% !TeX program = pdflatex %===================================================================== % VII-J.tex % % Standalone source for pdfLaTeX. No external files are required. % Compile twice: % pdflatex -interaction=nonstopmode -halt-on-error VII-J.tex % pdflatex -interaction=nonstopmode -halt-on-error VII-J.tex % % Scope: % Finite quantum networks; operational influence; timed dependency % graphs; independent causal relations; directional reachability; % exact and approximate simulation; and a buffered realization % across VII-F's fixed-cone particle-link event. % % Principal corrections: % 1. Scheduled ancestors upper-bound influence; they need not equal it. % 2. Joint outputs require joint support certificates, not merely % separate no-signalling checks for individual outputs. % 3. All classical control, routing, ancillas, resets, and records % belong to the dependency ledger. % 4. Full-diamond error epsilon gives an added signalling error % epsilon; half-diamond error epsilon gives 2 epsilon. % 5. The cardinal-hop diamond lies inside the equal-speed Euclidean % disk. The reverse inclusion fails: safety and numerical % coverage have opposite directions. % 6. A velocity polytope is asymptotic. A finite-time offset remains. % 7. VII-F's abstract cell transport is not a physical trajectory. % 8. The event changes slicing incidence, not the spacetime metric. % 9. A positive-duration implementation is buffered around the event. % The germ timestamp is data, not the availability time of its code. % 10. The Hamiltonian route has quantified tails, not strict support. % 11. The cone example supplies a causal discrete target and an exact % implementation; it is not a scalar-field discretization theorem. % % Evidence: % Standard reference metadata was checked against primary sources. % Cheung's supplied texts are cited as manuscripts. % No compiler, proof assistant, numerical solver, or hardware % experiment was executed in preparing this source. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=26mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{booktabs,array,tabularx,longtable} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={VII-J: Scheduled dependencies and certified physical causality}, pdfsubject={Localized quantum simulation, directional reachability, and a fixed-cone event} } \allowdisplaybreaks[2] \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \renewcommand{\arraystretch}{1.13} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{example}[theorem]{Example} \newtheorem{contract}[theorem]{Contract} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\Hil}{\mathcal H} \newcommand{\Alg}{\mathcal A} \newcommand{\Enc}{\mathcal E} \newcommand{\Dec}{\mathcal D} \newcommand{\Comp}{\mathcal C} \newcommand{\Sig}{\operatorname{Sig}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\id}{\operatorname{id}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\diam}{\operatorname{diam}} \newcommand{\conv}{\operatorname{conv}} \newcommand{\Sch}{\mathsf{Sch}} \newcommand{\Inf}{\mathsf{Inf}} \newcommand{\Phy}{\mathsf{Phys}} \newcommand{\Cone}{C_{\pi/2}} \newcommand{\eps}{\varepsilon} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\lvert #1\rangle\langle #2\rvert} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\one}{\mathbf 1} \newcommand{\scopebox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \title{\textbf{Landmark VII-J}\\[0.45em] Scheduled dependencies and certified physical causality\\ in localized quantum simulation\\[0.25em] \large Exact support, directional reachability, approximation,\\ and continuation through a fixed-cone slicing event} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We distinguish scheduled reachability, intervention-based quantum influence, and an independently supplied physical causal relation. For finite quantum networks, including their classical controllers, we prove that the complete timed dependency graph bounds the Heisenberg support of every jointly observed output. Absence of a scheduled path implies absence of signalling, including with an untouched entangled reference. A causal embedding of the dependency edges then gives an exact physical containment theorem for the encoded--computed--decoded channel. The directional route is developed separately. A finite, phase-expanded periodic graph has a cycle-velocity polytope and an explicit finite-time remainder. Support-function comparison certifies convex velocity inclusion, but an asymptotic inclusion alone does not certify short-time communication. A spatially varying example gives an exact earliest-arrival law. For approximate simulation we prove the signalling bound with consistent diamond-norm constants and separate simulation accuracy from causal safety. A finite-volume Hamiltonian argument gives a reference-stable influence-tail certificate; a product-formula estimate supplies a corresponding simulation error. For VII-F's three-to-six particle-link event, the ambient spinless cone remains fixed. Its uniformly spacelike, ordered slices give a conservative causal gate speed \(1/(1+2\sqrt2/3)\) in the slicing parameter. We certify physical inclusion across the event using persistent spatially marked registers, rather than abstract cell labels. An explicitly buffered, nontrivial two-qubit transport target is implemented exactly by causal routes and local gates, together with the separately localized event-code permutation. The target preserves arbitrary reference entanglement and has a unit signalling witness along an allowed causal route. No spacetime metric, field equation, Lorentz invariance, or gravitational backreaction is derived. The finite causal carrier model is not a quantum-field discretization. The separate characteristic quantum-product programme is not advanced. \end{abstract} \scopebox{Release and scope}{% \textbf{Constructed:} exact network-support and physical-containment theorems; operational signalling criteria; localization and clock contracts; periodic directional bounds; robust finite-time margins; approximate signalling and simulation estimates; and a buffered causal realization through VII-F's event. \smallskip \textbf{Two targets remain distinct.} Target A concerns the complete decoded simulation channel. Target B additionally embeds the actual elementary communication and gates in the supplied causal geometry. \smallskip \textbf{Explicit model boundary.} The cone application is a finite carrier-and-controller model on a prescribed metric cone. It does not derive matter propagation from Einstein dynamics or provide a continuum-field simulation. \smallskip No compilation, numerical execution, experimental implementation, or machine verification is asserted.} \tableofcontents %===================================================================== \section{The problem and the predecessor ledger} \label{sec:introduction} %===================================================================== The central distinction is \begin{equation} \boxed{ \text{scheduled dependency} \neq \text{actual quantum influence} \neq \text{geometric causal possibility}.} \label{eq:three} \end{equation} The aim is to prove relations among these objects after defining each one, not to identify them by terminology. The finite-channel conventions are standard \cite{Watrous}. Operational causal notions require a stated intervention class; broader notions of influence need not agree in every probabilistic theory \cite{Perinotti}. Circuit paths can overestimate the influence of a unitary transformation \cite{LorenzBarrett}. We use an intervention-based notion throughout. \subsection{What VII-E and VII-F actually provide} The supplied texts permit exact, manuscript-specific attribution. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.31\linewidth}X@{}} \toprule Supplied result & Scope retained here\\ \midrule VII-E, \texttt{thm:locality}, \texttt{cor:local-readout} & Backward support for prescribed block channels and their readouts, with radius \((t-s)\Delta\), not radius \(t-s\) unless \(\Delta=1\)\\ VII-E, \texttt{thm:retiming} & Equality of the total channel under legal reordering of fixed labelled channels; no clock identification\\ VII-E, \texttt{prop:relay}, \texttt{prop:direction} & An attained relay speed and an anisotropic velocity diamond; these are schedule statements\\ VII-E, \texttt{thm:pullback}, \texttt{thm:residual-obstruction} & Record compatibility and residual distinction, respectively; the supplied labels are used rather than an unverified \texttt{thm:record} or \texttt{thm:residual}\\ VII-F, \texttt{thm:certificate} & One particle-supported three-to-six slicing event on a fixed cone\\ VII-F, \texttt{thm:implementation} & A forward-domain event-code circuit with retained history and identity action on internal matter; no causal embedding of its compiler gates\\ \bottomrule \end{tabularx} \end{center} These statements are cited as manuscript results \cite{CheungVIIE,CheungVIIF}. The network and causal arguments used below are proved here. VII-G's history-stability theorem is a compatible finite-channel comparison, not the physical event theorem \cite{CheungVIIG}. VII-H's response distribution is neither a schedule nor an instrument \cite{CheungVIIH}. The classical, domain, and numerical results of VII-A--D likewise do not construct the causal embedding studied here \cite{CheungVIIA,CheungVIIB,CheungVIIC,CheungVIID}. \subsection{Two meanings of causal fidelity} \begin{definition}[Observable simulation: Target A] Let \(\Psi_{t,s}\) be a declared target channel. Let \[ \Phi_{t,s} =\Dec_t\circ\Comp_{t,s}\circ\Enc_s \] be the implemented channel on the same model input and output spaces. Target A consists of: \begin{enumerate} \item a causal-support or signalling estimate for \(\Phi_{t,s}\); \item a separately proved accuracy estimate comparing \(\Phi_{t,s}\) with \(\Psi_{t,s}\). \end{enumerate} Intermediate compiler gates need not represent model events. \end{definition} \begin{definition}[Causal elementary realization: Target B] Target B additionally assigns all implementation ports, communications, gates, preparations, measurements, and control records to events or regions in the supplied physical model. Each possible dependency has a certified causal route. \end{definition} Target B is stronger. Endpoint causal support alone does not construct a local realization of an arbitrary channel. Our positive Target B constructions use a specified routing architecture and local gate primitives. %===================================================================== \section{Finite channels and operational influence} \label{sec:influence} %===================================================================== Let the input factors be indexed by a finite set \(X\): \[ \Hil_X=\bigotimes_{x\in X}\Hil_x. \] For \(S\subseteq X\), define \[ \Alg_X(S) = L\!\left(\bigotimes_{x\in S}\Hil_x\right)\otimes I_{X\setminus S}. \] Output factors may have a different index set \(Y\). A channel is completely positive and trace preserving. Its adjoint is defined by \[ \Tr(B\Phi(Z))=\Tr(\Phi^*(B)Z). \] Adjoints are unital. We use \[ d_{\mathrm{tr}}(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1, \qquad d_\diamond(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \] The diamond norm, without the factor \(1/2\), is denoted explicitly. \subsection{Admissible interventions} For input \(A\subseteq X\), an intervention is any CPTP map \[ \mathcal I_A:L(\Hil_A)\longrightarrow L(\Hil_A). \] It is deterministic and nonselective. A locally measured outcome may be retained as an additional declared output, but conditional normalization of a selected branch is not an intervention in this definition. For output \(B\subseteq Y\), put \[ \Phi_B=\Tr_{Y\setminus B}\circ\Phi. \] \begin{definition}[Reference-assisted signalling] \label{def:signal} Let \(R\) be an arbitrary finite untouched reference. Define \begin{equation} \begin{split} \Sig_\Phi(A\to B) =\frac12\sup_{R,\rho,\mathcal I_A,\mathcal J_A} \Big\| &(\Phi_B\otimes\id_R) \big[ (\mathcal I_A\otimes\id_{A^cR})(\rho)\\ &\hspace{19mm} -(\mathcal J_A\otimes\id_{A^cR})(\rho) \big] \Big\|_1 . \end{split} \label{eq:signal} \end{equation} The observer receives \(B\) and the untouched reference \(R\). Thus \(0\leq\Sig_\Phi\leq1\). \end{definition} The reference may initially be entangled with the input. It is not acted upon by either intervention. This definition measures deterministic signalling, not postselected steering. \begin{theorem}[Exact no-signalling criterion] \label{thm:nosignal} The following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item \(\Sig_\Phi(A\to B)=0\); \item there is a channel \(\Xi:L(\Hil_{A^c})\to L(\Hil_B)\) with \[ \Phi_B=\Xi\circ\Tr_A; \] \item \[ \Phi_B^*(L(\Hil_B)) \subseteq I_A\otimes L(\Hil_{A^c}). \] \end{enumerate} The factorization implies equality with every untouched reference. \end{theorem} \begin{proof} Factorization implies (i), since a local trace-preserving intervention leaves the \(A^cR\) reduction unchanged. Suppose (i) holds. Compare the identity intervention with the replacement channel \[ \mathcal T_A(Z)=\Tr(Z)I_A/\dim\Hil_A. \] With no reference, equality for every input state gives equality of the corresponding linear maps: \[ \Phi_B=\Phi_B(\mathcal T_A\otimes\id_{A^c}). \] Define \[ \Xi(Z)=\Phi_B(I_A/\dim\Hil_A\otimes Z). \] This is CPTP and gives (ii). Taking adjoints proves (ii) implies (iii). Conversely, (iii) makes the output expectation depend only on the \(A^c\) reduction. The same replacement construction gives (ii). Tensoring the factorization with \(\id_R\) proves the final assertion. \end{proof} For a joint output \(B\), its \emph{actual influence set} consists of input factors \(a\) for which \(\Sig_\Phi(\{a\}\to B)>0\). Equivalently, it is the union of tensor factors occurring in the minimal support of \(\Phi_B^*(L(\Hil_B))\). The equivalence follows from Theorem~\ref{thm:nosignal} and \[ \bigcap_{a\in A}\Alg_X(X\setminus\{a\}) =\Alg_X(X\setminus A). \] \begin{remark}[Joint outputs matter] Separate no-signalling tests for \(B_1\) and \(B_2\) do not automatically certify the joint output \(B_1B_2\). A classical bit can be encoded as the parity of two uniformly random shares: each share alone is independent of the bit, while their joint law reveals it. Our support certificates below apply to every declared jointly observed output. \end{remark} %===================================================================== \section{The complete timed dependency network} \label{sec:network} %===================================================================== \subsection{Quantum and classical ports} A finite network is a directed acyclic circuit of channels with input and output ports, fixed initial ancillas, and declared discarded outputs. Its vertices include: \begin{itemize} \item quantum and classical register states; \item local gates and routed wire segments; \item ancilla preparations; \item destructive measurements and their outcome registers; \item resets, discards, guards, and classical feed-forward; \item every classical variable read to select a later operation. \end{itemize} For a conservative dependency graph, connect every input port of a gate to every output port of that gate. A tighter graph is allowed only with a proved joint-output support certificate for that gate. Identity wires have their ordinary directed edges. Classical registers use a distinguished basis. Controller maps first read the declared classical blocks and implement fixed branch maps. They are channels independent of an unknown input density operator. A canonical pinching extension allows the finite matrix-channel arguments to apply on a common ambient matrix space. \begin{contract}[Complete control ledger] \label{contract:control} A controller may inspect only declared classical inputs. Every variable affecting the gate choice, routing, guard, availability time, or reset is included among the control ports. A finite adaptive execution is padded to a fixed finite instruction envelope. Its dependency graph contains every possible branch and the control reads selecting that branch. No unrecorded state-functional scheduling is permitted. Ancillas are supplied independently of the unknown input; correlated ancilla preparation and transport must be declared. \end{contract} A local measurement instrument is represented by its complete record-retaining channel. A fixed controlled continuation is then a channel on the record and the registers it reads. Padding unused branches with identities and retaining the controller gives the required finite envelope. \subsection{Scheduled ancestors} For a set of output ports \(B\), let \[ \Sch^-(B) = \{\text{input factors having a directed dependency path to }B\}. \] This graph depends on the prescribed implementation, not merely on the final channel. \begin{theorem}[Scheduled support bounds actual influence] \label{thm:schedule} For the channel \(\Comp\) of a complete finite network, \begin{equation} \Comp^*(\Alg_Y(B)) \subseteq\Alg_X(\Sch^-(B)). \label{eq:schedule-support} \end{equation} Consequently, \[ \Inf^-_{\Comp}(B)\subseteq\Sch^-(B), \] and every intervention on inputs disjoint from \(\Sch^-(B)\) has zero signalling to \(B\), including with an untouched reference. \end{theorem} \begin{proof} Pull an output observable backwards through a topological order of the network. If it is the identity on all outputs of a gate, unitality of the adjoint makes it the identity on that gate's inputs. Otherwise, its possible support expands only to the declared predecessors of the observed gate outputs. For the conservative all-to-all gate graph, these are exactly all gate inputs. Tensoring with untouched factors does not add other support. Fixed preparations introduce no unknown input. A discard pulls a scalar identity backwards; a reset is treated as its stated channel. Measurement records and controlled continuations are included as ordinary channel ports. Repeating this process leaves support only on the scheduled ancestors. This proves \eqref{eq:schedule-support}. Theorem~\ref{thm:nosignal} gives the remaining assertions. \end{proof} This proof extends the prescribed-block mechanism of VII-E \cite[\texttt{thm:locality}]{CheungVIIE} by explicitly including the controller and all communication interfaces. \begin{example}[A scheduled path with no surviving signal] Declare a two-register SWAP, followed by the same SWAP. Its dependency diagram contains cross-register paths, but the total channel is the identity. An intervention on the first input cannot change the second output. Thus scheduled reachability can strictly exceed actual influence. \end{example} \begin{example}[Hidden classical communication] Measure a computational bit at \(A\), and prepare a distant output bit \(B\) equal to the result. The complete channel has \(\Sig(A\to B)=1\). Omitting the classical outcome wire from a quantum-only diagram gives an invalid support certificate. \end{example} \subsection{Distances, arrivals, and three clocks} A fixed undirected register graph with positive edge weights has a shortest-path metric \[ d_{\mathrm{comp}}(u,v) =\inf_{\gamma:u\to v}\sum_{e\in\gamma}w_e. \] Directed weights instead give an asymmetric extended distance. Weights representing routing cost are not automatically communication times or geometric lengths. In a timed network, waiting and availability are represented explicitly. The earliest-arrival function is \[ T_{\mathrm{arr}}(u,v;s) = \inf\{t:\text{a time-respecting path from }(u,s) \text{ to }(v,t)\}. \] The value is \(+\infty\) if there is no path. On a finite time-expanded DAG, dynamic programming in topological order computes the arrivals exactly. The three clocks are \[ \tau_{\mathrm{depth}},\qquad t_{\mathrm{hardware}},\qquad t_{\mathrm{model}}. \] A certificate assigns model times to ports and, separately, hardware start and finish times. Reordering independent gates can change depth or hardware time without changing the channel. It need not preserve the assigned model-time embedding. %===================================================================== \section{Localization and the exact physical-containment theorem} \label{sec:embedding} %===================================================================== \subsection{Independent physical possibility} A physical comparison structure consists of a set of events \(\mathcal M\) and a reflexive transitive relation \(\preceq\). It is specified independently of the computational schedule. For point-localized ports, assign an event \(p_v\in\mathcal M\). A scheduled edge \(u\to v\) is physically certified when \[ p_u\preceq p_v. \] For uncertain or extended localization regions \(\mathcal L_u\), a sufficient robust condition is \[ p\preceq q \quad\text{for every }p\in\mathcal L_u,\ q\in\mathcal L_v. \label{eq:allpoint} \] This all-point requirement is conservative. A smaller allowance must specify the correlated endpoint choices and their routes. Encoding and decoding are additional network components: \[ \Phi=\Dec\circ\Comp\circ\Enc. \] Their spreads are included in the full network. There is no free nonlocal encoding before the support theorem begins. \begin{theorem}[Exact encoded causal containment: Target A] \label{thm:physical} Suppose: \begin{enumerate}[label=\textup{(\roman*)}] \item the entire encoded--computed--decoded experiment is a complete finite network; \item every dependency edge has a physical endpoint certificate; \item input and output localizations use the same independent causal relation. \end{enumerate} For a joint output \(B\), let \(\Phy^-(B)\) be the input factors whose assigned events can causally reach at least one output event in \(B\). Then \begin{equation} \boxed{ \Inf^-_{\Phi}(B) \subseteq\Sch^-_{\mathrm{full}}(B) \subseteq\Phy^-(B).} \label{eq:main-containment} \end{equation} Inputs outside \(\Phy^-(B)\) cannot signal to \(B\), even with an untouched entangled reference. \end{theorem} \begin{proof} Theorem~\ref{thm:schedule} gives the first inclusion. Every scheduled path is a chain of certified causal endpoint relations. Transitivity gives a causal relation between its first and last events. Hence every scheduled ancestor belongs to the physical past. The signalling conclusion follows from Theorem~\ref{thm:nosignal}. \end{proof} The theorem is sufficient, not necessary. Extra scheduled dependencies can cancel in the complete channel. \subsection{Elementary realization rather than endpoint interpretation} \begin{contract}[Routing architecture] \label{contract:routing} The implementation model permits: \begin{enumerate} \item exact carriage of a finite register along a prescribed future causal curve; \item local unitary gates and complete instruments on registers meeting at the same station; \item local preparation of a specified input-independent ancilla; \item local storage, resets, and discards; \item classical records carried along the same kind of causal communication routes. \end{enumerate} Gate durations are positive and explicitly assigned. These are ideal model primitives, not claims of laboratory realization or a quantum-field description of a point register. \end{contract} \begin{proposition}[Gather--operate--distribute realization] \label{prop:gather} Consider a gate with input events \(p_1,\ldots,p_m\) and output events \(q_1,\ldots,q_n\). Suppose there are stations \(r_-\preceq r_+\) such that \[ p_i\preceq r_-\preceq r_+\preceq q_j \quad\text{for all }i,j, \] with specified routes, and the gate has a finite local implementation between \(r_-\) and \(r_+\). Then Contract~\ref{contract:routing} gives a Target B realization of that gate. \end{proposition} \begin{proof} Carry the inputs to \(r_-\), retain them through the local implementation, and carry the outputs from \(r_+\) to their destinations. Include every ancilla and outcome register in the local implementation. Each wire and elementary gate has causal endpoints and a declared support. The endpoint channel is the intended gate, including with arbitrary references. \end{proof} A causal relation between every input and output does not by itself imply the common-rendezvous hypothesis. We do not use the proposition as a universal localization theorem. QCA localizability and quantum causal graph dynamics provide stronger structural theorems under their own hypotheses \cite{SchumacherWerner,ArrighiNesmeWerner,ArrighiMartiel}. They are comparison results, not unproved substitutes here. \subsection{Composing relations through an update} \begin{proposition}[Support composition through a transition] \label{prop:eventcomposition} Let \[ \Phi=\Phi_+\circ\mathcal R\circ\Phi_-, \] where the three channels may act on differently indexed tensor products. Suppose their support predecessors are \(P_-\), \(P_{\mathcal R}\), and \(P_+\), respectively: each output algebra pulls back into the algebra of its declared predecessor set. Then \[ \Phi^*(\Alg(B)) \subseteq \Alg\bigl(P_-(P_{\mathcal R}(P_+(B)))\bigr). \] If all three predecessor relations have physical causal certificates in one common event space, the composite does also. \end{proposition} \begin{proof} Compose the three adjoint inclusions. Compose the corresponding causal chains and use transitivity. \end{proof} A change in register number is allowed if allocation and discard are channels with stated local supports. A newly prepared independent factor has no model-input predecessors. A state-dependent remapping must include its controller. %===================================================================== \section{Directional paths and finite-time reachability} \label{sec:directional} %===================================================================== \subsection{A velocity-set physical model} Let \(K_{\mathrm{phys}}(x,t)\subset\R^d\) be a declared compact convex velocity set. Assume that the paths used in a certificate are absolutely continuous and satisfy \[ \dot x(t)\in K_{\mathrm{phys}}(x(t),t) \quad\text{almost everywhere}. \] Define \(p\preceq_K q\) when an admissible path joins the events with nondecreasing time. Concatenation makes this relation transitive. For constant \(K\), the reachable displacement set in time \(T>0\) is exactly \(TK\): averages of allowed velocities lie in \(K\), and a constant velocity realizes every point of \(TK\). \begin{theorem}[Local directional inclusion] \label{thm:local-velocity} Suppose each scheduled communication has a specified continuous route with velocity in \(K_{\mathrm{sched}}(x,t)\), and \[ K_{\mathrm{sched}}(x,t)\subseteq K_{\mathrm{phys}}(x,t) \] at every point visited by those routes. Include gate, encoding, decoding, and storage dependencies in the same path contract. Then the full schedule is physically contained, and Theorem~\ref{thm:physical} applies. \end{theorem} \begin{proof} Every scheduled route is an admissible physical path. Concatenated routes remain admissible piecewise almost everywhere. The physical relation of each dependency edge is therefore certified. \end{proof} The hypothesis is local along actual routes. A comparison of two average speeds is not a replacement for it. \subsection{Support functions and robust margins} For a compact convex set \(K\), write \[ h_K(n)=\sup_{v\in K}n\cdot v. \] \begin{proposition}[Directional certificate] \label{prop:supportfunction} For compact convex \(K,L\subset\R^d\), \[ K\subseteq L \quad\Longleftrightarrow\quad h_K(n)\leq h_L(n)\quad\text{for every }n. \] If \(L=\{v:n_j\cdot v\leq b_j,\ 1\leq j\leq q\}\) is a polytope, it suffices to check \(h_K(n_j)\leq b_j\). \end{proposition} \begin{proof} Inclusion gives the support inequalities. If \(v\in K\setminus L\), let \(z\) be the Euclidean nearest point in \(L\). The minimizing property gives \((v-z)\cdot(w-z)\leq0\) for every \(w\in L\). Thus the direction \(n=v-z\) strictly separates \(v\) from \(L\), contradicting the support inequalities. The half-space claim is immediate. \end{proof} \begin{corollary}[Uncertainty margin] \label{cor:velocitymargin} Suppose for all unit \(n\), \[ \begin{split} h_{K_{\mathrm{sched}}^{\mathrm{true}}}(n) &\leq h_{\widehat K_{\mathrm{sched}}}(n) +\delta_{\mathrm{sched}}(n),\\ h_{K_{\mathrm{phys}}^{\mathrm{true}}}(n) &\geq h_{\widehat K_{\mathrm{phys}}}(n) -\delta_{\mathrm{phys}}(n). \end{split} \] If \[ h_{\widehat K_{\mathrm{phys}}}(n) -h_{\widehat K_{\mathrm{sched}}}(n) \geq \delta_{\mathrm{phys}}(n)+\delta_{\mathrm{sched}}(n), \] then true velocity inclusion holds. Strict inequality gives a positive certified margin. \end{corollary} \begin{proof} Combine the inequalities and apply Proposition~\ref{prop:supportfunction}. \end{proof} \subsection{A phase-expanded periodic schedule} A periodic dependency model has a finite quotient graph \(Q\). Its vertices include register type and schedule phase. Each edge \(e\) has displacement \(\Delta_e\in\R^d\) and duration \(T_e\geq T_{\min}>0\). Its periodic lift permits translated copies of each transition. Waiting at a port is allowed. Assume \(Q\) is strongly connected. For a simple directed cycle \(c\), set \[ \Delta(c)=\sum_{e\in c}\Delta_e,\qquad T(c)=\sum_{e\in c}T_e,\qquad v(c)=\frac{\Delta(c)}{T(c)}. \] There are finitely many simple cycles. Define \[ K_{\Sch} = \conv\bigl(\{0\}\cup\{v(c):c\text{ simple cycle}\}\bigr). \] \begin{theorem}[Cycle velocities with a finite-time offset] \label{thm:polytope} Let \(q=|V(Q)|\) and \(D_{\max}=\max_e\norm{\Delta_e}\). Every dependency path completed within elapsed time \(T\) satisfies \begin{equation} \Delta_{\mathrm{path}} \in TK_{\Sch}+\overline B_{(q-1)D_{\max}}. \label{eq:finiteoffset} \end{equation} The set of limiting velocities along paths with elapsed times tending to infinity is exactly \(K_{\Sch}\). These are statements about potential scheduled reachability, not about the channel carried by every path. \end{theorem} \begin{proof} Remove directed cycles from a finite walk until its remaining walk has no repeated vertex. Decompose each removed cycle into simple cycles. The remainder has at most \(q-1\) edges. The cycle contribution is \[ \sum_c m_c\Delta(c)=\sum_c m_cT(c)v(c). \] The sum of those cycle times is at most the elapsed time \(T\). Including \(0\), the contribution belongs to \(TK_{\Sch}\). The remaining displacement has norm at most \((q-1)D_{\max}\), proving \eqref{eq:finiteoffset}. The offset divided by \(T\) tends to zero, so every limiting velocity lies in \(K_{\Sch}\). Conversely, choose any finite convex combination of cycle velocities and zero. Repeat each chosen cycle sufficiently many times in the prescribed time proportions, connecting their quotient base vertices by fixed paths. Strong connectivity supplies the connectors. Their total duration and displacement are negligible relative to the repeated cycles. Additional waiting realizes the zero contribution. Integer repetition counts approximate arbitrary real proportions. This realizes every point of \(K_{\Sch}\). \end{proof} For a quotient that is not strongly connected, one must respect the order of reachable strongly connected components. The convex hull of every cycle in the graph can overestimate attainability from a specified initial component. The theorem is the timed, phase-explicit version of the periodic-graph mechanism discussed by Fritz \cite{Fritz}. It does not infer finite-time physical containment from an asymptotic polytope. \begin{example}[An asymptotically stationary schedule can make a fast jump] Take two quotient states and edges with displacements \(L\) and \(-L\), each of duration one, together with waiting. Every nontrivial cycle has zero displacement, so \(K_{\Sch}=\{0\}\). The first edge nevertheless moves a dependency by \(L\) in one unit of time. For \(L>c\) it violates a physical speed \(c\). The finite offset in \eqref{eq:finiteoffset} cannot be omitted. \end{example} \subsection{The corrected diamond--disk comparison} For cardinal hops of length \(a\), one per model interval \(\tau\), or a wait, \[ K_{\Sch}=\{v:\abs{v_1}+\abs{v_2}\leq a/\tau\}. \] For an isotropic physical speed \(c=a/\tau\), \[ K_{\Phy}=\{v:\norm v_2\leq c\}. \] Then \begin{equation} K_{\Sch}\subseteq K_{\Phy}, \qquad K_{\Phy}\not\subseteq K_{\Sch}. \label{eq:diamond-disk} \end{equation} Indeed, \[ h_{K_{\Sch}}(n)=c\norm n_\infty \leq c\norm n_2=h_{K_{\Phy}}(n), \] whereas the disk velocity \((c/\sqrt2,c/\sqrt2)\) lies outside the diamond. This corrects an orientation ambiguity in the proposed plan: the diamond is safe relative to the equal-speed disk; it does not cover every physically possible direction. VII-E's directional example concerns that failed reverse coverage \cite[\texttt{prop:direction}]{CheungVIIE}. Thus \[ \text{implementation influence}\subseteq\text{physical possibility} \] is causal safety, while \[ \text{target dependency}\subseteq\text{available numerical stencil} \] is an information-coverage requirement. Neither proves simulation accuracy. %===================================================================== \section{A spatially varying causal benchmark} \label{sec:varying} %===================================================================== Choose \(c_0>0\) and \(0\leq t\leq T\). In one spatial dimension let \[ K_{\Phy}(t)=\left[-\frac{c_0}{1+t}, \frac{c_0}{1+t}\right]. \] This is also the coordinate causal velocity set of the regular metric \[ \mathrm ds^2 =-\frac{c_0^2}{(1+t)^2}\mathrm dt^2+\mathrm dx^2. \] \begin{proposition}[Exact reachable set and earliest arrival] \label{prop:varying} Communication from \((s,x)\) to \((t,y)\), \(t\geq s\), is possible exactly when \begin{equation} \abs{y-x} \leq c_0\log\frac{1+t}{1+s}. \label{eq:varyingreach} \end{equation} Its earliest arrival is \[ T_{\mathrm{arr}}(x,y;s) =(1+s)\exp(\abs{y-x}/c_0)-1, \] provided that value lies in the declared time window. \end{proposition} \begin{proof} Integrating the velocity bound proves necessity. For sufficiency, put \[ x(r)=x+(y-x) \frac{\int_s^r c_0(1+u)^{-1}\dd u} {\int_s^t c_0(1+u)^{-1}\dd u}. \] Its speed is bounded by \(c_0/(1+r)\), and it reaches \(y\). Solving the equality in \eqref{eq:varyingreach} gives the arrival formula. \end{proof} A register route with this printed profile is certifiable even when a constant-speed route with the same endpoints is not. A controller may select among finitely many such routes only after its selecting record reaches the route's departure port. Preloading a known function \(c(t)\) is not an instantaneous global feed-forward operation. Taking an independently declared finite sequence of transport and local gates along these routes gives a nontrivial exact discrete target. Applying Theorem~\ref{thm:physical} proves its causal safety; equality of the implemented gates with the declared target proves its accuracy. No wave-equation discretization is claimed. %===================================================================== \section{Approximate observable simulation} \label{sec:approximation} %===================================================================== \subsection{The norm constant} \begin{theorem}[Reference-assisted signalling stability] \label{thm:signal-stability} Let \(\Phi,\Psi\) have the same input and output spaces. For a fixed output \(B\), suppose \[ \norm{\Phi_B-\Psi_B}_\diamond\leq\epsilon_B. \] Then \begin{equation} \boxed{ \Sig_\Phi(A\to B) \leq \min\{1,\Sig_\Psi(A\to B)+\epsilon_B\}.} \label{eq:signal-stability} \end{equation} In particular, a half-diamond error bound \(d_\diamond(\Phi_B,\Psi_B)\leq\epsilon\) contributes \(2\epsilon\), not \(\epsilon\), on the right. \end{theorem} \begin{proof} For a fixed reference, input, and two interventions, let \(\rho_{\mathcal I}\) and \(\rho_{\mathcal J}\) be the resulting states. The two simulation-error terms obey \[ \norm{((\Phi_B-\Psi_B)\otimes\id_R) (\rho_{\mathcal I})}_1 \leq\epsilon_B, \] and the same holds for \(\rho_{\mathcal J}\). The triangle inequality for the output difference therefore adds \(2\epsilon_B\) in trace norm and \(\epsilon_B\) after the prefactor \(1/2\). Take the supremum and use \(\Sig_\Phi\leq1\). \end{proof} An expectation estimate for one state or one observable does not supply the diamond hypothesis. The local norm above still quantifies over every input and reference. \subsection{Composed error ledgers} \begin{proposition}[Channel telescope] \label{prop:telescope} For channels \(\Phi_j,\Psi_j\) between matching successive spaces, if \[ \norm{\Phi_j-\Psi_j}_\diamond\leq\epsilon_j, \] then \[ \norm{\Phi_m\cdots\Phi_1-\Psi_m\cdots\Psi_1}_\diamond \leq\sum_{j=1}^m\epsilon_j. \] This includes separate encoding, discretization, transition, compilation, decoding, and hardware stages when corresponding channel comparisons have actually been supplied. \end{proposition} \begin{proof} Insert one difference at a time between unchanged channel prefixes and suffixes. Channels have diamond norm one, and the norm is submultiplicative. Apply the triangle inequality \cite{Watrous}. \end{proof} If a target has leakage at most \(\epsilon_{\Phy}\), the implemented leakage is at most \[ \min\left\{1,\epsilon_{\Phy} +\epsilon_{\mathrm{enc}} +\epsilon_{\mathrm{disc}} +\epsilon_{\mathrm{event}} +\epsilon_{\mathrm{comp}} +\epsilon_{\mathrm{dec}} +\epsilon_{\mathrm{hw}}\right\} \] in the full-diamond conventions above. An uncertainty in a geometric arrival time is not automatically a channel-norm uncertainty. It can instead be handled by the robust geometric margins below, or by a separate dynamical continuity bound. \begin{proposition}[Finite Hamiltonian perturbation bound] \label{prop:duhamel} For bounded self-adjoint time-dependent generators \(H(t),\widetilde H(t)\), with the same initial unitary, \[ \norm{U(t,s)-\widetilde U(t,s)} \leq\int_s^t\norm{H(r)-\widetilde H(r)}\dd r. \] Consequently, \[ \norm{\Ad_U-\Ad_{\widetilde U}}_\diamond \leq2\int_s^t\norm{H(r)-\widetilde H(r)}\dd r. \] \end{proposition} \begin{proof} Differentiate \(U(t,r)\widetilde U(r,s)\) and integrate. All surrounding propagators are unitary. For channels, expand \(UXU^\dagger-\widetilde UX\widetilde U^\dagger\) into two terms, also after tensoring with an identity. \end{proof} This is one valid way to turn a specified finite-dimensional geometry-dependent generator estimate into a channel error. %===================================================================== \section{The Hamiltonian route: tails and a constructive error} \label{sec:hamiltonian} %===================================================================== Continuous-time local Hamiltonian evolution generally has tails, unlike an exact bounded-range circuit. Lieb--Robinson theory supplies sharper volume-uniform bounds under suitable interaction assumptions \cite{BravyiHastingsVerstraete}. We give a deliberately conservative finite-volume proof sufficient for this paper. \subsection{An explicit finite-volume tail} Let \(G=(X,E)\) be a finite graph and \[ H(t)=\sum_{x\in X}h_x(t)+\sum_{e\in E}h_e(t), \] where \(h_x\) acts at one site, \(h_e\) at the two endpoints of \(e\), and all terms are bounded, self-adjoint, and piecewise continuous. Put \[ \Lambda(s,t)=\int_s^t\sum_{e\in E}\norm{h_e(r)}\dd r. \] Let \(d_G(A,B)\) be the minimum graph distance between the two nonempty disjoint regions. \begin{theorem}[Reference-stable finite-volume influence tail] \label{thm:tail} For the unitary evolution channel \(\Psi_{t,s}\), and \(d=d_G(A,B)\geq1\), \begin{equation} \boxed{ \Sig_{\Psi_{t,s}}(A\to B) \leq \min\left\{1,\sum_{n=d}^{\infty} \frac{(2\Lambda(s,t))^n}{n!}\right\}.} \label{eq:tail} \end{equation} For every \(\mu>0\), the tail is also bounded by \[ \exp\bigl(-\mu d+2e^\mu\Lambda(s,t)\bigr). \] The bounds are independent of the dimension of an untouched reference. They are not claimed uniform in volume. \end{theorem} \begin{proof} Pass to the interaction picture for the onsite terms. Their propagator is a product of onsite unitaries and preserves every support. It conjugates each two-site interaction without changing its norm or support. For an output observable \(O_{BR}=O_{BR}^\dagger\) with \(\norm{O_{BR}}\leq1\), expand the interaction-picture Heisenberg evolution in its norm-convergent time-ordered nested-commutator series. An order-\(n\) term has norm at most \[ 2^n\prod_{j=1}^n\norm{h_{e_j}(r_j)}. \] Summing the ordered integrals gives \((2\Lambda)^n/n!\). A nonzero commutator with an edge term requires that edge to meet the current support. Each commutator can extend the graph distance from \(B\) by at most one. Thus every term of order \(n0\). Prepare \(B\) in \(\ket0\), and use replacement interventions preparing \(A\) in \(\ket+\) or \(\ket-\). The two \(B\) outputs are \[ e^{-\ii JtX}\ket0,\qquad e^{\ii JtX}\ket0. \] Their trace distance is \(\abs{\sin(2Jt)}\), nonzero for arbitrarily small positive \(t\). A finite-dimensional interaction of this type therefore does not have an exact geometric propagation cutoff. \end{example} \subsection{A product-formula simulation certificate} Let \(H=\sum_{j=1}^m H_j\) be time independent and define \[ V_n(t)=\left( e^{-\ii(t/n)H_m}\cdots e^{-\ii(t/n)H_1} \right)^n, \qquad C_H=\sum_{j0\), \[ F_{s+\delta s}(x(s+\delta s))-F_s(x(s)) \geq\delta s-\lambda_0 d_C(x(s+\delta s),x(s)). \] Taking the almost-everywhere metric derivative gives \[ \frac{\dd T}{\dd s}\geq1-\lambda_0|\dot x|. \] The speed restriction makes this at least \(|\dot x|\). Thus the lifted curve is causal. \end{proof} This is an inner causal bound, not the entire physical velocity set. No equality with a schedule cone is asserted. \subsection{Localization and timing uncertainty} \begin{proposition}[Robust cone margin] \label{prop:conemargin} A nominal dependency has slicing-time gap \(h\), nominal endpoint distance at most \(D\), spatial localization radii \(r_-,r_+\), slicing-time errors at most \(\epsilon_s\) at each endpoint, and additional absolute spacetime-time errors at most \(\epsilon_T\) at each endpoint. It remains causally admissible if \begin{equation} h-(1+\lambda_0)D \geq (1+\lambda_0)(r_-+r_+) +2\epsilon_s+2\epsilon_T. \label{eq:robustcone} \end{equation} Strict inequality is a positive robustness margin. \end{proposition} \begin{proof} The actual spatial distance is at most \(D+r_-+r_+\), and the actual slicing gap is at least \(h-2\epsilon_s\). The time-difference estimate of Theorem~\ref{thm:conegates} then loses at most another \(2\epsilon_T\). Rearrangement gives \eqref{eq:robustcone}. \end{proof} The event guard of VII-F is an exact stored-clock test. This margin concerns physical port placement and availability. It is not a replacement of that guard by a rounded edge report. %===================================================================== \section{The event transition: exact macro fidelity and buffered locality} \label{sec:event} %===================================================================== \subsection{What must be added to the event-code theorem} Fix any one finite forward-domain event circuit \(U_{\mathrm{code}}\) from VII-F. Its input, history, work, and code registers are finite; work is returned to zero on the forward domain, and history is retained \cite[\texttt{thm:implementation}]{CheungVIIF}. Two additional choices are essential: \begin{enumerate} \item all code and location-descriptor bits are placed at a declared controller station; \item a model-time interval is assigned to their actual finite computation. \end{enumerate} A location descriptor stored at the station is not a remote quantum register. No instantaneous acquisition of unknown remote location data is implied. For Target A, one may interpret \(U_{\mathrm{code}}\) as a macro event on local code data with identity matter channel. For Target B, positive-duration gates require a time buffer. The code's germ timestamp \(s=0\) is data describing the geometric event. It is not the time at which every computed code bit becomes physically available. A computation announced after the event is not an instantaneous global geometric update. \subsection{A nontrivial finite target through the event} Let \(o\) be the cone apex and choose the persistent ray mark \[ e=(r_*,0),\qquad r_*=\frac1{64}. \] Set \[ h_*=\frac14. \] Place one input qubit at each of \(p_o(-h_*)\) and \(p_e(-h_*)\). The independent target is: \begin{enumerate} \item update the local event code by its specified \(U_{\mathrm{code}}\); \item exchange the two quantum states at the output ports \(p_o(h_*)\), \(p_e(h_*)\); \item retain code history and return clean work. \end{enumerate} Thus its matter channel is \(\Ad_{\mathrm{SWAP}}\), not the identity. This is a declared discrete carrier target, not an Einstein or scalar-field evolution. The encoding is the identity on each input qubit, with local clean code/work initialization. The decoding reads the returned target factors and the declared event record; it has no nonlocal preprocessing. \begin{lemma}[Endpoint times and strict causal slack] \label{lem:event-times} The four matter-port times are \[ \begin{array}{c|cc} &o&e\\ \hline s=-h_*&-h_*&-h_*+2r_*/3\\ s=h_*&2h_*&2h_*. \end{array} \] Both cross-port dependencies are causal. Their conservative slicing-time margin obeys \[ 2h_*-(1+\lambda_0)r_*>\frac{15}{32}. \] \end{lemma} \begin{proof} On a ray, \(K=0\) and \(H=2r_*/3\). At negative \(s\), the \(H\) branch is active; at positive \(h_*>2r_*/3\), the \(s\) branch is active. The apex time is \(s\) before and \(2s\) after the event. Since \(d_C(o,e)=r_*\), the causal claims follow from Theorem~\ref{thm:conegates}. Finally, \(\lambda_0<1\) gives \[ \frac12-\frac{1+\lambda_0}{64} >\frac12-\frac1{32}=\frac{15}{32}. \] \end{proof} \subsection{An elementary causal realization} Use the apex as the controller and rendezvous station. The model admits local finite quantum registers on its timelike axis under Contract~\ref{contract:routing}; no quantum-field boundary condition at the singular axis is asserted. The qubit starting at \(e\) travels radially inward at unit spacetime speed. It can arrive at the apex at \[ -h_*+\frac{2r_*}{3}+r_* =-h_*+\frac{5r_*}{3}<0. \] The other qubit is already there. Both wait until \(T=0\). Let \(L\) be the finite number of elementary gates in the event-code circuit plus three CNOT gates implementing SWAP. Execute them at the apex, each with assigned spacetime duration \[ \delta_T=\frac{h_*}{2(L+1)}. \] All gates finish before \(T=h_*/2\). Route the appropriate output qubit to \(e\) and retain the other at the apex. Since \[ h_*/2+r_*<2h_*, \] both output ports are reached in time, with waiting as needed. The code remains at the apex. \begin{theorem}[Exact causal and accurate event benchmark] \label{thm:eventbenchmark} For this declared target and architecture: \begin{enumerate}[label=\textup{(\roman*)}] \item every route and every elementary gate is causally admissible in the independently supplied cone; \item the implemented endpoint channel equals \[ \Ad_{U_{\mathrm{code}}}\otimes\Ad_{\mathrm{SWAP}}, \] with histories retained and work returned on the canonical forward domain; \item accuracy is exact, including with every untouched entangled reference; \item the two cross-port matter signalling strengths are one; \item the model-time computation is compatible with the geometric event at \(s=0\), without assigning a geometric meaning to intermediate compiler code states. \end{enumerate} \end{theorem} \begin{proof} The incoming and outgoing radial trajectories obey \(|\dot x|\leq\dot T\). Storage is along the fixed timelike axis or the fixed ray station. Every elementary gate is local at the apex with positive duration. Thus Target B holds. Transport carries the same tensor factors without measurement. At the rendezvous, the three CNOT gates give the SWAP exactly, while the independent code gates give \(U_{\mathrm{code}}\). The two operations act on distinct factors, so the endpoint channel is their tensor product. Unitary identities remain identities with every reference. For signalling from the apex input to the ray output, compare replacement interventions preparing \(\ket0\) and \(\ket1\). SWAP gives orthogonal ray outputs, with trace distance one. The reverse direction is identical. These signals follow causal routes, rather than exceeding physical possibility. The code describes the event germ at timestamp zero and becomes available after its finite computation. The supplied slicing passes continuously through zero during the bracket. Its metric and holonomy do not depend on the compiler states. The decoder statement concerns the canonical completed code, exactly as in VII-F. \end{proof} A hardware implementation may give each local gate a hardware duration \(\delta_{\mathrm{hw}}>0\). The certificate records \[ t_{\mathrm{hardware},j}=j\delta_{\mathrm{hw}}, \qquad T_j=j\delta_T, \] during the local stage. Along the positive-time apex worldline, \(s_j=T_j/2\). Logical gate depth, hardware duration, and the model's slicing parameter are therefore explicitly different quantities. The buffer is not a change in the cone's event. It is an added implementation latency around that event. An exact event code at model timestamp zero does not justify exposing its arbitrary compiler intermediate states as geometric configurations. \subsection{Pre-event, transition, and post-event composition} \begin{corollary}[Certified continuation through the particle-link event] \label{cor:event-continuation} Place any finite pre-event and post-event networks on the slices \eqref{eq:F}, with complete control ledgers and the edge certificates \eqref{eq:conegate}. Insert either: \begin{enumerate} \item the local identity-on-matter macro event with independently certified code availability; or \item the buffered benchmark of Theorem~\ref{thm:eventbenchmark}. \end{enumerate} The complete decoded network satisfies \eqref{eq:main-containment}. If its three stage channels have full-diamond errors \(\epsilon_-,\epsilon_{\mathcal R},\epsilon_+\) against a declared target, the total error is at most their sum. \end{corollary} \begin{proof} Use Proposition~\ref{prop:eventcomposition}, Theorem~\ref{thm:conegates}, and Proposition~\ref{prop:telescope}. \end{proof} This certificate does not require the graph of cell incidence to remain fixed. It requires the registers, routes, and causal port relation to remain specified. A new graph edge cannot be used as an instantaneous communication link merely because a geometric cell has been born. \subsection{What an instantaneous remapping would get wrong} \begin{proposition}[Zero-duration remote SWAP is excluded] \label{prop:instant} Two distinct points on any one slice \(\Sigma_s\) are spacelike separated. A same-slice remote SWAP has unit signalling between them and is not a causally faithful transition for the cone relation. \end{proposition} \begin{proof} The slice estimate gives \[ |F_s(x)-F_s(y)|\leq\lambda_0d_C(x,y)