=\\Gamma_{sr}\\Gamma_{ts}.\n\\]\nThus the equivalences are compatible with every certified change of\nresolution.\n\\end{theorem}","ck15-matrix-category-equivalence}\nThe functor\n\\[\n \\mathfrak M_r:\\mathcal L_{\\mathsf R}\\longrightarrow\n \\operatorname{Mat}_r(\\mathcal L_{\\mathsf R}),\n \\qquad\n X\\longmapsto[X;(X_i^r)_i],\n \\qquad\n f\\longmapsto([f]_{a i}^{r,r}),\n\\]\nis an equivalence.  A quasi-inverse returns the lane-decorated object named by\nthe tag and sends a matrix to\n\\[\n \\sum_{a,i}\\iota_{Y,a}^r h_{a i}\\pi_{X,i}^r.\n\\]\nFor every \\(s\\succeq r\\), the coordinate changes are functors and satisfy\n\\[\n \\Gamma_{sr}\\mathfrak M_s=\\mathfrak M_r.\n\\]\nFor \\(t\\succeq s\\succeq r\\), they also satisfy\n\\[\n \\Gamma_{tr}=\\Gamma_{sr}\\Gamma_{ts}.\n\\]\nThus the equivalences are compatible with every certified change of\nresolution.\n\\end{theorem}","\\mathcal L_{\\mathsf R}\\longrightarrow\n \\operatorname{Mat}_r(\\mathcal L_{\\mathsf R}),\n \\qquad\n X\\longmapsto[X;(X_i^r)_i],\n \\qquad\n f\\longmapsto([f]_{a i}^{r,r}),\n\\]\nis an equivalence.  A quasi-inverse returns the lane-decorated object named by\nthe tag and sends a matrix to\n\\[\n \\sum_{a,i}\\iota_{Y,a}^r h_{a i}\\pi_{X,i}^r.\n\\]\nFor every \\(s\\succeq r\\), the coordinate changes are functors and satisfy\n\\[\n \\Gamma_{sr}\\mathfrak M_s=\\mathfrak M_r.\n\\]\nFor \\(t\\succeq s\\succeq r\\), they also satisfy\n\\[\n \\Gamma_{tr}=\\Gamma_{sr}\\Gamma_{ts}.\n\\]\nThus the equivalences are compatible with every certified change of\nresolution.\n\\end{theorem}","e{Mat}_r(\\mathcal L_{\\mathsf R}),\n \\qquad\n X\\longmapsto[X;(X_i^r)_i],\n \\qquad\n f\\longmapsto([f]_{a i}^{r,r}),\n\\]\nis an equivalence.  A quasi-inverse returns the lane-decorated object named by\nthe tag and sends a matrix to\n\\[\n \\sum_{a,i}\\iota_{Y,a}^r h_{a i}\\pi_{X,i}^r.\n\\]\nFor every \\(s\\succeq r\\), the coordinate changes are functors and satisfy\n\\[\n \\Gamma_{sr}\\mathfrak M_s=\\mathfrak M_r.\n\\]\nFor \\(t\\succeq s\\succeq r\\), they also satisfy\n\\[\n \\Gamma_{tr}=\\Gamma_{sr}\\Gamma_{ts}.\n\\]\nThus the equivalences are compatible with every certified change of\nresolution.\n\\end{theorem}","X\\longmapsto[X;(X_i^r)_i],\n \\qquad\n f\\longmapsto([f]_{a i}^{r,r}),\n\\]\nis an equivalence.  A quasi-inverse returns the lane-decorated object named by\nthe tag and sends a matrix to\n\\[\n \\sum_{a,i}\\iota_{Y,a}^r h_{a i}\\pi_{X,i}^r.\n\\]\nFor every \\(s\\succeq r\\), the coordinate changes are functors and satisfy\n\\[\n \\Gamma_{sr}\\mathfrak M_s=\\mathfrak M_r.\n\\]\nFor \\(t\\succeq s\\succeq r\\), they also satisfy\n\\[\n \\Gamma_{tr}=\\Gamma_{sr}\\Gamma_{ts}.\n\\]\nThus the equivalences are compatible with every certified change of\nresolution.\n\\end{theorem}","\\begin{proof}\nThe unique lane matrix and composition clauses of\n\\cref{thm:block15-universal-lane-transport} show that \\(\\mathfrak M_r\\) is\nfully faithful.  Every object of the displayed matrix category is, by\ndefinition, the lane family of an object of \\(\\mathcal L_{\\mathsf R}\\), so it is\nessentially surjective.  Analysis and synthesis furnish the unit and counit\nisomorphisms.  Each \\(\\Gamma_{sr}\\) is conjugation by exact analysis and\nsynthesis, so it preserves identities and matrix composition.  The two\ndisplayed identities follow from cancellation of adjacent analysis--synthesis\npairs, equivalently from functorial coarsening in\n\\cref{cor:block15-functorial-coarsening}.\n\\end{proof}","\\begin{remark}[No dimension claim]\n\\label{rem:block15-address-count-not-dimension}\nThe number \\(|I_{X,r}|\\) counts declared addresses.  A lane object can have\narbitrary dimension, rank, length, measure, or entropy, including zero where\nthe ambient doctrine permits it.  Matrix equivalence therefore does not\nidentify address count with any of those invariants.\n\\end{remark}","\\begin{theorem}[Module multilinear lane expansion]\n\\label{thm:block15-multilinear-lane-expansion}\nLet \\(\\kk\\) be a commutative ring.  Let\n\\[\n X^{(1)},\\ldots,X^{(d)},Y\n\\]\nbe \\(\\kk\\)-modules with finite split lane systems, and let\n\\[\n T:X^{(1)}\\times\\cdots\\times X^{(d)}\\longrightarrow Y\n\\]\nbe a specified \\(\\kk\\)-multilinear map.  Define\n\\[\n T_{a;i_1,\\ldots,i_d}\n :=\n \\pi_{Y,a}T\n (\\iota_{1,i_1}-,\\ldots,\\iota_{d,i_d}-).\n\\]\nThen\n\\[\n T(x_1,\\ldots,x_d)\n =\n \\sum_{a,i_1,\\ldots,i_d}\n \\iota_{Y,a}\n T_{a;i_1,\\ldots,i_d}\n (\\pi_{1,i_1}x_1,\\ldots,\\pi_{d,i_d}x_d).\n\\]\nSubstitution of multilinear maps is governed by finite contraction over every\nintermediate lane address.","inear lane expansion]\n\\label{thm:block15-multilinear-lane-expansion}\nLet \\(\\kk\\) be a commutative ring.  Let\n\\[\n X^{(1)},\\ldots,X^{(d)},Y\n\\]\nbe \\(\\kk\\)-modules with finite split lane systems, and let\n\\[\n T:X^{(1)}\\times\\cdots\\times X^{(d)}\\longrightarrow Y\n\\]\nbe a specified \\(\\kk\\)-multilinear map.  Define\n\\[\n T_{a;i_1,\\ldots,i_d}\n :=\n \\pi_{Y,a}T\n (\\iota_{1,i_1}-,\\ldots,\\iota_{d,i_d}-).\n\\]\nThen\n\\[\n T(x_1,\\ldots,x_d)\n =\n \\sum_{a,i_1,\\ldots,i_d}\n \\iota_{Y,a}\n T_{a;i_1,\\ldots,i_d}\n (\\pi_{1,i_1}x_1,\\ldots,\\pi_{d,i_d}x_d).\n\\]\nSubstitution of multilinear maps is governed by finite contraction over every\nintermediate lane address.","Separately, let \\(X,Y\\) be \\(\\kk\\)-modules with finite split lane systems and\nsuppose a polynomial map is presented with specified data\n\\[\n P(x)=T_0+T_1(x)+T_2(x,x)+\\cdots+T_D(x,\\ldots,x),\n\\]\nwhere\n\\[\n T_0\\in Y,\n \\qquad\n T_n:X^n\\longrightarrow Y\\quad(1\\le n\\le D)\n\\]\nand every \\(T_n\\) is a supplied multilinear map.  Then\n\\[\n T_0=\\sum_a\\iota_{Y,a}\\pi_{Y,a}(T_0),\n\\]\nand adding this degree-zero decomposition to the displayed multilinear\nexpansions gives an exact lane expansion of \\(P\\).\n\\end{theorem}","se a polynomial map is presented with specified data\n\\[\n P(x)=T_0+T_1(x)+T_2(x,x)+\\cdots+T_D(x,\\ldots,x),\n\\]\nwhere\n\\[\n T_0\\in Y,\n \\qquad\n T_n:X^n\\longrightarrow Y\\quad(1\\le n\\le D)\n\\]\nand every \\(T_n\\) is a supplied multilinear map.  Then\n\\[\n T_0=\\sum_a\\iota_{Y,a}\\pi_{Y,a}(T_0),\n\\]\nand adding this degree-zero decomposition to the displayed multilinear\nexpansions gives an exact lane expansion of \\(P\\).\n\\end{theorem}","\\begin{proof}\nInsert \\(x_\\ell=\\sum_{i_\\ell}p_{\\ell,i_\\ell}x_\\ell\\) in every input, use\nmultilinearity to distribute the finite sums, and insert\n\\(1_Y=\\sum_a p_{Y,a}\\) in the output.  The coordinate formula follows from\nthe chosen splittings.  Repeating the calculation after substitution gives\nthe contraction law.  The degree-zero identity is output synthesis applied\nto \\(T_0\\); the polynomial statement is then a finite sum of the specified\nhomogeneous pieces.\n\\end{proof}","For integers \\(N\\ge1\\) and \\(n\\ge0\\), define the step factorial\n\\[\n (a)_{n;N}:=\\prod_{k=0}^{n-1}(a+Nk),\n \\qquad (a)_{0;N}:=1.\n\\]\nFor a commutative ring \\(A\\) and a formal series\n\\(F(z)=\\sum_{m\\ge0}a_mz^m\\in A[[z]]\\), define its residue lane\n\\[\n \\mathcal P_{N,j}F(z)\n :=\n \\sum_{\\substack{m\\ge0","For integers \\(N\\ge1\\) and \\(n\\ge0\\), define the step factorial\n\\[\n (a)_{n;N}:=\\prod_{k=0}^{n-1}(a+Nk),\n \\qquad (a)_{0;N}:=1.\n\\]\nFor a commutative ring \\(A\\) and a formal series\n\\(F(z)=\\sum_{m\\ge0}a_mz^m\\in A[[z]]\\), define its residue lane\n\\[\n \\mathcal P_{N,j}F(z)\n :=\n \\sum_{\\substack{m\\ge0\\\\m\\equiv j\\ (\\mathrm{mod}\\ N)}}a_mz^m,\n \\qquad 0\\le j<N.\n\\]","\\begin{theorem}[Exact factorial--residue realization]\n\\label{thm:block15-factorial-residue-realization}\nLet \\(N,c,n\\in\\mathbb Z\\) with \\(N\\ge1\\), \\(c\\ge0\\), and \\(n\\ge0\\).\nThen\n\\[\n \\boxed{\n \\frac{(Nn+c)!}{c!}\n =\n \\prod_{r=c+1}^{c+N}(r)_{n;N}.}\n\\]\nEach integer factor from \\(c+1\\) through \\(c+Nn\\) occurs in exactly one\nlane.  For every commutative ring \\(A\\), on the \\(A\\)-module \\(A[[z]]\\) the\noperators\n\\((\\mathcal P_{N,j})_{0\\le j<N}\\) satisfy\n\\[\n \\mathcal P_{N,j}^2=\\mathcal P_{N,j},\n \\qquad\n \\mathcal P_{N,j}\\mathcal P_{N,k}=0\\ (j\\ne k),\n \\qquad\n \\sum_{j=0}^{N-1}\\mathcal P_{N,j}=1.\n\\]\nIf \\(M\\in\\mathbb Z\\) and \\(M\\ge1\\), their exact refinement law is\n\\[\n \\mathcal P_{N,j}\n =\n \\sum_{\\ell=0}^{M-1}\\mathcal P_{NM,j+\\ell N}.\n\\]","theorem}[Exact factorial--residue realization]\n\\label{thm:block15-factorial-residue-realization}\nLet \\(N,c,n\\in\\mathbb Z\\) with \\(N\\ge1\\), \\(c\\ge0\\), and \\(n\\ge0\\).\nThen\n\\[\n \\boxed{\n \\frac{(Nn+c)!}{c!}\n =\n \\prod_{r=c+1}^{c+N}(r)_{n;N}.}\n\\]\nEach integer factor from \\(c+1\\) through \\(c+Nn\\) occurs in exactly one\nlane.  For every commutative ring \\(A\\), on the \\(A\\)-module \\(A[[z]]\\) the\noperators\n\\((\\mathcal P_{N,j})_{0\\le j<N}\\) satisfy\n\\[\n \\mathcal P_{N,j}^2=\\mathcal P_{N,j},\n \\qquad\n \\mathcal P_{N,j}\\mathcal P_{N,k}=0\\ (j\\ne k),\n \\qquad\n \\sum_{j=0}^{N-1}\\mathcal P_{N,j}=1.\n\\]\nIf \\(M\\in\\mathbb Z\\) and \\(M\\ge1\\), their exact refinement law is\n\\[\n \\mathcal P_{N,j}\n =\n \\sum_{\\ell=0}^{M-1}\\mathcal P_{NM,j+\\ell N}.\n\\]","ion]\n\\label{thm:block15-factorial-residue-realization}\nLet \\(N,c,n\\in\\mathbb Z\\) with \\(N\\ge1\\), \\(c\\ge0\\), and \\(n\\ge0\\).\nThen\n\\[\n \\boxed{\n \\frac{(Nn+c)!}{c!}\n =\n \\prod_{r=c+1}^{c+N}(r)_{n;N}.}\n\\]\nEach integer factor from \\(c+1\\) through \\(c+Nn\\) occurs in exactly one\nlane.  For every commutative ring \\(A\\), on the \\(A\\)-module \\(A[[z]]\\) the\noperators\n\\((\\mathcal P_{N,j})_{0\\le j<N}\\) satisfy\n\\[\n \\mathcal P_{N,j}^2=\\mathcal P_{N,j},\n \\qquad\n \\mathcal P_{N,j}\\mathcal P_{N,k}=0\\ (j\\ne k),\n \\qquad\n \\sum_{j=0}^{N-1}\\mathcal P_{N,j}=1.\n\\]\nIf \\(M\\in\\mathbb Z\\) and \\(M\\ge1\\), their exact refinement law is\n\\[\n \\mathcal P_{N,j}\n =\n \\sum_{\\ell=0}^{M-1}\\mathcal P_{NM,j+\\ell N}.\n\\]","(Nn+c)!}{c!}\n =\n \\prod_{r=c+1}^{c+N}(r)_{n;N}.}\n\\]\nEach integer factor from \\(c+1\\) through \\(c+Nn\\) occurs in exactly one\nlane.  For every commutative ring \\(A\\), on the \\(A\\)-module \\(A[[z]]\\) the\noperators\n\\((\\mathcal P_{N,j})_{0\\le j<N}\\) satisfy\n\\[\n \\mathcal P_{N,j}^2=\\mathcal P_{N,j},\n \\qquad\n \\mathcal P_{N,j}\\mathcal P_{N,k}=0\\ (j\\ne k),\n \\qquad\n \\sum_{j=0}^{N-1}\\mathcal P_{N,j}=1.\n\\]\nIf \\(M\\in\\mathbb Z\\) and \\(M\\ge1\\), their exact refinement law is\n\\[\n \\mathcal P_{N,j}\n =\n \\sum_{\\ell=0}^{M-1}\\mathcal P_{NM,j+\\ell N}.\n\\]","_{r=c+1}^{c+N}(r)_{n;N}.}\n\\]\nEach integer factor from \\(c+1\\) through \\(c+Nn\\) occurs in exactly one\nlane.  For every commutative ring \\(A\\), on the \\(A\\)-module \\(A[[z]]\\) the\noperators\n\\((\\mathcal P_{N,j})_{0\\le j<N}\\) satisfy\n\\[\n \\mathcal P_{N,j}^2=\\mathcal P_{N,j},\n \\qquad\n \\mathcal P_{N,j}\\mathcal P_{N,k}=0\\ (j\\ne k),\n \\qquad\n \\sum_{j=0}^{N-1}\\mathcal P_{N,j}=1.\n\\]\nIf \\(M\\in\\mathbb Z\\) and \\(M\\ge1\\), their exact refinement law is\n\\[\n \\mathcal P_{N,j}\n =\n \\sum_{\\ell=0}^{M-1}\\mathcal P_{NM,j+\\ell N}.\n\\]","factor from \\(c+1\\) through \\(c+Nn\\) occurs in exactly one\nlane.  For every commutative ring \\(A\\), on the \\(A\\)-module \\(A[[z]]\\) the\noperators\n\\((\\mathcal P_{N,j})_{0\\le j<N}\\) satisfy\n\\[\n \\mathcal P_{N,j}^2=\\mathcal P_{N,j},\n \\qquad\n \\mathcal P_{N,j}\\mathcal P_{N,k}=0\\ (j\\ne k),\n \\qquad\n \\sum_{j=0}^{N-1}\\mathcal P_{N,j}=1.\n\\]\nIf \\(M\\in\\mathbb Z\\) and \\(M\\ge1\\), their exact refinement law is\n\\[\n \\mathcal P_{N,j}\n =\n \\sum_{\\ell=0}^{M-1}\\mathcal P_{NM,j+\\ell N}.\n\\]","Suppose further that \\(K\\) is a field, that \\(N\\) is invertible in \\(K\\),\nand that \\(K\\) contains a primitive \\(N\\)-th root of unity \\(\\omega\\).  Then\nfor every \\(F\\in K[[z]]\\),\n\\[\n \\mathcal P_{N,j}F(z)\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\omega^{-j\\ell}F(\\omega^\\ell z).\n\\]\nIf, in addition, \\(K\\) has characteristic zero, the exponential and\nhypergeometric series below are defined in \\(K[[z]]\\).  Put\n\\[\n E_{N,j}(z)\n :=\\mathcal P_{N,j}e^z\n =\\sum_{n\\ge0}\\frac{z^{Nn+j}}{(Nn+j)!}.\n\\]\nIf\n\\[\n \\mathbf b_{N,j}\n :=\n \\left\\{\n \\frac{j+r}{N}:1\\le r\\le N,\n \\ r\\ne N-j\n \\right\\},\n\\]\nthen\n\\[\n E_{N,j}(z)\n =\n \\frac{z^j}{j!}\n {}_0F_{N-1}\n \\left(\n \\begin{matrix}-","uppose further that \\(K\\) is a field, that \\(N\\) is invertible in \\(K\\),\nand that \\(K\\) contains a primitive \\(N\\)-th root of unity \\(\\omega\\).  Then\nfor every \\(F\\in K[[z]]\\),\n\\[\n \\mathcal P_{N,j}F(z)\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\omega^{-j\\ell}F(\\omega^\\ell z).\n\\]\nIf, in addition, \\(K\\) has characteristic zero, the exponential and\nhypergeometric series below are defined in \\(K[[z]]\\).  Put\n\\[\n E_{N,j}(z)\n :=\\mathcal P_{N,j}e^z\n =\\sum_{n\\ge0}\\frac{z^{Nn+j}}{(Nn+j)!}.\n\\]\nIf\n\\[\n \\mathbf b_{N,j}\n :=\n \\left\\{\n \\frac{j+r}{N}:1\\le r\\le N,\n \\ r\\ne N-j\n \\right\\},\n\\]\nthen\n\\[\n E_{N,j}(z)\n =\n \\frac{z^j}{j!}\n {}_0F_{N-1}\n \\left(\n \\begin{matrix}-","K\\) contains a primitive \\(N\\)-th root of unity \\(\\omega\\).  Then\nfor every \\(F\\in K[[z]]\\),\n\\[\n \\mathcal P_{N,j}F(z)\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\omega^{-j\\ell}F(\\omega^\\ell z).\n\\]\nIf, in addition, \\(K\\) has characteristic zero, the exponential and\nhypergeometric series below are defined in \\(K[[z]]\\).  Put\n\\[\n E_{N,j}(z)\n :=\\mathcal P_{N,j}e^z\n =\\sum_{n\\ge0}\\frac{z^{Nn+j}}{(Nn+j)!}.\n\\]\nIf\n\\[\n \\mathbf b_{N,j}\n :=\n \\left\\{\n \\frac{j+r}{N}:1\\le r\\le N,\n \\ r\\ne N-j\n \\right\\},\n\\]\nthen\n\\[\n E_{N,j}(z)\n =\n \\frac{z^j}{j!}\n {}_0F_{N-1}\n \\left(\n \\begin{matrix}-","\\[\n \\mathcal P_{N,j}F(z)\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\omega^{-j\\ell}F(\\omega^\\ell z).\n\\]\nIf, in addition, \\(K\\) has characteristic zero, the exponential and\nhypergeometric series below are defined in \\(K[[z]]\\).  Put\n\\[\n E_{N,j}(z)\n :=\\mathcal P_{N,j}e^z\n =\\sum_{n\\ge0}\\frac{z^{Nn+j}}{(Nn+j)!}.\n\\]\nIf\n\\[\n \\mathbf b_{N,j}\n :=\n \\left\\{\n \\frac{j+r}{N}:1\\le r\\le N,\n \\ r\\ne N-j\n \\right\\},\n\\]\nthen\n\\[\n E_{N,j}(z)\n =\n \\frac{z^j}{j!}\n {}_0F_{N-1}\n \\left(\n \\begin{matrix}-\\\\ \\mathbf b_{N,j}\\end{matrix};\n \\frac{z^N}{N^N}\n \\right).\n\\]\nThus coefficient-residue, root-of-unity, and hypergeometric descriptions are\nexact additive coordinate realizations of one residue-address decomposition.\nThe step-factorial identity is an exact multiplicative companion with the\nsame address arithmetic; it is not asserted to be an additive idempotent\ndecomposition of the factorial.\n\\end{theorem}","\\begin{proof}\nEvery integer \\(m\\) with \\(c<m\\le c+Nn\\) has a unique representation\n\\(m=r+Nk\\) with \\(c+1\\le r\\le c+N\\) and \\(0\\le k<n\\).  Multiplying these\ndisjoint factors proves the factorial identity.","The projector laws follow coefficient by coefficient.  The residue classes\nmodulo \\(NM\\) lying over \\(j\\pmod N\\) are precisely\n\\(j,j+N,\\ldots,j+(M-1)N\\), which proves refinement.  Character\northogonality gives the Fourier formula.  Finally,\n\\[\n (Nn+j)!\n =\n j!\\,N^{Nn}\n \\prod_{r=1}^{N}\\left(\\frac{j+r}{N}\\right)_n.\n\\]\nExactly one factor in the product is \\((1)_n=n!\\).  Removing it leaves the\nparameter set \\(\\mathbf b_{N,j}\\) and gives the hypergeometric series.\n\\end{proof}","ector laws follow coefficient by coefficient.  The residue classes\nmodulo \\(NM\\) lying over \\(j\\pmod N\\) are precisely\n\\(j,j+N,\\ldots,j+(M-1)N\\), which proves refinement.  Character\northogonality gives the Fourier formula.  Finally,\n\\[\n (Nn+j)!\n =\n j!\\,N^{Nn}\n \\prod_{r=1}^{N}\\left(\\frac{j+r}{N}\\right)_n.\n\\]\nExactly one factor in the product is \\((1)_n=n!\\).  Removing it leaves the\nparameter set \\(\\mathbf b_{N,j}\\) and gives the hypergeometric series.\n\\end{proof}","\\begin{proposition}[Typed polyphase realization]\n\\label{prop:block15-polyphase-specialization}\nLet \\(K\\) be a field, let \\(N\\in\\mathbb Z\\) with \\(N\\ge1\\) be invertible in \\(K\\), and suppose\n\\(K\\) contains a primitive \\(N\\)-th root of unity.  For\n\\(F(z)=\\sum_{n\\ge0}a_nz^n\\in K[[z]]\\), set\n\\[\n F_j(w):=\\sum_{n\\ge0}a_{Nn+j}w^n.\n\\]\nThen\n\\[\n F(z)=\\sum_{j=0}^{N-1}z^jF_j(z^N)\n\\]\nis exact analysis and synthesis.  The Fourier residue projectors and the\npolyphase components satisfy the explicit coordinate relation\n\\[\n \\mathcal P_{N,j}F(z)=z^jF_j(z^N).\n\\]\nFor every integer \\(M\\ge1\\), refinement from \\(N\\) to \\(NM\\) is the\nprojector-sum law of\n\\cref{thm:block15-factorial-residue-realization}.\n\\end{proposition}","ield, let \\(N\\in\\mathbb Z\\) with \\(N\\ge1\\) be invertible in \\(K\\), and suppose\n\\(K\\) contains a primitive \\(N\\)-th root of unity.  For\n\\(F(z)=\\sum_{n\\ge0}a_nz^n\\in K[[z]]\\), set\n\\[\n F_j(w):=\\sum_{n\\ge0}a_{Nn+j}w^n.\n\\]\nThen\n\\[\n F(z)=\\sum_{j=0}^{N-1}z^jF_j(z^N)\n\\]\nis exact analysis and synthesis.  The Fourier residue projectors and the\npolyphase components satisfy the explicit coordinate relation\n\\[\n \\mathcal P_{N,j}F(z)=z^jF_j(z^N).\n\\]\nFor every integer \\(M\\ge1\\), refinement from \\(N\\) to \\(NM\\) is the\nprojector-sum law of\n\\cref{thm:block15-factorial-residue-realization}.\n\\end{proposition}","\\(K\\), and suppose\n\\(K\\) contains a primitive \\(N\\)-th root of unity.  For\n\\(F(z)=\\sum_{n\\ge0}a_nz^n\\in K[[z]]\\), set\n\\[\n F_j(w):=\\sum_{n\\ge0}a_{Nn+j}w^n.\n\\]\nThen\n\\[\n F(z)=\\sum_{j=0}^{N-1}z^jF_j(z^N)\n\\]\nis exact analysis and synthesis.  The Fourier residue projectors and the\npolyphase components satisfy the explicit coordinate relation\n\\[\n \\mathcal P_{N,j}F(z)=z^jF_j(z^N).\n\\]\nFor every integer \\(M\\ge1\\), refinement from \\(N\\) to \\(NM\\) is the\nprojector-sum law of\n\\cref{thm:block15-factorial-residue-realization}.\n\\end{proposition}","ontains a primitive \\(N\\)-th root of unity.  For\n\\(F(z)=\\sum_{n\\ge0}a_nz^n\\in K[[z]]\\), set\n\\[\n F_j(w):=\\sum_{n\\ge0}a_{Nn+j}w^n.\n\\]\nThen\n\\[\n F(z)=\\sum_{j=0}^{N-1}z^jF_j(z^N)\n\\]\nis exact analysis and synthesis.  The Fourier residue projectors and the\npolyphase components satisfy the explicit coordinate relation\n\\[\n \\mathcal P_{N,j}F(z)=z^jF_j(z^N).\n\\]\nFor every integer \\(M\\ge1\\), refinement from \\(N\\) to \\(NM\\) is the\nprojector-sum law of\n\\cref{thm:block15-factorial-residue-realization}.\n\\end{proposition}","primitive \\(N\\)-th root of unity.  For\n\\(F(z)=\\sum_{n\\ge0}a_nz^n\\in K[[z]]\\), set\n\\[\n F_j(w):=\\sum_{n\\ge0}a_{Nn+j}w^n.\n\\]\nThen\n\\[\n F(z)=\\sum_{j=0}^{N-1}z^jF_j(z^N)\n\\]\nis exact analysis and synthesis.  The Fourier residue projectors and the\npolyphase components satisfy the explicit coordinate relation\n\\[\n \\mathcal P_{N,j}F(z)=z^jF_j(z^N).\n\\]\nFor every integer \\(M\\ge1\\), refinement from \\(N\\) to \\(NM\\) is the\nprojector-sum law of\n\\cref{thm:block15-factorial-residue-realization}.\n\\end{proposition}","\\begin{proof}\nEvery exponent has a unique quotient and remainder on division by \\(N\\).\nThe Fourier formula in\n\\cref{thm:block15-factorial-residue-realization} identifies the same\ncoefficient sections with character projectors.\n\\end{proof}","\\begin{proposition}[Typed Cartier realization]\n\\label{prop:block15-cartier-specialization}\nLet \\(K\\) be a field and let \\(k\\ge2\\).  Work with \\(K\\)-valued formal\nseries and fix the least-significant-digit-first convention.  Define\n\\[\n \\Lambda_d\\left(\\sum_{n\\ge0}a_nz^n\\right)\n :=\\sum_{n\\ge0}a_{kn+d}z^n,\n \\qquad\n S_d(G):=z^dG(z^k),\n \\quad 0\\le d<k.\n\\]\nThen\n\\[\n \\Lambda_dS_e=\\delta_{de}1,\n \\qquad\n \\sum_{d=0}^{k-1}S_d\\Lambda_d=1.\n\\]\nFor a digit word \\(w=d_0d_1\\cdots d_{a-1}\\), the iterated Cartier lane is\n\\[\n \\Lambda_{d_{a-1}}\\cdots\\Lambda_{d_1}\\Lambda_{d_0}F.\n\\]\nIt selects coefficients with index\n\\[\n k^an+[w]_k,\n \\qquad\n [w]_k=d_0+kd_1+\\cdots+k^{a-1}d_{a-1}.\n\\]\nConcatenation gives strict radix refinement and the lane path law.\n\\end{proposition}","el{prop:block15-cartier-specialization}\nLet \\(K\\) be a field and let \\(k\\ge2\\).  Work with \\(K\\)-valued formal\nseries and fix the least-significant-digit-first convention.  Define\n\\[\n \\Lambda_d\\left(\\sum_{n\\ge0}a_nz^n\\right)\n :=\\sum_{n\\ge0}a_{kn+d}z^n,\n \\qquad\n S_d(G):=z^dG(z^k),\n \\quad 0\\le d<k.\n\\]\nThen\n\\[\n \\Lambda_dS_e=\\delta_{de}1,\n \\qquad\n \\sum_{d=0}^{k-1}S_d\\Lambda_d=1.\n\\]\nFor a digit word \\(w=d_0d_1\\cdots d_{a-1}\\), the iterated Cartier lane is\n\\[\n \\Lambda_{d_{a-1}}\\cdots\\Lambda_{d_1}\\Lambda_{d_0}F.\n\\]\nIt selects coefficients with index\n\\[\n k^an+[w]_k,\n \\qquad\n [w]_k=d_0+kd_1+\\cdots+k^{a-1}d_{a-1}.\n\\]\nConcatenation gives strict radix refinement and the lane path law.\n\\end{proposition}","eld and let \\(k\\ge2\\).  Work with \\(K\\)-valued formal\nseries and fix the least-significant-digit-first convention.  Define\n\\[\n \\Lambda_d\\left(\\sum_{n\\ge0}a_nz^n\\right)\n :=\\sum_{n\\ge0}a_{kn+d}z^n,\n \\qquad\n S_d(G):=z^dG(z^k),\n \\quad 0\\le d<k.\n\\]\nThen\n\\[\n \\Lambda_dS_e=\\delta_{de}1,\n \\qquad\n \\sum_{d=0}^{k-1}S_d\\Lambda_d=1.\n\\]\nFor a digit word \\(w=d_0d_1\\cdots d_{a-1}\\), the iterated Cartier lane is\n\\[\n \\Lambda_{d_{a-1}}\\cdots\\Lambda_{d_1}\\Lambda_{d_0}F.\n\\]\nIt selects coefficients with index\n\\[\n k^an+[w]_k,\n \\qquad\n [w]_k=d_0+kd_1+\\cdots+k^{a-1}d_{a-1}.\n\\]\nConcatenation gives strict radix refinement and the lane path law.\n\\end{proposition}","it-first convention.  Define\n\\[\n \\Lambda_d\\left(\\sum_{n\\ge0}a_nz^n\\right)\n :=\\sum_{n\\ge0}a_{kn+d}z^n,\n \\qquad\n S_d(G):=z^dG(z^k),\n \\quad 0\\le d<k.\n\\]\nThen\n\\[\n \\Lambda_dS_e=\\delta_{de}1,\n \\qquad\n \\sum_{d=0}^{k-1}S_d\\Lambda_d=1.\n\\]\nFor a digit word \\(w=d_0d_1\\cdots d_{a-1}\\), the iterated Cartier lane is\n\\[\n \\Lambda_{d_{a-1}}\\cdots\\Lambda_{d_1}\\Lambda_{d_0}F.\n\\]\nIt selects coefficients with index\n\\[\n k^an+[w]_k,\n \\qquad\n [w]_k=d_0+kd_1+\\cdots+k^{a-1}d_{a-1}.\n\\]\nConcatenation gives strict radix refinement and the lane path law.\n\\end{proposition}","\\begin{proposition}[Typed holonomic realization]\n\\label{prop:block15-holonomic-specialization}\nLet \\(K\\) be a characteristic-zero field and let the \\(K\\)-valued sequence\n\\((u_n)\\) satisfy a linear\nrecurrence with polynomial coefficients.  Beyond a finite singular prefix,\nwrite the recurrence as a finite-dimensional rational nonautonomous state\nmodel\n\\[\n v_{n+1}=A(n)v_n,\n \\qquad A(n)\\in\\operatorname{Mat}_d(K(n)).\n\\]\nFor a fixed integer \\(N\\ge1\\) and residue \\(0\\le j<N\\), the lane state\n\\(w_m=v_{Nm+j}\\) evolves by the exact \\(N\\)-step rational transfer\n\\[\n w_{m+1}\n =\n A(Nm+j+N-1)\\cdots A(Nm+j)w_m\n\\]\nwhenever the displayed factors are defined.  Equivalently, residue\nmultisection of a D-finite generating series gives D-finite lanes.  The finite\nsingular prefix and any exceptional denominators re","\\begin{proposition}[Typed holonomic realization]\n\\label{prop:block15-holonomic-specialization}\nLet \\(K\\) be a characteristic-zero field and let the \\(K\\)-valued sequence\n\\((u_n)\\) satisfy a linear\nrecurrence with polynomial coefficients.  Beyond a finite singular prefix,\nwrite the recurrence as a finite-dimensional rational nonautonomous state\nmodel\n\\[\n v_{n+1}=A(n)v_n,\n \\qquad A(n)\\in\\operatorname{Mat}_d(K(n)).\n\\]\nFor a fixed integer \\(N\\ge1\\) and residue \\(0\\le j<N\\), the lane state\n\\(w_m=v_{Nm+j}\\) evolves by the exact \\(N\\)-step rational transfer\n\\[\n w_{m+1}\n =\n A(Nm+j+N-1)\\cdots A(Nm+j)w_m\n\\]\nwhenever the displayed factors are defined.  Equivalently, residue\nmultisection of a D-finite generating series gives D-finite lanes.  The finite\nsingular prefix and any exceptional denominators remain explicit data.\n\\end{proposition}","zation]\n\\label{prop:block15-holonomic-specialization}\nLet \\(K\\) be a characteristic-zero field and let the \\(K\\)-valued sequence\n\\((u_n)\\) satisfy a linear\nrecurrence with polynomial coefficients.  Beyond a finite singular prefix,\nwrite the recurrence as a finite-dimensional rational nonautonomous state\nmodel\n\\[\n v_{n+1}=A(n)v_n,\n \\qquad A(n)\\in\\operatorname{Mat}_d(K(n)).\n\\]\nFor a fixed integer \\(N\\ge1\\) and residue \\(0\\le j<N\\), the lane state\n\\(w_m=v_{Nm+j}\\) evolves by the exact \\(N\\)-step rational transfer\n\\[\n w_{m+1}\n =\n A(Nm+j+N-1)\\cdots A(Nm+j)w_m\n\\]\nwhenever the displayed factors are defined.  Equivalently, residue\nmultisection of a D-finite generating series gives D-finite lanes.  The finite\nsingular prefix and any exceptional denominators remain explicit data.\n\\end{proposition}","ock15-holonomic-specialization}\nLet \\(K\\) be a characteristic-zero field and let the \\(K\\)-valued sequence\n\\((u_n)\\) satisfy a linear\nrecurrence with polynomial coefficients.  Beyond a finite singular prefix,\nwrite the recurrence as a finite-dimensional rational nonautonomous state\nmodel\n\\[\n v_{n+1}=A(n)v_n,\n \\qquad A(n)\\in\\operatorname{Mat}_d(K(n)).\n\\]\nFor a fixed integer \\(N\\ge1\\) and residue \\(0\\le j<N\\), the lane state\n\\(w_m=v_{Nm+j}\\) evolves by the exact \\(N\\)-step rational transfer\n\\[\n w_{m+1}\n =\n A(Nm+j+N-1)\\cdots A(Nm+j)w_m\n\\]\nwhenever the displayed factors are defined.  Equivalently, residue\nmultisection of a D-finite generating series gives D-finite lanes.  The finite\nsingular prefix and any exceptional denominators remain explicit data.\n\\end{proposition}","n}\nLet \\(K\\) be a characteristic-zero field and let the \\(K\\)-valued sequence\n\\((u_n)\\) satisfy a linear\nrecurrence with polynomial coefficients.  Beyond a finite singular prefix,\nwrite the recurrence as a finite-dimensional rational nonautonomous state\nmodel\n\\[\n v_{n+1}=A(n)v_n,\n \\qquad A(n)\\in\\operatorname{Mat}_d(K(n)).\n\\]\nFor a fixed integer \\(N\\ge1\\) and residue \\(0\\le j<N\\), the lane state\n\\(w_m=v_{Nm+j}\\) evolves by the exact \\(N\\)-step rational transfer\n\\[\n w_{m+1}\n =\n A(Nm+j+N-1)\\cdots A(Nm+j)w_m\n\\]\nwhenever the displayed factors are defined.  Equivalently, residue\nmultisection of a D-finite generating series gives D-finite lanes.  The finite\nsingular prefix and any exceptional denominators remain explicit data.\n\\end{proposition}","field and let the \\(K\\)-valued sequence\n\\((u_n)\\) satisfy a linear\nrecurrence with polynomial coefficients.  Beyond a finite singular prefix,\nwrite the recurrence as a finite-dimensional rational nonautonomous state\nmodel\n\\[\n v_{n+1}=A(n)v_n,\n \\qquad A(n)\\in\\operatorname{Mat}_d(K(n)).\n\\]\nFor a fixed integer \\(N\\ge1\\) and residue \\(0\\le j<N\\), the lane state\n\\(w_m=v_{Nm+j}\\) evolves by the exact \\(N\\)-step rational transfer\n\\[\n w_{m+1}\n =\n A(Nm+j+N-1)\\cdots A(Nm+j)w_m\n\\]\nwhenever the displayed factors are defined.  Equivalently, residue\nmultisection of a D-finite generating series gives D-finite lanes.  The finite\nsingular prefix and any exceptional denominators remain explicit data.\n\\end{proposition}","d sequence\n\\((u_n)\\) satisfy a linear\nrecurrence with polynomial coefficients.  Beyond a finite singular prefix,\nwrite the recurrence as a finite-dimensional rational nonautonomous state\nmodel\n\\[\n v_{n+1}=A(n)v_n,\n \\qquad A(n)\\in\\operatorname{Mat}_d(K(n)).\n\\]\nFor a fixed integer \\(N\\ge1\\) and residue \\(0\\le j<N\\), the lane state\n\\(w_m=v_{Nm+j}\\) evolves by the exact \\(N\\)-step rational transfer\n\\[\n w_{m+1}\n =\n A(Nm+j+N-1)\\cdots A(Nm+j)w_m\n\\]\nwhenever the displayed factors are defined.  Equivalently, residue\nmultisection of a D-finite generating series gives D-finite lanes.  The finite\nsingular prefix and any exceptional denominators remain explicit data.\n\\end{proposition}","\\begin{proof}\nIterate the state equation \\(N\\) times.  Closure of D-finite series under\nscaling the variable, finite sums, and root-of-unity multisection gives the\nequivalent generating-series statement after a field extension containing the\nneeded roots.  No autonomous finite-state claim is used.\n\\end{proof}","\\begin{proposition}[Typed local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}","\\begin{proposition}[Typed local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\ma","\\begin{proposition}[Typed local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrighta","\\begin{proposition}[Typed local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]","\\begin{proposition}[Typed local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cr","\\begin{proposition}[Typed local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cref{thm:block08-kernel-composition}.","ion}[Typed local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cref{thm:block08-kernel-composition}.","local clopen realization]\n\\label{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cref{thm:block08-kernel-composition}.","l{prop:block15-local-specialization}\nFix a prime \\(p\\) and an integer \\(a\\ge0\\).  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cref{thm:block08-kernel-composition}.",").  For\n\\(f\\in C(\\mathbb Z_p,\\mathbb Z_p)\\), with \\(q\\in\\mathbb Z_p\\) the branch\ncoordinate, define\n\\[\n \\Analysis_{p^a}(f)\n :=\n \\bigl(q\\mapsto f(p^aq+s)\\bigr)_{0\\le s<p^a}.\n\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cref{thm:block08-kernel-composition}.","\\]\nPiecewise synthesis on the disjoint clopen cosets\n\\(s+p^a\\mathbb Z_p\\) is its inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cref{thm:block08-kernel-composition}.","s inverse.  Refining from \\(p^a\\) to\n\\(p^{a+b}\\), for every integer \\(b\\ge0\\), gives the exact child-address and\nmixed-radix composition law.\nLet\n\\[\n \\mathcal M:=\\mathcal M_p^{-1}:C(\\mathbb Z_p,\\mathbb Z_p)\n \\xrightarrow{\\sim}c_0(\\mathbb Z_p)\n\\]\nbe Mahler analysis from \\cref{thm:block08-mahler-coordinate}, and write\n\\(R_{p^a,s}f(q):=f(p^aq+s)\\).  Conjugation gives the exact coefficient\noperator\n\\[\n \\mathcal T_{p^a,s}\n :=\\mathcal M R_{p^a,s}\\mathcal M^{-1}:\n c_0(\\mathbb Z_p)\\longrightarrow c_0(\\mathbb Z_p),\n\\]\nwhose affine kernel is computed in \\cref{thm:block08-affine-transfer}.  Its\nmixed-radix composition is \\cref{thm:block08-kernel-composition}.","The assertion of complete clopen analysis is restricted to these\nprime-power packets.  More general affine coefficient kernels may still\ncompose according to their declared mixed-radix law, but an arbitrary integer\nmodulus does not automatically partition \\(\\mathbb Z_p\\) into that many\ndisjoint clopen branches.\n\\end{proposition}","\\begin{proof}\nThe \\(p^a\\) cosets are disjoint, clopen, and cover \\(\\mathbb Z_p\\).  Restricting\nafter piecewise synthesis and synthesizing after restriction are identities.\nWriting a child address as \\(s+p^at\\) proves refinement.  The cited Mahler\nisomorphism makes the displayed conjugation exact, and the two cited kernel\ntheorems give its coefficient formula and composition law.\n\\end{proof}","\\begin{proposition}[Typed Fuss--Catalan realization]\n\\label{prop:block15-tree-specialization}\nLet \\(K\\) be a characteristic-zero field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{","\\begin{proposition}[Typed Fuss--Catalan realization]\n\\label{prop:block15-tree-specialization}\nLet \\(K\\) be a characteristic-zero field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic","\\begin{proposition}[Typed Fuss--Catalan realization]\n\\label{prop:block15-tree-specialization}\nLet \\(K\\) be a characteristic-zero field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order","\\begin{proposition}[Typed Fuss--Catalan realization]\n\\label{prop:block15-tree-specialization}\nLet \\(K\\) be a characteristic-zero field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for","\\begin{proposition}[Typed Fuss--Catalan realization]\n\\label{prop:block15-tree-specialization}\nLet \\(K\\) be a characteristic-zero field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed","\\begin{proposition}[Typed Fuss--Catalan realization]\n\\label{prop:block15-tree-specialization}\nLet \\(K\\) be a characteristic-zero field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked","lization}\nLet \\(K\\) be a characteristic-zero field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[…","field, let \\(q\\in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\al…","in\\mathbb Z\\) with\n\\(q\\ge2\\), and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac…",", and let\n\\(\\alpha\\in K^\\times\\).  The uniquely marked formal solution of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulti…","tion of\n\\[\n U=z+\\alpha U^q,\n \\qquad U(0)=0,\n\\]\nhas support\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\no…","t\n\\[\n [z^{1+(q-1)n}]U\n =\\frac{1}{(q-1)n+1}\\binom{qn}{n}\\alpha^n\n \\ne0\n \\qquad(n\\ge0),\n\\]\nand every other coefficient is zero.  Thus its support is exactly the set of\ndegrees congruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct image…","gruent to \\(1\\pmod{q-1}\\).\nFor an integer packet width \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct images.\n\\end{proposition}","h \\(N\\ge1\\), set\n\\(g=\\gcd(N,q-1)\\).  Exactly \\(N/g\\) residue addresses are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct images.\n\\end{proposition}","are active, namely\nthe congruence class \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct images.\n\\end{proposition}","lass \\(1\\pmod g\\).  Packetizing the specified multilinear\nmap \\((U_1,\\ldots,U_q)\\mapsto U_1\\cdots U_q\\) gives the exact finite\ncarry-convolution system supplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct images.\n\\end{proposition}","upplied by\n\\cref{thm:block15-multilinear-lane-expansion}.  On a cyclic orbit of order\n\\(q-1\\), the power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct images.\n\\end{proposition}","power map used for packet synthesis has\n\\[\n \\frac{q-1}{\\gcd(N,q-1)}\n\\]\ndistinct images.  If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct images.\n\\end{proposition}","If a fixed embedding \\(K\\hookrightarrow\\mathbb C\\) sends\n\\(\\alpha\\) to a positive real number and the marked branch is interpreted in\nthe smooth implicit analytic schema, its critical points and values satisfy\n\\[\n q\\alpha\\tau^{q-1}=1,\n \\qquad\n \\rho=\\tau-\\alpha\\tau^q=\\frac{q-1}{q}\\tau.\n\\]\nThe resulting orbit \\(\\rho\\mu_{q-1}\\) is precisely the dominant singularity\norbit, and packetization leaves \\((q-1)/g\\) distinct images.\n\\end{proposition}","\\begin{proof}\nLagrange inversion gives the displayed coefficient formula and zero at every\nother degree.  Characteristic zero and \\(\\alpha\\ne0\\) make every displayed\ncoefficient nonzero.  Their residues modulo \\(N\\) form the coset of the subgroup\ngenerated by \\(q-1\\), whose size is \\(N/g\\).  Multilinear lane expansion of\nthe product gives the packet equation, including the power of the packet\nvariable produced by the quotient in residue addition.  On the cyclic group\nof order \\(q-1\\), the kernel of multiplication by \\(N\\) has order \\(g\\), so\nthe image has order \\((q-1)/g\\).  The critical equations give the displayed\nvalues, while \\cref{thm:block11-dominant-orbit} identifies their full orbit as\nthe dominant one under the stated analytic hypotheses.\n\\end{proof}","t every\nother degree.  Characteristic zero and \\(\\alpha\\ne0\\) make every displayed\ncoefficient nonzero.  Their residues modulo \\(N\\) form the coset of the subgroup\ngenerated by \\(q-1\\), whose size is \\(N/g\\).  Multilinear lane expansion of\nthe product gives the packet equation, including the power of the packet\nvariable produced by the quotient in residue addition.  On the cyclic group\nof order \\(q-1\\), the kernel of multiplication by \\(N\\) has order \\(g\\), so\nthe image has order \\((q-1)/g\\).  The critical equations give the displayed\nvalues, while \\cref{thm:block11-dominant-orbit} identifies their full orbit as\nthe dominant one under the stated analytic hypotheses.\n\\end{proof}","o and \\(\\alpha\\ne0\\) make every displayed\ncoefficient nonzero.  Their residues modulo \\(N\\) form the coset of the subgroup\ngenerated by \\(q-1\\), whose size is \\(N/g\\).  Multilinear lane expansion of\nthe product gives the packet equation, including the power of the packet\nvariable produced by the quotient in residue addition.  On the cyclic group\nof order \\(q-1\\), the kernel of multiplication by \\(N\\) has order \\(g\\), so\nthe image has order \\((q-1)/g\\).  The critical equations give the displayed\nvalues, while \\cref{thm:block11-dominant-orbit} identifies their full orbit as\nthe dominant one under the stated analytic hypotheses.\n\\end{proof}","e0\\) make every displayed\ncoefficient nonzero.  Their residues modulo \\(N\\) form the coset of the subgroup\ngenerated by \\(q-1\\), whose size is \\(N/g\\).  Multilinear lane expansion of\nthe product gives the packet equation, including the power of the packet\nvariable produced by the quotient in residue addition.  On the cyclic group\nof order \\(q-1\\), the kernel of multiplication by \\(N\\) has order \\(g\\), so\nthe image has order \\((q-1)/g\\).  The critical equations give the displayed\nvalues, while \\cref{thm:block11-dominant-orbit} identifies their full orbit as\nthe dominant one under the stated analytic hypotheses.\n\\end{proof}","ficient nonzero.  Their residues modulo \\(N\\) form the coset of the subgroup\ngenerated by \\(q-1\\), whose size is \\(N/g\\).  Multilinear lane expansion of\nthe product gives the packet equation, including the power of the packet\nvariable produced by the quotient in residue addition.  On the cyclic group\nof order \\(q-1\\), the kernel of multiplication by \\(N\\) has order \\(g\\), so\nthe image has order \\((q-1)/g\\).  The critical equations give the displayed\nvalues, while \\cref{thm:block11-dominant-orbit} identifies their full orbit as\nthe dominant one under the stated analytic hypotheses.\n\\end{proof}","\\begin{proposition}[Typed cyclic-radical realization]\n\\label{prop:block15-algebraic-specialization}\nLet \\(K\\) be a characteristic-zero field containing \\(\\mu_d\\) and a\nprimitive \\(N\\)-th root of unity \\(\\omega\\), let\n\\(d,m,N\\in\\mathbb Z_{\\ge1}\\), and choose\nthe formal branch\n\\[\n F(z)=(1-\\alpha z^m)^{-1/d},\n \\qquad \\alpha\\in K^\\times.\n\\]\nWith \\(g=\\gcd(N,m)\\), its nonzero coefficients occupy exactly the subgroup of\n\\(\\mathbb Z/N\\mathbb Z\\) generated by \\(m\\), which has \\(N/g\\) addresses.\nThe rotated radicands\n\\[\n 1-\\alpha\\omega^{jm}z^m,\n \\qquad 0\\le j<N,\n\\]\nalso have exactly \\(N/g\\) distinct values.","[Typed cyclic-radical realization]\n\\label{prop:block15-algebraic-specialization}\nLet \\(K\\) be a characteristic-zero field containing \\(\\mu_d\\) and a\nprimitive \\(N\\)-th root of unity \\(\\omega\\), let\n\\(d,m,N\\in\\mathbb Z_{\\ge1}\\), and choose\nthe formal branch\n\\[\n F(z)=(1-\\alpha z^m)^{-1/d},\n \\qquad \\alpha\\in K^\\times.\n\\]\nWith \\(g=\\gcd(N,m)\\), its nonzero coefficients occupy exactly the subgroup of\n\\(\\mathbb Z/N\\mathbb Z\\) generated by \\(m\\), which has \\(N/g\\) addresses.\nThe rotated radicands\n\\[\n 1-\\alpha\\omega^{jm}z^m,\n \\qquad 0\\le j<N,\n\\]\nalso have exactly \\(N/g\\) distinct values.","bel{prop:block15-algebraic-specialization}\nLet \\(K\\) be a characteristic-zero field containing \\(\\mu_d\\) and a\nprimitive \\(N\\)-th root of unity \\(\\omega\\), let\n\\(d,m,N\\in\\mathbb Z_{\\ge1}\\), and choose\nthe formal branch\n\\[\n F(z)=(1-\\alpha z^m)^{-1/d},\n \\qquad \\alpha\\in K^\\times.\n\\]\nWith \\(g=\\gcd(N,m)\\), its nonzero coefficients occupy exactly the subgroup of\n\\(\\mathbb Z/N\\mathbb Z\\) generated by \\(m\\), which has \\(N/g\\) addresses.\nThe rotated radicands\n\\[\n 1-\\alpha\\omega^{jm}z^m,\n \\qquad 0\\le j<N,\n\\]\nalso have exactly \\(N/g\\) distinct values.","algebraic-specialization}\nLet \\(K\\) be a characteristic-zero field containing \\(\\mu_d\\) and a\nprimitive \\(N\\)-th root of unity \\(\\omega\\), let\n\\(d,m,N\\in\\mathbb Z_{\\ge1}\\), and choose\nthe formal branch\n\\[\n F(z)=(1-\\alpha z^m)^{-1/d},\n \\qquad \\alpha\\in K^\\times.\n\\]\nWith \\(g=\\gcd(N,m)\\), its nonzero coefficients occupy exactly the subgroup of\n\\(\\mathbb Z/N\\mathbb Z\\) generated by \\(m\\), which has \\(N/g\\) addresses.\nThe rotated radicands\n\\[\n 1-\\alpha\\omega^{jm}z^m,\n \\qquad 0\\le j<N,\n\\]\nalso have exactly \\(N/g\\) distinct values.","characteristic-zero field containing \\(\\mu_d\\) and a\nprimitive \\(N\\)-th root of unity \\(\\omega\\), let\n\\(d,m,N\\in\\mathbb Z_{\\ge1}\\), and choose\nthe formal branch\n\\[\n F(z)=(1-\\alpha z^m)^{-1/d},\n \\qquad \\alpha\\in K^\\times.\n\\]\nWith \\(g=\\gcd(N,m)\\), its nonzero coefficients occupy exactly the subgroup of\n\\(\\mathbb Z/N\\mathbb Z\\) generated by \\(m\\), which has \\(N/g\\) addresses.\nThe rotated radicands\n\\[\n 1-\\alpha\\omega^{jm}z^m,\n \\qquad 0\\le j<N,\n\\]\nalso have exactly \\(N/g\\) distinct values.",")-th root of unity \\(\\omega\\), let\n\\(d,m,N\\in\\mathbb Z_{\\ge1}\\), and choose\nthe formal branch\n\\[\n F(z)=(1-\\alpha z^m)^{-1/d},\n \\qquad \\alpha\\in K^\\times.\n\\]\nWith \\(g=\\gcd(N,m)\\), its nonzero coefficients occupy exactly the subgroup of\n\\(\\mathbb Z/N\\mathbb Z\\) generated by \\(m\\), which has \\(N/g\\) addresses.\nThe rotated radicands\n\\[\n 1-\\alpha\\omega^{jm}z^m,\n \\qquad 0\\le j<N,\n\\]\nalso have exactly \\(N/g\\) distinct values.","If, in addition, their distinct Kummer classes generate a subgroup isomorphic\nto \\((\\mathbb Z/d\\mathbb Z)^{N/g}\\) in the Kummer quotient\n\\[\n K(z)^\\times/K(z)^{\\times d},\n\\]\nthen the compositum of the corresponding radical branch fields has degree\n\\[\n d^{N/g}\n\\]\nover \\(K(z)\\).  Without that independence hypothesis, the lane orbit count\nremains exact but the degree formula is only an upper bound.\n\\end{proposition}","\\begin{proof}\nThe binomial series is supported at exponents \\(mn\\).  The cyclic subgroup\ncalculation gives both address and rotation counts.  The degree statement is\nthe standard Kummer independence calculation for the explicitly listed\nclasses.  A relation among the classes produces a field collision not detected\nby address count alone.\n\\end{proof}","\\begin{proposition}[Typed semisimple Galois realization]\n\\label{prop:block15-galois-specialization}\nLet \\(G\\) be finite and let \\(K\\) be a characteristic-zero splitting field\nfor \\(G\\).  The primitive central idempotents\n\\[\n e_\\chi\n =\n \\frac{\\chi(1)}{|G|}\n \\sum_{g\\in G}\\chi(g^{-1})g\n \\in K[G]\n\\]\nfor \\(\\chi\\in\\operatorname{Irr}_K(G)\\) form a complete\npairwise-annihilating lane system on every finite-dimensional\nsemisimple \\(K[G]\\)-module \\(V\\):\n\\[\n V=\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)} e_\\chi V.\n\\]\nEvery \\(G\\)-equivariant map is block diagonal in these isotypic lanes, while\nan arbitrary \\(K\\)-linear map has the full lane matrix of\n\\cref{thm:block15-universal-lane-transport}.","emisimple Galois realization]\n\\label{prop:block15-galois-specialization}\nLet \\(G\\) be finite and let \\(K\\) be a characteristic-zero splitting field\nfor \\(G\\).  The primitive central idempotents\n\\[\n e_\\chi\n =\n \\frac{\\chi(1)}{|G|}\n \\sum_{g\\in G}\\chi(g^{-1})g\n \\in K[G]\n\\]\nfor \\(\\chi\\in\\operatorname{Irr}_K(G)\\) form a complete\npairwise-annihilating lane system on every finite-dimensional\nsemisimple \\(K[G]\\)-module \\(V\\):\n\\[\n V=\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)} e_\\chi V.\n\\]\nEvery \\(G\\)-equivariant map is block diagonal in these isotypic lanes, while\nan arbitrary \\(K\\)-linear map has the full lane matrix of\n\\cref{thm:block15-universal-lane-transport}.","realization]\n\\label{prop:block15-galois-specialization}\nLet \\(G\\) be finite and let \\(K\\) be a characteristic-zero splitting field\nfor \\(G\\).  The primitive central idempotents\n\\[\n e_\\chi\n =\n \\frac{\\chi(1)}{|G|}\n \\sum_{g\\in G}\\chi(g^{-1})g\n \\in K[G]\n\\]\nfor \\(\\chi\\in\\operatorname{Irr}_K(G)\\) form a complete\npairwise-annihilating lane system on every finite-dimensional\nsemisimple \\(K[G]\\)-module \\(V\\):\n\\[\n V=\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)} e_\\chi V.\n\\]\nEvery \\(G\\)-equivariant map is block diagonal in these isotypic lanes, while\nan arbitrary \\(K\\)-linear map has the full lane matrix of\n\\cref{thm:block15-universal-lane-transport}.","ation]\n\\label{prop:block15-galois-specialization}\nLet \\(G\\) be finite and let \\(K\\) be a characteristic-zero splitting field\nfor \\(G\\).  The primitive central idempotents\n\\[\n e_\\chi\n =\n \\frac{\\chi(1)}{|G|}\n \\sum_{g\\in G}\\chi(g^{-1})g\n \\in K[G]\n\\]\nfor \\(\\chi\\in\\operatorname{Irr}_K(G)\\) form a complete\npairwise-annihilating lane system on every finite-dimensional\nsemisimple \\(K[G]\\)-module \\(V\\):\n\\[\n V=\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)} e_\\chi V.\n\\]\nEvery \\(G\\)-equivariant map is block diagonal in these isotypic lanes, while\nan arbitrary \\(K\\)-linear map has the full lane matrix of\n\\cref{thm:block15-universal-lane-transport}.","e and let \\(K\\) be a characteristic-zero splitting field\nfor \\(G\\).  The primitive central idempotents\n\\[\n e_\\chi\n =\n \\frac{\\chi(1)}{|G|}\n \\sum_{g\\in G}\\chi(g^{-1})g\n \\in K[G]\n\\]\nfor \\(\\chi\\in\\operatorname{Irr}_K(G)\\) form a complete\npairwise-annihilating lane system on every finite-dimensional\nsemisimple \\(K[G]\\)-module \\(V\\):\n\\[\n V=\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)} e_\\chi V.\n\\]\nEvery \\(G\\)-equivariant map is block diagonal in these isotypic lanes, while\nan arbitrary \\(K\\)-linear map has the full lane matrix of\n\\cref{thm:block15-universal-lane-transport}.","entral idempotents\n\\[\n e_\\chi\n =\n \\frac{\\chi(1)}{|G|}\n \\sum_{g\\in G}\\chi(g^{-1})g\n \\in K[G]\n\\]\nfor \\(\\chi\\in\\operatorname{Irr}_K(G)\\) form a complete\npairwise-annihilating lane system on every finite-dimensional\nsemisimple \\(K[G]\\)-module \\(V\\):\n\\[\n V=\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)} e_\\chi V.\n\\]\nEvery \\(G\\)-equivariant map is block diagonal in these isotypic lanes, while\nan arbitrary \\(K\\)-linear map has the full lane matrix of\n\\cref{thm:block15-universal-lane-transport}.","For a subgroup \\(H\\le G\\), the Reynolds operator\n\\[\n e_H=\\frac1{|H|}\\sum_{h\\in H}h\n\\]\nis an idempotent projector onto \\(V^H\\); equivalently,\n\\((e_H,1-e_H)\\) is a two-lane system on the underlying \\(K\\)-vector space.\nIt is not in general a Boolean\nselector formed from the isotypic projectors above.  Projectors belonging to unrelated\nsubgroups need not annihilate one another, so they do not automatically form\none lane system.  Transport through induction, restriction, or subgroup\nrefinement requires the corresponding Mackey or double-coset data.\n\\end{proposition}","\\begin{proof}\nMaschke semisimplicity and the character idempotent relations give the first\ndecomposition.  An equivariant map commutes with the center of \\(K[G]\\), hence\nwith every \\(e_\\chi\\).  Averaging proves that \\(e_H\\) is idempotent with image\n\\(V^H\\).  Two subgroup averages generally fail the pairwise-annihilation\naxiom, which proves the stated boundary.\n\\end{proof}","\\begin{theorem}[Finite atlas descent for lane transport]\n\\label{thm:block15-lane-descent}\nLet \\(\\mathfrak R\\) be a finite profiled atlas whose participating chart\ncategories are additive and idempotent-complete, with additive transition\nfunctors \\(F_{UV}:\\mathfrak R(U)\\to\\mathfrak R(V)\\) on its declared\noverlaps.  Let\n\\[\n X=(X_U,\\theta^X_{UV}),\n \\qquad\n Y=(Y_U,\\theta^Y_{UV}),\n \\qquad\n Z=(Z_U,\\theta^Z_{UV})\n\\]\nbe three compatible descent objects.  Thus the transition isomorphisms satisfy\nthe identity and cocycle equations.  Let\n\\[\n f=(f_U):X\\longrightarrow Y,\n \\qquad\n g=(g_U):Y\\longrightarrow Z\n\\]\nbe compatible descended morphisms.","s whose participating chart\ncategories are additive and idempotent-complete, with additive transition\nfunctors \\(F_{UV}:\\mathfrak R(U)\\to\\mathfrak R(V)\\) on its declared\noverlaps.  Let\n\\[\n X=(X_U,\\theta^X_{UV}),\n \\qquad\n Y=(Y_U,\\theta^Y_{UV}),\n \\qquad\n Z=(Z_U,\\theta^Z_{UV})\n\\]\nbe three compatible descent objects.  Thus the transition isomorphisms satisfy\nthe identity and cocycle equations.  Let\n\\[\n f=(f_U):X\\longrightarrow Y,\n \\qquad\n g=(g_U):Y\\longrightarrow Z\n\\]\nbe compatible descended morphisms.","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bije","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local p","Fix one finite resolution shape \\(\\mathsf R\\).  Suppose every local object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda…","object\n\\(A_U\\), for \\(A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transitio…","A\\in\\{X,Y,Z\\}\\), carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve th…","carries a finite profiled lane system over\n\\(\\mathsf R\\).  For every overlap, every \\(r\\in\\mathsf R\\), and every such\n\\(A\\), suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittin…",", suppose there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittings,\nfinite sums, composition, and refinement equations, with coherent splitting\ncomparisons on triple overlaps.","there is an address bijection\n\\[\n \\lambda_{UV}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittings,\nfinite sums, composition, and refinement equations, with coherent splitting\ncomparisons on triple overlaps.","V}^{A,r}:\n I_{A_U,r}\\xrightarrow{\\sim}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittings,\nfinite sums, composition, and refinement equations, with coherent splitting\ncomparisons on triple overlaps.","}I_{A_V,r}\n\\]\nsuch that\n\\[\n \\lambda_{UU}^{A,r}=1,\n \\qquad\n \\lambda_{UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittings,\nfinite sums, composition, and refinement equations, with coherent splitting\ncomparisons on triple overlaps.","UT}^{A,r}\n =\\lambda_{VT}^{A,r}\\lambda_{UV}^{A,r},\n\\]\non every triple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittings,\nfinite sums, composition, and refinement equations, with coherent splitting\ncomparisons on triple overlaps.","iple overlap \\((U,V,T)\\), and, for \\(s\\succeq r\\),\n\\[\n \\rho_{sr}^{A_V}\\lambda_{UV}^{A,s}\n =\n \\lambda_{UV}^{A,r}\\rho_{sr}^{A_U}.\n\\]\nRequire trivial address holonomy: for every \\(A\\) and \\(r\\), there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittings,\nfinite sums, composition, and refinement equations, with coherent splitting\ncomparisons on triple overlaps.","there is a\nglobal finite address set \\(I_{A,r}\\), a global refinement map\n\\(\\rho_{sr}^{A}:I_{A,s}\\twoheadrightarrow I_{A,r}\\), and bijections\n\\[\n \\mu_U^{A,r}:I_{A_U,r}\\xrightarrow{\\sim}I_{A,r}\n\\]\nsuch that\n\\[\n \\mu_V^{A,r}\\lambda_{UV}^{A,r}=\\mu_U^{A,r},\n \\qquad\n \\mu_U^{A,r}\\rho_{sr}^{A_U}\n =\\rho_{sr}^{A}\\mu_U^{A,s}.\n\\]\nAssume also that the transition isomorphisms intertwine the local projectors:\n\\[\n \\theta^A_{UV}\n F_{UV}(p_{A_U,i}^r)\n =\n p_{A_V,\\lambda_{UV}^{A,r}(i)}^r\n \\theta^A_{UV},\n\\]\nand that the verified transition profiles preserve the chosen splittings,\nfinite sums, composition, and refinement equations, with coherent splitting\ncomparisons on triple overlaps.","Then:\n\\begin{enumerate}[label=\\textup{(D\\arabic*)}]\n\\item after relabeling by the \\(\\mu_U^{A,r}\\), each local lane and its chosen\nsplitting define a lane summand in \\(\\Desc(\\mathfrak R)\\), and the global\naddresses carry the declared projector-sum refinement;\n\\item the local matrices \\(([f_U]_{a i})\\), \\(([g_U]_{c a})\\), and\n\\(([g_Uf_U]_{c i})\\) glue after the prescribed address relabelings and\nsplitting comparisons;\n\\item the descended matrices satisfy\n\\[\n [gf]_{c i}=\\sum_a[g]_{c a}[f]_{a i};\n\\]\n\\item every compatible family of support selectors and named residuals\ndescends, and every compatible refinement square remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defin","Then:\n\\begin{enumerate}[label=\\textup{(D\\arabic*)}]\n\\item after relabeling by the \\(\\mu_U^{A,r}\\), each local lane and its chosen\nsplitting define a lane summand in \\(\\Desc(\\mathfrak R)\\), and the global\naddresses carry the declared projector-sum refinement;\n\\item the local matrices \\(([f_U]_{a i})\\), \\(([g_U]_{c a})\\), and\n\\(([g_Uf_U]_{c i})\\) glue after the prescribed address relabelings and\nsplitting comparisons;\n\\item the descended matrices satisfy\n\\[\n [gf]_{c i}=\\sum_a[g]_{c a}[f]_{a i};\n\\]\n\\item every compatible family of support selectors and named residuals\ndescends, and every compatible refinement square remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a","Then:\n\\begin{enumerate}[label=\\textup{(D\\arabic*)}]\n\\item after relabeling by the \\(\\mu_U^{A,r}\\), each local lane and its chosen\nsplitting define a lane summand in \\(\\Desc(\\mathfrak R)\\), and the global\naddresses carry the declared projector-sum refinement;\n\\item the local matrices \\(([f_U]_{a i})\\), \\(([g_U]_{c a})\\), and\n\\(([g_Uf_U]_{c i})\\) glue after the prescribed address relabelings and\nsplitting comparisons;\n\\item the descended matrices satisfy\n\\[\n [gf]_{c i}=\\sum_a[g]_{c a}[f]_{a i};\n\\]\n\\item every compatible family of support selectors and named residuals\ndescends, and every compatible refinement square remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a split-losslessness certificate\ndescends directly from its left-inverse identity.  Moreover, suppose the\nrelevant","Then:\n\\begin{enumerate}[label=\\textup{(D\\arabic*)}]\n\\item after relabeling by the \\(\\mu_U^{A,r}\\), each local lane and its chosen\nsplitting define a lane summand in \\(\\Desc(\\mathfrak R)\\), and the global\naddresses carry the declared projector-sum refinement;\n\\item the local matrices \\(([f_U]_{a i})\\), \\(([g_U]_{c a})\\), and\n\\(([g_Uf_U]_{c i})\\) glue after the prescribed address relabelings and\nsplitting comparisons;\n\\item the descended matrices satisfy\n\\[\n [gf]_{c i}=\\sum_a[g]_{c a}[f]_{a i};\n\\]\n\\item every compatible family of support selectors and named residuals\ndescends, and every compatible refinement square remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a split-losslessness certificate\ndescends directly from its left-inverse identity.  Moreover, suppose the\nrelevant local kernels \\(K_U","Then:\n\\begin{enumerate}[label=\\textup{(D\\arabic*)}]\n\\item after relabeling by the \\(\\mu_U^{A,r}\\), each local lane and its chosen\nsplitting define a lane summand in \\(\\Desc(\\mathfrak R)\\), and the global\naddresses carry the declared projector-sum refinement;\n\\item the local matrices \\(([f_U]_{a i})\\), \\(([g_U]_{c a})\\), and\n\\(([g_Uf_U]_{c i})\\) glue after the prescribed address relabelings and\nsplitting comparisons;\n\\item the descended matrices satisfy\n\\[\n [gf]_{c i}=\\sum_a[g]_{c a}[f]_{a i};\n\\]\n\\item every compatible family of support selectors and named residuals\ndescends, and every compatible refinement square remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a split-losslessness certificate\ndescends directly from its left-inverse identity.  Moreover, suppose the\nrelevant local kernels \\(K_U=\\ker(\\Phi_U)\\) exist, eve","fter relabeling by the \\(\\mu_U^{A,r}\\), each local lane and its chosen\nsplitting define a lane summand in \\(\\Desc(\\mathfrak R)\\), and the global\naddresses carry the declared projector-sum refinement;\n\\item the local matrices \\(([f_U]_{a i})\\), \\(([g_U]_{c a})\\), and\n\\(([g_Uf_U]_{c i})\\) glue after the prescribed address relabelings and\nsplitting comparisons;\n\\item the descended matrices satisfy\n\\[\n [gf]_{c i}=\\sum_a[g]_{c a}[f]_{a i};\n\\]\n\\item every compatible family of support selectors and named residuals\ndescends, and every compatible refinement square remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a split-losslessness certificate\ndescends directly from its left-inverse identity.  Moreover, suppose the\nrelevant local kernels \\(K_U=\\ker(\\Phi_U)\\) exist, every transition functor\npreserves them, and there are coherent comparison isomorphisms\n\\[\n \\eta_{UV}:F_{UV}(K_U)\\xrig…","parisons;\n\\item the descended matrices satisfy\n\\[\n [gf]_{c i}=\\sum_a[g]_{c a}[f]_{a i};\n\\]\n\\item every compatible family of support selectors and named residuals\ndescends, and every compatible refinement square remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a split-losslessness certificate\ndescends directly from its left-inverse identity.  Moreover, suppose the\nrelevant local kernels \\(K_U=\\ker(\\Phi_U)\\) exist, every transition functor\npreserves them, and there are coherent comparison isomorphisms\n\\[\n \\eta_{UV}:F_{UV}(K_U)\\xrightarrow{\\sim}K_V\n\\]\ncompatible with the kernel inclusions and satisfying the triple-overlap\ncocycle.  Then the global kernel exists and is computed chartwise:\n\\[\n \\bigl(\\ker_{\\Desc(\\mathfrak R)}\\Phi\\bigr)_U\n \\cong\n \\ker_{\\mathfrak R(U)}(\\Phi_U).\n\\]\n\\end{enumerate}\n\\end{theorem}","quare remains a projector-sum\nrefinement square in the descent category;\n\\item every fold whose source objects, target objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a split-losslessness certificate\ndescends directly from its left-inverse identity.  Moreover, suppose the\nrelevant local kernels \\(K_U=\\ker(\\Phi_U)\\) exist, every transition functor\npreserves them, and there are coherent comparison isomorphisms\n\\[\n \\eta_{UV}:F_{UV}(K_U)\\xrightarrow{\\sim}K_V\n\\]\ncompatible with the kernel inclusions and satisfying the triple-overlap\ncocycle.  Then the global kernel exists and is computed chartwise:\n\\[\n \\bigl(\\ker_{\\Desc(\\mathfrak R)}\\Phi\\bigr)_U\n \\cong\n \\ker_{\\mathfrak R(U)}(\\Phi_U).\n\\]\n\\end{enumerate}\n\\end{theorem}","et objects, and structure maps are\ncompatible descent data defines a fold in \\(\\Desc(\\mathfrak R)\\).  Compatible\nselectors and left inverses descend, so a split-losslessness certificate\ndescends directly from its left-inverse identity.  Moreover, suppose the\nrelevant local kernels \\(K_U=\\ker(\\Phi_U)\\) exist, every transition functor\npreserves them, and there are coherent comparison isomorphisms\n\\[\n \\eta_{UV}:F_{UV}(K_U)\\xrightarrow{\\sim}K_V\n\\]\ncompatible with the kernel inclusions and satisfying the triple-overlap\ncocycle.  Then the global kernel exists and is computed chartwise:\n\\[\n \\bigl(\\ker_{\\Desc(\\mathfrak R)}\\Phi\\bigr)_U\n \\cong\n \\ker_{\\mathfrak R(U)}(\\Phi_U).\n\\]\n\\end{enumerate}\n\\end{theorem}","cends directly from its left-inverse identity.  Moreover, suppose the\nrelevant local kernels \\(K_U=\\ker(\\Phi_U)\\) exist, every transition functor\npreserves them, and there are coherent comparison isomorphisms\n\\[\n \\eta_{UV}:F_{UV}(K_U)\\xrightarrow{\\sim}K_V\n\\]\ncompatible with the kernel inclusions and satisfying the triple-overlap\ncocycle.  Then the global kernel exists and is computed chartwise:\n\\[\n \\bigl(\\ker_{\\Desc(\\mathfrak R)}\\Phi\\bigr)_U\n \\cong\n \\ker_{\\mathfrak R(U)}(\\Phi_U).\n\\]\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFor a global address \\(i\\in I_{A,r}\\), use the local projector indexed by\n\\((\\mu_U^{A,r})^{-1}(i)\\).  The trivialization identity makes these projectors\ncorrespond under the transition isomorphisms on every overlap; its refinement\nidentity makes their local\nprojector-sum laws agree.  The coherent splitting comparisons make the\ncorresponding inclusions and projections descent morphisms.  Finite\nbiproducts in the descent category are computed chartwise, so local analysis\nand synthesis glue and remain mutually inverse.","Compatibility of \\(f\\) and \\(g\\), followed by the splitting comparisons,\nshows that their local coordinate blocks agree on overlaps after applying the\nspecified address relabelings.  Hence the matrices descend.  Matrix\nmultiplication is a finite sum of compositions, both computed chartwise, so\nthe local path law glues to the displayed global law.  The same argument\napplies to projector sums, selectors, and residual differences.","Finally, a compatible fold is a morphism in the relevant descent category.\nCompatible selectors and left inverses glue, and their identities can be\nchecked chartwise.  Under the additional kernel hypotheses, the objects\n\\((K_U,\\eta_{UV})\\) form a descent object; compatibility with the local kernel\ninclusions gives a global morphism into the fold source.  Its universal\nproperty is checked in each chart, which proves the displayed chartwise\nkernel identity.  The split case needs no kernel-preservation hypothesis:\nthe compatible local left inverses glue directly.\n\\end{proof}","\\begin{corollary}[Certified atlas lane transport]\n\\label{cor:block15-certified-atlas-lane-transport}\nLet \\(\\mathfrak A_{\\mathrm{cert}}\\) be the certified finite atlas of this\nmanuscript.  On every subatlas whose passports include additive split-lane\nsystems, compatible refinement maps, coherent address relabelings with\ntrivial address holonomy, and the required preservation profile, the\nUniversal Lane Transport Theorem holds in the descent category.  Kernel\nclaims additionally require the kernel-preservation and coherence data in\n\\cref{thm:block15-lane-descent}.  In particular, any of the typed\nspecializations above may be transported across a certified reversible\noverlap after its native hypotheses, address trivializations, and splitting\ncocycles have been checked.\n\\end{corollary}","\\begin{boundary}[Descent is conditional on address coherence]\n\\label{bnd:block15-descent-boundary}\nAn equivalence of underlying semantic objects does not by itself identify two\nlane systems.  Without the bijections \\(\\lambda_{UV}^{A,r}\\), their cocycle\nlaw, compatible global trivializations \\(\\mu_U^{A,r}\\), refinement\ncompatibility, and coherent splitting comparisons, the local matrices need\nnot glue as ordinary globally addressed lanes.  Nontrivial address holonomy\ncan permute lanes around a loop; then only invariant orbit sums, or lanes in a\nseparately defined local-system-valued category, may descend.  The theorem\nabove verifies ordinary descent under trivial holonomy.  It does not\nmanufacture missing lane passports for an arbitrary heterogeneous atlas.\n\\end{boundary}","\\begin{proposition}[Endpoint selection does not implement refinement]\n\\label{prop:block15-endpoint-counterexample}\nLet \\(K\\) be a field and let \\(X=K^2\\).  At the coarse resolution use the\nsingle projector \\(p=1_X\\).  At the fine resolution use\n\\[\n p_1=\n \\begin{pmatrix}1&0","\\begin{proposition}[Endpoint selection does not implement refinement]\n\\label{prop:block15-endpoint-counterexample}\nLet \\(K\\) be a field and let \\(X=K^2\\).  At the coarse resolution use the\nsingle projector \\(p=1_X\\).  At the fine resolution use\n\\[\n p_1=\n \\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\n \\qquad\n p_2=\n \\begin{pmatrix}0&0\\\\0&1\\end{pmatrix},\n \\qquad p=p_1+p_2.\n\\]\nFor\n\\[\n f=p_2,\n\\]\nthe unique coarse block is \\(pfp=f\\ne0\\), whereas selecting the single fine\nendpoint \\((p_1,p_1)\\) gives \\(p_1fp_1=0\\).  Therefore a coarse block is the\nsum over all compatible descendant source-target pairs.  It cannot in general\nbe represented by choosing one fine endpoint.\n\\end{proposition}","0&1\\end{pmatrix},\n \\qquad p=p_1+p_2.\n\\]\nFor\n\\[\n f=p_2,\n\\]\nthe unique coarse block is \\(pfp=f\\ne0\\), whereas selecting the single fine\nendpoint \\((p_1,p_1)\\) gives \\(p_1fp_1=0\\).  Therefore a coarse block is the\nsum over all compatible descendant source-target pairs.  It cannot in general\nbe represented by choosing one fine endpoint.\n\\end{proposition}","\\begin{proof}\nThe displayed matrix multiplication gives the two values immediately.  The\nfull refinement identity recovers\n\\(f=\\sum_{a,i\\in\\{1,2\\}}p_afp_i\\), whose only nonzero term here is\n\\(p_2fp_2\\).\n\\end{proof}","Let \\(\\mathscr G\\) be a finite connected groupoid of charts.  For an arrow\n\\(\\gamma:U\\to V\\), let\n\\[\n F_\\gamma:\\mathcal C_U\\longrightarrow\\mathcal C_V\n\\]\nbe an additive equivalence between additive idempotent-complete categories.\nThe transition functors carry specified identity and composition comparisons.\nTo keep the formulas readable, those comparison isomorphisms are suppressed;\nevery displayed cocycle equation is understood after inserting the declared\ncomparisons.","A descent object\n\\[\n A=(A_U,\\theta^A_\\gamma)\n\\]\nconsists of objects \\(A_U\\in\\mathcal C_U\\) and isomorphisms\n\\[\n \\theta^A_\\gamma:F_\\gamma(A_U)\\xrightarrow{\\sim}A_V\n\\]\nsatisfying the identity and cocycle equations.  A descended morphism\n\\(f:X\\to Y\\) is a compatible family \\((f_U)_U\\) satisfying\n\\[\n \\theta^Y_\\gamma F_\\gamma(f_U)=f_V\\theta^X_\\gamma.\n\\]","\\begin{definition}[Holonomy-profiled finite lane datum]\n\\label{def:block16-holonomy-profiled-lane-datum}\nFix a finite resolution shape \\(\\mathsf R\\).  A holonomy-profiled finite lane\ndatum on a descent object \\(A\\) consists of the following information.","For every chart \\(U\\) and resolution \\(r\\in\\mathsf R\\), the object \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lamb","For every chart \\(U\\) and resolution \\(r\\in\\mathsf R\\), the object \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\ga","For every chart \\(U\\) and resolution \\(r\\in\\mathsf R\\), the object \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\be","For every chart \\(U\\) and resolution \\(r\\in\\mathsf R\\), the object \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement ma","For every chart \\(U\\) and resolution \\(r\\in\\mathsf R\\), the object \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}","For every chart \\(U\\) and resolution \\(r\\in\\mathsf R\\), the object \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}.\n\\]\nNo global address set and no global trivialization of the address bijections is\nassumed…","tion \\(r\\in\\mathsf R\\), the object \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}.\n\\]\nNo global address set and no global trivialization of the address bijections is\nassumed.\n\\end{definition}","ject \\(A_U\\)\nhas a finite profiled lane system\n\\[\n \\left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},\n \\iota^r_{A,U,i},\\pi^r_{A,U,i}\\right)_{i\\in I_{A,U,r}}\n\\]\nin the sense of \\cref{def:block15-finite-profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}.\n\\]\nNo global address set and no global trivialization of the address bijections is\nassumed.\n\\end{definition}","profiled-lane-system}.  For every\narrow \\(\\gamma:U\\to V\\), there is an address bijection\n\\[\n \\lambda^A_{\\gamma,r}:I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}.\n\\]\nNo global address set and no global trivialization of the address bijections is\nassumed.\n\\end{definition}","I_{A,U,r}\\xrightarrow{\\sim}I_{A,V,r}\n\\]\nand, for every address \\(i\\), a splitting comparison\n\\[\n \\beta^A_{\\gamma,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}.\n\\]\nNo global address set and no global trivialization of the address bijections is\nassumed.\n\\end{definition}","a,r,i}:\n F_\\gamma(A^r_{U,i})\\xrightarrow{\\sim}\n A^r_{V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}.\n\\]\nNo global address set and no global trivialization of the address bijections is\nassumed.\n\\end{definition}","V,\\lambda^A_{\\gamma,r}(i)}.\n\\]\nThey satisfy the address and splitting cocycle laws and the two compatibility\nidentities\n\\[\n \\theta^A_\\gamma F_\\gamma(\\iota^r_{A,U,i})\n =\n \\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\n \\beta^A_{\\gamma,r,i},\n\\]\n\\[\n \\beta^A_{\\gamma,r,i}F_\\gamma(\\pi^r_{A,U,i})\n =\n \\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}\\theta^A_\\gamma.\n\\]\nFor \\(s\\succeq r\\), the refinement maps are equivariant:\n\\[\n \\rho^{A,V}_{sr}\\lambda^A_{\\gamma,s}\n =\n \\lambda^A_{\\gamma,r}\\rho^{A,U}_{sr}.\n\\]\nNo global address set and no global trivialization of the address bijections is\nassumed.\n\\end{definition}","\\begin{definition}[Matrix local system]\n\\label{def:block16-matrix-local-system}\nLet \\(X,Y\\) carry holonomy-profiled lane data.  At source resolution \\(r\\)\nand target resolution \\(u\\), define the matrix group over a chart \\(U\\) by\n\\[\n \\mathscr M^{u,r}_{Y,X}(U)\n :=\n \\bigoplus_{a\\in I_{Y,U,u}}\n \\bigoplus_{i\\in I_{X,U,r}}\n \\Hom_{\\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).\n\\]\nFor \\(h=(h_{ai})\\in\\mathscr M^{u,r}_{Y,X}(U)\\), define transport along\n\\(\\gamma:U\\to V\\) by\n\\[\n \\left(\\gamma_*h\\right)_{\n \\lambda^Y_{\\gamma,u}(a),\\lambda^X_{\\gamma,r}(i)}\n :=\n \\beta^Y_{\\gamma,u,a}\n F_\\gamma(h_{ai})\n \\left(\\beta^X_{\\gamma,r,i}\\right)^{-1}.\n\\]\nThe group of compatible sections is\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n :=\n \\left\\{(h_U)_U:\\gamma_*h_U=h_V\n \\text{ for every }\\gamma:U\\to V\\","\\begin{definition}[Matrix local system]\n\\label{def:block16-matrix-local-system}\nLet \\(X,Y\\) carry holonomy-profiled lane data.  At source resolution \\(r\\)\nand target resolution \\(u\\), define the matrix group over a chart \\(U\\) by\n\\[\n \\mathscr M^{u,r}_{Y,X}(U)\n :=\n \\bigoplus_{a\\in I_{Y,U,u}}\n \\bigoplus_{i\\in I_{X,U,r}}\n \\Hom_{\\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).\n\\]\nFor \\(h=(h_{ai})\\in\\mathscr M^{u,r}_{Y,X}(U)\\), define transport along\n\\(\\gamma:U\\to V\\) by\n\\[\n \\left(\\gamma_*h\\right)_{\n \\lambda^Y_{\\gamma,u}(a),\\lambda^X_{\\gamma,r}(i)}\n :=\n \\beta^Y_{\\gamma,u,a}\n F_\\gamma(h_{ai})\n \\left(\\beta^X_{\\gamma,r,i}\\right)^{-1}.\n\\]\nThe group of compatible sections is\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n :=\n \\left\\{(h_U)_U:\\gamma_*h_U=h_V\n \\text{ for every }\\gamma:U\\to V\\right\\}.\n\\]\n\\end{definition}","\\label{def:block16-matrix-local-system}\nLet \\(X,Y\\) carry holonomy-profiled lane data.  At source resolution \\(r\\)\nand target resolution \\(u\\), define the matrix group over a chart \\(U\\) by\n\\[\n \\mathscr M^{u,r}_{Y,X}(U)\n :=\n \\bigoplus_{a\\in I_{Y,U,u}}\n \\bigoplus_{i\\in I_{X,U,r}}\n \\Hom_{\\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).\n\\]\nFor \\(h=(h_{ai})\\in\\mathscr M^{u,r}_{Y,X}(U)\\), define transport along\n\\(\\gamma:U\\to V\\) by\n\\[\n \\left(\\gamma_*h\\right)_{\n \\lambda^Y_{\\gamma,u}(a),\\lambda^X_{\\gamma,r}(i)}\n :=\n \\beta^Y_{\\gamma,u,a}\n F_\\gamma(h_{ai})\n \\left(\\beta^X_{\\gamma,r,i}\\right)^{-1}.\n\\]\nThe group of compatible sections is\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n :=\n \\left\\{(h_U)_U:\\gamma_*h_U=h_V\n \\text{ for every }\\gamma:U\\to V\\right\\}.\n\\]\n\\end{definition}","atrix-local-system}\nLet \\(X,Y\\) carry holonomy-profiled lane data.  At source resolution \\(r\\)\nand target resolution \\(u\\), define the matrix group over a chart \\(U\\) by\n\\[\n \\mathscr M^{u,r}_{Y,X}(U)\n :=\n \\bigoplus_{a\\in I_{Y,U,u}}\n \\bigoplus_{i\\in I_{X,U,r}}\n \\Hom_{\\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).\n\\]\nFor \\(h=(h_{ai})\\in\\mathscr M^{u,r}_{Y,X}(U)\\), define transport along\n\\(\\gamma:U\\to V\\) by\n\\[\n \\left(\\gamma_*h\\right)_{\n \\lambda^Y_{\\gamma,u}(a),\\lambda^X_{\\gamma,r}(i)}\n :=\n \\beta^Y_{\\gamma,u,a}\n F_\\gamma(h_{ai})\n \\left(\\beta^X_{\\gamma,r,i}\\right)^{-1}.\n\\]\nThe group of compatible sections is\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n :=\n \\left\\{(h_U)_U:\\gamma_*h_U=h_V\n \\text{ for every }\\gamma:U\\to V\\right\\}.\n\\]\n\\end{definition}","\\bigoplus_{a\\in I_{Y,U,u}}\n \\bigoplus_{i\\in I_{X,U,r}}\n \\Hom_{\\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).\n\\]\nFor \\(h=(h_{ai})\\in\\mathscr M^{u,r}_{Y,X}(U)\\), define transport along\n\\(\\gamma:U\\to V\\) by\n\\[\n \\left(\\gamma_*h\\right)_{\n \\lambda^Y_{\\gamma,u}(a),\\lambda^X_{\\gamma,r}(i)}\n :=\n \\beta^Y_{\\gamma,u,a}\n F_\\gamma(h_{ai})\n \\left(\\beta^X_{\\gamma,r,i}\\right)^{-1}.\n\\]\nThe group of compatible sections is\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n :=\n \\left\\{(h_U)_U:\\gamma_*h_U=h_V\n \\text{ for every }\\gamma:U\\to V\\right\\}.\n\\]\n\\end{definition}","\\begin{theorem}[Holonomy-Twisted Universal Lane Transport Theorem]\n\\label{thm:block16-holonomy-twisted-lane-transport}\nLet \\(X,Y,Z\\) be descent objects over \\(\\mathscr G\\), each equipped with a\nholonomy-profiled finite lane datum over the same finite resolution shape.\nThen the following statements hold.","\\begin{enumerate}[label=\\textup{(H\\arabic*)}]\n\\item \\textbf{Matrix local system.}\nThe maps \\(\\gamma_*\\) of\n\\cref{def:block16-matrix-local-system} are additive isomorphisms and satisfy\nthe identity and cocycle laws.  Hence\n\\(\\mathscr M^{u,r}_{Y,X}\\) is a local system of abelian groups with finitely many matrix positions in every chart.","\\item \\textbf{Exact twisted analysis and synthesis.}\nLane analysis induces an isomorphism\n\\[\n \\mathfrak A^{u,r}_{Y,X}:\n \\Hom_{\\Desc(\\mathscr G)}(X,Y)\n \\xrightarrow{\\sim}\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right),\n \\qquad\n f\\longmapsto\n \\left(\\bigl([f_U]^{u,r}_{ai}\\bigr)_{a,i}\\right)_U.\n\\]\nIts inverse is chartwise lane synthesis:\n\\[\n (h_U)_U\\longmapsto\n \\left(\n \\sum_{a,i}\n \\iota^u_{Y,U,a}h_{U,ai}\\pi^r_{X,U,i}\n \\right)_U.\n\\]\nThus nontrivial address holonomy does not destroy the complete matrix record.\nIt changes the global type of that record from a constantly labeled matrix to\na compatible section of a matrix local system.","\\item \\textbf{Twisted matrix composition.}\nFor\n\\[\n h\\in\\Gamma(\\mathscr G,\\mathscr M^{u,r}_{Y,X}),\n \\qquad\n k\\in\\Gamma(\\mathscr G,\\mathscr M^{v,u}_{Z,Y}),\n\\]\ndefine chartwise contraction by\n\\[\n (k\\star h)_{U,ci}\n :=\n \\sum_{a\\in I_{Y,U,u}}k_{U,ca}h_{U,ai}.\n\\]\nThen \\(k\\star h\\) is a compatible section of\n\\(\\mathscr M^{v,r}_{Z,X}\\), and\n\\[\n \\mathfrak A^{v,r}_{Z,X}(gf)\n =\n \\mathfrak A^{v,u}_{Z,Y}(g)\n \\star\n \\mathfrak A^{u,r}_{Y,X}(f).\n\\]\nThe diagonal identity matrices are compatible sections.  More precisely, fix\none declared resolution \\(r_A\\) for each lane-decorated descent object \\(A\\).\nThe category with the same objects, morphism groups\n\\[\n \\Gamma\\!\\left(\\mathscr G,\n \\mathscr M^{r_B,r_A}_{B,A}\\right),\n\\]\nand composition \\(\\star\\) is isomorphic, by analysis and synthesis, to the\ncorresponding category of lane-decorated descent objects and their","\\item \\textbf{Twisted matrix composition.}\nFor\n\\[\n h\\in\\Gamma(\\mathscr G,\\mathscr M^{u,r}_{Y,X}),\n \\qquad\n k\\in\\Gamma(\\mathscr G,\\mathscr M^{v,u}_{Z,Y}),\n\\]\ndefine chartwise contraction by\n\\[\n (k\\star h)_{U,ci}\n :=\n \\sum_{a\\in I_{Y,U,u}}k_{U,ca}h_{U,ai}.\n\\]\nThen \\(k\\star h\\) is a compatible section of\n\\(\\mathscr M^{v,r}_{Z,X}\\), and\n\\[\n \\mathfrak A^{v,r}_{Z,X}(gf)\n =\n \\mathfrak A^{v,u}_{Z,Y}(g)\n \\star\n \\mathfrak A^{u,r}_{Y,X}(f).\n\\]\nThe diagonal identity matrices are compatible sections.  More precisely, fix\none declared resolution \\(r_A\\) for each lane-decorated descent object \\(A\\).\nThe category with the same objects, morphism groups\n\\[\n \\Gamma\\!\\left(\\mathscr G,\n \\mathscr M^{r_B,r_A}_{B,A}\\right),\n\\]\nand composition \\(\\star\\) is isomorphic, by analysis and synthesis, to the\ncorresponding category of lane-decorated descent objects and their underlying\ndescended morphisms.","{v,u}_{Z,Y}),\n\\]\ndefine chartwise contraction by\n\\[\n (k\\star h)_{U,ci}\n :=\n \\sum_{a\\in I_{Y,U,u}}k_{U,ca}h_{U,ai}.\n\\]\nThen \\(k\\star h\\) is a compatible section of\n\\(\\mathscr M^{v,r}_{Z,X}\\), and\n\\[\n \\mathfrak A^{v,r}_{Z,X}(gf)\n =\n \\mathfrak A^{v,u}_{Z,Y}(g)\n \\star\n \\mathfrak A^{u,r}_{Y,X}(f).\n\\]\nThe diagonal identity matrices are compatible sections.  More precisely, fix\none declared resolution \\(r_A\\) for each lane-decorated descent object \\(A\\).\nThe category with the same objects, morphism groups\n\\[\n \\Gamma\\!\\left(\\mathscr G,\n \\mathscr M^{r_B,r_A}_{B,A}\\right),\n\\]\nand composition \\(\\star\\) is isomorphic, by analysis and synthesis, to the\ncorresponding category of lane-decorated descent objects and their underlying\ndescended morphisms.","y\n\\[\n (k\\star h)_{U,ci}\n :=\n \\sum_{a\\in I_{Y,U,u}}k_{U,ca}h_{U,ai}.\n\\]\nThen \\(k\\star h\\) is a compatible section of\n\\(\\mathscr M^{v,r}_{Z,X}\\), and\n\\[\n \\mathfrak A^{v,r}_{Z,X}(gf)\n =\n \\mathfrak A^{v,u}_{Z,Y}(g)\n \\star\n \\mathfrak A^{u,r}_{Y,X}(f).\n\\]\nThe diagonal identity matrices are compatible sections.  More precisely, fix\none declared resolution \\(r_A\\) for each lane-decorated descent object \\(A\\).\nThe category with the same objects, morphism groups\n\\[\n \\Gamma\\!\\left(\\mathscr G,\n \\mathscr M^{r_B,r_A}_{B,A}\\right),\n\\]\nand composition \\(\\star\\) is isomorphic, by analysis and synthesis, to the\ncorresponding category of lane-decorated descent objects and their underlying\ndescended morphisms.","_{U,ai}.\n\\]\nThen \\(k\\star h\\) is a compatible section of\n\\(\\mathscr M^{v,r}_{Z,X}\\), and\n\\[\n \\mathfrak A^{v,r}_{Z,X}(gf)\n =\n \\mathfrak A^{v,u}_{Z,Y}(g)\n \\star\n \\mathfrak A^{u,r}_{Y,X}(f).\n\\]\nThe diagonal identity matrices are compatible sections.  More precisely, fix\none declared resolution \\(r_A\\) for each lane-decorated descent object \\(A\\).\nThe category with the same objects, morphism groups\n\\[\n \\Gamma\\!\\left(\\mathscr G,\n \\mathscr M^{r_B,r_A}_{B,A}\\right),\n\\]\nand composition \\(\\star\\) is isomorphic, by analysis and synthesis, to the\ncorresponding category of lane-decorated descent objects and their underlying\ndescended morphisms.","and\n\\[\n \\mathfrak A^{v,r}_{Z,X}(gf)\n =\n \\mathfrak A^{v,u}_{Z,Y}(g)\n \\star\n \\mathfrak A^{u,r}_{Y,X}(f).\n\\]\nThe diagonal identity matrices are compatible sections.  More precisely, fix\none declared resolution \\(r_A\\) for each lane-decorated descent object \\(A\\).\nThe category with the same objects, morphism groups\n\\[\n \\Gamma\\!\\left(\\mathscr G,\n \\mathscr M^{r_B,r_A}_{B,A}\\right),\n\\]\nand composition \\(\\star\\) is isomorphic, by analysis and synthesis, to the\ncorresponding category of lane-decorated descent objects and their underlying\ndescended morphisms.","\\item \\textbf{Equivariant refinement.}\nEvery certified fine-to-coarse coordinate change from\n\\cref{prop:block15-certified-coarsening} commutes with the transition maps\n\\(\\gamma_*\\).  It therefore defines an isomorphism of matrix local systems.\nThese isomorphisms compose strictly over chains of declared resolutions and\nintertwine twisted matrix contraction.","\\item \\textbf{Holonomy-compatible selectors and residuals.}\nLet\n\\[\n E_U\\subseteq I_{Y,U,u}\\times I_{X,U,r}\n\\]\nbe a family satisfying\n\\[\n E_V=\n (\\lambda^Y_{\\gamma,u}\\times\\lambda^X_{\\gamma,r})(E_U)\n\\]\nfor every arrow \\(\\gamma:U\\to V\\).  Then the local selectors\n\\[\n C_{E_U}(h_U)_{ai}\n =\n \\begin{cases}\n h_{U,ai},&(a,i)\\in E_U,","\\item \\textbf{Holonomy-compatible selectors and residuals.}\nLet\n\\[\n E_U\\subseteq I_{Y,U,u}\\times I_{X,U,r}\n\\]\nbe a family satisfying\n\\[\n E_V=\n (\\lambda^Y_{\\gamma,u}\\times\\lambda^X_{\\gamma,r})(E_U)\n\\]\nfor every arrow \\(\\gamma:U\\to V\\).  Then the local selectors\n\\[\n C_{E_U}(h_U)_{ai}\n =\n \\begin{cases}\n h_{U,ai},&(a,i)\\in E_U,\\\\\n 0,&(a,i)\\notin E_U\n \\end{cases}\n\\]\nassemble to an idempotent endomorphism \\(C_E\\) of the section group.  For two\ncompatible support families \\(E,F\\),\n\\[\n C_EC_F=C_{E\\cap F},\n \\qquad\n R_E:=1-C_E,\n \\qquad\n h=C_Eh+R_Eh.\n\\]\nHence named residuals remain exact under arbitrary finite address holonomy,\nprovided the retained support itself is holonomy compatible.\n\\end{enumerate}\n\\end{theorem}","0,&(a,i)\\notin E_U\n \\end{cases}\n\\]\nassemble to an idempotent endomorphism \\(C_E\\) of the section group.  For two\ncompatible support families \\(E,F\\),\n\\[\n C_EC_F=C_{E\\cap F},\n \\qquad\n R_E:=1-C_E,\n \\qquad\n h=C_Eh+R_Eh.\n\\]\nHence named residuals remain exact under arbitrary finite address holonomy,\nprovided the retained support itself is holonomy compatible.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFor (H1), additivity and invertibility follow from the corresponding\nproperties of \\(F_\\gamma\\) and the splitting comparisons \\(\\beta\\).  The\naddress and splitting cocycle laws give\n\\((\\delta\\gamma)_*=\\delta_*\\gamma_*\\) after the declared transition\ncomparisons.","For (H2), let \\(f=(f_U)_U\\) be descended.  For a transition\n\\(\\gamma:U\\to V\\), the transported \\((a,i)\\)-block is\n\\[\n \\begin{aligned}\n &\\beta^Y_{\\gamma,u,a}\n F_\\gamma\\!\\left(\n \\pi^u_{Y,U,a}f_U\\iota^r_{X,U,i}\n \\right)\n \\left(\\beta^X_{\\gamma,r,i}\\right)^{-1}","For (H2), let \\(f=(f_U)_U\\) be descended.  For a transition\n\\(\\gamma:U\\to V\\), the transported \\((a,i)\\)-block is\n\\[\n \\begin{aligned}\n &\\beta^Y_{\\gamma,u,a}\n F_\\gamma\\!\\left(\n \\pi^u_{Y,U,a}f_U\\iota^r_{X,U,i}\n \\right)\n \\left(\\beta^X_{\\gamma,r,i}\\right)^{-1}\\\\\n &\\quad=\n \\pi^u_{Y,V,\\lambda^Y(a)}\n \\theta^Y_\\gamma F_\\gamma(f_U)(\\theta^X_\\gamma)^{-1}\n \\iota^r_{X,V,\\lambda^X(i)}\\\\\n &\\quad=\n \\pi^u_{Y,V,\\lambda^Y(a)}\n f_V\n \\iota^r_{X,V,\\lambda^X(i)},\n \\end{aligned}\n\\]\nwhich is the corresponding block of \\(f_V\\).  Thus analysis produces a\ncompatible section.","&\\quad=\n \\pi^u_{Y,V,\\lambda^Y(a)}\n f_V\n \\iota^r_{X,V,\\lambda^X(i)},\n \\end{aligned}\n\\]\nwhich is the corresponding block of \\(f_V\\).  Thus analysis produces a\ncompatible section.","Conversely, let \\(h=(h_U)_U\\) be a compatible section and synthesize\n\\[\n f_U:=\\sum_{a,i}\\iota^u_{Y,U,a}h_{U,ai}\\pi^r_{X,U,i}.\n\\]\nTransporting this sum, applying the splitting comparison identities, and\nreindexing through the address bijections gives \\(f_V\\).  Thus \\((f_U)_U\\) is\na descended morphism.  Chartwise analysis and synthesis are mutually inverse\nby \\cref{thm:block15-universal-lane-transport}; therefore their restrictions\nto compatible families are mutually inverse.","For (H3), the local path law gives the displayed contraction formula in every\nchart.  Applying \\(\\gamma_*\\) to a contracted block distributes over the finite\nsum and carries each product to the product of the transported blocks.  Since\n\\(\\lambda^Y_{\\gamma,u}\\) is a bijection, reindexing the intermediate address\nshows\n\\[\n \\gamma_*(k_U\\star h_U)\n =\n (\\gamma_*k_U)\\star(\\gamma_*h_U)\n =\n k_V\\star h_V.\n\\]\nThe identity section is compatible by the splitting cocycle.  The equivalence\nthen follows from (H2).","For (H5), support compatibility implies\n\\(\\gamma_*C_{E_U}=C_{E_V}\\gamma_*\\), so the selectors preserve compatible\nsections.  Every Boolean and residual identity is then checked chartwise.\n\\end{proof}","\\begin{corollary}[Base-fiber invariant normal form]\n\\label{cor:block16-base-fiber-invariant-normal-form}\nFix a base chart \\(U_0\\).  Evaluation at \\(U_0\\) induces an isomorphism\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n \\xrightarrow{\\sim}\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)},\n\\]\nwhere the right-hand side is the subgroup fixed by every loop-transport map.\nConsequently,\n\\[\n \\Hom_{\\Desc(\\mathscr G)}(X,Y)\n \\cong\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)}.\n\\]\nAfter choosing one arrow \\(\\tau_U:U_0\\to U\\) for each chart, the inverse sends\na fixed base matrix \\(h_0\\) to the section\n\\(h_U=(\\tau_U)_*h_0\\).  The result is independent of the chosen arrows because\ntwo choices differ by loop transport at \\(U_0\\).\n\\end{corollary}","corollary}[Base-fiber invariant normal form]\n\\label{cor:block16-base-fiber-invariant-normal-form}\nFix a base chart \\(U_0\\).  Evaluation at \\(U_0\\) induces an isomorphism\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n \\xrightarrow{\\sim}\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)},\n\\]\nwhere the right-hand side is the subgroup fixed by every loop-transport map.\nConsequently,\n\\[\n \\Hom_{\\Desc(\\mathscr G)}(X,Y)\n \\cong\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)}.\n\\]\nAfter choosing one arrow \\(\\tau_U:U_0\\to U\\) for each chart, the inverse sends\na fixed base matrix \\(h_0\\) to the section\n\\(h_U=(\\tau_U)_*h_0\\).  The result is independent of the chosen arrows because\ntwo choices differ by loop transport at \\(U_0\\).\n\\end{corollary}","Evaluation at \\(U_0\\) induces an isomorphism\n\\[\n \\Gamma\\!\\left(\\mathscr G,\\mathscr M^{u,r}_{Y,X}\\right)\n \\xrightarrow{\\sim}\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)},\n\\]\nwhere the right-hand side is the subgroup fixed by every loop-transport map.\nConsequently,\n\\[\n \\Hom_{\\Desc(\\mathscr G)}(X,Y)\n \\cong\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)}.\n\\]\nAfter choosing one arrow \\(\\tau_U:U_0\\to U\\) for each chart, the inverse sends\na fixed base matrix \\(h_0\\) to the section\n\\(h_U=(\\tau_U)_*h_0\\).  The result is independent of the chosen arrows because\ntwo choices differ by loop transport at \\(U_0\\).\n\\end{corollary}",",\\mathscr M^{u,r}_{Y,X}\\right)\n \\xrightarrow{\\sim}\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)},\n\\]\nwhere the right-hand side is the subgroup fixed by every loop-transport map.\nConsequently,\n\\[\n \\Hom_{\\Desc(\\mathscr G)}(X,Y)\n \\cong\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)}.\n\\]\nAfter choosing one arrow \\(\\tau_U:U_0\\to U\\) for each chart, the inverse sends\na fixed base matrix \\(h_0\\) to the section\n\\(h_U=(\\tau_U)_*h_0\\).  The result is independent of the chosen arrows because\ntwo choices differ by loop transport at \\(U_0\\).\n\\end{corollary}","\\right)\n \\xrightarrow{\\sim}\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)},\n\\]\nwhere the right-hand side is the subgroup fixed by every loop-transport map.\nConsequently,\n\\[\n \\Hom_{\\Desc(\\mathscr G)}(X,Y)\n \\cong\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)}.\n\\]\nAfter choosing one arrow \\(\\tau_U:U_0\\to U\\) for each chart, the inverse sends\na fixed base matrix \\(h_0\\) to the section\n\\(h_U=(\\tau_U)_*h_0\\).  The result is independent of the chosen arrows because\ntwo choices differ by loop transport at \\(U_0\\).\n\\end{corollary}","subgroup fixed by every loop-transport map.\nConsequently,\n\\[\n \\Hom_{\\Desc(\\mathscr G)}(X,Y)\n \\cong\n \\mathscr M^{u,r}_{Y,X}(U_0)^{\\Aut_{\\mathscr G}(U_0)}.\n\\]\nAfter choosing one arrow \\(\\tau_U:U_0\\to U\\) for each chart, the inverse sends\na fixed base matrix \\(h_0\\) to the section\n\\(h_U=(\\tau_U)_*h_0\\).  The result is independent of the chosen arrows because\ntwo choices differ by loop transport at \\(U_0\\).\n\\end{corollary}","\\begin{proof}\nA compatible section evaluates to a loop-fixed element at \\(U_0\\).  Conversely,\nfor an arrow \\(\\gamma:U\\to V\\), the two arrows\n\\(\\gamma\\tau_U\\) and \\(\\tau_V\\) differ by a loop at \\(U_0\\); loop invariance\ntherefore gives \\(\\gamma_*h_U=h_V\\).  Evaluation and this reconstruction are\nmutually inverse.  Compose with the analysis isomorphism in (H2).\n\\end{proof}","\\begin{remark}[What has and has not been removed]\n\\label{rem:block16-removed-triviality-only}\nThe theorem removes only the global address trivializations\n\\(\\mu_U^{A,r}\\) from \\cref{thm:block15-lane-descent}.  It does not remove the\naddress cocycle, refinement naturality, projector compatibility, splitting\ncomparisons, or additive preservation hypotheses.  Without those data, even\nthe matrix transition maps are undefined or fail their cocycle law.\n\\end{remark}","Fix a base chart \\(U_0\\).  Every loop \\(\\gamma\\in\\Aut_{\\mathscr G}(U_0)\\)\nacts on an address set by \\(i\\mapsto\\lambda^A_{\\gamma,r}(i)\\).  Write\n\\[\n H_A^r\n :=\n \\operatorname{im}\\!\\left(\n \\Aut_{\\mathscr G}(U_0)\n \\longrightarrow\n \\operatorname{Sym}(I_{A,U_0,r})\n \\right)\n\\]\nfor the resulting finite monodromy group.  For a source-target pair, the same\nloop acts diagonally by\n\\[\n (a,i)\\longmapsto\n \\left(\\lambda^Y_{\\gamma,u}(a),\n \\lambda^X_{\\gamma,r}(i)\\right).\n\\]\nIts image in the symmetric group of\n\\(I_{Y,U_0,u}\\times I_{X,U_0,r}\\) is denoted\n\\(H_{Y,X}^{u,r}\\).  This is the simultaneous monodromy image; it need not be\nthe direct product of the two separate monodromy groups.","\\(i\\mapsto\\lambda^A_{\\gamma,r}(i)\\).  Write\n\\[\n H_A^r\n :=\n \\operatorname{im}\\!\\left(\n \\Aut_{\\mathscr G}(U_0)\n \\longrightarrow\n \\operatorname{Sym}(I_{A,U_0,r})\n \\right)\n\\]\nfor the resulting finite monodromy group.  For a source-target pair, the same\nloop acts diagonally by\n\\[\n (a,i)\\longmapsto\n \\left(\\lambda^Y_{\\gamma,u}(a),\n \\lambda^X_{\\gamma,r}(i)\\right).\n\\]\nIts image in the symmetric group of\n\\(I_{Y,U_0,u}\\times I_{X,U_0,r}\\) is denoted\n\\(H_{Y,X}^{u,r}\\).  This is the simultaneous monodromy image; it need not be\nthe direct product of the two separate monodromy groups.","[\n H_A^r\n :=\n \\operatorname{im}\\!\\left(\n \\Aut_{\\mathscr G}(U_0)\n \\longrightarrow\n \\operatorname{Sym}(I_{A,U_0,r})\n \\right)\n\\]\nfor the resulting finite monodromy group.  For a source-target pair, the same\nloop acts diagonally by\n\\[\n (a,i)\\longmapsto\n \\left(\\lambda^Y_{\\gamma,u}(a),\n \\lambda^X_{\\gamma,r}(i)\\right).\n\\]\nIts image in the symmetric group of\n\\(I_{Y,U_0,u}\\times I_{X,U_0,r}\\) is denoted\n\\(H_{Y,X}^{u,r}\\).  This is the simultaneous monodromy image; it need not be\nthe direct product of the two separate monodromy groups.","\\begin{enumerate}[label=\\textup{(O\\arabic*)}]\n\\item For every orbit \\(\\mathcal O\\subseteq I_{A,U_0,r}\\) of \\(H_A^r\\), path\ntransport defines a path-independent orbit subset\n\\(\\mathcal O_U\\subseteq I_{A,U,r}\\) in every chart.  The local projector sums\n\\[\n P^r_{A,U,\\mathcal O}\n :=\n \\sum_{i\\in\\mathcal O_U}p^r_{A,U,i}\n\\]\ndescend to a split idempotent on \\(A\\).  As \\(\\mathcal O\\) ranges over the\nmonodromy orbits, these projectors are pairwise annihilating and sum to\n\\(1_A\\).","\\(\\mathcal O\\subseteq I_{A,U_0,r}\\) of \\(H_A^r\\), path\ntransport defines a path-independent orbit subset\n\\(\\mathcal O_U\\subseteq I_{A,U,r}\\) in every chart.  The local projector sums\n\\[\n P^r_{A,U,\\mathcal O}\n :=\n \\sum_{i\\in\\mathcal O_U}p^r_{A,U,i}\n\\]\ndescend to a split idempotent on \\(A\\).  As \\(\\mathcal O\\) ranges over the\nmonodromy orbits, these projectors are pairwise annihilating and sum to\n\\(1_A\\).","\\item The orbit projectors form the finest ordinary globally labeled\ncoarsening whose local projectors are sums of the original lane projectors.\nMore precisely, any globally labeled projector of the form\n\\[\n Q_U=\\sum_{i\\in S_U}p^r_{A,U,i}\n\\]\nwith \\(S_V=\\lambda^A_{\\gamma,r}(S_U)\\) and with the same global label preserved\naround every loop has \\(S_{U_0}\\) equal to a union of \\(H_A^r\\)-orbits.","\\item A particular local lane descends as an individually labeled global lane\nexactly when its address is fixed by \\(H_A^r\\).  All original lane labels\ndescend simultaneously exactly when the monodromy action on the address set is\ntrivial.","\\item Holonomy-compatible support families in (H5) are in bijection with\nunions of \\(H_{Y,X}^{u,r}\\)-orbits in the base pair-address set.  The minimal\nnonempty compatible supports are therefore the single pair orbits.  Their\nassociated selectors need not act nontrivially on the section group: a\nsingle-orbit selector is nonzero exactly when that orbit supports a nonzero\ncompatible section.","\\item If \\(s\\succeq r\\), equivariance of the refinement map sends every fine\nmonodromy orbit onto a coarse monodromy orbit.  For a coarse orbit\n\\(\\mathcal O\\),\n\\[\n P^r_{A,\\mathcal O}\n =\n \\sum_{\\substack{\\mathcal Q\\text{ a fine orbit}","\\item If \\(s\\succeq r\\), equivariance of the refinement map sends every fine\nmonodromy orbit onto a coarse monodromy orbit.  For a coarse orbit\n\\(\\mathcal O\\),\n\\[\n P^r_{A,\\mathcal O}\n =\n \\sum_{\\substack{\\mathcal Q\\text{ a fine orbit}\\\\\n \\mathcal Q\\subseteq(\\rho^A_{sr})^{-1}(\\mathcal O)}}\n P^s_{A,\\mathcal Q}.\n\\]\nThus orbit descent and projector-sum refinement commute exactly.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nChoose a path from \\(U_0\\) to \\(U\\) and transport \\(\\mathcal O\\) along its\naddress bijection.  Two path choices differ by a loop at \\(U_0\\), which\npreserves \\(\\mathcal O\\) setwise.  Hence \\(\\mathcal O_U\\) is path independent.\nProjector compatibility gives\n\\[\n \\theta^A_\\gamma F_\\gamma(P^r_{A,U,\\mathcal O})\n =\n P^r_{A,V,\\mathcal O}\\theta^A_\\gamma,\n\\]\nso the orbit sums descend.  Their local splittings are the finite direct sums\n\\(\\bigoplus_{i\\in\\mathcal O_U}A^r_{U,i}\\); the direct sums of the \\(\\beta\\)-maps,\nwith the induced permutation of summands, provide their descent data.  This\nproves (O1).","(U\\) and transport \\(\\mathcal O\\) along its\naddress bijection.  Two path choices differ by a loop at \\(U_0\\), which\npreserves \\(\\mathcal O\\) setwise.  Hence \\(\\mathcal O_U\\) is path independent.\nProjector compatibility gives\n\\[\n \\theta^A_\\gamma F_\\gamma(P^r_{A,U,\\mathcal O})\n =\n P^r_{A,V,\\mathcal O}\\theta^A_\\gamma,\n\\]\nso the orbit sums descend.  Their local splittings are the finite direct sums\n\\(\\bigoplus_{i\\in\\mathcal O_U}A^r_{U,i}\\); the direct sums of the \\(\\beta\\)-maps,\nwith the induced permutation of summands, provide their descent data.  This\nproves (O1).","o path choices differ by a loop at \\(U_0\\), which\npreserves \\(\\mathcal O\\) setwise.  Hence \\(\\mathcal O_U\\) is path independent.\nProjector compatibility gives\n\\[\n \\theta^A_\\gamma F_\\gamma(P^r_{A,U,\\mathcal O})\n =\n P^r_{A,V,\\mathcal O}\\theta^A_\\gamma,\n\\]\nso the orbit sums descend.  Their local splittings are the finite direct sums\n\\(\\bigoplus_{i\\in\\mathcal O_U}A^r_{U,i}\\); the direct sums of the \\(\\beta\\)-maps,\nwith the induced permutation of summands, provide their descent data.  This\nproves (O1).","For (O2), a globally fixed label must return to itself after every loop, so\n\\(S_{U_0}\\) is invariant under \\(H_A^r\\).  Every invariant subset of a finite\ngroup action is a union of orbits.  Therefore every such globally labeled\ncoarsening is coarser than the orbit partition.  Statement (O3) is the\nsingleton case.","Finally, equivariance of \\(\\rho^A_{sr}\\) implies that the image of a fine orbit\nis the orbit of the image of any one of its points.  Hence the preimage of a\ncoarse orbit is a union of fine orbits.  Summing the local projector-refinement\nidentity over all addresses in the coarse orbit gives the formula in (O5).\n\\end{proof}","\\begin{remark}[Object orbits and matrix orbits differ]\n\\label{rem:block16-object-vs-pair-orbits}\nThe orbit lanes on an object are controlled by \\(H_A^r\\backslash I_{A,r}\\).\nTransport selectors are controlled by the generally finer orbit set\n\\[\n H_{Y,X}^{u,r}\\backslash\n \\left(I_{Y,u}\\times I_{X,r}\\right).\n\\]\nEven when an object has only one transitive address orbit, its endomorphism\nmatrices can have several pair orbits.  Holonomy can therefore erase individual\nobject labels while retaining a nontrivial global transport algebra.\n\\end{remark}","The one-chart groupoid with automorphism group \\(H\\) gives a concrete exact\nmodel.  It also identifies the algebra carried by the pair-orbit selectors.","\\begin{theorem}[Permutation-module orbital lane calculus]\n\\label{thm:block16-permutation-orbital-calculus}\nLet \\(\\kk\\) be a commutative ring, let a finite group \\(H\\) act on finite sets\n\\(I,J,K\\), and let \\(\\kk[I],\\kk[J],\\kk[K]\\) be the corresponding permutation\nmodules.  For an orbit\n\\(\\mathcal O\\subseteq J\\times I\\) under the diagonal action, let\n\\(B_{\\mathcal O}:\\kk[I]\\to\\kk[J]\\) be the matrix whose \\((j,i)\\)-entry is one\nwhen \\((j,i)\\in\\mathcal O\\) and zero otherwise.  Then\n\\[\n \\Hom_{\\kk H}(\\kk[I],\\kk[J])\n =\n \\bigoplus_{\\mathcal O\\in H\\backslash(J\\times I)}\n \\kk B_{\\mathcal O}.\n\\]\nIn particular, an equivariant matrix is exactly a matrix constant on every\npair orbit.","heorem}[Permutation-module orbital lane calculus]\n\\label{thm:block16-permutation-orbital-calculus}\nLet \\(\\kk\\) be a commutative ring, let a finite group \\(H\\) act on finite sets\n\\(I,J,K\\), and let \\(\\kk[I],\\kk[J],\\kk[K]\\) be the corresponding permutation\nmodules.  For an orbit\n\\(\\mathcal O\\subseteq J\\times I\\) under the diagonal action, let\n\\(B_{\\mathcal O}:\\kk[I]\\to\\kk[J]\\) be the matrix whose \\((j,i)\\)-entry is one\nwhen \\((j,i)\\in\\mathcal O\\) and zero otherwise.  Then\n\\[\n \\Hom_{\\kk H}(\\kk[I],\\kk[J])\n =\n \\bigoplus_{\\mathcal O\\in H\\backslash(J\\times I)}\n \\kk B_{\\mathcal O}.\n\\]\nIn particular, an equivariant matrix is exactly a matrix constant on every\npair orbit.","rbital lane calculus]\n\\label{thm:block16-permutation-orbital-calculus}\nLet \\(\\kk\\) be a commutative ring, let a finite group \\(H\\) act on finite sets\n\\(I,J,K\\), and let \\(\\kk[I],\\kk[J],\\kk[K]\\) be the corresponding permutation\nmodules.  For an orbit\n\\(\\mathcal O\\subseteq J\\times I\\) under the diagonal action, let\n\\(B_{\\mathcal O}:\\kk[I]\\to\\kk[J]\\) be the matrix whose \\((j,i)\\)-entry is one\nwhen \\((j,i)\\in\\mathcal O\\) and zero otherwise.  Then\n\\[\n \\Hom_{\\kk H}(\\kk[I],\\kk[J])\n =\n \\bigoplus_{\\mathcal O\\in H\\backslash(J\\times I)}\n \\kk B_{\\mathcal O}.\n\\]\nIn particular, an equivariant matrix is exactly a matrix constant on every\npair orbit.","ion-orbital-calculus}\nLet \\(\\kk\\) be a commutative ring, let a finite group \\(H\\) act on finite sets\n\\(I,J,K\\), and let \\(\\kk[I],\\kk[J],\\kk[K]\\) be the corresponding permutation\nmodules.  For an orbit\n\\(\\mathcal O\\subseteq J\\times I\\) under the diagonal action, let\n\\(B_{\\mathcal O}:\\kk[I]\\to\\kk[J]\\) be the matrix whose \\((j,i)\\)-entry is one\nwhen \\((j,i)\\in\\mathcal O\\) and zero otherwise.  Then\n\\[\n \\Hom_{\\kk H}(\\kk[I],\\kk[J])\n =\n \\bigoplus_{\\mathcal O\\in H\\backslash(J\\times I)}\n \\kk B_{\\mathcal O}.\n\\]\nIn particular, an equivariant matrix is exactly a matrix constant on every\npair orbit.","Let\n\\[\n \\mathcal P\\subseteq J\\times I,\n \\qquad\n \\mathcal Q\\subseteq K\\times J,\n \\qquad\n \\mathcal R\\subseteq K\\times I\n\\]\nbe diagonal \\(H\\)-orbits.  For any \\((k,i)\\in\\mathcal R\\), define\n\\[\n c^{\\mathcal R}_{\\mathcal Q,\\mathcal P}\n :=\n \\#\\left\\{\n j\\in J:(k,j)\\in\\mathcal Q,\n (j,i)\\in\\mathcal P\n \\right\\}.\n\\]\nThis integer is independent of the chosen \\((k,i)\\in\\mathcal R\\), and\n\\[\n B_{\\mathcal Q}B_{\\mathcal P}\n =\n \\sum_{\\mathcal R\\in H\\backslash(K\\times I)}\n c^{\\mathcal R}_{\\mathcal Q,\\mathcal P}B_{\\mathcal R}.\n\\]\nThus the lane path law descends to an exact orbit basis with nonnegative\ninteger structure constants.\n\\end{theorem}","\\begin{proof}\nWrite an arbitrary \\(\\kk\\)-linear map as a matrix \\(A=(A_{ji})\\).  The\nintertwining equations for the permutation actions are equivalent to\n\\[\n A_{h j,h i}=A_{j i}\n \\qquad(h\\in H).\n\\]\nHence the entries are constant on diagonal orbits.  The orbit-indicator\nmatrices have disjoint supports, so they are linearly independent over every\ncommutative ring and span all invariant matrices.","The \\((k,i)\\)-entry of\n\\(B_{\\mathcal Q}B_{\\mathcal P}\\) is the displayed cardinality.  If\n\\((k',i')=(hk,hi)\\), then \\(j\\mapsto hj\\) is a bijection between the two sets\nbeing counted.  The entry is therefore constant on \\(\\mathcal R\\), which gives\nthe orbit-basis expansion.\n\\end{proof}","\\begin{corollary}[Orbital endomorphism algebra]\n\\label{cor:block16-orbital-endomorphism-algebra}\nFor one finite \\(H\\)-set \\(I\\), the pair-orbit matrices form a basis of\n\\(\\End_{\\kk H}(\\kk[I])\\).  Matrix multiplication in this basis is exactly the\nfinite compatible-path contraction of\n\\cref{thm:block16-holonomy-twisted-lane-transport}, with the preceding\nintersection numbers as structure constants.\n\\end{corollary}","\\begin{example}[The smallest nontrivial holonomy]\n\\label{ex:block16-c2-swap}\nLet \\(H=C_2=\\{1,\\sigma\\}\\) act on \\(I=\\{1,2\\}\\) by swapping the two addresses.\nOn \\(X=\\kk^2\\), let\n\\[\n S=\n \\begin{pmatrix}0&1","\\begin{example}[The smallest nontrivial holonomy]\n\\label{ex:block16-c2-swap}\nLet \\(H=C_2=\\{1,\\sigma\\}\\) act on \\(I=\\{1,2\\}\\) by swapping the two addresses.\nOn \\(X=\\kk^2\\), let\n\\[\n S=\n \\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\n \\qquad\n p_1=\n \\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\n \\qquad\n p_2=\n \\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}.\n\\]\nThe individual projectors do not descend because\n\\(Sp_1S^{-1}=p_2\\), while their orbit sum \\(p_1+p_2=1\\) does descend.  The\npair-address set has two orbits,\n\\[\n \\Delta=\\{(1,1),(2,2)\\},\n \\qquad\n \\Omega=\\{(1,2),(2,1)\\}.\n\\]\nAccordingly,\n\\[\n \\End_{\\kk C_2}(\\kk^2)\n =\n \\left\\{\n \\begin{pmatrix}a&b","0&1\\end{pmatrix}.\n\\]\nThe individual projectors do not descend because\n\\(Sp_1S^{-1}=p_2\\), while their orbit sum \\(p_1+p_2=1\\) does descend.  The\npair-address set has two orbits,\n\\[\n \\Delta=\\{(1,1),(2,2)\\},\n \\qquad\n \\Omega=\\{(1,2),(2,1)\\}.\n\\]\nAccordingly,\n\\[\n \\End_{\\kk C_2}(\\kk^2)\n =\n \\left\\{\n \\begin{pmatrix}a&b","0&1\\end{pmatrix}.\n\\]\nThe individual projectors do not descend because\n\\(Sp_1S^{-1}=p_2\\), while their orbit sum \\(p_1+p_2=1\\) does descend.  The\npair-address set has two orbits,\n\\[\n \\Delta=\\{(1,1),(2,2)\\},\n \\qquad\n \\Omega=\\{(1,2),(2,1)\\}.\n\\]\nAccordingly,\n\\[\n \\End_{\\kk C_2}(\\kk^2)\n =\n \\left\\{\n \\begin{pmatrix}a&b\\\\b&a\\end{pmatrix}:a,b\\in\\kk\n \\right\\}\n =\n \\kk B_\\Delta\\oplus\\kk B_\\Omega.\n\\]\nThe singleton endpoint selector \\(\\{(1,1)\\}\\) is not holonomy compatible and\nsends the equivariant identity matrix to \\(p_1\\), which is not equivariant.\nThe orbit selectors \\(C_\\Delta\\) and \\(C_\\Omega\\) are compatible and split every\nequivariant endomorphism into its diagonal-orbit and off-diagonal-orbit parts.\n\\end{example}","b&a\\end{pmatrix}:a,b\\in\\kk\n \\right\\}\n =\n \\kk B_\\Delta\\oplus\\kk B_\\Omega.\n\\]\nThe singleton endpoint selector \\(\\{(1,1)\\}\\) is not holonomy compatible and\nsends the equivariant identity matrix to \\(p_1\\), which is not equivariant.\nThe orbit selectors \\(C_\\Delta\\) and \\(C_\\Omega\\) are compatible and split every\nequivariant endomorphism into its diagonal-orbit and off-diagonal-orbit parts.\n\\end{example}","\\begin{proposition}[Rational descent of split cyclic Fourier lanes]\n\\label{prop:block16-rational-cyclotomic-orbit-lanes}\nLet \\(N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =","\\begin{proposition}[Rational descent of split cyclic Fourier lanes]\n\\label{prop:block16-rational-cyclotomic-orbit-lanes}\nLet \\(N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(","\\begin{proposition}[Rational descent of split cyclic Fourier lanes]\n\\label{prop:block16-rational-cyclotomic-orbit-lanes}\nLet \\(N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\el","\\begin{proposition}[Rational descent of split cyclic Fourier lanes]\n\\label{prop:block16-rational-cyclotomic-orbit-lanes}\nLet \\(N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\s","\\begin{proposition}[Rational descent of split cyclic Fourier lanes]\n\\label{prop:block16-rational-cyclotomic-orbit-lanes}\nLet \\(N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","ional descent of split cyclic Fourier lanes]\n\\label{prop:block16-rational-cyclotomic-orbit-lanes}\nLet \\(N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","rier lanes]\n\\label{prop:block16-rational-cyclotomic-orbit-lanes}\nLet \\(N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","N\\ge1\\), let \\(C_N=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","=\\langle g\\rangle\\), and let\n\\(K=\\mathbb Q(\\zeta_N)\\).  In \\(K[C_N]\\), define\n\\[\n e_j\n :=\n \\frac1N\\sum_{\\ell=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","ll=0}^{N-1}\n \\zeta_N^{-j\\ell}g^\\ell,\n \\qquad j\\in\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","\\mathbb Z/N\\mathbb Z.\n\\]\nThese are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","are the primitive split Fourier idempotents.  For\n\\(\\sigma_a\\in\\Gal(K/\\mathbb Q)\\), where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q",", where\n\\(\\sigma_a(\\zeta_N)=\\zeta_N^a\\),\n\\[\n \\sigma_a(e_j)=e_{aj}.\n\\]\nThe Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q","Galois orbits are indexed by divisors \\(q\\mid N\\), with\n\\[\n \\mathcal O_q\n =\n \\left\\{j\\in\\mathbb Z/N\\mathbb Z:\n \\frac{N}{\\gcd(N,j)}=q\\right\\}.\n\\]\nTheir orbit sums\n\\[\n E_q:=\\sum_{j\\in\\mathcal O_q}e_j\n\\]\nbelong to \\(\\mathbb Q[C_N]\\) and satisfy\n\\[\n E_qE_{q'}=\\delta_{q,q'}E_q,\n \\qquad\n \\sum_{q\\mid N}E_q=1.\n\\]\nMore explicitly,\n\\[\n \\boxed{\n E_q\n =\n \\frac1N\\sum_{\\ell=0}^{N-1}c_q(\\ell)g^\\ell,}\n\\]\nwhere\n\\[\n c_q(\\ell)\n :=\n \\sum_{\\substack{1\\le a\\le q\\\\\\gcd(a,q)=1}}\n e^{2\\pi i a\\ell/q}\n =\n \\sum_{d\\mid\\gcd(q,\\ell)}d\\,\\mu(q/d)\n \\in\\mathbb Z\n\\]\nis the Ramanujan sum.  The projectors \\((E_q)_{q\\mid N}\\) form the finest\n\\(\\mathbb Q\\)-rational coarsening obtainable by summing the split Fourier\nidempotents, and\n\\[\n \\mathbb Q[C_N]E_q\\cong\\mathbb Q(\\zeta_q).\n\\]\n\\end{proposition}","gcd(a,q)=1}}\n e^{2\\pi i a\\ell/q}\n =\n \\sum_{d\\mid\\gcd(q,\\ell)}d\\,\\mu(q/d)\n \\in\\mathbb Z\n\\]\nis the Ramanujan sum.  The projectors \\((E_q)_{q\\mid N}\\) form the finest\n\\(\\mathbb Q\\)-rational coarsening obtainable by summing the split Fourier\nidempotents, and\n\\[\n \\mathbb Q[C_N]E_q\\cong\\mathbb Q(\\zeta_q).\n\\]\n\\end{proposition}","\\begin{proof}\nCharacter orthogonality gives the primitive idempotent relations for the\n\\(e_j\\).  Applying \\(\\sigma_a\\) to their coefficients gives\n\\(\\sigma_a(e_j)=e_{aj}\\).  The unit group\n\\((\\mathbb Z/N\\mathbb Z)^\\times\\) acts transitively on elements of\n\\(\\mathbb Z/N\\mathbb Z\\) having the same additive order, so its orbits are the\nsets \\(\\mathcal O_q\\).","Write every \\(j\\in\\mathcal O_q\\) uniquely as\n\\(j=(N/q)a\\) with \\(a\\in(\\mathbb Z/q\\mathbb Z)^\\times\\).  The coefficient of\n\\(g^\\ell\\) in \\(E_q\\) is then\n\\[\n \\frac1N\n \\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}\n \\zeta_q^{-a\\ell}\n =\n \\frac{c_q(\\ell)}{N},\n\\]\nwhere the last equality uses the integrality and evenness of the Ramanujan\nsum.  Hence \\(E_q\\in\\mathbb Q[C_N]\\).  Orthogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod","Write every \\(j\\in\\mathcal O_q\\) uniquely as\n\\(j=(N/q)a\\) with \\(a\\in(\\mathbb Z/q\\mathbb Z)^\\times\\).  The coefficient of\n\\(g^\\ell\\) in \\(E_q\\) is then\n\\[\n \\frac1N\n \\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}\n \\zeta_q^{-a\\ell}\n =\n \\frac{c_q(\\ell)}{N},\n\\]\nwhere the last equality uses the integrality and evenness of the Ramanujan\nsum.  Hence \\(E_q\\in\\mathbb Q[C_N]\\).  Orthogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta","Write every \\(j\\in\\mathcal O_q\\) uniquely as\n\\(j=(N/q)a\\) with \\(a\\in(\\mathbb Z/q\\mathbb Z)^\\times\\).  The coefficient of\n\\(g^\\ell\\) in \\(E_q\\) is then\n\\[\n \\frac1N\n \\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}\n \\zeta_q^{-a\\ell}\n =\n \\frac{c_q(\\ell)}{N},\n\\]\nwhere the last equality uses the integrality and evenness of the Ramanujan\nsum.  Hence \\(E_q\\in\\mathbb Q[C_N]\\).  Orthogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta_q),\n\\]\nand \\(E_q\\) is the identity project","Write every \\(j\\in\\mathcal O_q\\) uniquely as\n\\(j=(N/q)a\\) with \\(a\\in(\\mathbb Z/q\\mathbb Z)^\\times\\).  The coefficient of\n\\(g^\\ell\\) in \\(E_q\\) is then\n\\[\n \\frac1N\n \\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}\n \\zeta_q^{-a\\ell}\n =\n \\frac{c_q(\\ell)}{N},\n\\]\nwhere the last equality uses the integrality and evenness of the Ramanujan\nsum.  Hence \\(E_q\\in\\mathbb Q[C_N]\\).  Orthogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta_q),\n\\]\nand \\(E_q\\) is the identity projector of the \\(q\\)-th factor.\n\\end{","Write every \\(j\\in\\mathcal O_q\\) uniquely as\n\\(j=(N/q)a\\) with \\(a\\in(\\mathbb Z/q\\mathbb Z)^\\times\\).  The coefficient of\n\\(g^\\ell\\) in \\(E_q\\) is then\n\\[\n \\frac1N\n \\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}\n \\zeta_q^{-a\\ell}\n =\n \\frac{c_q(\\ell)}{N},\n\\]\nwhere the last equality uses the integrality and evenness of the Ramanujan\nsum.  Hence \\(E_q\\in\\mathbb Q[C_N]\\).  Orthogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta_q),\n\\]\nand \\(E_q\\) is the identity projector of the \\(q\\)-th factor.\n\\end{proof}","uniquely as\n\\(j=(N/q)a\\) with \\(a\\in(\\mathbb Z/q\\mathbb Z)^\\times\\).  The coefficient of\n\\(g^\\ell\\) in \\(E_q\\) is then\n\\[\n \\frac1N\n \\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}\n \\zeta_q^{-a\\ell}\n =\n \\frac{c_q(\\ell)}{N},\n\\]\nwhere the last equality uses the integrality and evenness of the Ramanujan\nsum.  Hence \\(E_q\\in\\mathbb Q[C_N]\\).  Orthogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta_q),\n\\]\nand \\(E_q\\) is the identity projector of the \\(q\\)-th factor.\n\\end{proof}","=\n \\frac{c_q(\\ell)}{N},\n\\]\nwhere the last equality uses the integrality and evenness of the Ramanujan\nsum.  Hence \\(E_q\\in\\mathbb Q[C_N]\\).  Orthogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta_q),\n\\]\nand \\(E_q\\) is the identity projector of the \\(q\\)-th factor.\n\\end{proof}","hogonality and completeness follow\nby grouping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta_q),\n\\]\nand \\(E_q\\) is the identity projector of the \\(q\\)-th factor.\n\\end{proof}","uping the primitive \\(e_j\\) by their disjoint Galois orbits.  Any rational\nsum of split primitive idempotents must be Galois invariant, so its support is a\nunion of these orbits; this proves fineness.  Finally,\n\\[\n \\mathbb Q[C_N]\n \\cong\n \\mathbb Q[x]/(x^N-1)\n \\cong\n \\prod_{q\\mid N}\\mathbb Q[x]/(\\Phi_q(x))\n \\cong\n \\prod_{q\\mid N}\\mathbb Q(\\zeta_q),\n\\]\nand \\(E_q\\) is the identity projector of the \\(q\\)-th factor.\n\\end{proof}","\\begin{boundary}[Cyclic character lanes are not coefficient-residue lanes]\n\\label{bnd:block16-cyclic-character-residue-distinction}\nThe preceding proposition concerns primitive character idempotents in the\nsplit group algebra \\(K[C_N]\\) and their rational Galois descent.  The\ncoefficient-residue projectors \\(\\mathcal P_{N,j}\\) on formal series from\n\\cref{thm:block15-factorial-residue-realization} are defined coefficientwise\nover every commutative ring and already have fixed residue labels.  A chosen\nrotation representation relates the two constructions after scalar extension,\nbut their descent data are not identical and should not be conflated.  Nothing\nin orbit descent turns the factorial step-product into an additive idempotent\nsplitting.\n\\end{boundary}","The accompanying verifier tests finite permutation actions, pair-orbit bases,\nequivariant matrix composition, orbit-compatible and deliberately incompatible\nselectors, equivariant refinement, and the exact Ramanujan-sum projectors in\n\\(\\mathbb Q[C_N]\\).  It uses exact integer arithmetic after clearing the common cyclotomic\ndenominators.  These tests are\nan implementation regression for the displayed formulas, not a substitute for\nthe proofs and not an external validation record.","\\begin{definition}[Semantic slice]\n\\label{def:block17-semantic-slice}\nA \\emph{semantic slice} is a tuple\n\\[\n \\Sigma=(\\mathcal A,\\mathcal G,\\mathcal V,\\mathcal O)\n\\]\nconsisting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves,","\\begin{definition}[Semantic slice]\n\\label{def:block17-semantic-slice}\nA \\emph{semantic slice} is a tuple\n\\[\n \\Sigma=(\\mathcal A,\\mathcal G,\\mathcal V,\\mathcal O)\n\\]\nconsisting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definit","\\begin{definition}[Semantic slice]\n\\label{def:block17-semantic-slice}\nA \\emph{semantic slice} is a tuple\n\\[\n \\Sigma=(\\mathcal A,\\mathcal G,\\mathcal V,\\mathcal O)\n\\]\nconsisting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","begin{definition}[Semantic slice]\n\\label{def:block17-semantic-slice}\nA \\emph{semantic slice} is a tuple\n\\[\n \\Sigma=(\\mathcal A,\\mathcal G,\\mathcal V,\\mathcal O)\n\\]\nconsisting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","tuple\n\\[\n \\Sigma=(\\mathcal A,\\mathcal G,\\mathcal V,\\mathcal O)\n\\]\nconsisting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","al A,\\mathcal G,\\mathcal V,\\mathcal O)\n\\]\nconsisting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","athcal V,\\mathcal O)\n\\]\nconsisting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","ting of:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","\\item a carrier doctrine \\(\\mathcal A\\), which specifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","pecifies how alternatives, routes, or packets combine;\n\\item a continuation grammar \\(\\mathcal G\\), presented here by a monoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","onoid \\(M\\) or by a chosen generating family of admissible endomorphisms;\n\\item an observation category \\(\\mathcal V\\), such as \\(\\mathbf{Set}\\), \\(R\\text{-}\\mathbf{Mod}\\), or \\(K\\text{-}\\mathbf{Vect}\\);\n\\item a declared family \\(\\mathcal O=(o_j)_{j\\in J}\\) of observations in \\(\\mathcal V\\).\n\\end{enumerate}\nThe slice belongs to the stratum passport.  It does not replace the preservation profile \\(\\Pi\\).  The profile specifies which constructions a bridge preserves, while the slice specifies which future distinctions define the current semantic task.\n\\end{definition}","\\begin{definition}[Resolved observational system]\n\\label{def:block17-observational-system}\nLet \\(X\\) be an object equipped with an action\n\\[\n \\rho:M\\longrightarrow \\End(X)\n\\]\nof a continuation monoid and with observations\n\\[\n o_j:X\\longrightarrow Y_j\\qquad(j\\in J).\n\\]\nThe tuple\n\\[\n \\mathsf S=(X,M,\\rho,(o_j)_{j\\in J})\n\\]\nis a \\emph{resolved observational system}.  When the indicated product exists, its behavior map is\n\\[\n \\Beh_{\\mathsf S}:X\\longrightarrow\n \\prod_{(m,j)\\in M\\times J}Y_j,\n \\qquad\n x\\longmapsto\\bigl(o_j(\\rho(m)x)\\bigr)_{m,j}.\n\\]\nThe \\emph{relative behavioral object} is the coimage\n\\[\n Q_{\\mathsf S}:=\\operatorname{Coim}(\\Beh_{\\mathsf S})\n\\]\nwhenever that coimage exists.\n\\end{definition}","lved observational system]\n\\label{def:block17-observational-system}\nLet \\(X\\) be an object equipped with an action\n\\[\n \\rho:M\\longrightarrow \\End(X)\n\\]\nof a continuation monoid and with observations\n\\[\n o_j:X\\longrightarrow Y_j\\qquad(j\\in J).\n\\]\nThe tuple\n\\[\n \\mathsf S=(X,M,\\rho,(o_j)_{j\\in J})\n\\]\nis a \\emph{resolved observational system}.  When the indicated product exists, its behavior map is\n\\[\n \\Beh_{\\mathsf S}:X\\longrightarrow\n \\prod_{(m,j)\\in M\\times J}Y_j,\n \\qquad\n x\\longmapsto\\bigl(o_j(\\rho(m)x)\\bigr)_{m,j}.\n\\]\nThe \\emph{relative behavioral object} is the coimage\n\\[\n Q_{\\mathsf S}:=\\operatorname{Coim}(\\Beh_{\\mathsf S})\n\\]\nwhenever that coimage exists.\n\\end{definition}","with observations\n\\[\n o_j:X\\longrightarrow Y_j\\qquad(j\\in J).\n\\]\nThe tuple\n\\[\n \\mathsf S=(X,M,\\rho,(o_j)_{j\\in J})\n\\]\nis a \\emph{resolved observational system}.  When the indicated product exists, its behavior map is\n\\[\n \\Beh_{\\mathsf S}:X\\longrightarrow\n \\prod_{(m,j)\\in M\\times J}Y_j,\n \\qquad\n x\\longmapsto\\bigl(o_j(\\rho(m)x)\\bigr)_{m,j}.\n\\]\nThe \\emph{relative behavioral object} is the coimage\n\\[\n Q_{\\mathsf S}:=\\operatorname{Coim}(\\Beh_{\\mathsf S})\n\\]\nwhenever that coimage exists.\n\\end{definition}","In \\(\\mathbf{Set}\\), the behavior map identifies states exactly when every declared future observation agrees.  In a module category, the same object can be constructed directly from the intersection of kernels, without assuming that the possibly infinite product is part of the ambient profile.","\\begin{definition}[Behavioral congruence and linear behavioral kernel]\n\\label{def:block17-behavioral-kernel}\nIn \\(\\mathbf{Set}\\) or a variety of algebras, put\n\\[\n x\\equiv_{\\mathsf S}y\n \\quad\\Longleftrightarrow\\quad\n o_j(\\rho(m)x)=o_j(\\rho(m)y)\n \\quad\\text{for every }m\\in M,\\ j\\in J.\n\\]\nWrite \\(\\Theta_{\\mathsf S}\\) for this relation.  In \\(R\\text{-}\\mathbf{Mod}\\), define\n\\[\n N_{\\mathsf S}\n :=\n \\bigcap_{m\\in M}\\bigcap_{j\\in J}\n \\ker(o_j\\rho(m)).\n\\]\nThen\n\\[\n Q_{\\mathsf S}=X/N_{\\mathsf S}.\n\\]\n\\end{definition}","oral kernel]\n\\label{def:block17-behavioral-kernel}\nIn \\(\\mathbf{Set}\\) or a variety of algebras, put\n\\[\n x\\equiv_{\\mathsf S}y\n \\quad\\Longleftrightarrow\\quad\n o_j(\\rho(m)x)=o_j(\\rho(m)y)\n \\quad\\text{for every }m\\in M,\\ j\\in J.\n\\]\nWrite \\(\\Theta_{\\mathsf S}\\) for this relation.  In \\(R\\text{-}\\mathbf{Mod}\\), define\n\\[\n N_{\\mathsf S}\n :=\n \\bigcap_{m\\in M}\\bigcap_{j\\in J}\n \\ker(o_j\\rho(m)).\n\\]\nThen\n\\[\n Q_{\\mathsf S}=X/N_{\\mathsf S}.\n\\]\n\\end{definition}","\\begin{theorem}[Universal behavioral coimage]\n\\label{thm:block17-universal-behavioral-coimage}\nAssume the relevant effective quotient or module quotient exists.  Then \\(Q_{\\mathsf S}\\) preserves every declared future observation.  Moreover, if\n\\[\n q:X\\twoheadrightarrow Z\n\\]\nis any surjective representation through which every map \\(o_j\\rho(m)\\) factors, there is a unique map\n\\[\n \\overline q:Z\\longrightarrow Q_{\\mathsf S}\n\\]\nwith \\(\\overline q\\,q:X\\to Q_{\\mathsf S}\\) equal to the canonical quotient map.  Thus \\(Q_{\\mathsf S}\\) is the coarsest exact representation for the declared semantic slice.\n\\end{theorem}","l coimage]\n\\label{thm:block17-universal-behavioral-coimage}\nAssume the relevant effective quotient or module quotient exists.  Then \\(Q_{\\mathsf S}\\) preserves every declared future observation.  Moreover, if\n\\[\n q:X\\twoheadrightarrow Z\n\\]\nis any surjective representation through which every map \\(o_j\\rho(m)\\) factors, there is a unique map\n\\[\n \\overline q:Z\\longrightarrow Q_{\\mathsf S}\n\\]\nwith \\(\\overline q\\,q:X\\to Q_{\\mathsf S}\\) equal to the canonical quotient map.  Thus \\(Q_{\\mathsf S}\\) is the coarsest exact representation for the declared semantic slice.\n\\end{theorem}","ffective quotient or module quotient exists.  Then \\(Q_{\\mathsf S}\\) preserves every declared future observation.  Moreover, if\n\\[\n q:X\\twoheadrightarrow Z\n\\]\nis any surjective representation through which every map \\(o_j\\rho(m)\\) factors, there is a unique map\n\\[\n \\overline q:Z\\longrightarrow Q_{\\mathsf S}\n\\]\nwith \\(\\overline q\\,q:X\\to Q_{\\mathsf S}\\) equal to the canonical quotient map.  Thus \\(Q_{\\mathsf S}\\) is the coarsest exact representation for the declared semantic slice.\n\\end{theorem}","\\begin{proof}\nIn the relational setting, the kernel pair of \\(q\\) is contained in \\(\\Theta_{\\mathsf S}\\), because equality under \\(q\\) forces equality under every factorized future observation.  The universal property of the quotient by \\(\\Theta_{\\mathsf S}\\) gives the unique map.  In the module setting,\n\\[\n \\ker q\\subseteq\n \\bigcap_{m,j}\\ker(o_j\\rho(m))=N_{\\mathsf S},\n\\]\nso the ordinary quotient universal property gives the same conclusion.\n\\end{proof}","\\begin{boundary}[No observer-independent minimum]\n\\label{bnd:block17-no-absolute-minimum}\nThe phrase ``minimal representation'' is meaningful only after the carrier, continuation grammar, observation category, and task have been declared.  Changing any one of these data can change the quotient.  In particular, equality of residual functions in \\(\\mathbf{Set}\\) and linear dependence of residual functions in \\(K\\text{-}\\mathbf{Vect}\\) are different minimization problems.\n\\end{boundary}","ary}[No observer-independent minimum]\n\\label{bnd:block17-no-absolute-minimum}\nThe phrase ``minimal representation'' is meaningful only after the carrier, continuation grammar, observation category, and task have been declared.  Changing any one of these data can change the quotient.  In particular, equality of residual functions in \\(\\mathbf{Set}\\) and linear dependence of residual functions in \\(K\\text{-}\\mathbf{Vect}\\) are different minimization problems.\n\\end{boundary}","Let\n\\[\n K_0:=\\bigcap_j\\kerpair(o_j)\n\\]\nin the relational setting and\n\\[\n N_0:=\\bigcap_j\\ker(o_j)\n\\]\nin the linear setting.","\\begin{theorem}[Behavioral core theorem]\n\\label{thm:block17-behavioral-core}\nThe behavioral relation \\(\\Theta_{\\mathsf S}\\) is the greatest continuation-stable congruence contained in \\(K_0\\).  In the module setting, \\(N_{\\mathsf S}\\) is the greatest \\(\\rho(M)\\)-invariant submodule contained in \\(N_0\\).\n\\end{theorem}","\\begin{proof}\nThe identity continuation gives \\(\\Theta_{\\mathsf S}\\subseteq K_0\\) and \\(N_{\\mathsf S}\\subseteq N_0\\).  If \\(x\\equiv_{\\mathsf S}y\\), then for every \\(n,m\\in M\\) and every \\(j\\),\n\\[\n o_j\\rho(m)\\rho(n)x\n =o_j\\rho(mn)x\n =o_j\\rho(mn)y\n =o_j\\rho(m)\\rho(n)y,\n\\]\nso the relation is stable.  The same calculation proves invariance of \\(N_{\\mathsf S}\\).","Conversely, let \\(R\\subseteq K_0\\) be continuation stable.  If \\(xRy\\), then \\(\\rho(m)x\\,R\\,\\rho(m)y\\) for every \\(m\\), and membership in \\(K_0\\) forces\n\\[\n o_j\\rho(m)x=o_j\\rho(m)y\n\\]\nfor every \\(j\\).  Hence \\(R\\subseteq\\Theta_{\\mathsf S}\\).  The submodule statement is identical with relations replaced by differences.\n\\end{proof}","Assume \\(M\\) is generated by a family \\(A\\).  On congruences or submodules define\n\\[\n \\Phi(R)\n :=\n K_0\\cap\\bigcap_{a\\in A}\\rho(a)^{-1}(R),\n\\]\nwith the corresponding submodule formula in the linear setting.","\\begin{theorem}[Coinductive characterization]\n\\label{thm:block17-coinductive-characterization}\nThe behavioral core is the greatest fixed point of \\(\\Phi\\):\n\\[\n \\Theta_{\\mathsf S}=\\nu R.\\Phi(R),\n \\qquad\n N_{\\mathsf S}=\\nu N.\\Phi(N).\n\\]\n\\end{theorem}","\\begin{proof}\nEvery fixed point is invisible and stable under every generator, hence under\nthe generated monoid, so it is contained in the behavioral core by\n\\cref{thm:block17-behavioral-core}.  Conversely, the behavioral core is a fixed\npoint: stability gives its inclusion in its image under \\(\\Phi\\), while\nmembership in \\(\\Phi\\) of that core makes the identity observation invisible\nand makes every first-generator successor behaviorally invisible, hence makes\nevery continuation word invisible.  Its maximality therefore proves the\nasserted greatest-fixed-point identities.\n\\end{proof}","\\begin{corollary}[Finite certification]\n\\label{cor:block17-finite-certification}\nFor finite \\(X\\), descending partition refinement stabilizes after finitely many strict refinements.  For finite-dimensional \\(X\\) over a field, the chain\n\\[\n N^{(0)}=N_0,\n \\qquad\n N^{(r+1)}=\n N_0\\cap\\bigcap_{a\\in A}\\rho(a)^{-1}(N^{(r)})\n\\]\nstabilizes after at most \\(\\dim X\\) strict dimension drops.  At stabilization it equals \\(N_{\\mathsf S}\\).\n\\end{corollary}","lary}[Finite certification]\n\\label{cor:block17-finite-certification}\nFor finite \\(X\\), descending partition refinement stabilizes after finitely many strict refinements.  For finite-dimensional \\(X\\) over a field, the chain\n\\[\n N^{(0)}=N_0,\n \\qquad\n N^{(r+1)}=\n N_0\\cap\\bigcap_{a\\in A}\\rho(a)^{-1}(N^{(r)})\n\\]\nstabilizes after at most \\(\\dim X\\) strict dimension drops.  At stabilization it equals \\(N_{\\mathsf S}\\).\n\\end{corollary}","\\begin{proof}\nThe iteration is descending and every strict inclusion of finite-dimensional subspaces lowers dimension by at least one.  The behavioral core lies in \\(N^{(0)}\\), and monotonicity of \\(\\Phi\\), together\nwith its fixed-point property, preserves this inclusion at every stage.  Thus\nthe stabilized fixed point contains the behavioral core.  Greatest-fixed-point\nmaximality gives the reverse inclusion, proving equality.\n\\end{proof}","Fix a universe \\(\\mathscr Q(X)\\) of admissible queries on \\(X\\).  For \\(\\mathcal Q\\subseteq\\mathscr Q(X)\\), define\n\\[\n \\Ind(\\mathcal Q)\n :=\n \\bigcap_{q\\in\\mathcal Q}\\kerpair(q).\n\\]\nFor an equivalence relation or congruence \\(R\\), define\n\\[\n \\Def(R)\n :=\n \\{q\\in\\mathscr Q(X):R\\subseteq\\kerpair(q)\\}.\n\\]","\\begin{theorem}[Task--congruence Galois connection]\n\\label{thm:block17-task-galois}\nFor every query family \\(\\mathcal Q\\) and relation \\(R\\),\n\\[\n \\mathcal Q\\subseteq\\Def(R)\n \\quad\\Longleftrightarrow\\quad\n R\\subseteq\\Ind(\\mathcal Q).\n\\]\nConsequently,\n\\[\n \\Cl(\\mathcal Q):=\\Def(\\Ind(\\mathcal Q))\n\\]\nis a closure operator on query families, and\n\\[\n \\operatorname{Sat}(R):=\\Ind(\\Def(R))\n\\]\nis the observational saturation of \\(R\\) relative to the admitted query universe.\n\\end{theorem}","ongruence Galois connection]\n\\label{thm:block17-task-galois}\nFor every query family \\(\\mathcal Q\\) and relation \\(R\\),\n\\[\n \\mathcal Q\\subseteq\\Def(R)\n \\quad\\Longleftrightarrow\\quad\n R\\subseteq\\Ind(\\mathcal Q).\n\\]\nConsequently,\n\\[\n \\Cl(\\mathcal Q):=\\Def(\\Ind(\\mathcal Q))\n\\]\nis a closure operator on query families, and\n\\[\n \\operatorname{Sat}(R):=\\Ind(\\Def(R))\n\\]\nis the observational saturation of \\(R\\) relative to the admitted query universe.\n\\end{theorem}","\\begin{proof}\nBoth conditions say that every query in \\(\\mathcal Q\\) is constant on every \\(R\\)-class.  The closure laws follow from the standard identities associated with an antitone Galois connection.\n\\end{proof}","\\begin{definition}[Extensional task]\n\\label{def:block17-extensional-task}\nAn \\emph{extensional task} is a closed query theory, equivalently a query family modulo equality of the closure \\(\\Cl\\).  When the admitted query universe separates the desired quotients, an extensional task can equivalently be represented by its observationally definable congruence.\n\\end{definition}","\\begin{corollary}[Task strengthening]\n\\label{cor:block17-task-strengthening}\nAdding continuations or observations can only refine behavioral indistinguishability.  Thus\n\\[\n M\\subseteq M',\\qquad \\mathcal O\\subseteq\\mathcal O'\n \\quad\\Longrightarrow\\quad\n \\Theta_{M',\\mathcal O'}\\subseteq\\Theta_{M,\\mathcal O},\n\\]\nand there is a canonical forgetting map\n\\[\n Q_{M',\\mathcal O'}\\longrightarrow Q_{M,\\mathcal O}.\n\\]\n\\end{corollary}","\\begin{theorem}[Semantic quotient classification, classical]\n\\label{thm:block17-semantic-quotient-classification}\nLet \\(X\\) be an algebra in a variety.  Surjective semantic maps\n\\[\n q:X\\twoheadrightarrow Q\n\\]\nup to isomorphism of codomains are classified by congruences on \\(X\\) through\n\\[\n q\\longmapsto\\ker q.\n\\]\nMoreover, a quotient \\(q\\) maps to a quotient \\(r\\) exactly when \\(\\ker q\\subseteq\\ker r\\).\n\\end{theorem}","\\begin{corollary}[Combined and shared semantics]\n\\label{cor:block17-combined-shared-semantics}\nFor semantic quotients with kernels \\(N\\) and \\(M\\):\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item the least common refinement has kernel \\(N\\cap M\\);\n\\item the greatest common coarsening has kernel the generated congruence join \\(N\\vee M\\);\n\\item in a module category, \\(N\\vee M=N+M\\).\n\\end{enumerate}\n\\end{corollary}","Let \\(X\\) be an \\(R\\)-module and let\n\\[\n s:X\\twoheadrightarrow S,\n \\qquad\n t:X\\twoheadrightarrow T\n\\]\nbe surjective linear semantic maps.  Put\n\\[\n N=\\ker s,\n \\qquad\n M=\\ker t,\n \\qquad\n J=X/(N\\cap M),\n \\qquad\n C=X/(N+M).\n\\]","\\begin{theorem}[Semantic diamond and exact compatibility sequence]\n\\label{thm:block17-semantic-diamond}\nThe canonical square\n\\[\n\\begin{CD}\n J @>>> T\\\\\n @VVV @VVV\\\\\n S @>>> C\n\\end{CD}\n\\]\nis a pullback in \\(R\\text{-}\\mathbf{Mod}\\).  Equivalently,\n\\[\n J\\cong S\\times_C T.\n\\]\nThere is also a canonical short exact sequence\n\\[\n 0\\longrightarrow X/(N\\cap M)\n \\xrightarrow{\\alpha}\n X/N\\oplus X/M\n \\xrightarrow{\\beta}\n X/(N+M)\n \\longrightarrow0,\n\\]\nwhere\n\\[\n \\alpha(x+N\\cap M)=(x+N,x+M)\n\\]\nand\n\\[\n \\beta(a+N,b+M)=a-b+(N+M).\n\\]\n\\end{theorem}","S @>>> C\n\\end{CD}\n\\]\nis a pullback in \\(R\\text{-}\\mathbf{Mod}\\).  Equivalently,\n\\[\n J\\cong S\\times_C T.\n\\]\nThere is also a canonical short exact sequence\n\\[\n 0\\longrightarrow X/(N\\cap M)\n \\xrightarrow{\\alpha}\n X/N\\oplus X/M\n \\xrightarrow{\\beta}\n X/(N+M)\n \\longrightarrow0,\n\\]\nwhere\n\\[\n \\alpha(x+N\\cap M)=(x+N,x+M)\n\\]\nand\n\\[\n \\beta(a+N,b+M)=a-b+(N+M).\n\\]\n\\end{theorem}","\\begin{proof}\nThe map \\(\\alpha\\) is injective because its kernel is \\(N\\cap M\\).  The formula for \\(\\beta\\) is well defined and surjective.  Clearly \\(\\beta\\alpha=0\\).  If \\(\\beta(a+N,b+M)=0\\), then\n\\[\n a-b=n+m\n\\]\nfor some \\(n\\in N\\) and \\(m\\in M\\).  Set \\(x=a-n=b+m\\).  Then\n\\[\n (a+N,b+M)=(x+N,x+M)=\\alpha(x+N\\cap M).\n\\]\nThus \\(\\ker\\beta=\\im\\alpha\\), proving exactness.  The kernel description of the fiber product is the same calculation.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.76\\textwidth]{block17_semantic_diamond.pdf}\n\\caption{The semantic diamond.  The joint representation \\(J\\) retains both views.  The common quotient \\(C\\) is the compatibility interface over which the separate coordinates glue.}\n\\label{fig:block17-semantic-diamond}\n\\end{figure}","\\begin{definition}[Relative semantic residual]\n\\label{def:block17-relative-residual}\nThe information needed to augment \\(S\\) so that it also retains \\(T\\) is\n\\[\n R_{T\\mid S}\n :=\n \\ker(J\\to S)\n \\cong\n N/(N\\cap M)\n \\cong\n (N+M)/M.\n\\]\n\\end{definition}","\\begin{corollary}[Factorization criterion]\n\\label{cor:block17-factorization-criterion}\nThe following are equivalent:\n\\[\n t\\text{ factors through }s,\n \\qquad\n N\\subseteq M,\n \\qquad\n R_{T\\mid S}=0,\n \\qquad\n J\\to S\\text{ is an isomorphism}.\n\\]\nMoreover,\n\\[\n 0\\longrightarrow R_{T\\mid S}\n \\longrightarrow J\n \\longrightarrow S\n \\longrightarrow0\n\\]\nis exact.\n\\end{corollary}","\\begin{boundary}[Residuals need not split]\n\\label{bnd:block17-residual-splitting}\nThe exact sequence above does not canonically identify \\(J\\) with \\(S\\oplus R_{T\\mid S}\\).  Vector-space extensions split noncanonically.  Module, integral, semiring, filtered, or topological extensions may fail to split.  A residual packet presented as an independent coordinate therefore requires an explicit splitting theorem or constructor passport.\n\\end{boundary}","\\begin{theorem}[Task-change exact sequence]\n\\label{thm:block17-task-change}\nLet a stronger task have behavioral kernel \\(N_s\\), and let a weaker task have behavioral kernel \\(N_w\\), with\n\\[\n N_s\\subseteq N_w.\n\\]\nThen\n\\[\n 0\\longrightarrow N_w/N_s\n \\longrightarrow X/N_s\n \\longrightarrow X/N_w\n \\longrightarrow0\n\\]\nis exact.  Thus \\(N_w/N_s\\) is the exact semantic information discarded when the task is weakened.\n\\end{theorem}","\\begin{proof}\nApply the third isomorphism theorem to the chain \\(N_s\\subseteq N_w\\subseteq X\\).\n\\end{proof}","\\begin{corollary}[Composition of task loss]\n\\label{cor:block17-task-loss-composition}\nFor\n\\[\n N_1\\subseteq N_2\\subseteq N_3,\n\\]\nthere is an exact sequence\n\\[\n 0\\longrightarrow N_2/N_1\n \\longrightarrow N_3/N_1\n \\longrightarrow N_3/N_2\n \\longrightarrow0.\n\\]\nHence task loss composes as extension data rather than as a canonically split sum.\n\\end{corollary}","Let \\(\\mathcal R\\) be a raw algebra of expressions or histories for a fixed operation signature.  Let\n\\[\n s:\\mathcal R\\longrightarrow S,\n \\qquad\n t:\\mathcal R\\longrightarrow T\n\\]\nbe homomorphisms into two semantic carriers.","\\begin{theorem}[Carrier factorization gate]\n\\label{thm:block18-carrier-factorization}\nThere is a unique homomorphism\n\\[\n \\varphi:\\im(s)\\longrightarrow T\n\\]\nwith \\(t=\\varphi s\\) if and only if\n\\[\n \\kerpair(s)\\subseteq\\kerpair(t).\n\\]\nWhen the condition fails, any pair \\(x,y\\in\\mathcal R\\) satisfying\n\\[\n s(x)=s(y),\n \\qquad\n t(x)\\neq t(y)\n\\]\nis an exact carrier-change obstruction witness.\n\\end{theorem}","\\begin{proof}\nFactorization immediately implies constancy of \\(t\\) on every fiber of \\(s\\).  Conversely, define\n\\[\n \\varphi(s(x)):=t(x).\n\\]\nThe fiber condition makes this well defined.  Since both \\(s\\) and \\(t\\) preserve every primitive operation, so does \\(\\varphi\\).  Uniqueness follows because \\(s\\) is surjective onto its image.\n\\end{proof}","\\begin{definition}[Joint semantic completion]\n\\label{def:block18-joint-completion}\nDefine\n\\[\n j=(s,t):\\mathcal R\\longrightarrow S\\times T,\n \\qquad\n J_{s,t}:=\\im(j).\n\\]\nThe projection\n\\[\n p_s:J_{s,t}\\longrightarrow\\im(s)\n\\]\nis the \\emph{relative completion map} from the joint semantics to the source semantics.\n\\end{definition}","\\begin{theorem}[Universal joint completion]\n\\label{thm:block18-universal-joint-completion}\nThe object \\(J_{s,t}\\) is the coarsest quotient of \\(\\mathcal R\\) preserving both \\(s\\) and \\(t\\).  More precisely, if\n\\[\n c:\\mathcal R\\twoheadrightarrow C\n\\]\nadmits maps \\(a:C\\to S\\) and \\(b:C\\to T\\) with \\(s=ac\\) and \\(t=bc\\), then there is a unique map\n\\[\n C\\longrightarrow J_{s,t}\n\\]\ncompatible with both semantics.  Furthermore,\n\\[\n t\\text{ factors through }s\n \\quad\\Longleftrightarrow\\quad\n p_s\\text{ is an isomorphism}.\n\\]\n\\end{theorem}","oint completion]\n\\label{thm:block18-universal-joint-completion}\nThe object \\(J_{s,t}\\) is the coarsest quotient of \\(\\mathcal R\\) preserving both \\(s\\) and \\(t\\).  More precisely, if\n\\[\n c:\\mathcal R\\twoheadrightarrow C\n\\]\nadmits maps \\(a:C\\to S\\) and \\(b:C\\to T\\) with \\(s=ac\\) and \\(t=bc\\), then there is a unique map\n\\[\n C\\longrightarrow J_{s,t}\n\\]\ncompatible with both semantics.  Furthermore,\n\\[\n t\\text{ factors through }s\n \\quad\\Longleftrightarrow\\quad\n p_s\\text{ is an isomorphism}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe pair \\((a,b)\\) factors \\(j\\) through \\(c\\), and the image universal property gives the map to \\(J_{s,t}\\).  If \\(t=\\varphi s\\), then \\(J_{s,t}\\) is the graph of \\(\\varphi\\), so \\(p_s\\) is invertible.  Conversely, if \\(p_s\\) is invertible, define \\(\\varphi=p_t p_s^{-1}\\).\n\\end{proof}","Let \\(f:X\\to Y\\) be a linear bridge between observational systems, with behavioral kernels \\(N_X\\subseteq X\\) and \\(N_Y\\subseteq Y\\).","\\begin{definition}[Task-compatible bridge]\n\\label{def:block18-task-compatible-bridge}\nThe bridge is \\emph{task compatible} when\n\\[\n f(N_X)\\subseteq N_Y.\n\\]\nEquivalently,\n\\[\n N_X\\subseteq f^{-1}(N_Y).\n\\]\nThen there is a unique induced map\n\\[\n \\overline f:X/N_X\\longrightarrow Y/N_Y.\n\\]\n\\end{definition}","\\begin{theorem}[Bridge-loss exact sequence]\n\\label{thm:block18-bridge-loss-exact}\nFor every task-compatible linear bridge, there is an exact sequence\n\\[\n 0\\longrightarrow\n \\frac{f^{-1}(N_Y)}{N_X}\n \\longrightarrow\n \\frac{X}{N_X}\n \\xrightarrow{\\overline f}\n \\frac{Y}{N_Y}\n \\longrightarrow\n \\frac{Y}{N_Y+f(X)}\n \\longrightarrow0.\n\\]\nThus\n\\[\n L_f:=\\frac{f^{-1}(N_Y)}{N_X}\n\\]\nmeasures source distinctions collapsed by the bridge, while\n\\[\n E_f:=\\frac{Y}{N_Y+f(X)}\n\\]\nmeasures target semantic states outside the essential image.\n\\end{theorem}","t sequence]\n\\label{thm:block18-bridge-loss-exact}\nFor every task-compatible linear bridge, there is an exact sequence\n\\[\n 0\\longrightarrow\n \\frac{f^{-1}(N_Y)}{N_X}\n \\longrightarrow\n \\frac{X}{N_X}\n \\xrightarrow{\\overline f}\n \\frac{Y}{N_Y}\n \\longrightarrow\n \\frac{Y}{N_Y+f(X)}\n \\longrightarrow0.\n\\]\nThus\n\\[\n L_f:=\\frac{f^{-1}(N_Y)}{N_X}\n\\]\nmeasures source distinctions collapsed by the bridge, while\n\\[\n E_f:=\\frac{Y}{N_Y+f(X)}\n\\]\nmeasures target semantic states outside the essential image.\n\\end{theorem}","\\begin{proof}\nThe kernel of \\(\\overline f\\) consists of classes \\(x+N_X\\) for which \\(f(x)\\in N_Y\\), hence equals \\(f^{-1}(N_Y)/N_X\\).  The cokernel of \\(\\overline f\\) is the quotient of \\(Y/N_Y\\) by \\((f(X)+N_Y)/N_Y\\), which is \\(Y/(N_Y+f(X))\\).\n\\end{proof}","\\begin{corollary}[Exact semantic equivalence gate]\n\\label{cor:block18-semantic-equivalence-gate}\nThe induced bridge \\(\\overline f\\) is an isomorphism exactly when\n\\[\n L_f=0\n \\qquad\\text{and}\\qquad\n E_f=0.\n\\]\nFaithfulness on the declared task and essential surjectivity on the declared task are therefore independent obligations.\n\\end{corollary}","\\begin{theorem}[Loss-composition extension]\n\\label{thm:block18-loss-composition}\nFor induced quotient maps\n\\[\n Q_X\\xrightarrow{\\overline f}Q_Y\\xrightarrow{\\overline g}Q_Z,\n\\]\nthere is a natural exact sequence\n\\[\n 0\\longrightarrow\\ker\\overline f\n \\longrightarrow\\ker(\\overline g\\,\\overline f)\n \\longrightarrow\\ker\\overline g\\cap\\im\\overline f\n \\longrightarrow0.\n\\]\nIf \\(\\overline f\\) is surjective, this reduces to\n\\[\n 0\\longrightarrow L_f\n \\longrightarrow L_{gf}\n \\longrightarrow L_g\n \\longrightarrow0.\n\\]\n\\end{theorem}","Loss-composition extension]\n\\label{thm:block18-loss-composition}\nFor induced quotient maps\n\\[\n Q_X\\xrightarrow{\\overline f}Q_Y\\xrightarrow{\\overline g}Q_Z,\n\\]\nthere is a natural exact sequence\n\\[\n 0\\longrightarrow\\ker\\overline f\n \\longrightarrow\\ker(\\overline g\\,\\overline f)\n \\longrightarrow\\ker\\overline g\\cap\\im\\overline f\n \\longrightarrow0.\n\\]\nIf \\(\\overline f\\) is surjective, this reduces to\n\\[\n 0\\longrightarrow L_f\n \\longrightarrow L_{gf}\n \\longrightarrow L_g\n \\longrightarrow0.\n\\]\n\\end{theorem}","\\begin{proof}\nMap \\(x\\in\\ker(\\overline g\\,\\overline f)\\) to \\(\\overline f(x)\\).  Its image is exactly \\(\\ker\\overline g\\cap\\im\\overline f\\), and its kernel is \\(\\ker\\overline f\\).  Surjectivity of \\(\\overline f\\) identifies the intersection with \\(\\ker\\overline g\\).\n\\end{proof}","\\begin{theorem}[Carrier-indexed observer packet theorem]\n\\label{thm:block18-carrier-indexed-observer}\nLet \\(S\\) be a finite continuation-state set, let \\((A,\\oplus,0)\\) be a commutative monoid, let \\(B\\) be a finite branch alphabet, and let\n\\[\n \\delta:S\\times B\\longrightarrow S\n\\]\nbe a state transition.  Suppose every branch \\(b\\in B\\) acts by a monoid homomorphism\n\\[\n L_b:A\\longrightarrow A.\n\\]\nTake an empty aggregation to be \\(0\\).  For a state packet \\(a=(a_s)_{s\\in S}\\), define\n\\[\n a'_t\n :=\n \\bigoplus_{\\delta(s,b)=t}L_b(a_s).\n\\]\nIf a finite family of raw histories has already been aggregated by their state using \\(\\oplus\\), then this update gives exactly the same result as advancing every raw history separately and aggregating afterward.\n\\end{theorem}","bserver packet theorem]\n\\label{thm:block18-carrier-indexed-observer}\nLet \\(S\\) be a finite continuation-state set, let \\((A,\\oplus,0)\\) be a commutative monoid, let \\(B\\) be a finite branch alphabet, and let\n\\[\n \\delta:S\\times B\\longrightarrow S\n\\]\nbe a state transition.  Suppose every branch \\(b\\in B\\) acts by a monoid homomorphism\n\\[\n L_b:A\\longrightarrow A.\n\\]\nTake an empty aggregation to be \\(0\\).  For a state packet \\(a=(a_s)_{s\\in S}\\), define\n\\[\n a'_t\n :=\n \\bigoplus_{\\delta(s,b)=t}L_b(a_s).\n\\]\nIf a finite family of raw histories has already been aggregated by their state using \\(\\oplus\\), then this update gives exactly the same result as advancing every raw history separately and aggregating afterward.\n\\end{theorem}","bel{thm:block18-carrier-indexed-observer}\nLet \\(S\\) be a finite continuation-state set, let \\((A,\\oplus,0)\\) be a commutative monoid, let \\(B\\) be a finite branch alphabet, and let\n\\[\n \\delta:S\\times B\\longrightarrow S\n\\]\nbe a state transition.  Suppose every branch \\(b\\in B\\) acts by a monoid homomorphism\n\\[\n L_b:A\\longrightarrow A.\n\\]\nTake an empty aggregation to be \\(0\\).  For a state packet \\(a=(a_s)_{s\\in S}\\), define\n\\[\n a'_t\n :=\n \\bigoplus_{\\delta(s,b)=t}L_b(a_s).\n\\]\nIf a finite family of raw histories has already been aggregated by their state using \\(\\oplus\\), then this update gives exactly the same result as advancing every raw history separately and aggregating afterward.\n\\end{theorem}","ed-observer}\nLet \\(S\\) be a finite continuation-state set, let \\((A,\\oplus,0)\\) be a commutative monoid, let \\(B\\) be a finite branch alphabet, and let\n\\[\n \\delta:S\\times B\\longrightarrow S\n\\]\nbe a state transition.  Suppose every branch \\(b\\in B\\) acts by a monoid homomorphism\n\\[\n L_b:A\\longrightarrow A.\n\\]\nTake an empty aggregation to be \\(0\\).  For a state packet \\(a=(a_s)_{s\\in S}\\), define\n\\[\n a'_t\n :=\n \\bigoplus_{\\delta(s,b)=t}L_b(a_s).\n\\]\nIf a finite family of raw histories has already been aggregated by their state using \\(\\oplus\\), then this update gives exactly the same result as advancing every raw history separately and aggregating afterward.\n\\end{theorem}","habet, and let\n\\[\n \\delta:S\\times B\\longrightarrow S\n\\]\nbe a state transition.  Suppose every branch \\(b\\in B\\) acts by a monoid homomorphism\n\\[\n L_b:A\\longrightarrow A.\n\\]\nTake an empty aggregation to be \\(0\\).  For a state packet \\(a=(a_s)_{s\\in S}\\), define\n\\[\n a'_t\n :=\n \\bigoplus_{\\delta(s,b)=t}L_b(a_s).\n\\]\nIf a finite family of raw histories has already been aggregated by their state using \\(\\oplus\\), then this update gives exactly the same result as advancing every raw history separately and aggregating afterward.\n\\end{theorem}","\\begin{proof}\nExpand the raw history update and group all terms by their predecessor state and new branch.  Associativity and commutativity permit the regrouping.  The identity\n\\[\n L_b\\!\\left(\\bigoplus_i x_i\\right)\n =\n \\bigoplus_i L_b(x_i)\n\\]\nthen moves aggregation through the branch action.  The displayed formula is the resulting statewise sum.\n\\end{proof}","Two important realizations have the same state skeleton but different carriers.\n\\begin{enumerate}\n\\item For convex value semantics, \\(A\\) is the family of compact convex sets, including the empty set, \\(\\oplus\\) is convex hull of union with identity \\(\\varnothing\\), and \\(L_b(C)=A_b^{\\mathsf T}C\\).\n\\item For linear polyphase routing, \\(A\\) is an additive convolution algebra, \\(\\oplus\\) is coefficient addition, and \\(L_b\\) multiplies route kernels by the declared filter or transition weight.\n\\end{enumerate}\nThe theorem transports the observer skeleton only after the carrier law and the homomorphism obligations are specified.","ations have the same state skeleton but different carriers.\n\\begin{enumerate}\n\\item For convex value semantics, \\(A\\) is the family of compact convex sets, including the empty set, \\(\\oplus\\) is convex hull of union with identity \\(\\varnothing\\), and \\(L_b(C)=A_b^{\\mathsf T}C\\).\n\\item For linear polyphase routing, \\(A\\) is an additive convolution algebra, \\(\\oplus\\) is coefficient addition, and \\(L_b\\) multiplies route kernels by the declared filter or transition weight.\n\\end{enumerate}\nThe theorem transports the observer skeleton only after the carrier law and the homomorphism obligations are specified.","\\begin{theorem}[Cancellation barrier]\n\\label{thm:block18-cancellation-barrier}\nLet \\(R\\) be a nonzero ring and let \\(S\\) be a nontrivial zerosumfree semiring, so\n\\[\n a+b=0\\Longrightarrow a=b=0.\n\\]\nThere is no unital semiring homomorphism \\(R\\to S\\).\n\\end{theorem}","\\begin{proof}\nA unital homomorphism \\(\\varphi\\) would give\n\\[\n 0=\\varphi(1+(-1))=1_S+\\varphi(-1).\n\\]\nZerosumfreeness forces \\(1_S=0\\), contradicting nontriviality.\n\\end{proof}","\\begin{theorem}[Idempotence barrier]\n\\label{thm:block18-idempotence-barrier}\nLet \\(S\\) be a nontrivial unital semiring with idempotent addition, and let \\(R\\) be an additively cancellative nontrivial semiring.  There is no unital semiring homomorphism \\(S\\to R\\).\n\\end{theorem}","\\begin{proof}\nIn \\(S\\), \\(1+1=1\\).  Its image in \\(R\\) would satisfy \\(1_R+1_R=1_R\\), and additive cancellation would imply \\(1_R=0\\).\n\\end{proof}","\\begin{example}[Haar cancellation]\n\\label{ex:block18-haar-cancellation}\nFor the normalized Haar bank,\n\\[\n \\ell=\\frac{x_0+x_1}{\\sqrt2},\n \\qquad\n d=\\frac{x_0-x_1}{\\sqrt2},\n\\]\nand\n\\[\n x_0=\\frac{\\ell+d}{\\sqrt2},\n \\qquad\n x_1=\\frac{\\ell-d}{\\sqrt2}.\n\\]\nAt \\(x=(1,0)\\), the two route contributions to \\(x_0\\) are \\(1/2,1/2\\), while those to \\(x_1\\) are \\(1/2,-1/2\\).  Additive reconstruction returns \\((1,0)\\).  Replacing route addition by maximum returns \\((1/2,1/2)\\).  The convex/max carrier is therefore not an exact replacement for the signed additive carrier.\n\\end{example}","8-haar-cancellation}\nFor the normalized Haar bank,\n\\[\n \\ell=\\frac{x_0+x_1}{\\sqrt2},\n \\qquad\n d=\\frac{x_0-x_1}{\\sqrt2},\n\\]\nand\n\\[\n x_0=\\frac{\\ell+d}{\\sqrt2},\n \\qquad\n x_1=\\frac{\\ell-d}{\\sqrt2}.\n\\]\nAt \\(x=(1,0)\\), the two route contributions to \\(x_0\\) are \\(1/2,1/2\\), while those to \\(x_1\\) are \\(1/2,-1/2\\).  Additive reconstruction returns \\((1,0)\\).  Replacing route addition by maximum returns \\((1/2,1/2)\\).  The convex/max carrier is therefore not an exact replacement for the signed additive carrier.\n\\end{example}","Let \\(E\\subseteq K^M\\) be the set of distinct residual future functions of a scalar series or observational system.  Set minimization retains the elements of \\(E\\) as distinct states.  Free linearization produces the vector space \\(K^{(E)}\\), while linear minimization retains only their span.","\\begin{theorem}[Observer base-change exact sequence]\n\\label{thm:block18-observer-base-change}\nThe evaluation map\n\\[\n \\pi:K^{(E)}\\longrightarrow\\Span_K(E),\n \\qquad\n \\sum_{e\\in E}a_e[e]\n \\longmapsto\n \\sum_{e\\in E}a_e e\n\\]\nis surjective and fits into the exact sequence\n\\[\n 0\\longrightarrow\\Rel_K(E)\n \\longrightarrow K^{(E)}\n \\xrightarrow{\\pi}\\Span_K(E)\n \\longrightarrow0,\n\\]\nwhere\n\\[\n \\Rel_K(E):=\\ker\\pi\n\\]\nis the module of linear relations among distinct discrete residual states.  Hence Set minimization commutes with free linearization exactly when \\(E\\) is linearly independent.\n\\end{theorem}","),\n \\qquad\n \\sum_{e\\in E}a_e[e]\n \\longmapsto\n \\sum_{e\\in E}a_e e\n\\]\nis surjective and fits into the exact sequence\n\\[\n 0\\longrightarrow\\Rel_K(E)\n \\longrightarrow K^{(E)}\n \\xrightarrow{\\pi}\\Span_K(E)\n \\longrightarrow0,\n\\]\nwhere\n\\[\n \\Rel_K(E):=\\ker\\pi\n\\]\nis the module of linear relations among distinct discrete residual states.  Hence Set minimization commutes with free linearization exactly when \\(E\\) is linearly independent.\n\\end{theorem}","\\begin{proof}\nSurjectivity is the definition of the linear span.  The sequence is the kernel-cokernel sequence of \\(\\pi\\).  The kernel vanishes exactly when the basis symbols \\([e]\\) remain linearly independent after evaluation.\n\\end{proof}","\\begin{theorem}[Exact polyphase separation]\n\\label{thm:block18-polyphase-hankel-separation}\nFix \\(N\\geq1\\) and \\(z\\neq0\\).  Let \\(S_z\\) be the cyclic shift on \\(K^N\\) satisfying\n\\[\n S_z e_j=e_{j+1}\\quad(j<N-1),\n \\qquad\n S_z e_{N-1}=z e_0,\n\\]\nso \\(S_z^N=zI\\).  Put\n\\[\n M_N=I+S_z,\n \\qquad\n f_N(n)=e_0^{\\mathsf T}M_N^n e_0.\n\\]\nThen the \\(N\\times N\\) Hankel block\n\\[\n H_{ij}=f_N(i+j),\n \\qquad 0\\leq i,j<N,\n\\]\nhas determinant\n\\[\n \\det H=\\pm z^{N-1}.\n\\]\nConsequently its linear residual space has dimension \\(N\\).  If \\(z>0\\), the residual functions \\(k\\mapsto f_N(n+k)\\) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is \\(N\\).\n\\end{theorem}","olyphase separation]\n\\label{thm:block18-polyphase-hankel-separation}\nFix \\(N\\geq1\\) and \\(z\\neq0\\).  Let \\(S_z\\) be the cyclic shift on \\(K^N\\) satisfying\n\\[\n S_z e_j=e_{j+1}\\quad(j<N-1),\n \\qquad\n S_z e_{N-1}=z e_0,\n\\]\nso \\(S_z^N=zI\\).  Put\n\\[\n M_N=I+S_z,\n \\qquad\n f_N(n)=e_0^{\\mathsf T}M_N^n e_0.\n\\]\nThen the \\(N\\times N\\) Hankel block\n\\[\n H_{ij}=f_N(i+j),\n \\qquad 0\\leq i,j<N,\n\\]\nhas determinant\n\\[\n \\det H=\\pm z^{N-1}.\n\\]\nConsequently its linear residual space has dimension \\(N\\).  If \\(z>0\\), the residual functions \\(k\\mapsto f_N(n+k)\\) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is \\(N\\).\n\\end{theorem}","1\\) and \\(z\\neq0\\).  Let \\(S_z\\) be the cyclic shift on \\(K^N\\) satisfying\n\\[\n S_z e_j=e_{j+1}\\quad(j<N-1),\n \\qquad\n S_z e_{N-1}=z e_0,\n\\]\nso \\(S_z^N=zI\\).  Put\n\\[\n M_N=I+S_z,\n \\qquad\n f_N(n)=e_0^{\\mathsf T}M_N^n e_0.\n\\]\nThen the \\(N\\times N\\) Hankel block\n\\[\n H_{ij}=f_N(i+j),\n \\qquad 0\\leq i,j<N,\n\\]\nhas determinant\n\\[\n \\det H=\\pm z^{N-1}.\n\\]\nConsequently its linear residual space has dimension \\(N\\).  If \\(z>0\\), the residual functions \\(k\\mapsto f_N(n+k)\\) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is \\(N\\).\n\\end{theorem}","\\neq0\\).  Let \\(S_z\\) be the cyclic shift on \\(K^N\\) satisfying\n\\[\n S_z e_j=e_{j+1}\\quad(j<N-1),\n \\qquad\n S_z e_{N-1}=z e_0,\n\\]\nso \\(S_z^N=zI\\).  Put\n\\[\n M_N=I+S_z,\n \\qquad\n f_N(n)=e_0^{\\mathsf T}M_N^n e_0.\n\\]\nThen the \\(N\\times N\\) Hankel block\n\\[\n H_{ij}=f_N(i+j),\n \\qquad 0\\leq i,j<N,\n\\]\nhas determinant\n\\[\n \\det H=\\pm z^{N-1}.\n\\]\nConsequently its linear residual space has dimension \\(N\\).  If \\(z>0\\), the residual functions \\(k\\mapsto f_N(n+k)\\) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is \\(N\\).\n\\end{theorem}","lic shift on \\(K^N\\) satisfying\n\\[\n S_z e_j=e_{j+1}\\quad(j<N-1),\n \\qquad\n S_z e_{N-1}=z e_0,\n\\]\nso \\(S_z^N=zI\\).  Put\n\\[\n M_N=I+S_z,\n \\qquad\n f_N(n)=e_0^{\\mathsf T}M_N^n e_0.\n\\]\nThen the \\(N\\times N\\) Hankel block\n\\[\n H_{ij}=f_N(i+j),\n \\qquad 0\\leq i,j<N,\n\\]\nhas determinant\n\\[\n \\det H=\\pm z^{N-1}.\n\\]\nConsequently its linear residual space has dimension \\(N\\).  If \\(z>0\\), the residual functions \\(k\\mapsto f_N(n+k)\\) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is \\(N\\).\n\\end{theorem}","ad\n f_N(n)=e_0^{\\mathsf T}M_N^n e_0.\n\\]\nThen the \\(N\\times N\\) Hankel block\n\\[\n H_{ij}=f_N(i+j),\n \\qquad 0\\leq i,j<N,\n\\]\nhas determinant\n\\[\n \\det H=\\pm z^{N-1}.\n\\]\nConsequently its linear residual space has dimension \\(N\\).  If \\(z>0\\), the residual functions \\(k\\mapsto f_N(n+k)\\) are pairwise distinct, so the exact discrete future quotient is infinite while the linear continuation dimension is \\(N\\).\n\\end{theorem}","\\begin{proof}\nFactor \\(H=OC\\), where row \\(i\\) of \\(O\\) is \\(e_0^{\\mathsf T}M_N^i\\), and column \\(j\\) of \\(C\\) is \\(M_N^j e_0\\).  For \\(j<N\\), the binomial expansion of \\((I+S_z)^j e_0\\) is a unit upper-triangular change from \\((e_0,\\ldots,e_{N-1})\\), hence \\(\\det C=1\\).  The rows \\(e_0^{\\mathsf T}S_z^k\\), \\(0\\leq k<N\\), are \\(e_0^{\\mathsf T}\\) followed by \\(z e_{N-k}^{\\mathsf T}\\), so their determinant is \\(\\pm z^{N-1}\\).  The binomial row change is also unit triangular, proving the determinant formula.","}\nFactor \\(H=OC\\), where row \\(i\\) of \\(O\\) is \\(e_0^{\\mathsf T}M_N^i\\), and column \\(j\\) of \\(C\\) is \\(M_N^j e_0\\).  For \\(j<N\\), the binomial expansion of \\((I+S_z)^j e_0\\) is a unit upper-triangular change from \\((e_0,\\ldots,e_{N-1})\\), hence \\(\\det C=1\\).  The rows \\(e_0^{\\mathsf T}S_z^k\\), \\(0\\leq k<N\\), are \\(e_0^{\\mathsf T}\\) followed by \\(z e_{N-k}^{\\mathsf T}\\), so their determinant is \\(\\pm z^{N-1}\\).  The binomial row change is also unit triangular, proving the determinant formula.","For positive \\(z\\),\n\\[\n f_N(n)=\\sum_{r=0}^{\\lfloor n/N\\rfloor}\\binom n{Nr}z^r.\n\\]\nThe sequence is strictly increasing from index \\(N-1\\) onward.  Prefixes below \\(N\\) are distinguished by choosing the first suffix that crosses index \\(N\\), and later prefixes differ already at suffix zero.  Thus all residual functions are distinct.\n\\end{proof}","Let \\(S\\) be a finite observer-state set, let \\(B\\) be a finite branch\nalphabet, and let \\(C_s\\subseteq\\mathbb R^d\\) be a compact convex packet for\neach state, with empty packets allowed.  Use the conventions\n\\[\n h_{\\varnothing}(x)=-\\infty,\\qquad\n \\operatorname{conv}(\\varnothing)=\\varnothing,\\qquad\n \\max\\varnothing=-\\infty.\n\\]\nFor branch matrices \\((A_b)_{b\\in B}\\) and a transition\n\\(\\delta:S\\times B\\to S\\), put\n\\[\n C'_t\n =\n \\operatorname{conv}\n \\bigcup_{\\delta(s,b)=t}A_b^{\\mathsf T}C_s.\n\\]","ite observer-state set, let \\(B\\) be a finite branch\nalphabet, and let \\(C_s\\subseteq\\mathbb R^d\\) be a compact convex packet for\neach state, with empty packets allowed.  Use the conventions\n\\[\n h_{\\varnothing}(x)=-\\infty,\\qquad\n \\operatorname{conv}(\\varnothing)=\\varnothing,\\qquad\n \\max\\varnothing=-\\infty.\n\\]\nFor branch matrices \\((A_b)_{b\\in B}\\) and a transition\n\\(\\delta:S\\times B\\to S\\), put\n\\[\n C'_t\n =\n \\operatorname{conv}\n \\bigcup_{\\delta(s,b)=t}A_b^{\\mathsf T}C_s.\n\\]","\\begin{theorem}[Observer-conditioned convex packet transport]\n\\label{thm:block18-convex-packet-transport}\nFor every state \\(t\\in S\\) and direction \\(x\\),\n\\[\n h_{C'_t}(x)\n =\n \\max_{\\delta(s,b)=t}h_{C_s}(A_bx).\n\\]\nRepeated updates compose exactly with branch words.  Within each fixed observer state, replacing a finite atom family by any other finite generating family of the same convex hull preserves all future value queries generated by linear branch actions.\n\\end{theorem}","\\begin{proof}\nFor a linear map \\(A\\),\n\\[\n h_{A^{\\mathsf T}C}(x)=h_C(Ax).\n\\]\nIf the incoming fiber is empty, both sides of the formula are \\(-\\infty\\).\nOtherwise, the support function of the convex hull of a finite union is the\nmaximum of the component support functions, with the same conventions for\nempty components.  These identities give the formula.  Finite unions of\ncompact sets have compact convex hulls in finite dimension, so every updated\npacket stays in the declared carrier.  Iteration follows from composition of the linear maps and of \\(\\delta\\).  Equality of convex hulls is preserved by linear images and convex hull of union, proving future value equivalence.\n\\end{proof}","therwise, the support function of the convex hull of a finite union is the\nmaximum of the component support functions, with the same conventions for\nempty components.  These identities give the formula.  Finite unions of\ncompact sets have compact convex hulls in finite dimension, so every updated\npacket stays in the declared carrier.  Iteration follows from composition of the linear maps and of \\(\\delta\\).  Equality of convex hulls is preserved by linear images and convex hull of union, proving future value equivalence.\n\\end{proof}","\\begin{definition}[Class-relative defect]\n\\label{def:block19-class-relative-defect}\nLet \\(X\\) be a represented system, let \\(\\Gamma\\) be a declared class of admissible auxiliary structures, and let\n\\[\n d(X,g)\\in[0,\\infty]\n\\]\nbe a defect functional.  Define\n\\[\n \\delta_\\Gamma(X):=\\inf_{g\\in\\Gamma}d(X,g).\n\\]\nThe pair consisting of the infimum and its attainment status is the \\emph{class-relative defect profile}.\n\\end{definition}","\\begin{theorem}[Admissibility monotonicity]\n\\label{thm:block19-admissibility-monotonicity}\nIf \\(\\Gamma\\subseteq\\Gamma'\\), then\n\\[\n \\delta_{\\Gamma'}(X)\\leq\\delta_\\Gamma(X).\n\\]\nThus an obstruction in a restricted realization class need not remain an obstruction after the class is enlarged.\n\\end{theorem}","\\begin{definition}[Completion, gap, and escape]\n\\label{def:block19-trichotomy}\nAssume \\(d\\geq0\\).  The defect profile is:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item a \\emph{completion} when \\(\\delta_\\Gamma(X)=0\\) and the infimum is attained;\n\\item a \\emph{gap obstruction} when \\(\\delta_\\Gamma(X)>0\\);\n\\item an \\emph{escape obstruction} when \\(\\delta_\\Gamma(X)=0\\) but the infimum is not attained.\n\\end{enumerate}\n\\end{definition}","\\begin{proposition}[Compactness certificate for escape]\n\\label{prop:block19-escape-compactness}\nLet \\(\\Gamma\\) be a Hausdorff parameter space, and let \\(d(X,-)\\) be continuous.  Suppose every sublevel set satisfying a declared conditioning bound is compact.  If \\(\\delta_\\Gamma(X)=0\\) is not attained, then every near-minimizing sequence must eventually violate every such uniform conditioning bound.\n\\end{proposition}","\\begin{proof}\nFix a declared conditioning bound \\(c\\), write \\(\\Gamma_c\\) for the\nparameters satisfying it, and let\n\\[\n K_c=\\{g\\in\\Gamma_c:d(X,g)\\leq 1\\}.\n\\]\nBy hypothesis \\(K_c\\) is compact.  If \\(K_c\\) is empty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-mi","\\begin{proof}\nFix a declared conditioning bound \\(c\\), write \\(\\Gamma_c\\) for the\nparameters satisfying it, and let\n\\[\n K_c=\\{g\\in\\Gamma_c:d(X,g)\\leq 1\\}.\n\\]\nBy hypothesis \\(K_c\\) is compact.  If \\(K_c\\) is empty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-minimizing.\n\\end{proof","\\begin{proof}\nFix a declared conditioning bound \\(c\\), write \\(\\Gamma_c\\) for the\nparameters satisfying it, and let\n\\[\n K_c=\\{g\\in\\Gamma_c:d(X,g)\\leq 1\\}.\n\\]\nBy hypothesis \\(K_c\\) is compact.  If \\(K_c\\) is empty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-minimizing.\n\\end{proof}","write \\(\\Gamma_c\\) for the\nparameters satisfying it, and let\n\\[\n K_c=\\{g\\in\\Gamma_c:d(X,g)\\leq 1\\}.\n\\]\nBy hypothesis \\(K_c\\) is compact.  If \\(K_c\\) is empty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-minimizing.\n\\end{proof}","ma_c\\) for the\nparameters satisfying it, and let\n\\[\n K_c=\\{g\\in\\Gamma_c:d(X,g)\\leq 1\\}.\n\\]\nBy hypothesis \\(K_c\\) is compact.  If \\(K_c\\) is empty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-minimizing.\n\\end{proof}","c:d(X,g)\\leq 1\\}.\n\\]\nBy hypothesis \\(K_c\\) is compact.  If \\(K_c\\) is empty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-minimizing.\n\\end{proof}","act.  If \\(K_c\\) is empty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-minimizing.\n\\end{proof}","mpty, every near-minimizing\nsequence eventually has defect below \\(1\\) and therefore lies outside\n\\(\\Gamma_c\\).  Otherwise continuity gives an attained minimum\n\\[\n a_c=\\min_{g\\in K_c}d(X,g).\n\\]\nSince the zero infimum is not attained anywhere in \\(\\Gamma\\), \\(a_c>0\\).\nEvery near-minimizing sequence eventually has defect below\n\\(\\min\\{1,a_c\\}\\), and therefore cannot belong to \\(\\Gamma_c\\).  Since \\(c\\)\nwas arbitrary, each fixed conditioning bound is eventually violated; in\nparticular no subsequence satisfying one fixed bound can be near-minimizing.\n\\end{proof}","Let \\(T:(V,Q_s)\\to(W,Q_t)\\) be invertible between real or complex spaces of common finite dimension \\(n\\geq1\\) with positive-definite metrics.  Choose factors \\(Q_s=R_s^*R_s\\) and \\(Q_t=R_t^*R_t\\), and put\n\\[\n B_Q(T):=R_tTR_s^{-1}.\n\\]\nDefine\n\\[\n D_Q(T)\n :=\n \\frac{\\sigma_{\\max}(B_Q(T))}{\\sigma_{\\min}(B_Q(T))}\n \\geq1,\n \\qquad\n \\Delta_Q(T):=\\log D_Q(T).\n\\]","\\begin{proposition}[Coordinate covariance and composition]\n\\label{prop:block19-distortion-cocycle}\nSimultaneous coordinate transport of \\(T,Q_s,Q_t\\) leaves \\(D_Q(T)\\) invariant.  For composable invertible maps,\n\\[\n D_Q(T_2T_1)\n \\leq\n D_Q(T_2)D_Q(T_1),\n\\]\nso\n\\[\n \\Delta_Q(T_2T_1)\n \\leq\n \\Delta_Q(T_2)+\\Delta_Q(T_1).\n\\]\nFor a closed word \\(H\\),\n\\[\n D_Q(H)\n \\geq\n \\frac{\\max_{\\lambda\\in\\operatorname{spec}(H)}|\\lambda|}\n      {\\min_{\\lambda\\in\\operatorname{spec}(H)}|\\lambda|}.\n\\]\n\\end{proposition}","\\begin{proof}\nCoordinate transport changes \\(B_Q(T)\\) by left and right unitary factors.  Singular-value condition numbers are submultiplicative.  The spectral lower bound follows because every eigenvalue modulus lies between the smallest and largest singular values.\n\\end{proof}","\\begin{theorem}[Conformal metric-completion criterion]\n\\label{thm:block19-metric-completion}\nLet a connected finite graph carry real or complex vector spaces of common\ndimension \\(n\\geq1\\) and invertible edge transports \\(T_e\\).  Reverse traversal\nof an edge uses \\(T_e^{-1}\\).  At a chosen root, determinant-normalized closed\nholonomy words are\n\\[\n \\widehat H:=|\\det H|^{-1/n}H.\n\\]\nThey form the determinant-normalized holonomy representation.  Positive-definite\nvertex metrics and positive edge scales satisfying\n\\[\n T_e^*Q_{t(e)}T_e=\\lambda_e^2Q_{s(e)}\n\\]\nexist if and only if the determinant-normalized holonomy representation preserves a positive-definite metric at one root vertex.  Equivalently, the normalized holonomy is unitarizable.  In finite dimension, this is equivalent to relative compactness of its image.\n\\end{theorem}","raph carry real or complex vector spaces of common\ndimension \\(n\\geq1\\) and invertible edge transports \\(T_e\\).  Reverse traversal\nof an edge uses \\(T_e^{-1}\\).  At a chosen root, determinant-normalized closed\nholonomy words are\n\\[\n \\widehat H:=|\\det H|^{-1/n}H.\n\\]\nThey form the determinant-normalized holonomy representation.  Positive-definite\nvertex metrics and positive edge scales satisfying\n\\[\n T_e^*Q_{t(e)}T_e=\\lambda_e^2Q_{s(e)}\n\\]\nexist if and only if the determinant-normalized holonomy representation preserves a positive-definite metric at one root vertex.  Equivalently, the normalized holonomy is unitarizable.  In finite dimension, this is equivalent to relative compactness of its image.\n\\end{theorem}","\\begin{proof}\nA compatible family makes every closed root holonomy \\(H\\) conformal for the\nroot metric \\(Q_o\\).  Taking determinants fixes its positive scale as\n\\(|\\det H|^{1/n}\\), so \\(\\widehat H\\) preserves \\(Q_o\\).","Conversely, suppose every normalized root holonomy preserves \\(Q_o\\).  Choose\nspanning-tree maps \\(G_v\\) from the root to each vertex and put\n\\[\n Q_v=(G_v^{-1})^*Q_oG_v^{-1}.\n\\]\nFor an edge \\(e:u\\to v\\), the root word \\(H_e=G_v^{-1}T_eG_u\\) satisfies\n\\[\n H_e^*Q_oH_e=|\\det H_e|^{2/n}Q_o.\n\\]\nConsequently\n\\[\n T_e^*Q_vT_e=|\\det H_e|^{2/n}Q_u,\n \\qquad \\lambda_e=|\\det H_e|^{1/n}>0.\n\\]\nThis gives the required compatible family directly.  A group preserving a\npositive-definite metric is conjugate into a compact orthogonal or unitary\ngroup.  Conversely, averaging an inner product over the compact closure of a\nrelatively compact image gives an invariant positive-definite metric.\n\\end{proof}","Consider\n\\[\n A_1=\\begin{pmatrix}1&0\\\\1&0\\end{pmatrix},\n \\qquad\n A_2=\\begin{pmatrix}0&1\\\\0&-1\\end{pmatrix}.\n\\]","\\begin{theorem}[Sharp geometry-class separation]\n\\label{thm:block19-convex-quadratic-separation}\nThe norm \\(\\|x\\|_\\infty\\) satisfies\n\\[\n \\max\\{\\|A_1x\\|_\\infty,\\|A_2x\\|_\\infty\\}\n =\\|x\\|_\\infty\n\\]\nfor every \\(x\\in\\mathbb R^2\\).  In contrast, for every positive-definite quadratic metric \\(Q\\),\n\\[\n \\max\\{\\|A_1\\|_Q,\\|A_2\\|_Q\\}\\geq\\sqrt2.\n\\]\nEquality holds for scalar multiples of the Euclidean metric.  Hence the exact optimal convex/Finsler bound is \\(1\\), while the exact optimal quadratic bound is \\(\\sqrt2\\).\n\\end{theorem}","label{thm:block19-convex-quadratic-separation}\nThe norm \\(\\|x\\|_\\infty\\) satisfies\n\\[\n \\max\\{\\|A_1x\\|_\\infty,\\|A_2x\\|_\\infty\\}\n =\\|x\\|_\\infty\n\\]\nfor every \\(x\\in\\mathbb R^2\\).  In contrast, for every positive-definite quadratic metric \\(Q\\),\n\\[\n \\max\\{\\|A_1\\|_Q,\\|A_2\\|_Q\\}\\geq\\sqrt2.\n\\]\nEquality holds for scalar multiples of the Euclidean metric.  Hence the exact optimal convex/Finsler bound is \\(1\\), while the exact optimal quadratic bound is \\(\\sqrt2\\).\n\\end{theorem}","norm \\(\\|x\\|_\\infty\\) satisfies\n\\[\n \\max\\{\\|A_1x\\|_\\infty,\\|A_2x\\|_\\infty\\}\n =\\|x\\|_\\infty\n\\]\nfor every \\(x\\in\\mathbb R^2\\).  In contrast, for every positive-definite quadratic metric \\(Q\\),\n\\[\n \\max\\{\\|A_1\\|_Q,\\|A_2\\|_Q\\}\\geq\\sqrt2.\n\\]\nEquality holds for scalar multiples of the Euclidean metric.  Hence the exact optimal convex/Finsler bound is \\(1\\), while the exact optimal quadratic bound is \\(\\sqrt2\\).\n\\end{theorem}","\\begin{proof}\nThe two images are\n\\[\n A_1(x_1,x_2)=(x_1,x_1),\n \\qquad\n A_2(x_1,x_2)=(x_2,-x_2),\n\\]\nso the first identity is immediate.","Write\n\\[\n Q=\\begin{pmatrix}a&c\\\\c&b\\end{pmatrix},\n \\qquad\n d=ab-c^2>0.\n\\]\nThe squared induced norms are\n\\[\n u:=\\|A_1\\|_Q^2\n =\\frac{b(a+b+2c)}{d},\n \\qquad\n v:=\\|A_2\\|_Q^2\n =\\frac{a(a+b-2c)}{d}.\n\\]\nA direct calculation gives\n\\[\n ab\\bigl((a+b)^2-4c^2\\bigr)-4(ab-c^2)^2\n =ab(a-b)^2+4c^2(ab-c^2)\\geq0.\n\\]\nThus \\(uv\\geq4\\), and therefore\n\\[\n \\max\\{\\sqrt u,\\sqrt v\\}\\geq(uv)^{1/4}\\geq\\sqrt2.\n\\]\nEquality forces \\(a=b\\) and \\(c=0\\), and then \\(u=v=2\\).\n\\end{proof}","c&b\\end{pmatrix},\n \\qquad\n d=ab-c^2>0.\n\\]\nThe squared induced norms are\n\\[\n u:=\\|A_1\\|_Q^2\n =\\frac{b(a+b+2c)}{d},\n \\qquad\n v:=\\|A_2\\|_Q^2\n =\\frac{a(a+b-2c)}{d}.\n\\]\nA direct calculation gives\n\\[\n ab\\bigl((a+b)^2-4c^2\\bigr)-4(ab-c^2)^2\n =ab(a-b)^2+4c^2(ab-c^2)\\geq0.\n\\]\nThus \\(uv\\geq4\\), and therefore\n\\[\n \\max\\{\\sqrt u,\\sqrt v\\}\\geq(uv)^{1/4}\\geq\\sqrt2.\n\\]\nEquality forces \\(a=b\\) and \\(c=0\\), and then \\(u=v=2\\).\n\\end{proof}","end{pmatrix},\n \\qquad\n d=ab-c^2>0.\n\\]\nThe squared induced norms are\n\\[\n u:=\\|A_1\\|_Q^2\n =\\frac{b(a+b+2c)}{d},\n \\qquad\n v:=\\|A_2\\|_Q^2\n =\\frac{a(a+b-2c)}{d}.\n\\]\nA direct calculation gives\n\\[\n ab\\bigl((a+b)^2-4c^2\\bigr)-4(ab-c^2)^2\n =ab(a-b)^2+4c^2(ab-c^2)\\geq0.\n\\]\nThus \\(uv\\geq4\\), and therefore\n\\[\n \\max\\{\\sqrt u,\\sqrt v\\}\\geq(uv)^{1/4}\\geq\\sqrt2.\n\\]\nEquality forces \\(a=b\\) and \\(c=0\\), and then \\(u=v=2\\).\n\\end{proof}","\\begin{proposition}[Unipotent escape]\n\\label{prop:block19-unipotent-escape}\nLet\n\\[\n H=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}.\n\\]\nThere is no positive-definite metric \\(Q\\) with \\(H^*QH=Q\\).  Nevertheless, for\n\\[\n Q_s=\\operatorname{diag}(s^2,1),\n\\]\nwe have\n\\[\n D_{Q_s}(H)\\longrightarrow1\n \\qquad(s\\downarrow0),\n\\]\nwhile\n\\[\n \\kappa_2(Q_s)=s^{-2}\\longrightarrow\\infty.\n\\]\nThus the metric-completion defect has zero infimum but is not attained.\n\\end{proposition}","0&1\\end{pmatrix}.\n\\]\nThere is no positive-definite metric \\(Q\\) with \\(H^*QH=Q\\).  Nevertheless, for\n\\[\n Q_s=\\operatorname{diag}(s^2,1),\n\\]\nwe have\n\\[\n D_{Q_s}(H)\\longrightarrow1\n \\qquad(s\\downarrow0),\n\\]\nwhile\n\\[\n \\kappa_2(Q_s)=s^{-2}\\longrightarrow\\infty.\n\\]\nThus the metric-completion defect has zero infimum but is not attained.\n\\end{proposition}","\\begin{proof}\nWriting \\(Q=\\begin{psmallmatrix}a&c\\\\c&b\\end{psmallmatrix}\\), the equation \\(H^*QH=Q\\) forces \\(a=0\\), contradicting positive definiteness.  For \\(Q_s\\), the metric-normalized matrix is\n\\[\n \\begin{pmatrix}s&0","c&b\\end{psmallmatrix}\\), the equation \\(H^*QH=Q\\) forces \\(a=0\\), contradicting positive definiteness.  For \\(Q_s\\), the metric-normalized matrix is\n\\[\n \\begin{pmatrix}s&0","c&b\\end{psmallmatrix}\\), the equation \\(H^*QH=Q\\) forces \\(a=0\\), contradicting positive definiteness.  For \\(Q_s\\), the metric-normalized matrix is\n\\[\n \\begin{pmatrix}s&0\\\\0&1\\end{pmatrix}\n H\n \\begin{pmatrix}s^{-1}&0\\\\0&1\\end{pmatrix}\n =\n \\begin{pmatrix}1&s\\\\0&1\\end{pmatrix},\n\\]\nwhose singular-value condition number tends to one.  The conditioning formula is immediate.\n\\end{proof}","Suppose richer categories \\(\\widetilde{\\mathcal C}\\) and \\(\\widetilde{\\mathcal D}\\) carry forgetful functors\n\\[\n U:\\widetilde{\\mathcal C}\\to\\mathcal C,\n \\qquad\n V:\\widetilde{\\mathcal D}\\to\\mathcal D.\n\\]","\\begin{proposition}[Forgetful descent and enhancement lifting]\n\\label{prop:block19-forgetful-lifting}\nA rich bridge\n\\[\n \\widetilde B:\\widetilde{\\mathcal C}\\to\\widetilde{\\mathcal D}\n\\]\nautomatically determines an underlying bridge\n\\[\n V\\widetilde B:\\widetilde{\\mathcal C}\\to\\mathcal D.\n\\]\nGiven only a bare bridge \\(B:\\mathcal C\\to\\mathcal D\\), constructing \\(\\widetilde B\\) together with a comparison\n\\[\n V\\widetilde B\\cong BU\n\\]\nis a lifting problem.  Its existence, uniqueness, and residual automorphisms are additional passport fields and may carry typed obstruction objects.\n\\end{proposition}","gin{proposition}[Forgetful descent and enhancement lifting]\n\\label{prop:block19-forgetful-lifting}\nA rich bridge\n\\[\n \\widetilde B:\\widetilde{\\mathcal C}\\to\\widetilde{\\mathcal D}\n\\]\nautomatically determines an underlying bridge\n\\[\n V\\widetilde B:\\widetilde{\\mathcal C}\\to\\mathcal D.\n\\]\nGiven only a bare bridge \\(B:\\mathcal C\\to\\mathcal D\\), constructing \\(\\widetilde B\\) together with a comparison\n\\[\n V\\widetilde B\\cong BU\n\\]\nis a lifting problem.  Its existence, uniqueness, and residual automorphisms are additional passport fields and may carry typed obstruction objects.\n\\end{proposition}","Task changes and admissibility changes have opposite-looking effects.  If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information:\n\\[\n N_{\\mathrm{strong}}\\subseteq N_{\\mathrm{weak}},\n \\qquad\n Q_{\\mathrm{strong}}\\twoheadrightarrow Q_{\\mathrm{weak}}.\n\\]\nIf an admissible realization class is enlarged, the optimized defect can only decrease:\n\\[\n \\Gamma\\subseteq\\Gamma'\n \\quad\\Longrightarrow\\quad\n \\delta_{\\Gamma'}\\leq\\delta_\\Gamma.\n\\]\nThese are different orders and should not be merged into one scalar notion of a stronger contract.","\\begin{definition}[Doctrine-indexed task fiber]\n\\label{def:block19-task-fiber}\nFor a doctrine \\(D\\) and represented object \\(X\\), let\n\\[\n \\mathsf{Task}_D(X)\n\\]\ndenote the ordered family of continuation-stable semantic quotients admitted by the declared query universe.  Inside one fiber, kernel or congruence calculus governs task change.  A change of doctrine\n\\[\n D\\longrightarrow D'\n\\]\nrequires a bridge passport specifying carrier transport, preserved constructions, comparison maps, and any lifting obstruction.\n\\end{definition}","ning are governed by the exact sequences of \\cref{sec:block17-semantic-quotients,sec:block18-carrier-observer}.\n\\item Carrier changes are legal exactly on the factorization loci certified by \\cref{thm:block18-carrier-factorization}; otherwise the joint semantic image is the canonical exact completion preserving both views.\n\\item Observer changes may introduce additional relation objects such as \\(\\Rel_K(E)\\); they are not inferred from carrier compatibility alone.\n\\item Enlarging an admissible realization class weakens optimized defect but need not produce an attained completion.\n\\item Crossing to richer structure is a typed lifting problem and does not follow from the underlying bridge without an enhancement certificate.\n\\end{enumerate}\n\\end{theorem}","LIN-01 & C-finite recurrence \\(\\leftrightarrow\\) rational OGF \\(\\leftrightarrow\\) finite state & \\StatusClassical & \\nolinkurl{Projector_Seed_Calculus_for_Integer_Sequences_Local_Polyphase_Extension.tex}","HOL-01 & P-recursive \\(\\leftrightarrow\\) D-finite coefficient translation & \\StatusClassical & \\nolinkurl{Projector_Seed_Calculus_for_Integer_Sequences_Local_Polyphase_Extension.tex}","DIG-01 & Finite digit realization \\(\\leftrightarrow\\) finite k-kernel & \\StatusInternal & \\nolinkurl{Projector_Seed_Calculus_for_Integer_Sequences_Local_Polyphase_Extension.tex}","ALG-01 & Selected algebraic branch \\(\\to\\) packet polynomial system & \\StatusInternal & \\nolinkurl{Projector_Seed_Calculus_for_Integer_Sequences_Local_Polyphase_Extension.tex}","TRN-01 & Invertible transform \\(\\leftrightarrow\\) transported packet resolution & \\StatusInternal & \\nolinkurl{Projector_Seed_Calculus_for_Integer_Sequences_Local_Polyphase_Extension.tex}","PRD-01 & Euler-product exponents \\(\\leftrightarrow\\) logarithmic-derivative recurrence & \\StatusInternal & \\nolinkurl{Projector_Seed_Calculus_for_Integer_Sequences_Local_Polyphase_Extension.tex}","LOC-02 & Exact Mahler \\(\\leftrightarrow\\) affine polyphase coordinate transport & \\StatusInternal & \\nolinkurl{Projector_Seed_Calculus_for_Integer_Sequences_Local_Polyphase_Extension.tex; Addendum 04 strengthens it}","FMH-01 & Rational functional-Mahler mask \\(\\to\\) Cartier representation & \\StatusInternal & \\nolinkurl{Komorebi_Custom_GF_Cartier_Padic_Research_Report_0.3.pdf}","FMH-02 & Functional-Mahler finite state \\(\\to\\) automaton & \\StatusInternal & \\nolinkurl{Komorebi_Custom_GF_Cartier_Padic_Research_Report_0.3.pdf}","FMH-03 & Finite-mask Cartier transitions \\(\\leftrightarrow\\) carry matrices & \\StatusInternal & \\nolinkurl{Komorebi_Custom_GF_Cartier_Padic_Research_Report_0.3.pdf}","[3pt]\nFunctional-Mahler/Cartier report 0.3 & \\nolinkurl{Komorebi_Custom_GF_Cartier_Padic_Research_Report_0.3.pdf} & \\texttt{\\seqsplit{4480786c49abcb60fb39e07acf0e250c9a3af68c38ac7b4801f34bdaefd5a936}} & Primary source for the rational functional-Mahler \\(\\to\\) Cartier \\(\\to\\) automaton direction and finite-mask Cartier/carry identification.","\\midrule\nEven indicator & $\\mathbb F_2$ / 10 & $1+x^2$ & $1+z^2$ & $1+z^2$ & 3 & 8","Alternating unit & $\\mathbb F_3$ / 12 & $1+x$ & $1+z$ & $1+2z+z^2$ & 2 & 9","Period-three word & $\\mathbb F_2$ / 110 & $1+x+x^2$ & $1+z+z^2$ & $1+z+z^2$ & 3 & 8","\\midrule\nEnumerated normalized objects & $p=2$ through period 8, $p=3$ through period 6; 583 primitive unit-periodic words.","\\begin{table}[htbp]\n\\centering\n\\caption{Named exact examples in the rational-mask digital atlas.}\n\\label{tab:block04-examples}\n\\scriptsize\n\\begin{tabularx}{\\textwidth}{@{}L{0.18\\textwidth}L{0.07\\textwidth}L{0.17\\textwidth}L{0.20\\textwidth}L{0.09\\textwidth}Y@{}}\n\\toprule\nExample & Field & Reduced mask $A/B$ & First coefficients & $r_p/\\kappa_p$ & \\shortstack[l]{Collapse status /","\\midrule\nCantor-digit indicator & $\\mathbb F_3$ & $(1+z)/(1)$ & \\texttt{\\seqsplit{110110000110110000}} & $1/2$ & \\shortstack[l]{non-collapse","bound $9$}","Square-root denominator & $\\mathbb F_3$ & $(1)/(1+z)$ & \\texttt{\\seqsplit{121121000121121000}} & $2/4$ & \\shortstack[l]{non-collapse","Alternating collapse & $\\mathbb F_3$ & $(1+2z+z^2)/(1)$ & \\texttt{\\seqsplit{121212121212121212}} & $1/2$ & \\shortstack[l]{collapse","Finite transient & $\\mathbb F_2$ & $(1)/(1+z+z^2)$ & \\texttt{\\seqsplit{111000000000000000}} & $3/4$ & \\shortstack[l]{collapse","bound $8$}","\\midrule\nEnumerated reduced unit masks & $p=2$ with $\\deg A,\\deg B\\leq 3$, $p=3$ with $\\deg A,\\deg B\\leq 2$, $p=5$ with $\\deg A,\\deg B\\leq 1$; 125 masks in total.","\\midrule\nTernary Cantor-digit indicator & 3 / $\\mathbb F_3$ & 1 & 2 & automatic and regular","Thue-Morse parity & 2 / $\\mathbb F_2$ & 2 & 2 & automatic and regular","Identity sequence $n$ & 2 / $\\mathbb Q$ & 2 & infinite & regular, not automatic","Binary digit sum $s_2(n)$ & 2 / $\\mathbb Q$ & 2 & infinite & regular, not automatic","Square sequence $n^2$ & 2 / $\\mathbb Q$ & 3 & infinite & regular, not automatic","\\midrule\nExhaustive finite-state enumeration & base 2 over $\\mathbb F_2$ through 3 raw states: 5,898 raw candidates, 2,830 leading-zero-consistent candidates, 94 distinct minimized automata; base 2 over $\\mathbb F_3$ through 3 raw states: 19,830 raw candidates, 6,990 leading-zero-consistent candidates, 297 distinct minimized automata; base 3 over $\\mathbb F_2$ through 2 raw states: 258 raw candidates, 170 leading-zero-consistent candidates, 26 distinct minimized automata.","Unbounded regular examples & Polynomial sequences $n^m$ for bases 2 and 3 through degree 6, plus digit sums in bases 2, 3, and 4; 2,438 exact checks.","\\midrule\nDigit sum $s_k$ & $(1,1)$ & none & Saturated rank-two lattice; no modular defect.","Doubled digit sum $2s_k$ & $(2,2)$ & 2 & Rank-two defect $(\\mathbb Z/2)^2$; total semantic collapse modulo $2$.","Constant sequence $2$ & $(2)$ & 2 & One raw ghost coordinate modulo $2$; one semantic state.","Constant sequence $6$ & $(6)$ & 2, 3 & Bad primes $2$ and $3$; good reduction at every other prime.","Fitting and prime-power profiles & 8840 Fitting cases and 39680 prime-power profile cases over $p=2,3,5,7$.","\\midrule\nConstant-one function $\\mathbf 1$ & $\\beta_{1}=1$ & $1,1,1,1,1,1,\\ldots$ & $1,1,1,1,1,1,\\ldots$","M\\\"obius $\\mu$ & $\\beta_{1}=-1$ & $1,-1,0,0,0,0,\\ldots$ & $-1,-1,-1,-1,-1,-1,\\ldots$","Liouville $\\lambda$ & $\\beta_{1}=-1$, $\\beta_{2}=1$ & $1,-1,1,-1,1,-1,\\ldots$ & $-1,1,-1,1,-1,1,\\ldots$","Squarefree indicator $\\mu^2$ & $\\beta_{1}=1$, $\\beta_{2}=-1$ & $1,1,0,0,0,0,\\ldots$ & $1,-1,1,-1,1,-1,\\ldots$","Three-fold divisor function $\\tau_3$ & $\\beta_{1}=3$ & $1,3,6,10,15,21,\\ldots$ & $3,3,3,3,3,3,\\ldots$","\\midrule\n\\(x^2\\), branch 2q+0 & \\( (0, 1, 2) \\) & \\( (0, 4, 8) \\)","\\(x^2\\), branch 2q+1 & \\( (0, 1, 2) \\) & \\( (1, 8, 8) \\)","\\((1+2)^x\\), branch 2q & \\(c_r=2^r,\\ 0\\leq r\\leq4\\) & \\(d_k=8^k,\\ 0\\leq k\\leq4\\)","\\((1+3)^x\\), branch 3q & \\(c_r=3^r,\\ 0\\leq r\\leq4\\) & \\(d_k=63^k,\\ 0\\leq k\\leq4\\)","\\((1+5)^x\\), branch 5q & \\(c_r=5^r,\\ 0\\leq r\\leq4\\) & \\(d_k=7775^k,\\ 0\\leq k\\leq4\\)","mod-2 rank distribution, M=2,d=3 & \\( (2, 2) \\) & \\( (4, 4) \\)","\\begin{table}[htbp]\n\\centering\n\\small\n\\caption{Representative exact order profiles in the binomial--Fuchsian packet family.}\n\\label{tab:block09-examples}\n\\begin{tabularx}{\\textwidth}{@{}L{0.17\\textwidth}L{0.13\\textwidth}cccccY@{}}\n\\toprule\nSource & Packet & $r_{\\rm tail}$ & $r_D$ & $r_{\\rm orb}$ & Alias & Cancel & Interpretation","\\midrule\n$(1-z)^{-1/2}$ & $N=2,\\ j=0$ & 1 & 2 & 2 & 0 & 0 & irreducible nonintegral packet","$(1-z)^{-1/3}$ & $N=3,\\ j=2$ & 1 & 3 & 3 & 0 & 0 & finite local exponent torsion","$(1-z)^{-2}$ & $N=3,\\ j=1$ & 1 & 1 & 3 & 0 & 2 & rational collapse","$(1-z)^2$ & $N=4,\\ j=1$ & 0 & 1 & 3 & 1 & 2 & finite-support polynomial","$(1-z)^2$ & $N=4,\\ j=3$ & 0 & 0 & 3 & 1 & 3 & zero address packet","\\begin{table}[H]\n\\centering\n\\scriptsize\n\\caption{Representative exact degree and collision profiles in the Algebraic atlas algebraic packet atlas.}\n\\label{tab:block10-examples}\n\\begin{tabularx}{\\textwidth}{@{}L{0.17\\textwidth}L{0.14\\textwidth}cccccY@{}}\n\\toprule\nSource & Packet & $d$ & $d^N$ & degree & tuple loss & stabilizer loss & Interpretation","\\midrule\n$C=1+zC^2$ & parity $(E,O)$ & 2 & 4 & 4 & 0 & 0 & sharp Weil-restriction bound","$\\sqrt{(1-z)(1-2z)}$ & parity even/odd & 2 & 4 & 4 & 0 & 0 & generic biquadratic packet","$\\sqrt{1-z^2}$ & even packet & 2 & 4 & 2 & 1 & 1 & rotation alias plus descent stabilizer","$\\sqrt{1-z^2}$ & odd packet & 2 & 4 & 1 & 1 & 2 & exact zero-address cancellation","$\\sqrt{(1-z)^2}=1-z$ & normalized source & 1 & 1 & 1 & 0 & 0 & nonminimal quadratic equation rejected","\\begin{table}[htbp]\n\\centering\n\\caption{Raw and compressed diagonal Fa\\`a di Bruno term counts for three intermediate packets.}\n\\label{tab:block11-faa-counts}\n\\small\n\\begin{tabular}{rrrr}\n\\toprule\nOrder $n$ & Raw packet terms & Colored profiles & Compression factor","\\begin{table}[htbp]\n\\centering\n\\caption{Exact packet signatures for the marked full $q$-ary tree family $u=z+u^q$.}\n\\label{tab:block11-qary-phases}\n\\small\n\\begin{tabular}{rrrrrr}\n\\toprule\n$q$ & $N$ & $g=\\gcd(N,q-1)$ & Active packets & Packet orbit & Active addresses","\\midrule\n2 & 5 & 1 & 5 & 1 & $\\{0,1,2,3,4\\}$","3 & 4 & 2 & 2 & 1 & $\\{1,3\\}$","3 & 5 & 1 & 5 & 2 & $\\{0,1,2,3,4\\}$","4 & 4 & 1 & 4 & 3 & $\\{0,1,2,3\\}$","4 & 6 & 3 & 2 & 1 & $\\{1,4\\}$","5 & 6 & 2 & 3 & 2 & $\\{1,3,5\\}$","5 & 7 & 1 & 7 & 4 & $\\{0,1,2,3,4,5,6\\}$","6 & 10 & 5 & 2 & 1 & $\\{1,6\\}$","egin{landscape}\n\\begin{table}[p]\n\\centering\n\\caption{Authoritative broad-stratum bridge matrix for the certified global synthesis atlas. Every decorated equivalence is restricted to the named normalized overlap; the undecorated broad categories are not thereby identified.}\n\\label{tab:block13-bridge-matrix}\n\\scriptsize\n\\setlength{\\tabcolsep}{3.7pt}\n\\begin{tabular}{@{}lcccccccc@{}}\n\\toprule\nSource $\\backslash$ target & LIN & HOL & DIG & ALG & TREE & PROD & LOC & GAL","\\midrule\nLIN  & $\\mathsf E$ & $\\mathsf{FF}$ & $\\mathsf E_{\\rm per}$ & -- & -- & -- & $\\mathsf E_{\\rm per}$ & --","HOL  & $\\mathsf P$ & $\\mathsf E$ & -- & $\\mathsf E_{\\rm Fuss}/\\mathsf P$ & $\\mathsf E_{\\rm Fuss}/\\mathsf P$ & -- & -- & --","DIG  & $\\mathsf E_{\\rm per}$ & $\\mathsf P$ & $\\mathsf E$ & $\\mathsf P$ & -- & -- & $\\mathsf E_{\\rm rm}$ & --","ALG  & -- & $\\mathsf E_{\\rm Fuss}/\\mathsf D$ & $\\mathsf P$ & $\\mathsf E$ & $\\mathsf E_{\\rm Fuss}/\\mathsf P$ & -- & -- & --","TREE & -- & $\\mathsf E_{\\rm Fuss}/\\mathsf D$ & -- & $\\mathsf E_{\\rm Fuss}/\\mathsf D$ & $\\mathsf E$ & -- & -- & --","PROD & $\\mathsf D$ & -- & -- & -- & -- & $\\mathsf E$ & $\\mathsf P$ & --","LOC  & $\\mathsf E_{\\rm per}$ & -- & $\\mathsf E_{\\rm rm}$ & -- & -- & $\\mathsf P$ & $\\mathsf E$ & --","GAL  & -- & -- & -- & -- & -- & -- & -- & $\\mathsf E$","\\bottomrule\n\\end{tabular}\n\\vspace{5pt}\n\\parbox{0.96\\linewidth}{\\scriptsize\n\\textbf{Legend.} $\\mathsf E$: equivalence; $\\mathsf{FF}$: fully faithful inclusion; $\\mathsf D$: information-losing directional functor; $\\mathsf P$: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts $\\mathrm{per}$, $\\mathrm{rm}$, and $\\mathrm{Fuss}$ denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form $\\mathsf E_{\\mathrm{Fuss}}/\\mathsf D$ means equivalence on the marked Fuss slice and directional loss on the ambi","\\bottomrule\n\\end{tabular}\n\\vspace{5pt}\n\\parbox{0.96\\linewidth}{\\scriptsize\n\\textbf{Legend.} $\\mathsf E$: equivalence; $\\mathsf{FF}$: fully faithful inclusion; $\\mathsf D$: information-losing directional functor; $\\mathsf P$: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts $\\mathrm{per}$, $\\mathrm{rm}$, and $\\mathrm{Fuss}$ denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form $\\mathsf E_{\\mathrm{Fuss}}/\\mathsf D$ means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahle","\\bottomrule\n\\end{tabular}\n\\vspace{5pt}\n\\parbox{0.96\\linewidth}{\\scriptsize\n\\textbf{Legend.} $\\mathsf E$: equivalence; $\\mathsf{FF}$: fully faithful inclusion; $\\mathsf D$: information-losing directional functor; $\\mathsf P$: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts $\\mathrm{per}$, $\\mathrm{rm}$, and $\\mathrm{Fuss}$ denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form $\\mathsf E_{\\mathrm{Fuss}}/\\mathsf D$ means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahler coefficient equations and Newton--Mah","\\bottomrule\n\\end{tabular}\n\\vspace{5pt}\n\\parbox{0.96\\linewidth}{\\scriptsize\n\\textbf{Legend.} $\\mathsf E$: equivalence; $\\mathsf{FF}$: fully faithful inclusion; $\\mathsf D$: information-losing directional functor; $\\mathsf P$: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts $\\mathrm{per}$, $\\mathrm{rm}$, and $\\mathrm{Fuss}$ denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form $\\mathsf E_{\\mathrm{Fuss}}/\\mathsf D$ means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahler coefficient equations and Newton--Mahler interpolation remain distinct despite sharing the","\\bottomrule\n\\end{tabular}\n\\vspace{5pt}\n\\parbox{0.96\\linewidth}{\\scriptsize\n\\textbf{Legend.} $\\mathsf E$: equivalence; $\\mathsf{FF}$: fully faithful inclusion; $\\mathsf D$: information-losing directional functor; $\\mathsf P$: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts $\\mathrm{per}$, $\\mathrm{rm}$, and $\\mathrm{Fuss}$ denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form $\\mathsf E_{\\mathrm{Fuss}}/\\mathsf D$ means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahler coefficient equations and Newton--Mahler interpolation remain distinct despite sharing the LOC row.}\n\\end{table}\n\\end{landscape}","} $\\mathsf E$: equivalence; $\\mathsf{FF}$: fully faithful inclusion; $\\mathsf D$: information-losing directional functor; $\\mathsf P$: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts $\\mathrm{per}$, $\\mathrm{rm}$, and $\\mathrm{Fuss}$ denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form $\\mathsf E_{\\mathrm{Fuss}}/\\mathsf D$ means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahler coefficient equations and Newton--Mahler interpolation remain distinct despite sharing the LOC row.}\n\\end{table}\n\\end{landscape}","Weighted $q$-ary tree--algebraic--holonomic and tagged-invariant suite & 119,453","LOC & continuous $\\mathbb Z_p$-valued functions; Newton--Mahler coefficients, complete clopen branch packets, and affine branch coordinates & canonical Mahler coefficient sequence; complete branch analysis and synthesis at each declared depth & functional-Mahler equations require a separately declared coefficient lift and interpolation constructor","Fuss & marked $U=z+\\alpha U^q$ tree system, selected algebraic branch, and compressed P-recursive coefficient chart & parameter pair $(q,\\alpha)$ with its support and singularity-orbit package & forgetting the branch mark or generator grammar is information-losing","Gal & subgroup, fixed field, and faithful Reynolds projector for one fixed finite Galois extension & actual subgroup $H\\leq G$ with functorial core, normal-closure, and double-coset data & projective Brauer obstruction, modular or wild failure, and the tensor-fiber gate remain separately typed","\\midrule\n$\\mathbb {F}_2$/ word 10 & 2 & 2 & 2 & 3 & 2","$\\mathbb {F}_2$/ word 110 & 3 & 2 & 2 & 3 & 2","$\\mathbb {F}_3$/ word 12 & 2 & 1 & 1 & 2 & 1","\\begin{table}[htbp]\n\\centering\n\\caption{Named weighted q-ary examples in the cross-stratum atlas.}\n\\label{tab:block13-qary-examples}\n\\scriptsize\n\\begin{tabularx}{\\textwidth}{@{}L{0.10\\textwidth}L{0.11\\textwidth}L{0.12\\textwidth}L{0.25\\textwidth}L{0.17\\textwidth}Y@{}}\n\\toprule\n$q$ & $\\lambda$ & Support period & First supported coefficients & Active addresses at $N=6$ & Packet-orbit size","\\midrule\n2 & $1$ & 1 & $(1, 1, 2, 5,\\ldots)$ & $\\{0,1,2,3,4,5\\}$ & 1","3 & $2$ & 2 & $(1, 2, 12, 96,\\ldots)$ & $\\{1,3,5\\}$ & 1","4 & $-1$ & 3 & $(1, -1, 4, -22,\\ldots)$ & $\\{1,4\\}$ & 1","5 & $1/2$ & 4 & $(1, 1/2, 5/4, 35/8,\\ldots)$ & $\\{1,3,5\\}$ & 2","Continuous local & $C(\\mathbb Z_p,\\mathbb Z_p)$ & Canonical Mahler coefficients and complete clopen branch packets & No universal local-analytic normal form or optimal mixed-radix tree","Marked Fuss & $U=z+\\alpha U^q$ & Marked parameters $(q,\\alpha)$, algebraic branch, recurrence, tree support and orbit data & Complete only on the marked nonlinear--algebraic--holonomic family","[3pt]\nPO-13 & Broaden local Mahler theory & \\StatusInternal & Closed for continuous \\(\\mathbb Z_p\\)-valued functions, canonical Newton--Mahler coefficients, complete clopen and coefficient packets, finite-degree affine channels, sharp standard packet sizes, and principal-unit precision ladders. Exact higher kernel valuations, optimal mixed-radix trees, nonlinear local systems, state recognition, and a combined spectral--local signature remain open.","[3pt]\nPO-15 & Broaden algebraic packet minimality & \\StatusInternal & Finite selected-branch normalization, exact packet polynomial systems, the generic \\(d^N\\) Weil-restriction bound, and marked collision signatures are closed. Minimal multivariate system dimension, optimal symmetry-reduced degree bounds, ramified Puiseux passports, and universal branch-recognition algorithms remain open."]}