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hull}\\\\[.45em]\n\\longrightarrow\\text{seed rank}\\longrightarrow\\text{packet coordinates}\\\\[.45em]\n\\longrightarrow\\text{transfer blocks}\\longrightarrow\\text{functorial realization}.\n\\end{gathered}\\)}","What can a finite seed family generate? & Evaluation map \\(A^{\\oplus r}\\to M\\) & The generated hull is a quotient of a finite free module.","How many seeds are required? & \\(\\srank_A(M)\\) & For a finitely generated module, pass to \\(M/\\Jac(A)M\\). When the quotient algebra is semisimple Artinian, compute its simple multiplicities.","How are vectors and operators resolved? & Complete idempotents \\((P_\\alpha)\\) & \\(X\\cong\\bigoplus_\\alpha P_\\alpha X\\) and \\(T\\leftrightarrow(P_\\beta TP_\\alpha)\\).","\\midrule\n\\endhead\n\\(\\mathfrak G=(Q,I)\\) & Typed grammar & A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.","\\(\\Lambda_{\\mathfrak G}\\) & Grammar algebra & The quotient path algebra \\(\\bbk Q/I\\).","\\(\\A_X\\) & Realized algebra & The image of the grammar algebra in \\(\\End_{\\bbk}(|X|)\\).","\\(S\\subseteq M\\) & Seed family & The vectors from which the operator orbit is generated.","\\(\\Hull_A(S)\\) & Generated hull & The submodule \\(AS\\), or its declared topological closure.","\\(\\srank_A(M)\\) & Seed rank & The smallest cardinality of a generating family. A finite value is the minimum number of seeds.","\\(\\Pcal=(P_\\alpha)\\) & Packet system & Pairwise annihilating idempotents summing to the identity.","\\(X_\\alpha\\) & Packet space & The range \\(P_\\alpha X\\).","\\(\\mathcal A_{\\Pcal}\\) & Analysis & The map \\(x\\mapsto(P_\\alpha x)_\\alpha\\).","\\(\\mathcal S_{\\Pcal}\\) & Synthesis & The map \\((x_\\alpha)\\mapsto\\sum_\\alpha x_\\alpha\\).","\\(T_{\\beta\\alpha}\\) & Transfer block & The source and target block \\(P_\\beta TP_\\alpha:X_\\alpha\\to X_\\beta\\).","\\(\\Supp_{\\Pcal}(x)\\) & Packet support & The packet indices on which \\(x\\) has a nonzero coordinate.","\\(\\psrank\\) & Rank for seeds confined to individual packets & The minimal number of generating seeds each supported in a single packet.","\\begin{microexample}[title={R1: one operator with several primary blocks}]\nLet \\(T\\) act on a finite dimensional vector space.  Regard \\(X\\) as a \\(\\bbk[t]\\)-module by \\(t\\cdot x=Tx\\).  If \\(T\\) has three Jordan blocks for the same eigenvalue, then three seeds are necessary and sufficient for that primary component.  The largest number of blocks over any irreducible polynomial is the seed rank.\n\\end{microexample}","\\begin{microexample}[title={R2: a matrix algebra with multiplicity}]\nLet \\(A=M_2(\\bbk)\\) act diagonally on \\(M=(\\bbk^2)^{\\oplus3}\\).  One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module.  Hence\n\\[\n\\srank_A(M)=\\left\\lceil\\frac{3}{2}\\right\\rceil=2.\n\\]\nThis is the simplest nontrivial instance of the semisimple multiplicity formula.\n\\end{microexample}","\\begin{microexample}[title={R3: an exact but oblique packet split}]\nOn \\(X=\\bbk^2\\), set\n\\[\nP_1=\\begin{pmatrix}1&-1","\\begin{microexample}[title={R3: an exact but oblique packet split}]\nOn \\(X=\\bbk^2\\), set\n\\[\nP_1=\\begin{pmatrix}1&-1\\\\0&0\\end{pmatrix},\n\\qquad\nP_2=\\begin{pmatrix}0&1\\\\0&1\\end{pmatrix}.\n\\]\nThen \\(P_i^2=P_i\\), \\(P_1P_2=P_2P_1=0\\), and \\(P_1+P_2=I\\).  The ranges form a direct sum, but over \\(\\mathbb R\\) the projectors are not orthogonal in the Euclidean inner product.  Exact reconstruction and numerical conditioning are therefore separate questions.\n\\end{microexample}","0&1\\end{pmatrix}.\n\\]\nThen \\(P_i^2=P_i\\), \\(P_1P_2=P_2P_1=0\\), and \\(P_1+P_2=I\\).  The ranges form a direct sum, but over \\(\\mathbb R\\) the projectors are not orthogonal in the Euclidean inner product.  Exact reconstruction and numerical conditioning are therefore separate questions.\n\\end{microexample}","\\begin{microexample}[title={R4: a directed packet system}]\nLet \\(X=X_1\\oplus X_2\\oplus X_3\\).  Suppose the generating operators permit transfers \\(1\\to2\\) and \\(2\\to3\\), with no path entering packet \\(1\\).  Packet \\(1\\) is the unique source component.  If the corner action on \\(X_1\\) is cyclic and the propagation hypotheses of \\cref{thm:source-component-rank} hold, one seed in \\(X_1\\) can generate all three packets.\n\\end{microexample}","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$.","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finit","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partial","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the th","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the","\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder cou","begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder counts strict steps.","vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder counts strict steps.","f continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}\n\\RecordID{PSC-I-D-COEFFICIENT-REGIME}\n\\paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\\max\\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder counts strict steps.","\\begin{definition}[Canonical theorem record]\n\\label{def:theorem-record}\nA canonical theorem record is the tuple\n\\[\n(\\mathrm{ID},\\mathrm{status},\\mathrm{hypotheses},\\mathrm{statement},\n\\mathrm{dependencies},\\mathrm{proof},\\mathrm{exports},\\mathrm{boundary}).\n\\]\nA downstream paper cites the stable record ID.  A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.\n\\end{definition}\n\\RecordID{PSC-I-D-THEOREM-RECORD}","\\begin{definition}[Typed operator grammar]\n\\label{def:typed-grammar}\nLet $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","ion}[Typed operator grammar]\n\\label{def:typed-grammar}\nLet $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","r]\n\\label{def:typed-grammar}\nLet $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","k G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","l on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","linear combinations over $\\bbk$ of paths having common source and common target.  Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$.  A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TYPED-GRAMMAR}","\\begin{definition}[Realization on a single space]\n\\label{def:collapsed-realization}\nFor a typed realization $X$, define its realization on a single space by\n\\[\n|X|:=\\bigoplus_{v\\in Q_0}X_v.\n\\]\nEvery vertex idempotent acts as a coordinate projector on $|X|$, and every path acts as a block operator from its source space to its target space.\n\\end{definition}\n\\RecordID{PSC-I-D-COLLAPSED-REALIZATION}","\\begin{theorem}[Typed realization and module equivalence]\n\\label{thm:typed-module-equivalence}\nThe category of realizations of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","egin{theorem}[Typed realization and module equivalence]\n\\label{thm:typed-module-equivalence}\nThe category of realizations of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","ealizations of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","ns of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","unital left modules over $\\Lambda_{\\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}\n\\RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}\n\\Depends{PSC-I-D-TYPED-GRAMMAR.}","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constr","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original verte","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These","\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}",", the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","ambda_{\\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","n a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","fine the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","ts arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","linearly.  Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","ment of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$.  The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct.  For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}","\\begin{microexample}[title={A grammar with two vertices}]\nTake two vertices and one arrow \\(a:1\\to2\\).  A realization is simply a pair of spaces and a linear map\n\\[\nX_1\\xrightarrow{\\,X_a\\,}X_2.\n\\]\nAfter collapse, \\(|X|=X_1\\oplus X_2\\), the vertex idempotents are the two coordinate projectors, and \\(a\\) is a single block outside the diagonal.  This elementary case already contains the viewpoint of packets and transfer blocks used later.\n\\end{microexample}","\\subsection{The realized action and faithful reduction}\n\\begin{definition}[Realized operator algebra]\n\\label{def:realized-algebra}\nFor a realization $X$, the realized operator algebra is\n\\[\n\\A_X:=\\im\\!\\left(\\Lambda_{\\mathfrak G}\\longrightarrow\\End_{\\bbk}(|X|)\\right).\n\\]\nIt records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.\n\\end{definition}\n\\RecordID{PSC-I-D-REALIZED-ALGEBRA}","\\begin{proposition}[Action factorization and hull invariance]\n\\label{prop:action-factorization}\nLet $S\\subseteq |X|$.  The generated submodule under the grammar algebra equals the generated submodule under the realized operator algebra:\n\\[\n\\Lambda_{\\mathfrak G}S=\\A_XS.\n\\]\nConsequently, every statement about seed rank depends only on the realized action, not on redundant relations in a chosen presentation of the grammar.\n\\end{proposition}\n\\RecordID{PSC-I-P-ACTION-FACTORIZATION}\n\\Depends{PSC-I-D-REALIZED-ALGEBRA.}","\\begin{proof}\nThe action map $\\rho:\\Lambda_{\\mathfrak G}\\twoheadrightarrow\\A_X$ is surjective by definition and satisfies $\\lambda s=\\rho(\\lambda)s$.  Hence every grammar orbit vector lies in $\\A_XS$.  The reverse inclusion holds because every element of $\\A_X$ is $\\rho(\\lambda)$ for some $\\lambda$.\n\\end{proof}","\\begin{keyidea}\nReplacing the abstract grammar algebra by its image \\(\\A_X\\) removes syntax that acts as zero.  This is a faithful reduction of the action.  It does not change any orbit, hull, or computation of seed rank.\n\\end{keyidea}","\\subsection{Seeds, addressed hulls, and natural presentations}\n\\begin{definition}[Linear seed system]\n\\label{def:seed-system}\nA linear seed system is a tuple\n\\[\n\\mathfrak S=(X,\\A,S,\\Pcal,\\tau),\n\\]\nwhere $X$ is a unital left module over $\\A$, $\\A$ is fixed independently of the target element, $S=(s_1,\\ldots,s_r)$ is a finite distinguished seed family, $\\Pcal$ is either empty or a declared packet system, and $\\tau$ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-SYSTEM}","{Seeds, addressed hulls, and natural presentations}\n\\begin{definition}[Linear seed system]\n\\label{def:seed-system}\nA linear seed system is a tuple\n\\[\n\\mathfrak S=(X,\\A,S,\\Pcal,\\tau),\n\\]\nwhere $X$ is a unital left module over $\\A$, $\\A$ is fixed independently of the target element, $S=(s_1,\\ldots,s_r)$ is a finite distinguished seed family, $\\Pcal$ is either empty or a declared packet system, and $\\tau$ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-SYSTEM}","\\begin{definition}[Generated hull]\n\\label{def:generated-hull}\nFor $S\\subseteq X$, define\n\\[\n\\Hull_{\\A}(S):=\\sum_{s\\in S}\\A s.\n\\]\nIn the topological regime define\n\\[\n\\Hull_{\\A,\\tau}(S):=\\cl{\\Hull_{\\A}(S)}^{\\,\\tau}.\n\\]\nFor a singleton $S=\\{s\\}$, write $\\Hull_{\\A}(s)$.\n\\end{definition}\n\\RecordID{PSC-I-D-GENERATED-HULL}","\\begin{proposition}[Algebra generation and module generation are distinct]\n\\label{prop:algebra-module-generation}\nEvery nonzero unital algebra $A$, regarded as a left module over itself, has\n\\[\n\\srank_A(A)=1,\n\\]\nwith cyclic seed $1_A$.  This conclusion is independent of the number of algebra generators required to present $A$ as an algebra.\n\\end{proposition}\n\\RecordID{PSC-I-P-ALGEBRA-MODULE-GENERATION}\n\\Depends{PSC-I-D-GENERATED-HULL.}","\\begin{proof}\nFor every $a\\in A$, one has $a=a1_A$.  Hence $A1_A=A$.  The number of operations needed to generate $A$ as an algebra is a different invariant and does not enter the module calculation.\n\\end{proof}","\\begin{boundary}[Operators chosen for a target]\n\\label{bnd:target-dependent-operators}\nFor a fixed seed $s$ and target $x$, the existence of a specially defined operator $A_x$ satisfying $A_xs=x$ does not establish a natural seed calculus unless $A_x$ belongs to a grammar fixed independently of the target satisfying \\cref{def:natural-presentation}.\n\\end{boundary}\n\\RecordID{PSC-I-B-TARGET-DEPENDENT-OPERATORS}","tions by seeds and relations}\n\\begin{readerroadmap}\nA seed family gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","ns}\n\\begin{readerroadmap}\nA seed family gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","A seed family gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","nerated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","ed hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.\n\\end{readerroadmap}\n\\subsection{Finite free covers}\n\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$.  Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-EVALUATION}","\\begin{theorem}[Finite free presentation]\n\\label{thm:free-presentation}\nFor the map in \\cref{def:seed-evaluation},\n\\[\n\\im\\varepsilon_S=\\Hull_A(S),\n\\qquad\nN_S:=\\Ker\\varepsilon_S\\leq A^{\\oplus r},\n\\]\nand the induced map is an isomorphism of left modules over $A$\n\\[\n\\boxed{A^{\\oplus r}/N_S\\cong\\Hull_A(S).}\n\\]\nConsequently, $S$ generates $M$ if and only if $\\varepsilon_S$ is surjective.\n\\end{theorem}\n\\RecordID{PSC-I-T-FREE-PRESENTATION}\n\\Depends{PSC-I-D-SEED-EVALUATION and the first isomorphism theorem.}\n\\Exports{All later free seed presentations.}","\\begin{proof}\nThe image consists exactly of finite sums $\\sum_ja_js_j$, which is $\\Hull_A(S)$.  Because $\\varepsilon_S$ is linear over $A$, its kernel is a left submodule.  The first isomorphism theorem gives\n\\[\nA^{\\oplus r}/N_S\\cong\\im\\varepsilon_S=\\Hull_A(S).\n\\]\nThe final equivalence is the equality $\\im\\varepsilon_S=M$.\n\\end{proof}","\\begin{microexample}[title={A cyclic nilpotent chain}]\nLet \\(A=\\bbk[t]\\) and \\(M=\\bbk[t]/(t^3)\\).  The residue class \\(s=[1]\\) is one seed and\n\\[\n\\varepsilon_s:A\\to M,\n\\qquad\nf(t)\\mapsto f(t)[1]\n\\]\nhas kernel \\((t^3)\\).  Hence \\(M\\cong A/(t^3)\\), and the addressed vectors \\(s,ts,t^2s\\) are the visible orbit chain.\n\\end{microexample}","\\begin{corollary}[Characterization by minimal free covers]\n\\label{cor:minimal-free-cover}\nFor every finitely generated left module $M$ over $A$,\n\\[\n\\boxed{\n\\srank_A(M)=\\min\\{r\\geq0:\\text{there exists a surjection }A^{\\oplus r}\\twoheadrightarrow M\\}.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-MINIMAL-FREE-COVER}\n\\Depends{PSC-I-T-FREE-PRESENTATION.}","\\begin{proof}\nA seed family of size $r$ gives a surjection by \\cref{thm:free-presentation}.  Conversely, the images of the standard basis vectors under any surjection $A^{\\oplus r}\\twoheadrightarrow M$ form a generating family of size at most $r$.\n\\end{proof}","\\begin{corollary}[Presentation with one seed]\n\\label{cor:one-seed-presentation}\nFor $s\\in M$, the annihilator\n\\[\n\\operatorname{Ann}_A(s):=\\{a\\in A:as=0\\}\n\\]\nis a left ideal and\n\\[\n\\boxed{A/\\operatorname{Ann}_A(s)\\cong\\Hull_A(s).}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-ONE-SEED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION.}","\\begin{proof}\nThe evaluation map for one seed is $a\\mapsto as$.  Its kernel is $\\operatorname{Ann}_A(s)$, which is therefore a left ideal, and \\cref{thm:free-presentation} gives the quotient isomorphism.\n\\end{proof}","\\subsection{One seed and the distinction between module and algebra relations}\n\\begin{proposition}[Criterion for the annihilator to be an ideal on both sides]\n\\label{prop:two-sided-annihilator}\nFor $s\\in M$, the left ideal $L=\\operatorname{Ann}_A(s)$ is an ideal on both sides if and only if\n\\[\nL\\,\\Hull_A(s)=0.\n\\]\nIf $s$ is cyclic, these conditions are equivalent to\n\\[\n\\operatorname{Ann}_A(s)=\\operatorname{Ann}_A(M).\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-ANNIHILATOR-TWO-SIDED}\n\\Depends{PSC-I-C-ONE-SEED-PRESENTATION.}","\\begin{proof}\nThe left ideal $L$ is an ideal on both sides exactly when $ab\\in L$ for every $a\\in L$ and $b\\in A$.  This is equivalent to\n\\[\n0=(ab)s=a(bs)\n\\qquad(a\\in L,\\ b\\in A).\n\\]\nBy additivity, the last identity is equivalent to $L\\Hull_A(s)=0$.","If $s$ is cyclic, then $\\Hull_A(s)=M$.  Hence the preceding condition is $L\\subseteq\\operatorname{Ann}_A(M)$.  The reverse inclusion always holds because an operator annihilating all of $M$ annihilates $s$.  Thus equality follows.  Conversely, $\\operatorname{Ann}_A(M)$ is an ideal on both sides, so equality implies that $L$ is an ideal on both sides.\n\\end{proof}","\\begin{microexample}[title={Why a seed annihilator need not be an ideal on both sides}]\nLet \\(A=M_2(\\bbk)\\) act on \\(M=\\bbk^2\\), and let \\(s=e_1\\).  Then \\(\\operatorname{Ann}_A(s)\\) consists of matrices whose first column is zero.  It is a left ideal because left multiplication preserves a zero first column.  It is not generally a right ideal.  Thus the natural quotient for one seed is a quotient of the left regular module, not automatically a quotient algebra.\n\\end{microexample}","\\subsection{Topological completion of a presentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPL","\\subsection{Topological completion of a presentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of","\\subsection{Topological completion of a presentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","sentation}\n\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","$X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","ausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","_1,\\ldots,s_r)$.  Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","ically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","ilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$.  Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$.  If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COMPLETED-PRESENTATION}\n\\Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and c","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\wid","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  The","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every de","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to ze","\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in t","f}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\wideh…","f modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal prope…","$\\Hull_{\\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly bec","orted seminorms make it a topological isomorphism onto that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was","that subspace.  For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","tinuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","$X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","o X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","us and extends uniquely to $\\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","ypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","$, which converges because $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","ause $X$ is complete.  This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","This defines a continuous extension $\\widehat j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","j:\\widehat E\\to X$.  For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","lambda)\\to x$.  The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","xi)=x$.  Thus $\\widehat j$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","$ is surjective.  If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$.  Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero.  Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$.  Hence $\\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}","\\begin{definition}[Seed rank]\n\\label{def:seed-rank}\nFor a unital left module $M$ over $A$, define\n\\[\n\\srank_A(M):=\\min\\{|S|:AS=M\\}.\n\\]\nFor a topological module $(X,\\tau)$, define\n\\[\n\\srank_{\\A,\\tau}(X):=\\min\\{|S|:\\cl{\\A S}^{\\,\\tau}=X\\}.\n\\]\nSet $\\srank_A(0)=0$.  When the value is finite, it is the minimal number of module generators.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-RANK}","\\begin{proposition}[Cyclic characterization]\n\\label{prop:cyclic-characterization}\nFor $M\\neq0$,\n\\[\n\\srank_A(M)=1\n\\quad\\Longleftrightarrow\\quad\nM\\text{ contains a cyclic vector}.\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-CYCLIC-CHARACTERIZATION}\n\\Depends{PSC-I-D-SEED-RANK.}","\\begin{proof}\nBoth conditions say that there is $m\\in M$ with $Am=M$.\n\\end{proof}","\\begin{proposition}[Monotonicity under enlargement of the acting algebra]\n\\label{prop:acting-algebra-monotonicity}\nLet $B\\subseteq A\\subseteq\\End_{\\bbk}(X)$ be unital operator algebras with the same identity.  Then\n\\[\n\\boxed{\\srank_A(X)\\leq\\srank_B(X).}\n\\]\nThe same statement holds for topological seed rank when both algebras act continuously.\n\\end{proposition}\n\\RecordID{PSC-I-P-ACTING-ALGEBRA-MONOTONICITY}\n\\Depends{PSC-I-D-SEED-RANK.}","\\begin{proof}\nFor every $S\\subseteq X$, one has $BS\\subseteq AS$.  Hence every generating family over $B$ is also a generating family over $A$.  Taking closures gives the topological statement.\n\\end{proof}","\\subsection{Quotients, isomorphisms, and exact sequences}\n\\begin{theorem}[Quotient monotonicity]\n\\label{thm:quotient-monotonicity}\nLet $\\varphi:A\\to B$ be a unital ring homomorphism and let\n\\[\n\\Phi:M\\twoheadrightarrow N\n\\]\nbe a surjective additive map satisfying\n\\[\n\\Phi(am)=\\varphi(a)\\Phi(m)\n\\qquad(a\\in A,\\ m\\in M).\n\\]\nThen\n\\[\n\\boxed{\\srank_B(N)\\leq\\srank_A(M).}\n\\]\nIf $M$ and $N$ are topological modules, $\\Phi$ is continuous and surjective, and the actions are continuous, then\n\\[\n\\srank_{B,\\tau_N}(N)\\leq\\srank_{A,\\tau_M}(M).\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-MONOTONICITY}\n\\Depends{PSC-I-D-GENERATED-HULL and PSC-I-D-SEED-RANK.}\n\\Exports{All quotient lower bounds and realization inequalities.}","on{Quotients, isomorphisms, and exact sequences}\n\\begin{theorem}[Quotient monotonicity]\n\\label{thm:quotient-monotonicity}\nLet $\\varphi:A\\to B$ be a unital ring homomorphism and let\n\\[\n\\Phi:M\\twoheadrightarrow N\n\\]\nbe a surjective additive map satisfying\n\\[\n\\Phi(am)=\\varphi(a)\\Phi(m)\n\\qquad(a\\in A,\\ m\\in M).\n\\]\nThen\n\\[\n\\boxed{\\srank_B(N)\\leq\\srank_A(M).}\n\\]\nIf $M$ and $N$ are topological modules, $\\Phi$ is continuous and surjective, and the actions are continuous, then\n\\[\n\\srank_{B,\\tau_N}(N)\\leq\\srank_{A,\\tau_M}(M).\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-MONOTONICITY}\n\\Depends{PSC-I-D-GENERATED-HULL and PSC-I-D-SEED-RANK.}\n\\Exports{All quotient lower bounds and realization inequalities.}","onotonicity]\n\\label{thm:quotient-monotonicity}\nLet $\\varphi:A\\to B$ be a unital ring homomorphism and let\n\\[\n\\Phi:M\\twoheadrightarrow N\n\\]\nbe a surjective additive map satisfying\n\\[\n\\Phi(am)=\\varphi(a)\\Phi(m)\n\\qquad(a\\in A,\\ m\\in M).\n\\]\nThen\n\\[\n\\boxed{\\srank_B(N)\\leq\\srank_A(M).}\n\\]\nIf $M$ and $N$ are topological modules, $\\Phi$ is continuous and surjective, and the actions are continuous, then\n\\[\n\\srank_{B,\\tau_N}(N)\\leq\\srank_{A,\\tau_M}(M).\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-MONOTONICITY}\n\\Depends{PSC-I-D-GENERATED-HULL and PSC-I-D-SEED-RANK.}\n\\Exports{All quotient lower bounds and realization inequalities.}","\\begin{proof}\nLet $S$ generate $M$.  For every $n\\in N$, choose $m\\in M$ with $\\Phi(m)=n$ and write $m=\\sum_ia_is_i$.  Then\n\\[\nn=\\sum_i\\varphi(a_i)\\Phi(s_i),\n\\]\nso $\\Phi(S)$ generates $N$.","For the topological statement, let $S$ generate $M$ densely.  Given $n=\\Phi(m)$, choose a net $m_\\lambda\\in AS$ converging to $m$.  Continuity gives\n\\[\n\\Phi(m_\\lambda)\\longrightarrow n,\n\\]\nand every $\\Phi(m_\\lambda)$ lies in $B\\Phi(S)$.  Thus $B\\Phi(S)$ is dense in $N$.\n\\end{proof}","\\begin{microexample}[title={A quotient can strictly lower seed rank}]\nLet \\(A=\\bbk\\) act by scalars on \\(M=\\bbk^3\\).  Then \\(\\srank_A(M)=3\\).  Quotienting by a subspace of dimension one gives a scalar module of dimension two, whose seed rank is \\(2\\).  Quotient monotonicity is therefore often strict.\n\\end{microexample}","\\begin{theorem}[Short exact sequence bounds]\n\\label{thm:exact-sequence-bounds}\nLet\n\\[\n0\\longrightarrow L\\xrightarrow{\\iota}M\\xrightarrow{\\pi}N\\longrightarrow0\n\\]\nbe a short exact sequence of left modules over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(N)\\leq\\srank_A(M)\n\\leq\\srank_A(L)+\\srank_A(N).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-EXACT-SEQUENCE-BOUNDS}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nThe lower inequality is quotient monotonicity. For the upper inequality, choose generating families $(\\ell_i)_{i\\in I}$ of $L$ and $(n_j)_{j\\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","f}\nThe lower inequality is quotient monotonicity. For the upper inequality, choose generating families $(\\ell_i)_{i\\in I}$ of $L$ and $(n_j)_{j\\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","e upper inequality, choose generating families $(\\ell_i)_{i\\in I}$ of $L$ and $(n_j)_{j\\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","st possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","e a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}","\\begin{hypothesisguide}\nFor a short exact sequence, a generating family for \\(M\\) always maps to a generating family for the quotient.  It need not generate the submodule \\(L\\).  For the upper bound, choose generators of \\(L\\) separately and combine them with lifts of quotient generators.\n\\end{hypothesisguide}","\\subsection{Direct sums under shared and independent actions}\n\\begin{proposition}[Direct sum bounds for a shared action]\n\\label{prop:shared-direct-sum}\nIf one algebra $A$ acts diagonally on $M\\oplus N$, then\n\\[\n\\boxed{\n\\max\\{\\srank_A(M),\\srank_A(N)\\}\n\\leq\\srank_A(M\\oplus N)\n\\leq\\srank_A(M)+\\srank_A(N).\n}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-SHARED-DIRECT-SUM}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nThe coordinate projections are intertwining quotients and give the lower bound.  For the upper bound, embed minimal generating families of $M$ as $(m_i,0)$ and of $N$ as $(0,n_j)$. Their union generates by finite sums and has cardinality at most the sum of the two seed ranks.\n\\end{proof}","\\begin{theorem}[Direct sum formula for independent actions]\n\\label{thm:independent-direct-sum}\nLet $M$ be a left module over $A$ and $N$ a left module over $B$.  Under the action of the product algebra\n\\[\n(a,b)(m,n)=(am,bn),\n\\]\none has\n\\[\n\\boxed{\n\\srank_{A\\times B}(M\\oplus N)\n=\n\\max\\{\\srank_A(M),\\srank_B(N)\\}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-INDEPENDENT-DIRECT-SUM}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nThe coordinate projections give the lower bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{pro","\\begin{proof}\nThe coordinate projections give the lower bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","dinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","inimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","ros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}","\\subsection{Tensor generation and witnesses for lower bounds}\n\\begin{proposition}[Tensor product bound]\n\\label{prop:tensor-bound}\nLet $A$ and $B$ be unital algebras over $\\bbk$, let $M$ be a left module over $A$, and let $N$ be a left module over $B$.  Under the natural $A\\otimes_{\\bbk}B$ action,\n\\[\n\\boxed{\n\\srank_{A\\otimes B}(M\\otimes_{\\bbk}N)\n\\leq\n\\srank_A(M)\\srank_B(N).\n}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-TENSOR-BOUND}\n\\Depends{PSC-I-D-SEED-RANK.}","\\begin{proof}\nChoose minimal generating families $(m_i)_{i\\in I}$ for $M$ and $(n_j)_{j\\in J}$ for $N$. Each vector in either module is a finite linear combination of its generators. Bilinearity therefore expresses every simple tensor as a finite sum of terms\n\\[\n(am_i)\\otimes(bn_j)=(a\\otimes b)(m_i\\otimes n_j).\n\\]\nSimple tensors span the algebraic tensor product, so the family $(m_i\\otimes n_j)_{(i,j)\\in I\\times J}$ generates it. This family has cardinality at most $|I||J|$, which is the claimed bound. The proof also covers an empty generating family, when the corresponding module and tensor product are zero.\n\\end{proof}","M$ and $(n_j)_{j\\in J}$ for $N$. Each vector in either module is a finite linear combination of its generators. Bilinearity therefore expresses every simple tensor as a finite sum of terms\n\\[\n(am_i)\\otimes(bn_j)=(a\\otimes b)(m_i\\otimes n_j).\n\\]\nSimple tensors span the algebraic tensor product, so the family $(m_i\\otimes n_j)_{(i,j)\\in I\\times J}$ generates it. This family has cardinality at most $|I||J|$, which is the claimed bound. The proof also covers an empty generating family, when the corresponding module and tensor product are zero.\n\\end{proof}","\\begin{proposition}[Monotonicity under extension of scalars]\n\\label{prop:scalar-extension}\nLet \\(K/\\bbk\\) be a field extension, let \\(A\\) be a unital algebra over \\(\\bbk\\), and let \\(M\\) be a finitely generated left module over \\(A\\).  Set\n\\[\nA_K:=K\\otimes_{\\bbk}A,\n\\qquad\nM_K:=K\\otimes_{\\bbk}M.\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{A_K}(M_K)\\leq\\srank_A(M).\n}\n\\]\nEvery generating family \\(S\\subseteq M\\) extends to the generating family \\(1\\otimes S\\subseteq M_K\\).\n\\end{proposition}\n\\RecordID{PSC-I-P-SCALAR-EXTENSION-MONOTONICITY}\n\\Depends{PSC-I-D-SEED-RANK.}\n\\Exports{Extension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.}","ty under extension of scalars]\n\\label{prop:scalar-extension}\nLet \\(K/\\bbk\\) be a field extension, let \\(A\\) be a unital algebra over \\(\\bbk\\), and let \\(M\\) be a finitely generated left module over \\(A\\).  Set\n\\[\nA_K:=K\\otimes_{\\bbk}A,\n\\qquad\nM_K:=K\\otimes_{\\bbk}M.\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{A_K}(M_K)\\leq\\srank_A(M).\n}\n\\]\nEvery generating family \\(S\\subseteq M\\) extends to the generating family \\(1\\otimes S\\subseteq M_K\\).\n\\end{proposition}\n\\RecordID{PSC-I-P-SCALAR-EXTENSION-MONOTONICITY}\n\\Depends{PSC-I-D-SEED-RANK.}\n\\Exports{Extension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.}","\\begin{proof}\nLet \\(S=\\{s_1,\\ldots,s_r\\}\\) generate \\(M\\).  Every pure tensor \\(\\lambda\\otimes m\\in M_K\\) has an expression\n\\[\nm=\\sum_{i=1}^r a_is_i,\n\\]\nso\n\\[\n\\lambda\\otimes m\n=\n\\sum_{i=1}^r(\\lambda\\otimes a_i)(1\\otimes s_i).\n\\]\nPure tensors span \\(M_K\\), hence \\(1\\otimes S\\) generates \\(M_K\\) over \\(A_K\\).  Taking the minimum over \\(S\\) proves the inequality.\n\\end{proof}","\\subsection{Quotient witnesses for lower bounds}\n\\begin{definition}[Witness for seed rank]\n\\label{def:seed-rank-witness}\nA surjective intertwiner $\\Phi:M\\twoheadrightarrow N$ is a witness for seed rank of order $r$ when $\\srank(N)=r$.  It certifies $\\srank(M)\\geq r$.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-RANK-WITNESS}","\\begin{proposition}[Scalar quotient witness]\n\\label{prop:scalar-quotient}\nLet $M$ be a module over $A$ with coefficient field $\\bbk$.  Suppose $M$ has a quotient $N$ of dimension $r$ on which $A$ acts through a character\n\\[\n\\chi:A\\longrightarrow\\bbk.\n\\]\nThen\n\\[\n\\boxed{\\srank_A(M)\\geq r.}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-SCALAR-QUOTIENT-WITNESS}\n\\Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}","\\begin{proof}\nFor every $n\\in N$, the cyclic submodule $An$ is contained in the scalar span $\\bbk n$.  Therefore $q$ seed images span a subspace of dimension at most $q$.  Generating the quotient of dimension $r$ requires at least $r$ seeds.  Quotient monotonicity transfers the lower bound to $M$.\n\\end{proof}","\\begin{classicalinput}[Nakayama lemma]\n\\label{ci:nakayama}\nLet $A$ be a unital ring, let $J\\subseteq\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  If $M=N+JM$ for a submodule $N\\leq M$, then $M=N$.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-NAKAYAMA}","\\begin{theorem}[Generator number is detected modulo the Jacobson radical]\n\\label{thm:module-top}\nLet $A$ be a unital ring, let $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  A family $S\\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$.  When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top.  No semisimplicity of $A/J$ is required for the general equality.  Consequently,\n\\[\n\\boxed{\n\\srank_A(M)=\\srank_{A/J}(M/JM).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-MODULE-TOP}\n\\Depends{PSC-I-CI-NAKAYAMA and PSC-I-T-QUOTIENT-MONOTONICITY.}\n\\Exports{Generator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modules.}","n{theorem}[Generator number is detected modulo the Jacobson radical]\n\\label{thm:module-top}\nLet $A$ be a unital ring, let $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  A family $S\\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$.  When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top.  No semisimplicity of $A/J$ is required for the general equality.  Consequently,\n\\[\n\\boxed{\n\\srank_A(M)=\\srank_{A/J}(M/JM).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-MODULE-TOP}\n\\Depends{PSC-I-CI-NAKAYAMA and PSC-I-T-QUOTIENT-MONOTONICITY.}\n\\Exports{Generator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modules.}","radical]\n\\label{thm:module-top}\nLet $A$ be a unital ring, let $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  A family $S\\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$.  When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top.  No semisimplicity of $A/J$ is required for the general equality.  Consequently,\n\\[\n\\boxed{\n\\srank_A(M)=\\srank_{A/J}(M/JM).\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-MODULE-TOP}\n\\Depends{PSC-I-CI-NAKAYAMA and PSC-I-T-QUOTIENT-MONOTONICITY.}\n\\Exports{Generator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modules.}","\\begin{proof}\nIf $S$ generates $M$, its image generates every quotient.  Conversely, suppose the image of $S$ generates $M/JM$ and put $N=AS$.  Then\n\\[\nM=N+JM.\n\\]\nNakayama's lemma gives $M=N$.  Thus the same finite families generate before and after passage to the top, and the minimal cardinalities agree.\n\\end{proof}","\\begin{interpretation}\nThe theorem says that nilpotent or radical directions can lengthen orbit chains but cannot reduce the number of independent directions in the module top that must be seeded.  Once the images of the seeds generate \\(M/JM\\), Nakayama's lemma forces the entire module to be generated.\n\\end{interpretation}","\\begin{microexample}[title={A local nilpotent algebra}]\nLet \\(A=\\bbk[\\varepsilon]/(\\varepsilon^3)\\) and \\(M=A^{\\oplus2}\\).  The radical is \\(J=(\\varepsilon)\\), and\n\\[\nM/JM\\cong\\bbk^2.\n\\]\nTherefore two seeds are necessary and sufficient.  The extra nilpotent layers \\(\\varepsilon M\\) and \\(\\varepsilon^2M\\) are generated automatically once the two top directions are seeded.\n\\end{microexample}","\\begin{corollary}[Local algebra formula]\n\\label{cor:local-algebra-formula}\nLet $A$ be a local ring with Jacobson radical $J$ and residue division ring $D=A/J$.  For every finitely generated left module $M$ over $A$,\n\\[\n\\boxed{\n\\srank_A(M)=\\dim_D(M/JM),\n}\n\\]\nwhere the dimension is that of a left vector space over $D$.\n\\end{corollary}\n\\RecordID{PSC-I-C-LOCAL-ALGEBRA-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP.}","\\begin{proof}\nThe module $M/JM$ over $A/J$ is a left vector space over $D$.  Its minimal number of generators is its dimension over $D$.\n\\end{proof}","\\subsection{Semisimple multiplicity calculus}\n\\begin{classicalinput}[Wedderburn and Artin decomposition]\n\\label{ci:wedderburn}\nEvery semisimple Artinian ring is isomorphic to a finite product\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nof full matrix rings over division rings.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-WEDDERBURN-ARTIN}","\\begin{lemma}[Multiplicities in the regular module]\n\\label{lem:regular-multiplicities}\nLet\n\\[\nA=\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nand let $S_i=D_i^{d_i}$ denote the natural simple left module for the $i$th factor, extended by zero across the other factors.  Then, as a left module over $A$,\n\\[\n\\boxed{A\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus d_i}.}\n\\]\n\\end{lemma}\n\\RecordID{PSC-I-L-REGULAR-MODULE-MULTIPLICITIES}\n\\Depends{PSC-I-CI-WEDDERBURN-ARTIN.}","\\begin{proof}\nFor one factor $M_d(D)$, let $e_{jj}$ be the diagonal matrix units.  The left regular module decomposes as\n\\[\nM_d(D)=\\bigoplus_{j=1}^{d}M_d(D)e_{jj}.\n\\]\nEach minimal left ideal $M_d(D)e_{jj}$ is isomorphic to the natural column module $D^d$.  Taking the direct product of factors gives the stated decomposition.\n\\end{proof}","\\begin{theorem}[Semisimple seed rank formula over arbitrary division rings]\n\\label{thm:semisimple-seed-rank}\nLet\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","l{thm:semisimple-seed-rank}\nLet\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","eed-rank}\nLet\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","ng\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$.  Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}\n\\Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}\n\\Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}","\\begin{proof}\nBy \\cref{cor:minimal-free-cover}, $M$ is generated by $r$ elements exactly when it is a quotient of $A^{\\oplus r}$.  By \\cref{lem:regular-multiplicities},\n\\[\nA^{\\oplus r}\\cong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","\\cref{cor:minimal-free-cover}, $M$ is generated by $r$ elements exactly when it is a quotient of $A^{\\oplus r}$.  By \\cref{lem:regular-multiplicities},\n\\[\nA^{\\oplus r}\\cong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","\\[\nA^{\\oplus r}\\cong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","ong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}","\\begin{interpretation}\nFor the factor \\(M_{d_i}(D_i)\\), one free generator contributes \\(d_i\\) copies of the corresponding simple module.  If the top of \\(M\\) contains \\(m_i\\) copies, at least \\(\\lceil m_i/d_i\\rceil\\) free generators are required.  The worst simple type controls the global answer.","\\begin{microexample}[title={Multiplicity consumes seed capacity}]\nFor \\(A=M_2(\\bbk)\\) acting on \\((\\bbk^2)^{\\oplus3}\\), the regular module contributes two copies of the natural simple module per generator.  One generator cannot cover all three multiplicity directions.  Two can, giving seed rank \\(2\\).\n\\end{microexample}","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnot","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-C","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEM","\\Needspace{16\\baselineskip}\n\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","la]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","on over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","lus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}\n\\Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\begin{proof}\nBy \\cref{thm:module-top}, $\\srank_A(M)=\\srank_{A/J}(M/JM)$.  Apply \\cref{thm:semisimple-seed-rank} to the displayed decompositions of $A/J$ and $M/JM$, with every $D_i=\\bbk$.  This gives the stated maximum.  If $q=0$, the unital $A/J$-module $M/JM$ is zero, and \\cref{thm:module-top} gives seed rank zero.\n\\end{proof}","\\begin{corollary}[Full matrix algebra]\n\\label{cor:full-matrix}\nFor the natural module $S=D^d$ over $M_d(D)$,\n\\[\n\\srank_{M_d(D)}(S)=1.\n\\]\nMore strongly, every nonzero vector of $S$ is cyclic.\n\\end{corollary}\n\\RecordID{PSC-I-C-FULL-MATRIX}\n\\Depends{PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\begin{proof}\nThe formula gives $\\lceil1/d\\rceil=1$.  Directly, let $v\\neq0$ and choose a coordinate $v_j\\neq0$.  For any target $w=(w_i)$, the matrix whose $j$th column has entries $w_iv_j^{-1}$ and whose other columns vanish sends $v$ to $w$.\n\\end{proof}","\\begin{corollary}[Multiplicity obstruction]\n\\label{cor:multiplicity-obstruction}\nFor the diagonal action of $M_d(D)$ on $S^{\\oplus m}$,\n\\[\n\\boxed{\n\\srank_{M_d(D)}(S^{\\oplus m})\n=\n\\left\\lceil\\frac{m}{d}\\right\\rceil.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-MULTIPLICITY-OBSTRUCTION}\n\\Depends{PSC-I-T-SEMISIMPLE-SEED-RANK.}","\\begin{proof}\nApply \\cref{thm:semisimple-seed-rank} to the single simple type with multiplicity $m$.\n\\end{proof}","\\begin{sectionrecap}\nFor finitely generated modules, the minimal number of seeds is detected on the quotient by the Jacobson radical.  In the semisimple top, each generator has a fixed multiplicity capacity \\(d_i\\) in simple type \\(i\\), and the largest capacity ratio determines the exact seed rank.\n\\end{sectionrecap}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{04_semisimple_seed_rank.pdf}\n\\caption{The semisimple multiplicity formula for one simple block. A seed can expose at most $d$ independent multiplicity directions, so $m$ copies require $\\lceil m/d\\rceil$ seeds.}\n\\label{fig:semisimple}\n\\end{figure}","\\begin{readerroadmap}\nOne operator is equivalent to a module over \\(\\bbk[t]\\).  Primary decomposition isolates the irreducible polynomial sectors, and the quotient by \\(p(T)\\) counts how many cyclic chains begin in each sector.\n\\end{readerroadmap}\n\\subsection{Primary components and generator counts}","\\begin{classicalinput}[Primary decomposition for polynomial modules]\n\\label{ci:primary-decomposition}\nEvery module $X$ over $\\bbk[t]$ with finite dimension over the coefficient field decomposes as\n\\[\nX=\\bigoplus_{p\\in\\Irr(\\bbk[t])}X_p,\n\\]\nwhere $\\Irr(\\bbk[t])$ denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero $X_p$ is a finite direct sum of modules $\\bbk[t]/(p^e)$.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-PID-PRIMARY-DECOMPOSITION}","{classicalinput}[Primary decomposition for polynomial modules]\n\\label{ci:primary-decomposition}\nEvery module $X$ over $\\bbk[t]$ with finite dimension over the coefficient field decomposes as\n\\[\nX=\\bigoplus_{p\\in\\Irr(\\bbk[t])}X_p,\n\\]\nwhere $\\Irr(\\bbk[t])$ denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero $X_p$ is a finite direct sum of modules $\\bbk[t]/(p^e)$.\n\\end{classicalinput}\n\\RecordID{PSC-I-CI-PID-PRIMARY-DECOMPOSITION}","\\begin{theorem}[Seed rank for a single operator]\n\\label{thm:single-operator}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subs","\\begin{theorem}[Seed rank for a single operator]\n\\label{thm:single-operator}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","d rank for a single operator]\n\\label{thm:single-operator}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","T\\in\\End_{\\bbk}(X)$.  Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}\n\\RecordID{PSC-I-T-SINGLE-OPERATOR}\n\\Depends{\\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},\n\\nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and\n\\nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}\n\\Exports{Companion, recurrence, and systems of subsequences in residue classes.}","\\begin{proof}\nLet the minimal polynomial be\n\\[\nm_T(t)=\\prod_{p}p(t)^{e_p}.\n\\]\nThe acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the el","\\begin{proof}\nLet the minimal polynomial be\n\\[\nm_T(t)=\\prod_{p}p(t)^{e_p}.\n\\]\nThe acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","the minimal polynomial be\n\\[\nm_T(t)=\\prod_{p}p(t)^{e_p}.\n\\]\nThe acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","The acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","(p^{e_p}),\n\\]\nby the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","d annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","ent.  Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$.  By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}","\\begin{interpretation}\nFor one operator, the quotient \\(X_p/p(T)X_p\\) counts the number of independent primary chains for the irreducible polynomial \\(p\\).  Over an algebraic closure these become Jordan chains.  The seed rank is the largest chain count among the primary sectors, not the largest chain length.\n\\end{interpretation}","\\begin{microexample}[title={Block count versus block size}]\nA nilpotent operator with Jordan block sizes \\((5,1,1)\\) has seed rank \\(3\\), even though its largest block has length \\(5\\).  Three seeds are needed because there are three independent chains.  Once their initial vectors are chosen, repeated application of \\(T\\) generates every position in those chains.\n\\end{microexample}","\\subsection{Invariant factors, Jordan blocks, and cyclicity}\n\\begin{corollary}[Invariant factor count]\n\\label{cor:invariant-factor-count}\nIf $X\\neq0$ and the rational canonical decomposition is written with nonconstant invariant factors\n\\[\nX\\cong\\bigoplus_{j=1}^{r}\\bbk[t]/(f_j),\n\\qquad\nf_1\\mid f_2\\mid\\cdots\\mid f_r,\n\\]\nthen\n\\[\n\\boxed{\\srank_{\\bbk[T]}(X)=r.}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-INVARIANT-FACTOR-COUNT}\n\\Depends{PSC-I-T-SINGLE-OPERATOR.}","\\begin{proof}\nFor each irreducible $p$, the number of invariant factors divisible by $p$ equals the number of elementary divisors for $p$.  Because $f_1\\mid\\cdots\\mid f_r$ and $f_1\\neq1$ after zero summands are removed, some irreducible divisor of $f_1$ divides all $r$ invariant factors.  Hence $\\max_p\\mu_p(T)=r$.\n\\end{proof}","\\begin{corollary}[Formula for geometric multiplicity]\n\\label{cor:geometric-multiplicity}\nIf $\\bbk$ is algebraically closed, then\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)\n=\n\\max_{\\lambda\\in\\operatorname{Spec}(T)}\n\\dim_{\\bbk}\\Ker(T-\\lambda I).\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-SINGLE-OPERATOR}\n\\Depends{PSC-I-T-SINGLE-OPERATOR.}","\\begin{proof}\nThe irreducibles are $p=t-\\lambda$.  On the primary component for $\\lambda$, the quotient by $(T-\\lambda I)X_\\lambda$ has dimension equal to the number of Jordan blocks, which is also $\\dim\\Ker(T-\\lambda I)$.\n\\end{proof}","\\begin{corollary}[Criterion for a cyclic matrix]\n\\label{cor:cyclic-matrix}\nFor $X\\ne0$ of finite dimension, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $T$ has a cyclic vector.\n\\item $\\srank_{\\bbk[T]}(X)=1$.\n\\item Every nonzero primary component for $p$ is cyclic.\n\\item The rational canonical form has one invariant factor.\n\\item The minimal polynomial equals the characteristic polynomial.\n\\item After scalar extension to an algebraic closure, every eigenvalue has geometric multiplicity one.\n\\end{enumerate}\n\\end{corollary}\n\\RecordID{PSC-I-C-CYCLIC-MATRIX}\n\\Depends{PSC-I-T-SINGLE-OPERATOR, PSC-I-C-INVARIANT-FACTOR-COUNT, and PSC-I-P-CYCLIC-CHARACTERIZATION.}","\\begin{proof}\nThe equivalence of (i) and (ii) is \\cref{prop:cyclic-characterization}.  By \\cref{thm:single-operator}, (ii) holds exactly when every $\\mu_p(T)$ belonging to a nonzero primary component equals one, which is (iii).  Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv).  In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)$\\Leftrightarrow$(v).  Scalar extension preserves the equality of minimal and characteristic polynomials.  Over an algebraic closure, that equality is equivalent to one Jordan block for each eigenvalue, which is (vi).\n\\end{proof}","ii) holds exactly when every $\\mu_p(T)$ belonging to a nonzero primary component equals one, which is (iii).  Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv).  In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)$\\Leftrightarrow$(v).  Scalar extension preserves the equality of minimal and characteristic polynomials.  Over an algebraic closure, that equality is equivalent to one Jordan block for each eigenvalue, which is (vi).\n\\end{proof}","\\begin{definition}[Finite packet system]\n\\label{def:packet-system}\nA finite packet system on a vector space $X$ over $\\bbk$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\subseteq\\End_{\\bbk}(X)\n\\]\nsatisfying\n\\[\nP_\\alpha^2=P_\\alpha,\n\\qquad\nP_\\alpha P_\\beta=0\\quad(\\alpha\\neq\\beta),\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nSet $X_\\alpha=P_\\alpha X$.  Zero packet spaces may be removed without changing any identity.  No orthogonality is assumed unless stated.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-SYSTEM}","\\begin{definition}[Packet support and packet pure vectors]\n\\label{def:packet-support}\nFor a finite packet system \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(x\\in X\\), define\n\\[\n\\Supp_{\\Pcal}(x)\n:=\n\\{\\alpha\\in I:P_\\alpha x\\neq0\\}.\n\\]\nThe vector \\(x\\) is \\emph{packet pure} if \\(|\\Supp_{\\Pcal}(x)|\\leq1\\).  A finite family is packet pure if each of its members is packet pure, although different members may lie in different packets.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-SUPPORT}","\\begin{lemma}[Finite packet aggregation]\n\\label{lem:packet-aggregation}\nLet $\\sigma:I\\twoheadrightarrow J$ be a surjection and define\n\\[\nQ_j:=\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nThen $(Q_j)_{j\\in J}$ is a finite packet system and\n\\[\nQ_jX=\\bigoplus_{\\sigma(\\alpha)=j}X_\\alpha.\n\\]\n\\end{lemma}\n\\RecordID{PSC-I-L-PACKET-AGGREGATION}\n\\Depends{PSC-I-D-PACKET-SYSTEM.}","\\begin{proof}\nPairwise annihilation of the $P_\\alpha$ gives\n\\[\nQ_j^2=Q_j,\n\\qquad\nQ_jQ_k=0\\quad(j\\neq k).\n\\]\nSumming over $j$ recovers $\\sum_\\alpha P_\\alpha=I$.  The range statement follows from the directness forced by pairwise annihilation.\n\\end{proof}","\\begin{definition}[Packet analysis and synthesis]\n\\label{def:analysis-synthesis}\nFor a finite packet system, define\n\\[\n\\mathcal A_{\\Pcal}:X\\longrightarrow\\bigoplus_{\\alpha\\in I}X_\\alpha,\n\\qquad\nx\\longmapsto(P_\\alpha x)_\\alpha,\n\\]\nand\n\\[\n\\mathcal S_{\\Pcal}:\\bigoplus_{\\alpha\\in I}X_\\alpha\\longrightarrow X,\n\\qquad\n(x_\\alpha)_\\alpha\\longmapsto\\sum_\\alpha x_\\alpha.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-ANALYSIS-SYNTHESIS}","\\begin{theorem}[Exact packet coordinate isomorphism]\n\\label{thm:exact-packet-coordinates}\nFor every finite packet system,\n\\[\n\\boxed{\n\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}=I_X,\n\\qquad\n\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I_{\\oplus X_\\alpha}.\n}\n\\]\nHence\n\\[\n\\boxed{X\\cong\\bigoplus_{\\alpha\\in I}X_\\alpha.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-EXACT-PACKET-COORDINATES}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-ANALYSIS-SYNTHESIS.}\n\\Exports{Every exact packet reconstruction in the series.}","\\begin{proof}\nFor $x\\in X$,\n\\[\n\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x\n=\n\\sum_\\alpha P_\\alpha x\n=x.\n\\]\nIf $x_\\alpha\\in X_\\alpha$, then $P_\\beta x_\\alpha=\\delta_{\\alpha\\beta}x_\\alpha$, so\n\\[\n\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}(x_\\alpha)_\\alpha\n=\n\\left(P_\\beta\\sum_\\alpha x_\\alpha\\right)_\\beta\n=(x_\\beta)_\\beta.\n\\]\nThus analysis and synthesis are inverse linear maps.\n\\end{proof}","\\begin{corollary}[Packetwise equality]\n\\label{cor:packetwise-equality}\nFor $x,y\\in X$,\n\\[\nx=y\n\\quad\\Longleftrightarrow\\quad\nP_\\alpha x=P_\\alpha y\\text{ for every }\\alpha.\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKETWISE-EQUALITY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}","\\begin{proof}\nThe forward implication is immediate.  For the reverse implication, sum the packet equalities and use $\\sum_\\alpha P_\\alpha=I$.\n\\end{proof}","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of pa","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, co","\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","pha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","\\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}","\\begin{proof}\nFor \\(x=\\sum_\\alpha P_\\alpha x\\), any extension of the \\(f_\\alpha\\) must satisfy\n\\[\nf(x)=\\sum_\\alpha f_\\alpha(P_\\alpha x),\n\\]\nwhich proves both existence and uniqueness of the first formula.  For the second, the sum \\(g=\\sum_\\alpha g_\\alpha\\) satisfies\n\\[\nP_\\beta g\n=\\sum_\\alpha P_\\beta g_\\alpha\n=g_\\beta\n\\]\nbecause \\(g_\\alpha(Y)\\subseteq X_\\alpha\\).  Uniqueness follows from \\(\\sum_\\alpha P_\\alpha=I\\).\n\\end{proof}","\\begin{theorem}[Tensor product packet systems]\n\\label{thm:tensor-packet-system}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realization","\\begin{theorem}[Tensor product packet systems]\n\\label{thm:tensor-packet-system}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realizations.}","ct packet systems]\n\\label{thm:tensor-packet-system}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realizations.}","lpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\).  Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realizations.}","\\begin{proof}\nFor every pair of packet indices,\n\\[\n(P_\\alpha\\otimes Q_\\gamma)^2\n=P_\\alpha^2\\otimes Q_\\gamma^2\n=P_\\alpha\\otimes Q_\\gamma.\n\\]\nIf \\((\\alpha,\\gamma)\\neq(\\beta,\\delta)\\), then either \\(P_\\alpha P_\\beta=0\\) or \\(Q_\\gamma Q_\\delta=0\\), so the corresponding tensor idempotents annihilate each other.  Finally,\n\\[\n\\sum_{\\alpha,\\gamma}P_\\alpha\\otimes Q_\\gamma\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\otimes\n\\left(\\sum_\\gamma Q_\\gamma\\right)\n=I_X\\otimes I_Y.\n\\]\nThe range is generated by pure tensors \\(P_\\alpha x\\otimes Q_\\gamma y\\), hence equals \\(X_\\alpha\\otimes Y_\\gamma\\).  Exact reconstruction follows from \\cref{thm:exact-packet-coordinates}.\n\\end{proof}","ma^2\n=P_\\alpha\\otimes Q_\\gamma.\n\\]\nIf \\((\\alpha,\\gamma)\\neq(\\beta,\\delta)\\), then either \\(P_\\alpha P_\\beta=0\\) or \\(Q_\\gamma Q_\\delta=0\\), so the corresponding tensor idempotents annihilate each other.  Finally,\n\\[\n\\sum_{\\alpha,\\gamma}P_\\alpha\\otimes Q_\\gamma\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\otimes\n\\left(\\sum_\\gamma Q_\\gamma\\right)\n=I_X\\otimes I_Y.\n\\]\nThe range is generated by pure tensors \\(P_\\alpha x\\otimes Q_\\gamma y\\), hence equals \\(X_\\alpha\\otimes Y_\\gamma\\).  Exact reconstruction follows from \\cref{thm:exact-packet-coordinates}.\n\\end{proof}",",\\gamma)\\neq(\\beta,\\delta)\\), then either \\(P_\\alpha P_\\beta=0\\) or \\(Q_\\gamma Q_\\delta=0\\), so the corresponding tensor idempotents annihilate each other.  Finally,\n\\[\n\\sum_{\\alpha,\\gamma}P_\\alpha\\otimes Q_\\gamma\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\otimes\n\\left(\\sum_\\gamma Q_\\gamma\\right)\n=I_X\\otimes I_Y.\n\\]\nThe range is generated by pure tensors \\(P_\\alpha x\\otimes Q_\\gamma y\\), hence equals \\(X_\\alpha\\otimes Y_\\gamma\\).  Exact reconstruction follows from \\cref{thm:exact-packet-coordinates}.\n\\end{proof}","\\begin{proposition}[Dual packet systems]\n\\label{prop:dual-packet-system}\nLet \\(X\\) be finite dimensional and let \\(\\Pcal=(P_\\alpha)\\) be a finite packet system.  The algebraic dual maps\n\\[\nP_\\alpha^*:X^*\\to X^*,\n\\qquad\nP_\\alpha^*(\\varphi)=\\varphi\\circ P_\\alpha,\n\\]\nform a finite packet system on \\(X^*\\).  Its packet with index \\(\\alpha\\) is canonically isomorphic to \\(X_\\alpha^*\\), and\n\\[\nX^*\\cong\\bigoplus_\\alpha X_\\alpha^*.\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-DUAL-PACKET-SYSTEM}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Dual observables, covectors, trace pairings, and adjoint packet calculations.}","\\begin{proof}\nBecause dualization reverses products,\n\\[\nP_\\alpha^*P_\\beta^*=(P_\\beta P_\\alpha)^*\n=\\delta_{\\alpha\\beta}P_\\alpha^*.\n\\]\nAlso \\(\\sum_\\alpha P_\\alpha^*=I_{X^*}\\).  Thus the dual maps form a packet system.  Restriction gives a map from \\(P_\\alpha^*X^*\\) to \\(X_\\alpha^*\\).  It is injective because a covector in the range of \\(P_\\alpha^*\\) vanishes on every other packet, and it is surjective because any covector on \\(X_\\alpha\\) extends to \\(X\\) through the direct sum decomposition.\n\\end{proof}","tion reverses products,\n\\[\nP_\\alpha^*P_\\beta^*=(P_\\beta P_\\alpha)^*\n=\\delta_{\\alpha\\beta}P_\\alpha^*.\n\\]\nAlso \\(\\sum_\\alpha P_\\alpha^*=I_{X^*}\\).  Thus the dual maps form a packet system.  Restriction gives a map from \\(P_\\alpha^*X^*\\) to \\(X_\\alpha^*\\).  It is injective because a covector in the range of \\(P_\\alpha^*\\) vanishes on every other packet, and it is surjective because any covector on \\(X_\\alpha\\) extends to \\(X\\) through the direct sum decomposition.\n\\end{proof}","\\[\nP_\\alpha^*P_\\beta^*=(P_\\beta P_\\alpha)^*\n=\\delta_{\\alpha\\beta}P_\\alpha^*.\n\\]\nAlso \\(\\sum_\\alpha P_\\alpha^*=I_{X^*}\\).  Thus the dual maps form a packet system.  Restriction gives a map from \\(P_\\alpha^*X^*\\) to \\(X_\\alpha^*\\).  It is injective because a covector in the range of \\(P_\\alpha^*\\) vanishes on every other packet, and it is surjective because any covector on \\(X_\\alpha\\) extends to \\(X\\) through the direct sum decomposition.\n\\end{proof}","\\begin{microexample}[title={Tensoring the oblique packet split}]\nTensor the two oblique packets in Running Example R3 with the coordinate packets of \\(\\bbk^3\\).  The result has six exact packets.  No orthogonality is created by tensoring, but the analysis and synthesis identities remain exact because they are idempotent identities.\n\\end{microexample}","\\begin{theorem}[Finite Banach packet isomorphism]\n\\label{thm:banach-packets}\nLet $X$ be Banach and let $(P_\\alpha)_{\\alpha\\in I}$ be a finite family of bounded pairwise annihilating idempotents with sum $I_X$.  Then every range $X_\\alpha=P_\\alpha X$ is closed.  For any finite product norm equivalent to the maximum norm on $\\bigoplus_\\alpha X_\\alpha$, packet analysis and synthesis are bounded inverse operators.\n\\end{theorem}\n\\RecordID{PSC-I-T-BANACH-PACKET-ISOMORPHISM}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}","\\begin{proof}\nThe range of a bounded idempotent is its space of fixed points,\n\\[\nP_\\alpha X=\\Ker(I-P_\\alpha),\n\\]\nwhich is closed.  Because $I$ is finite, analysis and synthesis are bounded for the maximum and sum norms, respectively, and therefore for every equivalent finite product norm.  Their inverse identities are algebraic and were proved in \\cref{thm:exact-packet-coordinates}.\n\\end{proof}","\\begin{definition}[Packet coordinate condition number]\n\\label{def:packet-condition-number}\nFix a finite product norm on $\\bigoplus_\\alpha X_\\alpha$ equivalent to the maximum norm, as in \\cref{thm:banach-packets}. Define\n\\[\n\\kappa(\\Pcal):=\n\\|\\mathcal A_{\\Pcal}\\|\\,\n\\|\\mathcal S_{\\Pcal}\\|.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-CONDITION-NUMBER}","\\begin{proposition}[Conditioned coordinate transport]\n\\label{prop:conditioned-coordinates}\nFor a finite Banach packet system with $X\\ne0$,\n\\[\n\\kappa(\\Pcal)\\geq1.\n\\]\nIf $T\\in\\mathcal B(X)$ and\n\\[\n[T]_{\\Pcal}:=\\mathcal A_{\\Pcal}T\\mathcal S_{\\Pcal},\n\\]\nthen\n\\[\n\\boxed{\n\\|[T]_{\\Pcal}\\|\n\\leq\\kappa(\\Pcal)\\|T\\|,\n\\qquad\n\\|T\\|\n\\leq\\kappa(\\Pcal)\\|[T]_{\\Pcal}\\|.\n}\n\\]\n\\end{proposition}\n\\RecordID{PSC-I-P-CONDITIONED-COORDINATES}\n\\Depends{PSC-I-T-BANACH-PACKET-ISOMORPHISM.}","\\begin{proof}\nSince $I=\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}$,\n\\[\n1=\\|I\\|\\leq\\|\\mathcal S_{\\Pcal}\\|\\|\\mathcal A_{\\Pcal}\\|.\n\\]\nThe first displayed estimate follows from submultiplicativity.  The inverse identity\n\\[\nT=\\mathcal S_{\\Pcal}[T]_{\\Pcal}\\mathcal A_{\\Pcal}\n\\]\ngives the second estimate.\n\\end{proof}","\\begin{interpretation}\nThe quantity \\(\\kappa(\\Pcal)\\) measures how strongly the chosen packet coordinates can amplify perturbations.  An orthogonal Hilbert decomposition has the best possible behavior in the natural \\(\\ell^2\\) sum norm.  An oblique packet system can reconstruct every vector exactly and still have a large condition number.\n\\end{interpretation}","\\begin{corollary}[Completion preserves a finite packet split]\n\\label{cor:completion-packet-split}\nLet $M$ be a dense normed subspace of a Banach space $X$.  Suppose every bounded $P_\\alpha$ leaves $M$ invariant and the restricted projectors form a packet system on $M$.  Then\n\\[\nP_\\alpha X=\\cl{P_\\alpha M}\n\\]\nand the packet decomposition of $M$ extends uniquely to the completion $X$.\n\\end{corollary}\n\\RecordID{PSC-I-C-COMPLETION-PACKET-SPLIT}\n\\Depends{PSC-I-T-BANACH-PACKET-ISOMORPHISM.}","\\begin{proof}\nFor $x\\in X$, choose $m_n\\in M$ with $m_n\\to x$.  Continuity gives $P_\\alpha m_n\\to P_\\alpha x$, so $P_\\alpha X\\subseteq\\cl{P_\\alpha M}$.  The reverse inclusion follows because $P_\\alpha X$ is closed and contains $P_\\alpha M$.  The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.\n\\end{proof}","\\begin{definition}[Unconditional packet sum space]\n\\label{def:unconditional-packet-space}\nLet $(X_n)_{n\\geq1}$ be closed subspaces of a real or complex Banach space $X$.  Define\n\\[\n\\bigoplus_{n\\geq1}^{u}X_n\n\\]\nto be the space of sequences $x=(x_n)$ with $x_n\\in X_n$ for which $\\sum_nx_n$ converges unconditionally in $X$.  Equip it with\n\\[\n\\|x\\|_u\n:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-UNCONDITIONAL-PACKET-SPACE}","\\begin{lemma}[Completeness of the unconditional packet sum]\n\\label{lem:unconditional-completeness}\nThe normed space $\\bigoplus_{n\\geq1}^{u}X_n$ in \\cref{def:unconditional-packet-space} is Banach.\n\\end{lemma}\n\\RecordID{PSC-I-L-UNCONDITIONAL-COMPLETENESS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE.}","\\begin{proof}\nLet $x^{(k)}=(x_n^{(k)})$ be Cauchy in $\\|\\cdot\\|_u$.  For every fixed $n$,\n\\[\n\\|x_n^{(k)}-x_n^{(\\ell)}\\|\n\\leq\n\\|x^{(k)}-x^{(\\ell)}\\|_u,\n\\]\nso $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","k)}=(x_n^{(k)})$ be Cauchy in $\\|\\cdot\\|_u$.  For every fixed $n$,\n\\[\n\\|x_n^{(k)}-x_n^{(\\ell)}\\|\n\\leq\n\\|x^{(k)}-x^{(\\ell)}\\|_u,\n\\]\nso $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","}\\|\n\\leq\n\\|x^{(k)}-x^{(\\ell)}\\|_u,\n\\]\nso $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","so $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed.  Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.","Choose such a $k\\geq k_0$.  Since $x^{(k)}$ is unconditionally summable, there is a finite $F_0$ such that every finite $F\\subseteq\\mathbb N\\setminus F_0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","s a finite $F_0$ such that every finite $F\\subseteq\\mathbb N\\setminus F_0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","thbb N\\setminus F_0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}","\\begin{theorem}[Countable unconditional packet isomorphism]\n\\label{thm:countable-packets}\nLet $X$ be a real or complex Banach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions unde","\\begin{theorem}[Countable unconditional packet isomorphism]\n\\label{thm:countable-packets}\nLet $X$ be a real or complex Banach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions under explicit convergence control.}","sm]\n\\label{thm:countable-packets}\nLet $X$ be a real or complex Banach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions under explicit convergence control.}","ach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$.  Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}\n\\Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}\n\\Exports{Countable analytic packet decompositions under explicit convergence control.}","\\begin{proof}\nEach $P_nX=\\Ker(I-P_n)$ is closed.  For $x\\in X$,\n\\[\n\\|\\mathcal A_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end","\\begin{proof}\nEach $P_nX=\\Ker(I-P_n)$ is closed.  For $x\\in X$,\n\\[\n\\|\\mathcal A_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","roof}\nEach $P_nX=\\Ker(I-P_n)$ is closed.  For $x\\in X$,\n\\[\n\\|\\mathcal A_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","For $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$.  Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}","\\begin{remark}[Uniform boundedness]\n\\label{rem:uniform-boundedness}\nIf every expansion $\\sum_nP_nx$ is unconditional, the operators obtained from finite partial sums with signs form a pointwise bounded family.  The uniform boundedness principle therefore yields a finite constant $K$.  We retain this constant explicitly in \\cref{thm:countable-packets} because later estimates use it to bound the coordinate map.\n\\end{remark}\n\\RecordID{PSC-I-R-UNIFORM-BOUNDEDNESS}","\\begin{boundary}[Infinite packet sums]\n\\label{bnd:infinite-packet-topology}\nFor an infinite family, an identity such as $\\sum_\\alpha P_\\alpha=I$ requires a declared convergence mode and the corresponding uniform boundedness statement.  Pointwise, strong, norm, weak, and unconditional convergence are not interchangeable.\n\\end{boundary}\n\\RecordID{PSC-I-B-INFINITE-PACKET-TOPOLOGY}","\\begin{definition}[Transfer block]\n\\label{def:transfer-block}\nFor $T\\in\\End_{\\bbk}(X)$ and a finite packet system $\\Pcal$, define\n\\[\nT_{\\beta\\alpha}:=P_\\beta TP_\\alpha\n\\in\\Hom_{\\bbk}(X_\\alpha,X_\\beta).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-BLOCK}","\\begin{theorem}[Exact operator reconstruction]\n\\label{thm:operator-reconstruction}\nEvery $T\\in\\End_{\\bbk}(X)$ satisfies\n\\[\n\\boxed{T=\\sum_{\\alpha,\\beta\\in I}T_{\\beta\\alpha}.}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-OPERATOR-RECONSTRUCTION}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}","\\begin{proof}\nInsert the finite packet identity on both sides:\n\\[\nT=\n\\left(\\sum_\\beta P_\\beta\\right)\nT\n\\left(\\sum_\\alpha P_\\alpha\\right)\n=\n\\sum_{\\alpha,\\beta}P_\\beta TP_\\alpha.\n\\]\n\\end{proof}","\\begin{interpretation}\nOperator reconstruction inserts the identity on both sides of \\(T\\).  The source index \\(\\alpha\\) records where a vector begins, and the target index \\(\\beta\\) records where its image lands.  The resulting matrix is exact, and each entry is itself a linear map.\n\\end{interpretation}","\\begin{theorem}[Packet matrix multiplication]\n\\label{thm:block-multiplication}\nFor $S,T\\in\\End_{\\bbk}(X)$,\n\\[\n\\boxed{\n(ST)_{\\gamma\\alpha}\n=\n\\sum_{\\beta\\in I}S_{\\gamma\\beta}T_{\\beta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-MULTIPLICATION}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}\n\\Exports{All packet path expansions.}","\\begin{proof}\nInsert $I=\\sum_\\beta P_\\beta$ between $S$ and $T$:\n\\[\nP_\\gamma STP_\\alpha\n=\n\\sum_\\beta\nP_\\gamma SP_\\beta TP_\\alpha.\n\\]\n\\end{proof}","\\begin{corollary}[Packet powers and paths]\n\\label{cor:block-powers}\nFor $m\\geq1$,\n\\[\n(T^m)_{\\beta_m\\beta_0}\n=\n\\sum_{\\beta_1,\\ldots,\\beta_{m-1}}\nT_{\\beta_m\\beta_{m-1}}\n\\cdots\nT_{\\beta_1\\beta_0}.\n\\]\nThe same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order.\n\\end{corollary}\n\\RecordID{PSC-I-C-BLOCK-POWERS}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION.}","\\begin{proof}\nInduct on $m$ using \\cref{thm:block-multiplication}.  For a product of different operators, use the same induction with the specified operator at each step.\n\\end{proof}","\\begin{corollary}[Adjoint reversal]\n\\label{cor:adjoint-reversal}\nIf $X$ is Hilbert, every $P_\\alpha$ is orthogonal, and $T\\in\\mathcal B(X)$, then\n\\[\n\\boxed{(T_{\\beta\\alpha})^*=(T^*)_{\\alpha\\beta}.}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-ADJOINT-REVERSAL}\n\\Depends{PSC-I-D-TRANSFER-BLOCK.}","\\begin{proof}\nBecause $P_\\alpha^*=P_\\alpha$,\n\\[\n(P_\\beta TP_\\alpha)^*=P_\\alpha T^*P_\\beta.\n\\]\n\\end{proof}","\\begin{corollary}[Tensor product transfer blocks]\n\\label{cor:tensor-transfer-blocks}\nLet \\(S\\in\\End(X)\\) and \\(T\\in\\End(Y)\\), with packet systems \\(\\Pcal\\) and \\(\\Qcal\\).  Relative to the tensor product packet system of \\cref{thm:tensor-packet-system},\n\\[\n\\boxed{\n(S\\otimes T)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n=\nS_{\\beta\\alpha}\\otimes T_{\\delta\\gamma}.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-TENSOR-TRANSFER-BLOCKS}\n\\Depends{PSC-I-T-TENSOR-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}\n\\Exports{Tensor network, quantum circuit, and multivariate packet calculations.}","\\begin{proof}\nBy definition,\n\\begin{align*}\n(S\\otimes T)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n&=(P_\\beta\\otimes Q_\\delta)(S\\otimes T)(P_\\alpha\\otimes Q_\\gamma)\\\\\n&=(P_\\beta SP_\\alpha)\\otimes(Q_\\delta TQ_\\gamma)\\\\\n&=S_{\\beta\\alpha}\\otimes T_{\\delta\\gamma}.\n\\end{align*}\n\\end{proof}","\\begin{theorem}[Blockwise intertwining equations]\n\\label{thm:block-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representat","\\begin{theorem}[Blockwise intertwining equations]\n\\label{thm:block-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","ons]\n\\label{thm:block-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","k-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","et \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear.  Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-INTERTWINING}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}","\\begin{proof}\nApply \\(Q_\\gamma\\) on the left and \\(P_\\alpha\\) on the right.  Inserting \\(I_X=\\sum_\\beta P_\\beta\\) gives\n\\[\nQ_\\gamma UTP_\\alpha\n=\n\\sum_\\beta Q_\\gamma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","begin{proof}\nApply \\(Q_\\gamma\\) on the left and \\(P_\\alpha\\) on the right.  Inserting \\(I_X=\\sum_\\beta P_\\beta\\) gives\n\\[\nQ_\\gamma UTP_\\alpha\n=\n\\sum_\\beta Q_\\gamma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","P_\\beta\\) gives\n\\[\nQ_\\gamma UTP_\\alpha\n=\n\\sum_\\beta Q_\\gamma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","amma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish.  Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}","\\begin{theorem}[Full block algebra representation]\n\\label{thm:block-algebra}\nThe map\n\\[\n\\Theta_{\\Pcal}:\\End_{\\bbk}(X)\n\\longrightarrow\n\\prod_{\\beta,\\alpha\\in I}\\Hom_{\\bbk}(X_\\alpha,X_\\beta),\n\\qquad\nT\\longmapsto(T_{\\beta\\alpha})_{\\beta,\\alpha},\n\\]\nis an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators.\n\\end{theorem}\n\\RecordID{PSC-I-T-BLOCK-ALGEBRA}\n\\Depends{\\nolinkurl{PSC-I-T-OPERATOR-RECONSTRUCTION} and\n\\nolinkurl{PSC-I-T-BLOCK-MULTIPLICATION}.}","\\begin{proof}\nInjectivity follows from exact reconstruction.  Given arbitrary linear maps $B_{\\beta\\alpha}:X_\\alpha\\to X_\\beta$, define\n\\[\nT\\left(\\sum_\\alpha x_\\alpha\\right)\n:=\n\\sum_{\\alpha,\\beta}B_{\\beta\\alpha}x_\\alpha.\n\\]\nThe packet decomposition is unique and the index set is finite, so this defines a linear operator whose blocks are the prescribed maps.  Thus $\\Theta_{\\Pcal}$ is surjective.  Compatibility with multiplication is \\cref{thm:block-multiplication}.\n\\end{proof}","\\begin{corollary}[Propagation of packet support]\n\\label{cor:packet-support-propagation}\nLet \\(T\\in\\End(X)\\) and \\(x\\in X\\).  Then\n\\[\n\\boxed{\n\\Supp_{\\Pcal}(Tx)\n\\subseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reacha","\\begin{corollary}[Propagation of packet support]\n\\label{cor:packet-support-propagation}\nLet \\(T\\in\\End(X)\\) and \\(x\\in X\\).  Then\n\\[\n\\boxed{\n\\Supp_{\\Pcal}(Tx)\n\\subseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reachability, leakage","\\begin{corollary}[Propagation of packet support]\n\\label{cor:packet-support-propagation}\nLet \\(T\\in\\End(X)\\) and \\(x\\in X\\).  Then\n\\[\n\\boxed{\n\\Supp_{\\Pcal}(Tx)\n\\subseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reachability, leakage bounds, and support pruning.}","bseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}\n\\RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}\n\\Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Graph reachability, leakage bounds, and support pruning.}","\\begin{proof}\nFor every target packet \\(\\beta\\),\n\\[\nP_\\beta Tx\n=\n\\sum_{\\alpha\\in\\Supp_{\\Pcal}(x)}T_{\\beta\\alpha}P_\\alpha x.\n\\]\nIf every block in this sum is zero, then the coordinate with index \\(\\beta\\) vanishes.  This proves the inclusion.  Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right.  The final sufficient condition leaves at most one nonzero summand at each target, and kernel avoidance makes that summand nonzero.\n\\end{proof}","\\begin{corollary}[Dimension count]\n\\label{cor:dimension-count}\nIf $X$ is finite dimensional and $d_\\alpha=\\dim X_\\alpha$, then\n\\[\n\\dim\\End_{\\bbk}(X)\n=\n\\sum_{\\alpha,\\beta}d_\\alpha d_\\beta\n=\n\\left(\\sum_\\alpha d_\\alpha\\right)^2.\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-DIMENSION-COUNT}\n\\Depends{PSC-I-T-BLOCK-ALGEBRA.}","\\begin{proof}\nTake dimensions in the block algebra isomorphism and use $\\dim\\Hom(X_\\alpha,X_\\beta)=d_\\alpha d_\\beta$.\n\\end{proof}","\\begin{proposition}[Block diagonal projection and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","begin{proposition}[Block diagonal projection and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","n}[Block diagonal projection and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","bel{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra.  For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}\n\\RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}\n\\Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}\n\\Exports{Symmetry sector invariance and packet leakage decompositions.}","\\begin{proof}\nPairwise annihilation gives $\\Delta_{\\Pcal}^2=\\Delta_{\\Pcal}$ and identifies its range with diagonal block families.  If $T$ commutes with all $P_\\alpha$, then for $\\beta\\neq\\alpha$,\n\\[\nP_\\beta TP_\\alpha=P_\\beta P_\\alpha T=0.\n\\]\nThus (i) implies (ii).  Exact reconstruction shows (ii) implies (iii).  If $T=\\sum_\\alpha P_\\alpha TP_\\alpha$, then\n\\[\nTP_\\gamma=P_\\gamma TP_\\gamma=P_\\gamma T,\n\\]\nso (iii) implies (i).\n\\end{proof}","\\subsection{Support patterns, preorders, and triangular structure}\n\\begin{definition}[Support relation and support space]\n\\label{def:support-relation}\nFor a relation $R\\subseteq I\\times I$, with $(\\beta,\\alpha)$ interpreted as an allowed transfer $\\alpha\\to\\beta$, define\n\\[\n\\mathfrak M_R\n:=\n\\{T\\in\\End(X):T_{\\beta\\alpha}=0\\text{ whenever }(\\beta,\\alpha)\\notin R\\}.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-SUPPORT-RELATION}","\\begin{theorem}[Support pattern algebra criterion]\n\\label{thm:support-pattern}\nAssume $X_\\alpha\\neq0$ for every $\\alpha\\in I$.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $I_X\\in\\mathfrak M_R$ if and only if $R$ is reflexive.\n\\item $\\mathfrak M_R\\mathfrak M_R\\subseteq\\mathfrak M_R$ if and only if $R$ is transitive.\n\\item $\\mathfrak M_R$ is a unital subalgebra if and only if $R$ is a preorder.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-SUPPORT-PATTERN-ALGEBRA}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION.}","\\begin{proof}\nThe identity has diagonal blocks $I_{X_\\alpha}$ and no blocks outside the diagonal, so it belongs to $\\mathfrak M_R$ exactly when $(\\alpha,\\alpha)\\in R$ for every $\\alpha$.","Assume $R$ is transitive.  If a summand\n\\[\nS_{\\gamma\\beta}T_{\\beta\\alpha}\n\\]\nin the $(\\gamma,\\alpha)$ block of $ST$ is nonzero, then $(\\gamma,\\beta)$ and $(\\beta,\\alpha)$ lie in $R$.  Transitivity gives $(\\gamma,\\alpha)\\in R$, so every forbidden block of $ST$ vanishes.","Conversely, suppose transitivity fails.  Choose $\\alpha,\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}",", suppose transitivity fails.  Choose $\\alpha,\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","ails.  Choose $\\alpha,\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","lpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$.  Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$.  Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).\n\\end{proof}","\\begin{corollary}[Bounded support patterns closed under $*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","corollary}[Bounded support patterns closed under $*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","support patterns closed under $*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","$*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}\n\\RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}\n\\Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}","\\begin{proof}\nAdjoint reversal sends an allowed $(\\beta,\\alpha)$ block to an $(\\alpha,\\beta)$ block.  Symmetry is therefore sufficient.  If symmetry fails, choose a nonzero map of rank one in an allowed nonsymmetric block.  Its adjoint occupies the forbidden reverse block, proving necessity.  A reflexive, transitive, symmetric relation is an equivalence relation.\n\\end{proof}","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorp","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathf","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREO","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered","\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$.  Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular …","ad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","lpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","arries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","rtial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","k M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}\n\\Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}\n\\Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}","\\begin{proof}\nA nonzero block in $\\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","A nonzero block in $\\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","$\\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","s to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist.  This proves nilpotence.","Let $P_C=\\sum_{\\alpha\\in C}P_\\alpha$.  Define\n\\[\n\\rho:\\mathfrak M_R\\longrightarrow\\prod_C\\End(X_C),\n\\qquad\n\\rho(T)=(P_CTP_C)_C.\n\\]\nIf an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","C=\\sum_{\\alpha\\in C}P_\\alpha$.  Define\n\\[\n\\rho:\\mathfrak M_R\\longrightarrow\\prod_C\\End(X_C),\n\\qquad\n\\rho(T)=(P_CTP_C)_C.\n\\]\nIf an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","C),\n\\qquad\n\\rho(T)=(P_CTP_C)_C.\n\\]\nIf an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","rmediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset.  Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective.  Its kernel is exactly $\\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.\n\\end{proof}","\\begin{definition}[Transfer graph of a family generating the algebra]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","efinition}[Transfer graph of a family generating the algebra]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","rating the algebra]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","l=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","t system on $X$ with every $P_\\alpha\\in\\A$.  Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$.  The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$.  Reachability includes paths of length zero.\n\\end{definition}\n\\RecordID{PSC-I-D-TRANSFER-GRAPH}","\\begin{theorem}[Reachability support of the generated algebra]\n\\label{thm:algebra-reachability-support}\nFor every $A\\in\\A$,\n\\[\nP_\\beta AP_\\alpha=0\n\\qquad\\text{whenever }\\beta\\notin\\Reach(\\alpha).\n\\]\nEquivalently,\n\\[\n\\A\\subseteq\\mathfrak M_{R_\\Gamma},\n\\qquad\nR_\\Gamma:=\\{(\\beta,\\alpha):\\beta\\in\\Reach(\\alpha)\\}.\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-ALGEBRA-REACHABILITY-SUPPORT}\n\\Depends{PSC-I-D-TRANSFER-GRAPH and PSC-I-C-BLOCK-POWERS.}","\\begin{proof}\nIt is enough to treat a word in the generators and packet projectors.  Insert\n\\[\nI=\\sum_{\\gamma\\in I}P_\\gamma\n\\]\nbetween consecutive nonprojector generators.  Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","oof}\nIt is enough to treat a word in the generators and packet projectors.  Insert\n\\[\nI=\\sum_{\\gamma\\in I}P_\\gamma\n\\]\nbetween consecutive nonprojector generators.  Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","ve nonprojector generators.  Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps.  Thus a word has no block from $\\alpha$ to an unreachable $\\beta$.  Linear combinations preserve the same zero pattern.\n\\end{proof}","\\begin{interpretation}\nThe graph belongs to the declared generating family within \\(\\End(X)\\).  Multiplying its generators concatenates paths, so a word in them cannot create a block outside graph reachability.  Reachability alone gives less information in the other direction.  A nonzero edge block may reach only a proper subspace of its target packet.\n\\end{interpretation}","\\subsection{Reachability obstructions for seeds}\n\\begin{proposition}[Reachability obstruction for generated hulls]\n\\label{prop:reachability-obstruction}\nLet $I_0\\subseteq I$ and let\n\\[\nS\\subseteq\\bigoplus_{\\alpha\\in I_0}X_\\alpha.\n\\]\nThen\n\\[\n\\boxed{\n\\Hull_{\\A}(S)\n\\subseteq\n\\bigoplus_{\\beta\\in\\Reach(I_0)}X_\\beta.\n}\n\\]\nIn particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component.\n\\end{proposition}\n\\RecordID{PSC-I-P-REACHABILITY-OBSTRUCTION}\n\\Depends{PSC-I-T-ALGEBRA-REACHABILITY-SUPPORT.}","\\begin{proof}\nFor $s\\in S$, write $s=\\sum_{\\alpha\\in I_0}P_\\alpha s$.  If $\\beta$ is unreachable from $I_0$, then for every $A\\in\\A$,\n\\[\nP_\\beta As\n=\n\\sum_{\\alpha\\in I_0}P_\\beta AP_\\alpha s\n=0\n\\]\nby \\cref{thm:algebra-reachability-support}.  The same holds for every finite linear combination of orbit vectors.\n\\end{proof}","\\Needspace{15\\baselineskip}\n\\begin{definition}[Packet pure seed rank]\n\\label{def:directed-seed-rank}\nA vector is packet pure when it is zero or lies in one packet space $X_\\alpha$, equivalently when its packet support has cardinality at most one.  Define\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n:=\n\\min\\{|S|:S\\text{ is packet-pure and }\\A S=X\\}.\n\\]\nFor $C\\subseteq I$, put\n\\[\nP_C:=\\sum_{\\alpha\\in C}P_\\alpha,\n\\qquad\nX_C:=P_CX,\n\\qquad\n\\A_C:=P_C\\A P_C,\n\\]\nand define the local packet pure rank\n\\[\nr_C:=\\psrank_{\\A_C,\\Pcal|_C}(X_C).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-DIRECTED-SEED-RANK}","f:directed-seed-rank}\nA vector is packet pure when it is zero or lies in one packet space $X_\\alpha$, equivalently when its packet support has cardinality at most one.  Define\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n:=\n\\min\\{|S|:S\\text{ is packet-pure and }\\A S=X\\}.\n\\]\nFor $C\\subseteq I$, put\n\\[\nP_C:=\\sum_{\\alpha\\in C}P_\\alpha,\n\\qquad\nX_C:=P_CX,\n\\qquad\n\\A_C:=P_C\\A P_C,\n\\]\nand define the local packet pure rank\n\\[\nr_C:=\\psrank_{\\A_C,\\Pcal|_C}(X_C).\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-DIRECTED-SEED-RANK}","\\begin{proposition}[Basic packet pure rank bounds]\n\\label{prop:packet-pure-rank-bounds}\nFor every $A$ module resolved into packets with $P_\\alpha\\in A$,\n\\[\n\\boxed{\n\\srank_A(X)\n\\leq\n\\psrank_{A,\\Pcal}(X)\n\\leq\n\\sum_{\\alpha\\in I}\n\\srank_{P_\\alpha A P_\\alpha}(X_\\alpha).\n}\n\\]\nThe corner algebra on the right is unital on $X_\\alpha$ with identity $P_\\alpha$.\n\\end{proposition}\n\\RecordID{PSC-I-P-PACKET-PURE-RANK-BOUNDS}\n\\Depends{PSC-I-D-SEED-RANK and PSC-I-D-DIRECTED-SEED-RANK.}","\\begin{proof}\nEvery packet pure generating family is an unrestricted generating family, which gives the first inequality.  For each $\\alpha$, choose a minimal generating family $S_\\alpha\\subseteq X_\\alpha$ for the corner action $P_\\alpha A P_\\alpha$.  The union $S=\\bigcup_\\alpha S_\\alpha$ is packet pure.  Since every $x\\in X$ has the unique decomposition $x=\\sum_\\alpha x_\\alpha$ and every $x_\\alpha$ lies in $(P_\\alpha A P_\\alpha)S_\\alpha\\subseteq AS$, the union generates $X$.  Its cardinality is at most the displayed sum.\n\\end{proof}","}\nEvery packet pure generating family is an unrestricted generating family, which gives the first inequality.  For each $\\alpha$, choose a minimal generating family $S_\\alpha\\subseteq X_\\alpha$ for the corner action $P_\\alpha A P_\\alpha$.  The union $S=\\bigcup_\\alpha S_\\alpha$ is packet pure.  Since every $x\\in X$ has the unique decomposition $x=\\sum_\\alpha x_\\alpha$ and every $x_\\alpha$ lies in $(P_\\alpha A P_\\alpha)S_\\alpha\\subseteq AS$, the union generates $X$.  Its cardinality is at most the displayed sum.\n\\end{proof}","ng family, which gives the first inequality.  For each $\\alpha$, choose a minimal generating family $S_\\alpha\\subseteq X_\\alpha$ for the corner action $P_\\alpha A P_\\alpha$.  The union $S=\\bigcup_\\alpha S_\\alpha$ is packet pure.  Since every $x\\in X$ has the unique decomposition $x=\\sum_\\alpha x_\\alpha$ and every $x_\\alpha$ lies in $(P_\\alpha A P_\\alpha)S_\\alpha\\subseteq AS$, the union generates $X$.  Its cardinality is at most the displayed sum.\n\\end{proof}","\\begin{definition}[Packet system surjective along edges]\n\\label{def:edge-surjective}\nThe module resolved into packets is surjective along edges relative to $\\Gamma(\\mathcal G,\\Pcal)$ when, for every graph edge $\\alpha\\to\\beta$, there exists $B\\in\\A$ such that the pure block\n\\[\nP_\\beta BP_\\alpha:X_\\alpha\\twoheadrightarrow X_\\beta\n\\]\nis surjective.\n\\end{definition}\n\\RecordID{PSC-I-D-EDGE-SURJECTIVE}","\\begin{theorem}[Source component formula for packet pure seed rank]\n\\label{thm:source-component-rank}\nAssume $I$ is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges.  Let $\\mathfrak C_0$ be the set of source strongly connected components of the condensation graph of $\\Gamma(\\mathcal G,\\Pcal)$.  Then\n\\[\n\\boxed{\n\\psrank_{\\A,\\Pcal}(X)\n=\n\\sum_{C\\in\\mathfrak C_0}r_C.\n}\n\\]\n\\end{theorem}\n\\RecordID{PSC-I-T-SOURCE-COMPONENT-SEED-RANK}\n\\Depends{PSC-I-P-REACHABILITY-OBSTRUCTION and PSC-I-D-EDGE-SURJECTIVE.}\n\\Exports{Residual transport forests and directed source rank calculations.}","\\begin{proof}\nLet $S$ be a packet pure generating family for $X$.  Discard zero seeds, which do not change the generated hull, and fix a source component $C$.  Seeds outside $C$ cannot reach $C$ by \\cref{prop:reachability-obstruction}.  Moreover, for $s\\in X_C$ and $A\\in\\A$,\n\\[\nP_CAs=(P_CAP_C)s\\in\\A_Cs.\n\\]\nThus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\\A_C$, and there are at least $r_C$ of them.  Summing over disjoint source components gives\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n\\geq\n\\sum_{C\\in\\mathfrak C_0}r_C.\n\\]","gin{proof}\nLet $S$ be a packet pure generating family for $X$.  Discard zero seeds, which do not change the generated hull, and fix a source component $C$.  Seeds outside $C$ cannot reach $C$ by \\cref{prop:reachability-obstruction}.  Moreover, for $s\\in X_C$ and $A\\in\\A$,\n\\[\nP_CAs=(P_CAP_C)s\\in\\A_Cs.\n\\]\nThus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\\A_C$, and there are at least $r_C$ of them.  Summing over disjoint source components gives\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n\\geq\n\\sum_{C\\in\\mathfrak C_0}r_C.\n\\]","Discard zero seeds, which do not change the generated hull, and fix a source component $C$.  Seeds outside $C$ cannot reach $C$ by \\cref{prop:reachability-obstruction}.  Moreover, for $s\\in X_C$ and $A\\in\\A$,\n\\[\nP_CAs=(P_CAP_C)s\\in\\A_Cs.\n\\]\nThus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\\A_C$, and there are at least $r_C$ of them.  Summing over disjoint source components gives\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n\\geq\n\\sum_{C\\in\\mathfrak C_0}r_C.\n\\]","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\ga","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, cho","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed pa","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected compon","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive sur","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subsete","For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$,","e component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ily $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}",", and put $S=\\bigcup_CS_C$.  Then every source component lies in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","s in $\\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ical order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ponents preceding a component $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","ent $D$ that is not a source have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","e have already been shown to lie in $\\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","hoose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","$\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$.  Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","es a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","a}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","teq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$.  For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$.  Hence $X_D\\subseteq\\A S$.  Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}","\\begin{definition}[Packet propagating system]\n\\label{def:packet-propagating}\nA packet system that is surjective along edges is called packet propagating when every packet space $X_\\alpha$ is cyclic under its diagonal corner algebra\n\\[\nP_\\alpha\\A P_\\alpha.\n\\]\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-PROPAGATING}","\\begin{corollary}[Unweighted source component count]\n\\label{cor:source-component-count}\nAssume the module resolved into packets is packet propagating and $X_\\alpha\\neq0$ for every $\\alpha\\in I$.  If $c_0$ denotes the number of source strongly connected components, then\n\\[\n\\boxed{\n\\psrank_{\\A,\\Pcal}(X)=c_0.\n}\n\\]\n\\end{corollary}\n\\RecordID{PSC-I-C-SOURCE-COMPONENT-COUNT}\n\\Depends{PSC-I-T-SOURCE-COMPONENT-SEED-RANK.}","\\begin{proof}\nFix a source component $C$ and a vertex $\\alpha\\in C$.  Choose a cyclic vector $s\\in X_\\alpha$ for $P_\\alpha\\A P_\\alpha$.  It generates all of $X_\\alpha$.  Strong connectivity and surjectivity along edges then propagate the whole packet space to every vertex of $C$.  Thus $r_C=1$ for every source component.  Apply \\cref{thm:source-component-rank}.\n\\end{proof}","\\begin{boundary}[Mixed support seeds can merge source requirements]\n\\label{bnd:packet-pure-boundary}\nThe source component formulas concern packet pure seeds.  They do not give the unrestricted seed rank.  For example, let\n\\[\n\\A=\\bbk\\times\\bbk,\n\\qquad\nX=\\bbk\\oplus\\bbk\n\\]\nwith the coordinate action and the two coordinate packet projectors.  The transfer graph has two source components and\n\\[\n\\psrank_{\\A,\\Pcal}(X)=2,\n\\]\nbut the mixed vector $(1,1)$ is cyclic, so\n\\[\n\\srank_{\\A}(X)=1.\n\\]\n\\end{boundary}\n\\RecordID{PSC-I-B-PACKET-PURE-BOUNDARY}","rge source requirements]\n\\label{bnd:packet-pure-boundary}\nThe source component formulas concern packet pure seeds.  They do not give the unrestricted seed rank.  For example, let\n\\[\n\\A=\\bbk\\times\\bbk,\n\\qquad\nX=\\bbk\\oplus\\bbk\n\\]\nwith the coordinate action and the two coordinate packet projectors.  The transfer graph has two source components and\n\\[\n\\psrank_{\\A,\\Pcal}(X)=2,\n\\]\nbut the mixed vector $(1,1)$ is cyclic, so\n\\[\n\\srank_{\\A}(X)=1.\n\\]\n\\end{boundary}\n\\RecordID{PSC-I-B-PACKET-PURE-BOUNDARY}","]\n\\label{bnd:packet-pure-boundary}\nThe source component formulas concern packet pure seeds.  They do not give the unrestricted seed rank.  For example, let\n\\[\n\\A=\\bbk\\times\\bbk,\n\\qquad\nX=\\bbk\\oplus\\bbk\n\\]\nwith the coordinate action and the two coordinate packet projectors.  The transfer graph has two source components and\n\\[\n\\psrank_{\\A,\\Pcal}(X)=2,\n\\]\nbut the mixed vector $(1,1)$ is cyclic, so\n\\[\n\\srank_{\\A}(X)=1.\n\\]\n\\end{boundary}\n\\RecordID{PSC-I-B-PACKET-PURE-BOUNDARY}","\\begin{microexample}[title={Why mixed seeds change the answer}]\nLet the acting algebra be diagonal on \\(X=X_1\\oplus X_2\\), with no transfer between the packets.  Generation by packet pure seeds requires one seed in each source component, so \\(\\psrank=2\\).  A single mixed vector \\(x_1+x_2\\) may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can be \\(1\\).\n\\end{microexample}","\\begin{theorem}[Coordinate transport]\n\\label{thm:coordinate-transport}\nLet $U:X\\to Y$ be a vector space isomorphism and let $\\Pcal=(P_\\alpha)$ be a packet system on $X$.  Define\n\\[\nQ_\\alpha:=UP_\\alpha U^{-1}.\n\\]\nThen $\\Qcal=(Q_\\alpha)$ is a packet system on $Y$.  For $\\widetilde T=UTU^{-1}$,\n\\[\n\\boxed{\nQ_\\beta\\widetilde TQ_\\alpha\n=\nU(P_\\beta TP_\\alpha)U^{-1}.\n}\n\\]\nHence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport.  If a declared algebra generating family $\\mathcal G$ is transported to $U\\mathcal G U^{-1}$, then its transfer graph and reachability relation are also preserved.\n\\end{theorem}\n\\RecordID{PSC-I-T-COORDINATE-TRANSPORT}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}","sm and let $\\Pcal=(P_\\alpha)$ be a packet system on $X$.  Define\n\\[\nQ_\\alpha:=UP_\\alpha U^{-1}.\n\\]\nThen $\\Qcal=(Q_\\alpha)$ is a packet system on $Y$.  For $\\widetilde T=UTU^{-1}$,\n\\[\n\\boxed{\nQ_\\beta\\widetilde TQ_\\alpha\n=\nU(P_\\beta TP_\\alpha)U^{-1}.\n}\n\\]\nHence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport.  If a declared algebra generating family $\\mathcal G$ is transported to $U\\mathcal G U^{-1}$, then its transfer graph and reachability relation are also preserved.\n\\end{theorem}\n\\RecordID{PSC-I-T-COORDINATE-TRANSPORT}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}","e a packet system on $X$.  Define\n\\[\nQ_\\alpha:=UP_\\alpha U^{-1}.\n\\]\nThen $\\Qcal=(Q_\\alpha)$ is a packet system on $Y$.  For $\\widetilde T=UTU^{-1}$,\n\\[\n\\boxed{\nQ_\\beta\\widetilde TQ_\\alpha\n=\nU(P_\\beta TP_\\alpha)U^{-1}.\n}\n\\]\nHence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport.  If a declared algebra generating family $\\mathcal G$ is transported to $U\\mathcal G U^{-1}$, then its transfer graph and reachability relation are also preserved.\n\\end{theorem}\n\\RecordID{PSC-I-T-COORDINATE-TRANSPORT}\n\\Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}","\\begin{proof}\nConjugation preserves products and sums, so\n\\[\nQ_\\alpha Q_\\beta\n=UP_\\alpha P_\\beta U^{-1}\n=\\delta_{\\alpha\\beta}Q_\\alpha,\n\\qquad\n\\sum_\\alpha Q_\\alpha=I_Y.\n\\]\nThe block identity is direct multiplication:\n\\[\nQ_\\beta\\widetilde TQ_\\alpha\n=UP_\\beta TP_\\alpha U^{-1}.\n\\]\nAll listed invariants are determined by these transported blocks.\n\\end{proof}","\\begin{definition}[Packet compatible operator morphism]\n\\label{def:packet-morphism}\nLet $(X,\\A,\\Pcal)$ and $(Y,\\B,\\Qcal)$ be packet resolved operator modules with packet index sets $I$ and $J$.  A packet compatible operator morphism is a triple\n\\[\n(\\Phi,\\varphi,\\sigma)\n\\]\nconsisting of a unital algebra homomorphism $\\varphi:\\A\\to\\B$, a linear map $\\Phi:X\\to Y$ satisfying\n\\[\n\\Phi(Ax)=\\varphi(A)\\Phi(x),\n\\]\nand a map $\\sigma:I\\to J$ satisfying\n\\[\nQ_j\\Phi\n=\n\\Phi P_{\\sigma^{-1}(j)},\n\\qquad\nP_{\\sigma^{-1}(j)}:=\n\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nAn empty fiber is interpreted as the zero projector.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-MORPHISM}","Packet compatible operator morphism]\n\\label{def:packet-morphism}\nLet $(X,\\A,\\Pcal)$ and $(Y,\\B,\\Qcal)$ be packet resolved operator modules with packet index sets $I$ and $J$.  A packet compatible operator morphism is a triple\n\\[\n(\\Phi,\\varphi,\\sigma)\n\\]\nconsisting of a unital algebra homomorphism $\\varphi:\\A\\to\\B$, a linear map $\\Phi:X\\to Y$ satisfying\n\\[\n\\Phi(Ax)=\\varphi(A)\\Phi(x),\n\\]\nand a map $\\sigma:I\\to J$ satisfying\n\\[\nQ_j\\Phi\n=\n\\Phi P_{\\sigma^{-1}(j)},\n\\qquad\nP_{\\sigma^{-1}(j)}:=\n\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nAn empty fiber is interpreted as the zero projector.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-MORPHISM}","orphism]\n\\label{def:packet-morphism}\nLet $(X,\\A,\\Pcal)$ and $(Y,\\B,\\Qcal)$ be packet resolved operator modules with packet index sets $I$ and $J$.  A packet compatible operator morphism is a triple\n\\[\n(\\Phi,\\varphi,\\sigma)\n\\]\nconsisting of a unital algebra homomorphism $\\varphi:\\A\\to\\B$, a linear map $\\Phi:X\\to Y$ satisfying\n\\[\n\\Phi(Ax)=\\varphi(A)\\Phi(x),\n\\]\nand a map $\\sigma:I\\to J$ satisfying\n\\[\nQ_j\\Phi\n=\n\\Phi P_{\\sigma^{-1}(j)},\n\\qquad\nP_{\\sigma^{-1}(j)}:=\n\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nAn empty fiber is interpreted as the zero projector.\n\\end{definition}\n\\RecordID{PSC-I-D-PACKET-MORPHISM}","\\begin{definition}[Morphism of seeded systems]\n\\label{def:seed-morphism}\nA morphism\n\\[\n(X,\\A,S,\\Pcal)\\longrightarrow(Y,\\B,S',\\Qcal)\n\\]\nis a packet compatible operator morphism satisfying the seed condition\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S').\n\\]\nIn the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.\n\\end{definition}\n\\RecordID{PSC-I-D-SEED-MORPHISM}","\\begin{theorem}[Naturality of generated hulls and transfer blocks]\n\\label{thm:naturality}\nLet $(\\Phi,\\varphi,\\sigma)$ be a packet compatible operator morphism.  For every subset $S\\subseteq X$,\n\\[\n\\boxed{\n\\Phi(\\Hull_{\\A}(S))\n=\n\\Hull_{\\varphi(\\A)}(\\Phi(S))\n\\subseteq\n\\Hull_{\\B}(\\Phi(S)).\n}\n\\]\nFor $T\\in\\A$ and $i,j\\in J$,\n\\[\n\\boxed{\nQ_j\\varphi(T)Q_i\\Phi\n=\n\\sum_{\\substack{\\sigma(\\beta)=j","\\begin{theorem}[Naturality of generated hulls and transfer blocks]\n\\label{thm:naturality}\nLet $(\\Phi,\\varphi,\\sigma)$ be a packet compatible operator morphism.  For every subset $S\\subseteq X$,\n\\[\n\\boxed{\n\\Phi(\\Hull_{\\A}(S))\n=\n\\Hull_{\\varphi(\\A)}(\\Phi(S))\n\\subseteq\n\\Hull_{\\B}(\\Phi(S)).\n}\n\\]\nFor $T\\in\\A$ and $i,j\\in J$,\n\\[\n\\boxed{\nQ_j\\varphi(T)Q_i\\Phi\n=\n\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n}\n\\]\nIf $\\Phi$ is surjective and $S$ generates $X$, then $\\Phi(S)$ generates $Y$ under $\\varphi(\\A)$ and hence under $\\B$.  If, in addition, the morphism is seeded, then $S'$ generates $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-NATURALITY}\n\\Depends{PSC-I-D-PACKET-MORPHISM and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{The canonical naturality theorem for every later realization.}","sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n}\n\\]\nIf $\\Phi$ is surjective and $S$ generates $X$, then $\\Phi(S)$ generates $Y$ under $\\varphi(\\A)$ and hence under $\\B$.  If, in addition, the morphism is seeded, then $S'$ generates $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-NATURALITY}\n\\Depends{PSC-I-D-PACKET-MORPHISM and PSC-I-T-BLOCK-MULTIPLICATION.}\n\\Exports{The canonical naturality theorem for every later realization.}","\\begin{proof}\nEvery element of $\\Phi(\\Hull_{\\A}(S))$ is a finite sum of vectors\n\\[\n\\Phi(As)=\\varphi(A)\\Phi(s),\n\\]\nso it belongs to $\\Hull_{\\varphi(\\A)}(\\Phi(S))$.  Conversely, every generator $\\varphi(A)\\Phi(s)$ equals $\\Phi(As)$, giving equality.","For the block formula, use intertwining and packet compatibility twice:\n\\begin{align*}\nQ_j\\varphi(T)Q_i\\Phi\n&=Q_j\\varphi(T)\\Phi P_{\\sigma^{-1}(i)}\\\\\n&=Q_j\\Phi TP_{\\sigma^{-1}(i)}\\\\\n&=\\Phi P_{\\sigma^{-1}(j)}TP_{\\sigma^{-1}(i)}\\\\\n&=\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n\\end{align*}\nIf $\\Phi$ is surjective and $\\Hull_{\\A}(S)=X$, the hull equality gives $Y=\\Hull_{\\varphi(\\A)}(\\Phi(S))$.  If $\\Phi(S)\\subseteq\\Hull_{\\B}(S')$, then $\\Hull_{\\B}(\\Phi(S))\\subseteq\\Hull_{\\B}(S')$, so $S'$ generates $Y$.\n\\end{proof}","sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n\\end{align*}\nIf $\\Phi$ is surjective and $\\Hull_{\\A}(S)=X$, the hull equality gives $Y=\\Hull_{\\varphi(\\A)}(\\Phi(S))$.  If $\\Phi(S)\\subseteq\\Hull_{\\B}(S')$, then $\\Hull_{\\B}(\\Phi(S))\\subseteq\\Hull_{\\B}(S')$, so $S'$ generates $Y$.\n\\end{proof}","\\begin{proposition}[Category of seeded packet systems]\n\\label{prop:seed-category}\nPacket compatible operator morphisms compose, seeded morphisms compose, and identity morphisms satisfy both definitions.  Thus packet resolved operator modules form a category $\\mathbf{PktMod}$ and seeded packet systems form a category $\\mathbf{SeedSys}$.\n\\end{proposition}\n\\RecordID{PSC-I-P-SEED-SYSTEM-CATEGORY}\n\\Depends{\\nolinkurl{PSC-I-D-PACKET-MORPHISM},\n\\nolinkurl{PSC-I-D-SEED-MORPHISM}, and\n\\nolinkurl{PSC-I-T-NATURALITY}.}","\\begin{proof}\nLet $(\\Phi,\\varphi,\\sigma):(X,\\A,\\Pcal)\\to(Y,\\B,\\Qcal)$ and $(\\Psi,\\psi,\\tau):(Y,\\B,\\Qcal)\\to(Z,\\Ccal,\\Rcal)$ be morphisms.  Intertwining is preserved because\n\\[\n\\Psi\\Phi(Ax)\n=\n\\Psi(\\varphi(A)\\Phi x)\n=\n(\\psi\\circ\\varphi)(A)\\Psi\\Phi x.\n\\]\nFor $k$ in the target packet set,\n\\begin{align*}\nR_k\\Psi\\Phi\n&=\\Psi Q_{\\tau^{-1}(k)}\\Phi","\\begin{proof}\nLet $(\\Phi,\\varphi,\\sigma):(X,\\A,\\Pcal)\\to(Y,\\B,\\Qcal)$ and $(\\Psi,\\psi,\\tau):(Y,\\B,\\Qcal)\\to(Z,\\Ccal,\\Rcal)$ be morphisms.  Intertwining is preserved because\n\\[\n\\Psi\\Phi(Ax)\n=\n\\Psi(\\varphi(A)\\Phi x)\n=\n(\\psi\\circ\\varphi)(A)\\Psi\\Phi x.\n\\]\nFor $k$ in the target packet set,\n\\begin{align*}\nR_k\\Psi\\Phi\n&=\\Psi Q_{\\tau^{-1}(k)}\\Phi\\\\\n&=\\Psi\\Phi\n\\sum_{\\tau(\\sigma(\\alpha))=k}P_\\alpha.\n\\end{align*}\nThus the composite packet map is $\\tau\\circ\\sigma$.  If the morphisms are seeded, then\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S'),\n\\qquad\n\\Psi(S')\\subseteq\\Hull_{\\Ccal}(S'').\n\\]\nBy \\cref{thm:naturality},\n\\[\n\\Psi\\Phi(S)\n\\subseteq\n\\Psi(\\Hull_{\\B}(S'))\n\\subseteq\n\\Hull_{\\Ccal}(\\Psi(S'))\n\\subseteq\n\\Hull_{\\Ccal}(S'').\n\\]\nIdentity maps satisfy the definitions with identity algebra and packet maps.\n\\end{proof}","&=\\Psi\\Phi\n\\sum_{\\tau(\\sigma(\\alpha))=k}P_\\alpha.\n\\end{align*}\nThus the composite packet map is $\\tau\\circ\\sigma$.  If the morphisms are seeded, then\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S'),\n\\qquad\n\\Psi(S')\\subseteq\\Hull_{\\Ccal}(S'').\n\\]\nBy \\cref{thm:naturality},\n\\[\n\\Psi\\Phi(S)\n\\subseteq\n\\Psi(\\Hull_{\\B}(S'))\n\\subseteq\n\\Hull_{\\Ccal}(\\Psi(S'))\n\\subseteq\n\\Hull_{\\Ccal}(S'').\n\\]\nIdentity maps satisfy the definitions with identity algebra and packet maps.\n\\end{proof}","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\R","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PAC","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and q","\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","n invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","q X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$.  Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","line P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}\n\\RecordID{PSC-I-T-PACKET-EXACT-SEQUENCE}\n\\Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}\n\\Exports{Packetwise subobject and quotient constructions.}","\\begin{proof}\nRestriction preserves idempotency, pairwise annihilation, and the identity sum on $K$.  If $x\\in K\\cap X_\\alpha$, then $x=P_\\alpha x\\in P_\\alpha K$, proving $K_\\alpha=K\\cap X_\\alpha$.","The quotient operator is well defined because replacing $x$ by $x+k$ changes $P_\\alpha x$ by $P_\\alpha k\\in K$.  The idempotent and annihilation identities descend, and the sum is the identity on $X/K$.  The map $X_\\alpha\\to X/K$ has image $(X_\\alpha+K)/K$ and kernel $X_\\alpha\\cap K$, giving the short exact sequence and the quotient isomorphisms.  Exact packet coordinates identify the original sequence with the direct sum over $\\alpha$.\n\\end{proof}","fined because replacing $x$ by $x+k$ changes $P_\\alpha x$ by $P_\\alpha k\\in K$.  The idempotent and annihilation identities descend, and the sum is the identity on $X/K$.  The map $X_\\alpha\\to X/K$ has image $(X_\\alpha+K)/K$ and kernel $X_\\alpha\\cap K$, giving the short exact sequence and the quotient isomorphisms.  Exact packet coordinates identify the original sequence with the direct sum over $\\alpha$.\n\\end{proof}","\\begin{microexample}[title={Packet descent to a quotient}]\nLet \\(X=X_1\\oplus X_2\\) and let \\(K\\subseteq X_1\\) be a subspace.  The quotient has packets \\(X_1/K\\) and \\(X_2\\).  Any operator preserving \\(K\\) descends with the same organization by source and target packets, with blocks entering or leaving the first packet replaced by their induced quotient maps.  An induced block can become zero even when its original block is nonzero, so the pattern of nonzero blocks can shrink.  Any resulting zero packet is removed from the quotient packet system.\n\\end{microexample}","\\begin{theorem}[Packet descent through an invariant quotient]\n\\label{thm:quotient-packet-descent}\nUnder the hypotheses of \\cref{thm:packet-exact-sequence}, the operators\n\\[\n\\overline P_\\alpha(x+K)=P_\\alpha x+K\n\\]\nare well defined pairwise annihilating idempotents with sum $I_{X/K}$, and the quotient map satisfies\n\\[\n\\boxed{\\pi P_\\alpha=\\overline P_\\alpha\\pi.}\n\\]\nAfter zero packets are removed, they form the canonical quotient packet system.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-PACKET-DESCENT}\n\\Depends{PSC-I-T-PACKET-EXACT-SEQUENCE.}","\\begin{proof}\nAll assertions are the quotient part of \\cref{thm:packet-exact-sequence}.  The intertwining identity follows directly from the definition:\n\\[\n\\overline P_\\alpha\\pi(x)=P_\\alpha x+K=\\pi P_\\alpha(x).\n\\]\n\\end{proof}","\\begin{corollary}[Restriction and quotient of transfer blocks]\n\\label{cor:block-descent}\nUnder the hypotheses of \\cref{thm:packet-exact-sequence}, let $T\\in\\End(X)$ preserve $K$.  Then every block $T_{\\beta\\alpha}$ maps $K_\\alpha$ into $K_\\beta$, and the induced quotient operator $\\overline T$ satisfies\n\\[\n\\boxed{\n\\overline T_{\\beta\\alpha}\\,\\pi\n=\n\\pi T_{\\beta\\alpha},\n}\n\\]\nwhere $\\pi:X\\twoheadrightarrow X/K$.\n\\end{corollary}\n\\RecordID{PSC-I-C-QUOTIENT-PACKET-DESCENT}\n\\Depends{PSC-I-T-PACKET-EXACT-SEQUENCE.}","\\begin{proof}\nIf $x\\in K_\\alpha$, then $Tx\\in K$, so\n\\[\nT_{\\beta\\alpha}x=P_\\beta Tx\\in K\\cap X_\\beta=K_\\beta.\n\\]\nOn the quotient,\n\\[\n\\overline T_{\\beta\\alpha}\\pi\n=\n\\overline P_\\beta\\overline T\\overline P_\\alpha\\pi\n=\n\\pi P_\\beta TP_\\alpha.\n\\]\n\\end{proof}","\\begin{corollary}[Topological packet exact sequence]\n\\label{cor:topological-packet-exact-sequence}\nIf $X$ is Banach, the $P_\\alpha$ are bounded, and $K$ is closed and packet invariant, then every packet sequence in \\cref{thm:packet-exact-sequence} is a short exact sequence of Banach spaces with bounded maps, and the quotient projectors are bounded.\n\\end{corollary}\n\\RecordID{PSC-I-C-TOPOLOGICAL-PACKET-EXACT-SEQUENCE}\n\\Depends{\\nolinkurl{PSC-I-T-PACKET-EXACT-SEQUENCE} and\n\\nolinkurl{PSC-I-T-BANACH-PACKET-ISOMORPHISM}.}","\\begin{proof}\nClosedness of $K$ makes $X/K$ Banach.  The restricted projectors are bounded, and the quotient projectors are bounded because\n\\[\n\\|\\overline P_\\alpha(x+K)\\|_{X/K}\n\\leq\n\\|P_\\alpha\\|\\,\\|x+K\\|_{X/K}.\n\\]\nThe packet spaces are closed ranges of bounded idempotents.\n\\end{proof}","\\begin{theorem}[Dense realization theorem]\n\\label{thm:dense-realization}\nLet $X$ and $Y$ be Hausdorff topological vector spaces, let $\\Phi:X\\to Y$ be continuous, surjective, and intertwining with respect to $\\varphi$, and let $S$ topologically generate $X$.  Then $\\Phi(S)$ topologically generates $Y$.  If finite continuous packet systems satisfy\n\\[\nQ_j\\Phi=\\Phi P_{\\sigma^{-1}(j)},\n\\]\nthen\n\\[\n\\boxed{\nQ_jY=\\Phi(P_{\\sigma^{-1}(j)}X)\n}\n\\]\nfor every $j$.\n\\end{theorem}\n\\RecordID{PSC-I-T-DENSE-REALIZATION}\n\\Depends{PSC-I-T-NATURALITY.}","thm:dense-realization}\nLet $X$ and $Y$ be Hausdorff topological vector spaces, let $\\Phi:X\\to Y$ be continuous, surjective, and intertwining with respect to $\\varphi$, and let $S$ topologically generate $X$.  Then $\\Phi(S)$ topologically generates $Y$.  If finite continuous packet systems satisfy\n\\[\nQ_j\\Phi=\\Phi P_{\\sigma^{-1}(j)},\n\\]\nthen\n\\[\n\\boxed{\nQ_jY=\\Phi(P_{\\sigma^{-1}(j)}X)\n}\n\\]\nfor every $j$.\n\\end{theorem}\n\\RecordID{PSC-I-T-DENSE-REALIZATION}\n\\Depends{PSC-I-T-NATURALITY.}","\\begin{proof}\nGiven $y=\\Phi(x)$, choose a net $x_\\lambda\\in\\Hull_{\\A}(S)$ converging to $x$.  Then\n\\[\n\\Phi(x_\\lambda)\\to y,\n\\]\nand naturality places every $\\Phi(x_\\lambda)$ in $\\Hull_{\\B}(\\Phi(S))$.  Hence the latter hull is dense.  For packets, one inclusion follows from compatibility.  Conversely, if $y=Q_jy$, choose $x$ with $\\Phi(x)=y$.  Then\n\\[\ny=Q_j\\Phi(x)=\\Phi P_{\\sigma^{-1}(j)}x.\n\\]\n\\end{proof}","\\begin{theorem}[Quotient topology and completed realization]\n\\label{thm:quotient-topology}\nLet $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","m}[Quotient topology and completed realization]\n\\label{thm:quotient-topology}\nLet $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","leted realization]\n\\label{thm:quotient-topology}\nLet $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","et $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","e a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$.  Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism.  If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}\n\\RecordID{PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION}\n\\Depends{PSC-I-T-COMPLETED-PRESENTATION, the definition of quotient topology, and uniqueness of completion.}","\\begin{proof}\nThe algebraic first isomorphism theorem gives a bijection $\\widetilde\\Phi$.  By definition of the quotient topology, a set $U\\subseteq X/K$ is open exactly when its preimage in $X$ is open.  Because $\\Phi$ is a quotient map, this is equivalent to $\\widetilde\\Phi(U)$ being open in $Y$.  Thus $\\widetilde\\Phi$ is a homeomorphism.  The second statement applies the dense completion argument of \\cref{thm:completed-presentation} to the injective map $D/(D\\cap K)\\to Y$.\n\\end{proof}","bijection $\\widetilde\\Phi$.  By definition of the quotient topology, a set $U\\subseteq X/K$ is open exactly when its preimage in $X$ is open.  Because $\\Phi$ is a quotient map, this is equivalent to $\\widetilde\\Phi(U)$ being open in $Y$.  Thus $\\widetilde\\Phi$ is a homeomorphism.  The second statement applies the dense completion argument of \\cref{thm:completed-presentation} to the injective map $D/(D\\cap K)\\to Y$.\n\\end{proof}","\\begin{boundary}[Continuous surjections and quotient topology]\n\\label{bnd:topological-quotient}\nContinuity and surjectivity transport density as in \\cref{thm:dense-realization}.  They do not by themselves identify the given topology on $Y$ with the quotient topology from $X$.  The stronger conclusion in \\cref{thm:quotient-topology} requires the quotient map hypothesis or an equivalent open mapping condition.\n\\end{boundary}\n\\RecordID{PSC-I-B-TOPOLOGICAL-QUOTIENT}","\\begin{definition}[Realization functor and preservation profile]\n\\label{def:realization-functor}\nLet $\\mathbf{SeedSys}_{\\mathrm{abs}}$ be an abstract category of seed systems and let $\\mathbf C_D$ be a domain category.  A realization functor is a functor\n\\[\n\\mathsf{Ev}_D:\\mathbf{SeedSys}_{\\mathrm{abs}}\\longrightarrow\\mathbf C_D\n\\]\nwith a declared preservation profile specifying which of the following structures are preserved:\n\\[\n\\begin{gathered}\n0,\\ I,\\ \\text{finite sums},\\ \\text{composition},\\ \\text{biproducts},","unctor and preservation profile]\n\\label{def:realization-functor}\nLet $\\mathbf{SeedSys}_{\\mathrm{abs}}$ be an abstract category of seed systems and let $\\mathbf C_D$ be a domain category.  A realization functor is a functor\n\\[\n\\mathsf{Ev}_D:\\mathbf{SeedSys}_{\\mathrm{abs}}\\longrightarrow\\mathbf C_D\n\\]\nwith a declared preservation profile specifying which of the following structures are preserved:\n\\[\n\\begin{gathered}\n0,\\ I,\\ \\text{finite sums},\\ \\text{composition},\\ \\text{biproducts},\\\\\n\\text{kernels},\\ \\text{cokernels},\\ \\text{quotients},\\ \\text{tensor products},\\\\\n\\text{packet idempotents},\\ \\text{transfer blocks},\\ \\text{topological completions}.\n\\end{gathered}\n\\]\nPreservation of a structure means preservation of its defining identity or universal property, not merely a similarity of notation.\n\\end{definition}\n\\RecordID{PSC-I-D-REALIZATION-FUNCTOR}","\\begin{theorem}[Equational realization principle]\n\\label{thm:realization-export}\nLet $F:\\mathbf C\\to\\mathbf D$ be an additive functor between preadditive categories.  Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATIO","\\begin{theorem}[Equational realization principle]\n\\label{thm:realization-export}\nLet $F:\\mathbf C\\to\\mathbf D$ be an additive functor between preadditive categories.  Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","label{thm:realization-export}\nLet $F:\\mathbf C\\to\\mathbf D$ be an additive functor between preadditive categories.  Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","rom declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","reserved by $F$.  Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment.  If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}\n\\RecordID{PSC-I-T-REALIZATION-EXPORT}\n\\Depends{PSC-I-D-REALIZATION-FUNCTOR.}\n\\Exports{The formal criterion for semantic transfer across domains.}","\\begin{proof}\nFunctoriality gives\n\\[\nF(I)=I,\n\\qquad\nF(g\\circ f)=F(g)\\circ F(f),\n\\]\nand additivity gives\n\\[\nF(0)=0,\n\\qquad\nF(f+g)=F(f)+F(g),\n\\qquad\nF(-f)=-F(f).\n\\]\nIf the declared enrichment is present, linearity additionally gives $F(af)=aF(f)$ for every coefficient $a$.  A structural induction on the construction of a morphism expression therefore carries every permitted equational proof to its image under $F$.  If a step invokes a universal construction, preservation means that the image diagram satisfies the same universal property.  The uniquely induced morphism in the source proof is therefore carried to the uniquely induced morphism in the target proof.  Repeating this argument through the finite proof diagram transports the conclusion.\n\\end{proof}","}\nFunctoriality gives\n\\[\nF(I)=I,\n\\qquad\nF(g\\circ f)=F(g)\\circ F(f),\n\\]\nand additivity gives\n\\[\nF(0)=0,\n\\qquad\nF(f+g)=F(f)+F(g),\n\\qquad\nF(-f)=-F(f).\n\\]\nIf the declared enrichment is present, linearity additionally gives $F(af)=aF(f)$ for every coefficient $a$.  A structural induction on the construction of a morphism expression therefore carries every permitted equational proof to its image under $F$.  If a step invokes a universal construction, preservation means that the image diagram satisfies the same universal property.  The uniquely induced morphism in the source proof is therefore carried to the uniquely induced morphism in the target proof.  Repeating this argument through the finite proof diagram transports the conclusion.\n\\end{proof}","\\begin{definition}[Paper I export object]\n\\label{def:paper-one-export}\nThe canonical Paper I export is\n\\[\n\\boxed{\n\\mathsf{PSC}_I=\n\\left(\n\\begin{array}{c}\n\\text{typed grammars and realized operator algebras},\\\\\n\\text{seed systems, free presentations, and seed rank},\\\\\n\\text{module-top, semisimple, and single-operator formulas},\\\\\n\\text{finite and unconditional packet coordinates},\\\\\n\\text{transfer blocks and support algebras},\\\\\n\\text{reachability and packet-pure source generation},\\\\\n\\text{morphisms, invariant exact sequences, quotients, completions},\\\\\n\\text{realization functors and equational theorem transport}\n\\end{array}\n\\right).\n}\n\\]\n\\end{definition}\n\\RecordID{PSC-I-EXPORT}","Let the quiver have vertices $U,V$ and arrows\n\\[\nR:U\\to U,\n\\qquad\nA:U\\to V,\n\\qquad\nB:V\\to U,\n\\]\nwith the relation in the path algebra\n\\[\nBA=e_U,\n\\]\nwhose realization is the operator identity $BA=I_U$.\nRealize\n\\[\nU=\\bbk^2,\n\\qquad\nV=\\bbk^3,\n\\]\nby\n\\[\nR=\n\\begin{pmatrix}0&-1","Let the quiver have vertices $U,V$ and arrows\n\\[\nR:U\\to U,\n\\qquad\nA:U\\to V,\n\\qquad\nB:V\\to U,\n\\]\nwith the relation in the path algebra\n\\[\nBA=e_U,\n\\]\nwhose realization is the operator identity $BA=I_U$.\nRealize\n\\[\nU=\\bbk^2,\n\\qquad\nV=\\bbk^3,\n\\]\nby\n\\[\nR=\n\\begin{pmatrix}0&-1\\\\1&0\\end{pmatrix},\n\\qquad\nA(u_1,u_2)=(u_1,u_2,0),\n\\qquad\nB(v_1,v_2,v_3)=(v_1,v_2).\n\\]\nStarting from the seed $e_1\\in U$, repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.","1&0\\end{pmatrix},\n\\qquad\nA(u_1,u_2)=(u_1,u_2,0),\n\\qquad\nB(v_1,v_2,v_3)=(v_1,v_2).\n\\]\nStarting from the seed $e_1\\in U$, repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.","2,0),\n\\qquad\nB(v_1,v_2,v_3)=(v_1,v_2).\n\\]\nStarting from the seed $e_1\\in U$, repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.",", repeated application of $R$ spans $U$, and $A$ carries that span onto the first two coordinates of $V$. Any path from $U$ to $V$ ends with $A$ and therefore lands in $\\im A$. The third coordinate direction, which spans $\\ker B$, is unreachable from a seed supported only at $U$. Therefore the generated representation is\n\\[\nU\\oplus\\Span\\{(1,0,0),(0,1,0)\\}\n\\subsetneq U\\oplus V.\n\\]\nAdding the seed $(0,0,1)\\in V$ generates the full representation.","\\begin{proposition}[Typed reachability certificate]\n\\label{prop:typed-reachability}\nIn the preceding quiver realization, no seed supported at $U$ can generate the full representation, whereas the two seeds\n\\[\ne_1\\in U,\n\\qquad\ne_3\\in V\n\\]\ndo generate it.\n\\end{proposition}","\\begin{proof}\nThe first claim follows because every path that starts at $U$ and ends in $V$ has final arrow $A$, so its image lies in $\\im A$. For the second, $e_1$ and $Re_1$ span $U$. Their images under $A$ span $\\im A$, and $e_3$ spans $\\ker B$, the missing complementary direction in $V$.\n\\end{proof}","This example shows why a typed grammar is more than a notational convenience. The obstruction lies in the subspace $\\ker B\\subset V$, which is unreachable from seeds supported at $U$, and is visible before any coordinate calculation.","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of de","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concr","Let $\\operatorname{char}\\bbk\\ne2$ and let\n\\[\nA=\\bbk[x,y]/(y^2-x^3+x).\n\\]\nThe affine equation $y^2=x^3-x$ has classical discriminant $64$, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","c curve \\cite{silverman2009}. The Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","Lean development checks the algebraic implication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","mplication $2\\ne0\\Rightarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","ghtarrow64\\ne0$, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","ities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","alize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","ctive smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","e geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring.\nEvery class has a unique normal form\n\\[\nf(x,y)=a(x)+y b(x).\n\\]\nIndeed, $y^2-(x^3-x)$ is monic of degree two in $y$, so division in $\\bbk[x][y]$ gives a unique remainder of degree less than two in $y$. Concretely, every power $y^m$ with $m\\ge2$ can be reduced using\n\\[\ny^2=x^3-x.\n\\]","Take the seed\n\\[\ns=1\n\\]\nand let $M_x,M_y$ be multiplication by $x$ and $y$. Then\n\\[\nM_x^iM_y^js=x^iy^j,\n\\]\nso the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","The two packet spaces are\n\\[\nA_0=\\bbk[x],\n\\qquad\nA_1=y\\bbk[x].\n\\]\nLet\n\\[\nP_0(a+yb)=a,\n\\qquad\nP_1(a+yb)=yb.\n\\]\nThese idempotents are linear over $\\bbk$, annihilate one another, and satisfy $P_0+P_1=I$. They are not module maps over $A$, because multiplication by $y$ exchanges the two packets.","Identify the packet coefficients by\n\\[\nA_0\\oplus A_1\\longrightarrow\\bbk[x]\\oplus\\bbk[x],\n\\qquad\n(a,yb)\\longmapsto(a,b).\n\\]\nIn these ordered coefficient coordinates, with $M_x$ and $M_{x^3-x}$ denoting multiplication on $\\bbk[x]$,\n\\[\n{\nM_x=\n\\begin{pmatrix}\nM_x&0","Identify the packet coefficients by\n\\[\nA_0\\oplus A_1\\longrightarrow\\bbk[x]\\oplus\\bbk[x],\n\\qquad\n(a,yb)\\longmapsto(a,b).\n\\]\nIn these ordered coefficient coordinates, with $M_x$ and $M_{x^3-x}$ denoting multiplication on $\\bbk[x]$,\n\\[\n{\nM_x=\n\\begin{pmatrix}\nM_x&0\\\\\n0&M_x\n\\end{pmatrix},\n\\qquad\nM_y=\n\\begin{pmatrix}\n0&M_{x^3-x}\\\\\nI&0\n\\end{pmatrix}.\n}\n\\]\nThe two rows of the second matrix record the outgoing packet coefficients. An even input enters the odd packet without an additional factor. An odd input returns to the even packet through the factor imposed by the curve equation. Applying the matrix twice therefore preserves each packet and multiplies its coefficient by that factor.","The relation is reconstructed as an operator identity:\n\\[\nM_y^2=M_x^3-M_x.\n\\]","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{08_elliptic_packet_transfer.pdf}\n\\caption{Multiplication by $y$ exchanges the even and odd packets. The transfer from the odd packet to the even packet carries the relation weight $x^3-x$.}\n\\label{fig:elliptic-transfer}\n\\end{figure}","Write\n\\[\nf=a_0+ya_1,\n\\qquad\ng=b_0+yb_1.\n\\]\nThen\n\\[\n{\nfg\n=\n\\bigl(a_0b_0+(x^3-x)a_1b_1\\bigr)\n+\ny\\bigl(a_0b_1+a_1b_0\\bigr).\n}\n\\]\nThis is a convolution law for the two packets. It is already visible at the linear operator level because multiplication by a fixed element is a transfer operator. The even output collects the product of the two even coefficients and the weighted product of the two odd coefficients. The odd output collects the two products with one even and one odd coefficient. The quotient relation is thus retained in the weight rather than discarded during the decomposition.","\\begin{example}[Worked reduction]\nTake\n\\[\nf=x^2+y,\n\\qquad\ng=x-y.\n\\]\nThen\n\\[\n\\begin{aligned}\nfg\n&=x^3-x^2y+xy-y^2","\\begin{example}[Worked reduction]\nTake\n\\[\nf=x^2+y,\n\\qquad\ng=x-y.\n\\]\nThen\n\\[\n\\begin{aligned}\nfg\n&=x^3-x^2y+xy-y^2\\\\\n&=x^3-x^2y+xy-(x^3-x)\\\\\n&=x+(x-x^2)y.\n\\end{aligned}\n\\]\nThus\n\\[\nP_0(fg)=x,\n\\qquad\nP_1(fg)=(x-x^2)y.\n\\]\nThe symbolic certificate in the companion code performs the same quotient reduction exactly.\n\\end{example}","&=x+(x-x^2)y.\n\\end{aligned}\n\\]\nThus\n\\[\nP_0(fg)=x,\n\\qquad\nP_1(fg)=(x-x^2)y.\n\\]\nThe symbolic certificate in the companion code performs the same quotient reduction exactly.\n\\end{example}","The coordinate ring illustrates four distinct parts of the calculus:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the unit is a module seed.\n\\item $M_x$ and $M_y$ are address operators.\n\\item the curve equation is an operator relation.\n\\item parity in $y$ gives exact packet analysis, transfer, and reconstruction.\n\\end{enumerate}\nIt does not imply that every algebra can be presented as an algebra generated by one element. It shows instead that unital algebras are cyclic as modules over their full multiplication algebra.","Let $\\mathbb{F}\\in\\{\\R,\\C\\}$ and let\n\\[\nX=\\ell^2(\\Z;\\mathbb F)\n\\]\nwith a family of bounded reindexing operators indexed by the integers\n\\[\n(S_mx)[n]=x[n-m]\n\\qquad(m,n\\in\\Z).\n\\]\nThus $S_0=I$, $S_{m+r}=S_mS_r$, and $S_1$ is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of \\cite{conway1990}.\nThe impulse $\\delta_0$ satisfies\n\\[\nS_m\\delta_0=\\delta_m\n\\]\nfor every $m\\in\\Z$. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in $\\ell^2(\\Z)$.","\\in\\{\\R,\\C\\}$ and let\n\\[\nX=\\ell^2(\\Z;\\mathbb F)\n\\]\nwith a family of bounded reindexing operators indexed by the integers\n\\[\n(S_mx)[n]=x[n-m]\n\\qquad(m,n\\in\\Z).\n\\]\nThus $S_0=I$, $S_{m+r}=S_mS_r$, and $S_1$ is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of \\cite{conway1990}.\nThe impulse $\\delta_0$ satisfies\n\\[\nS_m\\delta_0=\\delta_m\n\\]\nfor every $m\\in\\Z$. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in $\\ell^2(\\Z)$.","dexed by the integers\n\\[\n(S_mx)[n]=x[n-m]\n\\qquad(m,n\\in\\Z).\n\\]\nThus $S_0=I$, $S_{m+r}=S_mS_r$, and $S_1$ is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of \\cite{conway1990}.\nThe impulse $\\delta_0$ satisfies\n\\[\nS_m\\delta_0=\\delta_m\n\\]\nfor every $m\\in\\Z$. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in $\\ell^2(\\Z)$.","\\begin{proposition}[Shift seed]\n\\label{prop:shift-seed}\nFor $n\\in\\N$, let $G(n)$ mean that some finite seed set with $n$ elements has dense hull under the declared family $\\{S_m:m\\in\\Z\\}$. Then\n\\[\n{\n1=\\min\\{n\\in\\N:G(n)\\}.\n}\n\\]\nEquivalently, one seed suffices and no empty seed family generates densely.\n\\end{proposition}","\\begin{proof}\nThe orbit of $\\delta_0$ contains every standard basis vector. Their finite linear span is dense, so $G(1)$ holds. The hull of the empty seed set is zero, while $\\ell^2(\\Z;\\mathbb F)$ is nonzero, so $G(0)$ fails. Therefore $1$ is the least admissible finite cardinality.\n\\end{proof}","For $N\\ge1$, define residue projectors\n\\[\n(P_{N,j}x)[n]\n=\n\\begin{cases}\nx[n],&n\\equiv j\\pmod N,","For $N\\ge1$, define residue projectors\n\\[\n(P_{N,j}x)[n]\n=\n\\begin{cases}\nx[n],&n\\equiv j\\pmod N,\\\\0,&\\text{otherwise.}\\end{cases}\n\\]\nThey form an orthogonal packet system and satisfy\n\\[\nS_1P_{N,j}=P_{N,j+1}S_1,\n\\]\nwith indices modulo $N$. The shift is therefore a cyclic transfer between residue packets.","0,&\\text{otherwise.}\\end{cases}\n\\]\nThey form an orthogonal packet system and satisfy\n\\[\nS_1P_{N,j}=P_{N,j+1}S_1,\n\\]\nwith indices modulo $N$. The shift is therefore a cyclic transfer between residue packets.","Let $X=\\bbk^n$ and let $\\cA=M_n(\\bbk)=\\End_{\\bbk}(X)$, with the full matrix algebra declared before a seed or target is chosen. By \\cref{cor:full-matrix}, every nonzero seed is cyclic. This is the maximal linear grammar on $X$. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.","For one operator $T$, \\cref{thm:single-operator} turns verified primary data\ninto an exact certificate for the minimum number of seeds.  The numerical companion compares:\n\\begin{enumerate}[label=(\\alph*)]\n\\item a nilpotent companion operator with one Jordan chain, for which the Krylov rank is $1,2,\\ldots,n$.\n\\item two equal Jordan blocks, for which one diagonal seed produces a chain only as long as one block and then stalls.\n\\end{enumerate}","Let $G$ be a nonempty finite graph with adjacency matrix $A_G$, and regard\n$\\bbk^{V(G)}$ as a module over $\\bbk[A_G]$.  The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.","\\begin{corollary}[Certified graph seed number]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","gin{corollary}[Certified graph seed number]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","aph seed number]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","r]\n\\label{cor:graph-rank}\nLet $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","et $\\bbk$ be a field and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","ld and let $\\Lambda$ be a nonempty finite set.  Suppose\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}","se\nthere are pairwise distinct scalars $a_{\\lambda}\\in\\bbk$, multiplicities\n$m_{\\lambda}\\in\\N$, and an eigenbasis\n\\[\n\\{b_{\\lambda,j}:\\lambda\\in\\Lambda,\\ 1\\le j\\le m_{\\lambda}\\}\n\\]\nof $\\bbk^{V(G)}$ satisfying\n\\[\nA_Gb_{\\lambda,j}=a_{\\lambda}b_{\\lambda,j}.\n\\]\nIf $m_{\\lambda}\\le R$ for all $\\lambda$ and\n$m_{\\lambda_*}=R$ for some $\\lambda_*$, then $R$ is the least natural number\nof adjacency seeds:\n\\[\n{\nR=\\min\\left\\{r\\in\\N:\n\\begin{array}{l}\n\\text{there exist }s_1,\\ldots,s_r\\text{ with}\\\\[2pt]\n\\bbk^{V(G)}=\\Span_{\\bbk}\n  \\{p(A_G)s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\end{array}\n\\right\\}.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nThe distinct eigenvalue labels give Lagrange polynomials that project onto\nthe individual eigenspace packets.  Padding one basis vector from every\npacket into each of $R$ seeds gives the upper bound.  Projection to the\n$m_{\\lambda}$ eigenspace coordinates is a surjective scalar quotient, so any\ngenerating family has at least $m_{\\lambda}$ members for every $\\lambda$.\nThe label $\\lambda_*$ gives the matching lower bound $R$.\n\\end{proof}","labels give Lagrange polynomials that project onto\nthe individual eigenspace packets.  Padding one basis vector from every\npacket into each of $R$ seeds gives the upper bound.  Projection to the\n$m_{\\lambda}$ eigenspace coordinates is a surjective scalar quotient, so any\ngenerating family has at least $m_{\\lambda}$ members for every $\\lambda$.\nThe label $\\lambda_*$ gives the matching lower bound $R$.\n\\end{proof}","\\begin{boundary}[Input from spectral classification]\nFor a finite undirected graph over $\\C$, the spectral theorem for real symmetric matrices\nclassically supplies a certificate of the displayed kind, and the numbers\n$m_{\\lambda}$ are the usual eigenvalue multiplicities. See\n\\cite{brouwerHaemers2012}.  The formal declaration verifies the\nformula for the number of seeds from the supplied eigenbasis and distinct eigenvalue\nlabels.  It does not derive diagonalizability from symmetry or implement an\neigensolver.\n\\end{boundary}","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex t","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for e","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path s","For every field $\\bbk$ and every $n\\ge1$, an endpoint of $P_n$ is a cyclic\nvector for the adjacency operator.  Consequently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","quently one is the least finite\nnumber of adjacency seeds:\n\\[\n{\n1=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","\\ldots,s_r,\\quad\n\\bbk^n=\\Span_{\\bbk}\\{p(A_{P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","P_n})s_j:p\\in\\bbk[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","\\right\\}.\n}\n\\]\nOver $\\C$, the familiar spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","spectral description has the distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","distinct eigenvalues\n\\[\n\\lambda_k=2\\cos\\frac{k\\pi}{n+1},\n\\qquad 1\\le k\\le n,\n\\]\nand the normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","normalized eigenvectors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","ctors have endpoint coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","point coordinate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","inate proportional to\n\\[\n\\sin\\frac{k\\pi}{n+1}\\ne0.\n\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","\\]\nThere is also a combinatorial proof valid over every field.  Let\n\\[\nK_n=[e_1,Ae_1,\\ldots,A^{n-1}e_1].\n\\]\nIts $(i,j)$ entry counts walks of length $j-1$ from the first vertex to vertex $i$. This entry is zero when $i>j$, while the diagonal entry is one because there is a unique shortest walk of length $i-1$ from vertex $1$ to vertex $i$. Hence $K_n$ is upper triangular with unit diagonal, so\n\\[\n\\det K_n=1\n\\qquad(n\\ge1).\n\\]\nThus the endpoint is cyclic for every path size, not merely for the tested range.","Two adjacent vertex vectors generate the adjacency module of every simple\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\","Two adjacent vertex vectors generate the adjacency module of every simple\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for","Two adjacent vertex vectors generate the adjacency module of every simple\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","le\ncycle.  Over $\\C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","C$, a Fourier quotient built from a primitive root of unity supplies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","plies a matching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","tching\nscalar obstruction of dimension two.  Hence, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","ce, for every $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","ery $n\\ge3$, two is the\nleast finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","least finite number of adjacency seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","y seeds:\n\\[\n{\n2=\\min\\left\\{r\\in\\N:\n\\exists s_1,\\ldots,s_r,\\quad\n\\C^n=\\Span_{\\C}\\{p(A_{C_n})s_j:p\\in\\C[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","[t],\\ 1\\le j\\le r\\}\n\\right\\}.\n}\n\\]\nTo see the upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","upper bound directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","d directly, index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.",", index the vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","vertices modulo $n$.\nThe identity $Ae_j=e_{j-1}+e_{j+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","+1}$ recovers successive vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","vertex vectors\nfrom $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","from $e_0$ and $e_1$, over any field. For the lower bound over $\\C$, let\n$\\zeta$ be a primitive $n$th root of unity and define\n\\[\nq(x)=\\left(\\sum_{j=0}^{n-1}\\zeta^j x_j,\n             \\sum_{j=0}^{n-1}\\zeta^{-j}x_j\\right).\n\\]\nThen $q(Ax)=(\\zeta+\\zeta^{-1})q(x)$. The images of $e_0$ and $e_1$\nare $(1,1)$ and $(\\zeta,\\zeta^{-1})$, which are independent because\n$\\zeta\\ne\\zeta^{-1}$ for $n\\ge3$. Thus $q$ is a scalar quotient\nof dimension two, and \\cref{prop:scalar-quotient} gives the\nmatching lower bound.","The usual spectral description writes the eigenvalues as\n\\[\n\\lambda_k=2\\cos\\frac{2\\pi k}{n},\n\\qquad 0\\le k<n.\n\\]\nThe values for $k$ and $n-k$ coincide. Apart from the simple characters at $k=0$ and, for even $n$, $k=n/2$, the remaining Fourier characters occur in pairs. Therefore the maximum eigenspace multiplicity is two, although some eigenspaces have multiplicity one, and\nthe certified minimum above is two.","\\midrule\n\\endhead\nCoordinate ring & $1$ & $M_x,M_y$ & even/odd $y$ degree & quotient by $y^2-x^3+x$","Shift system & $\\delta_0$ & $(S_m)_{m\\in\\Z}$ & residue classes of $n$ & $\\ell^2$ closure","Full matrix module & $s\\ne0$ & $M_n(\\bbk)$ & chosen idempotents & finite dimensional","Single operator & $s_1,\\ldots,s_r$ & $\\bbk[T]$ & primary or spectral & finite dimensional","Graph adjacency & vertex seeds & $\\bbk[A_G]$ & eigenspaces & finite dimensional","Typed grammar & vertex seeds & path algebra $\\bbk Q/I$ & vertex idempotents & relation quotient","\\Needspace{16\\baselineskip}\nFor a finite ordered seed family $S=(s_1,\\ldots,s_r)$, let $\\varepsilon_S:\\cA^{\\oplus r}\\to X$ be the seed evaluation map of \\cref{def:seed-evaluation}.  Its kernel is the relation module.  The free module modulo that kernel presents the generated hull by \\cref{thm:free-presentation}.  The invariant content is:\n\\[\n{\n\\begin{aligned}\n&\\text{generated hull: }\\Hull_{\\cA}(S),","\\Needspace{16\\baselineskip}\nFor a finite ordered seed family $S=(s_1,\\ldots,s_r)$, let $\\varepsilon_S:\\cA^{\\oplus r}\\to X$ be the seed evaluation map of \\cref{def:seed-evaluation}.  Its kernel is the relation module.  The free module modulo that kernel presents the generated hull by \\cref{thm:free-presentation}.  The invariant content is:\n\\[\n{\n\\begin{aligned}\n&\\text{generated hull: }\\Hull_{\\cA}(S),\\\\\n&\\text{seed complexity: }\\srank_{\\cA}(X),\\\\\n&\\text{packet coordinates: }x\\leftrightarrow(P_{\\alpha}x)_{\\alpha},\\\\\n&\\text{operator coordinates: }T\\leftrightarrow(P_{\\beta}TP_{\\alpha})_{\\beta,\\alpha},\\\\\n&\\text{relation module: }N_S:=\\Ker\\varepsilon_S\\leq\\cA^{\\oplus r},\\\\\n&\\text{free presentation: }\\cA^{\\oplus r}/N_S\\cong\\Hull_{\\cA}(S),\\\\\n&\\text{coordinate invariance: }P_{\\alpha}\\mapsto UP_{\\alpha}U^{-1},\\\\\n&\\text{realization: }\\Phi A=\\varphi(A)\\Phi.\n\\end{aligned}\n}\n\\]","\\midrule\nProjector idempotence residual & $0$","Projector annihilation residual & $0$","Projector completeness residual & $0$","Operator reconstruction residual & $0$","Residual for transported projectors, maximum & $1.85\\times10^{-15}$","Companion seed rank & $1$","Two equal Jordan blocks, seed rank & $2$","$M_3(\\C)$ acting on seven copies, seed rank & $3$","Sampled path graph seed ranks, $2\\le n\\le12$ & $1$","Sampled cycle graph seed ranks, $3\\le n\\le12$ & $2$","Full $M_4(\\R)$ orbit dimensions for two sampled nonzero seeds & $4,4$","Automated tests & $33/33$ passed","The finite shift pattern recovered by the experiment is\n\\[\n\\begin{pmatrix}\n0&0&0&1\\\\\n1&0&0&0\\\\\n0&1&0&0\\\\\n0&0&1&0\n\\end{pmatrix},\n\\]\nwhich is the exact packet adjacency of a cyclic transfer among four phases.","\\begin{construction}[Closure specification for the orbit hull]\n\\label{con:orbit-hull-algorithm}\nLet $V$ have finite dimension, let $(O_i)_{i\\in I}$ be a finite operator\nfamily, and let $S\\subseteq V$.  Define\n\\[\nW_0=\\Span_{\\bbk}S,\n\\qquad\nW_{n+1}=W_n+\\sum_{i\\in I}O_i(W_n).\n\\]\nThen there is an $n\\in\\N$ such that\n\\[\nW_{n+1}=W_n\n\\qquad\\text{and}\\qquad\nW_n=\\Hull_{(O_i)}(S),\n\\]\nwhere the right side is the least subspace containing $S$ and invariant under\nevery $O_i$.  Thus the ascending subspace iteration terminates and its stable\nvalue is exactly the finite operator hull.","pecification for the orbit hull]\n\\label{con:orbit-hull-algorithm}\nLet $V$ have finite dimension, let $(O_i)_{i\\in I}$ be a finite operator\nfamily, and let $S\\subseteq V$.  Define\n\\[\nW_0=\\Span_{\\bbk}S,\n\\qquad\nW_{n+1}=W_n+\\sum_{i\\in I}O_i(W_n).\n\\]\nThen there is an $n\\in\\N$ such that\n\\[\nW_{n+1}=W_n\n\\qquad\\text{and}\\qquad\nW_n=\\Hull_{(O_i)}(S),\n\\]\nwhere the right side is the least subspace containing $S$ and invariant under\nevery $O_i$.  Thus the ascending subspace iteration terminates and its stable\nvalue is exactly the finite operator hull.","l]\n\\label{con:orbit-hull-algorithm}\nLet $V$ have finite dimension, let $(O_i)_{i\\in I}$ be a finite operator\nfamily, and let $S\\subseteq V$.  Define\n\\[\nW_0=\\Span_{\\bbk}S,\n\\qquad\nW_{n+1}=W_n+\\sum_{i\\in I}O_i(W_n).\n\\]\nThen there is an $n\\in\\N$ such that\n\\[\nW_{n+1}=W_n\n\\qquad\\text{and}\\qquad\nW_n=\\Hull_{(O_i)}(S),\n\\]\nwhere the right side is the least subspace containing $S$ and invariant under\nevery $O_i$.  Thus the ascending subspace iteration terminates and its stable\nvalue is exactly the finite operator hull.","\\begin{construction}[Table specification for transfer blocks]\n\\label{con:transfer-block-algorithm}\nGiven finite projector systems $(P_i)_{i\\in I}$ and $(Q_j)_{j\\in J}$ and a\nlinear map $T$, form the table\n\\[\nB_{ji}=Q_jTP_i.\n\\]\nIts assembly reconstructs the input map exactly,\n\\[\n\\sum_{j\\in J}\\sum_{i\\in I}B_{ji}=T,\n\\]\nand its reported support relation is exactly\n$\\{(i,j):B_{ji}\\ne0\\}$.  The latter equality is the specification of the\ntransfer graph output. It does not claim that Lean extracts a routine that decides the edge list for every coefficient field.\n\\end{construction}","\\begin{construction}[Exact graph seed rank from a certificate]\n\\label{con:exact-graph-seed-rank}\nFor a finite square matrix $A$ over a field $K$, define the noncomputable exact\nspectral quantity\n\\[\nr_{\\mathrm{spec}}(A)\n=\n\\max_{\\mu\\in\\Spec_K(A)}\n\\dim_K\\ker(A-\\mu I),\n\\]\nwith value zero for an empty spectrum.  Suppose an exact certificate supplies\na finite eigenbasis indexed by labels and multiplicities, the eigenvalue\nequations, a multiplicity bound attained by one label, projector polynomials\nthat extract the padded eigenbasis generators, coverage of every spectral\nvalue by a label, equality of the declared and actual eigenspace dimensions,\nand membership of the maximizing eigenvalue in the spectrum.  Then\n$r_{\\mathrm{spec}}(A)$ is the least natural number of seeds under polynomial orbits\nfor $A$.","an exact certificate supplies\na finite eigenbasis indexed by labels and multiplicities, the eigenvalue\nequations, a multiplicity bound attained by one label, projector polynomials\nthat extract the padded eigenbasis generators, coverage of every spectral\nvalue by a label, equality of the declared and actual eigenspace dimensions,\nand membership of the maximizing eigenvalue in the spectrum.  Then\n$r_{\\mathrm{spec}}(A)$ is the least natural number of seeds under polynomial orbits\nfor $A$.","the eigenvalue\nequations, a multiplicity bound attained by one label, projector polynomials\nthat extract the padded eigenbasis generators, coverage of every spectral\nvalue by a label, equality of the declared and actual eigenspace dimensions,\nand membership of the maximizing eigenvalue in the spectrum.  Then\n$r_{\\mathrm{spec}}(A)$ is the least natural number of seeds under polynomial orbits\nfor $A$.","\\subsection{Theorem spine at a glance}\n\\ConceptChain{\n\\(\n\\begin{array}{c}\n\\text{free seed presentation}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{minimal generator number}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{reduction modulo the Jacobson radical}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{semisimple or primary computation}\n\\end{array}\n\\qquad\\qquad\n\\begin{array}{c}\n\\text{packet idempotents}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{exact analysis/synthesis}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{transfer block algebra}\\\\[0.25em]\n\\Downarrow\\\\[-0.1em]\n\\text{graphs, quotients, and realization}\n\\end{array}\n\\)\n}","\\item[Block support.] The set of packet pairs \\((\\beta,\\alpha)\\) for which \\(P_\\beta TP_\\alpha\\neq0\\).\n\\item[Collapsed realization.] The direct sum of all typed vertex spaces.\n\\item[Condensation graph.] The acyclic graph obtained by collapsing strongly connected components.\n\\item[Generated hull.] The submodule generated by a seed family, or its declared closure.\n\\item[Grammar.] A typed quiver and relation ideal describing legal operator words.\n\\item[Packet.] The range of one idempotent in a complete annihilating family.\n\\item[Packet pure.] Supported in at most one packet.\n\\item[Preservation profile.] The operations a realization functor is required to preserve.\n\\item[Realized algebra.] The image of the grammar algebra in the endomorphism algebra of a r","\\item[Block support.] The set of packet pairs \\((\\beta,\\alpha)\\) for which \\(P_\\beta TP_\\alpha\\neq0\\).\n\\item[Collapsed realization.] The direct sum of all typed vertex spaces.\n\\item[Condensation graph.] The acyclic graph obtained by collapsing strongly connected components.\n\\item[Generated hull.] The submodule generated by a seed family, or its declared closure.\n\\item[Grammar.] A typed quiver and relation ideal describing legal operator words.\n\\item[Packet.] The range of one idempotent in a complete annihilating family.\n\\item[Packet pure.] Supported in at most one packet.\n\\item[Preservation profile.] The operations a realization functor is required to preserve.\n\\item[Realized algebra.] The image of the grammar algebra in the endomorphism algebra of a realization.\n\\item[Seed rank.] The sma","trary division rings, with the conventions for left and right modules kept explicit in \\cref{thm:semisimple-seed-rank}.\n\\item The single operator formula and its cyclic criteria apply over arbitrary fields in \\cref{thm:single-operator,cor:cyclic-matrix}.\n\\item The countable unconditional packet theorem is proved in \\cref{thm:countable-packets}.\n\\item The classification of support compatible with $*$ is proved in \\cref{cor:star-support}.\n\\item The weighted source component formula gives the packet pure generation requirement under the propagation hypotheses in \\cref{thm:source-component-rank}.\n\\item Seed system composition, packet exact sequences, quotient block descent, dense realization, quotient completion, and semantic theorem transport are proved in \\cref{prop:seed-category,thm:packet-exact-sequence,cor:block-descent,thm:dense-realization,thm:quotient-topology,thm:realization-export}.\n\\item Exact sequence bounds, monotonicity under enlargement of the acting algebra, coordinate stability, projection onto diagonal blocks, and preorder decomposition are established in \\cref{thm:exac…","\\nolinkurl{PSC-I-C-SUPPORT-PATTERN-STAR} & \\ref{cor:star-support} & Bounded support patterns closed under $*$","Use the same space of dimension six throughout the first comparison:\n\\[\nX=\\QQ^6,\\qquad N=J_3(0)\\oplus J_3(0),\\qquad\ns=e_0+f_0.\n\\]\nHere each $J_3(0)$ moves a basis vector forward along a chain of three basis vectors. The projectors $P_e,P_f$ select the two chains. Their formulas and the proofs of the minimum seed counts appear on \\pagerefV{3} and \\pagerefV{4}.","[4pt]\n\\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\\\[4pt]\n\\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?\n\\end{tabularx}\n\nA packet projector can be available as a way to \\emph{inspect} a vector without belonging to the \\emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.\n\n\\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}\n\n\\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \\cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}\nMost of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.\n\n{\\small\n\\begin{tabularx}{\\linewidth}{@{}>{\\raggedright\\arraybackslash}p{.15\\linewidth}Y>{\\raggedright…","\\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \\cref{thm:single-operator}. Matrices over $\\QQ$}\nUse the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define\n\\[\nNe_0=e_1,\\quad Ne_1=e_2,\\quad Ne_2=0,\n\\qquad Nf_0=f_1,\\quad Nf_1=f_2,\\quad Nf_2=0.\n\\]\nStarting with $s=e_0+f_0$, the two chains advance together:\n\\[\ns=e_0+f_0,\\qquad Ns=e_1+f_1,\\qquad N^2s=e_2+f_2,\\qquad N^3s=0.\n\\]","\\vfig[.94]{linked_chains}\n\\atlasCaption{The vertical dotted pairings record equal coefficients, not an additional allowed operator. Every polynomial in $N$ uses the same coefficients on both chains.}","Since $N^3=0$, every polynomial orbit vector has the form\n\\[\np(N)s=a(e_0+f_0)+b(e_1+f_1)+c(e_2+f_2).\n\\]\nThe generated hull therefore has dimension three, even though the seed has a nonzero component in each packet of dimension three. The Krylov ranks are $1,2,3,3,3,3$.","\\head{What changes when a projector becomes admissible?}\nLet $P_e$ keep the coordinates along the $e$ chain and $P_f$ those along the $f$ chain. If these are merely measurement maps, the preceding hull is unchanged. If they become allowed generators, then\n\\[\nP_es=e_0,\\qquad P_fs=f_0,\n\\]\nand powers of $N$ now recover all six basis vectors. The same seed becomes cyclic for the larger algebra $\\QQ\\langle N,P_e,P_f\\rangle$.","Define a surjective linear map\n\\[\nq:X\\longrightarrow\\QQ^2,\\qquad\nq\\Bigl(\\sum a_je_j+\\sum b_jf_j\\Bigr)=(a_0,b_0).\n\\]","\\vfig[.78]{quotient_heads}\n\\atlasCaption{An actual $2\\times6$ quotient matrix. It forgets the advanced positions in both chains and retains two independent head coordinates.}","Because $qN=0$, one has, for every polynomial $p$,\n\\[\nq\\bigl(p(N)x\\bigr)=p(0)q(x).\n\\]\nThus every operator acts as a \\emph{scalar} on the quotient. The image of the orbit of one seed stays on one line in $\\QQ^2$. It cannot span this quotient of dimension two.","\\head{Lower and upper bounds meet}\nIf $r$ seeds generated $X$, their images would generate the quotient. Since scalar multiplication cannot create a new independent direction, $r\\ge2$. Conversely, the two seeds $e_0,f_0$ generate their respective chains, so\n\\[\n\\boxed{\\srank_{\\QQ[N]}(X)=2.}\n\\]\nThis proof uses the displayed operator and quotient. It does not ask a computer to search every possible seed, and it does not need an existence theorem for Jordan form.","\\head{A general pattern worth recognizing}\nA surjective intertwiner $q:X\\twoheadrightarrow Y$ turns any generating family of $X$ into a generating family of $Y$. Therefore a quotient on which the grammar loses mixing power can certify a lower bound in the original, much larger space.","\\vpage{5}{Relations identify words, not just outputs}{PRESENTATIONS / WHY A LEFT IDEAL APPEARS}{Main text: \\cref{thm:free-presentation,cor:one-seed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","uts}{PRESENTATIONS / WHY A LEFT IDEAL APPEARS}{Main text: \\cref{thm:free-presentation,cor:one-seed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","IDEAL APPEARS}{Main text: \\cref{thm:free-presentation,cor:one-seed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","eed-presentation,prop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","rop:two-sided-annihilator}}\nLet $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","et $F=\\kk\\langle z_1,\\ldots,z_r\\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is\n\\[\n\\Phi_s:F\\to X,\\qquad f\\mapsto f(T_1,\\ldots,T_r)s.\n\\]\nTwo words represent the same generated vector exactly when their difference lies in $I_s=\\Ker\\Phi_s$. Consequently\n\\[\nF/I_s\\cong\\Hull_F(s)\n\\]\nas \\emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.","\\head{The distinction is visible in two dimensions}\nUse $X=\\QQ^2$, seed $e_1=(1,0)^\\mathsf T$, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column:\n\\[\nI_{e_1}=\\left\\{\\mat{0&a","\\head{The distinction is visible in two dimensions}\nUse $X=\\QQ^2$, seed $e_1=(1,0)^\\mathsf T$, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column:\n\\[\nI_{e_1}=\\left\\{\\mat{0&a\\\\0&b}:a,b\\in\\QQ\\right\\}.\n\\]","\\vfig[.90]{left_ideal}\n\\atlasCaption{The relation $Ce_1=0$ is destroyed by multiplying $C$ on the right by $B$. The product $CB$ sends the seed to itself.}","For arbitrary $D$, $(DC)e_1=D(Ce_1)=0$. But $(CB)e_1=e_1\\ne0$. Thus $I_{e_1}$ is a left ideal, but it is not stable under multiplication on the right. The quotient retains the action on generated vectors. Calling it a ring would lose the distinction that makes the presentation correct.","\\head{Several seeds give a module of relations}\nFor $(s_1,\\ldots,s_m)$, evaluation is $(f_1,\\ldots,f_m)\\mapsto\\sum_j f_j(T)s_j$. Its kernel is a submodule of $F^m$ and can include relations \\emph{between} seeds, not only relations annihilating each seed separately.","\\vpage{6}{A reached type can still hide a direction}{TYPED MODEL / $U=\\QQ^2$ AND $V=\\QQ^3$}{Main text: \\cref{prop:typed-reachability,def:directed-seed-rank}}\nThe paper's typed model has a rotation $R:U\\to U$, an inclusion $A:U\\to V$, and a retraction $B:V\\to U$:\n\\[\nR=\\mat{0&-1","\\vpage{6}{A reached type can still hide a direction}{TYPED MODEL / $U=\\QQ^2$ AND $V=\\QQ^3$}{Main text: \\cref{prop:typed-reachability,def:directed-seed-rank}}\nThe paper's typed model has a rotation $R:U\\to U$, an inclusion $A:U\\to V$, and a retraction $B:V\\to U$:\n\\[\nR=\\mat{0&-1\\\\1&0},\\qquad\nA=\\mat{1&0\\\\0&1\\\\0&0},\\qquad\nB=\\mat{1&0&0\\\\0&1&0}.\n\\]\nThe typed relation is $BA=I_U$. In the opposite order,\n\\[\nAB=\\diag(1,1,0)\\ne I_V.\n\\]\nThe missing direction is $\\Ker B=\\QQ v_3$, not an entirely missing vertex.","0&1&0}.\n\\]\nThe typed relation is $BA=I_U$. In the opposite order,\n\\[\nAB=\\diag(1,1,0)\\ne I_V.\n\\]\nThe missing direction is $\\Ker B=\\QQ v_3$, not an entirely missing vertex.","\\head{One seed supported at $U$}\nThe vectors $u_1,Ru_1$ span $U$. Their images under $A$ span $\\QQ v_1+\\QQ v_2$. Every path from $U$ to $V$ ends with $A$, so it cannot supply $v_3$. The hull has dimension four. Adding the pure seed $v_3$ gives all five dimensions.","\\head{Why this does not prove an unrestricted minimum of two}\nThe bound path algebra includes the vertex idempotents $P_U,P_V$. A mixed seed\n\\[\ns=u_1+v_3\\in U\\oplus V\n\\]\nsatisfies $P_Us=u_1$ and $P_Vs=v_3$. It therefore generates the entire representation by itself. The paper proves that no seed \\emph{supported at $U$} is enough and that two displayed pure seeds suffice. It does not claim that two arbitrary mixed seeds are necessary.","\\vpage{7}{Exact does not mean orthogonal}{PACKET COORDINATES / AN OBLIQUE SPLITTING}{Main text: \\cref{thm:exact-packet-coordinates,cor:adjoint-reversal}}\nIn $\\RR^2$, take\n\\[\nP=\\mat{1&-1","\\vpage{7}{Exact does not mean orthogonal}{PACKET COORDINATES / AN OBLIQUE SPLITTING}{Main text: \\cref{thm:exact-packet-coordinates,cor:adjoint-reversal}}\nIn $\\RR^2$, take\n\\[\nP=\\mat{1&-1\\\\0&0},\\qquad Q=\\mat{0&1\\\\0&1}.\n\\]\nThese maps satisfy $P^2=P$, $Q^2=Q$, $PQ=QP=0$ and $P+Q=I$. Their images are the horizontal line and the diagonal line, respectively. They are not selfadjoint in the usual Euclidean inner product.","0&1}.\n\\]\nThese maps satisfy $P^2=P$, $Q^2=Q$, $PQ=QP=0$ and $P+Q=I$. Their images are the horizontal line and the diagonal line, respectively. They are not selfadjoint in the usual Euclidean inner product.","\\vfig[.68]{oblique_vectors}\n\\atlasCaption{Actual Euclidean vector geometry for $x=(3,2)$. The parallelogram reconstructs $x$ from $Px=(1,0)$ and $Qx=(2,2)$. Neither component is an orthogonal projection of $x$.}","Analysis records $(Px,Qx)$ and synthesis adds the two vectors. For a packet tuple $(u,v)\\in\\im P\\times\\im Q$,\n\\[\nP(u+v)=u,\\qquad Q(u+v)=v.\n\\]\nBoth inverse identities follow from annihilation and completeness, without any angle condition.","\\head{What the algebra does not preserve}\nFor the displayed $x$,\n\\[\n\\norm{x}^2=13,\\qquad \\norm{Px}^2+\\norm{Qx}^2=1+8=9.\n\\]\nThere is no Parseval identity here. The cross term is nonzero because the packet directions are not perpendicular. Likewise, the simple formula for reversing adjoints of packet blocks requires selfadjoint projectors. It is not justified for this oblique pair.","\\vpage{8}{A transfer block has an input and an output}{OPERATOR COORDINATES / MULTIPLICATION AS A SUM}{Main text: \\cref{thm:operator-reconstruction,thm:block-multiplication,cor:block-powers}}\nUse the three coordinate packets of dimension one in $\\QQ^3$. The block $T_{ij}=P_iTP_j$ sends input packet $j$ into output packet $i$. For this example, its nonzero scalar is the $(i,j)$ matrix entry.","\\head{Reconstruction retains every block}\nThe identity $P_1+P_2+P_3=I$ gives\n\\[\nT=\\sum_{i,j=1}^3 P_iTP_j.\n\\]\nSelecting only the diagonal would change the operator. The entries $2,3,4$ carry transfers between different packets.","\\head{Composition sums over intermediate packets}\nThe $(1,2)$ entry of the square has two contributing routes:\n\\[\n(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}\n=1\\cdot2+2\\cdot1+0=4.\n\\]\nFor input $3$ and output $1$, only the route through packet $2$ survives:\n\\[\n(T^2)_{13}=T_{12}T_{23}=2\\cdot3=6.\n\\]\nFor packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.","\\vpage{9}{Two visible routes can cancel exactly}{REACHABILITY / NECESSARY IS NOT SUFFICIENT}{Main text: \\cref{cor:block-powers,prop:reachability-obstruction}}\nLet a single operator $D$ on $\\QQ^4$ act by\n\\[\nDe_1=e_2+e_3,\\qquad De_2=e_4,\\qquad De_3=-e_4,\\qquad De_4=0.\n\\]\nEvery vertex of its transfer graph for the coordinate packets is reachable from vertex $1$. Nevertheless,\n\\[\nD^2e_1=e_4-e_4=0,\\qquad D^2=0.\n\\]","\\vfig[.84]{cancelled_paths}\n\\atlasCaption{The diamond is the actual weighted action of $D$. The two paths of length two exist, but their contributions to the same transfer block sum to zero.}","\\head{What the seed generates}\nWith the fixed algebra $\\QQ[D]$ and seed $e_1$,\n\\[\n\\Hull_{\\QQ[D]}(e_1)=\\Span\\{e_1,e_2+e_3\\}.\n\\]\nIt has dimension two, not four. Both the independent difference $e_2-e_3$ and the last coordinate $e_4$ are missing.","\\head{Why projecting in the middle changes the experiment}\nIf $P_2$ becomes an admissible primitive, the word $DP_2D$ satisfies\n\\[\nDP_2De_1=DP_2(e_2+e_3)=De_2=e_4.\n\\]\nBut $P_2$ was not an allowed generator of the original $\\QQ[D]$ grammar. Using it to argue that the original seed already reached $e_4$ would silently enlarge the algebra.","\\vpage{10}{Which zero patterns survive multiplication?}{SUPPORT ALGEBRAS / A UNIVERSAL TEST}{Main text: \\cref{thm:support-pattern}. All packet spaces are nonzero}\nSuppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let $E_{ij}$ denote the matrix that sends $e_j$ to $e_i$ and kills the other basis vectors.","\\vfig[.96]{support_closure}\n\\atlasCaption{Here $1$ means ``allowed,'' not that every admissible operator has that numerical value. The middle matrix is an actual product that violates the original pattern.}","The first pattern contains the diagonal and permits $1\\to2$ and $2\\to3$, but forbids $1\\to3$. Yet\n\\[\nE_{32}E_{21}=E_{31}\\ne0.\n\\]\nBoth factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","\\head{Reflexivity and transitivity have different jobs}\nThe identity has a nonzero diagonal block on every nonzero packet, so including the identity requires every packet to be related to itself. Multiplicative closure requires every composite pair. Consequently, for arbitrary algebraic block maps on nonzero packet spaces,\n\\[\n\\boxed{\\text{unital universal support algebra}\n\\quad\\Longleftrightarrow\\quad\n\\text{reflexive, transitive support relation}.}\n\\]\nThe transitive closure in the figure is the lower triangular pattern. It does define a unital algebra.","\\vpage{11}{Move the packets with the coordinates}{CONJUGACY / STRUCTURE VERSUS RAW SPARSITY}{Main text: \\cref{thm:coordinate-transport,cor:isomorphism-invariance,thm:naturality}}\nStart with $T=\\diag(1,2)$ and the two coordinate projections. Apply the invertible shear $U$. The transformed operator is $T^U=UTU^{-1}$, and the transformed projectors are $Q_i=UP_iU^{-1}$.","Indeed,\n\\[\nQ_1T^UQ_2=Q_2T^UQ_1=0,\n\\qquad\nQ_iT^UQ_j=U(P_iTP_j)U^{-1}.\n\\]\nA change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.","\\head{When the map is not invertible}\nFor an intertwiner $\\Phi A=\\varphi(A)\\Phi$, the hull identity uses the \\emph{image algebra}:\n\\[\n\\Phi\\bigl(\\Hull_{\\AA}(S)\\bigr)=\\Hull_{\\varphi(\\AA)}(\\Phi S).\n\\]\nReplacing $\\varphi(\\AA)$ with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not automatic equality.","\\vpage{12}{Shared control and independent control differ}{DIRECT SUMS / ONE SEED CAN CARRY TWO COMPONENTS}{Main text: \\cref{prop:shared-direct-sum,thm:independent-direct-sum}}\nTake $X=\\kk\\oplus\\kk$ and $s=(1,1)$. With one shared scalar acting on both coordinates, every orbit vector is $(a,a)$. With independent scalars, the orbit contains every $(a,b)$.","\\vfig[.94]{shared_independent}\n\\atlasCaption{The geometric picture uses $\\kk=\\RR$. On the right, shading shows a finite window of the whole plane. On the left, only the diagonal line is generated. The algebraic distinction holds over any field.}","The corresponding minimum seed numbers are\n\\[\n\\srank_{\\kk I}(\\kk^2)=2,\n\\qquad\n\\srank_{\\kk\\times\\kk}(\\kk\\oplus\\kk)=1.\n\\]\nIn the independent case, the algebra contains $(1,0)$ and $(0,1)$, which recover the two coordinates of $s$ separately.","\\head{This is why the independent formula uses a maximum}\nFor independent actions of $\\AA$ on $X$ and $\\BB$ on $Y$, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving\n\\[\n\\srank_{\\AA\\times\\BB}(X\\oplus Y)\n=\\max\\{\\srank_{\\AA}(X),\\srank_{\\BB}(Y)\\}.\n\\]\nWith a shared action, this selection need not be available. The general bound is the sum.","\\head{One operator can supply independent control}\nOver $\\QQ$, $T=\\diag(1,2)$ has polynomial projectors $2I-T$ and $T-I$. Thus $\\QQ[T]$ already separates the two eigenspaces. If the eigenvalues coincide, these formulas no longer separate them. Distinct primary labels play the same role on \\pagerefV{14}.","\\vpage{13}{How much row space can one seed supply?}{SPLIT SEMISIMPLE MODEL / A CONSTRUCTIVE FORMULA}{Main text: \\cref{thm:semisimple-seed-rank,cor:algebraically-closed-formula}}\nLet $M_2(\\QQ)$ act on $X=M_{2\\times5}(\\QQ)$ by left multiplication. Think of the five columns as five copies of the same natural module of dimension two. One seed is a $2\\times5$ matrix $V$.","Every row of $AV$ is a linear combination of the two rows of $V$. Hence one seed exposes at most two independent directions in the multiplicity space of dimension five. Two seeds expose at most four. At least three are necessary.","\\vfig[.98]{row_budget}\n\\atlasCaption{The three fixed seeds cover multiplicity directions $1,2$, then $3,4$, then $5$. The exact dimensions of their cumulative matrix orbits are $4,8,10$.}","\\head{An explicit synthesis, not just dimension counting}\nFor\n\\[\nW=\\mat{1&2&3&4&5\\\\6&7&8&9&10},\n\\]\nchoose $A_1$ from columns $1,2$ of $W$, $A_2$ from columns $3,4$, and $A_3$ with first column equal to column $5$ of $W$ and second column zero. Then\n\\[\nA_1V_1+A_2V_2+A_3V_3=W.\n\\]\nThe same construction works for arbitrary entries of $W$, so the lower bound is attained:\n\\[\n\\boxed{\\srank_{M_2(\\QQ)}M_{2\\times5}(\\QQ)=\\lceil5/2\\rceil=3.}\n\\]","6&7&8&9&10},\n\\]\nchoose $A_1$ from columns $1,2$ of $W$, $A_2$ from columns $3,4$, and $A_3$ with first column equal to column $5$ of $W$ and second column zero. Then\n\\[\nA_1V_1+A_2V_2+A_3V_3=W.\n\\]\nThe same construction works for arbitrary entries of $W$, so the lower bound is attained:\n\\[\n\\boxed{\\srank_{M_2(\\QQ)}M_{2\\times5}(\\QQ)=\\lceil5/2\\rceil=3.}\n\\]","\\head{Several split simple blocks}\nIf the acting algebra and module are supplied in compatible product coordinates, the row budget is computed separately in each block and the formula for independent actions takes the maximum. For block data $(d,m)=(2,5)$ and $(3,4)$, the result is $\\max(3,2)=3$, not $3+2$.","\\vpage{14}{Distinct labels let one polynomial separate chains}{PRIMARY DATA / AN ACTUAL EXTRACTION CERTIFICATE}{Main text: \\cref{thm:single-operator,cor:invariant-factor-count}}\nKeep the model in six dimensions, but compare two different operators:\n\\[\nN=J\\oplus J,\\qquad T=J\\oplus(2I+J),\\qquad J=J_3(0).\n\\]\nFor the same mixed seed $s=e_0+f_0$, the Krylov matrices are different:","\\vfig[.89]{primary_krylov}\n\\atlasCaption{Exact rational calculations. Changing the primary label from $0$ to $2$ changes the operator, not merely its coordinates. The determinant on the right is $512\\ne0$.}","\\head{The missing separating operation is now a polynomial}\nDefine\n\\[\np_2(t)=\\frac{3t^5-15t^4+20t^3}{16},\\qquad p_0(t)=1-p_2(t).\n\\]\nThese polynomials satisfy\n\\[\np_2\\equiv0\\pmod{t^3},\\qquad p_2\\equiv1\\pmod{(t-2)^3}.\n\\]\nConsequently $p_0(T)=P_e$ and $p_2(T)=P_f$. Applying them to $s$ extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on \\pagerefV{3}, the separating maps already belong to the polynomial grammar.","\\vpage{15}{Walk counts make an endpoint cyclic}{GRAPH MODEL / THE PATH CERTIFICATE}{Main text: Worked graph model. Endpoint Krylov proof for every path size}\nLet $A$ be the adjacency matrix of the path with five vertices. The $(i,j)$ entry of\n\\[\nK=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1]\n\\]\ncounts walks of length $j-1$ from vertex $1$ to vertex $i$.","A walk cannot reach vertex $i$ in fewer than $i-1$ steps. Therefore entries below the diagonal vanish. There is exactly one walk of length $i-1$ from the first vertex to vertex $i$: the shortest path. The diagonal entries are all one, so $\\det K=1$.","\\head{The same reason works at every size}\nFor a path with any $n\\ge1$, the endpoint Krylov matrix is upper triangular with unit diagonal. This is a combinatorial proof of cyclicity over \\emph{every field}. It does not rely on numerical eigenvalues. The appendix checks the finite cases $1\\le n\\le12$ as implementation examples, separately from the paper's proof for every path size.","\\vpage{16}{Closing a path changes the seed obstruction}{GRAPH MODEL / A CYCLE WITH SIX VERTICES}{Main text: Worked graph model. Proof of the minimum seed count for complex cycles}\nFor the adjacency matrix $A$ of $C_6$, the distinct eigenvalues are $2,1,-1,-2$, with multiplicities $1,2,2,1$. A single vertex seed spans only four directions. Two adjacent vertex seeds span all six.","\\head{A lower bound without a numerical eigensolver}\nThe two rows of the following matrix are independent and obey $qA=q$:\n\\[\nq=\\mat{2&1&-1&-2&-1&1","\\head{A lower bound without a numerical eigensolver}\nThe two rows of the following matrix are independent and obey $qA=q$:\n\\[\nq=\\mat{2&1&-1&-2&-1&1\\\\0&1&1&0&-1&-1}.\n\\]\nThus $q:\\QQ^6\\twoheadrightarrow\\QQ^2$ is a scalar quotient for the eigenvalue $1$. Its extension over $\\CC$ gives the same obstruction of dimension two. The orbit of any single seed projects to a line, so no single seed can generate the full module.","0&1&1&0&-1&-1}.\n\\]\nThus $q:\\QQ^6\\twoheadrightarrow\\QQ^2$ is a scalar quotient for the eigenvalue $1$. Its extension over $\\CC$ gives the same obstruction of dimension two. The orbit of any single seed projects to a line, so no single seed can generate the full module.","\\head{The matching construction}\nFrom adjacent vertex vectors $e_0,e_1$, the identity\n\\[\ne_{j+1}=Ae_j-e_{j-1}\n\\]\nrecovers successive vertices around the cycle. The lower and upper bounds therefore give two seeds for this example. The paper proves the corresponding minimum for every complex simple cycle of size at least three.","\\vpage{17}{A relation becomes a packet product rule}{COORDINATE ALGEBRA / NORMAL FORM RATHER THAN A CURVE SKETCH}{Main text: Worked coordinate ring model. Normal form and exact multiplication}\nWork over $\\QQ$ in\n\\[\nA=\\QQ[x,y]/(y^2-x^3+x).\n\\]\nBecause the relation is monic of degree two in $y$, each element has one unique form $a(x)+yb(x)$. The two packet coefficients are polynomials, not scalars.","\\vfig[.97]{normal_form}\n\\atlasCaption{A finite window of the two infinite coefficient lanes. Higher powers of $y$ are rewritten using the relation. They are not simply discarded. No geometric genus or smoothness claim is encoded by this picture.}","\\head{Multiplication by $y$ transfers between the lanes}\nIn the ordered coefficient coordinates $(a,b)$,\n\\[\nM_y=\\mat{0&x^3-x","\\head{Multiplication by $y$ transfers between the lanes}\nIn the ordered coefficient coordinates $(a,b)$,\n\\[\nM_y=\\mat{0&x^3-x\\\\1&0},\\qquad M_y^2=(x^3-x)I.\n\\]\nThe entries denote multiplication maps on $\\QQ[x]$. For $f=a_0+ya_1$ and $g=b_0+yb_1$,\n\\[\nfg=\\bigl(a_0b_0+(x^3-x)a_1b_1\\bigr)\n+y\\bigl(a_0b_1+a_1b_0\\bigr).\n\\]\nIn particular,\n\\[\n(x^2+y)(x-y)=x+(x-x^2)y.\n\\]","1&0},\\qquad M_y^2=(x^3-x)I.\n\\]\nThe entries denote multiplication maps on $\\QQ[x]$. For $f=a_0+ya_1$ and $g=b_0+yb_1$,\n\\[\nfg=\\bigl(a_0b_0+(x^3-x)a_1b_1\\bigr)\n+y\\bigl(a_0b_1+a_1b_0\\bigr).\n\\]\nIn particular,\n\\[\n(x^2+y)(x-y)=x+(x-x^2)y.\n\\]","\\head{One module seed does not mean one algebra generator}\nStarting from $1$, the fixed multiplication operators $M_x,M_y$ generate every monomial in normal form. Thus the algebra is cyclic as a module over its multiplication algebra. This does not remove either address operator, and does not say that the algebra is generated by one algebra element.","\\takeaway{The relation controls an exact weighted transfer between coefficient lanes. Packet reconstruction is $a+yb$, while packet multiplication retains the weight $x^3-x$.}","\\vpage{18}{Density is not finite generation of each vector}{INFINITE MODEL / THE BILATERAL SHIFT}{Main text: \\cref{prop:shift-seed,thm:completed-presentation}}\nOn $\\ell^2(\\ZZ)$, declare every integer shift\n\\[\n(S_mx)[n]=x[n-m].\n\\]\nThe seed $\\delta_0$ generates $\\delta_m=S_m\\delta_0$. Finite linear combinations give precisely the sequences with finite support. Their norm closure is all of $\\ell^2(\\ZZ)$.","\\vfig[.93]{shift_tail}\n\\atlasCaption{A finite window of the actual sequence $x[n]=2^{-|n|}$. Outside every finite window, coefficients remain nonzero. The shaded window illustrates the truncation $x^{(3)}$, not a periodic wraparound.}","This $x$ cannot be a finite linear combination of shifted impulses: it has infinite support. But if $x^{(N)}$ agrees with $x$ on $|n|\\le N$ and is zero elsewhere, then\n\\[\n\\norm{x-x^{(N)}}_2^2\n=2\\sum_{n>N}4^{-n}\n=\\frac{2}{3}\\,4^{-N}\\longrightarrow0.\n\\]\nThe exact squared errors for $N=1,3,5$ are $1/6$, $1/96$, and $1/1536$. These are identities from a geometric series, not floating residuals.","\\vpage{19}{Residue packets retain an address carry}{INFINITE MODEL / THREE EXACT SUBSEQUENCES}{Main text: \\cref{thm:banach-packets}. Residue packets}\nFor $j=0,1,2$, let $P_j$ keep the entries with $n\\equiv j\\pmod3$. These are bounded orthogonal projectors whose sum is the identity. The packet coordinates can be written as\n\\[\nx_j[k]=x[3k+j],\\qquad k\\in\\ZZ.\n\\]","\\head{What one shift does in these coordinates}\nLet $S=S_1$ and $z=Sx$. Substituting $3k+j$ into the shift definition gives\n\\[\nz_1[k]=x_0[k],\\qquad z_2[k]=x_1[k],\\qquad z_0[k]=x_2[k-1].\n\\]\nThe passage from phase $2$ to phase $0$ carries a shift of the coarse address. If $Lh[k]=h[k-1]$, the exact packet matrix is\n\\[\nS_{\\mathrm{packets}}=\\mat{0&0&L","\\head{What one shift does in these coordinates}\nLet $S=S_1$ and $z=Sx$. Substituting $3k+j$ into the shift definition gives\n\\[\nz_1[k]=x_0[k],\\qquad z_2[k]=x_1[k],\\qquad z_0[k]=x_2[k-1].\n\\]\nThe passage from phase $2$ to phase $0$ carries a shift of the coarse address. If $Lh[k]=h[k-1]$, the exact packet matrix is\n\\[\nS_{\\mathrm{packets}}=\\mat{0&0&L\\\\I&0&0\\\\0&I&0}.\n\\]\nReplacing $L$ by a scalar $1$ would preserve the phase graph but would lose the operator's action on the infinite sequences.","0&I&0}.\n\\]\nReplacing $L$ by a scalar $1$ would preserve the phase graph but would lose the operator's action on the infinite sequences.","\\head{Synthesis keeps the positions}\nTo reconstruct $x$, place $x_j[k]$ at its original address $3k+j$. In the ambient space formulation, this is the finite sum $x=P_0x+P_1x+P_2x$. Boundedness makes the same formulas valid after taking the $\\ell^2$ completion.","\\vpage{20}{Stop when the subspace stops growing}{EXACT COMPUTATION / AN INVARIANCE CERTIFICATE}{Main text: \\cref{con:orbit-hull-algorithm}}\nFor a fixed finite operator family $(O_i)$, start from $W_0=\\Span S$ and iterate\n\\[\nW_{n+1}=W_n+\\sum_i O_iW_n.\n\\]\nThis is an ascending chain. When two consecutive subspaces agree, every generator maps the stable subspace into itself. The stable value is therefore the entire hull of operator words, not just the span of the words tested so far.","\\vfig[.58]{hull_iteration}\n\\atlasCaption{Exact dimensions for the same mixed seed in Model A. The dashed curve allows only $N$. The solid curve also admits the two chain projectors. Plateau values are repeated to show the certified stop.}","\\head{Why the process stops in finite dimensions}\nIf an inclusion is strict, the dimension rises by at least one. It cannot rise beyond $\\dim X$. At equality, invariance and containment of the seeds prove both inclusions between the stable space and the least invariant hull.","\\vpage{21}{Tensor products require every cross term}{OPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONAL}{Main text: \\cref{prop:tensor-bound}. Algebraic tensor products}\nIf $(x_i)$ generates $X$ under $\\AA$ and $(y_j)$ generates $Y$ under $\\BB$, then the candidate tensor seeds are \\emph{all} $x_i\\otimes y_j$. For finite expansions,\n\\[\nx=\\sum_i A_ix_i,\\qquad y=\\sum_j B_jy_j,\n\\]\nbilinearity gives the required double sum:\n\\[\nx\\otimes y=\\sum_i\\sum_j(A_i\\otimes B_j)(x_i\\otimes y_j).\n\\]","require every cross term}{OPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONAL}{Main text: \\cref{prop:tensor-bound}. Algebraic tensor products}\nIf $(x_i)$ generates $X$ under $\\AA$ and $(y_j)$ generates $Y$ under $\\BB$, then the candidate tensor seeds are \\emph{all} $x_i\\otimes y_j$. For finite expansions,\n\\[\nx=\\sum_i A_ix_i,\\qquad y=\\sum_j B_jy_j,\n\\]\nbilinearity gives the required double sum:\n\\[\nx\\otimes y=\\sum_i\\sum_j(A_i\\otimes B_j)(x_i\\otimes y_j).\n\\]","\\vfig[.95]{tensor_rectangle}\n\\atlasCaption{The exact expansion of $(2e_1+3e_2)\\otimes(5f_1+7f_2)$. The terms away from the diagonal $14$ and $15$ cannot be omitted.}","Keeping only the tensors with matching indices $e_1\\otimes f_1$ and $e_2\\otimes f_2$ spans a subspace of dimension two in a tensor product of dimension four. It misses both cross directions. Under scalar actions on $\\QQ^2$ and $\\QQ^2$, all four basis tensors are necessary and sufficient.","\\head{The resulting upper bound}\nSince simple tensors span the algebraic tensor product, the complete rectangular family proves\n\\[\n\\boxed{\\srank_{\\AA\\otimes\\BB}(X\\otimes Y)\n\\le\\srank_{\\AA}(X)\\,\\srank_{\\BB}(Y).}\n\\]\nThis is not a universal equality claim. Changing the algebra changes the answer: the full independent matrix action can make a single nonzero pure tensor cyclic. 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