% En06-B.tex % Standalone source: no external bibliography or figures are required. % Compile twice: % pdflatex -interaction=nonstopmode -halt-on-error En06-B.tex % pdflatex -interaction=nonstopmode -halt-on-error En06-B.tex % % Scope: % This paper proves a transport contract for the explicitly specified % increasing-stencil family below. It does not prove the contract for % an arbitrary, unspecified predecessor substrate. % % No successful compilation, proof-assistant verification, independent % referee review, or journal acceptance is asserted. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{enumitem} \usepackage{booktabs,tabularx} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{3em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\B}{\mathcal B} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\conv}{\operatorname{conv}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\dH}{d_{\mathrm H}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|} \newcommand{\proj}[1]{\ket{#1}\bra{#1}} \newcommand{\ip}[2]{\left\langle#1,#2\right\rangle} \newcommand{\Alg}{\mathfrak A} \newcommand{\Poin}{\mathsf P^\uparrow_+} \newcommand{\eps}{\varepsilon} \newcommand{\CP}{\mathrm{CP}} \newcommand{\CPTP}{\mathrm{CPTP}} \title{\textbf{En06-B}\\[3pt] Constructive Spacetime Transport for Finite Instrument Networks\\ and an Exact Bell Instrument--Observable Limit} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We construct a payload-independent spacetime realization of every finite future-causal fixed-wire instrument network on an explicitly specified increasing-stencil lattice family. A residue-class construction supplies private transport lanes inside an ordinary lattice, including in one spatial dimension. Rounded affine paths give an every-prefix pacing estimate. Local gathering, unchanged payload instruments, and local dispersal take a vanishing amount of physical time; long routes operate in parallel. The complete instrument is preserved exactly, including every untouched reference. For a calibrated finite source experiment at scale \(j\), and a fixed proper orthochronous Poincare transformation \(g\), the construction gives, at every sufficiently fine target scale \(n\), \[ \dH(K_{j\to n},gK_j) \le C_g\eps_j+r_{\Gamma,j,g}(n), \qquad r_{\Gamma,j,g}(n)\longrightarrow0, \] with \(C_g\) independent of source-network complexity. Acquisition events, individual retained paths, and matched payload-modification footprints satisfy corresponding estimates. A full-tail diagonal argument follows. We then construct a Bell-pair measurement network with local setting inputs, retained quantum outputs, and an untouched quantum memory. Its logical instrument is exactly scale-independent. An explicit finite unitary apparatus dilation defines an isotone spacetime observable net with spacelike commutation, product-preserving comparison maps, and an exact state--instrument pairing. The prepared Bell specialization has CHSH value \(2\sqrt2\), while its unconditioned marginals are independent of the remote setting. The result is a finite-network continuum realization. It is not a quantum field theory, a derivation of physical light speed, or an exact microscopic boost symmetry. \end{abstract} \tableofcontents \section{Introduction and the precise scope of the result} \label{sec:intro} The fixed-wire interface recorded in Cheung's En06-A distinguishes a finite deterministic schedule, complete payload instruments, locally acquired reports, and a worldline-closed declared support \cite{CheungEn06A}. Its transport contract requires two different types of error: a coarse source error whose coefficient does not grow with the source graph, and a fine implementation error which may depend on that fixed graph and must vanish. This paper supplies such a construction for the target family defined in Section~\ref{sec:substrate}. The family is part of the theorem, not an unnamed inherited assumption. Its increasing directional resolution is essential. Fixed-periodic velocity polytopes and their anisotropy are discussed by Fritz \cite{Fritz,FritzCorrigendum} and, within the supplied manuscript series, by Cheung in En05-A and VII-J \cite{CheungEn05A,CheungVIIJ}. The construction uses ordinary lattice sites and singleton or two-site blocks. It does not introduce coincident copies of a site. Instead, finitely many residue classes reserve disjoint site sets for the wires and payload junctions of a fixed network. This gives a particularly direct repair of the lane-collision problem. \subsection{Three necessary corrections} \begin{enumerate}[label=\textup{(\roman*)}] \item \textbf{The target substrate must be specified.} A Euclidean upper speed bound alone does not supply isotropic routing. Our target generator set is stated explicitly. \item \textbf{Future-directed transport uses a time-orientation-preserving group.} We use the proper orthochronous Poincare group \(\Poin\). A time reversal does not preserve the directed network class. Spatial parity could be added without changing the proof, but time reversal is not included. \item \textbf{Channel comparison and observable comparison are different.} An arbitrary channel's Heisenberg adjoint need not preserve products. The observable comparison below is instead an explicit \(*\)-isomorphism on a selected logical dilation algebra. \end{enumerate} Cheung's En05-B is a documentary comparison for separated lanes, junction interfaces, and reference-preserving identity transport \cite{CheungEn05B}. En05-C is a comparison for complete-device specification and product-preserving observable identifications \cite{CheungEn05C}. None of their unseen proofs is used below. Discrete Lorentz covariance for particular quantum walks and cellular automata is a distinct construction \cite{ArrighiFacchiniForets}. Likewise, the scaling-algebra framework of Buchholz and Verch starts with a local quantum field theory and addresses its short-distance limits \cite{BuchholzVerch}. Neither result is substituted for the finite measurement-and-record construction proved here. \section{Geometry, instruments, and admitted networks} \label{sec:definitions} \subsection{Spacetime conventions} Fix a spatial dimension \(D\ge1\) and a constant \(c>0\). On \(\R^{1+D}\), write points as \(z=(t,x)\) and use \[ \norm{(t,x)}_c=\sqrt{c^2t^2+\norm{x}_2^2}, \qquad d_c(z,z')=\norm{z-z'}_c. \] All Hausdorff distances and neighborhoods in this paper use \(d_c\). For a set \(K\), let \[ K_\rho=\{z:\dist_{d_c}(z,K)\le\rho\}. \] An edge from \(z_v=(t_v,x_v)\) to \(z_w=(t_w,x_w)\) is future causal when \[ t_w\ge t_v, \qquad \norm{x_w-x_v}_2\le c(t_w-t_v). \label{eq:causalcondition} \] If its duration is zero, its endpoints coincide. For \(g\in\Poin\), let \(L_g\) be the operator norm of the linear part of \(g\) with respect to \(\norm{\cdot}_c\). Then \[ d_c(gz,gz')\le L_gd_c(z,z'), \qquad \dH(gK,gK')\le L_g\dH(K,K'). \label{eq:Lipschitz} \] Translations do not enter \(L_g\). \subsection{Complete instruments} For finite-dimensional Hilbert spaces, an instrument is a finite family of completely positive maps \[ \{\Phi_a:\B(\mathcal H_{\mathrm{in}}) \longrightarrow\B(\mathcal H_{\mathrm{out}})\}_{a\in A} \] such that \(\sum_a\Phi_a\) is trace preserving. Its complete record-retaining channel is \[ \widehat\Phi(X) =\sum_{a\in A}\Phi_a(X)\otimes\proj{a}. \label{eq:completeinstrument} \] Classical registers are represented by their diagonal matrix algebras. When a full-matrix extension is needed, input classical registers are dephased in their declared bases. We compare complete channels using \[ d_\diamond(\Phi,\Psi) =\frac12\norm{\Phi-\Psi}_\diamond. \] The diamond norm and finite-dimensional channel representations are standard; see Watrous \cite{Watrous}. The instrument distinction originates in the operational framework of Davies and Lewis \cite{DaviesLewis}. \begin{definition}[Finite embedded fixed-wire instrument network] \label{def:network} An admitted network consists of the following data. \begin{enumerate}[label=\textup{(\roman*)}] \item A finite directed acyclic graph, with typed quantum or classical wire ports and explicitly named input and output interfaces. \item A spacetime event \(z_v\) for every vertex. Every directed wire edge has future-causal endpoints and is represented geometrically by the straight segment between them. \item For every payload vertex, a complete finite-dimensional instrument on its entire declared input interface, with the entire output interface specified. Locally controlled payloads may read declared classical input wires. Their output types and wire routing do not depend on the outcomes. \item Explicit allocation, preparation, consumption, and terminal interfaces. A default experiment has a product preparation on its declared initial registers and no undeclared shared resource. \item Labelled report acquisitions. Each label names the local payload readout which first exposes that report. A report used downstream has a declared physical wire. A tuple retained only for statistical bookkeeping is not thereby a communication wire. \item All retained quantum, classical, apparatus, and environmental storage segments which belong to the chosen physical realization. \item A nonempty compact declared support \(K\) containing the physical network support \(F\), and a specified class of matched payload modifications. \end{enumerate} The skeleton consists of all these data except the variable payload maps. In particular, skeleton data include register types, routing, acquisitions, and allowed modification locations. \end{definition} Complete maps, rather than restrictions to a smaller regional algebra, are specified. This is the distinction emphasized in the extension-gap comparison in En05-C \cite{CheungEn05C}. The external channel is the contraction of this finite wired network. Input and output tensor products collect named ports mathematically; they do not assert that differently located ports are physically reunited at one instant. \subsection{Capacity and admissible local operations} We use the following explicit capacity convention. \begin{assumption}[Finite declared capacity, without a uniform bound] \label{ass:capacity} Each finite experiment may reserve finitely many finite-dimensional register factors at each of its finitely many used sites. There is no bound, uniform over all experiments, on this finite number. A singleton block may implement an arbitrary complete instrument on its declared factors. A two-site block may implement typed swaps and act as the identity on all other factors. Unused factors have fixed product blank states. \end{assumption} No gauge constraint or shared central sector is included in this tensor-product model. Extending the construction to constrained regional algebras would require additional interfaces; the tensor-free comparisons in En05-C do not automatically provide such a routing construction. \subsection{Calibrated discrete sources} \begin{definition}[Calibrated source experiment] \label{def:source} A source experiment of resolution \(\eps>0\) has tick duration \(\tau=\eps/c\). At each tick, its active blocks are singletons or pairs \(\{x,y\}\) with \(\norm{x-y}_2\le\eps\), completed by identity blocks. The closed footprint of a block firing during \([k\tau,(k+1)\tau]\) is \[ H_{k,B} = [k\tau,(k+1)\tau]\times\conv(B). \label{eq:blockfootprint} \] For a singleton, \(\conv(B)\) is that site. Logical registers use fixed typed output ports. Between consecutive uses they remain at their designated sites; a register transfer inside a pair block is contained in that block's footprint. All such retained segments are included in the physical support \(F\). Initial and terminal port events are also included. A report acquired by a block is acquired at that block's declared output tick and output site. Its actual acquisition event is part of the source data. The declared support is any nonempty compact \(K\supseteq F\). \end{definition} The source need not use the target substrate introduced below. Its calibration and fixed-wire geometry, however, are required. \section{The causal midpoint replacement} \label{sec:midpoint} For a pair block, use its spatial midpoint; for a singleton, use its site. The event time is the midpoint of its firing interval. Thus a block \(B=\{x,y\}\) during tick \(k\) has midpoint \[ p_{k,B} = \left((k+\tfrac12)\tau,\frac{x+y}{2}\right). \] Connect consecutive uses of each logical register by a straight segment between the corresponding midpoint events. Add the initial and terminal port events and their half-tick connecting segments. Let \(F^{\mathrm{mid}}\) be the resulting finite geometric graph. \begin{lemma}[Causal midpoint replacement with uniform coarse error] \label{lem:midpoint} For every calibrated source, \[ \dH(F^{\mathrm{mid}},F)\le2\eps. \label{eq:midpointHausdorff} \] Every directed edge of the replacement graph is future causal. For a report acquired at an event \(r\) by a block with midpoint \(p\), \[ d_c(r,p)\le\eps. \label{eq:acquisitionmidpoint} \] A block footprint is at Hausdorff distance at most \(\eps\) from its midpoint singleton. The constants are independent of the number of vertices, registers, or ticks. \end{lemma} \begin{proof} Suppose a logical register leaves one block at site \(q\) and next enters another block at the same site \(q\). If their spatial midpoints are \(u\) and \(v\), then \[ \norm{u-q}_2\le\frac{\eps}{2}, \qquad \norm{v-q}_2\le\frac{\eps}{2}. \] Consecutive distinct block uses are separated by at least one tick. Consequently \[ \norm{v-u}_2\le\eps \le c(t_v-t_u), \] so their midpoint segment is future causal. The same argument for an initial or terminal half-tick gives displacement at most \(\eps/2\) in time \(\tau/2\). During the storage interval between the two block uses, the interpolated midpoint segment is a convex combination of \(u\) and \(v\), both within \(\eps/2\) of \(q\). It is therefore within \(\eps/2\) spatial distance of the retained vertical segment at the same time. During the adjacent half-block intervals, comparison with an endpoint or with the block footprint costs at most \(2\eps\) in \(d_c\). Conversely, each retained storage point has the corresponding midpoint-segment point at the same time, except within those half-block intervals, where the same bound applies. Every point of a block footprint lies within \[ \sqrt{(c\tau/2)^2+(\eps/2)^2} =\frac{\eps}{\sqrt2} \] of its midpoint. This also covers footprint points not selected by a particular logical wire. Taking unions uses a maximum, not a sum, proving \eqref{eq:midpointHausdorff}. A report output event has time displacement \(\tau/2\) and spatial displacement at most \(\eps/2\) from the block midpoint, proving \eqref{eq:acquisitionmidpoint}. The block-footprint statement follows from the same estimate. \end{proof} \begin{remark}[No straightening across an entire long zigzag] The construction puts a vertex at every active register use. It does not replace a many-tick zigzag by one endpoint chord. The uniform coarse bound would not follow from that different operation. \end{remark} After applying \(g\in\Poin\), Lemma~\ref{lem:midpoint} gives \[ \dH(gF^{\mathrm{mid}},gF)\le2L_g\eps. \label{eq:transformedmidpoint} \] The transformed graph remains future causal. Extra points of \(K\setminus F\) are support metadata. They need not be physically populated and will not be confused with the minimal physical support. \section{An explicit increasing-stencil substrate} \label{sec:substrate} Fix a length \(\ell_0>0\). At target scale \(n\ge1\), put \[ a_n=\frac{\ell_0}{n}, \qquad h_n=\frac{\ell_0}{n^3}, \qquad \tau_n=\frac{a_n}{c}, \qquad R_n=\frac{a_n}{h_n}=n^2. \label{eq:scales} \] The sites are \(h_n\Z^D\). The allowed generators are \[ S_n=\{s\in\Z^D:0<\norm{s}_2\le n^2\}. \label{eq:stencil} \] A two-site block is legal whenever its displacement is \(h_ns\) with \(s\in S_n\). Singleton blocks are legal. At each integer tick, the active pairs must be a matching. Every remaining used site is a singleton. The partition may depend on the deterministic experiment skeleton, but not on the payload values, input state, outcomes, or external reference. The distinction between a skeleton-dependent schedule and a fixed autonomous periodic schedule is part of the model. \begin{lemma}[Microscopic speed and directional refinement] \label{lem:substrate} Every register hop has displacement at most \(a_n\) and duration \(\tau_n\), hence speed at most \(c\). Block footprints have \(d_c\)-diameter at most \(\sqrt2a_n\). The convex hulls \[ V_n=\conv\left(\{0\}\cup \left\{\frac{c}{n^2}s:s\in S_n\right\}\right) \] converge in Hausdorff distance to the Euclidean ball \(\overline B_c(0)\). \end{lemma} \begin{proof} The speed and footprint statements follow from \eqref{eq:scales}--\eqref{eq:stencil}. The inclusion \(V_n\subseteq\overline B_c(0)\) is immediate. For a unit vector \(u\), round \((n^2-\sqrt D)u\) coordinatewise to an integer vector \(s\). For sufficiently large \(n\), \[ \norm{s}_2 \le n^2-\sqrt D+\frac{\sqrt D}{2}E+V. \] Assign a different residue \(r\in\{0,\ldots,L-1\}\) to every edge and every macrovertex. For each assigned residue, define \[ \Lambda_{r,n} =Lh_n\Z^D+r h_ne_1. \label{eq:residuelane} \] These are pairwise disjoint subsets of \(h_n\Z^D\). Let \(Q_{r,n}\) be a fixed nearest-point rounding map onto \(\Lambda_{r,n}\), with deterministic tie breaking. Then \[ \norm{Q_{r,n}(x)-x}_2\le b_n, \qquad b_n=\frac{\sqrt D}{2}Lh_n. \label{eq:roundingbound} \] \begin{lemma}[Private lanes and local access] \label{lem:residues} Different assigned residues never share a physical lattice site, including when \(D=1\). For a vertex \(v\), let its hub be \(Q_{r_v,n}(x_v)\). For an incident edge \(e\), its endpoint port is \(Q_{r_e,n}(x_v)\). Their distance is at most \(2b_n\). For sufficiently large \(n\), one legal pair-block swap transfers a register between that port and the hub. \end{lemma} \begin{proof} The first coordinate of a site in \(\Lambda_{r,n}\), divided by \(h_n\), is congruent to \(r\) modulo \(L\). Distinct residues are therefore disjoint. This argument does not use a transverse spatial dimension. The access distance is at most \(2b_n\) by \eqref{eq:roundingbound}. Since \[ \frac{2b_n}{a_n}=\frac{\sqrt D L}{n^2}\longrightarrow0, \] the access displacement lies in the allowed stencil for all sufficiently large \(n\). \end{proof} \begin{remark}[What a private lane is] A private lane is a reserved subset of ordinary sites, not a separate coincident substrate. Geometric segments joining its sites may cross segments of another lane. Such a crossing does not identify site labels and does not introduce an interaction. The only interactions are the declared block maps. \end{remark} \subsection{Rounded affine pacing} \begin{lemma}[Every-prefix paced path] \label{lem:pacing} Let \(x,y\in\R^D\), let \(N\ge1\), and reserve one residue \(r\). Define \[ y_k=Q_{r,n}\left(x+\frac{k}{N}(y-x)\right), \qquad k=0,\ldots,N. \label{eq:roundedpath} \] If \[ \frac{\norm{y-x}_2}{N}+2b_n\le a_n, \label{eq:stepcriterion} \] then successive distinct positions are connected by legal generator-pair blocks. Equal successive positions are waits. For every prefix \(k\), \[ \norm{y_k-y_0-\frac{k}{N}(y-x)}_2\le2b_n. \label{eq:prefixestimate} \] The piecewise affine interpolation stays within \(b_n\) of the affine target path under the common affine parametrization. \end{lemma} \begin{proof} Write \[ y_k=x+\frac{k}{N}(y-x)+e_k, \qquad \norm{e_k}_2\le b_n. \] Then \[ \norm{y_{k+1}-y_k}_2 \le\frac{\norm{y-x}_2}{N}+2b_n, \] which proves legality. Subtracting the expression for \(y_0\) gives \eqref{eq:prefixestimate}. Between successive samples, the error is a convex combination of \(e_k\) and \(e_{k+1}\), so its norm is at most \(b_n\). \end{proof} In the notation of digital generator steps, if the nonzero steps are \(h_ns_i\), and zero steps represent waits, then \[ \sup_{k\le N} \norm{\sum_{i=1}^kh_ns_i -\frac{k}{N}(y-x)}_2 \le2b_n. \label{eq:generatorprefix} \] This is an every-prefix estimate, not an endpoint-only estimate. \section{Local junctions and the global partition schedule} \label{sec:schedule} Fix the finite causal graph after the zero-duration contraction. Let \(p_v\) be the number of original payload maps at macrovertex \(v\), and let \(d_v^-\), \(d_v^+\) be its incoming and outgoing wire counts. Choose \[ B\ge \max_v(d_v^-+p_v+d_v^+)+1. \label{eq:framebudget} \] All constants \(L\) and \(B\) depend only on this fixed graph. Let \[ t_*=\min_v t_v, \qquad T_\Gamma=\max_v t_v-t_*, \qquad \eta_n=n^{-1/2}. \] Retard the geometry slightly by \[ f_n(t)=t_*+(1+\eta_n)(t-t_*). \] The hub frame at vertex \(v\) begins at \[ k_v\tau_n, \qquad k_v=\left\lfloor\frac{f_n(t_v)}{\tau_n}\right\rfloor. \label{eq:vertexclock} \] Its duration is \(B\tau_n\). \subsection{The local frame} During a hub frame, execute: \begin{enumerate}[label=\textup{(\roman*)}] \item one incoming port-to-hub swap per incoming wire; \item the original payload maps in their fixed topological order, one per singleton tick, with local allocations and consumptions; \item one hub-to-outgoing-port swap per outgoing wire; \item identity waits to fill the frame to \(B\) ticks. \end{enumerate} All maps act as the identity on undeclared factors. Acquisitions occur at the actual output tick of their corresponding local payload map. \begin{lemma}[Finite-register junction implementation] \label{lem:junction} The local frame is legal for every sufficiently large \(n\). It gathers, processes, and separates any fixed finite set of typed registers without measuring them for routing. The routing parts are tensor-factor permutations and preserve the transported state jointly with every untouched reference. Their time overhead is at most \(B\tau_n\), and their geometric support shrinks to the vertex. \end{lemma} \begin{proof} Each incoming or outgoing transfer is a legal access swap by Lemma~\ref{lem:residues}. Transfers involving the same hub are serialized. The singleton payload acts only after all its input factors are present and before its required outputs are dispersed. The chosen topological order implements the same internal zero-duration subnetwork. A swap is unitary and merely relocates a typed factor. Its action on a logical state and an arbitrary reference is exactly the corresponding tensor-factor identification. No routing operation reads a quantum state or a measurement outcome. The frame has \(B\) ticks by construction. Every involved site is within \(b_n\) of \(x_v\), and all frame times are within \(B\tau_n+\tau_n+c^{-1}c\eta_nT_\Gamma\) of the intended vertex time. These bounds tend to zero. \end{proof} \subsection{Parallel routes between frames} For an edge \(e:v\to w\), put \[ T_e=t_w-t_v>0, \qquad N_e=k_w-k_v-B. \] Route from its outgoing port at time \((k_v+B)\tau_n\) to its incoming port at time \(k_w\tau_n\), using \eqref{eq:roundedpath} with \(N=N_e\). \begin{lemma}[Vanishing slack is sufficient] \label{lem:slack} For every fixed finite graph, all sufficiently large \(n\) satisfy \(N_e\ge1\) and the step criterion \eqref{eq:stepcriterion} for every edge. \end{lemma} \begin{proof} The rounding in \eqref{eq:vertexclock} gives \[ N_e\tau_n \ge (1+\eta_n)T_e-(B+1)\tau_n. \] Since the graph is fixed and has finitely many positive-duration edges, for sufficiently large \(n\), \[ (B+1)\tau_n\le\frac{\eta_nT_e}{2} \] simultaneously for all edges. Hence \[ N_e\tau_n\ge(1+\eta_n/2)T_e. \label{eq:durationmargin} \] Causality implies \[ \frac{\norm{x_w-x_v}_2}{N_e} \le\frac{a_n}{1+\eta_n/2}. \] The remaining step margin is \[ a_n-\frac{a_n}{1+\eta_n/2} =\frac{a_n\eta_n}{2+\eta_n}. \] On the other hand, \[ \frac{2b_n}{a_n}=\frac{\sqrt D L}{n^2} =o(\eta_n). \] Thus \(2b_n\le a_n\eta_n/(2+\eta_n)\) for all sufficiently large \(n\), proving \eqref{eq:stepcriterion}. Equation \eqref{eq:durationmargin} also proves \(N_e\ge1\). \end{proof} \begin{theorem}[Legal global scheduling without long-route serialization] \label{thm:schedule} For every fixed finite future-causal network, all sufficiently large target scales admit one deterministic legal partition schedule implementing all its routes and hub frames. Long wire routes execute in parallel. Only the finite local access operations at each hub are serialized. The schedule is independent of the input state, payload values, outcomes, and external reference. \end{theorem} \begin{proof} At each tick, a moving edge requests at most one pair, whose two sites lie in its assigned residue class. Different edge routes have disjoint residue classes, so their requested pairs are disjoint. A hub-frame access pair joins its private hub residue to an incident private edge residue. An incoming route has ended before its gathering swap; an outgoing route begins after its dispersal swap. Thus an access pair never competes with movement on that same edge. Different hub frames use different hub residues. An edge cannot be undergoing access at both endpoints at the same tick: Lemma~\ref{lem:slack} leaves at least one route tick between the two frames. Therefore all active pairs at a tick form a matching. Payload singleton ticks cannot compete with access at their own hub because the local frame serializes them. Complete the matching by singleton identities at all remaining used sites. The experiment uses finitely many sites and factors. The unused lattice may equivalently be completed by singleton identities without constructing an infinite active tensor product. Every decision was determined by the graph, assigned residues, rounding rules, and clock. No quantum or classical outcome was consulted. \end{proof} \begin{remark}[Crossings, overlaps, and dimension one] Coincident geometric wires can be transported as distinct private residue lanes, or as a bundled typed register when their skeletons allow it. Crossing geometric segments need no special interaction because their site sets are disjoint. At an actual payload junction, gathering and dispersal are exactly Lemma~\ref{lem:junction}. In particular, the proof does not rely on a false positive equal-time separation between crossing curves. It applies unchanged in \(D=1\). \end{remark} \section{Complete spacetime estimates and instrument preservation} \label{sec:transport} \subsection{A fine geometric error} Let \(F_\Gamma\) be the support of the finite point-vertex causal graph, including its straight wire and storage edges. Let \(F_{\Gamma,n}\) be the actual physical support of the constructed realization, including active block prisms, waits, hub frames, and retained paths. Define \[ \delta_{\Gamma,n} = c\eta_nT_\Gamma+(B+5)a_n+3b_n. \label{eq:fineerror} \] For the fixed graph, \(\delta_{\Gamma,n}\to0\). \begin{lemma}[Two-sided fine support and labelled estimates] \label{lem:finegeometry} For every sufficiently large \(n\), \[ \dH(F_{\Gamma,n},F_\Gamma)\le\delta_{\Gamma,n}. \label{eq:fineHausdorff} \] The same upper bound applies to each individually labelled wire support, including its local endpoint waits. Every actual payload acquisition is within \(\delta_{\Gamma,n}\) of its assigned graph vertex. The footprint of each original payload map, including a map inside a contracted macrovertex, is at Hausdorff distance at most \(\delta_{\Gamma,n}\) from its graph-vertex singleton. \end{lemma} \begin{proof} For an edge \(v\to w\), compare the original and implemented paths under the common affine parameter \(\lambda\in[0,1]\). The original time is \[ t(\lambda)=(1-\lambda)t_v+\lambda t_w. \] The implemented route time is \[ t_n(\lambda) =(1-\lambda)(k_v+B)\tau_n+\lambda k_w\tau_n. \] Its time error satisfies \[ c\abs{t_n(\lambda)-t(\lambda)} \le c\eta_nT_\Gamma+(B+1)a_n. \label{eq:timecomparison} \] Lemma~\ref{lem:pacing} bounds the spatial error of the interpolated route by \(b_n\). An active block prism is within \(2a_n\) in \(d_c\) of the corresponding interpolated route. Endpoint waits, gathering, and dispersal occur within \(b_n\) of the vertex position and within the frame's time bound. Payload singleton intervals have the same bound. Conversely, every original edge point has the corresponding implemented route point, and every graph vertex is close to its hub frame. These comparisons establish both Hausdorff directions. The deliberately enlarged constant in \eqref{eq:fineerror} covers all the displayed bounds. The argument applies edge by edge and vertex by vertex. Taking the union takes the maximum of their errors. The named wire estimates include the access and wait pieces attached to that wire. Acquisition times are actual frame output ticks, not replacement metadata, and satisfy the same estimate. \end{proof} \begin{remark}[The pacing acceptance test] A transformed idle wire is a causal affine segment with some velocity \(v\), generally nonzero after a boost. The construction samples the whole affine segment and obeys \eqref{eq:generatorprefix}. It does not travel to its final position early and then remain there for a macroscopic wait. All waits and accesses are included in Lemma~\ref{lem:finegeometry}. \end{remark} \subsection{Exact complete-channel identity} Let \(\widehat\Phi_\Gamma\) be the source network channel on its named logical ports. Let \(\widehat\Phi^{\mathrm{phys}}_{\Gamma,n}\) be the constructed physical channel. Let \(\mathcal E_n\) insert the logical inputs into their reserved input factors and insert all required product blanks. Let \(\mathcal D_n\) select and identify the named output factors, tracing unused factors. These are mathematical port comparison channels. \begin{theorem}[Payload naturality and complete instrument preservation] \label{thm:identity} For every payload assignment on the fixed skeleton, \[ \mathcal D_n\circ \widehat\Phi^{\mathrm{phys}}_{\Gamma,n}\circ \mathcal E_n = \widehat\Phi_\Gamma. \label{eq:logicalidentity} \] The identity holds after tensoring both sides with the identity on every finite-dimensional untouched reference. Matched payload modifications use exactly the same routes, residues, schedule, encodings, and decodings. \end{theorem} \begin{proof} Follow a topological ordering of the logical network. Every access or route operation moves a typed factor by swaps with reserved blank factors. On the logical factors this is the specified identity wire under tensor-factor relabelling. When the inputs of a payload vertex have arrived, its original complete map is applied to precisely those factors. In a macrovertex, the original finite zero-duration subnetwork is applied in a topological order. This has the same contraction as the source subnetwork because operations on independent factors commute. Induction over the vertices therefore gives the same complete map on all logical wires and records. Discarding unused blanks and identifying external ports gives \eqref{eq:logicalidentity}. The argument is an equality of linear maps, so its tensor extension by \(\id_R\) is immediate. The schedule was constructed without inspecting payload maps. Replacing any specified complete payload map leaves every routing and comparison interface unchanged. \end{proof} The decoding in \eqref{eq:logicalidentity} is not destructive readout. Readout followed by forgetting is a partial-trace operation, not an inverse of arbitrary quantum transport; this distinction is also recorded in En04-A \cite{CheungEn04A}. \begin{corollary}[Approximate payload implementation] \label{cor:approximate} Suppose the exact payload channel at vertex \(v\) is replaced by a channel with half-diamond error at most \(e_{v,n}\), including all its classical controls and output records. Then \[ d_\diamond\left( \mathcal D_n\widehat\Phi^{\mathrm{phys}}_{\Gamma,n}\mathcal E_n, \widehat\Phi_\Gamma \right) \le\sum_v e_{v,n}. \label{eq:telescoping} \] \end{corollary} \begin{proof} Replace payloads one at a time. The maps before and after a replacement are channels, including identity extensions by the other live factors. Composition with channels cannot increase the diamond norm. The triangle inequality gives \eqref{eq:telescoping}. \end{proof} \subsection{The spacetime transport contract} \begin{theorem}[Constructed transport contract] \label{thm:T} Fix a calibrated source experiment of scale \(j\), resolution \(\eps_j\), skeleton \(\Gamma_j\), physical support \(F_j\), declared support \(K_j\supseteq F_j\), and labelled acquisition tuple \(R_j\). Fix \(g\in\Poin\). For every sufficiently large target scale \(n\), the substrate of Section~\ref{sec:substrate} admits a payload-independent realization with the following properties. \begin{enumerate}[label=\textup{(\roman*)}] \item Its logical complete instrument equals the source complete instrument exactly, as in \eqref{eq:logicalidentity}, with every untouched reference. \item Its physical support satisfies \[ \dH(F_{j\to n},gF_j) \le C_g\eps_j+r_{\Gamma_j,j,g}(n), \label{eq:Tphysical} \] where one may take \[ C_g=2L_g, \qquad r_{\Gamma_j,j,g}(n) =\delta_{g\Gamma_j^{\mathrm{mid}},n}\longrightarrow0. \label{eq:Tconstants} \] \item A legal declared target support is \[ K_{j\to n}=gK_j\cup F_{j\to n}, \label{eq:declaredtarget} \] and it obeys \[ \dH(K_{j\to n},gK_j) \le C_g\eps_j+r_{\Gamma_j,j,g}(n). \label{eq:Tdeclared} \] \item For the actual acquisition tuples, \[ \max_\alpha d_c(R_{j\to n,\alpha},gR_{j,\alpha}) \le C_g\eps_j+r_{\Gamma_j,j,g}(n). \label{eq:Tacquisition} \] \item The corresponding named retained-wire supports obey the same form of estimate. For every matched set of changed payload vertices, the source and target changed-payload footprint unions also obey that form of two-sided Hausdorff estimate. \end{enumerate} The coefficient \(C_g\) is independent of the source graph's complexity. The threshold and the fine error may depend on the entire fixed finite source. \end{theorem} \begin{proof} Apply Lemma~\ref{lem:midpoint} to the source. Apply \(g\) to the resulting causal graph. Since \(g\) preserves future causality, Sections~\ref{sec:routing}--\ref{sec:schedule} construct a legal target realization at every sufficiently large \(n\). The triangle inequality, \eqref{eq:transformedmidpoint}, and Lemma~\ref{lem:finegeometry} give \[ \dH(F_{j\to n},gF_j) \le \delta_{g\Gamma_j^{\mathrm{mid}},n} +2L_g\eps_j. \] This proves \eqref{eq:Tphysical}. The set in \eqref{eq:declaredtarget} is compact and contains the actual target support. Every point of \(gK_j\) belongs to it. Every additional point lies in \(F_{j\to n}\), hence is within the displayed bound of \(gF_j\subseteq gK_j\). This proves \eqref{eq:Tdeclared}. It does not assert that the extra declared points are active physical sites. Each source acquisition lies within \(\eps_j\) of its source payload midpoint. The target acquisition is the actual output event of the corresponding payload map in its hub frame. Lemma~\ref{lem:finegeometry} gives \[ d_c(R_{j\to n,\alpha},gR_{j,\alpha}) \le \delta_{g\Gamma_j^{\mathrm{mid}},n} +L_g\eps_j, \] which is bounded by \eqref{eq:Tacquisition}. The same argument applies to individual retained supports. A source payload footprint is within \(\eps_j\) of its midpoint. Its target singleton payload footprint is within the fine error of the transformed midpoint. Taking unions over changed vertices again uses a maximum. Empty changed sets are handled as the unchanged case, without assigning a Hausdorff distance to an empty set. Theorem~\ref{thm:identity} proves the channel and reference statements. It also proves matched-skeleton naturality. \end{proof} \begin{remark}[The exact quantifier separation] The quantities \(B\), \(L\), \(T_\Gamma\), and the refinement threshold may grow with a fixed source's complexity. They appear only in \(r_{\Gamma_j,j,g}(n)\), which vanishes after that source is fixed. The coarse coefficient is \(2L_g\), not a graph-dependent sum. \end{remark} \begin{remark}[Identification with a predecessor family] Theorem~\ref{thm:T} discharges the stated transport contract for the target family \eqref{eq:scales}--\eqref{eq:stencil}. Applying it to a different prescribed target family requires checking that family's generator availability, clock, capacity, and partition rules. A documentary relationship between manuscripts does not perform that check. \end{remark} \section{Full-tail diagonal transport} \label{sec:diagonal} \begin{definition}[Full-tail instrument realization] A full-tail family consists of experiments at every sufficiently large scale \(j\), on fixed finite external logical interfaces, such that \[ K_j\longrightarrow K, \qquad R_j\longrightarrow R, \qquad d_\diamond(\widehat\Phi_j,\widehat\Phi)\longrightarrow0, \qquad \eps_j\longrightarrow0. \label{eq:fulltail} \] The finite skeleton may vary with \(j\). \end{definition} \begin{theorem}[Full-tail diagonal realization after transformation] \label{thm:diagonal} Let \eqref{eq:fulltail} hold for calibrated sources and fix \(g\in\Poin\). There are experiments on the target substrate at every sufficiently large scale \(n\), with \[ K_n'\longrightarrow gK, \qquad R_n'\longrightarrow gR, \qquad d_\diamond(\widehat\Phi_n',\widehat\Phi)\longrightarrow0. \label{eq:diagonallimit} \] If two source families have matched skeletons and differ only on payload footprints contained in \((K')_{\rho_j}\), where \(\rho_j\to0\), their transformed target families can use matched skeletons and differ only in a neighborhood of \(gK'\) whose radius tends to zero. \end{theorem} \begin{proof} For each fixed source \(j\), Theorem~\ref{thm:T} supplies a threshold beyond which its fine error is at most \(1/j\). Choose strictly increasing integers \(M_j\), with \(M_j\ge j\), so that this statement holds for every \(n\ge M_j\). For \(n\ge M_1\), define \[ j(n)=\max\{j:M_j\le n\}. \] The maximum is finite, and \(j(n)\to\infty\). Transport source \(j(n)\) to target scale \(n\). Then \[ \dH(K_n',gK) \le 2L_g\eps_{j(n)} +\frac1{j(n)} +L_g\dH(K_{j(n)},K), \] which tends to zero. The acquisition estimate is identical in form. Exact instrument preservation gives \[ d_\diamond(\widehat\Phi_n',\widehat\Phi) = d_\diamond(\widehat\Phi_{j(n)},\widehat\Phi)\longrightarrow0. \] For matched modifications, choose each \(M_j\) to work for the common skeleton and its changed-footprint union. The target changed footprints lie within \[ 2L_g\eps_{j(n)} +\frac1{j(n)} +L_g\rho_{j(n)} \] of \(gK'\). This radius tends to zero. Payload-independent transport supplies the common target skeleton. \end{proof} \begin{corollary}[Action on realizable finite operational objects] \label{cor:action} On full-tail objects with common logical interfaces, define \[ g\cdot(K,R,\widehat\Phi)=(gK,gR,\widehat\Phi). \label{eq:operationalaction} \] Within the explicit source--target class of this paper, Theorem~\ref{thm:diagonal} supplies full-tail realizations of these transformed objects and preserves witnessed matched modifications. The transformations satisfy the group law on these limit data. \end{corollary} \begin{proof} Realizability and modifications follow from Theorem~\ref{thm:diagonal}. On geometric data, \(g(hK)=(gh)K\) and similarly for acquisitions. Logical channels are unchanged under the fixed port identifications, giving the group law. \end{proof} This action is not an exact action on a single microscopic lattice or schedule. Nor does it reinstate unrestricted same-scale device boost closure. \section{A nontrivial finite quantum instrument} \label{sec:bell} We now construct one fixed continuum network, then realize it at every sufficiently fine target scale using the preceding theorem. \subsection{Geometry and local interfaces} Fix \(L_0>0\), \(T>L_0/c\), and \(H>0\). Let \[ A=(0,0), \qquad B_L=(T,-L_0e_1), \qquad B_R=(T,L_0e_1). \] The source-to-wing paths are \[ \gamma_L(t)=\left(t,-\frac{L_0}{T}te_1\right), \qquad \gamma_R(t)=\left(t,\frac{L_0}{T}te_1\right), \qquad 0\le t\le T. \label{eq:bellpaths} \] They are strictly timelike. Each wing has a vertical retained storage path from time \(T\) to \(T+H\). At \(A\), there are qubits \(Q_L,Q_R\) and a memory qubit \(W\). The memory accompanies the left-moving register and is never measured or acted on by the payload. Its purpose is to make the retained quantum-information content completely explicit. At \(B_L\) and \(B_R\), respectively, local input registers \(S_L,S_R\) select between two measurement settings. These are local input ports, not remotely supplied control wires. Default preparations are product preparations. The named quantum output is \(Q_LQ_RW\), and the named classical outputs include the retained settings and the two locally acquired outcomes. \subsection{Source gate and local projectors} Let \[ U_{\mathrm B} =\mathrm{CNOT}_{L\to R}(H_L\otimes I_R), \qquad U_{\mathrm B}\ket{00} =\ket{\Phi^+} =\frac{\ket{00}+\ket{11}}{\sqrt2}. \] Write \(X,Z\) for Pauli matrices and define \[ A_\theta=\cos\theta\,Z+\sin\theta\,X, \qquad P_{a,\theta}=\frac{I+aA_\theta}{2}, \qquad a\in\{+1,-1\}. \label{eq:projectors} \] Since \(A_\theta^2=I\), the \(P_{a,\theta}\) are orthogonal projectors with sum \(I\). Choose local setting tables \[ \theta_0=0,\quad \theta_1=\frac{\pi}{2}, \qquad \varphi_0=\frac{\pi}{4},\quad \varphi_1=-\frac{\pi}{4}. \label{eq:settingtable} \] \subsection{The complete channel} The input space is \[ \mathcal H_{\mathrm{in}} = Q_L\otimes Q_R\otimes W\otimes S_L\otimes S_R. \] For a matrix \(\rho\) on this space, let \[ \rho_{st}=\bra{st}_{S_LS_R}\rho\ket{st}_{S_LS_R}. \] Define \[ K_{ab\mid st} = (P_{a,\theta_s}\otimes P_{b,\varphi_t}\otimes I_W) (U_{\mathrm B}\otimes I_W). \label{eq:BellKraus} \] Using the label \(a=\pm1\) for a classical basis vector, set \[ \begin{split} \widehat\Phi(\rho) = \sum_{\substack{s,t\in\{0,1\}\\a,b\in\{\pm1\}}} &K_{ab\mid st}\rho_{st}K_{ab\mid st}^\dagger\\ &\otimes\proj{st}_{S_LS_R} \otimes\proj{a,b}_{R_LR_R}. \end{split} \label{eq:Bellchannel} \] This specifies the canonical full-matrix extension which dephases the setting inputs. On the declared classical input algebra, it is the controlled local instrument. \begin{proposition}[Complete instrument and nontrivial quantum output] \label{prop:Bellinstrument} Equation \eqref{eq:Bellchannel} is CPTP. Its selective quantum maps are the declared local Lueders maps after the source gate. It is not a constant classical sampler. Moreover, for product inputs between \(W\) and the other input factors, its action on \(W\) is exactly \(\id_W\). With an external reference entangled with \(W\), that entanglement is retained. Consequently the complete channel is not entanglement breaking. \end{proposition} \begin{proof} The Kraus operators of the full channel are the operators which read the setting basis \(\bra{st}\), apply \(K_{ab\mid st}\), and write the output setting and outcome basis vectors. They satisfy \[ \sum_{a,b}K_{ab\mid st}^\dagger K_{ab\mid st} =I_{Q_LQ_RW} \] for every \(s,t\). Summing over the orthogonal input setting projectors gives the full identity. Thus the channel is CPTP. The operators in \eqref{eq:BellKraus} contain \(I_W\), so the channel factors as the displayed Bell instrument tensored with \(\id_W\), with the conventional tensor ordering understood. This preserves every state of \(W\) and its untouched reference when the other factors are fixed independently. For an explicit nonconstant record example, take \(s=1\), \(t\) replaced, if desired, by an admitted right \(X\)-setting payload modification. The source inputs \(\ket{00}\) and \(\ket{10}\) produce \(\Phi^+\) and \(\Phi^-\), whose \(X\otimes X\) correlations have opposite signs. More generally the projectors in \eqref{eq:BellKraus} directly distinguish input coherences in their admitted measurement bases. Thus the selective channel contains more than one fixed output law. Finally, fix all other inputs and put \(W\) in a Bell state with an untouched reference. The \(W\)--reference marginal remains that Bell state after the channel. An entanglement-breaking channel could not have this property. \end{proof} The memory \(W\) is not needed for the Bell correlations. It is included to prevent any ambiguity about the survival of unknown quantum information in the complete output. \subsection{Bell correlations and unconditioned no-signalling} \begin{proposition}[Bell law and CHSH value] \label{prop:Belllaw} For source input \(\ket{00}_{Q_LQ_R}\), any fixed memory state, and fixed local setting inputs \(s,t\), \[ p(a,b\mid s,t) =\frac14\left(1+ab\cos(\theta_s-\varphi_t)\right). \label{eq:Belllaw} \] Both one-wing outcome marginals are uniform. The correlators \[ E_{st}=\sum_{a,b}ab\,p(a,b\mid s,t) \] satisfy \[ E_{00}+E_{01}+E_{10}-E_{11}=2\sqrt2. \label{eq:CHSH} \] \end{proposition} \begin{proof} The state \(\Phi^+\) has \[ \langle Z\otimes Z\rangle=1,\qquad \langle X\otimes X\rangle=1,\qquad \langle Z\otimes X\rangle =\langle X\otimes Z\rangle=0, \] and vanishing one-qubit Pauli expectations. Expanding the two projectors in \eqref{eq:projectors} gives \eqref{eq:Belllaw}. Summing over either outcome gives \(1/2\). The setting table gives \[ E_{00}=E_{01}=E_{10}=\frac1{\sqrt2}, \qquad E_{11}=-\frac1{\sqrt2}, \] proving \eqref{eq:CHSH}. The CHSH comparison is the standard one introduced in \cite{CHSH}; the value here has been computed directly. \end{proof} \begin{proposition}[Reference-stable unconditioned no-signalling] \label{prop:nosignalling} For a fixed state reaching the two wing interfaces, replacement of the left complete local instrument by any other trace-preserving left local instrument does not change the unconditioned right output, including any untouched reference. The analogous right-to-left statement holds. \end{proposition} \begin{proof} Let \(\Lambda_L\) be the complete left channel, including its record. Its adjoint is unital. For any observable \(B\) on the right output and untouched reference, \[ (\Lambda_L^*\otimes\id)(I_L\otimes B)=I_L\otimes B. \] Hence every such expectation is independent of \(\Lambda_L\). Equality of expectations for all output observables proves equality of the right reduced state. \end{proof} This is an unconditional statement. Conditional remote states may depend on a supplied local outcome. It also holds with local setting changes, provided the source state and remote input marginal are kept fixed. \section{An explicit local apparatus dilation} \label{sec:dilation} At each wing add three blank qubits: the record carrier \(R\), an outcome environment \(E\), and a setting environment \(F\). The setting input is \(S\). For \(a=\pm1\), let \(r(a)=(1-a)/2\in\{0,1\}\). For each setting \(\theta\), define a unitary on \(QRE\) by \[ C_\theta = \sum_{a=\pm1} P_{a,\theta}\otimes X_R^{r(a)}\otimes X_E^{r(a)}. \label{eq:measurementunitary} \] It is unitary because the projectors are orthogonal and the conditional bit flips are unitary. Let \(D_{SF}\) be \(\mathrm{CNOT}_{S\to F}\), and put \[ U_L = D_{S_LF_L} \left( \sum_{s=0}^1\proj{s}_{S_L}\otimes C_{\theta_s} \right)\otimes I_W, \label{eq:UL} \] with tensor order adjusted to the named factors. Define \(U_R\) analogously using \(\varphi_t\). The two wing unitaries act on disjoint factor sets. Their action on initialized apparatus factors is \[ \ket{s}_S\ket{\psi}_Q\ket{000}_{REF} \longmapsto \sum_{a=\pm1} \ket{s}_S P_{a,\theta_s}\ket{\psi}_Q \ket{r(a)}_R\ket{r(a)}_E\ket{s}_F. \label{eq:dilationaction} \] \begin{proposition}[Exact finite dilation] \label{prop:dilation} On the common finite Hilbert space \[ \mathcal H_0 = Q_L\otimes Q_R\otimes W \otimes S_L\otimes S_R \otimes R_L\otimes E_L\otimes F_L \otimes R_R\otimes E_R\otimes F_R, \label{eq:H0} \] let \[ U=(U_LU_R)(U_{\mathrm B}\otimes I_{\mathrm{rest}}). \label{eq:globalunitary} \] Insert the six apparatus blank qubits, apply \(U\), and trace \(E_LF_LE_RF_R\). The resulting operational channel is exactly \eqref{eq:Bellchannel}. \end{proposition} \begin{proof} Equation \eqref{eq:dilationaction} copies the outcome into two orthogonal carriers \(R,E\) and the setting into \(S,F\). Tracing \(E\) removes cross-outcome terms; tracing \(F\) removes off-diagonal setting terms. The remaining terms have precisely the Kraus expression in \eqref{eq:Bellchannel}. The identity on \(W\) is retained throughout. \end{proof} This is a concrete finite-dimensional instance of dilation \cite{Stinespring,Watrous}, not an appeal to a localization theorem. Its locations have already been fixed: \(U_{\mathrm B}\) acts at \(A\), and \(U_L,U_R\) act at the respective wing events. The environments are inaccessible to the operational output but are retained in the complete dilation ledger. For its spacetime support, they have declared local storage segments to the terminal horizon. Operational tracing is not physical annihilation. \section{The finite spacetime observable net} \label{sec:net} \subsection{Point-labelled local observable algebras} All observables are represented on the common pre-source Hilbert space \(\mathcal H_0\). Dormant apparatus factors are not assigned a local observable algebra before their allocation events. At \(A\), assign the algebra \[ \Alg_A=\B(Q_LQ_RW)\otimes I_{\mathrm{rest}}. \] Before the wing measurements, assign along the left path \[ \Alg_L^{\mathrm{pre}} = \Ad_{U_{\mathrm B}^\dagger} \bigl(\B(Q_LW)\otimes I_{\mathrm{rest}}\bigr), \label{eq:leftpre} \] and along the right path \[ \Alg_R^{\mathrm{pre}} = \Ad_{U_{\mathrm B}^\dagger} \bigl(\B(Q_R)\otimes I_{\mathrm{rest}}\bigr). \label{eq:rightpre} \] Let \[ \mathcal H_L=Q_L\otimes W\otimes S_L\otimes R_L\otimes E_L\otimes F_L, \qquad \mathcal H_R=Q_R\otimes S_R\otimes R_R\otimes E_R\otimes F_R. \] At and after \(B_L\), the full local dilation algebra is \[ \Alg_L^{\mathrm{post}} = \Ad_{U_{\mathrm B}^\dagger} \bigl(\B(\mathcal H_L)\otimes I_R\bigr), \label{eq:leftpost} \] and analogously \[ \Alg_R^{\mathrm{post}} = \Ad_{U_{\mathrm B}^\dagger} \bigl(I_L\otimes\B(\mathcal H_R)\bigr). \label{eq:rightpost} \] The source unitary is understood as acting trivially on all other factors. These expressions include the full input and output algebra at a local measurement event. Conjugating a full wing algebra by \(U_L\) or \(U_R\) leaves that full wing algebra unchanged. Specific postmeasurement observables nevertheless have their specific Heisenberg representatives. For example, an output pointer projection is represented by \[ U^\dagger\Pi_a^{R_L}U. \] It belongs to \(\Alg_L^{\mathrm{post}}\). \begin{definition}[Observable net] \label{def:net} For an open spacetime region \(O\), let \[ \Alg(O) = C^*\{\Alg_p:p\in O\cap F_{\mathrm{dil}}\} \subseteq\B(\mathcal H_0), \label{eq:netdefinition} \] where \(F_{\mathrm{dil}}\) is the declared source, branch, measurement, and retained-storage geometry, and \(\Alg_p\) is the applicable point-labelled algebra above. If there are no such points, set \(\Alg(O)=\C I\). \end{definition} This net is finite dimensional. Apparatus observables in the complete dilation need not all be accessible observables of the reduced operational instrument. \subsection{Isotony and microcausality} \begin{theorem}[Finite-net isotony and spacelike commutation] \label{thm:microcausality} The assignment \(O\mapsto\Alg(O)\) is isotone. If \(O_1\) and \(O_2\) are spacelike separated, then \[ [\Alg(O_1),\Alg(O_2)]=0. \label{eq:netmicrocausality} \] The net is noncommutative and contains nontrivial local quantum algebras. \end{theorem} \begin{proof} Isotony follows immediately from the inclusion of generating point sets. Every point of either active branch is causally related to \(A\). Every two points on the same branch, including its vertical storage continuation, are causally related because that branch is future timelike. Thus two spacelike-separated regions cannot both contain active points of the same branch. Nor can one contain \(A\) and the other an active branch point. It remains to compare left and right generators. At the common cut immediately after the source gate, the entire left factor set \(\mathcal H_L\) and right factor set \(\mathcal H_R\) are disjoint tensor factors. Their full algebras commute. Conjugating both by the same source unitary preserves commutators: \[ [U_{\mathrm B}^\dagger AU_{\mathrm B}, U_{\mathrm B}^\dagger BU_{\mathrm B}] = U_{\mathrm B}^\dagger[A,B]U_{\mathrm B}=0. \] The premeasurement algebras are subalgebras of these full wing algebras. Every left generator therefore commutes with every right generator. Products, sums, adjoints, and norm closure preserve this commutation, proving \eqref{eq:netmicrocausality}. A small region meeting the interior of the left branch contains a represented copy of \(\B(Q_LW)\), and similarly for the right branch. These are noncommutative matrix algebras. \end{proof} \begin{remark}[Shared ancestors do not invalidate the proof] The left and right branches share the entangling source. Their backward ancestor sets are not disjoint. The proof uses disjoint factors at one common circuit cut and a common preceding unitary, not disjoint ancestor supports. \end{remark} \begin{remark}[Finite circuit net, not a Haag--Kastler field theory] We prove isotony and spacelike commutation for the stated finite net. We do not assert a vacuum, positive-energy translation representation, time-slice axiom, field reconstruction, or arbitrarily many local degrees of freedom. \end{remark} \section{Exact instrument and observable refinement} \label{sec:limit} \subsection{Physical realizations at every sufficiently fine scale} Apply Theorem~\ref{thm:schedule} directly to the fixed continuum Bell dilation graph. This gives physical realizations at every sufficiently large \(n\). Let \(K_{\mathrm B}\) be its compact declared support and \(K_{\mathrm B,n}\) the target declared support. Actual acquisition events are denoted \(B_{L,n}^{\mathrm{acq}},B_{R,n}^{\mathrm{acq}}\). The encodings and decodings of Theorem~\ref{thm:identity} give \[ \widehat\Phi_n^{\mathrm{log}} = \mathcal D_n\widehat\Phi_n^{\mathrm{phys}}\mathcal E_n = \widehat\Phi \label{eq:exactBelllimit} \] for every sufficiently large \(n\). \begin{theorem}[Nontrivial complete-instrument limit] \label{thm:instrumentlimit} The Bell network has a full-tail continuum realization with \[ K_{\mathrm B,n}\to K_{\mathrm B}, \qquad B_{\nu,n}^{\mathrm{acq}}\to B_\nu \quad(\nu=L,R), \qquad d_\diamond(\widehat\Phi_n^{\mathrm{log}},\widehat\Phi)=0. \label{eq:Belllimits} \] The equality includes every untouched reference. It also holds for matched local setting or measurement-payload modifications. \end{theorem} \begin{proof} Lemma~\ref{lem:finegeometry} gives the geometric and actual acquisition convergence. Theorem~\ref{thm:identity} gives \eqref{eq:exactBelllimit} for each payload assignment on the fixed skeleton, together with its reference extension and matched-modification statement. \end{proof} \subsection{A product-preserving comparison, not arbitrary compression} At scale \(n\), use the finite unitary dilation realization with all its reserved factors. The initial physical Hilbert space can be tensor-factor identified as \[ \mathcal H_n^{\mathrm{phys}} \cong\mathcal H_0\otimes\mathcal H_n^{\mathrm{blank}}. \] Let \(P_n\) be this tensor-factor relabelling unitary and define \[ \iota_n(A) = P_n(A\otimes I_{\mathrm{blank}})P_n^\dagger, \qquad A\in\B(\mathcal H_0). \label{eq:iota} \] Set \[ \mathfrak B_n^{\mathrm{log}} =\iota_n(\B(\mathcal H_0)), \qquad j_n=\iota_n^{-1}. \label{eq:comparisonmaps} \] \begin{lemma}[Faithful algebraic comparison and physical representatives] \label{lem:obscomparison} The maps \(\iota_n\) and \(j_n\) are mutually inverse unital \(*\)-isomorphisms between \(\B(\mathcal H_0)\) and \(\mathfrak B_n^{\mathrm{log}}\). In particular, \[ j_n(\iota_n(A)\iota_n(B))=AB, \qquad j_n(\iota_n(A)^*)=A^*, \qquad \norm{\iota_n(A)}=\norm{A}. \label{eq:productcomparison} \] For every point-labelled continuum observable, its actual live-register representative at scale \(n\), pulled back through the physical unitary circuit, is \(\iota_n(A)\). Its physical localization approaches the declared continuum point or retained path location. \end{lemma} \begin{proof} Equation \eqref{eq:iota} is tensoring by the identity followed by unitary conjugation. It is a faithful unital \(*\)-homomorphism and is isometric, proving \eqref{eq:productcomparison}. For the second assertion, track all finite logical dilation factors, including the dormant apparatus factors. At any circuit cut, routing swaps implement a tensor permutation of those factors and the other reserved blank factors. Each payload unitary acts on the current logical factors as its prescribed logical unitary and acts trivially on the rest. Inductively, the unitary evolution to that cut is, under the initial and current tensor identifications, the logical prefix unitary tensored with the identity on the other factors. Pulling a current live-factor observable back therefore gives exactly \(\iota_n(A)\). Lemma~\ref{lem:finegeometry} controls its current physical location. This argument uses unitary factor tracking on a selected subalgebra. It does not assert that a general encoding compression or a general channel adjoint preserves products. \end{proof} \begin{theorem}[Instrument--observable refinement] \label{thm:observablelimit} For each admitted logical observable \(A\in\B(\mathcal H_0)\), the sequence \(A_n=\iota_n(A)\) has the exact comparison image \[ j_n(A_n)=A. \label{eq:exactobservablelimit} \] Finite products and adjoints converge through the same maps exactly, and the same statement holds for bounded logical sequences whose comparison images converge in norm. For an observable generated by finitely many point-labelled generators inside an open region \(O\), its physical representative is localized in \(O\) for every sufficiently large \(n\), provided those locations have positive distance from \(O^c\). For positively spacelike-separated compact localization sets, the corresponding admitted representatives commute at all sufficiently fine scales, and their limit algebras commute. \end{theorem} \begin{proof} The exact assertions follow from Lemma~\ref{lem:obscomparison}. If \(j_n(A_n)\to A\) and \(j_n(B_n)\to B\) in norm, uniform boundedness and \[ j_n(A_nB_n)=j_n(A_n)j_n(B_n) \] give norm convergence to \(AB\). Adjoints are continuous. For localization, take the minimum positive distance from the finitely many declared generator locations to \(O^c\). The geometric error tends to zero, so all representatives eventually lie in \(O\). The physical routed dilation factors and their local operations provide the asserted representatives. For two compact localization sets with a positive spacelike margin, the same geometric convergence preserves spacelike separation for all sufficiently fine scales. The logical representatives commute by Theorem~\ref{thm:microcausality}, and their physical pullbacks commute by the \(*\)-homomorphism \(\iota_n\). Their products therefore retain the same zero commutator under \(j_n\). No equality of full regional algebras at regions tangent to localization boundaries is asserted. \end{proof} Because the algebras are finite dimensional, the algebra generated by any collection of point algebras is already generated by a finite subcollection: successive strict increases of subalgebra dimension can occur only finitely often. Thus every observable of \(\Alg(O)\) can be treated by the finite-generator localization statement when its selected generators lie positively inside \(O\). This logical-identification construction is the finite analogue of the comparison discipline used in En05-C \cite{CheungEn05C}. It avoids identifying a moving physical register address with one fixed observable in an ambient infinite tensor product. \section{Operational--algebraic compatibility} \label{sec:pairing} For any logical input state \(\rho\), let \[ \sigma_\rho = \rho\otimes\proj{0}^{\otimes6}_{R_LE_LF_LR_RE_RF_R} \] on \(\mathcal H_0\), and define \[ \omega_\rho(A)=\Tr(\sigma_\rho A). \] Let \(\Pi_a^{R_L}\) and \(\Pi_b^{R_R}\) be the output record projections, and set \[ M_{ab} = U^\dagger \bigl(\Pi_a^{R_L}\Pi_b^{R_R}\bigr)U. \label{eq:reporteffects} \] Identities on all omitted factors are understood. \begin{theorem}[Exact state--instrument pairing] \label{thm:pairing} For every logical input state \(\rho\), \[ \begin{split} \Pr_\rho(a,b) &= \Tr\left[ \widehat\Phi(\rho) \bigl(I_{\mathrm q,S}\otimes\proj{a,b}_{R_LR_R}\bigr) \right]\\ &=\omega_\rho(M_{ab}). \end{split} \label{eq:pairing} \] More generally, for every observable \(B\) on the retained quantum and setting outputs, \[ \Tr\left[ \widehat\Phi(\rho) \bigl(B\otimes\proj{a,b}\bigr) \right] = \omega_\rho\left( U^\dagger(B\otimes\Pi_a^{R_L}\Pi_b^{R_R} \otimes I_{\mathrm{env}})U \right). \label{eq:selectivepairing} \] The same identities hold at every sufficiently fine scale under the instrument and observable comparison maps. The effects \(M_{ab}\) are products of commuting represented local pointer projections in the two measurement algebras. \end{theorem} \begin{proof} Proposition~\ref{prop:dilation} identifies the operational channel with the partial trace of \(U\sigma_\rho U^\dagger\). The defining property of partial trace and cyclicity of trace give \eqref{eq:pairing} and \eqref{eq:selectivepairing}. Each single-wing pointer projection pulls back into its wing algebra. These two algebras commute by Theorem~\ref{thm:microcausality}. Their product is \eqref{eq:reporteffects}. At scale \(n\), insert the physical blank state in the remaining factors. Lemma~\ref{lem:obscomparison} gives \[ \omega_{\rho,n}(\iota_n(A))=\omega_\rho(A). \] The exact channel identity \eqref{eq:exactBelllimit} and exact observable comparison \eqref{eq:exactobservablelimit} then reproduce the same pairings. \end{proof} Equation \eqref{eq:selectivepairing} is stronger than agreement of the terminal outcome law: it identifies every selective retained-output expectation. Conditional normalized states are obtained only on positive-probability branches. Theorem~\ref{thm:pairing}, Proposition~\ref{prop:nosignalling}, and Proposition~\ref{prop:Belllaw} establish simultaneously: \begin{enumerate}[label=\textup{(\roman*)}] \item spacelike commutation of local observable algebras; \item unconditioned independence from the remote setting; \item nonclassical joint Bell correlations; \item an exact complete-instrument refinement, not just a law limit. \end{enumerate} \section{Poincare transport of the worked limit} \label{sec:transformedexample} For \(g\in\Poin\), transform all source, branch, measurement, allocation, and storage geometry by \(g\), while retaining the same named finite logical gates and instruments. Theorem~\ref{thm:schedule} realizes this transformed finite causal graph on every sufficiently fine target scale. The two measurement events need not be simultaneous in the target coordinate time. Their actual acquisitions converge separately to \(gB_L\) and \(gB_R\). Define the transported net by \[ \Alg_g(O)=\Alg(g^{-1}O). \label{eq:transportednet} \] Since \(g\) preserves spacelike separation, it is isotone and locally commuting. The logical channel and all state--effect pairings remain the ones already proved. \begin{corollary}[Transported finite instrument--observable specimen] \label{cor:transportedBell} For each fixed \(g\in\Poin\), the Bell specimen has a full-tail realization with transformed compact support and separately transformed acquisition events, the same complete logical instrument, and the product-preserving observable comparison for \(\Alg_g\). \end{corollary} \begin{proof} Apply the finite graph realization to the transformed dilation skeleton. The proofs of Theorems~\ref{thm:identity}, \ref{thm:observablelimit}, and \ref{thm:pairing} concern the unchanged logical factor network and therefore apply under the same logical identifications. Its geometry converges by Lemma~\ref{lem:finegeometry}. The definition \eqref{eq:transportednet} transfers local commutativity. \end{proof} This is a transported family of finite experiments. It is not a claim that one fixed finite net and one fixed state carry a vacuum Poincare representation. \section{Debugging audit and exact mathematical release} \label{sec:release} \subsection{What has been repaired} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.25\linewidth}X@{}} \toprule Potential gap & Repair proved here\\ \midrule Endpoint-only routing & Rounded affine trajectories satisfy the every-prefix estimate \eqref{eq:generatorprefix}.\\[3pt] Boosted idle-wire pacing & The entire transformed causal segment is sampled; all route, wait, and access support is estimated.\\[3pt] False global lane separation & Disjoint residue classes give disjoint actual site sets, including in one dimension.\\[3pt] Unproved junction capacity & An explicit finite swap--payload--swap frame is charged \(B\tau_n\) physical time.\\[3pt] Directional service delay & Each long wire has its own site residue and moves in parallel; no factor proportional to the number of wires slows all routes.\\[3pt] Complexity-dependent coarse error & The coarse coefficient is \(2L_g\); graph-dependent constants occur only in a vanishing fixed-source fine error.\\[3pt] Declared versus active support & Both physical Hausdorff convergence and the separate declared support construction are stated.\\[3pt] Report-law versus instrument equality & The complete logical channel is equal at every sufficiently fine scale, with arbitrary references.\\[3pt] Channel adjoints versus algebra maps & Observable comparison is a faithful \(*\)-isomorphism on an explicit dilation subalgebra, not arbitrary compression.\\[3pt] Shared backward ancestors & Microcausality is proved at a common tensor-factor cut.\\[3pt] Input-independent sampling & The Bell instrument acts on arbitrary inputs and retains an exact quantum-memory channel.\\ \bottomrule \end{tabularx} \end{center} \subsection{Why the statement is not substrate-universal} \begin{proposition}[A fixed cardinal stencil does not satisfy the theorem] \label{prop:obstruction} In \(D=2\), consider only cardinal hops \(\pm ae_1,\pm ae_2\), each taking time \(a/c\). Such a family cannot approximate every Euclidean future-null segment with vanishing macroscopic time error. \end{proposition} \begin{proof} In elapsed time \(T\), a cardinal path has \[ \norm{\Delta x}_1\le cT. \] The Euclidean null displacement \[ \Delta x=\frac{cT}{\sqrt2}(1,1) \] has \(\ell^1\)-norm \(\sqrt2cT\). Approximating that displacement requires elapsed time at least \(\sqrt2T+o(1)\), rather than \(T+o(1)\). The discrepancy is macroscopic. \end{proof} Thus the availability of increasing short lattice generators cannot be removed from the theorem merely because every individual cardinal hop obeys the same Euclidean speed bound. \subsection{Release theorem} \begin{theorem}[En06-B release] \label{thm:release} Under the explicit finite-capacity convention and the target family \eqref{eq:scales}--\eqref{eq:stencil}, the following are constructed and proved. \begin{enumerate}[label=\textup{(\roman*)}] \item A deterministic payload-independent realization of every fixed finite future-causal typed instrument network at every sufficiently fine scale, using legal singleton and generator-pair partitions. \item Every-prefix paced routes, finite-register local junctions, parallel long-route scheduling, and two-sided full-support estimates. \item The spacetime transport contract \eqref{eq:Tphysical}--\eqref{eq:Tacquisition}, with a complexity-independent coarse coefficient and vanishing fixed-source fine error. \item Exact complete-instrument preservation, reference stability, matched payload naturality, and the full-tail diagonal theorem. \item One nontrivial Bell instrument with local setting inputs, retained quantum outputs, an exact untouched quantum memory, and a common finite unitary apparatus dilation. \item An isotone locally commuting finite spacetime observable net, product-preserving comparison maps, and an exact operational--algebraic pairing. \item Bell correlations with CHSH value \(2\sqrt2\), alongside reference-stable unconditioned no-signalling. \end{enumerate} No result here asserts a quantum field theory, an invariant microscopic boost action, an autonomous substrate independent of the experiment skeleton, a derivation of the chosen constant \(c\), or transport on every other proposed refinement family. \end{theorem} \begin{proof} Items (i)--(iv) are Theorems~\ref{thm:schedule}, \ref{thm:identity}, \ref{thm:T}, and \ref{thm:diagonal}, together with their routing and geometry lemmas. Items (v)--(vii) are Propositions~\ref{prop:Bellinstrument}, \ref{prop:dilation}, \ref{prop:Belllaw}, and \ref{prop:nosignalling}, and Theorems~\ref{thm:microcausality}, \ref{thm:instrumentlimit}, \ref{thm:observablelimit}, and \ref{thm:pairing}. The final exclusions delimit the stated construction and are not additional positive conclusions. \end{proof} \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Source-record discipline.} Cheung's manuscripts are cited by the identifiers and versions supplied with the question. Those citations document comparison interfaces, stated obstructions, and terminology. Every mathematical premise used in the new construction is stated in this paper. No unseen manuscript theorem is imported as a proof premise. \paragraph{Model discipline.} The source class is calibrated fixed-wire instrumentation. The target is the explicit increasing-stencil lattice family. Site capacity is finite in each experiment but is not uniformly bounded over all experiments. The deterministic partition schedule is chosen from the skeleton. These choices are substantive hypotheses, not claims about every physical model. \paragraph{Observable discipline.} The net is a finite complete-dilation net. Its local algebras may include apparatus and inaccessible environmental observables. The reduced instrument has its separately specified accessible output interface. Product preservation is established only on the explicitly identified logical dilation algebra. \paragraph{Transformation discipline.} Only time-orientation-preserving Poincare transformations are used. The continuum operational action does not imply exact boost closure at one lattice scale. \paragraph{Standing series boundaries.} This finite construction makes no claim concerning closure of O.1, opening of XV-B, discharge of XVI-N, or admission of a quantum coefficient in the analytic characteristic-product programme. The partial references and unresolved admission boundaries reported in XIV-J and XVI-N are not replaced by a finite Bell instrument \cite{CheungXIVJ,CheungXVIN}. No unrestricted adaptive-wiring compilation is asserted. \paragraph{Verification status.} This source uses standard packages and an inline bibliography and is intended for \texttt{pdflatex}. No successful compilation or external numerical execution is reported. No machine-checked proof, independent referee verification, or acceptance claim is made. \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{CheungEn06A} N.~Cheung, \emph{En06-A}, version 1, unpublished manuscript, supplied excerpts. In particular: \texttt{def:experiment}, \texttt{hyp:T}, \texttt{lem:midpoint}, \texttt{lem:diagonal}, \texttt{thm:algmicro}, and \texttt{thm:withdrawal}. Documentary comparison; no unseen proof is imported. \bibitem{CheungEn05B} N.~Cheung, \emph{En05-B}, version 1, unpublished manuscript, supplied excerpts. In particular: \texttt{def:schedule}, \texttt{lem:partition}, \texttt{lem:frame}, \texttt{thm:routing}, and \texttt{thm:identity}. Documentary comparison for one junction-and-lane construction. \bibitem{CheungEn05A} N.~Cheung, \emph{En05-A}, version 1, unpublished manuscript, supplied excerpts. In particular: \texttt{thm:isotropyobstruction}, \texttt{thm:polytope}, and \texttt{thm:boostobstruction}. Documentary comparison for velocity and closure obstructions. \bibitem{CheungEn05C} N.~Cheung, \emph{En05-C}, version 1, unpublished manuscript, supplied excerpts. In particular: \texttt{thm:comparison}, \texttt{cor:reference}, \texttt{prop:extensiongap}, \texttt{prop:sharedcenter}, \texttt{thm:sequence}, and \texttt{thm:refinementnet}. Documentary comparison for complete devices and observable maps. \bibitem{CheungVIIJ} N.~Cheung, \emph{VII-J}, version 1, unpublished manuscript, supplied excerpts. In particular: \texttt{thm:polytope}, ``Cycle velocities with a finite-time offset.'' Documentary comparison for scheduled reachability. \bibitem{CheungEn04A} N.~Cheung, \emph{En04-A}, version 1, unpublished manuscript, supplied excerpts. In particular: \texttt{thm:drop}, \texttt{prop:readoutdilation}, and \texttt{thm:classical}. Documentary comparison for readout, forgetting, and retained quantum outputs. \bibitem{CheungXIVJ} N.~Cheung, \emph{Landmark XIV-J}, version 1, unpublished manuscript, supplied excerpts. In particular: \texttt{thm:J2}, \texttt{thm:release}, and \texttt{cor:noadmission}. Recorded analytic scope boundary only. \bibitem{CheungXVIN} N.~Cheung, \emph{XVI-N}, version r12, unpublished manuscript, supplied excerpts. In particular: \texttt{thm:massless-obstruction} and the stated Route B boundary. Recorded scope boundary only. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}, Cambridge University Press, 2018. Finite-dimensional channels, instruments, dilations, and completely bounded trace norms. \bibitem{DaviesLewis} E.~B.~Davies and J.~T.~Lewis, ``An operational approach to quantum probability,'' \emph{Communications in Mathematical Physics} \textbf{17} (1970), 239--260. \bibitem{Stinespring} W.~F.~Stinespring, ``Positive functions on \(C^*\)-algebras,'' \emph{Proceedings of the American Mathematical Society} \textbf{6} (1955), 211--216. DOI: \texttt{10.1090/S0002-9939-1955-0069403-4}. \bibitem{Fritz} T.~Fritz, ``Velocity polytopes of periodic graphs and a no-go theorem for digital physics,'' \emph{Discrete Mathematics} \textbf{313} (2013), 1289--1301. arXiv:1109.1963. DOI: \texttt{10.1016/j.disc.2013.02.010}. \bibitem{FritzCorrigendum} T.~Fritz, ``Corrigendum to `Velocity polytopes of periodic graphs and a no-go theorem for digital physics','' \emph{Discrete Mathematics} \textbf{313} (2013), 2380. DOI: \texttt{10.1016/j.disc.2013.07.011}. 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