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Suppose an operator has the form\n\\[\nT=D+N,\n\\]\nwhere $D$ is invertible and every packet transfer in $N$ raises the filtration by a positive amount.  The ordinary Neumann series can then be expanded into weighted paths.  The weight is not the number of edges.  It is the total filtration gain along the path.  At precision $L$, every path of weight at least $L$ vanishes.  A finite transfer graph also gives a rational generating function for the number of paths of each total weight.  The relevant generating function facts are classical and are discussed in \\cite{Stanley2012}.","inversion in a complete filtered algebra.  Suppose an operator has the form\n\\[\nT=D+N,\n\\]\nwhere $D$ is invertible and every packet transfer in $N$ raises the filtration by a positive amount.  The ordinary Neumann series can then be expanded into weighted paths.  The weight is not the number of edges.  It is the total filtration gain along the path.  At precision $L$, every path of weight at least $L$ vanishes.  A finite transfer graph also gives a rational generating function for the number of paths of each total weight.  The relevant generating function facts are classical and are discussed in \\cite{Stanley2012}.","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation als","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves tha","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation we","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on l","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matri","The third part concerns integral separation.  Let $V$ be a vector space over the fraction field of a discrete valuation ring $\\OO$, and let $\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matrix polynomials may be found in \\cite{Serre1979,","$\\Lambda$ be an $\\OO$ lattice in $V$.  A rational projector $E$ need not preserve $\\Lambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matrix polynomials may be found in \\cite{Serre1979,Newman1972,GohbergLancasterRodman2009}.","ambda$.  Its separation weight is the least $s$ such that\n\\[\n\\pi^sE\\Lambda\\subseteq\\Lambda.\n\\]\nA Bézout construction gives an upper bound through a polynomial resultant.  That bound need not be exact because it does not see the position of the lattice.  The exact answer is controlled by the finite length module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda\\Big/\\bigoplus_{\\alpha}(\\Lambda\\cap V_\\alpha),\n\\]\nwhere $V=\\bigoplus_\\alpha V_\\alpha$ is the rational packet decomposition.  The largest Smith exponent of this gluing module is exactly the common denominator needed to make every packet projector integral.  This interpretation also proves that separation weight cannot decrease under refinement.  Background on local fields, Smith normal form, and matrix polynomials may be found in \\cite{Serre1979,Newman1972,GohbergLancasterRodman2009}.","The last part records how a defect between fine and coarse operators propagates.  If\n\\[\n\\Delta=CT_f-T_cC,\n\\]\nthen exact formulas describe the corresponding defect for powers, resolvents, holomorphic functions, exponentials, and a chain of several refinement levels.  These formulas are elementary, but collecting them in one place makes the later analytic uses transparent.  The spectral and resolvent background follows the usual operator theory in \\cite{Kato1995}.","The second is filtration depth. A transfer in $F^r\\Bcal$ vanishes modulo $F^r\\Bcal$. This membership gives a lower bound on its depth. A nonzero transfer may first become visible only at a higher precision. Two paths with the same number of edges can have different filtration depths. A longer path can appear earlier when its edges have smaller weights.","The third is separation depth.  A rational packet projector may require multiplication by $\\pi^s$ before it preserves the chosen lattice.  This number depends on both the rational decomposition and the lattice.  It can be zero even when the corresponding eigenvalues are close, and it can be positive even when the rational projector is simple.","The fourth notion is the compatibility defect. A fine operator and a separately chosen coarse operator need not commute with coarsening. Their discrepancy is\n\\[\n\\Delta=CT_f-T_cC.\n\\]\nThis operator records the failure of one model to represent another exactly. A chosen norm measures its size, while a chosen filtration can assign a depth. No integer compatibility depth is assigned without that additional structure.","A simple two packet model already contains all four notions.  Let\n\\[\nX=X_0\\oplus X_1,\n\\qquad\nD=\\begin{pmatrix}D_0&0\\\\0&D_1\\end{pmatrix},\n\\qquad\nN=\\begin{pmatrix}0&N_{01}\\\\N_{10}&0\\end{pmatrix}.\n\\]\nA finer family may split $X_0$ and $X_1$ into smaller summands.  The blocks $N_{01}$ and $N_{10}$ may enter at different filtration weights.  Over a fraction field, the rational packet spaces may fail to split the chosen lattice.  A coarse operator on one selected summand may also fail to intertwine with the full matrix.  The later sections isolate these four effects one at a time.","N_{10}&0\\end{pmatrix}.\n\\]\nA finer family may split $X_0$ and $X_1$ into smaller summands.  The blocks $N_{01}$ and $N_{10}$ may enter at different filtration weights.  Over a fraction field, the rational packet spaces may fail to split the chosen lattice.  A coarse operator on one selected summand may also fail to intertwine with the full matrix.  The later sections isolate these four effects one at a time.","\\begin{definition}[Packet system]\\label{def:packet_system}\nLet $X$ be a module over a unital ring.  A finite packet system on $X$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\n\\subseteq\\End(X)\n\\]\nof nonzero idempotents such that\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{when }\\alpha\\ne\\beta,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nWrite\n\\[\nX_\\alpha=P_\\alpha X,\n\\qquad\nX^{\\Pcal}=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nThe analysis and synthesis maps are\n\\[\n\\Acal_{\\Pcal}x=(P_\\alpha x)_\\alpha,\n\\qquad\n\\Scal_{\\Pcal}(x_\\alpha)_\\alpha=\\sum_\\alpha x_\\alpha.\n\\]\nThey are inverse isomorphisms.\n\\end{definition}","ket_system}\nLet $X$ be a module over a unital ring.  A finite packet system on $X$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\n\\subseteq\\End(X)\n\\]\nof nonzero idempotents such that\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{when }\\alpha\\ne\\beta,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nWrite\n\\[\nX_\\alpha=P_\\alpha X,\n\\qquad\nX^{\\Pcal}=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nThe analysis and synthesis maps are\n\\[\n\\Acal_{\\Pcal}x=(P_\\alpha x)_\\alpha,\n\\qquad\n\\Scal_{\\Pcal}(x_\\alpha)_\\alpha=\\sum_\\alpha x_\\alpha.\n\\]\nThey are inverse isomorphisms.\n\\end{definition}","\\begin{definition}[Refinement]\\label{def:refinement}\\label{def:refinement-map}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on the same module.  We say that $\\Qcal$ refines $\\Pcal$ when there is a surjection\n\\[\n\\rho:J\\longrightarrow I\n\\]\nsuch that\n\\[\nP_\\alpha=\\sum_{\\rho(\\beta)=\\alpha}Q_\\beta\n\\]\nfor every $\\alpha\\in I$.  The index $\\beta$ is then called a child of $\\rho(\\beta)$.\n\\end{definition}","ition}[Refinement]\\label{def:refinement}\\label{def:refinement-map}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on the same module.  We say that $\\Qcal$ refines $\\Pcal$ when there is a surjection\n\\[\n\\rho:J\\longrightarrow I\n\\]\nsuch that\n\\[\nP_\\alpha=\\sum_{\\rho(\\beta)=\\alpha}Q_\\beta\n\\]\nfor every $\\alpha\\in I$.  The index $\\beta$ is then called a child of $\\rho(\\beta)$.\n\\end{definition}","\\begin{theorem}[Children, parents, and transitivity]\\label{thm:children_transitivity}\\label{thm:nested-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nP_\\alpha Q_\\beta\n=\nQ_\\beta P_\\alpha\n=\n\\begin{cases}\nQ_\\beta & \\rho(\\beta)=\\alpha,","\\begin{theorem}[Children, parents, and transitivity]\\label{thm:children_transitivity}\\label{thm:nested-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nP_\\alpha Q_\\beta\n=\nQ_\\beta P_\\alpha\n=\n\\begin{cases}\nQ_\\beta & \\rho(\\beta)=\\alpha,\\\\\n0 & \\rho(\\beta)\\ne\\alpha.\n\\end{cases}\n\\]\nHence $Q_\\beta X\\subseteq P_{\\rho(\\beta)}X$.  If a third packet system $\\Rcal$ refines $\\Qcal$, then $\\Rcal$ refines $\\Pcal$ through the composite parent map.\n\\end{theorem}","0 & \\rho(\\beta)\\ne\\alpha.\n\\end{cases}\n\\]\nHence $Q_\\beta X\\subseteq P_{\\rho(\\beta)}X$.  If a third packet system $\\Rcal$ refines $\\Qcal$, then $\\Rcal$ refines $\\Pcal$ through the composite parent map.\n\\end{theorem}","\\begin{proof}\nMultiply the identity\n\\[\nP_\\alpha=\\sum_{\\rho(\\gamma)=\\alpha}Q_\\gamma\n\\]\nby $Q_\\beta$ on either side.  Pairwise annihilation in the fine family leaves $Q_\\beta$ when $\\rho(\\beta)=\\alpha$ and leaves zero otherwise.  For transitivity, substitute the child expansion of each $Q_\\beta$ into the expansion of $P_\\alpha$ and group the resulting terms by the composite parent.\n\\end{proof}","A refinement tower is a sequence $(\\Pcal_\\ell)_{\\ell\\geq0}$ with parent maps\n\\[\n\\rho_{m,\\ell}:I_m\\longrightarrow I_\\ell\n\\qquad(m\\geq\\ell)\n\\]\nsuch that $\\Pcal_m$ refines $\\Pcal_\\ell$ through $\\rho_{m,\\ell}$ for every $m\\geq\\ell$, and\n\\[\n\\rho_{\\ell,\\ell}=\\id_{I_\\ell},\n\\qquad\n\\rho_{n,\\ell}=\\rho_{m,\\ell}\\circ\\rho_{n,m}\n\\qquad(n\\geq m\\geq\\ell).\n\\]\nThus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","\\begin{definition}[Coordinate change across levels]\\label{def:coordinate_change}\\label{def:coordinate-refinement}\nFor two packet systems $\\Pcal$ and $\\Qcal$ on $X$, define\n\\[\nJ_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Qcal}\\Scal_{\\Pcal}:X^{\\Pcal}\\longrightarrow X^{\\Qcal}\n\\]\nand\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}:X^{\\Qcal}\\longrightarrow X^{\\Pcal}.\n\\]\nWhen $\\Qcal$ refines $\\Pcal$, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.\n\\end{definition}","\\begin{theorem}[Exact coordinate change and coherence]\\label{thm:coordinate_coherence}\\label{prop:perfect-reconstruction}\nFor any two packet systems on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","herence]\\label{thm:coordinate_coherence}\\label{prop:perfect-reconstruction}\nFor any two packet systems on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","wo packet systems on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","s on $X$,\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","w\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}=I_{X^{\\Pcal}},\n\\qquad\nJ_{\\Qcal\\leftarrow\\Pcal}C_{\\Pcal\\leftarrow\\Qcal}=I_{X^{\\Qcal}}.\n\\]\nFor three packet systems $\\Pcal$, $\\Qcal$, and $\\Rcal$ on $X$,\n\\[\nJ_{\\Rcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\nJ_{\\Rcal\\leftarrow\\Pcal},\n\\]\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}C_{\\Qcal\\leftarrow\\Rcal}\n=\nC_{\\Pcal\\leftarrow\\Rcal}.\n\\]\nIf $\\Qcal$ refines $\\Pcal$ through $\\rho$, then\n\\[\n\\bigl(J_{\\Qcal\\leftarrow\\Pcal}x\\bigr)_\\beta\n=\nQ_\\beta x_{\\rho(\\beta)},\n\\qquad\n\\bigl(C_{\\Pcal\\leftarrow\\Qcal}y\\bigr)_\\alpha\n=\n\\sum_{\\rho(\\beta)=\\alpha}y_\\beta.\n\\]\n\\end{theorem}","\\begin{proof}\nThe packet analysis and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","n{proof}\nThe packet analysis and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","ket analysis and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","and synthesis maps are inverse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","rse.  Therefore\n\\[\nC_{\\Pcal\\leftarrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","arrow\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Qcal}\\Acal_{\\Qcal}\\Scal_{\\Pcal}\n=\n\\Acal_{\\Pcal}\\Scal_{\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","\\Pcal}\n=\nI.\n\\]\nThe other inverse identity is the same calculation with the two systems exchanged.  The coherence identities follow by inserting\n\\[\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_X.\n\\]\nNow assume that $\\Qcal$ refines $\\Pcal$.  If $x=(x_\\alpha)_\\alpha$ with $x_\\alpha\\in P_\\alpha X$, then\n\\[\nQ_\\beta\\sum_\\alpha x_\\alpha\n=\nQ_\\beta x_{\\rho(\\beta)}\n\\]\nby \\cref{thm:children_transitivity}.  The formula for $C$ follows by synthesis followed by projection onto the parent packet.\n\\end{proof}","\\begin{remark}[Refinement and reduction]\\label{rem:refinement_reduction}\nThe maps in \\cref{thm:coordinate_coherence} are isomorphisms because both coordinate spaces represent all of $X$.  They do not reduce dimension and they do not discard packets.  Approximation begins only after one chooses a truncation or compression rule.\n\\end{remark}","\\begin{corollary}[Orthogonal Hilbert packets]\\label{cor:orthogonal_isometry}\nIf $X$ is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product $\\langle(x_\\alpha),(y_\\alpha)\\rangle=\\sum_\\alpha\\langle x_\\alpha,y_\\alpha\\rangle$. Then analysis and synthesis are unitary. Every coordinate change in \\cref{thm:coordinate_coherence} is unitary.\n\\end{corollary}","\\begin{proof}\nFor an orthogonal packet system,\n\\[\n\\|x\\|^2=\\sum_\\alpha\\|P_\\alpha x\\|^2.\n\\]\nThus analysis is an isometry onto the full packet direct sum.  Synthesis is its inverse.  A coordinate change is a composition of two unitary maps.\n\\end{proof}","Every coarse packet is also uniquely reconstructed from its fine children.  If vectors $x_\\beta\\in Q_\\beta X$ sum to a coarse packet $x$, applying $Q_\\gamma$ recovers $x_\\gamma=Q_\\gamma x$.  This is the uniqueness clause of the canonical nested-refinement statement.","\\subsection{Typed coordinate naturality}\nThese coordinate changes are coherent through every finite tower. If three\ncomplete packet systems are denoted by $a,b,c$, then\n\\[\nJ_{c,b}J_{b,a}=J_{c,a},\n\\qquad J_{a,a}=I,\n\\qquad J_{b,a}^{-1}=J_{a,b}.\n\\]\nThese laws hold for arbitrary exact packet systems representing the same\nambient module. The parent maps supply the component formulas linking\nchildren to their parents. Coordinate coherence itself does not require them.","The same coherence applies to operators. Let $P_a,P_b$ be packet systems on $X$, let $Q_a,Q_b$ be packet\nsystems on $Y$, where $X$ and $Y$ are modules over the same scalar ring, and let $T:X\\to Y$ be a linear map over that ring. Its exact coordinate\nrepresentation\n\\[\n\\widehat T_{Q,P}=\\cA_QT\\cS_P\n\\]\npreserves identities and typed composition, and the naturality square is\n\\[\n\\boxed{\nJ^Y_{Q_b,Q_a}\\,\\widehat T_{Q_a,P_a}\n=\n\\widehat T_{Q_b,P_b}\\,J^X_{P_b,P_a}.\n}\n\\]\nThe identity, cocycle, composition, inverse, and naturality laws are all proved\nin Lean. The separate refinement hypotheses supply the local component\nformulas linking each child to its parent.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{01_nested_dyadic_refinement.pdf}\n\\caption{A nested residue hierarchy. Each residue modulo $2$ is the union of two residues modulo $4$, and each residue modulo $4$ is the union of two residues modulo $8$.}\n\\label{fig:refinement-tree}\n\\end{figure}\n\\subsection{Operator blocks across refinement}","For $T\\in\\End(X)$ and a packet system $\\Pcal$, write\n\\[\nT^{\\Pcal}_{\\beta\\alpha}=P_\\beta T P_\\alpha.\n\\]","\\begin{theorem}[Refinement of operator blocks]\\label{thm:block_refinement}\\label{thm:transfer-block-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nT^{\\Pcal}_{ba}\n=\n\\sum_{\\substack{\\rho(\\beta)=b","\\begin{theorem}[Refinement of operator blocks]\\label{thm:block_refinement}\\label{thm:transfer-block-refinement}\nSuppose that $\\Qcal$ refines $\\Pcal$ through $\\rho$.  Then\n\\[\nT^{\\Pcal}_{ba}\n=\n\\sum_{\\substack{\\rho(\\beta)=b\\\\\\rho(\\alpha)=a}}\nT^{\\Qcal}_{\\beta\\alpha}.\n\\]\nThe matrix of $T$ in fine coordinates is related to the matrix in coarse coordinates by\n\\[\n[T]_{\\Pcal}\n=\nC_{\\Pcal\\leftarrow\\Qcal}[T]_{\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}.\n\\]\n\\end{theorem}","rho(\\alpha)=a}}\nT^{\\Qcal}_{\\beta\\alpha}.\n\\]\nThe matrix of $T$ in fine coordinates is related to the matrix in coarse coordinates by\n\\[\n[T]_{\\Pcal}\n=\nC_{\\Pcal\\leftarrow\\Qcal}[T]_{\\Qcal}J_{\\Qcal\\leftarrow\\Pcal}.\n\\]\n\\end{theorem}","\\begin{proof}\nExpand the two coarse projectors into their children:\n\\[\nP_bTP_a\n=\n\\left(\\sum_{\\rho(\\beta)=b}Q_\\beta\\right)\nT\n\\left(\\sum_{\\rho(\\alpha)=a}Q_\\alpha\\right).\n\\]\nDistributing the finite sums gives the first identity.  For the coordinate identity, use\n\\[\n[T]_{\\Pcal}=\\Acal_{\\Pcal}T\\Scal_{\\Pcal}\n\\]\nand insert\n\\[\nI_X=\\Scal_{\\Qcal}\\Acal_{\\Qcal}.\n\\]\n\\end{proof}","\\begin{corollary}[Support under refinement]\\label{cor:block_support_refinement}\nUnder the hypotheses of \\cref{thm:block_refinement}, a coarse block $T^{\\Pcal}_{ba}$ is zero if and only if every fine descendant block\n\\[\nT^{\\Qcal}_{\\beta\\alpha}\n\\qquad\n\\bigl(\\rho(\\beta)=b,\\ \\rho(\\alpha)=a\\bigr)\n\\]\nis zero.\n\\end{corollary}","\\begin{proof}\nIf every fine block is zero, the coarse block is zero by \\cref{thm:block_refinement}.  Conversely, suppose $P_bTP_a=0$.  Multiply on the left by $Q_\\beta$ and on the right by $Q_\\alpha$.  The child identities in \\cref{thm:children_transitivity} give\n\\[\nQ_\\beta TQ_\\alpha=0.\n\\]\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{03_refinement_incidence.pdf}\n\\caption{The incidence matrix $M_{r,s}=1$ when the fine residue $s$ belongs to the coarse residue $r$. Its rows implement exact aggregation from fine packets to coarse packets.}\n\\label{fig:incidence}\n\\end{figure}\n\\subsection{Common refinements}","\\begin{theorem}[Common refinement of commuting packet systems]\\label{thm:common_refinement}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on $X$.  Assume\n\\[\nP_\\alpha Q_\\beta=Q_\\beta P_\\alpha\n\\]\nfor all $\\alpha$ and $\\beta$.  Remove the zero operators from the family\n\\[\nR_{\\alpha\\beta}=P_\\alpha Q_\\beta.\n\\]\nThe remaining operators form a packet system that refines both $\\Pcal$ and $\\Qcal$.  The same statement holds for any finite collection of pairwise commuting packet systems.\n\\end{theorem}","[Common refinement of commuting packet systems]\\label{thm:common_refinement}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on $X$.  Assume\n\\[\nP_\\alpha Q_\\beta=Q_\\beta P_\\alpha\n\\]\nfor all $\\alpha$ and $\\beta$.  Remove the zero operators from the family\n\\[\nR_{\\alpha\\beta}=P_\\alpha Q_\\beta.\n\\]\nThe remaining operators form a packet system that refines both $\\Pcal$ and $\\Qcal$.  The same statement holds for any finite collection of pairwise commuting packet systems.\n\\end{theorem}","nement of commuting packet systems]\\label{thm:common_refinement}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ and $\\Qcal=(Q_\\beta)_{\\beta\\in J}$ be packet systems on $X$.  Assume\n\\[\nP_\\alpha Q_\\beta=Q_\\beta P_\\alpha\n\\]\nfor all $\\alpha$ and $\\beta$.  Remove the zero operators from the family\n\\[\nR_{\\alpha\\beta}=P_\\alpha Q_\\beta.\n\\]\nThe remaining operators form a packet system that refines both $\\Pcal$ and $\\Qcal$.  The same statement holds for any finite collection of pairwise commuting packet systems.\n\\end{theorem}","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, u","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family wh","\\begin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","egin{proof}\nCommutativity gives\n\\[\nR_{\\alpha\\beta}^2=P_\\alpha^2Q_\\beta^2=R_{\\alpha\\beta}.\n\\]\nIf $(\\alpha,\\beta)\\ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","ne(\\alpha',\\beta')$, then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","then either $P_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","_\\alpha P_{\\alpha'}=0$ or $Q_\\beta Q_{\\beta'}=0$, so the corresponding products annihilate one another.  Finally,\n\\[\n\\sum_{\\alpha,\\beta}R_{\\alpha\\beta}\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\left(\\sum_\\beta Q_\\beta\\right)\n=I_X.\n\\]\nFor a fixed $\\alpha$,\n\\[\nP_\\alpha\n=P_\\alpha\\sum_\\beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","beta Q_\\beta\n=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","=\n\\sum_\\beta R_{\\alpha\\beta},\n\\]\nand the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","nd the same argument works for a fixed $\\beta$. Since each $P_\\alpha$ and $Q_\\beta$ is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite family statement. For the empty collection, use the supported identity family: $(I_X)$ when $X\\ne0$ and the empty family when $X=0$.\n\\end{proof}","\\begin{proposition}[Coarsest common refinement]\\label{prop:coarsest_common_refinement}\nUnder the hypotheses of \\cref{thm:common_refinement}, let $\\Rcal$ be a packet system that refines both $\\Pcal$ and $\\Qcal$.  Then $\\Rcal$ refines the nonzero product family $(P_\\alpha Q_\\beta)$.  Hence that product family is the coarsest packet system that refines both $\\Pcal$ and $\\Qcal$.\n\\end{proposition}","\\begin{proof}\nLet $R_\\gamma$ be a packet in $\\Rcal$.  Because $\\Rcal$ refines $\\Pcal$, there is a unique $\\alpha$ with\n\\[\nP_\\alpha R_\\gamma=R_\\gamma.\n\\]\nLikewise there is a unique $\\beta$ with\n\\[\nQ_\\beta R_\\gamma=R_\\gamma.\n\\]\nTherefore\n\\[\nP_\\alpha Q_\\beta R_\\gamma=R_\\gamma.\n\\]\nEvery packet of $\\Rcal$ lies in one product packet. Multiplying $\\sum_\\gamma R_\\gamma=I_X$ by $P_\\alpha Q_\\beta$ and using the parent identities gives\n\\[\nP_\\alpha Q_\\beta\n=\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma","a packet in $\\Rcal$.  Because $\\Rcal$ refines $\\Pcal$, there is a unique $\\alpha$ with\n\\[\nP_\\alpha R_\\gamma=R_\\gamma.\n\\]\nLikewise there is a unique $\\beta$ with\n\\[\nQ_\\beta R_\\gamma=R_\\gamma.\n\\]\nTherefore\n\\[\nP_\\alpha Q_\\beta R_\\gamma=R_\\gamma.\n\\]\nEvery packet of $\\Rcal$ lies in one product packet. Multiplying $\\sum_\\gamma R_\\gamma=I_X$ by $P_\\alpha Q_\\beta$ and using the parent identities gives\n\\[\nP_\\alpha Q_\\beta\n=\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma\\\\Q_\\beta R_\\gamma=R_\\gamma}}R_\\gamma.\n\\]\nThus the packets with the same pair of parents reconstruct the product packet. If that product packet is nonzero, the sum has a child, so the parent map onto the supported product family is surjective.\n\\end{proof}","\\begin{example}[A two level split]\\label{ex:two_level_split}\nLet\n\\[\nX=X_{00}\\oplus X_{01}\\oplus X_{10}\\oplus X_{11}.\n\\]\nAssume that all four summands are nonzero. The four coordinate projections form a fine packet system.  The two sums\n\\[\nP_0=Q_{00}+Q_{01},\n\\qquad\nP_1=Q_{10}+Q_{11}\n\\]\nform a coarse packet system.  If $x=(x_{00},x_{01},x_{10},x_{11})$, the coarse coordinates are\n\\[\n(x_{00}+x_{01},x_{10}+x_{11})\n\\]\nwhen the sums are interpreted inside the two coarse direct summands.  Fine analysis recovers the four entries exactly.  Nothing has been approximated.\n\\end{example}","\\begin{definition}[Residue selector]\\label{def:residue_selector}\nLet $R$ be a commutative ring and let\n\\[\nF(z)=\\sum_{n\\geq0}a_nz^n\\in R[[z]].\n\\]\nFor $N\\geq1$ and $0\\leq j<N$, define\n\\[\n(P_{N,j}F)(z)\n=\n\\sum_{n\\equiv j\\pmod N}a_nz^n.\n\\]\n\\end{definition}","\\begin{proposition}[Residue packet algebra]\\label{prop:residue_algebra}\\label{prop:residue-algebra}\nFor every commutative ring $R$, the operators $(P_{N,j})_{0\\leq j<N}$ satisfy the identities below.  If $R$ is nonzero, they form a packet system on $R[[z]]$:\n\\[\nP_{N,i}P_{N,j}=\\delta_{ij}P_{N,j},\n\\qquad\n\\sum_{j=0}^{N-1}P_{N,j}=I.\n\\]\n\\end{proposition}","\\begin{proof}\nEvery nonnegative integer belongs to one residue class modulo $N$.  A coefficient survives both $P_{N,i}$ and $P_{N,j}$ exactly when the two labels are equal to its residue.  Summing all selectors preserves every coefficient once.\n\\end{proof}","The coefficient-mask definition of $P_{N,j}$ also makes sense over a commutative semiring: it only retains or sets coefficients to zero. We use that same definition in the following semiring statement.","Let $U$ act on coefficient sequences by\n\\[\n(UF)(z)=F(\\omega z).\n\\]\nIf $\\omega^N=1$, then $U^N=I$.","\\begin{theorem}[Root-of-unity realization]\\label{thm:root-of-unity}\nLet $R$ be a commutative semiring, let $N>0$, and suppose\n$\\omega,\\omega^\\vee,\\eta\\in R$ satisfy $\\omega^N=1$ and the finite\n$N\\times N$ Fourier orthogonality table\n\\[\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}\\omega^{rk}\n=\n\\begin{cases}\n1,&r=j,","\\begin{theorem}[Root-of-unity realization]\\label{thm:root-of-unity}\nLet $R$ be a commutative semiring, let $N>0$, and suppose\n$\\omega,\\omega^\\vee,\\eta\\in R$ satisfy $\\omega^N=1$ and the finite\n$N\\times N$ Fourier orthogonality table\n\\[\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}\\omega^{rk}\n=\n\\begin{cases}\n1,&r=j,\\\\\n0,&r\\ne j,\n\\end{cases}\n\\qquad 0\\le j,r<N.\n\\]\nThen\n\\[\n\\boxed{\nP_{N,j}\n=\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}U^k.\n}\n\\]\nIn particular, if $R$ is a field, $N>0$, and $\\omega$ is a primitive\n$N$th root of unity, the certificate holds with\n$\\omega^\\vee=\\omega^{-1}$ and $\\eta=N^{-1}$.\n\\end{theorem}","0,&r\\ne j,\n\\end{cases}\n\\qquad 0\\le j,r<N.\n\\]\nThen\n\\[\n\\boxed{\nP_{N,j}\n=\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}U^k.\n}\n\\]\nIn particular, if $R$ is a field, $N>0$, and $\\omega$ is a primitive\n$N$th root of unity, the certificate holds with\n$\\omega^\\vee=\\omega^{-1}$ and $\\eta=N^{-1}$.\n\\end{theorem}","\\begin{proof}\nOn the monomial $z^n$, the right-hand side acts by\n\\[\n\\eta\\sum_{k=0}^{N-1}(\\omega^\\vee)^{jk}\\omega^{nk}.\n\\]\nThe identity $\\omega^N=1$ reduces this expression to the table entry indexed\nby $r=n\\bmod N$. The finite certificate therefore gives exactly the\ncoefficient mask defining $P_{N,j}$.\nOver a field, the primitive-root geometric sum supplies the certificate.\n\\end{proof}","The field or orthogonality condition is essential. A primitive element of\norder $N$ in a ring with zero divisors does not by itself force the geometric\nsums away from the diagonal to vanish. For example, in $\\mathbb F_3\\times\\mathbb F_3$ the\nelement $(-1,1)$ has order two, but the second component of its nontrivial\ncharacter sum is nonzero. The coefficient selector uses no division and remains valid\nwithout any Fourier certificate. This distinction becomes decisive at the\ndyadic prime.","\\subsection{The field specialization}\n\\begin{theorem}[Character realization]\\label{thm:character_realization}\nLet $\\bbk$ be a field, assume that $N$ is nonzero in $\\bbk$, and suppose that $\\bbk$ contains a primitive $N$th root of unity $\\omega$.  Define\n\\[\n(UF)(z)=F(\\omega z).\n\\]\nThen\n\\[\nP_{N,j}\n=\n\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k.\n\\]\n\\end{theorem}","\\begin{proof}\nOn $z^n$, the right side acts by the scalar\n\\[\n\\frac1N\\sum_{k=0}^{N-1}\\omega^{k(n-j)}.\n\\]\nThis is one when $n\\equiv j\\pmod N$ and zero otherwise by the geometric sum identity in the field $\\bbk$.\n\\end{proof}","\\begin{remark}[Two distinct constructions]\\label{rem:selector_character}\nThe coefficient selector of \\cref{def:residue_selector} uses no division.  The character formula in \\cref{thm:character_realization} divides by $N$.  At a prime that divides $N$, the coefficient selector may remain integral even when the character average does not.\n\\end{remark}","\\begin{theorem}[Nested residue refinement]\\label{thm:residue_refinement}\\label{thm:polyphase-refinement}\nLet $N,M$ be positive integers with $N$ dividing $M$, and let $0\\leq j<N$.  Then\n\\[\nP_{N,j}\n=\n\\sum_{\\substack{0\\leq s<M","\\begin{theorem}[Nested residue refinement]\\label{thm:residue_refinement}\\label{thm:polyphase-refinement}\nLet $N,M$ be positive integers with $N$ dividing $M$, and let $0\\leq j<N$.  Then\n\\[\nP_{N,j}\n=\n\\sum_{\\substack{0\\leq s<M\\\\s\\equiv j\\pmod N}}P_{M,s}.\n\\]\nWhen $R$ is nonzero, the packet system modulo $M$ refines the packet system modulo $N$.\n\\end{theorem}","s\\equiv j\\pmod N}}P_{M,s}.\n\\]\nWhen $R$ is nonzero, the packet system modulo $M$ refines the packet system modulo $N$.\n\\end{theorem}","\\begin{proof}\nAn exponent $n$ satisfies $n\\equiv j\\pmod N$ exactly when its unique residue $s$ modulo $M$ satisfies $s\\equiv j\\pmod N$.\n\\end{proof}","\\begin{example}[The shift graph of residue packets]\\label{ex:residue_shift}\nAssume that $R$ is nonzero and $N>0$. Multiplication by $z$ sends the residue packet $P_{N,j}R[[z]]$ into $P_{N,j+1}R[[z]]$, where the index is read modulo $N$.  Its packet graph is the directed cycle\n\\[\n0\\longrightarrow1\\longrightarrow\\cdots\\longrightarrow N-1\\longrightarrow0.\n\\]\nMultiplication by a polynomial supported on a set of residues gives the corresponding finite set of phase shifts.  This is the simplest transfer graph behind a polyphase representation.\n\\end{example}","\\begin{theorem}[Coprime factorization]\\label{thm:crt_factorization}\nLet $A$ and $B$ be coprime positive integers.  For each $j$ modulo $AB$, write\n\\[\nj_A=j\\pmod A,\n\\qquad\nj_B=j\\pmod B.\n\\]\nThen\n\\[\nP_{AB,j}=P_{A,j_A}P_{B,j_B}.\n\\]\nEvery pair $(j_A,j_B)$ occurs for one residue $j$ modulo $AB$.\n\\end{theorem}","\\begin{proof}\nA coefficient survives the product on the right exactly when its exponent satisfies both congruences.  The Chinese remainder theorem gives one residue class modulo $AB$ with those two coordinates.  The needed form of the theorem is standard and may be found in \\cite{Lang2002}.\n\\end{proof}","\\begin{corollary}[Dyadic and odd coordinates]\\label{cor:dyadic_odd_coordinates}\\label{thm:dyadic-odd}\nWrite $N=2^aM$ with $M$ odd.  Every residue packet modulo $N$ is the product of one packet modulo $2^a$ and one packet modulo $M$.\n\\end{corollary}","\\begin{proof}\nApply \\cref{thm:crt_factorization} to the coprime moduli $2^a$ and $M$.\n\\end{proof}","\\begin{example}[Residues modulo twelve]\\label{ex:mod_twelve}\nThe pair\n\\[\n(j\\pmod4,j\\pmod3)=(3,1)\n\\]\ncorresponds to $j=7$ modulo $12$.  Hence\n\\[\nP_{12,7}=P_{4,3}P_{3,1}.\n\\]\nThis factorization concerns addresses.  It does not say that the two factors have the same integral spectral behavior.\n\\end{example}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{02_crt_dyadic_odd_grid.pdf}\n\\caption{A composite residue packet is addressed by its dyadic and odd coordinates. The displayed mod-$24$ system is the product of mod $8$ and mod $3$.}\n\\label{fig:crt-grid}\n\\end{figure}","\\begin{definition}[Complete multiplicative filtration]\\label{def:filtered_algebra}\nA complete filtered algebra is a unital algebra $\\Bcal$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","ef:filtered_algebra}\nA complete filtered algebra is a unital algebra $\\Bcal$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","bra is a unital algebra $\\Bcal$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","$ with a decreasing sequence of two-sided ideals\n\\[\n\\Bcal=F^0\\Bcal\\supseteq F^1\\Bcal\\supseteq F^2\\Bcal\\supseteq\\cdots\n\\]\nsuch that\n\\[\nF^r\\Bcal\\,F^s\\Bcal\\subseteq F^{r+s}\\Bcal,\n\\qquad\n\\bigcap_{L\\geq0}F^L\\Bcal=0,\n\\]\nand the natural map\n\\[\n\\Bcal\\longrightarrow\\varprojlim_L\\Bcal/F^L\\Bcal\n\\]\nis an isomorphism.  A transfer term $N_e$ has a declared positive integer weight $w(e)\\in\\mathbb Z_{>0}$ when\n\\[\nN_e\\in F^{w(e)}\\Bcal.\n\\]\n\\end{definition}","In the filtration topology, reduction to $\\Bcal/F^L\\Bcal$ is continuous when the quotient has the discrete topology. Its kernel $F^L\\Bcal$ is therefore closed. In particular, a convergent sum of terms in $F^L\\Bcal$ remains in that ideal.","If $\\OO$ is a complete discrete valuation ring with uniformizer $\\pi$ and $\\Bcal$ is a separated and complete $\\OO$ algebra for the filtration\n\\[\nF^L\\Bcal=\\pi^L\\Bcal,\n\\]\nthen this is a standard example. Finite matrix algebras over $\\OO$ satisfy these conditions. Completeness of $\\OO$ alone does not imply completeness of every $\\OO$ algebra. Basic facts about completeness in this setting are reviewed in \\cite{Serre1979}.","\\begin{theorem}[Weighted inverse]\\label{thm:weighted_inverse}\nLet $\\Bcal$ be a complete filtered algebra.  Let $D\\in\\Bcal$ be invertible with\n\\[\nD,D^{-1}\\in F^0\\Bcal.\n\\]\nLet $E$ be a finite set and suppose\n\\[\nT=D+\\sum_{e\\in E}N_e,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","red algebra.  Let $D\\in\\Bcal$ be invertible with\n\\[\nD,D^{-1}\\in F^0\\Bcal.\n\\]\nLet $E$ be a finite set and suppose\n\\[\nT=D+\\sum_{e\\in E}N_e,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","\\Bcal$ be invertible with\n\\[\nD,D^{-1}\\in F^0\\Bcal.\n\\]\nLet $E$ be a finite set and suppose\n\\[\nT=D+\\sum_{e\\in E}N_e,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","N_e\\in F^{w(e)}\\Bcal,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","l,\n\\qquad\nw(e)\\geq1.\n\\]\nSet\n\\[\nA_e=D^{-1}N_e.\n\\]\nFor a word $\\gamma=(e_1,\\ldots,e_r)$, write\n\\[\nA_\\gamma=A_{e_r}\\cdots A_{e_1},\n\\qquad\nW(\\gamma)=\\sum_{j=1}^rw(e_j).\n\\]\nThe empty word has weight zero and contributes the identity.  Then $T$ is invertible and\n\\[\nT^{-1}\n=\n\\sum_\\gamma(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThe series converges in the filtration, and the term indexed by $\\gamma$ belongs to $F^{W(\\gamma)}\\Bcal$.\n\\end{theorem}","\\begin{proof}\nPut\n\\[\nA=\\sum_{e\\in E}A_e.\n\\]\nSince left multiplication by $D^{-1}$ preserves the filtration, $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}","nce left multiplication by $D^{-1}$ preserves the filtration, $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}","e filtration, $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}",", $A\\in F^1\\Bcal$.  Completeness gives the convergent Neumann series\n\\[\n(I+A)^{-1}=\\sum_{r\\geq0}(-A)^r.\n\\]\nBecause $T=D(I+A)$,\n\\[\nT^{-1}=(I+A)^{-1}D^{-1}.\n\\]\nExpanding each power of $A$ gives the sum over words, with the rightmost edge acting first.  Multiplicativity of the filtration gives\n\\[\nA_\\gamma\\in F^{W(\\gamma)}\\Bcal.\n\\]\nFor each $L$, only finitely many words have weight below $L$ because $E$ is finite and every edge has positive weight.  Thus the word expansion converges to the same inverse.\n\\end{proof}","The preceding formula does not yet use packet blocks. When $D$ is packet diagonal and the terms $N_e$ are individual transfers, only words with compatible packet endpoints can contribute. These words become directed paths.","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\be","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\a","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and","\\begin{theorem}[Block path formula]\\label{thm:block_path_formula}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ket system in a complete filtered algebra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ra of operators.  Suppose\n\\[\nD=\\sum_\\alpha P_\\alpha D P_\\alpha\n\\]\nbelongs to $\\Bcal$ and is invertible on every packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ery packet.  Assume that the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","the assembled inverse\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","P_\\alpha\\in\\Bcal=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","=F^0\\Bcal,\n\\qquad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","ad D_\\alpha=P_\\alpha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","pha DP_\\alpha,\n\\]\nwhere each packet inverse is extended by zero to the other packets.  Let the edge set be finite, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","e, assume $w(e)\\geq1$ for every edge, and write\n\\[\nN=\\sum_{e:\\alpha\\to\\beta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","eta}N_e,\n\\qquad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","uad\nN_e=P_\\beta N_eP_\\alpha,\n\\qquad\nN_e\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","\\in F^{w(e)}\\Bcal.\n\\]\nFor an edge $e:\\alpha\\to\\beta$, put\n\\[\nA_e=(P_\\beta DP_\\beta)^{-1}N_e.\n\\]\nThen the block from $P_\\alpha X$ to $P_\\beta X$ of $(D+N)^{-1}$ is\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n=\n\\sum_{\\gamma:\\alpha\\leadsto\\beta}\n(-1)^{|\\gamma|}A_\\gamma D_\\alpha^{-1},\n\\]\nwhere $D_\\alpha=P_\\alpha DP_\\alpha$ and the sum runs over directed paths from $\\alpha$ to $\\beta$, including the empty path when $\\alpha=\\beta$.\n\\end{theorem}","\\begin{proof}\nThe diagonal inverse is\n\\[\nD^{-1}=\\sum_\\alpha D_\\alpha^{-1}P_\\alpha.\n\\]\nThe block $P_\\beta D^{-1}NP_\\alpha$ is nonzero only when there is a declared edge from $\\alpha$ to $\\beta$.  Repeated block multiplication shows that the $r$th power of $D^{-1}N$ is the sum over all directed paths of length $r$, with the operator factors ordered along the path.  Apply \\cref{thm:weighted_inverse} and multiply on the right by the packet diagonal inverse.\n\\end{proof}","\\begin{corollary}[Minimum path weight]\\label{cor:minimum_path_weight}\nFor packets $\\alpha$ and $\\beta$, let $d(\\alpha,\\beta)$ be the least total weight of a directed path from $\\alpha$ to $\\beta$, with $d=\\infty$ when no path exists.  Use the convention $F^\\infty\\Bcal=\\bigcap_{L\\geq0}F^L\\Bcal=0$.  If $\\alpha\\ne\\beta$, then\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n\\in\nF^{d(\\alpha,\\beta)}\\Bcal.\n\\]\nIf there is no path from $\\alpha$ to $\\beta$, the block is zero.\n\\end{corollary}","ary}[Minimum path weight]\\label{cor:minimum_path_weight}\nFor packets $\\alpha$ and $\\beta$, let $d(\\alpha,\\beta)$ be the least total weight of a directed path from $\\alpha$ to $\\beta$, with $d=\\infty$ when no path exists.  Use the convention $F^\\infty\\Bcal=\\bigcap_{L\\geq0}F^L\\Bcal=0$.  If $\\alpha\\ne\\beta$, then\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n\\in\nF^{d(\\alpha,\\beta)}\\Bcal.\n\\]\nIf there is no path from $\\alpha$ to $\\beta$, the block is zero.\n\\end{corollary}","path weight]\\label{cor:minimum_path_weight}\nFor packets $\\alpha$ and $\\beta$, let $d(\\alpha,\\beta)$ be the least total weight of a directed path from $\\alpha$ to $\\beta$, with $d=\\infty$ when no path exists.  Use the convention $F^\\infty\\Bcal=\\bigcap_{L\\geq0}F^L\\Bcal=0$.  If $\\alpha\\ne\\beta$, then\n\\[\nP_\\beta(D+N)^{-1}P_\\alpha\n\\in\nF^{d(\\alpha,\\beta)}\\Bcal.\n\\]\nIf there is no path from $\\alpha$ to $\\beta$, the block is zero.\n\\end{corollary}","\\begin{proof}\nEvery term in the block path formula is indexed by a path from $\\alpha$ to $\\beta$ and lies in the filtration level given by its total weight.  If no such path exists, the sum is empty.\n\\end{proof}","\\begin{example}[A longer path that appears earlier]\\label{ex:weighted_route}\nConsider three packets $\\alpha$, $\\beta$, and $\\gamma$.  Suppose there is a direct edge from $\\alpha$ to $\\gamma$ of weight three, together with edges\n\\[\n\\alpha\\longrightarrow\\beta,\n\\qquad\n\\beta\\longrightarrow\\gamma\n\\]\nof weight one each.  The two edge route has total weight two, so it contributes to the inverse before the direct edge does.  This is why the truncation rule uses total weight rather than path length. The lower bound in \\cref{cor:minimum_path_weight} need not be an equality because several paths of the same weight may cancel.\n\\end{example}","\\begin{corollary}[Acyclic transfer graph]\\label{cor:acyclic_inverse}\nIf the packet transfer graph is acyclic, then $D^{-1}N$ is nilpotent.  The inverse in \\cref{thm:block_path_formula} is a finite sum and does not require completeness of the filtration.\n\\end{corollary}","\\begin{corollary}[Stability of inversion at finite precision]\\label{cor:inverse_stability}\nLet $T$ and $\\widetilde T$ be invertible elements of $F^0\\Bcal$.  If\n\\[\nT-\\widetilde T\\in F^L\\Bcal,\n\\]\nthen\n\\[\nT^{-1}-\\widetilde T^{-1}\\in F^L\\Bcal.\n\\]\n\\end{corollary}","\\begin{proof}\nUse the resolvent identity at zero in the form\n\\[\nT^{-1}-\\widetilde T^{-1}\n=\nT^{-1}(\\widetilde T-T)\\widetilde T^{-1}.\n\\]\nBoth inverse factors belong to $F^0\\Bcal$, so the product lies in $F^L\\Bcal$.\n\\end{proof}","\\begin{corollary}[Exact finite precision truncation]\\label{cor:finite_precision}\nUnder the hypotheses of \\cref{thm:weighted_inverse}, define\n\\[\nR_L\n=\n\\sum_{W(\\gamma)<L}(-1)^{|\\gamma|}A_\\gamma D^{-1}.\n\\]\nThen\n\\[\nT^{-1}-R_L\\in F^L\\Bcal,\n\\]\n\\[\nTR_L-I\\in F^L\\Bcal,\n\\qquad\nR_LT-I\\in F^L\\Bcal.\n\\]\nThus $R_L$ is the exact inverse in $\\Bcal/F^L\\Bcal$.\n\\end{corollary}","\\begin{proof}\nEvery omitted word has weight at least $L$, so the difference from the full inverse lies in $F^L\\Bcal$.  Multiplication by $T$ or by its inverse preserves membership in $F^L\\Bcal$ under the stated filtration assumptions.  The two residual statements follow.\n\\end{proof}","\\Needspace{4\\baselineskip}\n\\begin{example}[A triangular transfer]\\label{ex:triangular_transfer}\nLet $\\OO$ be complete with uniformizer $\\pi$, and consider\n\\[\nT=\n\\begin{pmatrix}\n1&\\pi^a","\\Needspace{4\\baselineskip}\n\\begin{example}[A triangular transfer]\\label{ex:triangular_transfer}\nLet $\\OO$ be complete with uniformizer $\\pi$, and consider\n\\[\nT=\n\\begin{pmatrix}\n1&\\pi^a\\\\\n0&1\n\\end{pmatrix}\n\\]\nwith $a\\geq1$.  The only nontrivial transfer has weight $a$, and its square is zero.  Hence\n\\[\nT^{-1}\n=\n\\begin{pmatrix}\n1&-\\pi^a","0&1\n\\end{pmatrix}\n\\]\nwith $a\\geq1$.  The only nontrivial transfer has weight $a$, and its square is zero.  Hence\n\\[\nT^{-1}\n=\n\\begin{pmatrix}\n1&-\\pi^a","0&1\n\\end{pmatrix}\n\\]\nwith $a\\geq1$.  The only nontrivial transfer has weight $a$, and its square is zero.  Hence\n\\[\nT^{-1}\n=\n\\begin{pmatrix}\n1&-\\pi^a\\\\\n0&1\n\\end{pmatrix}.\n\\]\nAt precision $L\\leq a$, the transfer vanishes and the inverse is the identity modulo $\\pi^L$.  At depth $a+1$, the transfer between the two packets first becomes visible.\n\\end{example}","0&1\n\\end{pmatrix}.\n\\]\nAt precision $L\\leq a$, the transfer vanishes and the inverse is the identity modulo $\\pi^L$.  At depth $a+1$, the transfer between the two packets first becomes visible.\n\\end{example}","Let $\\OO$ be a complete discrete valuation ring with uniformizer $\\pi$, fraction field $K$, and filtration\n\\[\nF^m\\cB=\\pi^m\\cB\n\\]\non an $\\OO$-algebra $\\cB$. The finite quotient statements below do not require $\\cB$ itself to be complete. Suppose\n\\[\nT=D+H,\n\\]\nwhere $D$ is invertible and the perturbation $H$ raises filtration. The symbol\n$K$ is reserved for the fraction field.","A packet decomposition resolves $H=\\sum_{e\\in E}H_e$ and assigns a\ncertified filtration gain\n\\[\nD^{-1}H_e\\in F^{w(e)}\\cB,\n\\qquad w(e)\\in\\Z_{>0}.\n\\]\nEach $e$ is a directed transfer edge. For an ordered transfer word\n\\[\n\\gamma=(e_1,\\ldots,e_r),\n\\]\ndefine\n\\[\nW(\\gamma)=\\sum_{j=1}^rw(e_j),\n\\qquad\nA_\\gamma=A_{e_1}\\cdots A_{e_r},\n\\qquad A_e=D^{-1}H_e.\n\\]","\\Needspace{8\\baselineskip}\n\\begin{theorem}[Certified weighted inverse]\\label{thm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","lineskip}\n\\begin{theorem}[Certified weighted inverse]\\label{thm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","[Certified weighted inverse]\\label{thm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","hm:weighted-inverse}\nLet $R$ be a possibly noncommutative ring, let $D^{-1}$ be a supplied\ntwo-sided inverse of $D$, and put $A=D^{-1}H$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item If $A^r=0$, then $D+H$ has the two-sided inverse\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)D^{-1}.}\n\\]\n\\item If $R$ is a complete normed ring and $\\lVert A\\rVert<1$, then\n$\\sum_{k\\ge0}(-A)^k$ converges and\n\\[\n\\boxed{(D+H)^{-1}=\\left(\\sum_{k\\ge0}(-A)^k\\right)D^{-1}.}\n\\]\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFactor\n\\[\nT=D(I+A),\n\\qquad A=D^{-1}H.\n\\]\nFor a finite cutoff $r$, multiplication on either side gives the\nnoncommutative telescoping identity\n\\[\n(I+A)\\sum_{k=0}^{r-1}(-A)^k\n=\n\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)(I+A)\n=1-(-A)^r.\n\\]\nThis proves (a). In (b), the strict norm bound makes the geometric series\nsummable. Continuity of multiplication passes the same identities to the\nlimit. Multiplication by $D^{-1}$ gives the displayed inverse in both orders.\n\\end{proof}","tor\n\\[\nT=D(I+A),\n\\qquad A=D^{-1}H.\n\\]\nFor a finite cutoff $r$, multiplication on either side gives the\nnoncommutative telescoping identity\n\\[\n(I+A)\\sum_{k=0}^{r-1}(-A)^k\n=\n\\left(\\sum_{k=0}^{r-1}(-A)^k\\right)(I+A)\n=1-(-A)^r.\n\\]\nThis proves (a). In (b), the strict norm bound makes the geometric series\nsummable. Continuity of multiplication passes the same identities to the\nlimit. Multiplication by $D^{-1}$ gives the displayed inverse in both orders.\n\\end{proof}","\\Needspace{4\\baselineskip}\n\\begin{corollary}[Finite precision]\\label{cor:finite-precision}\nLet $F^\\bullet R$ be a multiplicative decreasing filtration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","egin{corollary}[Finite precision]\\label{cor:finite-precision}\nLet $F^\\bullet R$ be a multiplicative decreasing filtration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","llet R$ be a multiplicative decreasing filtration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","tration and let\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n","et\n$q_L:R\\to R_L$ be a ring homomorphism that kills $F^L R$. Suppose\n\\[\nH=\\sum_{e\\in E}H_e,\n\\qquad D^{-1}H_e\\in F^{w(e)}R,\n\\qquad w(e)\\ge1,\n\\]\nwith $E$ finite. Then $q_L(D^{-1}H)^L=0$, the finite sum in\n\\cref{thm:weighted-inverse}(a) is a two-sided inverse after applying $q_L$,\nand its word expansion may discard every word of weight at least $L$.\nMore explicitly, for a length cutoff $N$, define\n\\[\nS^{\\mathrm{pr}}_{L,N}\n=\n\\sum_{n=0}^{N-1}(-1)^n\n\\sum_{\\substack{\\gamma\\in E^n\\\\W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le …","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{coroll","W(\\gamma)<L}}A_\\gamma.\n\\]\nLean proves directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","s directly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","ly that $q_L(S^{\\mathrm{pr}}_{L,N})$ equals the mapped\ngeometric inverse whenever $q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","$q_L(A^N)=0$, and hence is a two-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","wo-sided inverse of\n$q_L(I+A)$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","$, where $A=\\sum_eA_e=D^{-1}H$. With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","With the supplied two-sided\ninverse $D^{-1}$ of $D$, this gives the explicit two-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","-sided inverse\n\\[\n\\boxed{q_L\\!\\left(S^{\\mathrm{pr}}_{L,N}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","}D^{-1}\\right)}\n\\quad\\text{of}\\quad q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","d q_L(D+H).\n\\]\nIf every edge weight is at least $m$ and $L\\le mN$, then the nilpotence\ncondition follows from the filtration data because\n$m|\\gamma|\\le W(\\gamma)$. For $m>0$, the executable choice\n$N=\\lceil L/m\\rceil$ satisfies $L\\le mN$. This elementary ceiling inequality\nis also proved in Lean. The choice $m=1,N=L$ gives the standard cutoff.\nIn the DVR example $q_L:R\\to R/\\pi^L R$, this reads\n\\[\nq_L(T)^{-1}\n=\nq_L\\!\\left(\\sum_{W(\\gamma)<L}\n(-1)^{|\\gamma|}A_\\gamma D^{-1}\\right).\n\\]\n\\end{corollary}","Here $A_\\gamma=(D^{-1}H_{e_1})\\cdots(D^{-1}H_{e_r})$ in the written\norder. Multiplicativity places it in $F^{W(\\gamma)}R$, so the quotient kills\nit when $W(\\gamma)\\ge L$. Positive weights also imply\n$|\\gamma|\\le W(\\gamma)$, making the retained search finite.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{06_filtered_inverse_depth.pdf}\n\\caption{At precision $L$, weighted inversion requires only transfer words whose cumulative weight is below $L$.}\n\\label{fig:inverse-depth}\n\\end{figure}","\\begin{definition}[Weighted transfer series]\\label{def:transfer_series}\nLet $G$ be a finite directed multigraph.  Give every edge $e$ a positive integer weight $w(e)$ and a scalar multiplicity $m(e)$ in a field $\\bbk$.  Define the polynomial adjacency matrix\n\\[\nA(t)_{ij}\n=\n\\sum_{e:i\\to j}m(e)t^{w(e)}.\n\\]\nFor column vectors $u$ and $v$, define\n\\[\nZ_{u,v}(t)\n=\n\\sum_{n\\geq0}a_nt^n,\n\\]\nwhere\n\\[\na_n=\\sum_{i,j}\\ \\sum_{\\substack{\\gamma:i\\leadsto j","\\begin{definition}[Weighted transfer series]\\label{def:transfer_series}\nLet $G$ be a finite directed multigraph.  Give every edge $e$ a positive integer weight $w(e)$ and a scalar multiplicity $m(e)$ in a field $\\bbk$.  Define the polynomial adjacency matrix\n\\[\nA(t)_{ij}\n=\n\\sum_{e:i\\to j}m(e)t^{w(e)}.\n\\]\nFor column vectors $u$ and $v$, define\n\\[\nZ_{u,v}(t)\n=\n\\sum_{n\\geq0}a_nt^n,\n\\]\nwhere\n\\[\na_n=\\sum_{i,j}\\ \\sum_{\\substack{\\gamma:i\\leadsto j\\\\W(\\gamma)=n}}\n u_i v_j\\prod_{\\nu=1}^{|\\gamma|}m(e_\\nu).\n\\]\nHere $\\gamma=(e_1,\\ldots,e_{|\\gamma|})$ is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.\n\\end{definition}","W(\\gamma)=n}}\n u_i v_j\\prod_{\\nu=1}^{|\\gamma|}m(e_\\nu).\n\\]\nHere $\\gamma=(e_1,\\ldots,e_{|\\gamma|})$ is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.\n\\end{definition}","\\begin{theorem}[Rationality of the transfer series]\\label{thm:transfer_rationality}\nThe weighted transfer series satisfies\n\\[\nZ_{u,v}(t)\n=\nu^{\\mathsf T}(I-A(t))^{-1}v.\n\\]\nIt is a rational function whose denominator divides\n\\[\n\\det(I-A(t)).\n\\]\n\\end{theorem}","\\begin{proof}\nSince every entry of $A(t)$ has zero constant term, the inverse exists in the formal power series ring and\n\\[\n(I-A(t))^{-1}=\\sum_{r\\geq0}A(t)^r.\n\\]\nThe $(i,j)$ entry of $A(t)^r$ is the sum of the monomials attached to all paths of length $r$ from $i$ to $j$.  Summing over $r$ records paths of every length.  Multiplication by the endpoint vectors gives the stated series.  The adjugate formula gives\n\\[\n(I-A(t))^{-1}\n=\n\\frac{\\operatorname{adj}(I-A(t))}{\\det(I-A(t))},\n\\]\nso every entry is rational.\n\\end{proof}","ce every entry of $A(t)$ has zero constant term, the inverse exists in the formal power series ring and\n\\[\n(I-A(t))^{-1}=\\sum_{r\\geq0}A(t)^r.\n\\]\nThe $(i,j)$ entry of $A(t)^r$ is the sum of the monomials attached to all paths of length $r$ from $i$ to $j$.  Summing over $r$ records paths of every length.  Multiplication by the endpoint vectors gives the stated series.  The adjugate formula gives\n\\[\n(I-A(t))^{-1}\n=\n\\frac{\\operatorname{adj}(I-A(t))}{\\det(I-A(t))},\n\\]\nso every entry is rational.\n\\end{proof}","\\begin{corollary}[Recurrence for path counts]\\label{cor:path_recurrence}\nWrite\n\\[\nZ_{u,v}(t)=\\frac{P(t)}{Q(t)},\n\\qquad\nQ(t)=1+q_1t+\\cdots+q_dt^d.\n\\]\nUse the convention $\\deg 0=-\\infty$. Then, for every nonnegative integer $n>\\deg P$,\n\\[\na_n+q_1a_{n-1}+\\cdots+q_da_{n-d}=0.\n\\]\n\\end{corollary}","\\begin{proof}\nCompare coefficients in the identity\n\\[\nQ(t)Z_{u,v}(t)=P(t).\n\\]\nBeyond the degree of $P$, the coefficient on the right is zero.\n\\end{proof}","In the recurrence, set $a_k=0$ when $k<0$. This is the coefficient convention for the formal power series $Z_{u,v}(t)$ and makes the first recurrence equations meaningful even when $n<d$.","\\begin{corollary}[Several independent weights]\\label{cor:multigraded_series}\nSuppose every edge has a weight vector\n\\[\nw(e)=(w_1(e),\\ldots,w_r(e))\\in\\mathbb N^r\\setminus\\{0\\}.\n\\]\nReplace $t^{w(e)}$ by\n\\[\nt_1^{w_1(e)}\\cdots t_r^{w_r(e)}.\n\\]\nThe resulting multigraded transfer series is rational in $t_1,\\ldots,t_r$.\n\\end{corollary}","\\begin{example}[A transfer series with one loop]\\label{ex:simple_transfer_series}\nTake vertices $0$, $1$, and $2$.  Give the edge $0\\to1$ weight one, the loop $1\\to1$ weight two, and the edge $1\\to2$ weight one.  Every path from $0$ to $2$ uses the first and last edges and may traverse the loop any number of times.  Hence\n\\[\nZ_{0,2}(t)\n=\nt^2(1+t^2+t^4+\\cdots)\n=\n\\frac{t^2}{1-t^2}.\n\\]\nThere is one path at every positive even weight and no path at an odd weight.  The denominator records the increase of two units produced by the loop.\n\\end{example}","Let $G$ be a finite directed graph with positive integer edge weights. Define the polynomial adjacency matrix\n\\[\nA(t)_{ij}\n=\n\\sum_{e:i\\to j}t^{w(e)},\n\\]\nwhere parallel edges are distinct elements of the finite edge type and are\ntherefore counted with their full multiplicity.","For fixed start and target vertices $s,t$, define\n\\[\nZ_{s,t}(t)\n=\n\\sum_{n\\ge0}a_n(s,t)t^n,\n\\]\nwhere $a_n(s,t)$ counts directed walks from $s$ to $t$ with total weight $n$.\nVertices and edges may repeat. Thus ``path zeta'' is retained as the name of\nthe series and Lean API, but the walks are not required to visit each vertex only once.","\\begin{theorem}[Automatic finite-graph transfer-zeta rationality]\\label{thm:transfer-zeta}\nAssume the vertex and edge sets are finite and every edge has positive integer\nweight. For every $n$, exact enumeration of edge tuples of lengths at most\n$n$, followed by endpoint and total-weight filtering, returns $a_n(s,t)$.\nThese coefficients obey the recurrence obtained by removing the first edge\n\\[\na_n(s,t)\n=\n\\mathbf 1_{s=t,\\,n=0}\n+\\sum_{\\substack{e:\\,\\operatorname{source}(e)=s","\\begin{theorem}[Automatic finite-graph transfer-zeta rationality]\\label{thm:transfer-zeta}\nAssume the vertex and edge sets are finite and every edge has positive integer\nweight. For every $n$, exact enumeration of edge tuples of lengths at most\n$n$, followed by endpoint and total-weight filtering, returns $a_n(s,t)$.\nThese coefficients obey the recurrence obtained by removing the first edge\n\\[\na_n(s,t)\n=\n\\mathbf 1_{s=t,\\,n=0}\n+\\sum_{\\substack{e:\\,\\operatorname{source}(e)=s\\\\w(e)\\le n}}\na_{n-w(e)}(\\operatorname{target}(e),t).\n\\]\nLet\n\\[\nT(t)=I-A(t),\\qquad\nQ(t)=\\det T(t),\\qquad\nP_{s,t}(t)=\\operatorname{adj}(T(t))_{s,t}.\n\\]\nThen $Q(0)=1$, hence $Q\\ne0$ and $Q$ is a unit in $\\Z[[t]]$, and the\nindependently enumerated walk series satisfies\n\\[\n\\boxed{\nZ_{s,t}(t)=Q(t)^{-1}P_{s,t}(t),\n\\qquad\nQ(t)Z_{s,t}(t)=P_{s,t}(t)\n}\n\\quad\\text{in }\\Z[[t]].\n\\]\n\\end{theorem}","w(e)\\le n}}\na_{n-w(e)}(\\operatorname{target}(e),t).\n\\]\nLet\n\\[\nT(t)=I-A(t),\\qquad\nQ(t)=\\det T(t),\\qquad\nP_{s,t}(t)=\\operatorname{adj}(T(t))_{s,t}.\n\\]\nThen $Q(0)=1$, hence $Q\\ne0$ and $Q$ is a unit in $\\Z[[t]]$, and the\nindependently enumerated walk series satisfies\n\\[\n\\boxed{\nZ_{s,t}(t)=Q(t)^{-1}P_{s,t}(t),\n\\qquad\nQ(t)Z_{s,t}(t)=P_{s,t}(t)\n}\n\\quad\\text{in }\\Z[[t]].\n\\]\n\\end{theorem}","\\begin{proof}\nEvery edge has weight at least one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that con","\\begin{proof}\nEvery edge has weight at least one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the inde","\\begin{proof}\nEvery edge has weight at least one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","t one, so a word of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ord of weight $n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","$n$ has length at\nmost $n$. The enumerator therefore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","refore searches a finite set containing every\ncontributing walk, and its filters are exactly the defining predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ning predicates for\n$a_n$. Removing the first edge gives the displayed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ed disjoint decomposition\ninto the empty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","ty diagonal walk and the first-edge tails. In matrix form this is\n\\[\nZ=I+AZ,\n\\qquad (I-A)Z=I.\n\\]\nPositive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","tive weights give $A(0)=0$, so $T(0)=I$ and\n$Q(0)=\\det(I)=1$. The two adjugate identities\n\\[\nT\\operatorname{adj}(T)=Q I,\n\\qquad\n\\operatorname{adj}(T)T=Q I\n\\]\nremain valid after inclusion into $\\Z[[t]]$. Multiplying the adjugate by the\nformal-series inverse of $Q$ constructs a two-sided inverse of $T$.\nUniqueness of the inverse identifies that construction with the independently\nenumerated matrix $Z$, yielding the entrywise formulas.\n\\end{proof}","This is the finite transfer-matrix method of \\cite{Stanley2012}, but here its\ndeterminant, adjugate, normalization, recurrence from the first edge, and identification\nwith the enumerated series are all formalized. The older API accepting an\nindependently supplied normalized coefficient identity $QZ=P$ is retained for\nexternal certificates. It is no longer needed to obtain rationality for the\nfinite graph defined above. The Python companion performs an exact symbolic\nconstruction as an independent executable check.","\\begin{corollary}[Linear recurrence]\\label{cor:zeta-recurrence}\nWith the canonical polynomials from \\cref{thm:transfer-zeta}, write\n\\[\nQ(t)=\\sum_kq_kt^k,\n\\qquad\nZ_{s,t}(t)=\\sum_{n\\ge0}a_n(s,t)t^n.\n\\]\nThen the walk counts satisfy the recurrence with constant coefficients determined by\n$Q$ after the finite initial range contributed by $P_{s,t}$, with $\\deg 0=-\\infty$:\n\\[\n\\boxed{\n\\sum_{k=0}^{n}q_k a_{n-k}(s,t)=0\n\\qquad\\bigl(n>\\deg P_{s,t}\\bigr).\n}\n\\]\n\\end{corollary}","\\begin{example}[A computed transfer zeta]\nFor the graph used in the numerical companion, the walk series from packet $0$ to packet $2$ is\n\\[\nZ_{0,2}(t)\n=\n\\frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.\n\\]\nThe first nonzero coefficients are\n\\[\n1,1,1,3,1,5,4,10,7,19,17,37,\\ldots\n\\]\nat their corresponding weights. Direct enumeration through weight $30$ agrees coefficient by coefficient with the rational expansion.\n\\end{example}","Let $\\OO$ be a discrete valuation ring with fraction field $K$ and uniformizer $\\pi$.  Let $\\Lambda$ be a finite free $\\OO$ module and put\n\\[\nV=K\\otimes_{\\OO}\\Lambda.\n\\]\nSuppose $A\\in\\End_{\\OO}(\\Lambda)$.","\\begin{theorem}[Bézout spectral projector]\\label{thm:bezout_projector}\nAssume that\n\\[\n\\chi_A(x)=f(x)g(x)\n\\]\nwith monic polynomials $f,g\\in\\OO[x]$ that are coprime in $K[x]$.  Choose $u,v\\in\\OO[x]$ and a nonzero $r\\in\\OO$ such that\n\\[\nu(x)f(x)+v(x)g(x)=r.\n\\]\nThen\n\\[\nE_f=r^{-1}v(A)g(A)\n\\]\nis the projector of $V$ onto $\\Ker f(A)$ along $\\Ker g(A)$.  In particular,\n\\[\nE_f^2=E_f,\n\\qquad\nrE_f\\in\\End_{\\OO}(\\Lambda).\n\\]\n\\end{theorem}","\\begin{proof}\nCayley Hamilton gives\n\\[\nf(A)g(A)=0.\n\\]\nIf $x\\in\\Ker f(A)$, then the Bézout identity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","(A)=0.\n\\]\nIf $x\\in\\Ker f(A)$, then the Bézout identity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","$, then the Bézout identity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","tity gives\n\\[\nv(A)g(A)x=rx,\n\\]\nso $E_fx=x$.  If $x\\in\\Ker g(A)$, then $E_fx=0$.  The same identity gives the direct sum decomposition\n\\[\nV=\\Ker f(A)\\oplus\\Ker g(A).\n\\]\nIndeed, every $x\\in V$ is the sum of\n\\[\nr^{-1}v(A)g(A)x\\in\\Ker f(A)\n\\]\nand\n\\[\nr^{-1}u(A)f(A)x\\in\\Ker g(A).\n\\]\nThus $E_f$ is the stated projector and is idempotent.  The formula for $rE_f$ has coefficients in $\\OO$, so it preserves $\\Lambda$.\n\\end{proof}","\\begin{theorem}[Several coprime spectral packets]\\label{thm:several_projectors}\nAssume\n\\[\n\\chi_A(x)=\\prod_{i=1}^rf_i(x),\n\\]\nwhere the monic factors $f_i\\in\\OO[x]$ are pairwise coprime in $K[x]$.  Put\n\\[\ng_i=\\prod_{j\\ne i}f_j.\n\\]\nThere are complementary projectors $E_i$ with\n\\[\n\\im E_i=\\Ker f_i(A),\n\\qquad\n\\sum_iE_i=I,\n\\qquad\nE_iE_j=0\n\\quad(i\\ne j).\n\\]\nOne may choose $E_i$ so that\n\\[\n\\Res(f_i,g_i)E_i\\in\\End_{\\OO}(\\Lambda)\n\\]\nup to multiplication of the resultant by a unit.\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:bezout_projector} to the factorization $\\chi_A=f_ig_i$.  The resulting projector is one on $\\Ker f_i(A)$ and zero on every $\\Ker f_j(A)$ with $j\\ne i$.  The direct sum decomposition follows from the polynomial Chinese remainder theorem.  The orthogonality and sum identities follow by evaluating the corresponding idempotents in $K[x]/(\\chi_A)$.  The Sylvester Bézout relation for $(f_i,g_i)$ gives the stated denominator bound.\n\\end{proof}","}\nApply \\cref{thm:bezout_projector} to the factorization $\\chi_A=f_ig_i$.  The resulting projector is one on $\\Ker f_i(A)$ and zero on every $\\Ker f_j(A)$ with $j\\ne i$.  The direct sum decomposition follows from the polynomial Chinese remainder theorem.  The orthogonality and sum identities follow by evaluating the corresponding idempotents in $K[x]/(\\chi_A)$.  The Sylvester Bézout relation for $(f_i,g_i)$ gives the stated denominator bound.\n\\end{proof}","\\begin{corollary}[Resultant denominator bound]\\label{cor:resultant_bound}\nUnder the hypotheses of \\cref{thm:bezout_projector}, one may take\n\\[\nr=\\Res(f,g)\n\\]\nup to multiplication by a unit.  Therefore\n\\[\n\\pi^{\\val_\\pi(\\Res(f,g))}E_f\\Lambda\\subseteq\\Lambda.\n\\]\n\\end{corollary}","\\begin{example}[A resultant bound that is not exact]\\label{ex:loose_resultant}\nLet\n\\[\nA=\\diag(0,\\pi^s)\n\\]\non the standard lattice $\\Lambda=\\OO^2$.  The factors $x$ and $x-\\pi^s$ have resultant $-\\pi^s$, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors.  Their exact separation weight is zero.  The resultant sees the closeness of the two eigenvalues.  It does not see that the chosen lattice already splits along their eigenspaces.\n\\end{example}","Let $\\OO$ be a DVR, $K=\\Frac(\\OO)$, and $A\\in\\End_\\OO(\\Lambda)$ for a finite free lattice $\\Lambda$. Suppose the characteristic polynomial factors over $\\OO$ as\n\\[\n\\chi_A(x)=f(x)g(x),\n\\]\nwhere $f$ and $g$ are coprime over $K[x]$.","\\begin{theorem}[Evaluated B\\'ezout spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","begin{theorem}[Evaluated B\\'ezout spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","}[Evaluated B\\'ezout spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","out spectral projector]\\label{thm:bezout-projector}\nLet $F,G,U,V\\in\\End_K(\\Lambda\\otimes_\\OO K)$ and let $r\\in K^\\times$.\nSuppose\n\\[\nUF+VG=rI,\n\\qquad FG=0,\n\\qquad FG=GF,\n\\qquad GU=UG.\n\\]\nPut $N=VG$. Then\n\\[\n\\boxed{N^2=rN.}\n\\]\nConsequently the operator\n\\[\n\\boxed{\nE=r^{-1}VG\n}\n\\]\nis idempotent, acts as the identity on $\\ker F$, and acts as zero on\n$\\ker G$. If $V$ and $G$ preserve $\\Lambda$, then\n\\[\nrE=VG\\in\\End_\\OO(\\Lambda).\n\\]\n\\end{theorem}","\\begin{proof}\nThe B\\'ezout identity gives $N=rI-UF$. The two stated commutation relations\nand $FG=0$ give\n\\[\nN(UF)=VGUF=VUGF=VUFG=0.\n\\]\nTherefore\n\\[\nN^2=N(rI-UF)=rN.\n\\]\nSince $r\\ne0$, $E^2=r^{-2}N^2=E$. If $Fx=0$, the evaluated B\\'ezout\nidentity gives $VGx=rx$, hence $Ex=x$. If $Gx=0$, then $Ex=0$.\nThe lattice assertion follows directly from $rE=VG$.\n\\end{proof}","The equality $N^2=rN$ before division uses neither $r\\ne0$ nor the other\npairwise commutation relations that hold in the polynomial application.\nNormalizing to $E$ uses $r\\ne0$. Its action on $\\ker F$ uses only the B\\'ezout\nidentity and $r\\ne0$, while vanishing on $\\ker G$ follows immediately from $N=VG$.\nThe bundled Lean certificate records all pairwise commutations as convenient\nevaluated polynomial data, but the minimal theorems expose only the hypotheses\nthey consume.","For the polynomial application, one supplies $F=f(A)$, $G=g(A)$,\n$U=u(A)$, and $V=v(A)$. A B\\'ezout or resultant identity\n$uf+vg=r$ evaluates to the first certificate, while\n$(fg)(A)=0$ supplies the second. When $fg=\\chi_A$, this last equality is\nCayley Hamilton. Standard B\\'ezout, resultant, and DVR facts are recalled\nfrom \\cite{Lang2002,Serre1979}. The inherited certificate theorem begins with the evaluated\ncertificate. It does not infer exact action merely from nilpotence of\n$f(A)$ or $g(A)$, which would be insufficient.","A)$. A B\\'ezout or resultant identity\n$uf+vg=r$ evaluates to the first certificate, while\n$(fg)(A)=0$ supplies the second. When $fg=\\chi_A$, this last equality is\nCayley Hamilton. Standard B\\'ezout, resultant, and DVR facts are recalled\nfrom \\cite{Lang2002,Serre1979}. The inherited certificate theorem begins with the evaluated\ncertificate. It does not infer exact action merely from nilpotence of\n$f(A)$ or $g(A)$, which would be insufficient.","ezout or resultant identity\n$uf+vg=r$ evaluates to the first certificate, while\n$(fg)(A)=0$ supplies the second. When $fg=\\chi_A$, this last equality is\nCayley Hamilton. Standard B\\'ezout, resultant, and DVR facts are recalled\nfrom \\cite{Lang2002,Serre1979}. The inherited certificate theorem begins with the evaluated\ncertificate. It does not infer exact action merely from nilpotence of\n$f(A)$ or $g(A)$, which would be insufficient.","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothes","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nt","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is ob","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\","\\begin{definition}[Separation weight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","ight]\\label{def:separation-weight}\nFor a rational projector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","rojector $E$ and a specified $\\OO$ lattice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","ice $\\Lambda$, provided\nat least one clearing exponent exists, define\n\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","\\[\ns_{\\pi,\\Lambda}(E)\n=\n\\min\\{s\\ge0:\\pi^sE(\\Lambda)\\subseteq\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","\\Lambda\\}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","}.\n\\]\nIf the evaluated certificate arises from integral $v,g$ and\n$r=u\\pi^s$ with $u^{-1}$ preserving the lattice, then\n\\cref{thm:bezout-projector} gives the upper bound\n\\[\ns_{\\pi,\\Lambda}(E)\\le s=\\val_\\pi(r).\n\\]\nIndeed,\n\\[\n\\boxed{\\pi^sE=u^{-1}VG}\n\\]\npreserves $\\Lambda$. The minimal Lean theorem needs only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","only the lattice action of\n$u^{-1}$. The bundled wrapper for lattice units records both $u$ and $u^{-1}$.\nA nonzero scalar in the fraction field need not be an integral unit.\nExistence of a clearing exponent is an explicit hypothesis. Once one exists,\nthe least one is obtained from the least element property of the natural numbers.\nClearing exponents are upward closed provided multiplication by $\\pi$\npreserves the lattice.\n\\end{definition}","\\begin{theorem}[Projector precision delay]\\label{thm:precision-delay}\nLet $E$ be a rational projector with $\\pi^sE(\\Lambda)\\subseteq\\Lambda$. If\n\\[\nx\\equiv y\\pmod{\\pi^{m+s}\\Lambda},\n\\]\nthen\n\\[\nEx\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $m+s$ input digits suffice to determine $m$ output digits of the projected packet.\n\\end{theorem}","\\begin{proof}\nWrite $x-y=\\pi^{m+s}z$. Since $\\pi^sE(z)\\in\\Lambda$,\n\\[\nE(x-y)=\\pi^m\\bigl(\\pi^sE(z)\\bigr)\\in\\pi^m\\Lambda.\n\\]\n\\end{proof}","The theorem gives a precision budget for delayed separation. A packet with weight $s$ can be reconstructed at the requested output precision by retaining $s$ additional input digits.","\\Needspace{8\\baselineskip}\n\\begin{definition}[Lattice packet data]\\label{def:lattice_packet}\nLet\n\\[\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha\n\\]\nbe a finite direct sum over $K$, with packet projectors $E_\\alpha$.  Let $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","ttice_packet}\nLet\n\\[\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha\n\\]\nbe a finite direct sum over $K$, with packet projectors $E_\\alpha$.  Let $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","et\n\\[\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha\n\\]\nbe a finite direct sum over $K$, with packet projectors $E_\\alpha$.  Let $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","et $\\Lambda\\subseteq V$ be an $\\OO$ lattice.  Put\n\\[\n\\Lambda_\\alpha=\\Lambda\\cap V_\\alpha,\n\\qquad\n\\Lambda_{\\mathrm{split}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","it}}=\\bigoplus_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","s_\\alpha\\Lambda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","mbda_\\alpha,\n\\]\nand define the gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","e gluing module\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","e\n\\[\nG_{\\Lambda,\\Pcal}\n=\n\\Lambda/\\Lambda_{\\mathrm{split}}.\n\\]\nThe separation weight of $E_\\alpha$ is\n\\[\ns_\\Lambda(E_\\alpha)\n=\n\\min\\{s\\geq0:\\pi^sE_\\alpha\\Lambda\\subseteq\\Lambda\\}.\n\\]\nThe global separation weight is\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{s_\\Lambda(E_\\alpha):\\alpha\\in I\\}\\bigr).\n\\]\nFor a finitely generated torsion $\\OO$ module $G$, write $\\expn_\\pi(G)$ for the least $s$ such that $\\pi^sG=0$.\n\\end{definition}","The gluing module $G_{\\Lambda,\\Pcal}$ is a finitely generated torsion $\\OO$ module, hence has finite length, because $\\Lambda_{\\mathrm{split}}$ and $\\Lambda$ are full lattices in the same vector space. To see that the split lattice is full, clear denominators of bases of the rational summands. Each $\\Lambda\\cap V_\\alpha$ then spans $V_\\alpha$, so their direct sum spans $V$.","\\begin{theorem}[Gluing module and exact separation weight]\\label{thm:gluing_weight}\nWith the notation of \\cref{def:lattice_packet}, the map\n\\[\n\\Phi:G_{\\Lambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","ght]\\label{thm:gluing_weight}\nWith the notation of \\cref{def:lattice_packet}, the map\n\\[\n\\Phi:G_{\\Lambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","_packet}, the map\n\\[\n\\Phi:G_{\\Lambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","ambda,\\Pcal}\n\\longrightarrow\n\\bigoplus_{\\alpha\\in I}E_\\alpha\\Lambda/\\Lambda_\\alpha,\n\\qquad\n[x]\\longmapsto(E_\\alpha x+\\Lambda_\\alpha)_\\alpha\n\\]\nis injective, and every coordinate projection of its image is surjective.  We also have\n\\[\n\\boxed{\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\n=\ns_\\Lambda(\\Pcal).\n}\n\\]\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO\n\\]\nwith $a_1\\leq\\cdots\\leq a_r$, then\n\\[\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\n\\end{theorem}","\\begin{proof}\nIf $x\\in\\Lambda$ maps to zero, then $E_\\alpha x\\in\\Lambda_\\alpha$ for every $\\alpha$.  Since\n\\[\nx=\\sum_\\alpha E_\\alpha x,\n\\]\nwe have $x\\in\\Lambda_{\\mathrm{split}}$.  Thus $\\Phi$ is injective.  The coordinate map onto $E_\\alpha\\Lambda/\\Lambda_\\alpha$ is surjective by the definition of $E_\\alpha\\Lambda$.","Let $s=s_\\Lambda(\\Pcal)$.  For every $x\\in\\Lambda$ and every $\\alpha$,\n\\[\n\\pi^sE_\\alpha x\\in\\Lambda_\\alpha.\n\\]\nHence\n\\[\n\\pi^sx=\\sum_\\alpha\\pi^sE_\\alpha x\\in\\Lambda_{\\mathrm{split}},\n\\]\nso $\\pi^sG_{\\Lambda,\\Pcal}=0$.  This proves\n\\[\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\\leq s.\n\\]\nIf $d<s$, then for some $\\alpha$ there is an $x\\in\\Lambda$ with\n\\[\n\\pi^dE_\\alpha x\\notin\\Lambda_\\alpha.\n\\]\nThe class $\\pi^d[x]$ is therefore nonzero in the gluing module.  Thus $\\pi^d$ does not annihilate the gluing module, and equality follows.  The final formula is the usual description of the exponent through Smith normal form.\n\\end{proof}","$ and every $\\alpha$,\n\\[\n\\pi^sE_\\alpha x\\in\\Lambda_\\alpha.\n\\]\nHence\n\\[\n\\pi^sx=\\sum_\\alpha\\pi^sE_\\alpha x\\in\\Lambda_{\\mathrm{split}},\n\\]\nso $\\pi^sG_{\\Lambda,\\Pcal}=0$.  This proves\n\\[\n\\expn_\\pi(G_{\\Lambda,\\Pcal})\\leq s.\n\\]\nIf $d<s$, then for some $\\alpha$ there is an $x\\in\\Lambda$ with\n\\[\n\\pi^dE_\\alpha x\\notin\\Lambda_\\alpha.\n\\]\nThe class $\\pi^d[x]$ is therefore nonzero in the gluing module.  Thus $\\pi^d$ does not annihilate the gluing module, and equality follows.  The final formula is the usual description of the exponent through Smith normal form.\n\\end{proof}","\\begin{corollary}[Total gluing and worst precision]\\label{cor:length_and_weight}\nIf\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^r\\OO/\\pi^{a_j}\\OO,\n\\]\nthen\n\\[\n\\length_{\\OO}(G_{\\Lambda,\\Pcal})\n=\n\\sum_{j=1}^ra_j,\n\\qquad\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq r\\}\\bigr).\n\\]\nThus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.\n\\end{corollary}","\\begin{corollary}[Two complementary packets]\\label{cor:binary_gluing}\nSuppose\n\\[\nV=V_1\\oplus V_2\n\\]\nwith projector $E$ onto $V_1$ and complementary projector $I-E$.  Then\n\\[\n\\Lambda/(\\Lambda_1\\oplus\\Lambda_2)\n\\cong\nE\\Lambda/\\Lambda_1\n\\cong\n(I-E)\\Lambda/\\Lambda_2.\n\\]\nConsequently\n\\[\ns_\\Lambda(E)=s_\\Lambda(I-E)\n=\n\\expn_\\pi\\bigl(\\Lambda/(\\Lambda_1\\oplus\\Lambda_2)\\bigr).\n\\]\n\\end{corollary}","\\begin{proof}\nMap $x\\in\\Lambda$ to $Ex+\\Lambda_1$.  Its kernel consists of the $x$ for which $Ex\\in\\Lambda_1$.  In that case\n\\[\n(I-E)x=x-Ex\\in\\Lambda\\cap V_2=\\Lambda_2,\n\\]\nso the kernel is $\\Lambda_1\\oplus\\Lambda_2$.  The map is surjective by definition.  The complementary map is handled in the same way.  The equality of weights follows from \\cref{thm:gluing_weight}.\n\\end{proof}","\\begin{corollary}[Integral splitting criterion]\\label{cor:integral_split}\nEvery packet projector preserves $\\Lambda$ if and only if\n\\[\n\\Lambda=\\bigoplus_\\alpha(\\Lambda\\cap V_\\alpha).\n\\]\nEquivalently,\n\\[\ns_\\Lambda(\\Pcal)=0\n\\quad\\Longleftrightarrow\\quad\nG_{\\Lambda,\\Pcal}=0.\n\\]\n\\end{corollary}","\\begin{proof}\nIf every projector preserves $\\Lambda$, then\n\\[\nx=\\sum_\\alpha E_\\alpha x\n\\]\nexpresses every $x\\in\\Lambda$ as a sum of vectors in $\\Lambda_\\alpha$.  The converse is immediate because each projector is the identity on one direct summand and zero on the others.  The final equivalence follows from \\cref{thm:gluing_weight}.\n\\end{proof}","\\begin{theorem}[Gluing exact sequence under refinement]\\label{thm:gluing_refinement}\nLet $\\Qcal=(Q_\\beta)$ refine $\\Pcal=(P_\\alpha)$ on $V$, and let $\\Lambda$ be a lattice.  Define\n\\[\n\\Lambda_{\\Pcal}\n=\n\\bigoplus_\\alpha(\\Lambda\\cap P_\\alpha V),\n\\qquad\n\\Lambda_{\\Qcal}\n=\n\\bigoplus_\\beta(\\Lambda\\cap Q_\\beta V).\n\\]\nThen\n\\[\n\\Lambda_{\\Qcal}\\subseteq\\Lambda_{\\Pcal}\\subseteq\\Lambda\n\\]\nand there is a short exact sequence\n\\[\n0\n\\longrightarrow\n\\Lambda_{\\Pcal}/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Pcal}\n\\longrightarrow0.\n\\]\nIn particular,\n\\[\n\\boxed{\ns_\\Lambda(\\Pcal)\\leq s_\\Lambda(\\Qcal).\n}\n\\]\n\\end{theorem}","luing exact sequence under refinement]\\label{thm:gluing_refinement}\nLet $\\Qcal=(Q_\\beta)$ refine $\\Pcal=(P_\\alpha)$ on $V$, and let $\\Lambda$ be a lattice.  Define\n\\[\n\\Lambda_{\\Pcal}\n=\n\\bigoplus_\\alpha(\\Lambda\\cap P_\\alpha V),\n\\qquad\n\\Lambda_{\\Qcal}\n=\n\\bigoplus_\\beta(\\Lambda\\cap Q_\\beta V).\n\\]\nThen\n\\[\n\\Lambda_{\\Qcal}\\subseteq\\Lambda_{\\Pcal}\\subseteq\\Lambda\n\\]\nand there is a short exact sequence\n\\[\n0\n\\longrightarrow\n\\Lambda_{\\Pcal}/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Pcal}\n\\longrightarrow0.\n\\]\nIn particular,\n\\[\n\\boxed{\ns_\\Lambda(\\Pcal)\\leq s_\\Lambda(\\Qcal).\n}\n\\]\n\\end{theorem}","ambda_{\\Pcal}\n=\n\\bigoplus_\\alpha(\\Lambda\\cap P_\\alpha V),\n\\qquad\n\\Lambda_{\\Qcal}\n=\n\\bigoplus_\\beta(\\Lambda\\cap Q_\\beta V).\n\\]\nThen\n\\[\n\\Lambda_{\\Qcal}\\subseteq\\Lambda_{\\Pcal}\\subseteq\\Lambda\n\\]\nand there is a short exact sequence\n\\[\n0\n\\longrightarrow\n\\Lambda_{\\Pcal}/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Qcal}\n\\longrightarrow\n\\Lambda/\\Lambda_{\\Pcal}\n\\longrightarrow0.\n\\]\nIn particular,\n\\[\n\\boxed{\ns_\\Lambda(\\Pcal)\\leq s_\\Lambda(\\Qcal).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nEvery fine packet space lies in its parent packet space, so every element of $\\Lambda_{\\Qcal}$ belongs to $\\Lambda_{\\Pcal}$.  The displayed exact sequence is the standard quotient sequence for nested submodules.  The exponent of a quotient cannot exceed the exponent of the module from which it is obtained.  Apply \\cref{thm:gluing_weight} to the two gluing modules.\n\\end{proof}","\\begin{example}[Strict growth under refinement]\\label{ex:strict_weight_growth}\nTake the standard lattice $\\mathbb Z_2^2$ and the involution from \\cref{prop:sign_projectors}.  The one packet system $(I)$ has weight zero.  Refining it into the two rational sign packets produces weight one.  The coarse lattice is already split because there is only one summand.  The fine eigenspaces are glued together by one dyadic digit.\n\\end{example}","% Keep the final proof marker with its displayed conclusion.\n\\enlargethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","Keep the final proof marker with its displayed conclusion.\n\\enlargethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","splayed conclusion.\n\\enlargethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","largethispage{\\baselineskip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","skip}\n\\Needspace{8\\baselineskip}\n\\begin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","gin{theorem}[Necessary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","ary and sufficient precision delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","n delay]\\label{thm:precision_delay}\nLet $E$ be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","be a rational projector on $V$, let $\\Lambda$ be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","be a lattice, and let\n\\[\ns=s_\\Lambda(E).\n\\]\nFor every $m\\geq0$, the rule\n\\[\n[x]_{m+s}\\longmapsto[Ex]_m\n\\]\nfrom $\\Lambda/\\pi^{m+s}\\Lambda$ to $V/\\pi^m\\Lambda$ is well defined.  Here $m$ is an integer.  If $d$ is an integer with $0\\leq d<s$, then for every integer $m\\geq0$ there are $x,y\\in\\Lambda$ such that\n\\[\nx\\equiv y\\pmod{\\pi^{m+d}\\Lambda}\n\\]\nbut\n\\[\nEx\\not\\equiv Ey\\pmod{\\pi^m\\Lambda}.\n\\]\nThus $s$ extra digits are both sufficient and necessary for a uniform precision guarantee.\n\\end{theorem}","\\begin{proof}\nIf\n\\[\nx-y=\\pi^{m+s}z,\n\\]\nthen\n\\[\nE(x-y)=\\pi^m(\\pi^sEz)\\in\\pi^m\\Lambda.\n\\]\nThis proves that the rule is well defined.  Now take $d<s$.  By the minimality of $s$, there is a $z\\in\\Lambda$ with\n\\[\n\\pi^dEz\\notin\\Lambda.\n\\]\nSet\n\\[\nx=\\pi^{m+d}z,\n\\qquad\ny=0.\n\\]\nThen $x\\equiv y$ modulo $\\pi^{m+d}\\Lambda$, while\n\\[\nE(x-y)=\\pi^m(\\pi^dEz)\\notin\\pi^m\\Lambda.\n\\]\n\\end{proof}","\\begin{example}[The same rational split with two lattices]\\label{ex:lattice_dependence}\nLet\n\\[\nV=Ke_1\\oplus Ke_2\n\\]\nand let $E$ project onto $Ke_1$.  On the split lattice\n\\[\n\\Lambda_0=\\OO e_1\\oplus\\OO e_2,\n\\]\nthe weight is zero.  Fix $s\\geq1$ and consider\n\\[\n\\Lambda_s\n=\n\\OO e_1+\n\\OO(\\pi^{-s}e_1+e_2).\n\\]\nThen\n\\[\n\\Lambda_s\\cap Ke_1=\\OO e_1,\n\\qquad\n\\Lambda_s\\cap Ke_2=\\pi^s\\OO e_2.\n\\]\nThe gluing module is isomorphic to $\\OO/\\pi^s\\OO$, so\n\\[\ns_{\\Lambda_s}(E)=s.\n\\]\nThe rational decomposition has not changed.  Only the lattice has changed.\n\\end{example}","same rational split with two lattices]\\label{ex:lattice_dependence}\nLet\n\\[\nV=Ke_1\\oplus Ke_2\n\\]\nand let $E$ project onto $Ke_1$.  On the split lattice\n\\[\n\\Lambda_0=\\OO e_1\\oplus\\OO e_2,\n\\]\nthe weight is zero.  Fix $s\\geq1$ and consider\n\\[\n\\Lambda_s\n=\n\\OO e_1+\n\\OO(\\pi^{-s}e_1+e_2).\n\\]\nThen\n\\[\n\\Lambda_s\\cap Ke_1=\\OO e_1,\n\\qquad\n\\Lambda_s\\cap Ke_2=\\pi^s\\OO e_2.\n\\]\nThe gluing module is isomorphic to $\\OO/\\pi^s\\OO$, so\n\\[\ns_{\\Lambda_s}(E)=s.\n\\]\nThe rational decomposition has not changed.  Only the lattice has changed.\n\\end{example}","with two lattices]\\label{ex:lattice_dependence}\nLet\n\\[\nV=Ke_1\\oplus Ke_2\n\\]\nand let $E$ project onto $Ke_1$.  On the split lattice\n\\[\n\\Lambda_0=\\OO e_1\\oplus\\OO e_2,\n\\]\nthe weight is zero.  Fix $s\\geq1$ and consider\n\\[\n\\Lambda_s\n=\n\\OO e_1+\n\\OO(\\pi^{-s}e_1+e_2).\n\\]\nThen\n\\[\n\\Lambda_s\\cap Ke_1=\\OO e_1,\n\\qquad\n\\Lambda_s\\cap Ke_2=\\pi^s\\OO e_2.\n\\]\nThe gluing module is isomorphic to $\\OO/\\pi^s\\OO$, so\n\\[\ns_{\\Lambda_s}(E)=s.\n\\]\nThe rational decomposition has not changed.  Only the lattice has changed.\n\\end{example}","The example can be realized by an integral operator.  Define\n\\[\nAe_1=0,\n\\qquad\nAe_2=\\pi^se_2.\n\\]\nIn the basis\n\\[\ne_1,\n\\quad\n\\pi^{-s}e_1+e_2,\n\\]\nthe matrix of $A$ is\n\\[\n\\begin{pmatrix}\n0&-1","The example can be realized by an integral operator.  Define\n\\[\nAe_1=0,\n\\qquad\nAe_2=\\pi^se_2.\n\\]\nIn the basis\n\\[\ne_1,\n\\quad\n\\pi^{-s}e_1+e_2,\n\\]\nthe matrix of $A$ is\n\\[\n\\begin{pmatrix}\n0&-1\\\\\n0&\\pi^s\n\\end{pmatrix}.\n\\]\nIts eigenvalue factors have resultant equal to $\\pi^s$ up to a unit, and the exact projector weight on $\\Lambda_s$ is also $s$.","0&\\pi^s\n\\end{pmatrix}.\n\\]\nIts eigenvalue factors have resultant equal to $\\pi^s$ up to a unit, and the exact projector weight on $\\Lambda_s$ is also $s$.","\\begin{proposition}[Sign projectors over a lattice]\\label{prop:sign_projectors}\nLet\n\\[\nA=\n\\begin{pmatrix}\n1&1\\\\\n0&-1\n\\end{pmatrix}\n\\]\nact on $\\mathbb Z_p^2$.  Over $\\mathbb Q_p$, the sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2","0&-1\n\\end{pmatrix}\n\\]\nact on $\\mathbb Z_p^2$.  Over $\\mathbb Q_p$, the sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2","0&-1\n\\end{pmatrix}\n\\]\nact on $\\mathbb Z_p^2$.  Over $\\mathbb Q_p$, the sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2\\\\\n0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2","0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2\\\\\n0&1\n\\end{pmatrix}.\n\\]\nFor every odd prime $p$, both projectors preserve $\\mathbb Z_p^2$.  At $p=2$,\n\\[\ns_2(E_+)=s_2(E_-)=1.\n\\]\nThe factors $x-1$ and $x+1$ have resultant $2$.\n\\end{proposition}","0&1\n\\end{pmatrix}.\n\\]\nFor every odd prime $p$, both projectors preserve $\\mathbb Z_p^2$.  At $p=2$,\n\\[\ns_2(E_+)=s_2(E_-)=1.\n\\]\nThe factors $x-1$ and $x+1$ have resultant $2$.\n\\end{proposition}","\\begin{proof}\nThe displayed matrices are complementary idempotents.  The scalar $2$ is a unit in $\\mathbb Z_p$ for odd $p$, so both matrices are integral there.  Over $\\mathbb Z_2$, multiplying by $2$ clears the denominators, while neither projector preserves the standard lattice.  Thus both weights are one.  The resultant of the two linear factors is their root difference, which is $2$.\n\\end{proof}","Modulo $2$, the two eigenvalues become equal.  The rational sign packets remain distinct over $\\mathbb Q_2$, but the standard lattice does not split into their intersections until one more dyadic digit is retained.","\\Needspace{4\\baselineskip}\n\\begin{proposition}[Integral character averages away from the group order]\\label{prop:character_integrality}\nLet $\\OO$ be a discrete valuation ring, let $N$ be a unit in $\\OO$, and suppose $\\OO$ contains a primitive $N$th root of unity $\\omega$.  If an integral operator $U$ satisfies $U^N=I$, then\n\\[\nE_j=\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k\n\\]\npreserves every $U$ stable $\\OO$ lattice.  At primes dividing $N$, this formula alone gives no integral projector.\n\\end{proposition}","edspace{4\\baselineskip}\n\\begin{proposition}[Integral character averages away from the group order]\\label{prop:character_integrality}\nLet $\\OO$ be a discrete valuation ring, let $N$ be a unit in $\\OO$, and suppose $\\OO$ contains a primitive $N$th root of unity $\\omega$.  If an integral operator $U$ satisfies $U^N=I$, then\n\\[\nE_j=\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k\n\\]\npreserves every $U$ stable $\\OO$ lattice.  At primes dividing $N$, this formula alone gives no integral projector.\n\\end{proposition}","osition}[Integral character averages away from the group order]\\label{prop:character_integrality}\nLet $\\OO$ be a discrete valuation ring, let $N$ be a unit in $\\OO$, and suppose $\\OO$ contains a primitive $N$th root of unity $\\omega$.  If an integral operator $U$ satisfies $U^N=I$, then\n\\[\nE_j=\\frac1N\\sum_{k=0}^{N-1}\\omega^{-jk}U^k\n\\]\npreserves every $U$ stable $\\OO$ lattice.  At primes dividing $N$, this formula alone gives no integral projector.\n\\end{proposition}","\\begin{proof}\nEvery scalar coefficient in the displayed sum belongs to $\\OO$ because $N$ is a unit and $\\omega$ is integral.  Each power of $U$ preserves the lattice, so their $\\OO$ linear combination does as well.  When $N$ is not a unit, division by $N$ is not an operation in $\\OO$, so the same argument is unavailable.\n\\end{proof}","At a prime dividing $N$, the factor $1/N$ no longer guarantees that the character average preserves the lattice. Cancellation may still make a particular projector integral. Coefficient residue selectors remain exact because they never divide by the group order.","Consider the integral involution\n\\[\nA=\n\\begin{pmatrix}\n1&1\\\\\n0&-1\n\\end{pmatrix},\n\\qquad\nA^2=I.\n\\]\nOver a field of characteristic different from two, its sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2","0&-1\n\\end{pmatrix},\n\\qquad\nA^2=I.\n\\]\nOver a field of characteristic different from two, its sign projectors are\n\\[\nE_+=\\frac{I+A}{2}\n=\n\\begin{pmatrix}\n1&1/2\\\\\n0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2","0&0\n\\end{pmatrix},\n\\]\n\\[\nE_-=\\frac{I-A}{2}\n=\n\\begin{pmatrix}\n0&-1/2\\\\\n0&1\n\\end{pmatrix}.\n\\]\nThey are orthogonal idempotents and sum to $I$.","0&1\n\\end{pmatrix}.\n\\]\nThey are orthogonal idempotents and sum to $I$.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{07_dyadic_delayed_separation.pdf}\n\\caption{The rational sign projectors for an integral involution. Their denominator has $2$-adic weight one.}\n\\label{fig:dyadic-projectors}\n\\end{figure}","\\begin{proposition}[Odd-adic versus dyadic behavior]\\label{prop:dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, exactly the two local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","sus dyadic behavior]\\label{prop:dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, exactly the two local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","behavior]\\label{prop:dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, exactly the two local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}",":dyadic-sign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, exactly the two local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","ign}\nFor every prime $p\\ne2$, every entry of $E_+$ and $E_-$ is $p$-integral,\nand each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, exactly the two local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","each projector has exact local denominator weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, exactly the two local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","weight zero. At $p=2$, each\nprojector has exact local denominator weight one. Multiplication by $2$ makes\nit integral, while the unscaled matrix is not integral:\n\\[\ns_{2,\\Z_2^2}(E_+)=s_{2,\\Z_2^2}(E_-)=1.\n\\]\nThe characteristic factors $x-1$ and $x+1$ have resultant $2$. Its $2$-adic\nvaluation is one, exactly the two local projector weights above, and its\n$p$-adic valuation is zero at every prime $p\\ne2$.\n\\end{proposition}","\\begin{proof}\nThe reduced denominators of the displayed entries are $1$ or $2$, which proves\nthe local integrality and minimality statements directly. The resultant is\n\\[\n\\Res(x-1,x+1)=2.\n\\]\nThus $v_2(2)=1$ and $v_p(2)=0$ for primes $p\\ne2$.\n\\end{proof}","The two characteristic factors $x-1$ and $x+1$ have the same reduction in\n$\\F_2[x]$. The rational matrices $E_+$ and $E_-$ themselves cannot be reduced\nmodulo $2$, because their entries contain $1/2$. Consequently the integral\nspecial fibre does not inherit their separate idempotent splitting, even\nthough the two projectors remain distinct over $\\Q_2$. This is the simplest\nmodel of delayed dyadic separation.","The failure of the character formula\n\\[\n\\frac12(I\\pm A)\n\\]\nto be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished:\n\\[\n\\boxed{\n\\text{integral address selector}\n\\quad\\neq\\quad\n\\text{character-average spectral projector}\n}\n\\]\nwhen the group order is not a unit. The first can remain exact over $\\Z_2$. The second can acquire a denominator weight.","\\begin{standardfact}[Smith normal form over a discrete valuation ring]\\label{fact:smith}\nLet $M\\in M_{m,n}(\\OO)$.  There are invertible matrices\n\\[\nU\\in\\operatorname{GL}_m(\\OO),\n\\qquad\nV\\in\\operatorname{GL}_n(\\OO)\n\\]\nsuch that\n\\[\nUMV\n=\n\\diag(\\pi^{s_1},\\ldots,\\pi^{s_r},0,\\ldots,0),\n\\]\nwhere\n\\[\n0\\leq s_1\\leq\\cdots\\leq s_r\n\\]\nand $r$ is the rank over $K$.  The exponents are uniquely determined.  See \\cite{Newman1972}.\n\\end{standardfact}","\\begin{proposition}[Computing a packet weight from an inclusion matrix]\\label{prop:weight_from_inclusion}\nChoose an $\\OO$ basis of $\\Lambda$ and an $\\OO$ basis of $\\Lambda_{\\mathrm{split}}$.  Let $B$ be the matrix of the inclusion\n\\[\n\\Lambda_{\\mathrm{split}}\\hookrightarrow\\Lambda.\n\\]\nIf the Smith form of $B$ is\n\\[\n\\diag(\\pi^{a_1},\\ldots,\\pi^{a_n}),\n\\qquad\n0\\leq a_1\\leq\\cdots\\leq a_n,\n\\]\nthen\n\\[\nG_{\\Lambda,\\Pcal}\n\\cong\n\\bigoplus_{j=1}^n\\OO/\\pi^{a_j}\\OO,\n\\qquad\ns_\\Lambda(\\Pcal)=\\max\\bigl(\\{0\\}\\cup\\{a_j:1\\leq j\\leq n\\}\\bigr).\n\\]\n\\end{proposition}","\\begin{proof}\nThe gluing module is the cokernel of the inclusion matrix $B$.  Smith normal form gives its invariant factors.  The weight formula follows from \\cref{thm:gluing_weight}.\n\\end{proof}","\\begin{theorem}[Smith weighted congruence solve]\\label{thm:smith_congruence}\nLet $M\\in M_{m,n}(\\OO)$ have Smith exponents $s_1,\\ldots,s_r$.  Assume the residue field is finite of cardinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","bel{thm:smith_congruence}\nLet $M\\in M_{m,n}(\\OO)$ have Smith exponents $s_1,\\ldots,s_r$.  Assume the residue field is finite of cardinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","e field is finite of cardinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","rdinality $q$, fix $L\\geq1$, and consider\n\\[\nMx\\equiv b\\pmod{\\pi^L}.\n\\]\nPut\n\\[\n\\widetilde b=Ub\n\\]\nfor a Smith matrix $U$ as in \\cref{fact:smith}.  The congruence is solvable if and only if\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^{\\min(s_i,L)}}\n\\qquad(1\\leq i\\leq r)\n\\]\nand\n\\[\n\\widetilde b_i\\equiv0\n\\pmod{\\pi^L}\n\\qquad(r<i\\leq m).\n\\]\nWhen it is solvable, the number of solutions in $(\\OO/\\pi^L\\OO)^n$ is\n\\[\n\\boxed{\nq^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nSet $x=Vy$.  Multiplying the congruence by $U$ reduces it to independent equations\n\\[\n\\pi^{s_i}y_i\\equiv\\widetilde b_i\\pmod{\\pi^L}\n\\qquad(1\\leq i\\leq r)\n\\]\nand zero equations in the remaining rows.  Multiplication by $\\pi^{s_i}$ on $\\OO/\\pi^L\\OO$ has image equal to the elements divisible by $\\pi^{\\min(s_i,L)}$ and kernel of size\n\\[\nq^{\\min(s_i,L)}.\n\\]\nEach of the $n-r$ unused variables is free and contributes $q^L$ choices.  Multiplying the independent counts gives the formula.\n\\end{proof}","ultiplying the congruence by $U$ reduces it to independent equations\n\\[\n\\pi^{s_i}y_i\\equiv\\widetilde b_i\\pmod{\\pi^L}\n\\qquad(1\\leq i\\leq r)\n\\]\nand zero equations in the remaining rows.  Multiplication by $\\pi^{s_i}$ on $\\OO/\\pi^L\\OO$ has image equal to the elements divisible by $\\pi^{\\min(s_i,L)}$ and kernel of size\n\\[\nq^{\\min(s_i,L)}.\n\\]\nEach of the $n-r$ unused variables is free and contributes $q^L$ choices.  Multiplying the independent counts gives the formula.\n\\end{proof}","\\begin{corollary}[Obstruction and branching data]\\label{cor:smith_obstruction}\nThe transformed entries of $\\widetilde b$ that fail the stated divisibility conditions are the complete finite congruence obstructions.  When all obstructions vanish, the Smith exponents determine the number of solution branches at depth $L$.\n\\end{corollary}","\\begin{example}[Three weighted directions]\\label{ex:smith_three}\nTake $p=2$, $L=6$, and the diagonal weights\n\\[\n(0,1,3).\n\\]\nThe right side\n\\[\n(1,2,8)\n\\]\nis admissible.  The three coordinate kernels have sizes\n\\[\n1,\n\\quad2,\n\\quad8,\n\\]\nso there are $16$ solutions.  Replacing the final entry $8$ by $4$ violates the required divisibility by $2^3$ and makes the system unsolvable.\n\\end{example}","Let $M$ be a matrix over a DVR. Smith normal form gives invertible matrices $U,V$ such that\n\\[\nUMV\n=\n\\diag(\\pi^{s_1}u_1,\\ldots,\\pi^{s_r}u_r,0,\\ldots,0),\n\\]\nwhere each $u_i$ is a unit and\n\\[\n0\\le s_1\\le\\cdots\\le s_r.\n\\]\nThe exponents $s_i$ are directional lifting weights.","\\begin{theorem}[Concrete Smith-weighted congruence solve]\\label{thm:smith-solve}\nLet $p>1$ and consider the diagonal system over\n$(\\Z/(p^L)\\Z)^d$\n\\[\n\\diag(p^{s_1},\\ldots,p^{s_d})x=b\n\\pmod{p^L}.\n\\]\nThe system is solvable if and only if\n\\[\np^{\\min(s_i,L)}\\mid (b_i)_{\\mathrm{val}}\n\\]\nfor every $i$, where $(b_i)_{\\mathrm{val}}$ is the canonical natural\nrepresentative. Conditional on this admissibility condition, the number of\nsolutions is\n\\[\n\\boxed{\np^{\\sum_{i=1}^d\\min(s_i,L)}.\n}\n\\]\nIf any coordinate is inadmissible, the solution set is empty and its\ncardinality is zero.\n\\end{theorem}","ongruence solve]\\label{thm:smith-solve}\nLet $p>1$ and consider the diagonal system over\n$(\\Z/(p^L)\\Z)^d$\n\\[\n\\diag(p^{s_1},\\ldots,p^{s_d})x=b\n\\pmod{p^L}.\n\\]\nThe system is solvable if and only if\n\\[\np^{\\min(s_i,L)}\\mid (b_i)_{\\mathrm{val}}\n\\]\nfor every $i$, where $(b_i)_{\\mathrm{val}}$ is the canonical natural\nrepresentative. Conditional on this admissibility condition, the number of\nsolutions is\n\\[\n\\boxed{\np^{\\sum_{i=1}^d\\min(s_i,L)}.\n}\n\\]\nIf any coordinate is inadmissible, the solution set is empty and its\ncardinality is zero.\n\\end{theorem}","\\begin{proof}\nLean proves the one-coordinate arithmetic directly over\n$\\mathbb Z/p^L\\mathbb Z$ for every $p>1$:\n\\[\n\\bigl(\\exists x\\in\\mathbb Z/p^L\\mathbb Z:\\quad p^s x=b\\pmod{p^L}\\bigr)\n\\quad\\Longleftrightarrow\\quad\np^{\\min(s,L)}\\mid b_{\\mathrm{val}},\n\\]\nwhere $b_{\\mathrm{val}}$ is the canonical natural representative, and every\nadmissible fibre has exactly $p^{\\min(s,L)}$ elements. The coordinates are\nindependent, so the product equivalence gives the displayed global count. If\none coordinate fibre is empty, the global fibre is empty. Lean proves both the\nglobal solvability equivalence and the conditional count, including\ncardinality zero on the inadmissible branch.\n\\end{proof}","For an arbitrary DVR $\\OO$ with residue field cardinality $q$, the analogous\nformula\n\\[\nq^{\\sum_i\\min(s_i,L)}\n\\]\nfollows from the solvability and fibre count for each coordinate. The current\nrevision adds direct proofs of this DVR coordinate arithmetic; the current\ncorrespondence records those targets and the rectangular transport separately.\nThe inherited abstract product interface remains available when the coordinate\nfacts are supplied as certificates. A Smith form and its changes of basis are\ninputs to the transport statement; it does not provide an extracted\ndiagonalization routine. See \\cite{Serre1979,Lang2002}.","If a Smith form is rank deficient, a literal zero diagonal factor requires\nthe corresponding target coordinate to vanish modulo $p^L$ and then\ncontributes all $p^L$ possible coordinate values. In the concrete quotient it\ncan equivalently be represented by any weight $s_i\\ge L$, since\n$p^{s_i}=0$ in $\\Z/(p^L)\\Z$. The theorem then gives exactly that condition and\nfactor. The numbered theorem takes the diagonal data as its input.","\\begin{example}\nAt $p=2$, depth $L=6$, and weights $(0,1,3)$, the right side $(1,2,8)$ is admissible. The coordinate kernels have sizes\n\\[\n1,2,8,\n\\]\nso the system has $16$ solutions. Replacing $8$ by $4$ in the final coordinate violates the divisibility condition and makes the system unsolvable.\n\\end{example}","\\begin{interpretation}\nThe gluing calculation takes the largest Smith weight because every packet must be separated to the requested precision. The congruence calculation retains the individual directional weights. A coordinate with weight $s_i$ contributes a solvability threshold and $\\min(s_i,L)$ to the branching exponent. The independent coordinate counts multiply, so these truncated weights add in the exponent.\n\\end{interpretation}","Let $X_f$ and $X_c$ be vector spaces or Banach spaces over a common field, and let\n\\[\nC:X_f\\longrightarrow X_c\n\\]\nbe a linear map,\nand let\n\\[\nT_f\\in\\End(X_f),\n\\qquad\nT_c\\in\\End(X_c).\n\\]\nDefine\n\\[\n\\Delta=CT_f-T_cC.\n\\]","\\begin{theorem}[Power defect identity]\\label{thm:power_defect}\\label{thm:power-defect}\nFor every integer $n\\geq1$,\n\\[\nCT_f^n-T_c^nC\n=\n\\sum_{k=0}^{n-1}T_c^{n-1-k}\\Delta T_f^k.\n\\]\n\\end{theorem}","\\begin{proof}\nThe statement is clear for $n=1$.  For the induction step, write\n\\[\nCT_f^{n+1}-T_c^{n+1}C\n=\n(CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC)\n\\]\nand apply the induction hypothesis.\n\\end{proof}","\\begin{corollary}[Power norm bound]\\label{cor:power_norm}\nFor bounded operators between Banach spaces,\n\\[\n\\|CT_f^n-T_c^nC\\|\n\\leq\n\\|\\Delta\\|\n\\sum_{k=0}^{n-1}\n\\|T_c\\|^{n-1-k}\\|T_f\\|^k.\n\\]\n\\end{corollary}","\\Needspace{4\\baselineskip}\n\\begin{corollary}[Norm bound]\\label{cor:defect-norm}\nFor a submultiplicative operator norm,\n\\[\n\\|CT_f^n-T_c^nC\\|\n\\le\n\\|\\Delta\\|\n\\sum_{k=0}^{n-1}\n\\|T_c\\|^{n-1-k}\\|T_f\\|^k.\n\\]\nIf $\\|T_f\\|,\\|T_c\\|\\le\\rho$, then\n\\[\n\\boxed{\n\\|CT_f^n-T_c^nC\\|\n\\le n\\rho^{n-1}\\|\\Delta\\|.\n}\n\\]\n\\end{corollary}","\\begin{corollary}[Polynomial defect]\\label{cor:polynomial_defect}\nFor\n\\[\np(z)=\\sum_{n=0}^Na_nz^n,\n\\]\nwe have\n\\[\nCp(T_f)-p(T_c)C\n=\n\\sum_{n=1}^Na_n\n\\sum_{k=0}^{n-1}T_c^{n-1-k}\\Delta T_f^k.\n\\]\n\\end{corollary}","\\begin{proof}\nThe constant terms cancel because $CI=IC$.  Apply \\cref{thm:power_defect} to every positive power and sum.\n\\end{proof}","\\begin{theorem}[Resolvent defect identity]\\label{thm:resolvent_defect}\\label{thm:resolvent-defect}\nIf $z-T_f$ and $z-T_c$ are invertible, then\n\\[\nC(z-T_f)^{-1}-(z-T_c)^{-1}C\n=\n(z-T_c)^{-1}\\Delta(z-T_f)^{-1}.\n\\]\n\\end{theorem}","\\begin{proof}\nMultiply the claimed identity on the left by $z-T_c$ and on the right by $z-T_f$.  The left side becomes\n\\[\n(z-T_c)C-C(z-T_f)\n=\nCT_f-T_cC\n=\n\\Delta.\n\\]\nMultiplying back by the two resolvents proves the formula.\n\\end{proof}","\\subsection{Inverse witnesses from one side and norm control}\nFor continuous linear maps, the same identity gives the certified norm bound\n\\[\n\\boxed{\n\\|C R_f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the m","ntinuous linear maps, the same identity gives the certified norm bound\n\\[\n\\boxed{\n\\|C R_f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the more general theory of refinement defects reserved for Paper III.","d norm bound\n\\[\n\\boxed{\n\\|C R_f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the more general theory of refinement defects reserved for Paper III.","f-R_c C\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|,\n}\n\\]\nfrom a left inverse witness for the coarse scalar shift and a right inverse\nwitness for the fine scalar shift. These witnesses suffice for the displayed\nidentity. On their own, they need not define inverses on both sides and hence\nneed not define resolvents. If $\\Delta=0$, Lean also proves, for every finite\npolynomial $p$, the exact compatibility\n\\[\n\\boxed{C\\,p(T_f)=p(T_c)\\,C,}\n\\]\nin both the algebraic setting and the setting of continuous linear maps. Thus\nall powers and polynomial actions intertwine with coarse reconstruction. The\nstated inverse witnesses also intertwine. In particular, genuine resolvents\ndo so when the scalar shifts are invertible. This is the linear prototype\nof the more general theory of refinement defects reserved for Paper III.","\\subsection{Analytic functional calculus}\n\\begin{theorem}[Holomorphic functional calculus defect]\\label{thm:holomorphic_defect}\nLet $X_f$ and $X_c$ be complex Banach spaces, and let $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded complex-linear maps.  Let $h$ be holomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|","lculus defect]\\label{thm:holomorphic_defect}\nLet $X_f$ and $X_c$ be complex Banach spaces, and let $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded complex-linear maps.  Let $h$ be holomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|","$X_c$ be complex Banach spaces, and let $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded complex-linear maps.  Let $h$ be holomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|","olomorphic on an open set $U$ containing both spectra.  Let $\\Gamma$ be an admissible finite rectifiable Cauchy cycle in $U$ and in both resolvent sets, with winding number one on the union of the spectra and zero outside $U$.  The length of a cycle counts multiplicities.  Then\n\\[\nCh(T_f)-h(T_c)C\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma\nh(z)(z-T_c)^{-1}\\Delta(z-T_f)^{-1}\\,dz.\n\\]\nConsequently,\n\\begin{align*}\n\\|Ch(T_f)-h(T_c)C\\|\n&\\leq\n\\frac{\\operatorname{length}(\\Gamma)}{2\\pi}\n\\sup_{z\\in\\Gamma}|h(z)|\\,\\|\\Delta\\|\\\\\n&\\quad\\times\n\\sup_{z\\in\\Gamma}\\|(z-T_c)^{-1}\\|\n\\sup_{z\\in\\Gamma}\\|(z-T_f)^{-1}\\|.\n\\end{align*}\n\\end{theorem}","\\begin{proof}\nThe assertion is immediate if either space is zero. Otherwise, use the rectifiable-cycle form of the holomorphic functional calculus in \\cite[Chapter~VII, \\S4, Theorem~4.1, formula~(4.5), and Proposition~4.6]{Conway1990}. For each operator $T$, compare $\\Gamma$ with a positive simple system around its spectrum. Their difference has zero winding outside $U\\setminus\\sigma(T)$, so Theorem~4.1 applied to the resolvent-valued integrand identifies the two integrals. Consequently,\n\\[\nh(T)\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma h(z)(z-T)^{-1}\\,dz.\n\\]\nSubtract the two formulas after composing with $C$, then use \\cref{thm:resolvent_defect} inside the integral.  The norm estimate follows from the standard contour bound.\n\\end{proof}","e-cycle form of the holomorphic functional calculus in \\cite[Chapter~VII, \\S4, Theorem~4.1, formula~(4.5), and Proposition~4.6]{Conway1990}. For each operator $T$, compare $\\Gamma$ with a positive simple system around its spectrum. Their difference has zero winding outside $U\\setminus\\sigma(T)$, so Theorem~4.1 applied to the resolvent-valued integrand identifies the two integrals. Consequently,\n\\[\nh(T)\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma h(z)(z-T)^{-1}\\,dz.\n\\]\nSubtract the two formulas after composing with $C$, then use \\cref{thm:resolvent_defect} inside the integral.  The norm estimate follows from the standard contour bound.\n\\end{proof}","nd Proposition~4.6]{Conway1990}. For each operator $T$, compare $\\Gamma$ with a positive simple system around its spectrum. Their difference has zero winding outside $U\\setminus\\sigma(T)$, so Theorem~4.1 applied to the resolvent-valued integrand identifies the two integrals. Consequently,\n\\[\nh(T)\n=\n\\frac{1}{2\\pi i}\n\\int_\\Gamma h(z)(z-T)^{-1}\\,dz.\n\\]\nSubtract the two formulas after composing with $C$, then use \\cref{thm:resolvent_defect} inside the integral.  The norm estimate follows from the standard contour bound.\n\\end{proof}","\\begin{theorem}[Exponential defect identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","onential defect identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","l defect identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","identity]\\label{thm:exponential_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","ntial_defect}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","t}\nLet $C:X_f\\to X_c$, $T_f$, and $T_c$ be bounded linear maps between Banach spaces over a common field.  For real $t$, use real or complex spaces. For complex $t$, use complex spaces and complex-linear maps.  Then\n\\[\nCe^{tT_f}-e^{tT_c}C\n=\n\\int_0^t e^{(t-s)T_c}\\Delta e^{sT_f}\\,ds,\n\\]\nwhere the integral is taken along the straight segment from $0$ to $t$ when $t$ is complex.  For real $t\\geq0$,\n\\[\n\\|Ce^{tT_f}-e^{tT_c}C\\|\n\\leq\n\\|\\Delta\\|\n\\int_0^t\n\\|e^{(t-s)T_c}\\|\n\\|e^{sT_f}\\|\\,ds.\n\\]\n\\end{theorem}","\\begin{proof}\nFor real $t$, differentiate\n\\[\nF(s)=e^{(t-s)T_c}Ce^{sT_f}.\n\\]\nWe obtain\n\\[\nF'(s)=e^{(t-s)T_c}\\Delta e^{sT_f}.\n\\]\nIntegration from $0$ to $t$ gives the identity.  The complex statement follows by the same calculation along the straight segment.  The norm bound follows from the integral formula.\n\\end{proof}","Let\n\\[\nC_\\ell:X_{\\ell+1}\\longrightarrow X_\\ell\n\\]\nbe linear maps for $0\\leq\\ell<L$, and let $T_\\ell\\in\\End(X_\\ell)$.  Put\n\\[\n\\Delta_\\ell=C_\\ell T_{\\ell+1}-T_\\ell C_\\ell.\n\\]\nFor $a<b$, write\n\\[\nC_{a,b}=C_aC_{a+1}\\cdots C_{b-1},\n\\]\nwith $C_{a,a}=I$.","\\begin{theorem}[Multilevel defect decomposition]\\label{thm:multilevel_defect}\nThe total defect from level $L$ to level zero is\n\\[\nC_{0,L}T_L-T_0C_{0,L}\n=\n\\sum_{\\ell=0}^{L-1}\nC_{0,\\ell}\\Delta_\\ell C_{\\ell+1,L}.\n\\]\nFor bounded maps,\n\\[\n\\|C_{0,L}T_L-T_0C_{0,L}\\|\n\\leq\n\\sum_{\\ell=0}^{L-1}\n\\|C_{0,\\ell}\\|\n\\|\\Delta_\\ell\\|\n\\|C_{\\ell+1,L}\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nInsert the intermediate terms\n\\[\nC_{0,\\ell}T_\\ell C_{\\ell,L}\n\\]\nfor $0\\leq\\ell\\leq L$.  Consecutive differences telescope.  The difference between the terms indexed by $\\ell+1$ and $\\ell$ is\n\\[\nC_{0,\\ell}(C_\\ell T_{\\ell+1}-T_\\ell C_\\ell)C_{\\ell+1,L}.\n\\]\nSumming gives the identity.  The norm estimate follows from the triangle inequality.\n\\end{proof}","\\begin{example}[A visible one step defect]\\label{ex:defect_example}\nLet $X_f=\\bbk^2$, let $X_c=\\bbk$, and let\n\\[\nC(x_1,x_2)=x_1.\n\\]\nTake\n\\[\nT_f=\n\\begin{pmatrix}\na&b","\\begin{example}[A visible one step defect]\\label{ex:defect_example}\nLet $X_f=\\bbk^2$, let $X_c=\\bbk$, and let\n\\[\nC(x_1,x_2)=x_1.\n\\]\nTake\n\\[\nT_f=\n\\begin{pmatrix}\na&b\\\\\nc&d\n\\end{pmatrix},\n\\qquad\nT_c=(a).\n\\]\nThen\n\\[\n\\Delta=CT_f-T_cC=(0,b).\n\\]\nThe coarse operator is exactly compatible with the fine operator precisely when the discarded coordinate does not feed into the retained coordinate.  The entry $c$ does not appear in the one step defect because it moves information in the opposite direction.  It can still affect later powers through the terms in \\cref{thm:power_defect}.\n\\end{example}","c&d\n\\end{pmatrix},\n\\qquad\nT_c=(a).\n\\]\nThen\n\\[\n\\Delta=CT_f-T_cC=(0,b).\n\\]\nThe coarse operator is exactly compatible with the fine operator precisely when the discarded coordinate does not feed into the retained coordinate.  The entry $c$ does not appear in the one step defect because it moves information in the opposite direction.  It can still affect later powers through the terms in \\cref{thm:power_defect}.\n\\end{example}","\\begin{corollary}[Exact compatibility]\\label{cor:exact_compatibility}\nIf every one step defect $\\Delta_\\ell$ vanishes, then\n\\[\nC_{0,L}p(T_L)=p(T_0)C_{0,L}\n\\]\nfor every polynomial $p$.  The analogous resolvent, holomorphic, and exponential identities also hold whenever the relevant expressions are defined.\n\\end{corollary}","Let $x=(x_n)_{n\\in\\Z}\\in c_{00}(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","\\in\\Z}\\in c_{00}(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","in c_{00}(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","(\\Z;R)$ and let the seed be $\\delta_0$\nunder the bilateral shifts $(S^r\\delta_0)_n=\\delta_{n,r}$. This makes the\nalgebraic span and every displayed sum finite. The $N$ packets are\n\\[\nx_j[m]=x[Nm+j].\n\\]\nSynthesis interleaves these sequences. A linear filter with a finite impulse\nresponse that is invariant under shifts becomes an $N\\times N$ matrix of\ntransfer blocks. Refining from $N$ to $M$ with $N\\mid M$ splits each coarse\nphase into its descendants without changing the reconstructed signal. The\nsame formulas extend to standard sequence completions when the synthesis\nseries converges. This is the usual polyphase viewpoint from multirate filter\nbanks \\cite{Vaidyanathan1993}.","A banded operator need not remain banded after residue reordering, but its\nnonzero couplings become a structured packet graph. With the convention\n$(S_rx)_n=x_{n-r}$, the only active transfer is\n\\[\nP_{N,j+r}S_rP_{N,j},\n\\]\nwith indices taken modulo $N$. More general finite stencils produce a finite set of active phase differences.","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges n","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpe","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these v","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sample","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two v","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Highe","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compa","Let $Z$ and $Z_0\\ne0$ be complex impedances, put $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","t $z=Z/Z_0$, and assume $z\\ne-1$. Write\n\\[\n\\Gamma=\\frac{z-1}{z+1}.\n\\]\nOn the open reflection disk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","isk $|\\Gamma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","ma|<1$, multiplication by any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","y any phase $\\omega_N^k$ stays in the disk and has a finite inverse-chart image. The phase action is\n\\[\n\\Gamma\\mapsto\\omega_N^k\\Gamma.\n\\]\nOutside the open disk, the same chart formula requires $1-\\omega_N^k\\Gamma\\ne0$. Pulling the phase action back on this domain gives\n\\[\nT_{N,k}(z)\n=\n\\frac{(1+\\omega_N^k)z+(1-\\omega_N^k)}\n{(1-\\omega_N^k)z+(1+\\omega_N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","N^k)}.\n\\]\nFor $N=2$ and $k=1$,\n\\phantomsection\n\\[\nT_{2,1}(z)=\\frac1z,\n\\qquad z\\ne0,-1.\n\\label{eq:smith-inversion}\n\\]\nThus the nontrivial action at the first phase level exchanges normalized\nimpedance and admittance. Calling these values ``packets'' is geometric\nterminology for the sampled orbit. It does not assert that the two values\nform linear idempotent ranges. Higher refinements at powers of $2$ sample the\nsame reflection circle at progressively finer electrical phases. Compare the\nstandard Smith chart conventions in \\cite{pozar2012}.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.96\\textwidth]{11_smith_polyphase_orbit.pdf}\n\\caption{Nested phase orbits at powers of $2$ on the Smith reflection disk. The full treatment of the Smith chart is deferred to the signal paper.}\n\\label{fig:smith-orbit}\n\\end{figure}","The symmetric weight\n\\[\nw_{\\mathrm{Smith}}(\\Gamma)\n=\n\\frac1{|1-\\Gamma|^2}\n+\n\\frac1{|1+\\Gamma|^2}\n\\]\nindicates where small errors in reflection coordinates produce large impedance or admittance errors.","\\midrule\nPacket synthesis residuals for moduli $2,3,4,6,8,12$ & $0$","Nested refinement residuals for $2\\to4$, $2\\to8$, $3\\to6$, $3\\to12$, $4\\to8$, $4\\to12$, $6\\to12$ & $0$","CRT residuals for dyadic and odd moduli for $4\\times3$, $8\\times3$, $4\\times5$, $8\\times5$ & $0$","Maximum finite Fourier projector residual & $6.60\\times10^{-15}$","Weighted inverse residual modulo $2^{10}$ & $0$","Weighted inverse residual modulo $3^8$ & $0$","Weighted inverse residual modulo $5^7$ & $0$","Difference between filtered and direct modular inverses & $0$","Transfer zeta coefficients versus direct walk enumeration through weight $30$ & exact match","Dyadic sign projector resultant weight & $v_2(2)=1$","Sign projector resultant weight at the odd prime $3$ & $v_3(2)=0$","Power defect telescoping identity residual & at most $6.94\\times10^{-18}$","Resolvent defect identity residual & $1.09\\times10^{-17}$","Smith $N=2$ phase versus $z\\mapsto1/z$ & residual below $1.4\\times10^{-16}$","\\begin{certificate}[Weighted inverse]\nFor the tested diagonal-plus-transfer matrices, the filtered inverse agrees exactly with direct modular inversion at all tested precisions:\n\\[\n2^{10},\\qquad3^8,\\qquad5^7.\n\\]\nThe maximum modular residual is zero in every case.\n\\end{certificate}","\\begin{certificate}[Transfer zeta]\nThe rational function\n\\[\n\\frac{-t^3}{t^9-t^7-t^6-t^5+2t^4+t^3+t^2-1}\n\\]\nwas expanded through degree $30$. Every coefficient agrees with direct weighted-walk enumeration.\n\\end{certificate}","\\begin{certificate}[Dyadic separation]\nThe code verifies exactly that the two rational sign projectors are idempotent, orthogonal, and complementary, that scaling by $2$ makes both integral, and that the same projectors are already integral modulo every tested odd prime.\n\\end{certificate}","\\midrule\n\\endhead\n$P_{\\ell,\\alpha}$ & packet projector at refinement level $\\ell$","$I_\\ell$ & packet address set at level $\\ell$","$\\beta\\succ\\alpha$ & fine packet $\\beta$ is a child of coarse packet $\\alpha$","$\\cA_\\ell,\\cS_\\ell$ & packet analysis and synthesis maps","$J_{\\ell+1,\\ell}$ & exact coordinate refinement map","$C_{\\ell,\\ell+1}$ & exact coordinate regrouping from fine to coarse","$T^{(\\ell)}_{\\beta\\alpha}$ & transfer block $P_{\\ell,\\beta}TP_{\\ell,\\alpha}$","$P_{N,j}$ & residue or phase selector for address $j\\pmod N$","$w(e)$ & positive filtration weight of a transfer edge","$W(\\gamma)$ & cumulative weight of a transfer word or graph walk","$A(t)$ & polynomial weighted adjacency matrix","$Z_{s,t}(t)$ & weighted transfer-zeta function from vertex $s$ to vertex $t$","$s_{\\pi,\\Lambda}(E)$ & separation weight of a rational projector on the chosen lattice, when a clearing exponent exists","$\\Delta$ & linear refinement defect for one step $CT_f-T_cC$","The verified contract is\n$\\operatorname{synthesize}(\\operatorname{analyze}(x,N))=x$. The Lean\ncounterpart uses packet coordinates restricted to projector ranges and proves\nboth inverse laws. The finite Python routine checks the concrete array\nrealization.","Lean proves that the mapped pruned sum is exactly the mapped finite geometric\ninverse. The lower bound $m$ gives $m|\\gamma|\\le W(\\gamma)$, so\n$L\\le mN$ derives $q_L(A^N)=0$ from the filtration certificate and makes the\nreturned value an inverse on both sides of $q_L(D+H)$. The Lean lemma\n$L\\le m\\lceil L/m\\rceil$ justifies the displayed executable cutoff for\n$m>0$. Taking $m=1$ and $N=L$ gives\nthe unconditional cutoff for positive weights. The Python implementation checks the\nDVR matrix specialization against direct modular inversion.","Positive weights make the coefficient loop finite because every contributing\nword has length at most $n$. Lean proves the enumeration correct, derives its\nrecurrence from the first edge, proves $(I-A)Z=I$, constructs the\ndeterminant/adjugate inverse, and identifies the two matrices by inverse\nuniqueness. Thus the normalized $QZ=P$ identity and its induced recurrence\nfollow from finite graph data. The Python companion repeats the exact symbolic\nconstruction as an executable check. The Lean proof does not use it as an\noracle.","The proof order is:\n\\[\n\\begin{aligned}\n&\\text{complete packet axioms}\n  \\Rightarrow\\text{finite-packet uniqueness and coordinate coherence}\\\\\n&\\qquad\\Rightarrow\\text{coordinate naturality and refined transfer blocks},\\\\\n&\\text{residue selectors and a finite Fourier table}\n  \\Rightarrow\\text{Fourier formulas},\\\\\n&\\text{coprime moduli}\\Rightarrow\\text{dyadic--odd formulas},\\\\\n&\\text{quotient, filtration, and edge-weight certificates}\n  \\Rightarrow\\text{explicit pruned inverse},\\\\\n&\\text{finite positive-weight graph}\n  \\Rightarrow\\text{semantic enumeration and first-edge recurrence}\\\\\n&\\qquad\\Rightarrow\\det/\\operatorname{adj}\\text{ rationality and recurrence},\\\\\n&\\text{minimal evaluated B\\'ezout data and inverse-unit lattice action}\\\\\n&\\qquad\\Rightarrow\\text{separation bound, upward closure, and precision delay},\\\\\n&\\text{concrete }\\mathbb Z/p^L\\mathbb Z\\text{ coordinate fibres}\n  \\Rightarrow\\text{finite Smith product count},\\\\\n&\\text{one-step typed defect}\n  \\Rightarrow\\text{powers, polynomial actions, and witnessed resolvents}.\n\\end{aligned}\n\\]","The walk branch for transfer zeta is logically independent of operator\ninversion. Its rationality conclusion follows from finite graph data without\na supplied $QZ=P$ premise. Likewise, CRT factorization follows directly from\nresidue selectors and coprimality. It does not require the abstract refinement\ntheorem, although it supplies a concrete refinement family. General Smith\ndiagonalization and the construction of polynomial B\\'ezout/resultant data\nremain inputs to the formalized branches that use those certificates.","\\head{Address resolution}\nA coefficient such as $x_6$ can sit in the even packet, then the residue-$2$ packet modulo $4$, then the residue-$6$ packet modulo $8$. Its value does not become more accurate. Its \\emph{address becomes more specific}, while the full object remains exactly reconstructible.","\\head{Arithmetic precision}\nAn operator word with total filtration weight at least $L$ vanishes in the specified depth-$L$ quotient. Increasing $L$ can make additional words relevant. This is a change in the precision of the ring in which the answer is required, not a lossless rearrangement of one fixed list.","\\head{Separation precision}\nFor the integral involution used later, $E_+=(I+A)/2$ is an exact rational projector but not an integral one at $2$. One extra input digit suffices to recover the projected difference to the requested precision. The packet is present over the fraction field even when its integral separation is delayed.","midrule\n$P_{N,j}$ & Retain coefficients with address $j$ modulo $N$. & Even/odd masks, then mod-$4$ and mod-$8$ masks.","$\\mathcal A_N,\\mathcal S_N$ & Split into range-restricted packet vectors, then sum them. & A permutation/interleaving of eight coordinates.","$J,C$ & Exact refinement and regrouping. In the defect chapter, $C$ may instead be any supplied coarsening map. & A permutation in V5 and a shear in V22.","$T_{ba}$ & Input packet $a$, output packet $b$: $P_bTP_a$. & Blocks of a finite cyclic filter.","$D,H,A$ & $T=D+H=D(I+A)$, with $A=D^{-1}H$. & Explicit integral $2\\times2$ matrices.","$w(e),W(\\gamma)$ & Positive certified edge gain and its additive word weight. & $w(a)=1$, $w(b)=2$.","$L$ & Precision depth in a quotient such as $\\ZZ/2^L\\ZZ$. & $L=5$ gives arithmetic modulo $32$.","$A(t),Z(t)$ & Polynomial graph adjacency and unsigned walk-count series. & Two parallel edges remain distinct.","$E,\\Lambda,s$ & A rational projector, a specified lattice, and a clearing exponent. & $E_+$ has local weight $1$ at $2$.","$(s_i)$ & Directional weights of a supplied Smith diagonal. & $(1,2)$ modulo $8$ or $(0,1,3)$ across depths.","$\\Delta$ & The failure of an independently chosen operator to intertwine: $CT_f-T_cC$. & One nonzero entry $1/16$.","\\vpage{3}{Follow one coefficient through the tower}{MODEL A / LOSSLESS ADDRESS REFINEMENT}{Traditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11)}\nTake a vector $x=(x_0,\\ldots,x_{15})$. A packet is not a single summary number: it retains every coefficient at its selected addresses. For example,\n\\[\nP_{2,0}x=(x_0,0,x_2,0,\\ldots,x_{14},0).\n\\]\nThe rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on V5 instead uses increasing residue order.","The address $6$ is even, is $2$ modulo $4$, and is $6$ modulo $8$. Its coefficient therefore lies in the successive ranges\n\\[\nP_{8,6}X\\subseteq P_{4,2}X\\subseteq P_{2,0}X.\n\\]\nThe corresponding identities hold as operators:\n\\[\nP_{2,0}=P_{4,0}+P_{4,2},\\qquad\nP_{4,2}=P_{8,2}+P_{8,6}.\n\\]\nApplying either composite parent selector to $P_{8,6}x$ leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.","For $x=(1,2,3,4)$, the fine packets are the four coordinate-supported vectors. Regrouping gives $(1,0,3,0)$ and $(0,2,0,4)$. Every original coordinate remains available in one of these two vectors.","\\vfig[.98]{regrouping_not_averaging}\n\\platecaption{The left and middle panels store the same four independent coordinates. The right panel instead applies a $2\\times4$ numerical incidence matrix to four scalar entries. That is not the range-restricted exact coordinate map.}","Define\n\\[\nX_N^{\\rm pkt}=\\bigoplus_j P_{N,j}X,\n\\qquad\n\\mathcal A_Nx=(P_{N,j}x)_j,\n\\qquad\n\\mathcal S_N(x_j)_j=\\sum_jx_j.\n\\]\nCompleteness and mutual annihilation give both inverse laws. Consequently,\n\\[\nJ=\\mathcal A_4\\mathcal S_2,\\qquad C=\\mathcal A_2\\mathcal S_4,\n\\qquad CJ=I,\\quad JC=I.\n\\]\nThe apparently larger list of fine packets simply has smaller individual packet spaces.","\\head{A kernel witnesses genuine information loss}\nFor the scalar incidence matrix $B$ drawn on the right,\n\\[\nB(1,0,-1,0)^\\mathsf T=0.\n\\]\nThe zero vector and $(1,0,-1,0)$ therefore have the same two scalar summaries. No inverse can recover the original four scalars from that output alone.","\\vpage{5}{The operator changes coordinates too}{MODEL A / COHERENCE AND NATURALITY}{Traditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6)}\nUse a finite periodic carrier $X=\\QQ^8$ for this example. Let $Se_j=e_{j+1\\bmod8}$ and let\n\\[\nT=I+2S+S^2.\n\\]\nThis is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write $R_N$ for the permutation that groups coordinates in increasing residue order modulo $N$.","COHERENCE AND NATURALITY}{Traditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6)}\nUse a finite periodic carrier $X=\\QQ^8$ for this example. Let $Se_j=e_{j+1\\bmod8}$ and let\n\\[\nT=I+2S+S^2.\n\\]\nThis is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write $R_N$ for the permutation that groups coordinates in increasing residue order modulo $N$.","\\vfig[.99]{operator_naturality}\n\\platecaption{Actual $8\\times8$ matrices. Dots are zero entries. The other entries are exact integers. Heavy lines mark the packet blocks: size four at level two and size two at level four.}","The reordered operators and change of coordinates are\n\\[\n\\widehat T_N=R_NTR_N^{-1},\\qquad J_{4,2}=R_4R_2^{-1}.\n\\]\nMultiplication cancels the intermediate coordinate maps:\n\\[\nJ_{4,2}\\widehat T_2=\\widehat T_4J_{4,2},\\qquad\nJ_{8,4}J_{4,2}=J_{8,2}.\n\\]\nThus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.","\\head{Refined blocks still reconstruct the same map}\nA coarse block expands as\n\\[\nP_bTP_a=\\sum_{\\beta\\succ b,\\,\\alpha\\succ a}P_\\beta TP_\\alpha.\n\\]\nThe terms are maps embedded in the original source and target spaces. Adding their norms, or adding their matrix entries after forgetting the embeddings, is not this operator identity.","\\vpage{6}{Selectors survive without division}{MODEL A / RESIDUE MASKS IN CHARACTERISTIC TWO}{Traditional paper: Proposition 4.2 (p. 9)}\nWork first in $\\mathbb F_2$, where $1+1=0$. The eight coefficients below are just a finite display of a coefficient sequence. The even and odd projectors are diagonal masks with entries zero and one.","\\vfig[.95]{division_free_masks}\n\\platecaption{Each nonzero input coefficient is retained by exactly one mask. Reconstruction is exact over $\\mathbb F_2$. No average and no root of unity is used.}","For a general coefficient ring and a positive integer $N$, the definition is\n\\[\n(P_{N,j}x)_n=\\begin{cases}x_n,&n\\equiv j\\pmod N,","For a general coefficient ring and a positive integer $N$, the definition is\n\\[\n(P_{N,j}x)_n=\\begin{cases}x_n,&n\\equiv j\\pmod N,\\\\0,&\\text{otherwise}.\\end{cases}\n\\]\nAn address belongs to one residue class, so applying two masks either keeps that same coefficient or kills it. This proves\n\\[\nP_{N,i}P_{N,j}=\\delta_{ij}P_{N,j},\\qquad \\sum_jP_{N,j}=I.\n\\]\nThe argument is coordinatewise and does not require the coefficients to form a field.","0,&\\text{otherwise}.\\end{cases}\n\\]\nAn address belongs to one residue class, so applying two masks either keeps that same coefficient or kills it. This proves\n\\[\nP_{N,i}P_{N,j}=\\delta_{ij}P_{N,j},\\qquad \\sum_jP_{N,j}=I.\n\\]\nThe argument is coordinatewise and does not require the coefficients to form a field.","\\head{Why the usual even/odd average is a different formula}\nOver a field of characteristic not two, parity extraction can be represented using $F(z)$ and $F(-z)$ and division by $2$. In characteristic two, $-z=z$ and $2=0$, so that formula cannot be used. The coefficient mask, however, still distinguishes even from odd \\emph{exponents}.","This distinction also explains the later involution example. Failure to reduce a rational spectral projector containing $1/2$ does not make every integral coefficient-address selector disappear.","\\vpage{7}{A Fourier certificate you can inspect}{MODEL A / AN EXACT NINE-ENTRY TABLE}{Traditional paper: Theorem 4.3 (p. 10)}\nTake $N=3$ and work in $\\mathbb F_7$. Set\n\\[\n\\omega=2,\\qquad \\omega^{-1}=4,\\qquad \\eta=3^{-1}=5.\n\\]\nHere $2^3=1$ and neither $2$ nor $2^2$ is one. The unnormalized geometric sums are the entries in the first matrix.","\\vfig[.97]{finite_fourier_table}\n\\platecaption{All entries are exact residues modulo seven. Multiplying the diagonal value $3$ by $5$ produces $1$. The lower calculation is a genuine failed Fourier certificate in a different coefficient ring.}","On the monomial $z^n$, define $UF(z)=F(2z)$. The proposed projector acts by\n\\[\n5\\sum_{k=0}^2 4^{jk}2^{nk}.\n\\]\nThe identity $2^3=1$ makes this depend only on $n\\bmod3$. The nine-entry table therefore establishes the formula for every exponent once periodicity is used, rather than requiring a separate test at every coefficient.","\\head{Order alone is not sufficient over every ring}\nIn $\\mathbb F_3\\times\\mathbb F_3$, the element $(-1,1)$ has order two, but one component is a trivial character. Its nontrivial Fourier sum does not vanish in that component. Thus an element of order $N$ in a ring with zero divisors is not, by itself, the full orthogonality certificate.","\\vpage{8}{One address, two independent residues}{MODEL A / THE DYADIC AND ODD RECTANGLE}{Traditional paper: Corollary 4.9 (p. 11)}\nBecause $4$ and $3$ are coprime, a residue modulo $12$ is specified by a column $a=n\\bmod4$ and a row $b=n\\bmod3$. Each cell contains the unique combined address.","\\vfig[.96]{crt_address_grid}\n\\platecaption{The twelve cells are a permutation of $0,\\ldots,11$, not twelve extra degrees of freedom. The selected cell contains every integer congruent to $7$ modulo $12$.}","An explicit reconstruction formula is\n\\[\nn\\equiv9a+4b\\pmod{12},\n\\]\nbecause $9$ is one modulo four and zero modulo three, while $4$ has the opposite pair of residues. For $(a,b)=(3,1)$ this gives $31\\equiv7$.","Applying the two masks in either order gives\n\\[\nP_{12,7}=P_{4,3}P_{3,1}=P_{3,1}P_{4,3}.\n\\]\nMore generally, an address survives exactly when it satisfies both congruences. The coprime reconstruction identifies this intersection with one combined residue class.","\\head{The hypothesis can be seen failing}\nReplace $4$ and $3$ by $4$ and $6$. The requirements $n\\equiv0\\pmod4$ and $n\\equiv1\\pmod6$ are incompatible: the first makes $n$ even and the second makes it odd. A full rectangular address product no longer exists. Compatible pairs must agree modulo the greatest common divisor.","\\vpage{9}{Which words can survive precision five?}{MODEL B / ORDERED TRANSFERS IN A FILTERED RING}{Traditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18)}\nLet $E_{ij}$ send the $j$th basis vector to the $i$th. In integral $2\\times2$ matrices set\n\\[\na=2E_{01},\\qquad b=4E_{10},\\qquad A=a+b.\n\\]\nTheir certified dyadic weights are $1$ and $2$. At precision $L=5$, every word of weight at least five vanishes modulo $32$.","can survive precision five?}{MODEL B / ORDERED TRANSFERS IN A FILTERED RING}{Traditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18)}\nLet $E_{ij}$ send the $j$th basis vector to the $i$th. In integral $2\\times2$ matrices set\n\\[\na=2E_{01},\\qquad b=4E_{10},\\qquad A=a+b.\n\\]\nTheir certified dyadic weights are $1$ and $2$. At precision $L=5$, every word of weight at least five vanishes modulo $32$.","\\vfig[.97]{weighted_word_cutoff}\n\\platecaption{All nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes $I$. Other words of weight below five can still be zero because $a^2=b^2=0$.}","The words are \\emph{written matrix products}: in $ab$, the map $b$ acts first. Therefore\n\\[\nab=8E_{00},\\qquad ba=8E_{11},\\qquad aba=16E_{01}.\n\\]\nThe two products $ab$ and $ba$ cannot be identified merely because they have equal weight. Both their order and their source/target information matter.","\\vpage{10}{An inverse computed in both orders}{MODEL B / A COMPLETE MODULO-32 CALCULATION}{Traditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18)}\nKeep the matrices from V9 and choose\n\\[\nD=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix},\\quad\nD^{-1}=\\begin{pmatrix}1&-1\\\\0&1\\end{pmatrix},\\quad\nT=D(I+A)=\\begin{pmatrix}5&3\\\\4&1\\end{pmatrix}.\n\\]\nThe six retained contributions sum to\n\\[\nS=I-a-b+ab+ba-aba\n=\\begin{pmatrix}9&-18","4&1\\end{pmatrix}.\n\\]\nThe six retained contributions sum to\n\\[\nS=I-a-b+ab+ba-aba\n=\\begin{pmatrix}9&-18\\\\-4&9\\end{pmatrix}.\n\\]\nThe required inverse is $K=SD^{-1}$ in the quotient, not $D^{-1}S$.","-4&9\\end{pmatrix}.\n\\]\nThe required inverse is $K=SD^{-1}$ in the quotient, not $D^{-1}S$.","\\vfig[.98]{modular_inverse}\n\\platecaption{The displayed $K$ uses canonical residues from $0$ to $31$. Both left and right products are the identity modulo $32$. These are exact modular equalities, not floating-point residuals.}","One can verify the claim without trusting the word expansion:\n\\[\nTK=\\begin{pmatrix}129&64\\\\64&33\\end{pmatrix}\\equiv I,\n\\qquad\nKT=\\begin{pmatrix}65&32\\\\192&97\\end{pmatrix}\\equiv I\\pmod{32}.\n\\]\nThe cancellation of the exterior factor $D$ explains the order. Since $T=D(I+A)$, multiply its inverse by $D^{-1}$ on the \\emph{right} of the inverse of $I+A$.","192&97\\end{pmatrix}\\equiv I\\pmod{32}.\n\\]\nThe cancellation of the exterior factor $D$ explains the order. Since $T=D(I+A)$, multiply its inverse by $D^{-1}$ on the \\emph{right} of the inverse of $I+A$.","\\vpage{11}{The residual tells you when to stop}{MODEL B / A CERTIFIED CUTOFF, NOT A HEURISTIC}{Traditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18), Section B.2 (p. 47)}\nFor $S_N=\\sum_{k=0}^{N-1}(-A)^k$, telescoping gives\n\\[\n(I+A)S_N=S_N(I+A)=I-(-A)^N.\n\\]\nThus the exact residual is known. To certify precision $L$, it is enough to show that every residual entry is divisible by $2^L$.","\\vfig[.94]{inverse_residual_depth}\n\\platecaption{The plotted values are the minimum dyadic valuations of the nonzero entries of $(-A)^N$. The dashed line is the requested cutoff $L=5$. This is an exact valuation profile, not a measured numerical convergence curve.}","In this particular model, $A^2=8I$. Therefore\n\\[\nv_2(A^{2k})=3k,\\qquad v_2(A^{2k+1})=3k+1,\n\\]\nwhere the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo $32$. Four terms leave valuation six and do suffice.","The generic positive-weight estimate is more conservative. If every edge weight is at least $m>0$, a word of length $N$ has weight at least $mN$. Hence the cutoff\n\\[\nN=\\left\\lceil\\frac Lm\\right\\rceil\n\\]\nis sufficient. Here the certified minimum weight is $m=1$, so the generic choice is five. The extra identity $A^2=8I$ justifies stopping at four in this example.","\\vpage{12}{Walks are not simple paths}{MODEL C / A GRAPH WITH PARALLEL EDGES}{Traditional paper: Theorem 6.6 (p. 21)}\nConsider the directed graph below. The loop $\\ell$ has weight one, the two distinct edges $a,b$ from $0$ to $1$ each have weight two, and the return edge $r$ has weight one.","A walk may revisit a vertex and reuse an edge. At total weight five there are six walks from $0$ to $1$. Two use three loops followed by either $a$ or $b$. Four travel to $1$, return to $0$, and travel to $1$ again, with two independent choices of parallel edge.","Their multiplicities enter the polynomial adjacency matrix:\n\\[\nA(t)=\\begin{pmatrix}t&2t^2\\\\t&0\\end{pmatrix}.\n\\]\nThe coefficient $2$ remembers the two distinct edges. Replacing it by $1$ would count a different graph.","t&0\\end{pmatrix}.\n\\]\nThe coefficient $2$ remembers the two distinct edges. Replacing it by $1$ would count a different graph.","\\head{The first-edge decomposition}\nLet $a_n(s,t)$ count walks of total weight $n$ from $s$ to $t$. Removing the first edge gives\n\\[\na_n(s,t)=\\mathbf1_{s=t,\\,n=0}\n +\\sum_{\\substack{e:\\,\\mathrm{source}(e)=s","\\head{The first-edge decomposition}\nLet $a_n(s,t)$ count walks of total weight $n$ from $s$ to $t$. Removing the first edge gives\n\\[\na_n(s,t)=\\mathbf1_{s=t,\\,n=0}\n +\\sum_{\\substack{e:\\,\\mathrm{source}(e)=s\\\\w(e)\\le n}}\n a_{n-w(e)}(\\mathrm{target}(e),t).\n\\]\nThe empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most $n$, so enumeration at each fixed weight is finite.","w(e)\\le n}}\n a_{n-w(e)}(\\mathrm{target}(e),t).\n\\]\nThe empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most $n$, so enumeration at each fixed weight is finite.","\\vpage{13}{A finite graph produces its denominator}{MODEL C / AUTOMATIC RATIONALITY AND RECURRENCE}{Traditional paper: Theorem 6.6 (p. 21), Corollary 6.7 (p. 22)}\nFor the graph on V12, compute the determinant and adjugate rather than guessing a recurrence:\n\\[\nI-A(t)=\\begin{pmatrix}1-t&-2t^2\\\\-t&1\\end{pmatrix},\n\\quad Q(t)=1-t-2t^3,\n\\quad \\operatorname{adj}(I-A)_{0,1}=2t^2.\n\\]\nSince $Q(0)=1$, its inverse exists as a formal power series with integer coefficients.","-t&1\\end{pmatrix},\n\\quad Q(t)=1-t-2t^3,\n\\quad \\operatorname{adj}(I-A)_{0,1}=2t^2.\n\\]\nSince $Q(0)=1$, its inverse exists as a formal power series with integer coefficients.","\\vfig[.99]{zeta_recurrence}\n\\platecaption{The coefficients are independently computed by the first-edge recurrence and checked against the rational series through weight $24$. Literal edge-walk enumeration supplies a third check through weight $10$.}","From $QZ_{0,1}=2t^2$ one obtains\n\\[\na_0=a_1=0,\\quad a_2=2,\\qquad a_n=a_{n-1}+2a_{n-3}\\quad(n\\ge3).\n\\]\nThe numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","\\head{Three boundaries of this calculation}\nThis is a \\emph{formal} series identity. No analytic convergence at a chosen real or complex value of $t$ is needed. The canonical determinant need not be the minimal denominator after cancellation. Finally, positive weights prevent infinitely many walks at a fixed weight: a single weight-zero loop would already give the infinitely many distinct walks $1,\\ell,\\ell^2,\\ldots$ at weight zero.","\\takeaway{Finite positive-weight graph data determine a rational series and an eventual recurrence. The proof does not begin by assuming a certificate $QZ=P$ that already contains that conclusion.}","\\vpage{14}{Separate before dividing}{MODEL D / AN EVALUATED B\\'EZOUT CERTIFICATE}{Traditional paper: Theorem 7.5 (p. 25), Definition 7.6 (p. 26)}\nUse the integral involution\n\\[\nA=\\begin{pmatrix}1&1\\\\0&-1\\end{pmatrix},\\qquad A^2=I.\n\\]\nWith $F=A-I$ and $G=A+I$, the identities\n\\[\n-F+G=2I,\\qquad FG=0\n\\]\nare an explicit evaluated B\\'ezout certificate. Here all the required commutations follow because these operators are polynomials in the same matrix.","0&-1\\end{pmatrix},\\qquad A^2=I.\n\\]\nWith $F=A-I$ and $G=A+I$, the identities\n\\[\n-F+G=2I,\\qquad FG=0\n\\]\nare an explicit evaluated B\\'ezout certificate. Here all the required commutations follow because these operators are polynomials in the same matrix.","\\vfig[.96]{bezout_normalization}\n\\platecaption{The denominator-free numerator is quasi-idempotent: $N^2=2N$. Division by the nonzero scalar $2$ gives an idempotent over the fraction field. It does not yet make the projector integral on the selected lattice.}","The numerator $N=G$ acts by $2$ on $\\ker(A-I)$ and by zero on $\\ker(A+I)$. Thus\n\\[\nE_+=\\frac12N\n=\\begin{pmatrix}1&1/2","The numerator $N=G$ acts by $2$ on $\\ker(A-I)$ and by zero on $\\ker(A+I)$. Thus\n\\[\nE_+=\\frac12N\n=\\begin{pmatrix}1&1/2\\\\0&0\\end{pmatrix}\n\\]\nis the required separator. The complementary projector is $E_-=I-E_+$.","0&0\\end{pmatrix}\n\\]\nis the required separator. The complementary projector is $E_-=I-E_+$.","\\head{A denominator bound need not be minimal}\nThe same projector can be presented with the inflated certificate $-2F+2G=4I$. Its numerator is $2G$ and its normalization is still $E_+$. This supplies a two-digit clearing bound at $2$, although one digit is enough. Thus the valuation of a supplied certificate gives an upper bound, not automatically the least possible weight.","For the original sign factors the concrete resultant is $\\operatorname{Res}(x-1,x+1)=2$ and the bound is sharp. That equality is special to this displayed calculation, not a universal minimality theorem for every resultant certificate.","\\vpage{15}{Where the dyadic digit is lost}{MODEL D / THE EIGENLATTICE HAS INDEX TWO}{Traditional paper: Proposition 9.3 (p. 32)}\nThe eigenspaces are spanned by $(1,0)$ and $(-1,2)$. Their integral eigenvectors do not form an integral basis of $\\ZZ^2$. The change-of-basis determinant is two.","\\vfig[.94]{dyadic_eigenlattice}\n\\platecaption{An ordinary integer-lattice picture, not a drawing of the topology of $\\QQ_2$. Filled dots have even second coordinate. Unfilled dots lie in the other coset. The hollow point at $(1/2,0)$ records the image of an integral vector under $E_+$.}","Write\n\\[\nB=\\begin{pmatrix}1&-1\\\\0&2\\end{pmatrix},\\qquad\nAB=B\\diag(1,-1),\\qquad\nE_+=B\\diag(1,0)B^{-1}.\n\\]\nThe index-two eigenlattice explains the denominator in $B^{-1}$ and in the projector. Over the localized lattice $\\Lambda_p=\\ZZ_{(p)}^2$, the number $2$ is a unit for every odd prime $p$. Hence the local denominator weight is zero there and one at $p=2$.","0&2\\end{pmatrix},\\qquad\nAB=B\\diag(1,-1),\\qquad\nE_+=B\\diag(1,0)B^{-1}.\n\\]\nThe index-two eigenlattice explains the denominator in $B^{-1}$ and in the projector. Over the localized lattice $\\Lambda_p=\\ZZ_{(p)}^2$, the number $2$ is a unit for every odd prime $p$. Hence the local denominator weight is zero there and one at $p=2$.","\\head{The special fiber does not retain the rational splitting}\nModulo two, $x-1=x+1$ and $A$ becomes a nontrivial unipotent matrix. The entries $1/2$ in $E_\\pm$ cannot be reduced modulo two. By contrast, the coefficient selectors on V6 use only zero and one and remain available.","\\vpage{16}{One extra digit is genuinely necessary}{MODEL D / A SHARP PRECISION-DELAY EXAMPLE}{Traditional paper: Theorem 7.7 (p. 26), Proposition 9.3 (p. 32)}\nThe projector acts by\n\\[\nE_+(x,y)=(x+y/2,0).\n\\]\nIts factor $1/2$ can consume one dyadic digit. The claim concerns differences: if the inputs agree modulo $2^{m+1}\\Lambda$, their projected differences lie in $2^m\\Lambda$.","\\vfig[.97]{precision_delay}\n\\platecaption{For $m=3$, inputs differing by $(0,8)$ agree modulo eight, but their projected outputs differ by $(4,0)$ and do not. Raising the input agreement to modulo sixteen makes the projected outputs agree modulo eight.}","The example works at every positive depth:\n\\[\nE_+(0,2^m)=(2^{m-1},0)\\notin2^m\\Lambda,\n\\qquad\nE_+(0,2^{m+1})=(2^m,0)\\in2^m\\Lambda.\n\\]\nThus a zero-delay promise fails, while one extra input digit is sufficient. This witnesses sharpness for the specified projector and lattice.","\\head{The general calculation is just as short}\nIf $\\pi^sE\\Lambda\\subseteq\\Lambda$ and $x-y=\\pi^{m+s}v$, then\n\\[\nE(x-y)=\\pi^m(\\pi^sEv)\\in\\pi^m\\Lambda.\n\\]\nNo rounding procedure is hiding in this identity. Nor does it authorize reducing $E$ modulo $\\pi$ when its entries are nonintegral. Individual projected points may lie in a fractional lattice. The theorem controls their difference at the stated integral scale.","\\vpage{17}{See every Smith fiber modulo eight}{MODEL E / TWO DIRECTIONAL WEIGHTS}{Traditional paper: Theorem 10.6 (p. 36)}\nConsider the actual map of finite sets\n\\[\n(\\ZZ/8\\ZZ)^2\\longrightarrow(\\ZZ/8\\ZZ)^2,\n\\qquad (x,y)\\longmapsto(2x,4y).\n\\]\nThe first coordinate of the output must be even. The second must be zero or four. Those are precisely the attainable targets in the right panel.","For a specific attainable target $(2,4)$,\n\\[\n2x\\equiv2\\pmod8\\iff x\\in\\{1,5\\},\\qquad\n4y\\equiv4\\pmod8\\iff y\\in\\{1,3,5,7\\}.\n\\]\nThe coordinate choices are independent, giving $2\\cdot4=8$ solutions. The entire carrier is partitioned consistently:\n\\[\n8\\text{ attainable targets}\\ \\times\\ 8\\text{ inputs per target}=64.\n\\]\nFor an inadmissible target, such as $(1,4)$ or $(2,2)$, the fiber is empty. The same nonzero cardinality formula must not be applied to it.","\\head{The arithmetic behind the picture}\nMultiplication by $p^s$ modulo $p^L$ has kernel size $p^{\\min(s,L)}$. Its image consists exactly of residues whose canonical representatives are divisible by that same power. The displayed model has $p=2$, $L=3$ and weights $(1,2)$.","\\vpage{18}{Transport the whole solution fiber}{MODEL E / EXPLICIT SMITH COORDINATES}{Traditional paper: Theorem 10.6 (p. 36)}\nThe preceding diagonal system is connected to a nondiagonal one by the unimodular change of variables\n\\[\nM=\\begin{pmatrix}2&2\\\\0&4\\end{pmatrix},\\quad\nV=\\begin{pmatrix}1&-1\\\\0&1\\end{pmatrix},\\quad\nMV=\\diag(2,4)=D.\n\\]\nSince $\\det V=1$, $x=Vy$ is bijective modulo eight. Thus $Mx=b$ is equivalent to $Dy=b$.","0&1\\end{pmatrix},\\quad\nMV=\\diag(2,4)=D.\n\\]\nSince $\\det V=1$, $x=Vy$ is bijective modulo eight. Thus $Mx=b$ is equivalent to $Dy=b$.","\\vfig[.95]{smith_coordinate_transport}\n\\platecaption{The eight solutions for the target $(2,4)$, before and after the coordinate shear. Coordinates are residues in $\\{0,\\ldots,7\\}$. No interpolation on this modular grid is intended.}","For example, the diagonal solution $y=(1,3)$ corresponds to\n\\[\nx=Vy=(-2,3)\\equiv(6,3)\\pmod8,\n\\]\nand $M(6,3)^\\mathsf T=(18,12)^\\mathsf T\\equiv(2,4)^\\mathsf T$. Applying the inverse shear recovers the original solution.","\\vpage{19}{Precision exposes directional freedom}{MODEL E / A DEPTH-BY-DEPTH FIBER BUDGET}{Traditional paper: Theorem 10.6 (p. 36)}\nTake $p=2$ and the supplied diagonal weights $(s_0,s_1,s_2)=(0,1,3)$. At depth $L$, an admissible target has\n\\[\n2^{W(L)}\\text{ solutions},\\qquad\nW(L)=\\min(0,L)+\\min(1,L)+\\min(3,L).\n\\]\nEach direction contributes only the amount visible at the current depth.","\\vfig[.93]{smith_depth_budget}\n\\platecaption{The stacked height records the exponent $W(L)$. The label above each bar records the actual number of solutions. The zero-weight direction contributes no branching. This is a finite cardinality calculation, not an information-theoretic assumption about random inputs.}","The values are\n\\[\n\\begin{array}{c|rrrrrr}\nL&1&2&3&4&5&6\\\\\\hline\nW(L)&2&3&4&4&4&4\\\\\n\\#\\text{ solutions}&4&8&16&16&16&16.\n\\end{array}\n\\]\nOnce $L\\ge3$, the nonzero weights have been fully exposed and the admissible fiber cardinality stabilizes at sixteen. The input space still grows as $2^{3L}$. The number of attainable targets grows accordingly.","\\#\\text{ solutions}&4&8&16&16&16&16.\n\\end{array}\n\\]\nOnce $L\\ge3$, the nonzero weights have been fully exposed and the admissible fiber cardinality stabilizes at sixteen. The input space still grows as $2^{3L}$. The number of attainable targets grows accordingly.","At depth six, $(1,2,8)$ is admissible and gives sixteen solutions. Replacing the last coordinate by four makes the target inadmissible and gives zero solutions, even though the weights and depth are unchanged.","\\head{What changes with a zero diagonal factor?}\nA literal zero factor contributes $L$, not a fixed eventual weight, and requires a zero target coordinate. Its fiber therefore keeps growing with the modulus. That case must not be mistaken for a finite nonzero Smith weight.","\\vpage{20}{Filter first or split first: same output}{MODEL F / A FINITE POLYPHASE REALIZATION}{Traditional paper: Section B.1 (p. 47)}\nUse the finitely supported signal $x=(1,2,3,4,5,6)$ and the filter $h=(1,2,1)$. Outside the displayed support, the input is zero. Let $e_m=x_{2m}$ and $o_m=x_{2m+1}$.","If $w$ denotes one delay on a phase sequence, the filter becomes\n\\[\n\\begin{pmatrix}y_{\\rm even}","If $w$ denotes one delay on a phase sequence, the filter becomes\n\\[\n\\begin{pmatrix}y_{\\rm even}\\\\y_{\\rm odd}\\end{pmatrix}\n=\n\\underbrace{\\begin{pmatrix}1+w&2w\\\\2&1+w\\end{pmatrix}}_{H_{\\rm phase}(w)}\n\\begin{pmatrix}e\\\\o\\end{pmatrix}.\n\\]\nThe cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","Here $e=1+3w+5w^2$ and $o=2+4w+6w^2$, so direct polynomial multiplication gives\n\\[\ny_{\\rm even}=1+8w+16w^2+17w^3,\n\\qquad\ny_{\\rm odd}=4+12w+20w^2+6w^3.\n\\]\nThis is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.","\\vpage{21}{Two Smith constructions, different roles}{MODEL G / A NONLINEAR PHASE ORBIT}{Traditional paper: Section 12.3 (p. 42), Section 12.3, Smith-phase identity (p. 43)}\nSmith \\emph{normal form} concerns linear congruences. The Smith \\emph{impedance chart} uses the nonlinear coordinate change\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThey share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points $\\Gamma_k=\\tfrac12 e^{ik\\pi/4}$.","AR PHASE ORBIT}{Traditional paper: Section 12.3 (p. 42), Section 12.3, Smith-phase identity (p. 43)}\nSmith \\emph{normal form} concerns linear congruences. The Smith \\emph{impedance chart} uses the nonlinear coordinate change\n\\[\n\\Gamma=\\frac{z-1}{z+1},\\qquad z=\\frac{1+\\Gamma}{1-\\Gamma}.\n\\]\nThey share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points $\\Gamma_k=\\tfrac12 e^{ik\\pi/4}$.","\\vfig[.96]{smith_chart_orbit}\n\\platecaption{Matching labels identify the same point in the two coordinates. All eight points are computed algebraically. Only their placement on the page is rendered numerically. The outer left circle is $|\\Gamma|=1$, not part of the sampled radius-$1/2$ orbit.}","The half-turn $\\Gamma\\mapsto-\\Gamma$ gives\n\\[\nz\\longmapsto\\frac1z,\\qquad z\\ne0,-1.\n\\]\nThus the labels $0$ and $4$ are $3$ and $1/3$. Label $2$ is $(3+4i)/5$, and the opposite point is its reciprocal. The domain restrictions come from the forward and inverse coordinate denominators.","\\takeaway{The chart is a geometric realization of phase action. Its nonlinear coordinate sensitivity near $\\Gamma=\\pm1$ is not the Smith congruence-fiber count and not a new linear projector theorem.}","\\vpage{22}{A small defect can grow under powers}{MODEL H / EXACT TELESCOPING, EXPLICIT GROWTH}{Traditional paper: Theorem 11.1 (p. 37), Corollary 11.3 (p. 38)}\nChoose an invertible coordinate shear but independently keep the same numerical operator at both levels:\n\\[\nC=\\begin{pmatrix}1&\\varepsilon\\\\0&1\\end{pmatrix},\\quad\nT_f=T_c=\\diag(1,2),\\quad \\varepsilon=\\frac1{16}.\n\\]\nThen $\\Delta=CT_f-T_cC=\\varepsilon E_{01}$. Exact coordinate invertibility has not forced these independently selected operators to intertwine.","0&1\\end{pmatrix},\\quad\nT_f=T_c=\\diag(1,2),\\quad \\varepsilon=\\frac1{16}.\n\\]\nThen $\\Delta=CT_f-T_cC=\\varepsilon E_{01}$. Exact coordinate invertibility has not forced these independently selected operators to intertwine.","\\vfig[.95]{power_defect_growth}\n\\platecaption{Actual defects and certified upper bounds in the induced infinity norm, both divided by $\\varepsilon$. All heights are exact integers. The upper estimate is valid but need not be tight.}","Direct multiplication gives\n\\[\nCT_f^n-T_c^nC=\\varepsilon(2^n-1)E_{01}.\n\\]\nThe general telescoping formula explains every contribution:\n\\[\nCT_f^n-T_c^nC\n=\\sum_{k=0}^{n-1}T_c^{n-1-k}\\Delta T_f^k.\n\\]\nSince both operator norms are two, the source theorem bounds the error by $n2^{n-1}\\varepsilon$. It does not promise a uniform-in-$n$ small error merely because $\\varepsilon$ is small.","Had we chosen the transported operator $T_c'=CT_fC^{-1}$, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.","\\vpage{23}{Resolvents amplify errors near poles}{MODEL H / A DOMAIN-SENSITIVE COMPARISON}{Traditional paper: Theorem 11.5 (p. 38)}\nFor the same model, take $z\\notin\\{1,2\\}$ and form the actual inverse matrices\n\\[\nR_f=(zI-T_f)^{-1},\\qquad R_c=(zI-T_c)^{-1}.\n\\]\nTheir off-diagonal intertwining error is\n\\[\nCR_f-R_cC=R_c\\Delta R_f\n=\\frac{\\varepsilon}{(z-1)(z-2)}E_{01}.\n\\]\nBoth inverse hypotheses matter. Neither resolvent is asserted at a pole.","\\vfig[.94]{resolvent_amplification}\n\\platecaption{The bars show exact amplification factors for five rational choices $z>2$. Rows are separate samples, not equally spaced positions on the $z$ axis. The coefficient $\\varepsilon=1/16$ has been divided out.}","At $z=3$ the amplification factor is only $1/2$. At $z=201/100$, it is $10000/101$, so a small one-step defect can create a much larger resolvent defect. This follows from the exact formula rather than from unstable floating-point subtraction of two nearly singular inverses.","The certified general norm estimate is\n\\[\n\\|CR_f-R_cC\\|\n\\le \\|R_c\\|\\,\\|\\Delta\\|\\,\\|R_f\\|.\n\\]\nThe paper also proves a one-sided-witness version: a left inverse for the coarse shift and a right inverse for the fine shift suffice for the algebraic identity. That does not turn arbitrary one-sided witnesses into a claim of two-sided resolvent existence.","Weighted inversion & A two-sided inverse of $D$, a multiplicative filtration, and positive certified gains. & Convergence in every possible complete filtered algebra.","Smith congruences & Concrete diagonal data modulo $p^L$, with $p>1$. & An automatically constructed general-DVR Smith normal form.","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round]\n\\node[] at (0.70000,3.05000) {$A$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (0.35000,2.15000) {$1$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (1.05000,2.15000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.00000,1.10000) rectangle ++(0.70000,0.70000);\n\\node[] at (0.35000,1.45000) {$0$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round]\n\\node[] at (0.70000,3.05000) {$A$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (0.35000,2.15000) {$1$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (1.05000,2.15000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.00000,1.10000) rectangle ++(0.70000,0.70000);\n\\node[] at (0.35000,1.45000) {$0$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++(0.70000,0.70000);\n\\node[] at (1.05000,1.45000) {$-1$};\n\\node[] at (4.30000,3.05000) {$N=I+A$};\n\\draw[fill=black!12,draw=black!45,line wid","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round]\n\\node[] at (0.70000,3.05000) {$A$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (0.35000,2.15000) {$1$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (1.05000,2.15000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.00000,1.10000) rectangle ++(0.70000,0.70000);\n\\node[] at (0.35000,1.45000) {$0$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++(0.70000,0.70000);\n\\node[] at (1.05000,1.45000) {$-1$};\n\\node[] at (4.30000,3.05000) {$N=I+A$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (3.60000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (3.95000,2.15000) {$2$};\n\\draw[fill=black!12,draw=black!45,line wid","ll=black!12,draw=black!45,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (0.35000,2.15000) {$1$};\n\\draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.80000) rectangle ++(0.70000,0.70000);\n\\node[] at (1.05000,2.15000) 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rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,2.70000) {$9$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,2.30000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,2.70000) {$6$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,2.30000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,2.70000) {$3$};\n\\node[anchor=east] at (-0.15000,1.85000) {$b=1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.85000) {$4$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.85000) {$1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.85000) {$10$};\n\\draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(","0000,2.30000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,2.70000) {$6$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,2.30000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,2.70000) {$3$};\n\\node[anchor=east] at (-0.15000,1.85000) {$b=1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.85000) {$4$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.85000) {$1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.85000) {$10$};\n\\draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[text=white] at (4.02500,1.85000) {$7$};\n\\node[anchor=east] at (-0.15000,1.00000) {$b=2$};\n\\draw[fill=black!9,draw","5000,2.30000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,2.70000) {$3$};\n\\node[anchor=east] at (-0.15000,1.85000) {$b=1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.85000) {$4$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.85000) {$1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.85000) {$10$};\n\\draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[text=white] at (4.02500,1.85000) {$7$};\n\\node[anchor=east] at (-0.15000,1.00000) {$b=2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.00000) {$8$};\n\\draw[fill=black!9,draw","fill=black!9,draw=black!45,line width=.35pt,] (0.00000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.85000) {$4$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.85000) {$1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.85000) {$10$};\n\\draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[text=white] at (4.02500,1.85000) {$7$};\n\\node[anchor=east] at (-0.15000,1.00000) {$b=2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.00000) {$8$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=",",draw=black!45,line width=.35pt,] (1.15000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.85000) {$1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.85000) {$10$};\n\\draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[text=white] at (4.02500,1.85000) {$7$};\n\\node[anchor=east] at (-0.15000,1.00000) {$b=2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.00000) {$8$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,li","00) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.85000) {$1$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.85000) {$10$};\n\\draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[text=white] at (4.02500,1.85000) {$7$};\n\\node[anchor=east] at (-0.15000,1.00000) {$b=2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.00000) {$8$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.",") rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.85000) {$10$};\n\\draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(1.15000,0.80000);\n\\node[text=white] at (4.02500,1.85000) {$7$};\n\\node[anchor=east] at (-0.15000,1.00000) {$b=2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.00000) {$8$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[]","00) rectangle ++(1.15000,0.80000);\n\\node[text=white] at (4.02500,1.85000) {$7$};\n\\node[anchor=east] at (-0.15000,1.00000) {$b=2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.00000) {$8$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.200","fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (0.57500,1.00000) {$8$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19","fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","+(1.15000,0.80000);\n\\node[] at (1.72500,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","0,1.00000) {$5$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","raw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","00,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","node[] at (2.87500,1.00000) {$2$};\n\\draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","l=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}",",0.80000);\n\\node[] at (4.02500,1.00000) {$11$};\n\\node[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","[] at (0.57500,3.55000) {$a=0$};\n\\node[] at (1.72500,3.55000) {$a=1$};\n\\node[] at (2.87500,3.55000) {$a=2$};\n\\node[] at (4.02500,3.55000) {$a=3$};\n\\node[] at (2.30000,4.15000) {$a=n\\bmod4$};\n\\node[] at (7.50000,3.20000) {$b=n\\bmod3$};\n\\node[] at (7.50000,2.25000) {$(a,b)=(3,1)\\ \\Longleftrightarrow\\ n=7\\pmod{12}$};\n\\node[] at (7.50000,1.25000) {$7,\\ 19,\\ 31,\\ldots$};\n\\node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round]\n\\node[anchor=east] at (-0.20000,3.70000) {$x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (2.25000,3.40000) rectangle ++","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round]\n\\node[anchor=east] at (-0.20000,3.70000) {$x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (2.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,3.40000) rectangle ++","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round]\n\\node[anchor=east] at (-0.20000,3.70000) {$x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (2.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,3.40000) rectangle ++(","ll=black!22,draw=black!45,line width=.35pt,] (0.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (2.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,3.40000) rectangle ++","ll=black!22,draw=black!45,line width=.35pt,] (0.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (2.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.87500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (5.25000,3.40000) rectangle ++","ll=black!0,draw=black!45,line width=.35pt,] (1.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (2.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.87500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (5.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (5.62500,3.70000) {$1$};\n\\node[anchor=east] at (-0.20000,2.85000) {$P_{2,0}x$};\n\\draw[fill=black!22,draw=bla","ll=black!22,draw=black!45,line width=.35pt,] (2.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.87500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (5.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (5.62500,3.70000) {$1$};\n\\node[anchor=east] at (-0.20000,2.85000) {$P_{2,0}x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=bla","ll=black!22,draw=black!45,line width=.35pt,] (3.00000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,3.70000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.87500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (5.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (5.62500,3.70000) {$1$};\n\\node[anchor=east] at (-0.20000,2.85000) {$P_{2,0}x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,2.85000) {$0$};\n\\draw[fill=black!0,draw=blac","ll=black!0,draw=black!45,line width=.35pt,] (3.75000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.87500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (5.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (5.62500,3.70000) {$1$};\n\\node[anchor=east] at (-0.20000,2.85000) {$P_{2,0}x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black","5000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,3.70000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.87500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (5.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (5.62500,3.70000) {$1$};\n\\node[anchor=east] at (-0.20000,2.85000) {$P_{2,0}x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (2.25000,2.55000) rectangle ++(0","40000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.87500,3.70000) {$1$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (5.25000,3.40000) rectangle ++(0.75000,0.60000);\n\\node[] at (5.62500,3.70000) {$1$};\n\\node[anchor=east] at (-0.20000,2.85000) {$P_{2,0}x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (2.25000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,2.85000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,2.55000) rectangle ++(0",".40000) rectangle ++(0.75000,0.60000);\n\\node[] at (5.62500,3.70000) {$1$};\n\\node[anchor=east] at (-0.20000,2.85000) {$P_{2,0}x$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (0.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (2.25000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,2.85000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,2.55000) rectangle ++(0","ill=black!22,draw=black!45,line width=.35pt,] (0.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (0.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (0.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (2.25000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,2.85000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (3.37500,2.85000) {$1$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (3.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (4.12500,2.85000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,2.55000) rectangle ++(","fill=black!0,draw=black!45,line width=.35pt,] (0.75000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.12500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (1.50000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (1.87500,2.85000) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (2.25000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at (2.62500,2.85000) {$0$};\n\\draw[fill=black!22,draw=black!45,line width=.35pt,] (3.00000,2.55000) rectangle ++(0.75000,0.60000);\n\\node[] at 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{$\\frac{1}{2}$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (9.10000,1.00000) rectangle ++(0.65000,0.65000);\n\\node[] at (9.42500,1.32500) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (9.75000,1.00000) rectangle ++(0.65000,0.65000);\n\\node[] at (10.07500,1.32500) {$0$};\n\\node[font=\\sffamily\\footnotesize] at (10.40000,-0.15000) {\\shortstack{$2E_+$ integral",",draw=black!45,line width=.35pt,] (9.75000,1.65000) rectangle ++(0.65000,0.65000);\n\\node[] at (10.07500,1.97500) {$\\frac{1}{2}$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (9.10000,1.00000) rectangle ++(0.65000,0.65000);\n\\node[] at (9.42500,1.32500) {$0$};\n\\draw[fill=black!0,draw=black!45,line width=.35pt,] (9.75000,1.00000) rectangle ++(0.65000,0.65000);\n\\node[] at (10.07500,1.32500) {$0$};\n\\node[font=\\sffamily\\footnotesize] at (10.40000,-0.15000) {\\shortstack{$2E_+$ integral","000);\n\\node[] at (10.07500,1.97500) {$\\frac{1}{2}$};\n\\draw[fill=black!0,draw=black!45,line 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{$2$};\n\\node[]","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round]\n\\node[anchor=west] at (0.10000,4.20000) {$1$};\n\\draw[fill=black!20,draw=black!45,line width=.35pt,] (1.30000,3.97000) rectangle ++(0.15000,0.46000);\n\\node[] at (5.90000,4.20000) {$0$};\n\\node[] at (7.85000,4.20000) {$I$};\n\\node[] at (10.30000,4.20000) {keep};\n\\node[anchor=west] at (0.10000,3.52000) {$a$};\n\\draw[fill=black!20,draw=black!45,line width=.35pt,] (1.30000,3.29000) rectangle ++(0.73000,0.46000);\n\\node[] at (5.90000,3.52000) {$1$};\n\\node[] at (7.85000,3.52000) {$-2E_{01}$};\n\\node[] at (10.30000,3.52000) {keep};\n\\node[anchor=west] at (0.10000,2.84000) {$b$};\n\\draw[fill=black!20,draw=black!45,line width=.35pt,] (1.30000,2.61000) rectangle ++(1.46000,0.46000);\n\\node[] at (5.90000,2.84000) {$2$};\n\\node[] at (7.85000,2.84000) {$-4E_{10}$};\n\\node[] at (10.30000,2.84000) {keep};\n\\node[ancho","{$1$};\n\\draw[fill=black!20,draw=black!45,line width=.35pt,] 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