% En03-A.tex % Self-contained pdfLaTeX source. % % Build: % pdflatex -interaction=nonstopmode -halt-on-error En03-A.tex % pdflatex -interaction=nonstopmode -halt-on-error En03-A.tex % % Verification status: % This source has been reviewed textually. No successful TeX compilation, % proof-assistant check, independent numerical execution, or referee review % is claimed. % % Revision scope: % - Retains the original shared statement numbering 1--69. % - Adds independently numbered Interface propositions. % - Makes complete control/storage accounting explicit. % - Proves the channel causal-cut and process-law statements locally. % - Separates record, residual-instrument, and mechanism identification. % - Adds an explicit classical-register memory witness. % - States and proves conditional approximation and sampling certificates. % - Retains the original fixed-cone four-face exchange; no new % gravitational construction or field equation is asserted. % % Bibliography: % Cheung's Enter article was checked against the publisher's accepted- % article page. Companion identifiers are cited as documentary source % records. Where only summaries were supplied, that limitation is stated. % No missing manuscript title, publication date, DOI, or theorem proof % is invented. % % Publisher status checked: % https://www.frontiersin.org/journals/physics/articles/ % 10.3389/fphy.2026.1954943/abstract % % Author and affiliation are deliberately left unspecified. \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage{microtype} \usepackage[margin=26mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,array,longtable,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \usepackage[nameinlink,noabbrev]{cleveref} \hypersetup{ pdftitle={Observation Boundaries for Destructive Quantum Readout and External Geometry}, pdfsubject={En03-A: typed interfaces, causal cuts, records, residual instruments, and certified geometric events}, pdfkeywords={destructive measurement, quantum channels, noninterference, deferred measurement, instruments, external geometry, causal cuts} } \setlength{\emergencystretch}{3em} \renewcommand{\arraystretch}{1.12} \setlist[enumerate]{itemsep=3pt,topsep=5pt} \setlist[itemize]{itemsep=3pt,topsep=5pt} % Original statements retain one shared counter. \newtheorem{theorem}{Theorem} \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{failure}[theorem]{Failure condition} \newtheorem{openproblem}[theorem]{Open problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem*{processremark}{Process remark} % Supplementary results do not change statements 1--69. \theoremstyle{plain} \newtheorem{interfaceprop}{Interface proposition}[section] \crefname{assumption}{Assumption}{Assumptions} \Crefname{assumption}{Assumption}{Assumptions} \crefname{failure}{Failure condition}{Failure conditions} \Crefname{failure}{Failure condition}{Failure conditions} \crefname{openproblem}{Open problem}{Open problems} \Crefname{openproblem}{Open problem}{Open problems} \crefname{interfaceprop}{Interface proposition}{Interface propositions} \Crefname{interfaceprop}{Interface proposition}{Interface propositions} \numberwithin{equation}{section} \newcommand{\Enter}{\operatorname{Enter}} \newcommand{\CPTP}{\mathrm{CPTP}} \newcommand{\Jsched}{J_{\mathrm{sched}}} \newcommand{\Lor}{\mathrm{SO}^{+}(2,1)} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator{\Cell}{Cell} \DeclareMathOperator{\Dec}{Dec} \DeclareMathOperator{\Err}{Err} \DeclareMathOperator{\loc}{loc} \newcommand{\id}{\operatorname{id}} \newcommand{\Copy}{\mathsf{Copy}} \newcommand{\Bit}{\mathsf{Bit}} \newcommand{\Qtype}{\mathsf Q} \newcommand{\mode}{\mathsf{mode}} \newcommand{\perm}{\mathsf{perm}} \newcommand{\cH}{\mathcal H} \newcommand{\cA}{\mathcal A} \newcommand{\cQ}{\mathcal Q} \newcommand{\cM}{\mathcal M} \newcommand{\cI}{\mathcal I} \newcommand{\cF}{\mathcal F} \newcommand{\cG}{\mathcal G} \newcommand{\cD}{\mathcal D} \newcommand{\cO}{\mathcal O} \newcommand{\cN}{\mathcal N} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Z}{\mathbb Z} \newcommand{\one}{\mathbf 1} \newcommand{\td}{d_{\mathrm{tr}}} \newcommand{\tv}{d_{\mathrm{TV}}} \newcommand{\dd}{\,\mathrm d} \title{Observation Boundaries for Destructive Quantum Readout\\ and External Geometry} \author{} \date{\small Manuscript En03-A} \begin{document} \maketitle \begin{abstract} We formulate a finite, linearly typed interface for destructive computational-basis measurement, external classical geometry, and declared matter--geometry coupling. The observation boundary includes not only coupling observables but also control, guards, capacity tests, locations, and retained storage. Under explicit noninterference hypotheses, arbitrary spectator extensions with the same accessible reduction preserve geometric traces. Immediate destructive readout and a restricted coherent-then-readout implementation agree on their common record--matter output. For a channel \(T\), coarse effects \(E_w\), and aggregated fine effects \(\Pi_w\), record compatibility for every input is equivalent to \(T^*(E_w)=\Pi_w\). Its exact unassisted worst-case total-variation error is an event operator norm. Residual compatibility requires a separate instrument certificate. Complete-record equality does not identify a hidden mechanism; a persistent classical register can produce carrier-only history dependence. Prescribed finite channel networks admit a causal-cut factorization through their scheduled ancestors, including in the presence of an untouched reference. This channel statement is not transferred to nonlinear geometric coupling. Conditional stability, integrated residual, and independent-run sampling certificates are stated with their additional hypotheses. An external conical witness, a static gauge flip, and one complete four-face adjacency exchange illustrate the interface. The exchange has explicit charts, edge laws, event germs, an exact guard, a compact collar, location transport, and a finite reversible event implementation. The ambient spacetime is fixed. No gravitational field equation, continuum limit, or quantum-memory-resource theorem is asserted. \end{abstract} \section{Introduction} A geometric extension of a quantum programming language must specify what geometry can observe. This requirement is independent of the choice of gravitational dynamics or a continuum limit. Linear ownership of quantum names does not determine a mass; a measurement record is not a coarse-graining map; and a circuit dependency region is not automatically a physical light cone. We use a finite circuit interface with destructive readout \begin{equation} \Enter:\Qtype^{\otimes n}\longrightarrow\Bit^{\otimes n}. \label{eq:enter-type} \end{equation} Its semantics is the Born destructive computational-basis measurement used by Cheung's Enter calculus~\cite{Cheung2026Enter,CheungI}. Measured quantum names are consumed, and their tensor factors are not returned. We do not reprove a language-wide type-safety theorem or derive the Born rule. Our mixed-state channel presentation is a finite-dimensional semantic interface for that measurement operation. Four additional structures are declared: \begin{enumerate} \item an external classical geometric configuration; \item a location map for active matter names; \item coupling algebras specifying accessible matter observables; \item a complete control and resource interface specifying update selection, applicability, support, and retained storage. \end{enumerate} Geometry is primitive metadata, not a quantity inferred from name cardinality, measurement count, or workspace size. Two distinctions organize the paper: \begin{equation} \boxed{\begin{gathered} \text{Record compatibility is a pullback;}\\ \text{residual compatibility is a separate instrument certificate.} \end{gathered}} \label{eq:record-boundary-summary} \end{equation} \begin{equation} \boxed{\begin{gathered} \text{Scheduled ancestors bound influence;}\\ \text{a classical register can retain carrier-only history dependence.} \end{gathered}} \label{eq:schedule-memory-summary} \end{equation} \subsection{Results and dependencies} The main results are: \begin{enumerate} \item exact finite-step support containment and causal-cut factorization for prescribed channel dynamics; \item retiming invariance for a fixed labelled gate partial order; \item spectator-extension invariance under complete control and resource blindness; \item restricted destructive deferred-readout equivalence; \item an effect-pullback criterion and exact unassisted record error; \item a residual-instrument obstruction and accessible-algebra comparison criteria; \item a complete fixed-spacetime certificate for one four-face adjacency exchange. \end{enumerate} Supplementary interface propositions give complete-process record laws, a classical-register memory witness, and conditional approximation and sampling bounds. Finite-dimensional channel conventions follow Watrous~\cite{Watrous2018}. The circuit precedent for deferred measurement is Gurevich and Blass~\cite{GurevichBlass2022}. The geometric setting uses conical point-particle spacetimes and polygonal developments~\cite{DeserJackiwHooft1984,tHooft1992, tHooft1993,Kadar2005}. Cheung's companion records motivate comparisons with observation, process, and schedule interfaces~\cite{CheungVIIE,CheungVIIJ, CheungPsy02A,CheungPsy02B}. They are not premises transferring a model identity or a gravitational solution. Every mathematical statement used here is formulated and proved locally. The source ledger in \cref{app:sources} distinguishes full supplied passages from statement summaries; no inspection of an entire manuscript corpus is claimed. \subsection{Scope} The configuration space may contain arbitrary real lengths and affine isometries. These remain external labels unless a finite code and decoder are supplied. The executable examples cover only declared finite families. A geometric coupling may be nonlinear in the matter state. Consequently the complete coupled evolution is not generally a quantum channel. Channel locality, no-signalling, distance contraction, and retiming are asserted only for the prescribed channel objects named in their statements. The four-face construction changes the cellulation of a prescribed slicing of an already fixed spacetime. It is not a new gravitational solution, a particle-supported transition, or an implementation of the full polygon model. No field equation, curvature-convergence theorem, Newtonian limit, or equivalence of matter clocks and propagation cones is proved. \section{Typed interfaces and semantic configurations} \subsection{Finite-dimensional conventions} All matter, code, and record spaces used in channel statements are finite-dimensional. Write \(L(\cH)\) for the operator space and \(D(\cH)\) for its density operators. A channel is a completely positive, trace-preserving linear map. Its adjoint satisfies \begin{equation} \Tr(A\Phi(\rho))=\Tr(\Phi^*(A)\rho). \label{eq:adjoint} \end{equation} Channel adjoints are unital and completely positive~\cite{Watrous2018}. A classical register has a distinguished basis. An admissible joint state is block diagonal in that register, not merely diagonal after taking its marginal. Declared classical operations preserve this joint block-diagonal state space. Their individual Kraus operators need not be diagonal; classical permutations are allowed. For a unital \(^{*}\)-subalgebra \(\cA\), define \begin{equation} \omega_\rho|_{\cA}:A\longmapsto\Tr(\rho A). \label{eq:state-functional} \end{equation} Equality of these functionals means equality on the named observable interface. Our distance conventions are \begin{equation} \tv(p,q)=\frac12\sum_w\abs{p(w)-q(w)}, \qquad \td(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1. \label{eq:distance-conventions} \end{equation} The trace norm itself has no factor \(1/2\). Outcome sets and complete history alphabets are finite unless explicitly stated otherwise. \subsection{A minimal linear circuit interface} Wire types are generated by \[ A::=I\mid\Qtype\mid\Bit\mid A\otimes A. \] A judgement \(f:A\to B\) consumes the entire input interface and returns the specified output interface. Formation includes identities, permutations, sequential composition, and tensor composition: \[ \frac{f:A\to B\qquad g:B\to C}{g\circ f:A\to C}, \qquad \frac{f:A\to B\qquad g:C\to D} {f\otimes g:A\otimes C\to B\otimes D}. \] The circuit constants used here are unitary gates, preparation of \(\ket0\), and destructive computational-basis measurement. Finite reversible code operations use NOT, CNOT, and Toffoli. Classical zero preparation and classical reversible gates act on \(\Bit\) interfaces; their basis-permutation unitary lifts act on \(\Qtype\) interfaces. No conversion between these types is implicit. Linearity concerns ownership. Successive gates may consume and return the same physical factor. No rule duplicates a live quantum wire. % 1 \begin{remark}[Relation to a larger calculus] \label{rem:calculus} The interface is specified independently of the complete language. An embedding into the full calculus of~\cite{Cheung2026Enter} must match its constants, typing, and circuit-adequacy interface. No theorem below relies on the Layer II measurement-uniqueness argument or asserts gate-library realization of every semantic channel. \end{remark} \subsection{Three measurement interfaces} Let \(S\) have computational basis \(\{\ket z:z\in Z\}\), and let \(B\) be its accessible complement. Put \[ M_z=\bra z_S\otimes I_B:\cH_{SB}\longrightarrow\cH_B. \] % 2 \begin{definition}[Destructive \(\Enter\)] \label{def:enter} The destructive measurement channel is \begin{equation} \cM_S(\rho) =\sum_z\ket z\bra z_R\otimes M_z\rho M_z^\dagger, \qquad \cM_S:L(\cH_{SB})\longrightarrow L(\cH_{RB}). \label{eq:destructive} \end{equation} The record \(R\) is classical. The factor \(S\) is absent. The circuit operation \(\Enter_S\) is interpreted by \(\cM_S\). \end{definition} This is destructive Born readout, not a returned measured quantum register~\cite{Cheung2026Enter,CheungI}. % 3 \begin{definition}[Coherent recording] \label{def:recording} A fresh quantum carrier \(A\), with basis indexed by \(Z\), is introduced by the isometry \begin{equation} V:\cH_{SB}\longrightarrow\cH_{SAB}, \qquad V(\ket z_S\otimes\ket b_B) =\ket z_S\otimes\ket z_A\otimes\ket b_B. \label{eq:copy-isometry} \end{equation} Its channel is \(\Copy_S(\rho)=V\rho V^\dagger\). For qubits it is zero preparation followed by CNOT. \end{definition} % 4 \begin{definition}[Represented nondestructive instrument] \label{def:nondestructive} For \(P_z=\ket z\bra z_S\otimes I_B\), set \begin{equation} \cI_S^{\mathrm{nd}}(\rho) =\sum_z\ket z\bra z_R\otimes P_z\rho P_z. \label{eq:nondestructive} \end{equation} Its residual interface is \(S\otimes B\). It is not the destructive primitive. \end{definition} \begin{center} \begin{tabular}{@{}llll@{}} \toprule Operation & \(S\) returned & Record & Role\\ \midrule \(\Enter_S\) & No & Classical & Destructive primitive\\ \(\Copy_S\) & Yes & Quantum carrier & Isometric recording\\ \(\cI_S^{\mathrm{nd}}\) & Yes & Classical & Residual instrument\\ \bottomrule \end{tabular} \end{center} Coherent recording does not clone arbitrary states: \[ V\ket+=\frac{\ket{00}+\ket{11}}{\sqrt2} \ne \ket+\ket+ =\frac{\ket{00}+\ket{01}+\ket{10}+\ket{11}}2. \] Deferred \(\Enter\) means recording, continuation on the declared complement, destructive carrier readout, and comparison after restriction away from retained \(S\). \subsection{External geometry and transition strata} % 5 \begin{definition}[External geometric interface] \label{def:external} A geometric interface specifies: \begin{enumerate} \item a set \(\cQ\) of classical configurations; \item an admissible subset \begin{equation} \cQ_{\mathrm{adm}}=\cQ_{\mathrm{reg}}\cup\cQ_{\mathrm{tr}}; \label{eq:admissible-strata} \end{equation} \item a finite label set \(\Cell(q)\) for each configuration; \item support, admissibility, and resource certificates for updates; \item optionally, a supplied gauge equivalence \(\sim_g\). \end{enumerate} Regular configurations satisfy the model's stated nondegeneracy conditions. Transition configurations require a model-specific event certificate and may retain one-sided germ labels whose common geometric realization is degenerate. Admissibility is semantic. A finite representation and a decision procedure are additional structures. \end{definition} In \cref{sec:geometry}, every bounded regular edge has positive length. Fixed unbounded exterior rays, when used, are described as rays rather than finite length reports. At the event, a vanishing central edge and coincident endpoint labels are germ metadata, not a regular embedded one-cell. % 6 \begin{definition}[Geometry--matter configuration] \label{def:configuration} A configuration is \begin{equation} C=(q;\Gamma;\rho;\loc), \qquad \loc:\Gamma\longrightarrow\Cell(q), \label{eq:configuration} \end{equation} where \(q\in\cQ_{\mathrm{adm}}\), \(\Gamma\) is a finite labelled interface, and \(\rho\in D(\cH_\Gamma)\). Classical names satisfy joint block diagonality. The geometric projection is \(\pi_g(C)=q\). Control and resource labels are either included in \(q\) or carried as separately declared classical data. Compared executions have equal initial declared control and resource data. Permitted transition-germ locations, their realizations, and their transport belong to the event certificate. \end{definition} Interfaces may change under commands. Destructive measurement replaces measured quantum names by fresh record names. An identity extension always refers to the current unacted-on interface. \subsection{Three rule kinds} % 7 \begin{definition}[Certified interface rules] \label{def:rules} A finite run uses: \begin{enumerate} \item A \emph{geometric rule}, or \((G)\)-rule, which replaces supported geometric labels by a specified function of those labels and declared resources. It does not inspect matter state or matter-name cardinality. Its complete output is admissible and includes location transport. A transition rule specifies incoming and outgoing germs, their common event realization, and both one-sided continuations. \item A \emph{matter rule}, which applies a declared channel \begin{equation} \Psi_q:L(\cH_{\Gamma_0})\longrightarrow L(\cH_{\Gamma'_0}) \label{eq:matter-rule} \end{equation} on a named local interface, tensored with the identity on the remaining names. It may depend on \(q\), but does not change \(q\). Allocation and output locations are part of its certificate. \item A \emph{coupling rule}, which declares a unital observable algebra \(\cA_v\) and updates geometry by \begin{equation} q'=F_v(q,\omega_\rho|_{\cA_v}). \label{eq:coupling} \end{equation} It returns a complete admissible configuration and a location transport map. It leaves the matter density operator unchanged. \end{enumerate} \end{definition} Changing one particle scalar while freezing incompatible chart data need not preserve constraints. A coupling therefore returns a complete certified update, not merely a new scalar. % 8 \begin{remark}[Nonlinear backreaction] \label{rem:nonlinear} The map in \cref{eq:coupling} need not be affine in \(\rho\). The complete coupled evolution is not generally \(\CPTP\). The noninterference proofs compare equal coupling inputs; they do not establish no-signalling for arbitrary nonlinear backreaction. \end{remark} \subsection{Complete control and resource accounting} The controller selects commands, guards, support, and capacity checks. Every state-dependent register, queue, timing signal, applicability predicate, and readable transport history is part of that interface. % 9 \begin{assumption}[Spectator-blind control] \label{ass:sched} For the compared executions, inaccessible spectators do not change: \begin{enumerate} \item the selected command or its declared matter channel; \item guard, applicability, or capacity-check results; \item the support, block membership, or transport of original names; \item successor control and declared resource data. \end{enumerate} The controller is a specified function of the complete declared inputs. Equal such inputs select the same update and certificate. No undeclared spectator-dependent input is available to it. The location transports are defined on all permitted spectator locations, and spectator storage and transport history have separately reserved resources sufficient for the compared run. Randomized controllers are aligned using a common input-independent seed, or a specified coupled law satisfying the same equality property. \end{assumption} Separate reservation is not a claim of unlimited physical memory. It is a hypothesis about which finite resource budget controls the geometric update. A global occupancy test may violate the hypothesis. % 10 \begin{definition}[Gauge and admissible geometric observables] \label{def:gauge} Gauge equivalence is a supplied equivalence relation on geometric descriptions. In the examples it includes chart changes and retriangulations representing the same metric and marked slice. An admissible geometric observable is constant on its equivalence classes. A fixed spacetime does not identify all marked-slice observables at different times. Circuit retiming is not presumed to be a gauge symmetry of nonlinear coupled evolution. \end{definition} \section{Kinematics and causal cuts on a prescribed schedule} \label{sec:schedules} Throughout this section the channel list, metric, and schedule are prescribed. State-dependent geometric selection is not hidden inside a channel list. \subsection{Scheduled blocks} % 11 \begin{definition}[Scheduled matter dynamics] \label{def:schedule} Let \((X,d)\) be a finite metric space with site spaces \(\cH_x\), and write \[ \cH_X=\bigotimes_{x\in X}\cH_x, \qquad \cA(Y)=L\left(\bigotimes_{x\in Y}\cH_x\right) \otimes I_{X\setminus Y}. \] For \(a,\tau>0\), let \(\mathcal B_t\) be a partition of \(X\) at tick \(t=1,\ldots,T\), with \[ \diam_d B\le R_{\mathrm{upd}}a. \] The dynamics is \begin{equation} \Phi_t=\bigotimes_{B\in\mathcal B_t}\Phi_{t,B}, \qquad c_{\mathrm{sched}}=\frac{R_{\mathrm{upd}}a}{\tau}, \label{eq:scheduled-dynamics} \end{equation} where each block map is a channel on its reserved site interface. \end{definition} Readout may write a reserved local record register. A fixed ambient description may contain an inactive placeholder after destruction; that placeholder is not an accessible surviving measured wire. % 12 \begin{definition}[Schedule past] \label{def:schedule-past} For \(Y\subseteq X\) and \(0\le s\le t\le T\), set \begin{equation} \begin{aligned} N_{t,t}(Y)&=Y,\\ N_{s,t}(Y)&= \bigcup\{B\in\mathcal B_{s+1}: B\cap N_{s+1,t}(Y)\ne\varnothing\}, \qquad s0, \label{eq:cone-metric} \end{equation} with \(\theta\) periodic modulo \(2\pi\). Its completion has a timelike conical axis. The spinless point-particle interpretation is standard~\cite{DeserJackiwHooft1984}; no field equation is derived. On the universal cover, \[ D(T,r,\widetilde\theta) =(T,r\cos(\alpha\widetilde\theta), r\sin(\alpha\widetilde\theta)). \] For the deck transformation \(\mathsf T(T,r,\widetilde\theta) =(T,r,\widetilde\theta+2\pi)\), define \begin{equation} R(\mu)= \begin{pmatrix} 1&0&0\\ 0&\cos\mu&-\sin\mu\\ 0&\sin\mu&\cos\mu \end{pmatrix}. \label{eq:rotation} \end{equation} Then \begin{equation} D\circ\mathsf T^{-1}=R(\mu)\circ D. \label{eq:developing} \end{equation} Choose \(0<\epsilon<\pi/4\), with angular lifts \[ U_1=(-\epsilon,\pi+\epsilon),\qquad U_2=(\pi-\epsilon,2\pi+\epsilon), \] restricted where necessary to injective developing charts. Their two overlap components have transitions \begin{equation} \Lambda_{2\leftarrow1}^{\pi}=I, \qquad \Lambda_{1\leftarrow2}^{0}=R(\mu). \label{eq:seam-transitions} \end{equation} The positively oriented particle word is \begin{equation} (\Lambda_{1\leftarrow2}^{0},0) (\Lambda_{2\leftarrow1}^{\pi},0) =(R(\mu),0). \label{eq:particle-word} \end{equation} Overlap-component indexing prevents suppression of the nontrivial seam. No smooth flat chart contains the axis. \subsection{Tangent holonomy} % 34 \begin{lemma}[Developing holonomy and tangent transport] \label{lem:soldering} Suppose \begin{equation} D\circ\mathsf T_\gamma^{-1}=h_\gamma\circ D, \qquad h_\gamma(y)=L_\gamma y+b_\gamma. \label{eq:affine-development} \end{equation} With this convention, parallel transport around \(\gamma\), in the starting developed frame, has matrix \(L_\gamma\). \end{lemma} \begin{proof} Lift the loop from \(\widetilde x\) to \(\mathsf T_\gamma\widetilde x\). Flat developed transport keeps the Minkowski vector constant: \[ dD_{\mathsf T_\gamma\widetilde x}(v') =dD_{\widetilde x}(v). \] Endpoint identification uses \(d\mathsf T_\gamma^{-1}\). Differentiating \cref{eq:affine-development} gives \[ dD_{\widetilde x}\,d\mathsf T_\gamma^{-1}(v') =L_\gamma dD_{\mathsf T_\gamma\widetilde x}(v') =L_\gamma dD_{\widetilde x}(v). \] \end{proof} The cone word acts on spacetime tangent vectors, not an unrelated internal fibre. Developments and their connection data must remain distinct~\cite{Kadar2005}. \subsection{Deficit and particle--slice data} % 35 \begin{lemma}[Axis and trace information] \label{lem:axis} For \(L=C R(\mu)C^{-1}\), \(C\in\Lor\), and \(0<\mu<2\pi\), \begin{equation} \Fix(L)=\R C e_0,\qquad p_L=C e_0,\qquad e_0=(1,0,0). \label{eq:particle-axis} \end{equation} The future unit fixed vector is unique, and \[ \Tr L=1+2\cos\mu. \] The trace alone does not distinguish \(\mu\) from \(2\pi-\mu\). \end{lemma} \begin{proof} The spatial rotation has no nonzero fixed vector in its spatial plane. Conjugation preserves trace and future orientation and carries the fixed line to \(\R C e_0\). \end{proof} For an oriented orthonormal basis \((u,v)\) of \(p_L^\perp\), with \((p_L,u,v)\) positively oriented, \[ \cos\mu=\eta(u,Lu),\qquad \sin\mu=\eta(v,Lu) \] recover the oriented deficit. For a spacelike piece with future unit normal \(n\), set \begin{equation} \chi(p,n)=\arccosh(-\eta(p,n)). \label{eq:rapidity} \end{equation} Both vectors must be expressed in the same development. % 36 \begin{proposition}[Gauge invariance of marked-slice data] \label{prop:slice-invariance} The oriented deficit and \(\chi(p,n)\) are invariant under proper, time-orientation-preserving chart changes. Rapidity is an invariant of the particle together with the marked slice piece, not of the particle alone. \end{proposition} \begin{proof} A chart change simultaneously conjugates \(L\) and transforms \(p,u,v,n\), preserving the orientations and inner products. \end{proof} \subsection{Equal name data, unequal marked-slice data} Choose \(k\ge3\) and \[ \vartheta=\frac{2\pi-\mu}{k}<\pi,\qquad u_j=\left( \cos((j+\tfrac12)\vartheta), \sin((j+\tfrac12)\vartheta)\right). \] On sector \([j\vartheta,(j+1)\vartheta]\), let \(\beta=\tanh\zeta\) and set \begin{equation} T=\beta\,u_j\cdot x. \label{eq:fan} \end{equation} The future unit normal is \[ n_j=\cosh\zeta\,e_0+\sinh\zeta\,(0,u_j). \] On a shared ray, both functions have value \(\beta r\cos(\vartheta/2)\); the wedge faces match similarly. The result is a continuous piecewise-flat spacelike neighbourhood. Joining equal-radius ray endpoints by straight chords gives a finite polygonal star. % 37 \begin{proposition}[A witness invisible to name data] \label{prop:fan-witness} Attach identical matter interfaces, states, and abstract cell assignments to the resting fan and to \cref{eq:fan} with \(\zeta\ne0\). Their particle holonomy is the same, but \begin{equation} \chi(e_0,n_j)= \begin{cases} 0,&\text{resting fan},\\ \abs{\zeta},&\text{tilted fan}. \end{cases} \label{eq:fan-rapidity} \end{equation} \end{proposition} \begin{proof} The relevant inner products are \(1\) and \(\cosh\zeta\), and \(\arccosh(\cosh\zeta)=\abs{\zeta}\). \end{proof} An arbitrary globally boosted plane need not descend through the cone identification. The sectorwise construction avoids that assumption. It does not identify rapidity with stress energy. \subsection{Affine intersection laws} % 38 \begin{proposition}[Constraint words and affine edge lengths] \label{prop:affine} A polygonal slice induced from a fixed flat conical spacetime and its component-indexed atlas has ordered transition products equal to the corresponding monodromies. On an interval of fixed incidence, suppose normals are constant, offsets are affine in the parameter, and endpoints solve nonsingular constant-coefficient linear intersection systems. If an edge has a fixed spacelike direction, its length is affine on subintervals where orientation does not reverse. \end{proposition} \begin{proof} The first statement is chart continuation. An endpoint satisfying \(Az=b_0+s b_1\), with \(A\) constant and invertible, is affine. An edge difference with fixed unit spacelike direction is \((c_0+s c_1)e\), of length \(\abs{c_0+s c_1}\). \end{proof} Affine endpoint motion alone does not make arbitrary segment lengths affine. Artificial subdivision edges are checked explicitly below. No estimate is continued through an event merely by assuming regular-stratum smoothness. \subsection{A static square gauge flip} \label{sec:square} Set \(\mu=\pi/2\). In a nonsingular developed sector take \[ a=(0,3,1),\quad b=(0,4,1),\quad c=(0,4,2),\quad d=(0,3,2). \] They bound a unit square with a matching certified exterior. \begin{center} \begin{tabularx}{\textwidth}{@{}lXX@{}} \toprule Field & Before & After\\ \midrule Faces & \(abc,acd\) & \(abd,bcd\)\\ Boundary & \(ab,bc,cd,da\) & unchanged\\ Internal edge & \(ac\) & \(bd\)\\ Boundary lengths & \(1,1,1,1\) & unchanged\\ Internal length & \(\sqrt2\) & \(\sqrt2\)\\ Patch affine transitions & \((I,0)\) & \((I,0)\)\\ Particle labels in patch & none & none\\ Mode & \(m=0\) & \(m=1\)\\ Permission & \(\kappa\ge1\) & \(\kappa-1\)\\ \bottomrule \end{tabularx} \end{center} % 39 \begin{definition}[Square guard and transport] \label{def:square-guard} The guard is \(m=0\), \(\kappa\ge1\), and membership in the fixed square family. Names on the old interior faces or diagonal move to the designated successor \(abd\). Boundary and exterior labels are retained. No matter channel or coupling fires. \end{definition} % 40 \begin{proposition}[Square certificate] \label{prop:square} The update is a certified \((G)\)-rule preserving the metric, marked resting slice, and vertex monodromies. Particle deficit and rapidity remain \(\pi/2\) and \(0\). \end{proposition} \begin{proof} Both triangulations cover the same square and boundary. Every internal transition word is \((I,0)\). The particle word remains \((R(\pi/2),0)(I,0)\), with axis and resting normal \(e_0\). \end{proof} This constant-length gauge flip is not an edge-collapse exchange. Its guard and permission accounting are spectator-blind. \subsection{Four affine faces} \label{sec:fourfaces} Let \(s=0\) be the event. In a common chart \(z=(T,x,y)\), choose \[ 00, \] and define \begin{equation} \begin{aligned} f_E&=px+as,& f_W&=-px+as,\\ f_N&=qy+bs,& f_S&=-qy+bs, \end{aligned} \qquad F_s=\max\{f_E,f_W,f_N,f_S\}. \label{eq:fourplanes} \end{equation} An active face attains the maximum; its interior is the strict-maximum region. The concrete certificate uses \begin{equation} p=\frac35,\quad q=\frac5{13},\quad a=\frac45,\quad b=\frac{12}{13},\quad \Delta=\frac8{65}. \label{eq:rational} \end{equation} % 41 \begin{proposition}[Spacelike graph and time order] \label{prop:R1} The graph of \(F_s\) is continuous, piecewise affine, and \(p\)-Lipschitz with \(p<1\). Every face is spacelike. For \(s_2>s_1\), \begin{equation} F_{s_2}(x,y)-F_{s_1}(x,y)\ge a(s_2-s_1)>0. \label{eq:time-ordering} \end{equation} \end{proposition} \begin{proof} The spatial gradients have norm \(p\) or \(q\); their maximum retains the common Lipschitz bound. A face tangent satisfies \[ \eta((v_i\cdot u,u),(v_i\cdot u,u)) =\abs u^2-(v_i\cdot u)^2>0 \] for \(u\ne0\). Every face function increases by at least \(a(s_2-s_1)\), so their maximum does too. \end{proof} % 42 \begin{proposition}[Incoming edge] \label{prop:R2} For \(s<0\), \(E,W\) share exactly \begin{equation} x=0,\quad \abs y\le\frac{\Delta(-s)}q,\quad T=as. \label{eq:incoming-edge} \end{equation} Its endpoints are \(EWN,EWS\), and its length is \begin{equation} \ell_-(s)=\frac{2\Delta}{q}(-s)=\frac{16}{25}(-s). \label{eq:incominglength} \end{equation} \end{proposition} \begin{proof} Equality gives \(x=0\). Dominance requires \(as\ge q\abs y+bs\), hence \(q\abs y\le-\Delta s\). The two equalities at its endpoints add \(N\) or \(S\). The displacement is purely in the \(y\)-direction. \end{proof} % 43 \begin{proposition}[Outgoing edge] \label{prop:R3} For \(s>0\), \(N,S\) share exactly \begin{equation} y=0,\quad \abs x\le\frac{\Delta s}p,\quad T=bs. \label{eq:outgoing-edge} \end{equation} Its endpoints are \(NSE,NSW\), and its length is \begin{equation} \ell_+(s)=\frac{2\Delta}{p}s=\frac{16}{39}s. \label{eq:outgoinglength} \end{equation} \end{proposition} \begin{proof} Equality gives \(y=0\); dominance requires \(p\abs x\le\Delta s\). The displacement is purely spatial. \end{proof} % 44 \begin{proposition}[A junction, not a patch collapse] \label{prop:R4} At \(s=0\), all four planes meet at the origin and the central lengths vanish, but every face contains a two-dimensional open region. The incidence change is \begin{equation} \{EWN,EWS\}\longrightarrow\{EWNS\} \longrightarrow\{NSE,NSW\}. \label{eq:exchange-incidence} \end{equation} \end{proposition} \begin{proof} Here \(F_0=\max\{p\abs x,q\abs y\}\). For example, \(E\) is strict on \(x>0,\ px>q\abs y\). The other regions follow by symmetry; the four-way intersection is only the origin. \end{proof} % 45 \begin{proposition}[Transversality and isolation] \label{prop:R5} The central one-sided derivatives are \begin{equation} \dot\ell_-(0^-)=-2\Delta/q<0,\qquad \dot\ell_+(0^+)=2\Delta/p>0. \label{eq:transversality} \end{equation} Mixed adjacencies \(EN,ES,WN,WS\) persist. No other core pair exchanges a positive-length edge near the event. \end{proposition} \begin{proof} Differentiate the central laws. In each quadrant a mixed adjacency lies on \[ p\abs x-q\abs y=\Delta s \] and extends from a central triple junction to the exterior. The only remaining pairs are \(EW\) and \(NS\), already calculated. \end{proof} The exchange is determined by the declared maximum slicing rule. It is not a uniqueness theorem over all foliations. \paragraph{Adapted frames.} For a slope \(v_i\), choose a Lorentz map with time coordinate \[ X_i^0=\frac{T-v_i\cdot(x,y)}{\sqrt{1-\abs{v_i}^2}}. \] On its face this equals \(s\). The concrete maps are \begin{equation} \begin{aligned} C_E&=\begin{pmatrix} 5/4&-3/4&0\\-3/4&5/4&0\\0&0&1 \end{pmatrix}, & C_W&=\begin{pmatrix} 5/4&3/4&0\\3/4&5/4&0\\0&0&1 \end{pmatrix},\\ C_N&=\begin{pmatrix} 13/12&0&-5/12\\0&1&0\\-5/12&0&13/12 \end{pmatrix}, & C_S&=\begin{pmatrix} 13/12&0&5/12\\0&1&0\\5/12&0&13/12 \end{pmatrix}. \end{aligned} \label{eq:boosts} \end{equation} They are proper future-preserving Lorentz maps. Set \begin{equation} \Lambda_{i\leftarrow j}=C_iC_j^{-1}. \label{eq:core-transitions} \end{equation} The opposite-pair entries are \begin{equation} \Lambda_{E\leftarrow W} =\begin{pmatrix}17/8&-15/8&0\\-15/8&17/8&0\\0&0&1\end{pmatrix}, \qquad \Lambda_{N\leftarrow S} =\begin{pmatrix}97/72&0&-65/72\\0&1&0\\-65/72&0&97/72\end{pmatrix}. \label{eq:opposite-transitions} \end{equation} For \(X_+=E,X_-=W,Y_+=N,Y_-=S\), \begin{equation} \Lambda_{X_\epsilon\leftarrow Y_\nu} = \begin{pmatrix} 65/48&-\epsilon\,3/4&\nu\,25/48\\ -\epsilon\,13/16&5/4&-\epsilon\nu\,5/16\\ \nu\,5/12&0&13/12 \end{pmatrix}. \label{eq:mixedtransitions} \end{equation} Reverse maps are inverses; all core translations are zero. % 46 \begin{proposition}[Exact ordered products] \label{prop:R6} Every closed contractible core word is the affine identity. In particular, \begin{align} \Lambda_{E\leftarrow W}\Lambda_{W\leftarrow N} \Lambda_{N\leftarrow E}&=I, & \Lambda_{E\leftarrow W}\Lambda_{W\leftarrow S} \Lambda_{S\leftarrow E}&=I, \label{eq:incoming-products}\\ \Lambda_{N\leftarrow S}\Lambda_{S\leftarrow E} \Lambda_{E\leftarrow N}&=I, & \Lambda_{N\leftarrow S}\Lambda_{S\leftarrow W} \Lambda_{W\leftarrow N}&=I. \label{eq:outgoing-products} \end{align} The event four-face word is also \(I\). \end{proposition} \begin{proof} Every word telescopes: \[ (C_{i_0}C_{i_1}^{-1})\cdots (C_{i_{m-1}}C_{i_0}^{-1})=I. \] \end{proof} Nontrivial adapted transitions here are frame changes in one flat development, not evidence of spacetime curvature. \subsection{Compact collar and frozen exterior} \label{sec:collar} For \(-1\le s\le1\) and \(H=2\), define \begin{equation} g_s(x,y)=\min\{F_s(x,y),H+s\}. \label{eq:capped} \end{equation} The uncapped interior region is \begin{equation} \mathscr R_s=[-X_s,X_s]\times[-Y_s,Y_s], \qquad X_s=\frac{10+s}{3},\quad Y_s=\frac{26+s}{5}, \label{eq:inner-rectangle} \end{equation} since \(F_s\le H+s\) is equivalent to these two inequalities. Choose \begin{equation} K=[-R_x,R_x]\times[-R_y,R_y], \qquad R_x=\frac{14}{3},\quad R_y=6. \label{eq:K} \end{equation} Then \begin{equation} R_x-X_s=\frac{4-s}{3}>0,\qquad R_y-Y_s=\frac{4-s}{5}>0. \label{eq:collar-margins} \end{equation} % 47 \begin{proposition}[Spacelike capped slices] \label{prop:cap} The capped graph is continuous, piecewise flat, and \(p\)-Lipschitz. It agrees with \(T=H+s\) near \(\partial K\), and retains the lower time increment \(a(s_2-s_1)\). Both central edge laws remain unchanged in the window. \end{proposition} \begin{proof} Minimum and maximum preserve a common Lipschitz bound and a common lower time increment. Strict collar margins give the resting boundary neighbourhood. The incoming central height margin is \[ H+s-as=2+s/5\ge9/5; \] the outgoing margin is \[ H+s-bs=2+s/13\ge25/13. \] Thus the cap does not clip the central edges. \end{proof} \paragraph{Finite collar cells.} There are four core faces and four resting collar polygons. The north-east collar has spatial vertices \begin{equation} (X_s,0),\ (R_x,0),\ (R_x,R_y),\ (0,R_y),\ (0,Y_s),\ (X_s,Y_s), \label{eq:collar-hexagon} \end{equation} all at \(T=H+s\). Reflections, with boundary order reversed when needed, give the other counterclockwise polygons. Inner and outer side midpoints divide each rectangle boundary into eight segments. At an inner corner, two core faces and one collar face meet. At an inner midpoint, one core face and two coplanar collar subdivisions meet. These subdivisions have fixed incidence and are not additional transitions. The collar chart is \begin{equation} A_0(z)=(I,-H e_0)z=(T-H,x,y). \label{eq:collar-chart} \end{equation} Core-to-collar transitions are \begin{equation} A_0(C_i,0)^{-1}=(C_i^{-1},-H e_0). \label{eq:core-collar-transition} \end{equation} For affine pairs, \begin{equation} (L,b)(L',b')=(LL',Lb'+b). \label{eq:affine-multiplication} \end{equation} % 48 \begin{proposition}[Collar words and polygon closure] \label{prop:collarproducts} Collar and mixed vertex words are the affine identity. All polygons close, and matching edges have equal induced lengths. \end{proposition} \begin{proof} All cell charts are globally defined affine isometries \(A_i\) of the same patch; transitions \(A_iA_j^{-1}\) telescope. The core polygons are intersections of their declared half-spaces with \(\mathscr R_s\), and the collar polygons have the listed boundaries. Shared endpoints and segments are identical. \end{proof} \paragraph{Attachment to the cone.} Use the wedge of angle \(3\pi/2\), and translate the patch by \[ (X,Y)=(20+x,20+y). \] The entire \(K\) lies inside its first quadrant, away from axis and seam. Set \(T=H+s\) on the rest of the cone. For an explicit finite exterior construction, divide the wedge into its three convex quadrants. In the first quadrant cut by the vertical lines through \(20-R_x,20,20+R_x\) and the horizontal lines through \(20-R_y,20,20+R_y\). Retain the cells outside the translated \(K\), and replace its interior by the eight patch cells. This finite line arrangement matches the eight boundary segments. Unbounded exterior cells and rays are permitted and remain fixed. Shift the cone development to \[ D_H=(T-H,r\cos(\alpha\widetilde\theta), r\sin(\alpha\widetilde\theta)). \] The time shift commutes with \(R(\mu)\). The collar-to-exterior transition is \begin{equation} (I,(0,20,20)). \label{eq:exterior-transition} \end{equation} % 49 \begin{proposition}[Boundary matching and particle word] \label{prop:R8} Incoming and outgoing descriptions use the same boundary labels, constant spatial boundary vertices and lengths, resting boundary plane \(T-H=s\), and fixed exterior transitions. The cone metric, exterior incidence, and resting particle neighbourhood are not rewritten. The particle word remains \begin{equation} (R(\pi/2),0)(I,0)=(R(\pi/2),0). \label{eq:frozen-particle-word} \end{equation} \end{proposition} \begin{proof} The cap is resting near \(\partial K\); all its boundary data and charts are common. A particle loop can be chosen in the unchanged axis neighbourhood. A path through the patch instead has a telescoping word determined by its endpoint charts, independent of its internal incidence. \end{proof} Frozen exterior means unchanged geometric and affine-time laws, not frozen spacetime time coordinates. \paragraph{Complete core polygon boundaries.} Write \[ U_\nu=(as,0,\nu\Delta(-s)/q)\quad(s<0),\qquad Z_\epsilon=(bs,\epsilon\Delta s/p,0)\quad(s>0), \] and \[ V_{\epsilon,\nu}=(H+s,\epsilon X_s,\nu Y_s), \quad M^x_\epsilon=(H+s,\epsilon X_s,0), \quad M^y_\nu=(H+s,0,\nu Y_s). \] Counterclockwise incoming boundaries are \[ \begin{aligned} E&:\ U_-,V_{+,-},M^x_+,V_{+,+},U_+,\\ N&:\ U_+,V_{+,+},M^y_+,V_{-,+}. \end{aligned} \] The outgoing boundaries are \[ \begin{aligned} E&:\ Z_+,V_{+,-},M^x_+,V_{+,+},\\ N&:\ Z_-,Z_+,V_{+,+},M^y_+,V_{-,+}. \end{aligned} \] Reflecting and reversing order gives \(W,S\). At \(s=0\), the central vertices coincide; remove repeated consecutive vertices in the geometric realization. The remaining vertices are the four outer corners \((H+s,\epsilon R_x,\nu R_y)\) and four outer side midpoints \((H+s,\epsilon R_x,0)\), \((H+s,0,\nu R_y)\). Each mixed arm ends at \(V_{\epsilon,\nu}\), and has length \begin{equation} L_{\mathrm{arm}}(s)= \begin{cases} \dfrac{10+s}{75}\sqrt{1921},&-1\le s\le0,\\[4pt] \dfrac{26+s}{195}\sqrt{1921},&0\le s\le1. \end{cases} \label{eq:arms} \end{equation} The two event values agree. % 50 \begin{proposition}[All lengths and no peripheral collapse] \label{prop:alllengths} The complete bounded patch lengths are in \cref{tab:certificate}. Every noncentral actual edge has length greater than \(1/2\) throughout the window. All eight faces remain two-dimensional. The central exchange is the only incidence event. \end{proposition} \begin{proof} The north-east incoming arm displacement is \(X_s(p,1,p/q)\), of spacelike norm \(X_s\sqrt{1921}/25\). The outgoing displacement is \(Y_s(q,q/p,1)\), of norm \(Y_s\sqrt{1921}/39\). Reflection gives the other arms. Resting lengths are \(X_s,Y_s,(4-s)/3,(4-s)/5,R_x,R_y\). Their minimum is \(3/5\); arm lengths are at least \(3\sqrt{1921}/25>1/2\). Each core face has a strict-maximum neighbourhood immediately inside its inner side midpoint. Every collar polygon has nonempty interior by the strict margins. Central endpoints remain strictly inside \(\mathscr R_s\), and all peripheral incidence is fixed. \end{proof} \begin{table}[p] \centering \small \begin{tabularx}{\textwidth}{@{}p{.24\textwidth}XXX@{}} \toprule Field & Incoming & Event & Outgoing\\ \midrule Core faces & \(E,W,N,S\) & same & same\\ Collar faces & four hexagons & same & same\\ Central adjacency & \(EW\) & four-face point & \(NS\)\\ Central triples & \(EWN,EWS\) & common junction & \(NSE,NSW\)\\ Central length & \(16(-s)/25\) & \(0\) & \(16s/39\)\\ Four arms & \(\frac{10+s}{75}\sqrt{1921}\) & \(\frac{2}{15}\sqrt{1921}\) & \(\frac{26+s}{195}\sqrt{1921}\)\\ Four horizontal cap segments & \(X_s\) & \(10/3\) & \(X_s\)\\ Four vertical cap segments & \(Y_s\) & \(26/5\) & \(Y_s\)\\ Two horizontal collar joins & \((4-s)/3\) & \(4/3\) & \((4-s)/3\)\\ Two vertical collar joins & \((4-s)/5\) & \(4/5\) & \((4-s)/5\)\\ Four outer horizontal segments & \(14/3\) & \(14/3\) & \(14/3\)\\ Four outer vertical segments & \(6\) & \(6\) & \(6\)\\ Core charts & \cref{eq:boosts} & unchanged & unchanged\\ Core transitions & \(C_iC_j^{-1}\) & same table & new incidence\\ Collar transitions & \(A_iA_j^{-1}\) & unchanged & unchanged\\ Boundary transitions & fixed & fixed & fixed\\ Particle labels & \((\pi/2,e_0)\) & unchanged & unchanged\\ Mode & \(0\) & \(0\to1\) & \(1\)\\ Permission & \(k\ge1\) & \(k\to k-1\) & \(k-1\)\\ \bottomrule \end{tabularx} \caption{Complete affected bounded data; the fixed exterior completes the configuration. A regular patch has 18 vertices, 25 edges, and 8 faces. The event realization has 17 vertices and 24 edges. Two stored central slots give 26 length reports.} \label{tab:certificate} \end{table} \begin{table}[htbp] \centering \small \setlength{\tabcolsep}{4pt} \begin{tabular}{@{}lccccc@{}} \toprule Length & \(-1\) & \(-1/2\) & \(0\) & \(1/2\) & \(1\)\\ \midrule Active central & \(16/25\) & \(8/25\) & \(0\) & \(8/39\) & \(16/39\)\\ Arm/\(\sqrt{1921}\) & \(3/25\) & \(19/150\) & \(2/15\) & \(53/390\) & \(9/65\)\\ \(X_s\) & \(3\) & \(19/6\) & \(10/3\) & \(7/2\) & \(11/3\)\\ \(Y_s\) & \(5\) & \(51/10\) & \(26/5\) & \(53/10\) & \(27/5\)\\ \((4-s)/3\) & \(5/3\) & \(3/2\) & \(4/3\) & \(7/6\) & \(1\)\\ \((4-s)/5\) & \(1\) & \(9/10\) & \(4/5\) & \(7/10\) & \(3/5\)\\ Outer horizontal & \(14/3\) & \(14/3\) & \(14/3\) & \(14/3\) & \(14/3\)\\ Outer vertical & \(6\) & \(6\) & \(6\) & \(6\) & \(6\)\\ \bottomrule \end{tabular} \caption{Exact reference snapshots. Report words are derived from these laws, not independent exact geometric labels.} \label{tab:snapshots} \end{table} \subsection{Event germs, guard, transport, and finite code} \label{sec:finite} Let \(\cQ_-\) be the regular incoming family for \(-1\le s<0\), and \(\cQ_+\) the outgoing family for \(0