%===================================================================== % En06-A.tex % A corrected synthesis of 085VvBp1.tex, 085VvBp2.tex, % 085VvBp3.tex, and the supplied En05-A comparison. % % Compile: % pdflatex En06-A.tex % pdflatex En06-A.tex % % Self-contained: no external .bib, figures, or shell escape. % A successful compilation or proof-assistant execution is not % asserted. Substantive corrections are documented in the appendices. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1.05in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{enumitem} \usepackage{booktabs,tabularx,array,longtable} \usepackage{microtype} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning} \usepackage{xurl} \usepackage[ colorlinks=true, linkcolor=blue, citecolor=blue, urlcolor=blue ]{hyperref} \hypersetup{ pdftitle={En06-A: Causal Cones, Increasing Stencils, and Operational Transport}, pdfauthor={Ngo Cheung}, pdfsubject={A corrected layer account of partitioned quantum dynamics} } \allowdisplaybreaks \setlength{\emergencystretch}{2em} \setlist[enumerate]{itemsep=3pt,topsep=5pt} \setlist[itemize]{itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} % ---------- Theorem environments ---------- \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{hypothesis}[theorem]{Hypothesis} \newtheorem{example}[theorem]{Example} \newtheorem{problem}[theorem]{Problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % ---------- Macros ---------- \newcommand{\eps}{\varepsilon} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}} \newcommand{\Lat}{\Lambda} \newcommand{\Hil}{\mathcal{H}} \newcommand{\Alg}{\mathcal{A}} \newcommand{\BAlg}{\mathfrak{B}} \newcommand{\AAlg}{\mathfrak{A}} \newcommand{\XX}{\mathcal{X}} \newcommand{\Theory}{\mathcal{T}} \newcommand{\Poin}{\mathcal{P}^{\uparrow}} \newcommand{\Ent}{\mathsf{Enter}} \newcommand{\Drop}{\mathsf{Drop}} \newcommand{\Sch}{\mathsf{Sch}} \newcommand{\Inf}{\mathsf{Inf}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\id}{\operatorname{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\diam}{\operatorname{diam}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\conv}{\operatorname{conv}} \newcommand{\Loc}{\operatorname{loc}} \newcommand{\FN}{\operatorname{fn}} \newcommand{\Ball}{B} \newcommand{\Btwo}{\overline{B}_2} \newcommand{\dH}{d_{\mathrm H}} \newcommand{\dc}{d_c} \newcommand{\dtv}{d_{\mathrm{TV}}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddia}{d_{\diamond}} \newcommand{\Fid}{F} \newcommand{\sem}[1]{[\![#1]\!]} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\shadow}{\pi_{\mathrm{sp}}} \newcommand{\Source}[1]{\texttt{#1}} \newcommand{\excess}{\operatorname{exc}} \title{En06-A: Causal Cones, Increasing Stencils,\\ and Operational Transport\\[3pt] \large A Corrected Layer Account of Partitioned Quantum Dynamics} \author{Ngo Cheung\\ Independent Researcher, Hong Kong SAR, China\\ \texttt{info@cheungngomedical.com}} \date{October 2026} \begin{document} \maketitle \begin{abstract} We combine three supplied manuscripts on partitioned quantum dynamics with a preceding snapshot--channel comparison, separating proved discrete results, geometric scaling results, and additional operational and algebraic requirements. Complete hybrid instruments admit equivalent snapshot-factorization and Heisenberg-support descriptions. This gives an order-theoretic domain of dependence, its exact metric envelope, and invariance under legal reorderings of one finite causal network. Scheduled ancestry bounds influence but need not equal it. We give a self-contained increasing-stencil construction and a uniform shape estimate \[ d_{\mathrm H}\!\left( h_n B_{S_n}(0,k),\,k\eps_n\overline B_2 \right) \le (D+1)\eps_n+2\delta_n k\eps_n. \] The changing word metrics therefore have Euclidean limiting cone envelopes, while admissible relays have limiting speed \(c\) in every direction. For an explicitly defined, full-tail, fixed-wire outcome-law theory, matched modifications satisfy continuum domain of dependence, operational no-signalling, and sharp maximal speed \(c\). The supplied transport manuscript is not treated as an unchecked premise. Its common-final-tick readout convention is incompatible with its asserted boost action, and its spatial routing estimates do not establish the required spacetime support estimates. Additional normalization, scheduling, and algebraic issues are isolated. A corrected transport contract yields, conditionally, a Poincare action on the coarse relational theory in every \(D\ge1\). Observable-algebra microcausality requires a separate localized, product-preserving realization; commuting channels alone do not supply it. Unrestricted device resynchronization closure remains withdrawn because it excludes ordinary bidirectional relays. No instrument-level scaling limit, relativistic field theory, gravitational dynamics, or invariant physical speed is derived. \end{abstract} \tableofcontents %===================================================================== \section{Purpose, sources, and the layer distinction} \label{sec:intro} %===================================================================== The supplied manuscripts address different objects: \begin{description}[style=nextline,leftmargin=2em] \item[CONE.] \Source{085VvBp1.tex} studies finite instrument networks, the order-theoretic and metric cone, semantic refinement, foliation independence, and conditional continuum and dynamical statements \cite{CheungCONE}. \item[STEN.] \Source{085VvBp2.tex} proposes changing word metrics with increasingly dense directional stencils and supplies the main geometric construction \cite{CheungSTEN}. \item[TRANSPORT.] \Source{085VvBp3.tex} proposes a five-part transport argument, a relational outcome-law limit, a tail-limit algebra, and a concrete Poincare action \cite{CheungTRANSPORT}. \item[En05-A.] The supplied comparison separates complete snapshots, channels, adjoint support, residual response, and device closure \cite{CheungEn05A}. \end{description} These are source records, not automatic theorem premises. The results asserted here are either proved below or explicitly conditional on displayed hypotheses. In particular, a missing mathematical argument is not repaired by describing it as awaiting machine verification. The designation of this synthesis is \emph{En06-A}, superseding the working designation En05-B used in the supplied combination plan. \subsection{The layer table} The following table is normative for the conclusions of this paper. ``Conditional'' means a mathematical hypothesis is still needed; it does not mean merely that a proof has not been machine checked. \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{.22\linewidth}p{.19\linewidth}X@{}} \toprule Layer & Source ownership & Status in En06-A\\ \midrule 0--1: complete discrete dynamics and causal cone & CONE; En05-A comparison & Proved for complete partitioned range-\(r\) instruments.\\ \addlinespace Legal reorderings and foliation independence & CONE; VII-E; En03-A & Proved for one finite wired network. Not a boost symmetry.\\ \addlinespace Euclidean limiting cone envelope & STEN & Proved for changing increasing-stencil word metrics.\\ \addlinespace Coarse operational domain of dependence and speed & TRANSPORT proposal & Proved below for the explicitly restricted full-tail, fixed-wire relational theory. Not an instrument-level limit.\\ \addlinespace General transport and coarse frame action & TRANSPORT & Conditional on the corrected transport contract. The literal common-final-tick source class is not boost closed.\\ \addlinespace Observable algebra and microcausality & TRANSPORT; En05-A & Discrete reversible commutation is proved. A matching algebraic limit requires a localized observable realization.\\ \addlinespace Unrestricted device resynchronization closure & CONE Stage 7 & Withdrawn: incompatible with ordinary bidirectional relays and well-founded executability.\\ \addlinespace Field content and curved-background dynamics & CONE, C8--C11 & Not constructed. Separate convergence and covariance obligations.\\ \bottomrule \end{tabularx} \end{center} \subsection{Comparison interfaces} The locality comparisons are with Cheung En03-A, VII-E, VII-G, VII-J, and Psy04-B \cite{CheungEn03A,CheungVIIE,CheungVIIG,CheungVIIJ,CheungPsy04B}. They concern complete interfaces, prescribed supports, and unconditioned laws. The optional program semantics are compared with Landmark I, En04-A, Hawk04-B, and En02-C \cite{CheungI,CheungEn04A,CheungHawk04B,CheungEn02C}. Cheung's publisher record describes the Enter calculus as a linear calculus with an operational characterization of destructive measurement, separating the Born-equipped calculus from the measurement-characterization layer \cite{CheungEC}. No syntax-specific metatheorem is needed to prove the geometric cone. The finite-speed geometry is related to periodic-graph velocity polytopes \cite{Fritz}, partitioned cellular automata \cite{ToffoliMargolus}, and reversible quantum cellular automata \cite{SchumacherWerner}. The present class includes irreversible instruments and does not inherit a reversible-QCA classification theorem. %===================================================================== \section{Complete local interfaces and snapshots} \label{sec:interfaces} %===================================================================== \subsection{Finite experiments on countable substrates} We work with finite time windows and finitely many active registers. A countable site set may have an unlimited reserve of names, but every experiment declares the finite registers it actually uses. Multiple declared registers may occupy one site. This is an explicit capacity convention. It is not the injective one-qubit-per-site convention of one supplied draft. Arbitrarily many names cannot be gathered into a uniformly bounded-size block under that injective convention. At a site and tick, a finite hybrid observable algebra has the form \begin{equation} \Alg_{x,s} = \bigoplus_{\lambda\in L_{x,s}} \mathcal B(\Hil_{x,s,\lambda}), \label{eq:hybridalgebra} \end{equation} where \(L_{x,s}\) contains classical record, controller, occupation, and placement labels. Different labels may have different quantum dimensions. An inactive reserved slot is not an accessible surviving measured system. For finite \(I\subseteq\Lat\), set \(\Alg_s(I)=\bigotimes_{x\in I}\Alg_{x,s}\). States are positive arrays \begin{equation} \omega_I=\bigoplus_\lambda\omega_{I,\lambda}, \qquad \sum_\lambda\Tr\omega_{I,\lambda}=1. \label{eq:hybridstate} \end{equation} Compatible finite-region states suffice on a countable substrate; an unspecified infinite tensor-product density operator is not used. \begin{definition}[Complete recorded instrument] A finite-outcome instrument is a family of completely positive maps \(\{\Phi_y\}_y\) for which \(\sum_y\Phi_y\) is trace preserving. Its recorded channel is \[ \widehat\Phi(\omega)=\bigoplus_y\Phi_y(\omega). \] Every outcome, including failure, belongs to this complete channel. Discard of a classical output sums blocks without normalization. \end{definition} We use the usual finite-dimensional CP-map and channel formalism \cite{Watrous}. \begin{assumption}[Complete local control]\label{ass:control} Every input used by a block is in its declared local interface: quantum systems, classical records, controller memory, random seeds, occupancy flags, and resource registers. The complete map is fixed independently of the unknown input state. Adaptivity is classical blockwise control on those local inputs. An unrepresented global scheduler or remote controller is not admitted. \end{assumption} This is the controller-support discipline emphasized in VII-G \cite{CheungVIIG}. Initially correlated resources are permissible in the general channel model only when their interfaces are declared. The product-default coarse model introduced later has a narrower preparation convention. \begin{definition}[Snapshot]\label{def:snapshot} Let \(R_{I,s}\) be restriction to the tick-\(s\) interface in \(I\): trace out external quantum systems and sum external classical labels. Define \[ \Sigma_s(I)=R_{I,s}(\omega_s). \] For \(p_\lambda=\Tr\omega_{I,\lambda}>0\), the conditional quantum state is \(\rho_{I\mid\lambda}=\omega_{I,\lambda}/p_\lambda\). No normalized state is assigned when \(p_\lambda=0\). \end{definition} \begin{remark}[The affine object] The affine snapshot is the unnormalized block array \((\omega_{I,\lambda})_\lambda\). The normalized conditional states and their probabilities determine it, but conditional normalization is not itself affine. This corrects the snapshot wording in the supplied predecessors. \end{remark} \subsection{The common comparison lemma} An observable supported in \(I\) is extended by identity on the complement. \begin{lemma}[Snapshot--channel--adjoint comparison] \label{lem:comparison} Let \(\Phi\) be a complete channel between finite tensor-product hybrid interfaces. Fix input region \(I\) and output region \(Y\). The following are equivalent: \begin{enumerate}[label=(\roman*)] \item There is a complete channel \(K\) such that \(R_Y\Phi=KR_I\). \item Equal input snapshots on \(I\) give equal output snapshots on \(Y\), for all input states. \item \[ \Phi^*(\Alg_{\mathrm{out}}(Y)) \subseteq\Alg_{\mathrm{in}}(I). \] \end{enumerate} The factorization is stable under tensoring with an untouched reference. To infer equality of joint output--reference states, the two inputs must agree on their joint \(I\)--reference reductions. \end{lemma} \begin{proof} (i) implies (ii). For (ii) implies (i), choose a fixed complementary state \(\sigma\), put \(E_\sigma(\omega_I)=\omega_I\otimes\sigma\), and define \(K=R_Y\Phi E_\sigma\). This is a complete channel. Condition (ii) identifies \(R_Y\Phi(\omega)\) with \(KR_I(\omega)\) on all states, and therefore on the whole linear input space. Taking adjoints gives (iii). Conversely, (iii) makes the expectation of every output observable on \(Y\) a function of the snapshot on \(I\); this is (ii). Tensor the linear channel identity in (i) with a reference identity. The resulting factorization involves \(I\) together with the reference, which proves the final statement. \end{proof} This is the common mathematical form of En05-A \Source{lem:comparison}, the factorization interface in VII-J \Source{thm:nosignal}, and the adjoint interface in En03-A \Source{lem:one-layer} \cite{CheungEn05A,CheungVIIJ,CheungEn03A}. The proof here does not import any of those manuscripts. %===================================================================== \section{Order-theoretic and metric domains of dependence} \label{sec:cone} %===================================================================== \begin{definition}[Substrate and schedule] A substrate is \((\Lat,d,r)\), where \(\Lat\) is countable, \(d\) is integer valued with finite balls, and \(r\ge1\). A schedule \(\mathcal P=(\mathcal P_s)_s\) partitions \(\Lat\) at each tick into finite blocks \(b\) with \(\diam_d b\le r\). Tick \(s\) maps its input interface to the tick-\((s+1)\) interface. \end{definition} \begin{definition}[Semantic admissibility] A semantically admissible update is \[ \Phi_s=\bigotimes_{b\in\mathcal P_s}\Phi_{s,b}, \] where every factor is a complete local channel satisfying Assumption~\ref{ass:control}. Names and records may be relocated only within the block. Allocation and record creation have declared local output interfaces. \end{definition} The radius bounds one block update, not the number of gates evaluated inside it. No hidden gate-depth or uniform storage bound is imposed. \subsection{One-layer support and scheduled ancestry} For finite \(Y\), set \[ U_s(Y)=\bigcup\{b\in\mathcal P_s:b\cap Y\ne\varnothing\}. \] \begin{lemma}[One-layer support]\label{lem:onelayer} \[ \Phi_s^*(\Alg_{s+1}(Y))\subseteq\Alg_s(U_s(Y)). \] The reduced output channel on \(Y\) depends only on the block channels meeting \(Y\). \end{lemma} \begin{proof} The complement of \(U_s(Y)\) is a union of blocks. The output observable is the identity on those blocks. Their adjoints are unital, since their complete channels are trace preserving. They contribute only input identities. The remaining factors act inside \(U_s(Y)\). \end{proof} Define backward sets by \begin{equation} N_T(Y)=Y,\qquad N_s(Y)=\bigcup\{b\in\mathcal P_s:b\cap N_{s+1}(Y)\ne\varnothing\}. \label{eq:scheduledpast} \end{equation} \begin{theorem}[Order-theoretic domain of dependence] \label{thm:order} For a finite complete wired instrument network, the output reduction on terminal ports \(Y\) factors through the initial reduction on their scheduled ancestors: \[ R_Y\Phi=K_YR_{\Sch^-(Y)}. \] Changing complete gates outside the ancestor subnetwork does not change that reduction, provided the wiring and the included gates agree. For a partitioned schedule, the initial ancestor region is \(N_{t_0}(Y)\) from \eqref{eq:scheduledpast}. No metric is required for the network statement. \end{theorem} \begin{proof} Pull an output observable backward in reverse topological order. If it is the identity on all outputs of a gate, unitality makes it the identity on that gate's inputs. Otherwise its possible support expands only to the declared input predecessors. Include all classical control, outcome, and resource ports in this graph. At the input cut the support lies in the scheduled ancestors. Lemma~\ref{lem:comparison} gives the channel factorization. Excluded gates act on identity factors, so replacing them by any complete gates on the same interfaces does not change the pullback. For a partitioned schedule the same induction is precisely \eqref{eq:scheduledpast}. \end{proof} Multiple readout ports, including ports acquired at different ticks, are covered by this finite-network statement. Report ports may be retained as formal outputs for calculating their joint law; this does not make their information available to a physical remote controller. \begin{corollary}[Ancestry bounds influence]\label{cor:influence} If operational influence means a change of unconditioned output law under a trace-preserving local intervention, then \[ \Inf^-_\Phi(Y)\subseteq\Sch^-(Y). \] The inclusion need not be equality. \end{corollary} \begin{proof} An intervention outside the ancestors preserves their reduced state, even for entangled initial states. Apply Theorem~\ref{thm:order}. For strict inclusion, declare a memory input but return the constant answer \(0\). The memory is scheduled but not influential. \end{proof} This is the distinction in Psy04-B and VII-J \cite{CheungPsy04B,CheungVIIJ}. Selective normalization and common-cause correlations are not the interventions quantified here. \subsection{The metric envelope} The triangle inequality gives \begin{equation} N_s(Y)\subseteq \{x:d(x,Y)\le r(T-s)\}. \label{eq:metricenvelope} \end{equation} For terminal site \(A\), let \(I_s=B_d(A,r(T-s))\). If a block meets \(I_{s+1}\), then it lies in \(I_s\): for \(z\in b\cap I_{s+1}\) and \(w\in b\), \[ d(w,A)\le d(w,z)+d(z,A)\le r+r(T-s-1). \] \begin{theorem}[Exact metric cone]\label{thm:metric} Two runs on the same schedule have equal terminal snapshots at \((A,T)\) if: \begin{enumerate}[label=(\roman*)] \item their initial snapshots agree on \(B_d(A,r(T-t_0))\); \item their block channels agree at each tick \(sr\tau\), the initial cone and every inner block exclude \(x\). Theorem~\ref{thm:metric} forces equal readout laws. \end{proof} \begin{theorem}[Foliation independence of one finite network] \label{thm:foliation} Every linear extension of the dependency partial order of a finite complete wired network gives the same complete instrument and the same joint recorded-history law. \end{theorem} \begin{proof} Incomparable events have disjoint current wire interfaces. Their extensions by identities commute, including on entangled inputs. Any two linear extensions of a finite partial order are connected by adjacent swaps of incomparable events. Every swap preserves the complete composition. \end{proof} The statement compares one labelled network under legal orderings. It does not remove dependencies on records, change its metric, or produce boost-related devices. Execution-dependent descriptions of the order of intermediate events should not be identified as the same ordered trajectory; the complete channel and joint labelled records are the invariants. \begin{remark}[Infinite structures] Local finiteness and acyclicity alone do not give a finite ancestral calculation or a well-founded initial-value problem. The chain \(\ZZ\) with edges \(j\to j+1\) is locally finite and acyclic but has infinite past at every vertex. The generalization of Theorem~\ref{thm:order} requires a finite ancestral network relative to an input cut, or a separately proved infinite-network construction and convergence theorem. \end{remark} %===================================================================== \section{Enter accounting and optional syntactic realization} \label{sec:enter} %===================================================================== For a measured register \(S\) and surviving register \(T\), destructive computational readout is \begin{equation} \Ent_S(\rho_{ST}) = \sum_x\kb{x}{x}_K\otimes (\bra{x}_S\otimes I_T)\rho_{ST}(\ket{x}_S\otimes I_T). \label{eq:enter} \end{equation} The measured names are consumed; \(K\) is a classical record. \begin{proposition}[Readout and operational discard]\label{prop:drop} \[ \Tr_K\Ent_S(\rho_{ST})=\Tr_S\rho_{ST}, \] including every untouched reference. If \(W\ket{x}_S=\ket{x}_S\ket{x}_K\), then \[ \Tr_S\!\left[ (W\otimes I_T)\rho_{ST}(W^\dagger\otimes I_T) \right]=\Ent_S(\rho_{ST}). \] \end{proposition} \begin{proof} Expand \(\rho_{ST}=\sum_{x,y}\kb{x}{y}_S\otimes\rho_{xy}\). The first partial trace gives \(\sum_x\rho_{xx}\). The second gives \(\sum_x\kb{x}{x}_K\otimes\rho_{xx}\). The calculation is unchanged by tensoring with a reference identity. \end{proof} Coherent recording with an accessible measured factor is not destructive Enter. Deferred readout is valid only for a stated record-compatible context. Operational discard is not physical annihilation. These distinctions agree with Landmark I, En04-A, and Hawk04-B \cite{CheungI,CheungEn04A,CheungHawk04B}. \begin{proposition}[Finite-program channel accounting]\label{prop:program} A finite acyclic instruction tree built from declared allocations, gates, destructive readouts, classical control, and discard has path maps \(K_h\) with \[ \sum_hK_h^\dagger K_h=I. \] Its reported selective maps are \[ \Phi_y(\rho)=\sum_{h:\operatorname{out}(h)=y}K_h\rho K_h^\dagger. \] Its complete denotation is CPTP and reference preserving. \end{proposition} \begin{proof} A complete history fixes the classical choices. Compose the corresponding allocation isometries, gates, and measurement bras. Branch normalizations cancel against branch probabilities. At a readout, \(\sum_xM_x^\dagger M_x=I\); gates and allocations are isometric. Backward induction over the finite tree proves the completeness identity. Group histories by reported output and tensor the path maps with a reference identity. \end{proof} This is the comparison with En04-A \Source{thm:soundness}, Hawk04-B \Source{thm:adequacy}, and En02-C \Source{thm:C1} \cite{CheungEn04A,CheungHawk04B,CheungEn02C}. A block is \emph{calculus generated} only if a well-typed program over the available library denotes its complete instrument. A finite network of such blocks realizes a composite program only when sequential composition, parallel ownership, and input/output wiring match. Those are additional syntactic requirements. Strong normalization concerns reductions of a fixed term, not externally inserted idle ticks or an indefinitely driven substrate. No exact finite-gate compilation of arbitrary specified instruments, no efficiency claim, and no support-preserving normalization of arbitrary adaptive wiring are asserted here. \begin{proposition}[Classical-record reconstruction]\label{prop:EB} A quantum-to-classical channel followed by a decoder whose only nontrivial input is that record is entanglement breaking. It cannot recover every unknown quantum state with its reference. \end{proposition} \begin{proof} The composite has form \[ \rho\longmapsto\sum_x\Tr(E_x\rho)\sigma_x. \] With a reference, its output is a sum of positive product operators, hence separable. The identity channel preserves a maximally entangled input, so the channels cannot agree. \end{proof} Compare Landmark I \Source{prop:classical-recovery} and the standard measure-and-prepare characterization \cite{CheungI,HorodeckiEB}. A receiver holding an additional entangled quantum resource has a different interface. \begin{definition}[Complete residual response] For an isometry \(V:C\to R\otimes B\), residual response means non-constancy on the code of \(\mathcal C(\rho)=\Tr_R(V\rho V^\dagger)\). The complement includes all inaccessible systems and preparation purifications of the stated dilation. \end{definition} \begin{proposition}[Complete restoration does not identify a residual] \label{prop:residual} A finite isometry has a CPTP left inverse on its complete output. This does not imply recovery from a chosen component or constancy of its complement. \end{proposition} \begin{proof} For a state \(\tau_C\), define \[ \mathcal L(X)=V^\dagger XV+ \Tr((I-VV^\dagger)X)\tau_C. \] It is CP and trace preserving, and \(\mathcal L(V\rho V^\dagger)=\rho\). For \(V\ket{\psi}=\ket0_R\otimes\ket{\psi}_B\), the \(R\)-channel is constant and the \(B\)-channel is the identity. \end{proof} A responding residual outside the cone of a constant readout is not a signalling path to that readout. The restore map returns a code from the complete output; it does not establish that the code is already in a selected residual. Nor is residual response geometric backreaction; a same-model geometric dictionary would be a separate construction \cite{CheungEn04A,CheungHawk04A}. %===================================================================== \section{Long-range primitives: exactly which guarantee fails} \label{sec:longrange} %===================================================================== Fix \(P,Q\) with \(d(P,Q)=\ell>r\). Remove \(P,Q\) from their ordinary blocks, discard empty remainders, and insert the exceptional block \(\{P,Q\}\). Channels on the shortened blocks must be specified anew; a channel on the original larger block does not automatically restrict to them. \begin{proposition}[Inert and transmissive exceptional blocks] \label{prop:exceptional} \begin{enumerate}[label=(\alph*)] \item If the exceptional channel factorizes as \(\Phi_P\otimes\Phi_Q\), with no cross relocation or classical flow, it can be replaced by singleton channels. The original-metric cone continues to apply. \item Lossless one-tick bit transport from \(P\) to \(Q\) violates the original-metric radius-\(r\) cone. \item The geometry change alone neither creates nonlinearity nor permits universal deterministic cloning. \end{enumerate} \end{proposition} \begin{proof} (a) is equality of the two product updates. For (b), prepare bit \(0\) or \(1\) at \(P\), transport it with the same rule, and read at \(Q\). The laws are \(\delta_0,\delta_1\). For (c), finite compositions of complete instruments are linear on unnormalized hybrid states. They remain CPTP after ignored systems are traced out. Using root fidelity \(\Fid(\rho,\sigma)=\norm{\sqrt\rho\sqrt\sigma}_1\), CPTP maps do not decrease fidelity \cite{Watrous}. A deterministic cloner of two nonorthogonal pure states of overlap \(0r(m+2)\). A Bell-distribution prehistory, channel capacity, and local gate availability must be declared. An \(O(1)\) protocol overhead does not prove super-cone transport at every distance \(\ell>r\). No metatheorem about an arbitrary new syntax is inferred merely from these physical-channel facts. %===================================================================== \section{Fixed-stencil and periodic obstructions} \label{sec:obstruction} %===================================================================== Two obstructions concern different objects and should be stated separately. \subsection{Fixed word metrics} Let \(S\subset\ZZ^D\setminus\{0\}\) be finite, symmetric, and generating. Write \[ B_S(k)=\left\{\sum_{i=1}^j s_i:j\le k,\ s_i\in S\right\}, \qquad C_S=\conv(S). \] \begin{lemma}[Uniform convex filling]\label{lem:filling} If \(L=\max_{s\in S}\norm{s}_2\), then \[ B_S(k)\subseteq kC_S,\qquad \dH(B_S(k),kC_S)\le(D+1)L \] for every integer \(k\ge0\). \end{lemma} \begin{proof} Symmetry gives \(0\in C_S\), proving the first inclusion. For \(z\in kC_S\), Caratheodory's theorem gives \(z/k=\sum_{i=1}^{D+1}\lambda_i s_i\), with nonnegative weights summing to one. This representation follows by eliminating positive weights along affine dependences until at most \(D+1\) remain. Set \(m_i=\lfloor k\lambda_i\rfloor\) and \(y=\sum_i m_is_i\). Then \(\sum_i m_i\le k\), so \(y\in B_S(k)\), and \[ \norm{z-y}_2\le\sum_i(k\lambda_i-m_i)\norm{s_i}_2 \le(D+1)L. \] The case \(k=0\) is immediate. \end{proof} \begin{theorem}[Fixed-stencil metric obstruction]\label{thm:fixedstencil} Under a fixed positive physical length-to-time normalization, the large-scale word-metric cone of a fixed finite \(S\) has a cross-section proportional to \(C_S\). For \(D\ge2\), it is not a nonzero Euclidean ball. \end{theorem} \begin{proof} Divide Lemma~\ref{lem:filling} by \(k\) and let \(k\to\infty\). The error tends to zero. The limiting body is a finite polytope; a Euclidean ball in \(D\ge2\) has infinitely many extreme points. \end{proof} \begin{remark}[What schedule averaging cannot and can change] A common input-independent mixture of admissible schedules retains the metric upper envelope, provided the two compared runs agree for each schedule realization and include the relevant control data. This does \emph{not} prove that the actual influence region of every averaged dynamics equals the envelope. The identity dynamics has no positive-distance influence. Randomization may suppress signals or alter typical propagation; it does not change a separately defined word-metric envelope. This is the corrected reading of STEN's schedule-averaging claim. \end{remark} \subsection{Timed periodic dependency models} Let \(Q\) be a finite strongly connected quotient graph. Each edge has displacement \(\Delta_e\) and duration \(u_e>0\); translated copies of its lift and waiting are allowed. For a simple directed cycle \(a\), put \[ v(a)=\frac{\sum_{e\in a}\Delta_e}{\sum_{e\in a}u_e}, \qquad K_{\Sch}=\conv\bigl(\{0\}\cup\{v(a)\}\bigr). \] \begin{proposition}[Velocity polytope and finite-time offset] \label{prop:polytope} With \(q=\abs{V(Q)}\) and \(D_{\max}=\max_e\norm{\Delta_e}_2\), every path completed within time \(u\) satisfies \[ \Delta_{\mathrm{path}} \in uK_{\Sch}+\Btwo(0,(q-1)D_{\max}). \] Its limiting velocity set is \(K_{\Sch}\). \end{proposition} \begin{proof} Remove cycles from a walk until the remainder has no repeated vertex. It has at most \(q-1\) edges. The removed cycles decompose into simple cycles, and their total displacement is a duration-weighted sum of their velocities. Since \(0\in K_{\Sch}\), that contribution belongs to \(uK_{\Sch}\). Conversely, repeat cycles in desired time proportions and use fixed connector walks, available by strong connectivity. Connector costs are negligible along increasingly long repetitions; waiting realizes the zero contribution. This approximates every convex combination. \end{proof} This is the comparison with Fritz and Cheung VII-J \cite{Fritz,CheungVIIJ}. A finite-time jump is not bounded by the asymptotic polytope alone. For the cardinal model with velocities \(\pm ce_i\), \[ K_{\Sch}=\{v:\norm v_1\le c\}. \] A Euclidean unit direction \(u\) has radial speed \(c/\norm u_1\): in two dimensions it is \(c\) on axes but \(c/\sqrt2\) on diagonals. Compare En03-A \Source{prop:directional} and VII-E's directional comparison \cite{CheungEn03A,CheungVIIE}. A fixed finite periodic model therefore has its own non-Euclidean attainability obstruction. It is not identical to an arbitrary schedule's metric envelope. If quotient complexity grows, the finite-time offset must be controlled along with the polytope shape. %===================================================================== \section{Increasing stencils: the geometric escape} \label{sec:stencils} %===================================================================== We now prove the geometric component proposed in STEN. The metric changes with the mesh; the theorem is not about one fixed substrate. \begin{definition}[Admissible stencil] For \(0<\delta\le1/2\) and \(L\ge4\sqrt D/\delta\), a \((\delta,L)\)-stencil is a finite set \(S\subset\ZZ^D\setminus\{0\}\) such that: \begin{enumerate}[label=(S\arabic*)] \item \(S=-S\) and \(S\) contains \(\pm e_i\); \item \(\norm s_2\le L\) for every \(s\in S\); \item directions of generators with \(\norm s_2\ge(1-\delta)L\) form a \(\delta\)-net of the unit sphere. \end{enumerate} \end{definition} \begin{lemma}[Stencil existence]\label{lem:stencil} Such stencils exist, with at most \(C_D\delta^{-(D-1)}+2D\) generators. \end{lemma} \begin{proof} Choose a symmetric \(\delta/2\)-net \(\{u_i\}\) of the sphere. For \(D\ge2\), a maximal separated set gives the stated count by packing disjoint spherical caps; for \(D=1\), the sphere has two points. Round \((L-\sqrt D)u_i\) to an integer vector \(s_i\). The error is at most \(\sqrt D/2\), so \[ L-\tfrac32\sqrt D\le\norm{s_i}_2\le L-\tfrac12\sqrt D. \] These vectors are long and nonzero. For \(\rho=L-\sqrt D\), normalization changes direction by at most \(\sqrt D/\rho\le\delta/2\). Together with the net error this gives a \(\delta\)-net. Adjoin negatives and the unit generators. \end{proof} Choose sequences \[ \eps_n\downarrow0,\qquad \delta_n\downarrow0,\qquad L_n\ge4\sqrt D/\delta_n, \] and corresponding stencils \(S_n\). Set \begin{equation} h_n=\eps_n/L_n,\quad \Lat_n=h_n\ZZ^D,\quad \Delta t_n=\eps_n/c,\quad P_n=\conv\{s/L_n:s\in S_n\}. \label{eq:family} \end{equation} The word metric is \[ d_n(h_nz,h_nz')=d_{S_n}(z,z'), \] and the combinatorial block radius is \(r=1\). \begin{remark}[Notation and normalization] The physical word ball is \[ B_{d_n}(0,k)=h_nB_{S_n}(k). \] It is not \(h_nB_{d_n}(0,k)\); the latter would rescale twice. The one-tick reach is at most \(\eps_n\), not necessarily exactly \(\eps_n\), since the stencil need not contain a vector of length exactly \(L_n\). Accordingly, \(c\) is an exact upper normalization and an asymptotically attained speed. \end{remark} \begin{lemma}[Polytope approximation]\label{lem:polyapprox} \[ (1-2\delta_n)\Btwo(0,1)\subseteq P_n\subseteq\Btwo(0,1), \qquad \dH(P_n,\Btwo(0,1))\le2\delta_n. \] \end{lemma} \begin{proof} The upper inclusion follows from the length bound. For unit \(u\), choose a long generator of direction within \(\delta_n\) of \(u\). Then \(\norm{s/L_n-u}_2\le2\delta_n\), so the support function of \(P_n\) in direction \(u\) is at least \(1-2\delta_n\). The supporting-half-space description of a convex body gives the lower inclusion. Radial projection gives the Hausdorff bound. \end{proof} \begin{theorem}[Uniform two-parameter shape theorem]\label{thm:shape} For every \(n\), \(k\ge0\), and \(x\in\Lat_n\), \begin{align} \dH(h_nB_{S_n}(k),k\eps_nP_n)&\le(D+1)\eps_n, \label{eq:shapepoly}\\ \dH(B_{d_n}(x,k),\Btwo(x,k\eps_n)) &\le(D+1)\eps_n+2\delta_n k\eps_n. \label{eq:shape} \end{align} In particular, the second estimate tends uniformly to zero for \(k\eps_n\le R<\infty\). \end{theorem} \begin{proof} Apply Lemma~\ref{lem:filling} with maximum generator length at most \(L_n\), then multiply by \(h_n\). This gives \eqref{eq:shapepoly}. Translation invariance and Lemma~\ref{lem:polyapprox} give \[ \dH(k\eps_nP_n,k\eps_n\Btwo)\le2\delta_n k\eps_n. \] Use the triangle inequality. \end{proof} The additive rounding error is independent of \(k\). The anisotropy term grows as \(k\eps_n\); the \emph{whole} error is not independent of \(k\). One tick reaches \(h_n(S_n\cup\{0\})\), not every lattice point in \(\eps_nP_n\). For example, with \(S=\{\pm1,\pm3\}\), the point \(2\) lies in \(\conv(S)\) but is not reachable in one word step. \subsection{Partitions, relays, and capacity} \begin{proposition}[Edge partitions and relay attainment] \label{prop:matchings} Every generator edge extends to a partition into generator pairs. Every finite word path can therefore be relayed one edge per tick by admissible partitions. The schedule class has \(c_{\mathrm{lat}}=1\). Its maximal Euclidean signalling speed is \[ c_n=c\,\frac{\max_{s\in S_n}\norm s_2}{L_n}, \qquad (1-\delta_n)c\le c_n\le c. \] \end{proposition} \begin{proof} For a generator \(s\ne0\), the graph of edges \(z\leftrightarrow z+s\) is a disjoint union of bi-infinite chains, the cosets of \(\ZZ s\). Choose one representative for each coset and write its sites as \(z_0+ks\). Pair \(2k+p\) with \(2k+p+1\), \(p=0,1\). Either phase is a full partition. Choose the phase containing each desired path edge and let other blocks act identically. Connectedness and finite balls on the infinite word graph give a vertex at every integer distance, so geodesic bit relays attain the combinatorial upper bound \(1\). Every block has physical diameter at most \(h_n\max_s\norm s_2\). The same one-layer argument bounds influence by that reach per tick. A record transport along a maximum-length edge attains the corresponding one-tick ratio. Long generators give the displayed lower bound. \end{proof} For a finite declared register collection, choose sufficient local capacity and route names one at a time using unit-generator edges, keeping already delivered names stationary. This gives a routing \emph{construction in the schedule class}. It does not show that every arbitrary fixed schedule routes. A fair fixed schedule that repeats both matching phases of every generator is tail routing, but may have a different physical throughput. The distinction between schedule-class attainment and a fixed schedule's speed remains essential. \subsection{Nested refinements} A useful explicit nested family is, for sufficiently large \(n\), \begin{equation} \eps_n=4^{-n},\qquad L_n=2^n,\qquad \delta_n=4\sqrt D\,2^{-n}. \label{eq:nested} \end{equation} Then \(h_n=8^{-n}\), the spatial and tick grids are nested, and \(L_n\eps_n\to0\). Nesting the site sets does not nest the generators or identify their dynamical channels. %===================================================================== \section{Euclidean cone convergence and microscopic compatibility} \label{sec:geometrylimit} %===================================================================== Use the spacetime comparison metric \[ \dc((t,x),(t',x'))=\max(c\abs{t-t'},\norm{x-x'}_2). \] This is a topology and error metric, not a Lorentz-invariant metric. Events embed as \((s\Delta t_n,x)\). Define the full discrete metric past of an apex \(a_n=(T_n\Delta t_n,A_n)\) by \[ J^-_n(a_n)= \{(s\Delta t_n,x):s\le T_n,\ d_n(x,A_n)\le T_n-s\}. \] Finite experiments use its truncation below their starting tick. Local convergence means that, on every compact comparison set, points of either set have distance tending to zero from the other \emph{full set}. It does not require approximating points to remain inside the same compact set. \begin{theorem}[Local event approximation and cone convergence] \label{thm:coneconv} The event grids become locally dense. If \(a_n\to a=(t_a,x_a)\), then \[ J^-_n(a_n)\longrightarrow J^-_2(a)=\{(t,x):t\le t_a,\ \norm{x-x_a}_2\le c(t_a-t)\} \] locally, uniformly for apexes in bounded sets. If \(e_n=\dc(a_n,a)\), then the excess of \(J^-_n(a_n)\cap K\) over \(J^-_2(a)\) is at most \(e_n\). For a bounded \(K\), the reverse excess is at most \[ e_n+(D+2)\eps_n+2\delta_nR_K \] for a fixed sufficiently large \(R_K\) and all large \(n\). \end{theorem} \begin{proof} The grid \(\Delta t_n\ZZ\times h_n\ZZ^D\) approximates every point with error at most \(\max(\eps_n/2,\sqrt D\,h_n/2)\) in \(\dc\). The exact upper word-ball containment gives \(J^-_n(a_n)\subseteq J^-_2(a_n)\). Translate a point by \(a-a_n\); it then lies in \(J^-_2(a)\), at distance at most \(e_n\). Conversely, translate \(p\in J^-_2(a)\cap K\) by \(a_n-a\) to a point \(p'\in J^-_2(a_n)\). Round its time downward to the tick grid. The time error is less than \(\Delta t_n\), and its spatial coordinate lies in the Euclidean ball of radius \(k\eps_n=c(T_n\Delta t_n-s\Delta t_n)\). For bounded \(K\) and bounded apexes, these radii are bounded by a fixed \(R_K\). Theorem~\ref{thm:shape} gives a word-ball point with spatial error at most \((D+1)\eps_n+2\delta_nR_K\). Together with the translation and time rounding this gives the bound. \end{proof} Thus the corrected geometric requirements C1 and C2 are theorems. The condition involving event grids intersected with \emph{every} compact set in CONE is too strong: a singleton absent from a given grid cannot be approximated by that grid's intersection with the singleton. \begin{proposition}[C2b with zero slack]\label{prop:C2b} For the increasing-stencil family, \[ \ell_n=\sup\{\norm{x-y}_2:d_n(x,y)\le1\} \le\eps_n=c\Delta t_n. \] Every scheduled dependency path of \(k\) ticks has Euclidean displacement at most \(ck\Delta t_n\). \end{proposition} \begin{proof} A distance-one displacement is a generator displacement. Sum the one-step bounds and use the triangle inequality. \end{proof} This is a bound on potential information paths as well as relocations of a named object. A geometric shape theorem and an operational path attainment statement remain logically distinct. \begin{figure}[t] \centering \begin{tikzpicture}[scale=1.45] \draw[->] (-1.4,0)--(1.45,0); \draw[->] (0,-1.4)--(0,1.45); \draw[blue,thick] (0,0) circle (1); \draw[orange,thick] (1,0)--(0,1)--(-1,0)--(0,-1)--cycle; \draw[green!50!black,thick] (0:0.96) \foreach \a in {30,60,...,360}{--(\a:0.96)}--cycle; \node[blue,anchor=west] at (1.05,0.8){Euclidean ball}; \node[orange!80!black,anchor=west] at (1.05,0.4){cardinal diamond}; \node[green!50!black,anchor=west] at (1.05,0.0){growing stencil}; \end{tikzpicture} \caption{The escape from the fixed-stencil obstruction is a sequence of changing finite polytopes. The ambient norm and the prescribed speed normalization are construction inputs.} \end{figure} %===================================================================== \section{A precise coarse relational outcome-law theory} \label{sec:coarse} %===================================================================== This section provides an unconditional operational result without using the disputed general transport theorem. \subsection{The admitted finite class} \begin{definition}[Fixed-wire local experiment]\label{def:experiment} At scale \(n\), an admitted experiment has: \begin{enumerate}[label=(\roman*)] \item a finite deterministic schedule of active singleton or generator-pair blocks, with identity completion; \item finite declared quantum and classical interfaces, deterministic wire routing and allocation interfaces, and complete locally controlled payload instruments; \item a product default state on all declared initial registers, with no external shared random seed or correlated undeclared resource; \item a fixed finite tuple of labelled report ports, each acquired by a local readout at its own tick and site; \item a nonempty compact declared support \(K_n\) containing all active input/output block footprints and all retained wire paths and dwell segments between their uses. \end{enumerate} The readout acquisition times need not coincide. A report retained solely to calculate the joint law is not an operational communication wire. If used downstream, its physical wire must be declared. The skeleton consists of everything except the locally controlled payload maps. The complete report law is \(\mu_n\). \end{definition} This is a deliberately restricted class. Adaptive payloads are included; arbitrary outcome-dependent wiring is not silently assumed to have a support-preserving fixed-wire compilation. \begin{definition}[Full-tail limit objects and modifications] \label{def:limit} A limit object is \(\xi=(K,R,\mu)\), where \(K\) is compact, \(R\) is a finite tuple of labelled acquisition events in \(K\), and \(\mu\) is a law on fixed finite alphabets. It belongs to \(\XX\) if experiments exist at every sufficiently large scale with \[ K_n\to K,\qquad R_n\to R,\qquad \dtv(\mu_n,\mu)\to0. \] Two objects differ by a modification in compact \(K'\) when matched full-tail families have the same skeletons and report designations, and payloads differ only on block footprints in \((K')_{\rho_n}\), with \(\rho_n\to0\). The initial default state is common. The theory \(\Theory\) consists of these objects, this witnessed modification relation, admitted disjoint composites, and conditioning of laws at positive-probability events. \end{definition} The relation is not determined by schedule membership alone. It requires matched local interventions. \subsection{Continuum domain of dependence} Let \(J^-_2(R)=\bigcup_{r\in R}J^-_2(r)\). \begin{theorem}[Coarse continuum domain of dependence] \label{thm:coarseDOD} If a matched modification pair has \[ \dist_{\dc}(K',J^-_2(R))>0, \] then its two complete report laws agree. \end{theorem} \begin{proof} Every discrete dependency path obeys the physical bound in Proposition~\ref{prop:C2b}. Report acquisition events converge to \(R\), and modified footprints approach \(K'\). The positive margin therefore places every modified gate outside the ancestor subnetwork of the report ports for all sufficiently large \(n\). Their initial inputs and included gates agree. Apply Theorem~\ref{thm:order} to the full report tuple, then pass to the total-variation limit. \end{proof} The same proof applies to any subset of report ports. It proves equality of their joint marginal, not merely equality of each one-port marginal. \begin{proposition}[Spacelike composition and no-signalling] \label{prop:coarseNS} Let \(\xi^1,\xi^2\in\XX\) have compact supports satisfying \[ \norm{x_p-x_q}_2\ge c\abs{t_p-t_q}+\delta \quad(p\in K^1,\ q\in K^2) \] for some \(\delta>0\). Their independent composite belongs to \(\XX\), with law \(\mu^1\otimes\mu^2\). Each marginal is independent of the choice of the other experiment. More generally, modifications outside the past of a chosen report subset leave that subset's joint marginal unchanged. \end{proposition} \begin{proof} The inequality gives positive separation of spatial shadows. Hausdorff convergence makes the active site sets of the two families disjoint for all large \(n\). Their independent product-default preparations, complete channels, and report ports factor. The product laws converge because \[ \dtv(\mu_n^1\otimes\mu_n^2,\mu^1\otimes\mu^2) \le\dtv(\mu_n^1,\mu^1)+\dtv(\mu_n^2,\mu^2). \] For the marginal statement apply Theorem~\ref{thm:coarseDOD} to that report subset. \end{proof} \begin{remark}[Product law is a preparation convention] Microcausality does not require spacelike outcomes to be independent. Entangled background states can correlate them. The product conclusion above uses the independent product-default experiment class; it is not a derivation of a product vacuum in a quantum field theory. The general channel no-signalling theorem permits entangled inputs. \end{remark} If a realized support is the union of two positively spacelike separated compact components, with one report in each, its law factorizes in this product-default class. At small mesh, no block or retained wire can connect the components, so the finite circuit splits. This is the restricted product-rigidity comparison with TRANSPORT. \subsection{Sharp speed and nondegenerate relays} \begin{theorem}[Directional geometric and coarse relays] \label{thm:relays} For every Euclidean unit \(v\), duration \(\sigma>0\), and source position \(x\), there are matched full-tail experiments with source modifications approaching \(A=(0,x)\), report events approaching \(B=(\sigma,x+c\sigma v)\), and laws exactly \(\delta_0,\delta_1\) at every scale. Their declared supports converge to the closed segment from \(A\) to \(B\). \end{theorem} \begin{proof} Choose a long generator \(s_n\) with direction within \(\delta_n\) of \(v\), a source site \(x_n\to x\), and \(k_n\) with \(k_n\Delta t_n\to\sigma\). A singleton source payload writes bit \(0\) or \(1\). Relay the bit along \(k_n\) successive \(s_n\)-edges using the matching phases of Proposition~\ref{prop:matchings}, then read it locally. The constant finite number of preparation/readout ticks adds \(O(\Delta t_n)\), which vanishes. The displacement is \(k_nh_ns_n\), with length between \((1-\delta_n)k_n\eps_n\) and \(k_n\eps_n\), and direction tending to \(v\). It tends to \(c\sigma v\). The footprints and paths lie within vanishing neighborhoods of the segment, and all source differences lie in a shrinking neighborhood of \(A\). The report laws remain deterministic and distinct. \end{proof} \begin{corollary}[Sharp coarse signalling speed]\label{cor:coarsespeed} Define \(c_{\mathrm{coarse}}\) as the supremum of \(\norm{x_B-x_A}_2/(t_B-t_A)\), \(t_B>t_A\), over matched modification signalling pairs localized arbitrarily near \(A,B\). Then \[ c_{\mathrm{coarse}}=c. \] \end{corollary} \begin{proof} For a ratio greater than \(c\), sufficiently small neighborhoods of \(A\) have positive distance from the past of neighborhoods of \(B\). Theorem~\ref{thm:coarseDOD} excludes a law difference. Theorem~\ref{thm:relays} attains the limiting boundary. \end{proof} Thus the geometric relay membership problem is solved for this \emph{defined} full-tail coarse class. This does not prove existence of an instrument-level realization satisfying the stronger diamond-norm C3 of CONE. \subsection{Conditioning and compactness} For positive-probability \(E\), total-variation convergence implies \[ \mu_n(\,\cdot\mid E)\longrightarrow\mu(\,\cdot\mid E). \] For example, if \(\dtv(\mu_n,\mu)=\eta_n<\mu(E)/2\), the conditional distance is at most \(4\eta_n/\mu(E)\). This is convergence of conditional \emph{laws}, not a deterministic physical postselection protocol. Conditioning a product law on equality of two independent fair bits creates correlated bits. It need not be realizable with the same disconnected support in the product-default class. Finite outcome simplices and the hyperspace of nonempty compact subsets of a compact metric space are compact. They give subsequential support--law limits \cite{Burago}. They do not, without more work: \begin{itemize} \item fill a subsequence into a family at every fine scale; \item preserve matched modifications and composition; \item provide common quantum input/output embeddings; \item establish diamond-norm convergence. \end{itemize} Full-tail filling is one purpose of the conditional transport contract below. It is not obtained merely by renaming all subsequential limits as experiments. %===================================================================== \section{The five-part transport proposal and its exact boundary} \label{sec:transport} %===================================================================== TRANSPORT proposes the following chain \cite{CheungTRANSPORT}: \[ \mathrm{P1}\longrightarrow\mathrm{P2} \longrightarrow\mathrm{P3}\longrightarrow\mathrm{P4} \longrightarrow\mathrm{P5}. \] The parts have different mathematical content. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.27\linewidth}X@{}} \toprule Part & Source statement & Treatment here\\ \midrule P1 & Support-preserving fixed-wire compilation & A separate normalization obligation for arbitrary adaptive wiring. We use fixed-wire experiments explicitly.\\ \addlinespace P2 & Midpoint-event causal embedding & The midpoint causality lemma is proved below. Causal endpoint placement is not yet digital realization.\\ \addlinespace P3 & Collision-managed routing, scheduling, and lateness & Quantitative transport obligation. The supplied proof does not establish its spacetime-support bound.\\ \addlinespace P4 & Exact relabelled payload simulation & Proved below whenever a valid realization with matching wires is given.\\ \addlinespace P5 & Coarse/fine diagonal transport & Proved below under an explicit transport contract. It cannot repair a failed P1--P3 hypothesis.\\ \bottomrule \end{tabularx} \end{center} \subsection{The midpoint lemma} \begin{lemma}[Midpoint-event causality]\label{lem:midpoint} For a fixed-wire experiment at scale \(j\), let \(e=(s,b)\) and \(e'=(s',b')\), \(s's\). Let \(m(b)\) be the midpoint of a pair or the site of a singleton. Then \[ \norm{m(b')-m(b)}_2\le\eps_j \le c(s'-s)\Delta t_j. \] The midpoint segment is within \(O(\eps_j)\) of the declared worldline-closed support. \end{lemma} \begin{proof} The register is static at a site \(p\) between its participations. Its deterministic output position from \(b\) and input position in \(b'\) are both \(p\). Thus \(p\in b\cap b'\). Each midpoint is within \(\eps_j/2\) of \(p\). The time difference is at least one tick. The retained dwell segment at \(p\) belongs to the declared support. Interpolating the midpoint offsets bounds the midpoint segment's distance from that dwell segment by at most \(\eps_j\). \end{proof} An orthochronous Poincare transformation preserves the future-causal endpoint displacement. Its Lipschitz constant in \(\dc\) bounds the image support error. This proves the geometric comparison, not the scheduler estimate. \subsection{Exact simulation once geometry is provided} \begin{proposition}[Reference-preserving relabelled simulation] \label{prop:simulation} Suppose a fixed finite complete payload network is realized on new carriers with the same wire graph and interfaces. Transport blocks act only as wire-address changes. At each payload event the original complete map is applied under the corresponding tensor-factor relabelling. Then the realized network has the original complete channel under the external-interface relabelling, including every reference. \end{proposition} \begin{proof} Choose a topological ordering of the original network. The new realization respects its dependencies. Initially identify corresponding wire spaces by a tensor relabelling. At every payload event, conjugacy of its local map carries the identified intermediate state to the next identified state. Address changes do not alter internal logical states. Induct over the finite network, retaining every outcome branch. Theorem~\ref{thm:foliation} removes the choice of legal ordering. Tensor the calculation with a reference identity. \end{proof} This is a channel-network simulation statement. It is comparable with En02-C's reference-preserving soundness \cite{CheungEn02C}, but does not assert availability or compilation of arbitrary payload instruments over a hardware library. \subsection{Why the literal source covariance cannot be retained} \begin{proposition}[Common-final-tick obstruction] \label{prop:simultaneous} The limit class defined literally in TRANSPORT, whose designated readouts all occur at one final tick, is not closed under boosts. This fails already for two independent deterministic readouts. \end{proposition} \begin{proof} Every realizing family in that class has one acquisition time \(a_n=t_n^+\Delta t_n\) for every designated readout. If two readout events converge, their limiting time coordinates must therefore agree. Two singleton deterministic readouts at \(r_1=(0,0)\) and \(r_2=(0,Le_1)\), \(L>0\), are realizable, with law \(\delta_{(0,0)}\). A boost of nonzero velocity \(v\) in direction \(e_1\) gives time coordinates \(0\) and \(-\gamma vL/c^2\), which differ. The transformed object has no realizing family satisfying the common-final-tick definition. \end{proof} Holding earlier records until a final tick preserves values, but changes the acquisition events and adds physical storage segments. It cannot prove covariance of the originally labelled spacetime data. Definition~\ref{def:experiment} removes this obstruction by allowing distinct actual acquisition times. \begin{proposition}[Spatial route control is not spacetime control] \label{prop:pacing} A route lying exactly on a spatial segment, with endpoints fired at their correct nominal times and no late arrival, need not stay near the intended spacetime segment. \end{proposition} \begin{proof} In one dimension take \(00. \] Indeed minimizing \(\max(c\abs{t'-\ell/c},\abs{\ell-vt'})\) balances the two terms at \(t'=2\ell/(c+v)\in[0,T]\). The positive distance does not disappear when the digital mesh tends to zero. \end{proof} This example applies to the image of a long idle source worldline. The supplied endpoint-lateness induction does not bound departure times, distributed waiting, or the spacetime location of early arrivals. A pacing estimate must be supplied. \begin{remark}[Verification gap versus mathematical gap] Propositions~\ref{prop:simultaneous} and~\ref{prop:pacing} are not absence-of-machine-checking observations. They identify an incompatible definition and an insufficient estimate. Additional problems with lane separation, charging, and local inner simulation are documented in Appendix~\ref{app:audit}. They do not establish that every alternative transport construction is impossible. \end{remark} \subsection{A corrected transport contract} \begin{hypothesis}[T: Payload-natural spacetime transport] \label{hyp:T} For the fixed-wire experiment class of Definition~\ref{def:experiment}, assume the following. For each \(g\in\Poin\) there are numbers \(a_j(g)\to0\), independent of the complexity and payloads of a source skeleton at scale \(j\). For every source experiment \(\xi\) at that scale, and every sufficiently large target scale \(m\), there is a transported experiment with: \begin{enumerate}[label=(T\arabic*)] \item exactly the same complete report law and labelled alphabets; \item declared support at Hausdorff distance at most \(a_j(g)+b_{\xi,g,m}\) from \(gK_\xi\), and each acquisition event within that bound of its corresponding \(g\)-image, where \(b_{\xi,g,m}\to0\) as \(m\to\infty\) at fixed \(\xi,g\); \item a skeleton and routing plan independent of payloads; \item matched transports of payload-modification pairs, with differences confined within the same error of the transformed modification footprints. \end{enumerate} The spacetime bound covers all paths, waits, allocation interfaces, and retained record storage, not just spatial routes and meeting endpoints. The identity case is required as well. \end{hypothesis} This contract is not asserted as a theorem for the full class. The source's desired form \(a_j(g)\le C_g\eps_j\) is one sufficient choice. The fine error may depend on the finite source complexity, because the diagonal argument fixes a source before refining it. \begin{lemma}[Diagonal transport]\label{lem:diagonal} Assume T. Suppose experiments \(\xi_j\), on an infinite sequence of source scales, satisfy \[ K_j\to K,\qquad R_j\to R,\qquad \mu_j\to\mu \] in Hausdorff, event, and total-variation topology. For every \(g\in\Poin\), there is a full-tail realizing family for \((gK,gR,\mu)\). The identity case supplies full-tail filling. \end{lemma} \begin{proof} For each source scale \(j\), choose an increasing threshold \(m(j)\) so that all target scales \(m\ge m(j)\) admit its transport and have \(b_{\xi_j,g,m}\le2^{-j}\). For \(m(j)\le N