\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,mathtools} \usepackage{aliascnt} \usepackage{microtype} \usepackage[margin=1.15in]{geometry} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \urlstyle{same} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{doi:#1}}} \usepackage[nameinlink,noabbrev]{cleveref} \hypersetup{pdftitle={Integral tail signatures and the Hilbert--Smith conjecture}, pdfauthor={A. Dabrowski's AI agents}, pdfsubject={Integral signature obstruction to effective p-adic manifold actions}, pdfkeywords={Hilbert-Smith conjecture, controlled algebra, equivariant signature}} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newaliascnt{proposition}{theorem} \newtheorem{proposition}[proposition]{Proposition} \aliascntresetthe{proposition} \newaliascnt{lemma}{theorem} \newtheorem{lemma}[lemma]{Lemma} \aliascntresetthe{lemma} \newaliascnt{corollary}{theorem} \newtheorem{corollary}[corollary]{Corollary} \aliascntresetthe{corollary} \theoremstyle{definition} \newaliascnt{definition}{theorem} \newtheorem{definition}[definition]{Definition} \aliascntresetthe{definition} \theoremstyle{remark} \newaliascnt{remark}{theorem} \newtheorem{remark}[remark]{Remark} \aliascntresetthe{remark} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\prop}{prop} \DeclareMathOperator{\Kar}{Kar} \DeclareMathOperator{\Free}{Free} \DeclareMathOperator{\Cone}{Cone} \DeclareMathOperator{\Sign}{Sign} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\tr}{tr} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\CP}{\mathbb {CP}} \newcommand{\Zp}{\mathbb Z_p} \newcommand{\Linf}{L^{\langle-\infty\rangle}} \newcommand{\Lspec}{\mathbf L^{\langle-\infty\rangle}} \title[Integral tail signatures and Hilbert--Smith]{Integral tail signatures and the Hilbert--Smith conjecture} % Author information: name, affiliation, and contact address. \author{A. Dabrowski's AI agents} \email{alex@dabrow.cc} \date{October 8, 2026} \begin{document} \begin{abstract} The obstruction to an effective $p$-adic manifold action is an integer signature tail. On asymptotically controlled rational modules over one fixed free $C_p$-space, finite-cover Mayer--Vietoris makes the difference between $p$ copies and quotient-permutation tensoring nilpotent. Every degree-four class therefore has eventually $p$-divisible signatures. An effective action gives a class of signature $1$: coordinate averaging provides degree-one sphere control, free sheets turn quotient representatives into a compatible homotopy idempotent, and its normalized Poincar\'e corner forgets to a scalar manifold germ. Successive localization boundaries recover one copy of $\CP^2$, up to sign. The resulting contradiction proves the Hilbert--Smith conjecture. \end{abstract} \maketitle \section{Introduction}\label{sec:introduction} The Hilbert--Smith conjecture extends Hilbert's fifth problem from locally Euclidean groups to groups acting on manifolds. The cases of one- and two-manifolds are classical \cite{Pa19}; Pardon proved the three-dimensional case \cite{Pa13}. In arbitrary dimension, the conjecture holds for Lipschitz actions \cite{RS97} and quasiconformal actions \cite{Mar99}. We prove the following. \begin{theorem}[$p$-adic exclusion]\label{thm:main} For every prime $p$, a continuous action of $\Zp$ on a connected finite-dimensional manifold has nontrivial kernel. \end{theorem} \begin{corollary}[Hilbert--Smith]\label{cor:HS} A locally compact Hausdorff group acting continuously and effectively on a connected finite-dimensional manifold is a Lie group with its given topology. \end{corollary} Manifolds are topological and Hausdorff, and may have boundary. Lie groups are allowed arbitrarily many connected components. The construction takes place in one coordinate chart; it requires no global orientation or triangulation of the acted-on manifold. The standard reduction to $p$-adic exclusion uses locally compact group structure and Newman's theorem \cite{Pa19}. The obstruction is an integer signature tail. For cyclic groups $G_i=C_{p^{a_i}}$ with index-$p$ subgroups $P_i$, and a fixed compact free $C_p$-space $T$, tensoring with the positive permutation form $\tau_i=\Q[G_i/P_i]$ is locally $p$ copies of the identity. Controlled Mayer--Vietoris therefore makes $p\,\mathrm{id}-\tau\otimes(-)$ nilpotent, and equivariant signature characters force every class in $L_4(\mathcal A_{\mathbf G}(T))$ to have eventually $p$-divisible ordinary signatures. An effective $\Zp$-action, however, would supply a class with signature $1$. Coordinate averaging gives an invariant degree-one sphere control map. With this control, fine local chain models turn finite quotient representatives into a homotopy action. Adding free sheets cancels the detector displacement, allowing averaging to produce a duality-compatible homotopy idempotent. Its normalized corner forgets to the scalar manifold germ of $S^n\times\CP^2$. Successive localization boundaries then recover one copy of $\CP^2$, up to sign. The resulting signature contradicts the divisibility above and proves the theorem. The construction and boundary evaluation use Karoubi localization \cite{CP95}, controlled Poincar\'e--Lefschetz duality \cite{Ran99}, and algebraic Poincar\'e pairs and unions \cite{Ran80I,Ran80II}. Idempotent splitting uses \cite{BS01}; the local carrier and normalization calculations are given below. \subsection*{Conventions} All actions are jointly continuous. Chain coefficients are rational, but the $L$-groups and signature targets are not rationalized. Complexes have a single finite degree range, independent of the sequence index; ranks and coefficient sizes are unrestricted. In the asymptotic categories below, equality means eventual equality of matrices, so finite initial segments of sequences may be discarded. In an exterior quotient, equality means that the defect factors through an exterior-supported object by asymptotically small maps. We use maximum product metrics and take empty suprema to be zero. We use ultimate lower quadratic $L$-theory, $L_k(\mathcal A)=\pi_k\Lspec(\mathcal A)$ \cite[Definition 4.16]{CP95}, \cite[\S17]{Ran92}. The properties used below are functoriality and additivity, Karoubi localization with natural connecting maps \cite[Theorem 4.2]{CP95}, projective-to-ultimate decoration maps, and invariance under idempotent completion \cite[Theorem 17.2 and Definition 17.7]{Ran92}. For localization boundaries we use the pair description in \cite[proof of Theorem 4.1]{CP95}. Small skeletons (one representative per isomorphism class) avoid set-theoretic size issues in $K$- and $L$-theory. \section{Asymptotic categories and support}\label{alg:categories} Fix $G_i=C_{p^{a_i}}$, $a_i\ge1$, index-$p$ subgroups $P_i$, and $G_i/P_i=C_p$. Let $X$ be a compact metric $C_p$-space with isometric action; each $G_i$ acts through $G_i/P_i$. \begin{definition}\label{alg:category} An object of $\mathcal A_{\mathbf G}(X)$ is a sequence of finite free based modules $M_i=\Q[B_i]$, with $B_i$ a free $G_i$-set and equivariant labels $\ell_i:B_i\to X$. Morphisms are sequences of equivariant rational matrices, modulo eventual equality, satisfying \begin{equation}\label{alg:propagation} \prop(u_i)=\sup_{(u_i)_{b'b}\ne0}d(\ell_i'(b'),\ell_i(b))\longrightarrow0. \end{equation} Duality fixes labeled bases and transposes matrices. Trivial groups give $\mathcal A_{\mathbf1}(X)$. Ranks and coefficients are unrestricted. \end{definition} The propagation bounds \begin{equation}\label{alg:propbounds} \prop(vu)\le\prop(v)+\prop(u),\qquad \prop(u+v)\le\max\{\prop(u),\prop(v)\} \end{equation} show that composition and addition are well defined. Forgetting equivariance and pushing labels through uniformly continuous equivariant maps give duality-preserving functors. For a closed invariant $S\subset X$, let $\mathcal A_S=\mathcal A_{\mathbf G,S}(X)$ consist of objects whose labels approach $S$ uniformly, with $\mathcal A_\varnothing=0$. Write $I_S$ for the ideal of maps factoring through $\mathcal A_S$. For a closed invariant exterior $Z\subset X$, put \begin{equation}\label{alg:exterior} \mathcal B_{\mathbf G,Z}(X)=\mathcal A_{\mathbf G}(X)/\mathcal A_Z. \end{equation} A map belongs to $I_S$ exactly when both endpoints of every nonzero entry approach $S$ uniformly: necessity follows from propagation, and sufficiency by restricting the source to its nonzero columns. \begin{lemma}[Support calculus]\label{alg:filtration}\label{alg:excision} \label{alg:relabel}\label{alg:separation} The inclusion $\mathcal A_S\subset\mathcal A_{\mathbf G}(X)$ is a duality-invariant Karoubi filtration \cite[Definition 1.27]{CP95}. For a closed invariant cover $X=A\cup B$, with $Y=A\cap B$, there are involution-preserving equivalences \begin{equation}\label{alg:excision-equivalence} \mathcal A_A\cap\mathcal A_B=\mathcal A_Y,\qquad \mathcal A_A/\mathcal A_Y\simeq\mathcal A_{\mathbf G}(X)/\mathcal A_B. \end{equation} For nonempty closed invariant $S$, $\mathcal A_{\mathbf G}(S)\simeq\mathcal A_S$. If $d(S,Z)>0$, $\mathcal A_S\to\mathcal B_{\mathbf G,Z}(X)$ is fully faithful. \end{lemma} \begin{proof} For a null sequence $r_i\ge0$, split each basis at distance $r_i$ from $S$. These equivariant orthogonal splits are filtered by maxima and compatible with sums. A map to or from an object supported within distance $a_i$ of $S$, with propagation $\epsilon_i$, factors through the near summand for $r=a+\epsilon$. This verifies the filtration axioms. By compactness, points approaching both $A$ and $B$ approach $Y$. Splitting labels into $A$ and $B\setminus A$ gives essential surjectivity of the quotient equivalence. For a factorization between $A$-supported objects through a $B$-supported object, retain the intermediate rows reached by the first factor. They approach both sets, so the factorization is through $\mathcal A_Y$. This proves full faithfulness, including $Y=\varnothing$ by compact separation. Relabel each free orbit using a nearest point of $S$ and extend equivariantly. The identity matrices are natural orthogonal isomorphisms with propagation tending to zero; relabeling changes a map's propagation by at most its two labeling errors. Finally, the endpoint description of $I_Z$ makes every map between $S$-supported objects in that ideal eventually zero when $d(S,Z)>0$. \end{proof} Splits precede idempotent completion, denoted $\Kar$; an object $(M,e)$ carries the support of $M$. \section{Integer signatures and tails}\label{alg:tails} For a finite group $G$, the correspondence between invariant rational forms $b$ and group-ring forms uses the unnormalized trace: \begin{equation}\label{alg:trace} \lambda(v,w)=\sum_{g\in G}b(v,gw)g,\qquad \tr_1\lambda(v,w)=b(v,w). \end{equation} Thus the coefficient-of-identity trace $\tr_1$ makes forgetting equivariance duality-preserving, including on projective summands. The half-unit comparison and \cite[Theorem 5.2]{Ran89} identify the projectively decorated group $L_4^p(\Q[G])$ with the Witt group of nonsingular invariant symmetric rational forms. Such a form has signature representation \begin{equation}\label{alg:signature} \Sign_G(V,b)=[V^+_{\C}]-[V^-_{\C}]\in R_{\C}(G), \end{equation} where $R_\C(G)$ is the complex representation ring and $V^\pm$ are the positive and negative spectral subspaces on $V\otimes_\Q\R$ for an invariant positive inner product. The representation is independent of this choice: projection along one negative subspace identifies any two positive subspaces, by inertia, and similarly for negative ones. To see that it descends to the Witt group, let $L$ be an invariant Lagrangian and choose an invariant complement $K$ by averaging. Replacing $u\in K$ by $u-c(u)/2$, where $c:K\to L$ satisfies $b(cu,v)=b(u,v)$ for $u,v\in K$, makes $K$ Lagrangian. Hence metabolic forms have zero signature representation. The character of $\Sign_G(V,b)$ at the identity is the ordinary integer signature $\sigma(V,b)$; tensoring with a positive form multiplies characters. \begin{lemma}[Cyclic arithmetic]\label{alg:cyclotomic} For $G=C_{p^a}=\langle g\rangle$, if $\Sign_G(V,b)(g)=0$, then $p\mid\sigma(V,b)$. \end{lemma} \begin{proof} Write the signature character as $\sum_{j=0}^{p^a-1}m_j\chi_j$, $m_j\in\Z$, with $\chi_j(g)=\zeta_{p^a}^j$, where $\zeta_{p^a}$ is a primitive $p^a$-th root of unity. Then $F(t)=\sum_jm_jt^j$ is divisible in $\Z[t]$ by the $p^a$-th cyclotomic polynomial $\Phi_{p^a}(t)$, so $F(1)=\sigma(V,b)$ is divisible by $\Phi_{p^a}(1)=p$. \end{proof} Write $\Free(R)$ for the category of finite free based $R$-modules. Put $\mathcal P=\prod_i\Free(\Q[G_i])$ and let $\mathcal S$ be its finite-support subcategory. Forgetting labels gives a functor to \begin{equation}\label{alg:Vcategory} \mathcal V_{\mathbf G}=\mathcal P/\mathcal S, \end{equation} whose morphisms are sequences modulo eventual equality. \begin{proposition}[Tail extraction]\label{alg:extraction} There is a natural homomorphism \begin{equation}\label{alg:tail-isomorphism} L_4(\mathcal V_{\mathbf G})\longrightarrow \prod_iL_4^p(\Q[G_i])/\bigoplus_iL_4^p(\Q[G_i]), \end{equation} given by the coordinate classes of any lift to $L_4(\mathcal P)$. It commutes with forgetting equivariance and positive permutation tensor functors. \end{proposition} \begin{proof} The semisimple rings $\Q[G_i]$ have vanishing negative $K$-groups \cite[III, \S4]{Wei13}, also as free-module categories \cite[\S11, p.~101]{Ran92}. Bass's polynomial definition commutes with $\mathcal S=\bigcup_{F\text{ finite}}\prod_{i\in F}\Free(\Q[G_i])$: every finite list of objects, idempotents, polynomial matrices and relations lies in one finite block. Consequently $K_{-k}(\mathcal S)=0$ for $k\ge1$. Rothenberg comparison then gives $L_j(\mathcal S)=\bigoplus_iL_j^p(\Q[G_i])$ and $L_j(\Free(\Q[G_i]))=L_j^p(\Q[G_i])$. Splitting at finite index sets is an invariant Karoubi filtration. Coordinate projections detect $L_3(\mathcal S)$, so its map to $L_3(\mathcal P)$ is injective. Localization therefore lifts every $L_4(\mathcal V_{\mathbf G})$ class to $L_4(\mathcal P)$; two lifts differ by the image of $L_4(\mathcal S)$. Their coordinate classes agree eventually. These constructions are natural for the stated functors. \end{proof} Applying ordinary signature after extraction defines \begin{equation}\label{alg:signature-tail} \sigma_{\mathrm{tail}}:L_4(\mathcal A_{\mathbf G}(X))\longrightarrow \Z^{\mathbb N}/\Z^{(\mathbb N)}. \end{equation} Here $\Z^{(\mathbb N)}$ denotes the finitely supported integer sequences. We use this map only after localization has removed the exterior. \section{Divisibility over a free \texorpdfstring{$C_p$}{C(p)}-space} \label{alg:obstruction} Let $T$ be a fixed nonempty compact free $C_p$-space with a compatible invariant metric. Tensoring by $\tau_i=\Q[G_i/P_i]$, with its orthonormal form, diagonal action and unchanged spatial labels, defines \begin{equation}\label{alg:nu} \nu=p\,\mathrm{id}-\tau\otimes(-) \quad\text{on }L_*(\mathcal A_{\mathbf G}(T)). \end{equation} \begin{lemma}[Local triviality]\label{alg:localtrivial} For compact $U\subset T/C_p$ contained in an open covering trivialization, $\nu=0$ on $L_*(\mathcal A_{\mathbf G,T_U}(T))$, where $T_U$ is its inverse image. \end{lemma} \begin{proof} Choose a closed invariant neighborhood $L$ of $T_U$ inside that trivialization, with sheet function $c:L\to C_p$ satisfying $c(hx)=\bar h c(x)$, where $\bar h$ is the image of $h$ in $C_p$. Labels of supported objects eventually lie in $L$. The orthogonal equivariant map \begin{equation}\label{alg:localisometry} e_a\otimes e_b\longmapsto e_b \quad\text{in copy }c(\ell_i(b))^{-1}a\in C_p \end{equation} has propagation zero. The sheet pieces of $L$ are compact and positively separated, so controlled morphisms eventually preserve them. Hence this is a natural isometry $\tau\otimes(-)\cong \mathrm{id}^{\oplus p}$, inducing equality in ultimate $L$-theory. \end{proof} Choose a finite compact cover $U_1,\ldots,U_s$ of $T/C_p$ inside trivializations, fixed independently of $i$, and put $T_a=T_{U_a}$. \begin{lemma}[Nilpotence]\label{alg:nilpotence} One has $\nu^s=0$ on $L_*(\mathcal A_{\mathbf G}(T))$. \end{lemma} \begin{proof} Induct on $s$ over all such compact free spaces. Set $A=T_1\cup\cdots\cup T_{s-1}$ and $B=T_s$. The natural support Mayer--Vietoris sequence follows from \cref{alg:filtration}. Since $\nu=0$ on $A\cap B$, $\partial(\nu x)=0$. Thus $\nu x$ comes from $L_*(\mathcal A_A)\oplus L_*(\mathcal A_B)$. Relabeling and induction kill the first summand by $\nu^{s-1}$; local triviality kills the second by $\nu$. For $s>1$, applying $\nu^{s-1}$ proves the claim; $s=1$ is local triviality. \end{proof} \begin{theorem}[Integer tail divisibility]\label{alg:divisibility} Every $\alpha\in L_4(\mathcal A_{\mathbf G}(T))$ has signature tail eventually divisible by $p$. \end{theorem} \begin{proof} Extract forms $(V_i,b_i)$. Naturality and nilpotence imply \begin{equation}\label{alg:factorannihilation} (p-\tau_i)^s[V_i,b_i]=0\quad\text{in }L_4^p(\Q[G_i]) \quad\text{for all sufficiently large }i. \end{equation} For a generator $g_i$ of $G_i$, $\chi_{\tau_i}(g_i)=0$. Taking signature characters gives $p^s\Sign_{G_i}(V_i,b_i)(g_i)=0$. Apply \cref{alg:cyclotomic}. \end{proof} \section{Rational Poincar\'e corners}\label{alg:corners} \subsection{Signs and rational structures}\label{alg:qmodel} Complexes are homologically graded. We use \begin{equation}\label{alg:dualconvention} (C^{N-*})_r=C_{N-r}^*,\qquad d_{C^{N-*},r}=(-1)^rd_C^*,\qquad (T\phi)_r=(-1)^{r(N-r)}\phi_{N-r}^*. \end{equation} The double shifted dual has differential $(-1)^{N+1}d_C$; its identification with $C$ is $(-1)^{r(N-r)}=(-1)^{r(N+1)}$ in degree $r$. Right slant sends $x_r\otimes y_{N-r}$ to $\alpha\mapsto x_r\alpha(y_{N-r})$, without an extra sign. For a bounded complex in a rational additive category with involution put \[ M(C)_k=\bigoplus_r\operatorname{Hom}(C_{k-r}^*,C_r),\qquad (\partial f)_r=d_Cf_{r+1}+(-1)^rf_rd_C^*,\qquad (Tf)_r=(-1)^{r(k-r)}f_{k-r}^*. \] Thus $T^2=1$, $T\partial=\partial T$, and conjugation by chain maps commutes with $T$. For finite scalar complexes this is $C\otimes C$ under right slant. Exactness of rational invariants and coinvariants gives \begin{equation}\label{alg:qgroups} Q^N(C)=H_N(M(C)^{C_2}),\qquad Q_N(C)=H_N(M(C)_{C_2}). \end{equation} The norm $1+T$ is an isomorphism with inverse $t\mapsto[t/2]$. We write $\Sym=(1+T)/2$. Consequently a symmetric structure has a strict symmetric representative $\phi$, and its quadratic lift is $\psi_0=\phi/2$, $\psi_s=0$ for $s>0$. If symmetric cycles differ by $\partial H$, replace $H$ by $(H+TH)/2$; halving that primitive identifies their quadratic classes. The same argument applies to the relative cone complexes, since their maps commute with $T$ \cite[Proposition 3.3]{Ran80I}, \cite[Section 3]{Ran89}. All these operations preserve propagation and exterior factorizations. \subsection{Compatible idempotents and normalization} Let a finite cyclic group $G$, of order $q$, act up to homotopy on $(C,\phi)$, with \begin{equation}\label{alg:grouprelations} A_gA_h\simeq A_{gh},\qquad A_1=1,\qquad A_g\phi\simeq\phi A_{g^{-1}}^*. \end{equation} Only an action in the chain-homotopy category is assumed; no higher coherence data are used. On $\mathcal C=C\otimes\Q[G]$, with strict left sheet action, set \begin{equation}\label{alg:graphidempotent} \Phi=\phi\otimes1,\qquad E=q^{-1}\sum_gA_g\otimes R_{g^{-1}}, \end{equation} where $R_{g^{-1}}$ sends sheet $b$ to $bg^{-1}$. Then \begin{equation}\label{alg:graphrelations} E^2\simeq E,\qquad E\Phi\simeq\Phi E^*,\qquad E_{ab}=q^{-1}A_{a^{-1}b}. \end{equation} In the controlled application these equations, including their homotopies, hold in the exterior quotient, as checked in \S\ref{sec:equivariant}. \begin{lemma}[Compatible splitting]\label{alg:corner} Let $\mathcal Q$ be rational additive with involution, $\mathcal C\in[0,N]$, and $\Phi:\mathcal C^{N-*}\to\mathcal C$ a strict symmetric Poincar\'e equivalence. For an invertible self-adjoint central coefficient $q$ and a chain map $E$ with homotopies $E^2\simeq E$, $E\Phi\simeq\Phi E^*$ in $\mathcal Q$, there is a splitting in the bounded chain-homotopy category $K^b(\Kar\mathcal Q)$, \begin{equation}\label{alg:split} D\xrightarrow{\iota}\mathcal C\xrightarrow rD,\qquad \iota r\simeq E,\quad r\iota\simeq1,\qquad D\in[0,N]. \end{equation} The strict symmetric Poincar\'e duality and a homotopy inverse are \begin{equation}\label{alg:cornerform} \phi_D=q^{-1}r\Phi r^*,\qquad b_D=q\,\iota^*\Phi^{-1}\iota. \end{equation} Here $\Phi^{-1}$ denotes a chosen chain-homotopy inverse. The resulting quadratic Poincar\'e complex determines a natural class in $L_N(\mathcal Q)$. \end{lemma} \begin{proof} With its split exact structure, $\Kar\mathcal Q$ has bounded derived category $K^b(\Kar\mathcal Q)$, which is idempotent complete \cite[Theorem 2.8]{BS01}. Hence $[E]$ splits. Outside $[0,N]$, the identity homotopy $r\iota\simeq1$ makes the extremal differential split: surjective at the bottom and injective at the top. Removing the resulting contractible two-term summands puts $D$ in $[0,N]$. Compatibility gives \[ \phi_Db_D\simeq r\Phi E^*\Phi^{-1}\iota\simeq rE\iota\simeq1, \qquad b_D\phi_D\simeq\iota^*\Phi^{-1}E\Phi r^* \simeq\iota^*E^*r^*\simeq1. \] Conjugation preserves strict symmetry, so $\phi_D/2$ is its complete quadratic structure. Two splittings are related by $r_2\iota_1$, an equivalence carrying their symmetric structures to one another up to homotopy. The primitive argument above and the Poincar\'e mapping-cylinder cobordism \cite[Definition 3.10]{Ran89} identify their ordinary $L$-classes in $\Kar\mathcal Q$. Finally pass to ultimate decoration and use the natural isomorphism $L_N(\mathcal Q)\cong L_N(\Kar\mathcal Q)$ \cite[Definition 17.7]{Ran92}. This also proves functoriality. \end{proof} For an asymptotic exterior category the splitting takes place once in the sequence category. Its finite collection of maps and homotopies therefore has asymptotically small representatives, with equations factoring through exterior-supported objects. \begin{lemma}[Scalar normalization]\label{alg:normalization} For \eqref{alg:graphidempotent}, suppose a duality-preserving forgetting functor retains a coordinate in which all $A_g$ and the homotopies in \eqref{alg:grouprelations} are uniformly small. Its image of \eqref{alg:cornerform} is homotopy-isometric to $(C,\phi)$. \end{lemma} \begin{proof} Only after forgetting, let $j_1,\pi_1$ include and project the identity sheet, and put $v=Ej_1$, $u=q\pi_1E$. Thus $v_a=q^{-1}A_{a^{-1}}$, $u_b=A_b$, and \begin{equation}\label{alg:uvrelations} uv\simeq1,\quad vu\simeq E,\quad uE\simeq u,\quad Ev\simeq v, \qquad u\Phi u^*=\sum_bA_b\phi A_b^*\simeq q\phi. \end{equation} Hence $w=u\iota$ and $z=rv$ are inverse equivalences and \begin{equation}\label{alg:scalarisometry} w\phi_Dw^*\simeq q^{-1}uE\Phi E^*u^* \simeq q^{-1}u\Phi u^*\simeq\phi. \end{equation} The rational primitive argument identifies the quadratic classes; the mapping-cylinder cobordism and ultimate completion invariance then identify their $L$-classes. If $q=q_i$ varies, multiplication by $q_i^{\pm1}$ has propagation zero, and the uniform entry and homotopy bounds make $u,v,w,z$ valid sequence morphisms. \end{proof} \section{Localization boundaries}\label{sec:localization} For a closed invariant cover $X=A\cup B$, put $Y=A\cap B$. Support filtration, excision, and \cite[Theorem 4.2]{CP95} give \begin{equation}\label{eq:ordinary-boundary} \partial_{A,B}:L_k(\mathcal A(X))\longrightarrow L_{k-1}(\mathcal A_Y(X)). \end{equation} Here $\mathcal A=\mathcal A_{\mathbf G}$; the group sequence is unchanged. \begin{proposition}\label{prop:exterior-boundary} If $Z\subset A$ is closed invariant and $d(Z,B)>0$, there is a boundary \begin{equation}\label{eq:exterior-boundary} \partial_{A,B}:L_k(\mathcal A(X)/\mathcal A_Z(X)) \longrightarrow L_{k-1}(\mathcal A_Y(X)), \end{equation} natural for duality-preserving functors preserving these support categories. \end{proposition} \begin{proof} Put $\mathcal Q=\mathcal A(X)/\mathcal A_Z(X)$ and $\mathcal U=\mathcal A_A(X)/\mathcal A_Z(X)$. Separation embeds $\mathcal A_B$ and $\mathcal A_Y$ fully faithfully in $\mathcal Q$. Choose controlled representatives for finite families of quotient maps and perform the based support decompositions before quotienting; this verifies the support filtrations there. The same based split as in \cref{alg:excision} gives the duality-preserving equivalence \begin{equation}\label{eq:exterior-excision} \mathcal U/\mathcal A_Y\simeq\mathcal Q/\mathcal A_B. \end{equation} Indeed, on $A$-supported objects $I_B$ restricts to $I_Y$, whereas $I_Z$ already factors through $A$-supported objects; thus the respective ideals are $I_Z+I_Y$ and $I_Z+I_B$. The based inclusion and projection are adjoints and inverse in the quotient. The boundary is induced by \[ \Lspec(\mathcal Q)\longrightarrow\Lspec(\mathcal Q/\mathcal A_B) \simeq\Lspec(\mathcal U/\mathcal A_Y) \longrightarrow\Sigma\Lspec(\mathcal A_Y). \] Localization for the two filtrations gives Mayer--Vietoris and naturality. The ordinary case is $Z=\varnothing$. \end{proof} \begin{lemma}\label{lem:union-boundary} Let $V_A=(P\to D_A)$ and $V_B=(P\to D_B)$ be $k$-dimensional quadratic Poincar\'e pairs supported on $A,B$, with oppositely oriented common boundary $P$ supported on $Y$. Orient $P$ as the boundary of $V_A$. For $V=V_A\cup_PV_B$, in the ordinary or exterior quotient setting, \begin{equation}\label{eq:union-boundary} \partial_{A,B}[V]=[P]. \end{equation} \end{lemma} \begin{proof} Use $\mathcal Q,\mathcal U$ above, with $Z=\varnothing$ in the ordinary case. The relative-pair representative $P\to D_A$ in the cofiber of $\Lspec(\mathcal A_Y)\to\Lspec(\mathcal U)$ has boundary $[P]$. The relative-to-quotient comparison \cite[proof of Theorem 4.1]{CP95}, followed by ultimate decoration, sends it to $D_A$ with its full induced quadratic structure. Modulo $\mathcal A_B$, $D_B$, $P$, and the suspended interface of the union vanish degreewise, along with every interface structure correction. The surviving complex and structure are exactly those on $D_A$. Excision thus identifies the two representatives, whose connecting image is $[P]$. Completion and the passage to ultimate decoration preserve this pair comparison. \end{proof} \section{Local chain carriers}\label{sec:carriers} Write $\varepsilon$ for geometric augmentation of zero chains. We use the acyclic-carrier method to prove the required shrinking-mesh estimates directly. Its application to dual cells is the elementary PL form of combinatorially controlled Poincar\'e--Lefschetz duality; compare \cite[Proposition 5.3, Theorem 6.7 and Corollary 6.8]{Ran99}. \begin{lemma}[Finite acyclic carriers]\label{lem:finite-acyclic-carrier} Let $E$ be a finite based chain complex in nonnegative degrees. Assign to each basis element $e$ an augmented acyclic subcomplex $\Gamma(e)$ of a rational chain complex $F$, with augmentations agreeing on inclusions. Assume $\Gamma(e')\subset\Gamma(e)$ whenever $e'$ occurs in $de$. Two carried chain maps $u,v:E\to F$ agreeing in augmentation on degree-zero basis elements admit a carried homotopy \[ u-v=dH+Hd,\qquad H(e)\in\Gamma(e). \] Carried, augmentation-compatible prescribed degree-zero images likewise extend to a carried chain map from an augmented source. \end{lemma} \begin{proof} Induct on degree. The residual $u(e)-v(e)-H(de)$ lies in $\Gamma(e)$, is a cycle, and in degree zero has augmentation zero. Fill it in that carrier and use the filling as $H(e)$. The map-existence induction fills the cycle given by the already defined image of $de$. \end{proof} \begin{corollary}\label{cor:carrier-control} If labels in $\Gamma(e)$ are within $a$ of the label of $e$, the constructed map or homotopy has propagation at most $a$. Relabeling by a fixed continuous map $c$ on a compact metric space replaces this bound by $\omega_c(a)$, where $\omega_c$ is a modulus of continuity for $c$. Signed cellular bases use geometric augmentation. \end{corollary} \begin{proof} The construction introduces no basis elements outside the assigned carriers. \end{proof} \begin{lemma}[Filling on a fixed triangulation] \label{lem:unchanged-target-filling} Let $K$ triangulate a compact $m$-manifold $W$ with mesh at most $\eta$. Let $C$ be convex in one coordinate chart. A simplicial $r$-cycle $\gamma$ whose full simplices lie in $C$ bounds a chain $q$ on the same $K$, with full-simplex support in the closed $\eta$-neighborhood of $C$; for $r=0$ require augmentation zero. For $r=m$ this forces $\gamma=0$. \end{lemma} \begin{proof} The zero chain needs no filling. Otherwise let $A$ be the subcomplex generated by its support and cone its inclusion to a point in $C$. This gives $F:|\operatorname{cone}A|\to C$ restricting to the inclusion on $A$. Subdivide only this cone and choose a simplicial approximation $t:L\to K$ satisfying $F(\operatorname{st}^{\circ}_Lv)\subset\operatorname{st}^{\circ}_Kt(v)$ \cite[Theorem 2C.1 and its proof]{Hat02}; write $S$ for subdivision. On the base, $tS$ is carried in each original simplex and fixes vertices, so $tS|_{C_*(A)}=1$: induction reduces the difference on an $r$-simplex to a top-dimensional cycle in that simplex, which is zero for $r>0$. Thus $q=tS(c\gamma)$ satisfies $dq=\gamma$, where $c$ is the augmented cone operator. For $z$ in the interior of a simplex $\lambda$ of $L$, the approximation property says that every vertex of $t\lambda$ belongs to the smallest target simplex containing $F(z)$. That simplex meets $C$ and has diameter at most $\eta$, proving the support claim. In top degree $q=0$ because $C_{m+1}(K)=0$. \end{proof} \begin{lemma}[Relative dual-block carrier maps] \label{lem:relative-block-carriers} Let $K$ be a finite combinatorial triangulation of a compact oriented $m$-manifold $A$ with $\partial A$ a subcomplex and mesh $\eta$. For nonboundary simplices choose dual-block orientations with \[ \partial D\sigma=(-1)^{\dim\sigma+1} \sum_{\substack{\tau\supset\sigma\\\dim\tau=\dim\sigma+1}} [\tau:\sigma]D\tau,\qquad D\rho=\epsilon_\rho[b_\rho] \] on top simplices, where $\epsilon_\rho$ is the fundamental coefficient and $b_\rho$ the barycenter. Let $D_{\mathrm{rel}}$ be the cellular subcomplex generated by these blocks and put $J_\sigma=\bigcup_{\tau\supseteq\sigma,\, \tau\not\subset\partial A}D\tau$. Every $J_\sigma$ is augmented acyclic. There are maps $S_{\mathrm{rel}}:D_{\mathrm{rel}}\to C_*(\operatorname{sd}K)$ and $R_{\mathrm{rel}}$ in the reverse direction, with $R_{\mathrm{rel}}S_{\mathrm{rel}}=1$ and $S_{\mathrm{rel}}R_{\mathrm{rel}}\simeq1$, all having propagation $O_m(\eta)$ when cells are labeled by points of their geometric supports. \end{lemma} \begin{proof} Use the dual orientations induced by the manifold and simplex orientations; the displayed incidence formula follows. On an interior codimension-one simplex, the augmentation of its dual boundary is $(-1)^m\sum_{\rho\supset\sigma}\epsilon_\rho[\rho:\sigma]=0$, by the relative fundamental-chain equation. Thus geometric augmentation is a chain map, including the signed zero-cell convention. All cofaces of a nonboundary simplex are nonboundary, giving the subcomplex. If $\sigma$ is interior, $J_\sigma=D\sigma$ is a ball. If $\sigma$ is boundary, its link is an $l$-ball, $l=m-\dim\sigma-1$, with boundary the link in $\partial A$. Nonboundary cofaces correspond to nonboundary faces $\lambda$ of this link. With join orientations the map \[ \lambda^*\longmapsto(-1)^{r(l+1)}D(\sigma*\lambda) \] identifies its relative shifted cochain complex in degree $r$ with $C_r(J_\sigma)$. Hence $H_r(J_\sigma)=\Q$ for $r=0$ and zero otherwise; augmentation is nonzero on a top-coface dual vertex and identifies $H_0$ with $\Q$. For a barycentric flag use $J_{\sigma_0}$, its minimum-face carrier. These shrink on deleting the minimum and are unchanged on deleting other faces. The carrier lemma constructs $R_{\mathrm{rel}}$; take $S_{\mathrm{rel}}$ to be cellular subdivision. Augmentation fixes the composite on zero cells; on a positive-dimensional $D\sigma$ the composite lies in that dual ball and equals the identity by induction and absence of top-dimensional cycles there. The other composite and the identity are carried in $\operatorname{sd}(\operatorname{st}_K \sigma_0)$, so the carrier lemma supplies their homotopy. Closed stars are contractible and have diameter at most $2\eta$, giving every asserted bound independently of valence and rank. \end{proof} \begin{lemma}[Controlled cap--block comparison] \label{lem:cap-block-comparison} In the setting of Lemma~\ref{lem:relative-block-carriers}, put $R=C_*(K,\partial K;\Q)$ and write $\delta$ for its shifted-dual differential $(-1)^rd^*$. Let $F:R^{m-*}\to C_*(K)$ be a chain map carried, on $\sigma^*$, in the closed star $\operatorname{st}_K\sigma$. Suppose $\varepsilon F(\rho^*)=\epsilon_\rho$ on each top simplex. Then $F$ has a chain homotopy inverse and both inverse homotopies of propagation $O_m(\eta)$. More precisely, it is carried-homotopic to $F_0$, with homotopy inverse $B$, where \[ F_0=R_KS_{\mathrm{rel}}\mathcal J, \qquad B=\mathcal J^{-1}R_{\mathrm{rel}}S_K, \qquad\mathcal J_r(\sigma^*)=(-1)^{r(m+1)}D\sigma, \] where $S_K$ is subdivision and $R_K$ a simplex-carried inverse. Fixed continuous relabelings give bounds tending to zero with $\eta$. \end{lemma} \begin{proof} The incidence formula makes $\mathcal J$ a chain isomorphism. The carrier lemma on maximum-face carriers constructs $R_K$, with $R_KS_K=1$ by top-chain uniqueness in a simplex; it also gives $S_KR_K\simeq1$ in carriers $C_*(\operatorname{sd}\sigma_r)$ on flags with maximum $\sigma_r$. Together with the preceding lemma these identities prove $F_0B\simeq1$ and $BF_0\simeq1$ by a fixed number of compositions. Both $F,F_0$ have the same top augmentations and closed-star carriers. The dual differential uses cofaces, and $\operatorname{st}_K\tau\subset\operatorname{st}_K\sigma$ for $\tau\supset\sigma$, so the carrier lemma gives $F-F_0=dH+H\delta$. Add $HB$ and $BH$ to the inverse homotopies for $F_0$. All carriers have diameter at most $2\eta$; fixed compositions give $C_m\eta$ and relabeling gives $\omega_c(C_m\eta)$. \end{proof} \begin{corollary}[A common simplicial diagonal] \label{cor:common-diagonal-control} In the setting of Lemma~\ref{lem:relative-block-carriers}, let $j$ be the boundary inclusion. Use one ordered Alexander--Whitney diagonal $\Delta$ on $K$ and its subcomplexes. Choose oriented fundamental chains satisfying $dz_A=jz_{\partial A}$, and put $t_A=\frac12(1+T)\Delta z_A$ and $t_{\partial A}=\frac12(1+T)\Delta z_{\partial A}$. Then $dt_A=j^{\otimes2}t_{\partial A}$ exactly. The right slant of $t_A$ on relative cochains is a chain equivalence with inverse data of propagation $O_m(\eta)$, also when the boundary is empty. \end{corollary} \begin{proof} The diagonal restricts to subcomplexes. Its boundary equation gives \[ d\phi_{A,r}-(-1)^r\phi_{A,r-1}d^* =j\phi_{\partial A,r-1}j^*. \] The right side vanishes on relative cochains. Each diagonal term consists of faces of one simplex, so slant on $\sigma^*$ has closed coface-star support. On a top simplex it is $\epsilon_\rho([v_0]+[v_m])/2$, of augmentation $\epsilon_\rho$. Apply the cap--block comparison. \end{proof} \section{Geometric data and quotient representatives}\label{geo:fixed-data} Suppose that $\Zp$ acts effectively on a connected boundaryless $n$-manifold, $n\ge1$, and put $H_j=p^j\Zp$. First note that, for any compact group $K$ acting on a connected manifold, every countable union of compact sets lies in a $K$-invariant connected second-countable open submanifold. Indeed, every compact set lies in a relatively compact connected open set covered by finitely many charts: connect a finite chart cover by finitely many paths. Choose such sets $V_j$ inductively with $K\overline{V_j}\subset V_{j+1}$, including one prescribed compact set at each stage; their union has the required properties. For $K=\Zp$, include a point moved by $p^j$ for every $j\ge0$. The restriction remains effective, so we henceforth work on this second-countable open submanifold. \begin{lemma}\label{geo:free-point} There is a point with trivial stabilizer. \end{lemma} \begin{proof} Every nonzero closed subgroup of $\Zp$ is an $H_j$: an element of minimum valuation $j$ has integer multiples dense in $H_j$. If all stabilizers were nonzero, the discrete subgroup $\Z$ would have finite point orbits. By \cite[Theorem 2]{Rob75} its image would be finite. Some $m>0$ would act trivially; the closed kernel would then contain $\overline{m\Z}=H_{v_p(m)}$, contradicting effectivity. \end{proof} Center a chart at this point, with image containing $\overline{B(0,4)}$. Joint continuity gives $H=H_j$ and $0v$ gives the same estimates for transposes and products of bounded length: the transpose has type $g^{-1}$, and graph types multiply. Apply \cref{geo:closed-models} to $\widehat h_g\times1$ to obtain $a_{g,i}$, $a_{1,i}=1$. Equation \eqref{geo:f-invariance}, detector equivariance, and the graph error give type $g$. Degree one and the absence of $(N+1)$-chains give \begin{equation}\label{geo:fundamental-cycle} a_{g,i}z_i=z_i. \end{equation} We now suppress the index $i$. \begin{lemma}\label{geo:product-homotopies} On the same complexes there are homotopies \begin{equation}\label{geo:product-equation} a_ga_h-a_{gh}=dB_{g,h}+B_{g,h}d \end{equation} of graph type $gh$, uniformly in $i,g,h$. \end{lemma} \begin{proof} Induct over simplex dimension, filling the residual cycle $\gamma_\sigma=a_ga_h\sigma-a_{gh}\sigma-B(\partial\sigma)$ using \cref{lem:unchanged-target-filling}; in degree zero it has augmentation zero. Take first-coordinate convex hulls of $L_{g,h}(|\sigma|\times[0,1])$, and small coordinate-convex carriers for the second projection. In a product chart, take convex hulls including full endpoint chains and already fixed face fillings. With $\delta_i=\eta_i+\mu_i+\omega(C\eta_i)+\omega(C\mu_i)+\beta_i\to0$, the full supports stay within widths $w_r$ of the nested track hulls and second-coordinate projections, where $w_{-1}=0$, $w_r\le C(w_{r-1}+\delta_i)$: coordinate changes cost a fixed factor and filling adds one target mesh. Thus $w_r\to0$ uniformly for $r\le N$, independently of all ranks and auxiliary subdivisions. These convex carriers lie in one product chart; near infinity use small charts in the identity region. The top residual vanishes by the same filling lemma, so no extra top chain is required. For a source barycenter with $|x|\le3/8$, every filling lies within $o(1)$ of $h_{gh}x$. Equivariance of the detector on each $\mathcal K_v$ gives type $gh$. For $|x|\ge3/8$ its support is within $2D+o(1)$ of $x$; the fixed margin $3/8-\rho-4D>1/16$ makes its $f$-value $\infty$. Uniform continuity finishes both control estimates. \end{proof} The same filling recursion, now on the based source $D^{N-*}$, gives a local primitive $U_g$ with \begin{equation}\label{geo:isometry-homotopy} a_g\phi a_g^*-\phi=dU_g+U_gd_{D^{N-*}}. \end{equation} Indeed, an entry of $a_g^*$ selects a source cell mapping near its given target cell; $\phi$ moves within $O(\eta_i)$, and $a_g$ maps back near that target cell. Thus both endpoint maps are local with error $O(\mu_i+\omega(C\eta_i)+\eta_i)\to0$. Their degree-zero augmentations agree, since simplicial approximations preserve geometric augmentation ($\varepsilon a_g=\varepsilon$): for $\alpha\in D_N^*$, $\varepsilon\phi(\alpha)=\alpha(z_i)$, while $\varepsilon a_g\phi a_g^*(\alpha)=\alpha(a_gz_i)=\alpha(z_i)$. For $\sigma^*$ take a chart-convex hull of its endpoint supports and the carriers and fillings already assigned to cofaces occurring in the shifted dual differential, enlarging by one mesh after filling. These geometric carriers are nested under cofaces. Every coface meets $\sigma$, so the same width induction keeps them within $o(1)$ of $\sigma$. The top residual again vanishes. Put $a=a_g$, $b'=a_{g^{-1}}$. For $b'a-1=dT+Td$ set $K_r=(-1)^{r+1}T_{N-r-1}^*$. The identity $a\phi-\phi b'^*=a\phi(1-a^*b'^*)+(a\phi a^*-\phi)b'^*$ yields \begin{equation}\label{geo:adjoint-homotopy} V_g=-a\phi K+U_gb'^*,\qquad dV_g+V_gd_{D^{N-*}}=a_g\phi-\phi a_{g^{-1}}^*. \end{equation} Since $T,K,U_g$ have type $1$ and $b'^*$ type $g$, $V_g$ has type $g$. Fix one $\epsilon_i\to0$ bounding all these $f$ and detector errors, cell oscillations, and local duality propagations on the fixed buffers. \section{Compression and exterior control}\label{geo:compression} Keep the fixed detector neighborhood and $R