\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={VII-F-ST: A particle-supported slicing certificate with a finite spacetime Enter interface}, pdfsubject={Prescribed conical geometry, transverse incidence, destructive clocks, and causal transport} } \setlength{\emergencystretch}{3em} \setlist{topsep=5pt,itemsep=3pt} \allowdisplaybreaks \numberwithin{equation}{section} \setcounter{tocdepth}{1} \urlstyle{same} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\HH}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\AA}{\mathcal A} \newcommand{\OO}{\mathcal O} \newcommand{\MM}{\mathcal M} \newcommand{\II}{\mathcal I} \newcommand{\NN}{\mathcal N} \newcommand{\DD}{\mathcal D} \newcommand{\WW}{\mathcal W} \newcommand{\dd}{\mathrm d} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\proj}[1]{\kb{#1}{#1}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\loc}[1]{\texttt{\detokenize{#1}}} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{#1}}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\Dec}{Dec} \DeclareMathOperator{\Hol}{Hol} \DeclareMathOperator{\Area}{Area} \DeclareMathOperator{\Vol}{Vol} \newcommand{\statementbox}[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{VII-F-ST: A particle-supported slicing certificate\\ with a finite spacetime Enter interface}\\[5pt] \large Transverse three-to-six incidence, framed affine holonomy,\\ destructive clock encoding, and causal payload transport} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We construct a self-contained local Grade S adaptation of VII-F's particle-supported slicing event to I-B's prescribed \(3+1\)-dimensional conical product. The background has fixed deficit \(\pi/2\); its singular set is a worldsheet. An explicit product lift preserves VII-F's transverse three-to-six incidence change, the exact birth laws for six vertices, twelve edges, and six faces, and one common framed affine peripheral holonomy. The construction distinguishes a developing atlas around the defect from the compact regular laboratory in which all quantum operations occur. The additional contribution is a complete finite operational certificate. An exact guarded metadata update has a forward decoder, a constructive NOT--CNOT--Toffoli realization with retained history, and an explicit storage and gate bound. A destructive finite clock returns both its classical code and every unnormalized conditional payload block, including arbitrary untouched references. Four exact classical evaluations are represented by finite diagonal matrices and a completely specified table-probe instrument. The resulting package separates event-time calibration from history labeling: reading an installed event code is not acquiring information from the singular worldsheet. A buffered, target-adapted causal route transports the surviving payload, location/history data, and classical control record without changing the complete instrument. Two useful diagnostic conclusions are proved explicitly. Intrinsic lengths and areas alone cannot determine the exact event guard, even before rounding, because they are constant throughout the incoming epoch. Also, matching classical records leaves a possible residual discrepancy of trace distance \(1/2\). These features make the package a reproducible finite test case for separating geometric consistency, finite readout, and physically scheduled implementation. The classical incidence event and the general channel principles are not claimed as new in themselves. The contribution is their dimensionally explicit, supported assembly with checked interfaces. No metric emergence, singular detector, Einstein dynamics, continuum-field acquisition, quantum Ward identity, or inherited quantum response coefficient is constructed. \end{abstract} \statementbox{Exact scope}{% The metric, deficit, conical worldsheet, lapse, clock calibration, preparations, finite tables, and communication permissions are inputs. The event changes a prescribed marking and its transverse cell incidence, not the ambient spacetime. The geometric certificate may meet the stipulated singular stratum. The quantum laboratory does not. A finite clock reads a prepared encoding of the event; it does not measure an unknown continuum geometry at the defect.} \tableofcontents %===================================================================== \section{Introduction, Enter philosophy, and the adaptation contract} \label{sec:introduction} %===================================================================== VII-F constructs one explicit particle-supported slicing event on a fixed \(2+1\)-dimensional spinless cone \cite{CheungVIIF}. I-B supplies a finite Enter interface on the regular part of a prescribed \(3+1\)-dimensional conical product \cite{CheungIB}. Their combination requires more than appending an axial coordinate to a transverse diagram. The singular set changes dimension, a meridian needs an atlas rather than one regular chart, and the finite apparatus must remain separated from the singular stratum. These points are addressed explicitly below. The conical and polygonal context is classical \cite{DJH,Hooft1993}. Neither particle-supported incidence changes nor the instrument formalism are introduced here as new general ideas. The purpose is a complete certificate for one precisely declared adaptation. \subsection{The Enter interface discipline} In Cheung's Enter calculus, a runtime quantum name is an access right to a tensor factor. Destructive computational-basis readout consumes the measured names and returns classical outcomes, while unmeasured factors may survive \cite{CheungEnter,CheungI}. The accompanying philosophy separates: \[ \text{destructive readout},\qquad \text{quantum-return recovery},\qquad \text{physical implementation},\qquad \text{geometric interpretation}. \] A classical code is not the return of an unknown quantum clock; a channel is not a thermal machine; a suggestive geometric label is not a derivation of a metric \cite{CheungPhilosophy}. We adopt a specified finite Born instrument and prove its complete channel formula. We do not rederive the measurement characterization of Manuscript I from abbreviated axioms, certify an arbitrary apparatus to satisfy that manuscript's strategy class, or extend its conclusion to quantum fields. \begin{definition}[Local Grade S] \label{def:grade} A local Grade S package comprises: \begin{enumerate}[label=\textup{(S\arabic*)}] \item a prescribed Lorentzian background, a supported classical presentation, and a compact regular operational laboratory; \item an explicitly proved classical certificate on its declared geometric domain; \item a complete finite destructive-clock instrument; \item a finite encoding of already-defined evaluations and labels, with its decoder domain and resources; \item reference-preserving transport whose complete dependencies are embedded in the prescribed causal relation. \end{enumerate} Grade E would additionally require a construction deriving the geometric data from operational records. It is not constructed. \end{definition} \paragraph{Source roles.} VII-F provides the source family and certificate discipline. I-B provides the operational pattern. VII-E, VII-G, VII-E-ST, En09-A, En05-A, and VII-J provide documentary comparisons \cite{CheungOperational,CheungEn09,CheungLocality}. Every positive result used in this adaptation is proved below. A documentary separation is not a proof transfer. %===================================================================== \section{Prescribed conical product and two different supports} \label{sec:background} %===================================================================== Fix a length unit \(\ell>0\), speed \(c>0\), and time origin \(T_*\). Write \[ t=\frac{c(T-T_*)}{\ell},\qquad z=\ell\zeta,\qquad \mu=\frac{\pi}{2},\qquad \alpha=\frac34. \] The regular physical spacetime is \begin{equation} M=\RR_T\times C_\mu^\circ\times\RR_z,\qquad \bar g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2, \label{eq:metric} \end{equation} where \(r>0\) and \(\phi\) has period \(2\pi\). Its stipulated conical completion \(\overline M\) has \begin{equation} \WW=\RR_T\times\{o\}\times\RR_z. \label{eq:worldsheet} \end{equation} No smooth field equation is imposed on \(\WW\). Use a cut transverse wedge of opening \(3\pi/2\), with developed directions \[ e_i=(\cos(i\pi/2),\sin(i\pi/2)),\qquad i=0,1,2,3. \] Sector \(i\), \(i=0,1,2\), has dimensionless coordinates \[ x=\xi e_i+\eta e_{i+1},\qquad \xi,\eta\ge0. \] Physical spatial coordinates in that sector are \(\ell x\). Cone-ray labels are cyclic modulo three, but \(e_0\) and \(e_3\) are distinct cut-space copies. The upper seam is identified with the lower by the affine map \begin{equation} J=(\diag(1,R(\pi/2),1),0),\qquad R(\pi/2)= \begin{pmatrix}0&-1\\1&0\end{pmatrix}, \label{eq:J} \end{equation} in dimensionless coordinates \((t,X,Y,\zeta)\). It sends \((t,0,-r,\zeta)\) to \((t,r,0,\zeta)\). This is wedge gluing, not a quotient of the entire plane by the finite rotation group. \begin{proposition}[Local geometry] \label{prop:geometry} The signature is \((-,+,+,+)\), with future orientation \(\partial_T\). Static observers have \(\dd\tau=\dd T\). On a lifted regular angular interval, \[ \operatorname{dev}(T,r,\phi,z) =(c(T-T_*),r\cos(\alpha\phi),r\sin(\alpha\phi),z) \] is a local Minkowski isometry. A future-causal curve satisfies \[ \text{spatial path length}\le c\,\Delta T. \] The map \(J\) fixes \(\WW\) pointwise and has neither time nor axial translation. \end{proposition} \begin{proof} The first two assertions follow directly from \eqref{eq:metric}. The identity \[ \dd X^2+\dd Y^2=\dd r^2+\alpha^2r^2\dd\phi^2 \] proves local isometry. The causal inequality follows by integrating the pointwise causal-speed bound. Equation~\eqref{eq:J} proves the last assertion. \end{proof} \begin{proposition}[The regular product is not globally hyperbolic] \label{prop:nonGH} The manifold \(M\) has a noncompact causal diamond. \end{proposition} \begin{proof} Fix \(r_0>0\), an angle, and \(z=0\). Take endpoints at that spatial position and times \(T_*\pm2r_0/c\). For \(r_n\downarrow0\), the events at time \(T_*\), radius \(r_n\), and the same angle and axial coordinate lie in the diamond, using radial causal paths. They have no convergent subsequence in \(M\). \end{proof} \begin{definition}[Supported presentation] \label{def:support} Fix \(Z>0\). The geometric presentation is the finite sector atlas on \[ \{\xi+\eta\le4\}\times[-Z,Z] \] together with its prescribed exterior and its stipulated conical stratum. It is compact in the spatial completion and is not a single regular chart. The operational laboratory is a relatively compact regular tube with closure disjoint from \(\WW\), containing all installed register ports, local gates, and communication paths. All preparations, ancillary inputs, and controller data are declared at the initial boundary. There is no undeclared incoming boundary input. No Green operator, field propagation problem, or boundary-value evolution is constructed. \end{definition} The distinction between the two supports is essential. A particle-supported geometric certificate cannot have its entire geometric closure disjoint from the defect. A finite apparatus can, and in this paper does. The axial construction below is a product on the declared slab. At its axial ends the marking is the restriction of that same specified product. We do not infer a compactly supported axial interpolation to a different exterior, since such an interpolation could introduce additional incidence parameters. %===================================================================== \section{The exact transverse family and its product lift} \label{sec:family} %===================================================================== The following functions reproduce the source family \cite[\loc{eq:family}]{CheungVIIF}: \begin{equation} \rho=\xi+\eta,\qquad h=\frac23(\xi+\eta),\qquad k=\frac35\min\{\xi,\eta\}, \qquad F_s=s+\min\{2,\max\{h,s+k\}\}, \quad -1\le s\le1. \label{eq:family} \end{equation} They agree on identified rays: there \(\rho=r/\ell\), \(h=2\rho/3\), and \(k=0\). Consequently \(F_s\) is a single-valued continuous function on the completed transverse cone. Define the \(3+1\)-dimensional markings \begin{equation} \widehat\Sigma_s= \left\{(T,x,z): T=T_*+\frac{\ell}{c}F_s(x/\ell),\quad z\in\RR\right\}. \label{eq:lift} \end{equation} Only the supported slab of Definition~\ref{def:support} and the separately specified regular laboratory are operationally used. \begin{lemma}[Active region] \label{lem:active} For all transverse points, \[ h-k\ge\frac{11}{30}\rho. \] Thus: \begin{enumerate}[label=\textup{(\roman*)}] \item for \(s\le0\), \(F_s=s+\min\{2,h\}\); \item for \(s>0\), the branch \(s+k\) can dominate \(h\) only where \(\rho\le30s/11<3\); \item for every parameter and \(\rho\ge3\), \(F_s=2+s\). \end{enumerate} \end{lemma} \begin{proof} Use \(\min\{\xi,\eta\}\le\rho/2\). For \(s\le0\), \(s+k\le h\). For \(s>0\), \(s+k\ge h\) implies \(s\ge11\rho/30\). Finally \(h\ge2\) when \(\rho\ge3\). \end{proof} \begin{theorem}[Embedded, uniformly spacelike ordered lift] \label{thm:spacelike} The markings \(\widehat\Sigma_s\) are embedded graphs. Every regular affine piece is spacelike, with dimensionless spatial Lipschitz constant \[ \lambda=\frac{2\sqrt2}{3}<1. \] Their induced metrics satisfy the lower bound \[ \bar g(v,v)\ge\frac{\ell^2}{9}|v|^2 \] for dimensionless spatial tangent coordinates. For \(s_2>s_1\), \begin{equation} s_2-s_1\le F_{s_2}(x)-F_{s_1}(x) \le2(s_2-s_1). \label{eq:ordering} \end{equation} Distinct points on one regular marking are not causally related. The extension to the conical stratum is understood in the stratified Lipschitz sense, not as a smooth hypersurface theorem at \(\WW\). \end{theorem} \begin{proof} The affine spatial gradients, including the axial component, are \[ (2/3,2/3,0),\qquad (3/5,0,0),\qquad (0,3/5,0),\qquad (0,0,0), \] up to sector rotations. Taking minima and maxima preserves a common Lipschitz bound. On an affine graph of gradient \(a\), the induced metric is \(\ell^2(I-aa^{\mathsf T})\), whose least eigenvalue is at least \(\ell^2(1-\lambda^2)=\ell^2/9\). Integrate the gradient bound along spatial paths and take the infimum of their lengths. The graph projection is a homeomorphism. The strict Lipschitz bound and Proposition~\ref{prop:geometry} exclude causal pairs of distinct graph points. For fixed \(x\), the term after the initial \(s\) in \eqref{eq:family} is nondecreasing and 1-Lipschitz in \(s\), which gives \eqref{eq:ordering}. \end{proof} The radial collar \[ 7/2\le\rho\le4,\qquad |\zeta|\le Z \] is regular and horizontal at \(t=2+s\). The exterior has fixed spatial cells and fixed gluing, translated according to this prescribed equation. It is not the same embedded subset for all \(s\). The singular intersection is \begin{equation} \widehat\Sigma_s\cap\WW =\{(T_*+\ell p(s)/c,o,z):z\in\RR\},\qquad p(s)= \begin{cases}s,&s\le0,\\2s,&s\ge0.\end{cases} \label{eq:intersection} \end{equation} In a transverse section this is the source's particle point. In \(3+1\) dimensions it is a line in a worldsheet. %===================================================================== \section{Cellulation, exact measures, and the unique event parameter} \label{sec:cells} %===================================================================== All coordinates and measures in this section are dimensionless unless a power of \(\ell\) is displayed. The transverse data are adapted from VII-F, with their formulas checked directly below \cite{CheungVIIF}. \subsection{Vertices and faces} Let \[ A_i=3e_i,\qquad D_i=4e_i,\qquad B_i=\frac32s\,e_i,\qquad C_i=\frac{15}{11}s(e_i+e_{i+1})\quad(s>0). \] The complete sector vertex data are \[ \begin{array}{c|c|c} \text{vertex}&(\xi,\eta)&t\\ \hline P&(0,0)&p(s)\\ A_i&(3,0)&s+2\\ A_{i+1}&(0,3)&s+2\\ D_i&(4,0)&s+2\\ D_{i+1}&(0,4)&s+2\\ B_i&(3s/2,0)&2s\\ B_{i+1}&(0,3s/2)&2s\\ C_i&(15s/11,15s/11)&31s/11 \end{array} \] The last three entries are regular vertices only for \(s>0\). For \(s\le0\), the core faces are \[ V_i=[P,A_i,A_{i+1}],\qquad t=s+\frac23(\xi+\eta). \] For \(s>0\), replace each core triangle by \begin{align} U_i^-&=[P,B_i,C_i],& t&=2s+\frac35\eta,\nonumber\\ U_i^+&=[P,C_i,B_{i+1}],& t&=2s+\frac35\xi,\nonumber\\ V_i&=[B_i,A_i,A_{i+1},B_{i+1},C_i],& t&=s+\frac23(\xi+\eta). \label{eq:faces} \end{align} The persistent \(V_i\) is a simple, generally nonconvex polygon. For every \(s\), \[ Q_i=[A_i,D_i,D_{i+1},A_{i+1}],\qquad t=s+2, \] and the region \(\rho\ge4\) retains its prescribed horizontal exterior face and exterior ray labels. \begin{lemma}[Exact decomposition] \label{lem:decomposition} These faces partition the transverse graph without overlapping face interiors or omitted regions. \end{lemma} \begin{proof} For \(s\le0\), apply Lemma~\ref{lem:active}. For \(s>0\), in \(\xi\ge\eta\) the equality \(h=s+k\) is \[ 10\xi+\eta=15s, \] joining \(B_i\) to \(C_i\). In \(\eta\ge\xi\) it is \[ \xi+10\eta=15s, \] joining \(C_i\) to \(B_{i+1}\). The inner dominance region is exactly the union of the two triangles \(U_i^\pm\). Since \(\rho(C_i)=30s/11<3\), it is separated from the crest \(\rho=3\). The remaining core region is the stated simple pentagon, and the cap gives the horizontal collar and exterior. \end{proof} \subsection{Edges and lengths} Persistent finite edges are \[ e_i= \begin{cases}[P,A_i],&s\le0,\\[B_i,A_i],&s>0,\end{cases} \quad c_i=[A_i,A_{i+1}],\quad t_i=[A_i,D_i],\quad o_i=[D_i,D_{i+1}]. \] Born edge labels are \[ u_i=[P,B_i],\quad v_i=[P,C_i],\quad w_i^-=[B_i,C_i],\quad w_i^+=[C_i,B_{i+1}]. \] \begin{proposition}[Complete finite-edge table] \label{prop:lengths} Each row has multiplicity three. The physical lengths are \[ \begin{array}{c|c|c} \text{edge}&s\le0&s>0\\ \hline e_i&\ell\sqrt5&\ell\sqrt5(1-s/2)\\ c_i&3\ell\sqrt2&3\ell\sqrt2\\ t_i&\ell&\ell\\ o_i&4\ell\sqrt2&4\ell\sqrt2\\ u_i&\text{inactive}&3\ell s/2\\ v_i&\text{inactive}&3\ell\sqrt{41}\,s/11\\ w_i^-&\text{inactive}&3\ell\sqrt{65}\,s/22\\ w_i^+&\text{inactive}&3\ell\sqrt{65}\,s/22 \end{array} \] At \(s=0\), the born labels are germ metadata, not regular zero-length edges. \end{proposition} \begin{proof} For a developed segment, \(L^2=\ell^2(|\Delta x|^2-(\Delta t)^2)\). On \(e_i\), the spatial length is \(3\) or \(3-3s/2\), with time slope \(2/3\), giving the first row. The collar rows have \(\Delta t=0\). For \(u_i\), both endpoint times equal \(2s\). For \(v_i\), put \(b=15s/11\); its displacement is \((3b/5,b,b)\), whose squared norm is \(41b^2/25\). For \(w_i^-\), \[ (\Delta t,\Delta\xi,\Delta\eta) =\frac{s}{22}(18,-3,30), \] whose squared norm is \(585s^2/484\). The remaining edge follows by symmetry. \end{proof} \begin{proposition}[Areas and one-sided laws] \label{prop:areas} The induced transverse areas are \[ \begin{array}{c|c|c} \text{face}&s\le0&s>0\\ \hline V_i&3\ell^2/2& \ell^2(3/2-15s^2/22)\\ U_i^-&\text{inactive}&9\ell^2s^2/11\\ U_i^+&\text{inactive}&9\ell^2s^2/11\\ Q_i&7\ell^2/2&7\ell^2/2 \end{array} \] Every active face has positive area. In particular, \(\Area(V_i)\ge9\ell^2/11\). The born edges vanish to first order and the born areas to second order at \(s=0^+\). The changing line in one half-sector has Euclidean parameter-normal speed \(15/\sqrt{101}\). \end{proposition} \begin{proof} The graph-area factors are \(1/3\) for the \(h\)-plane and \(4/5\) for a \(k\)-plane. The original projected core triangle has area \(9/2\). Each new projected triangle has area \[ \frac12\frac{3s}{2}\frac{15s}{11} =\frac{45s^2}{44}. \] Subtract both from \(9/2\) and multiply by \(1/3\), or multiply one by \(4/5\), to obtain the table. The collar is horizontal and has area \((16-9)/2\). The speed follows from \(10\xi+\eta=15s\). \end{proof} The endpoint derivatives are \[ B_i'=\frac32e_i,\qquad C_i'=\frac{15}{11}(e_i+e_{i+1}). \] Before the event all finite intrinsic lengths and areas are constant. These are slicing-parameter laws, not material velocity laws. \subsection{The transverse link and the lifted event locus} \begin{theorem}[Unique transverse incidence-event parameter] \label{thm:event} The declared family has exactly one incidence-event parameter, \(s=0\). Across it: \begin{enumerate}[label=\textup{(\roman*)}] \item the six vertices \(B_i,C_i\) emerge from \(P\); \item twelve edges and six faces emerge with the laws above; \item the transverse link at the singular intersection changes from three incident faces to six. \end{enumerate} There is no other collision, vanishing active cell, or contact with the crest in the parameter window. \end{theorem} \begin{proof} The incoming cyclic link is \[ e_0,V_0,e_1,V_1,e_2,V_2. \] The outgoing cyclic link is \[ u_0,U_0^-,v_0,U_0^+,\ u_1,U_1^-,v_1,U_1^+,\ u_2,U_2^-,v_2,U_2^+. \] The bisectors are genuine creases because the adjacent gradients are \((3/5,0)\) and \((0,3/5)\). Across a cone ray, unfolding changes the sign of the transverse gradient, so those creases are also genuine. For each open epoch, the combinatorics is fixed. Positivity follows from Propositions~\ref{prop:lengths} and \ref{prop:areas}; separation and complete coverage follow from Lemma~\ref{lem:decomposition}. All born coordinates tend to the apex at zero, while the persistent cells have their incoming nondegenerate limits. The different link valences exclude a mere relabeling. \end{proof} The transverse compact patch counts are \[ \begin{array}{c|ccc|c} &V&E&F&V-E+F\\ \hline s<0&7&12&6&1\\ s>0&13&24&12&1 \end{array} \] At \(s=0\), use the incoming regular cellulation and retain the outgoing formulas only as one-sided germ data. \begin{corollary}[Dimensionally explicit lift] \label{cor:product} On \(|\zeta|0\), the loop \(\rho=s\) lies inside the new star because \(h-k\le h=2s/30\), only \(m=1\) is admitted; at zero either mode denotes its stated germ. Exact formulas are reconstructed from the fixed family identifier and finite parameter, not from rounded lengths. For \(n\) original names, reserve six current location bits and six history bits per name. Axial port coordinates and any additional spectator locations are independently declared. Actual live quantum ports in this paper lie in the regular collar; the abstract singular label is not a quantum wire on \(\WW\). \subsection{Constructive reversible realization} For each valid event word \(g\), let \(f(g)\) be its outgoing word with permission decremented. There are \(2^r-1\) such input words. \begin{theorem}[Forward decoder and exact reversible extension] \label{thm:reversible} There is a finite NOT--CNOT--Toffoli circuit whose action on the declared forward domain is \begin{equation} \ket g_G\ket0_H \ket{x_1,\ldots,x_n}_L\ket{0,\ldots,0}_{L_H} \longmapsto \ket{f(g)}_G\ket g_H \ket{x_1,\ldots,x_n}_L\ket{x_1,\ldots,x_n}_{L_H}. \label{eq:reversible} \end{equation} It acts identically on internal matter and returns its clean work to zero. A sufficient total bit-carrier reserve is \begin{equation} M_E=4B-2+12n. \label{eq:event-storage} \end{equation} A constructive upper bound on elementary NOT, CNOT, and Toffoli gates is \begin{equation} G_E\le (2^r-1)(4B-1)(8B-5)+6n. \label{eq:event-gates} \end{equation} This bound is not asserted optimal. \end{theorem} \begin{proof} On the \(2B\) geometric/current-history bits, exchange each string \((g,0)\) with \((f(g),g)\), and fix all other strings. These pairs are disjoint: the first components have different modes, and the second component of the output identifies its input. This defines a permutation. A transposition of two \(N\)-bit strings can be implemented along a Gray path of at most \(N\) edges by at most \(2N-1\) adjacent-string transpositions. Each adjacent transposition is a NOT on one target controlled on the other \(N-1\) bits. Convert negative controls with NOT gates, compute the conjunction of the controls with \(N-2\) clean work bits, flip the target, and uncompute. This uses at most \(4N-5\) elementary gates for \(N\ge2\). Set \(N=2B\) and sum over the \(2^r-1\) pairs. Finally, six CNOTs per name copy its basis-coded current location into the blank location-history block. The current and retained-history blocks require \(2B+12n\) bits, and the clean conjunction reserve is \(2B-2\). The decoder reads \(f(g)\), which is precisely the stated outgoing germ with the spent permission. \end{proof} Only the forward canonical domain is given a geometric meaning. Intermediate compiler states and off-domain strings are not asserted to be geometric configurations. The reverse permutation does not move a physical readout backward in \(T\). Retained history is not clean work. The general reversible-computation context is standard \cite{Bennett}; the construction above provides the particular finite realization needed here. \subsection{Spectator blindness and a classical--quantum lift} \begin{assumption}[Complete spectator blindness] \label{ass:blind} The guard, fixed-family validity check, geometric successor, certificate, permission decision, and original-name transport do not inspect internal matter, occupancy counts, or inaccessible spectators. Spectator location/history storage is separately reserved. The controller has no hidden spectator input. \end{assumption} \begin{theorem}[Complete spectator consistency] \label{thm:spectators} The guarded geometry-only event satisfies Assumption~\ref{ass:blind}. For an entangled extension \(\widetilde\rho_{MN}\) with \(\Tr_N\widetilde\rho_{MN}=\rho_M\), its geometric trace, certificate, permission change, and original-name locations are unchanged, and its restricted matter output is \(\rho_M\). The same restriction statement holds history block by history block when prescribed local instruments are inserted. \end{theorem} \begin{proof} The guard reads only \(m,j,\nu\) and fixed geometric validity. The transport is the same fixed label function for each original name; it does not test how many names exist. The event's internal matter channel is the identity. For an inserted selective map \(\Phi\), \[ \Tr_N[(\Phi\otimes\id_N)(X)] =\Phi(\Tr_NX), \] by expansion in matrix units of \(N\). Apply this identity along each fixed history. \end{proof} For completeness, let \(\mathsf G\) be a finite set of all admitted coded labels and retained histories, with invalid strings assigned a fixed no-action convention. If \(F\) is the complete deterministic classical successor, its hybrid channel is \[ (\mathcal T X)_h=\sum_{g:F(g)=h}X_g,\qquad X\in\bigoplus_{g\in\mathsf G}\BB(\HH_M). \] Its full-matrix lift has Kraus operators \[ \ket{F(g)}\bra g\otimes I_M. \] Their squared moduli sum to the identity, proving complete positivity and trace preservation. The lift discards off-diagonal input label blocks; it is a declared classical interface, not a quantum superposition of geometries. %===================================================================== \section{Finite destructive clock and an exact evaluation table} \label{sec:clock} %===================================================================== \subsection{The complete instrument} Let \(C\) have dimension \(D=2^m\), with calibration basis \(\{\ket{\tau_a}\}\). Let \(S\) include all surviving payload registers and let \(E\) be any untouched reference. Write \(P_a=\proj{\tau_a}\) and \(D_a=\bra{\tau_a}_C\otimes I_S\). \begin{theorem}[Complete destructive-clock identity] \label{thm:instrument} The channel implemented by a calibrated basis change followed by destructive Enter on all \(m\) clock names is \begin{equation} \MM_C(X)= \sum_a\proj a_R\otimes D_aXD_a^\dagger = \sum_a\proj a_R\otimes \Tr_C[(P_a\otimes I)X(P_a\otimes I)]. \label{eq:instrument} \end{equation} It is completely positive and trace preserving on all operators, including after tensoring with \(\id_E\). It returns the code and the unnormalized conditional payload, and \begin{equation} \Tr_R\MM_C(X)=\Tr_CX. \label{eq:discard} \end{equation} The clock has no quantum output factor. \end{theorem} \begin{proof} The Kraus operators are \(\ket a_R\otimes D_a\), and \[ \sum_aD_a^\dagger D_a =\sum_aP_a\otimes I_S=I_{CS}. \] Expand \(X=\sum_{a,b}\kb{\tau_a}{\tau_b}\otimes X_{ab}\). Both expressions in \eqref{eq:instrument} have block \(X_{aa}\). Summing the blocks proves \eqref{eq:discard}. The calculation includes a reference without change. Consumption is the declared destructive interface: after the measurement the \(C\)-names are unavailable as quantum inputs. \end{proof} The distinction between record and conditional residual is standard instrument theory \cite{DaviesLewis,Watrous}. Equation~\eqref{eq:instrument} is the same explicit finite interface compared with I-B and En09-A \cite{CheungIB,CheungEn09}, not a field-measurement theorem. \subsection{Four exact classical evaluations} Take a two-qubit clock and the four labels \[ s_0=-1,\qquad s_1=0,\qquad s_2=\frac12,\qquad s_3=1. \] At zero, the table uses the incoming regular cellulation. The outgoing zero germ is an independently retained metadata option, not six positive-area faces. The exact values are \begin{equation} \begin{array}{c|c|c|c|c|c|c} a&s_a&p(s_a)&\text{link valence}& L_e^2/\ell^2&L_u^2/\ell^2& \Area(U_i^-)/\ell^2\\ \hline 0&-1&-1&3&5&0&0\\ 1&0&0&3&5&0&0\\ 2&1/2&1&6&45/16&9/16&9/44\\ 3&1&2&6&5/4&9/4&9/11 \end{array} \label{eq:evaluation-table} \end{equation} Zeros in inactive \(u,U\) slots are reporting conventions. Their cells are not active. The persistent-face area vector is \[ (3/2,\ 3/2,\ 117/88,\ 9/11). \] The exact incidence-event indicator is \[ \epsilon=(0,1,0,0). \] Every table column \(v_a\) defines a finite diagonal matrix \[ A_v=\sum_{a=0}^3v_aP_a. \] These matrices represent already calculated values of the prescribed family. They are not acquired observables of an unknown continuum field. For example, normalize the born-face area by \(9/11\): \[ q=(0,0,1/4,1). \] After clock readout, prepare a fresh pointer in \(\sqrt{1-q_a}\ket0+\sqrt{q_a}\ket1\), conditional on the classical code \(a\), then destroy that pointer. \begin{proposition}[Exact finite table probe] \label{prop:table-probe} The resulting complete channel is \begin{equation} \mathcal P(X)= \sum_{a,y}\proj a_R\otimes\proj y_Y\otimes p_y(a)D_aXD_a^\dagger,\qquad p_1(a)=q_a,\quad p_0(a)=1-q_a. \label{eq:table-probe} \end{equation} It is trace preserving, leaves the payload unacted upon within each clock block, and satisfies \[ \Pr(Y=1\mid a)=q_a. \] The formula holds with every reference. \end{proposition} \begin{proof} Use Kraus operators \(\sqrt{p_y(a)}\ket a_R\ket y_YD_a\). Completeness is \(\sum_y p_y(a)=1\) followed by clock completeness. The probability and reference identities follow. \end{proof} On the canonical preparation \(P_1\otimes\rho_S\), the clock reports the event code with certainty. A source supplying that preparation supplies a representation promise. No inference from unknown geometry to this preparation has been constructed. \subsection{Calibration is not singular acquisition} Declare distinct physical time labels \[ \theta_a=T_*+\frac{\ell}{c}p(s_a). \] The history dictionary can associate code \(a\) with \((s_a,\theta_a,\text{table row},J,\text{germ convention})\). It does not determine the apparatus event at which a history readout is performed. \begin{proposition}[Event-time and history-label calibration] \label{prop:calibration} At one actual readout event \(e\), exact event-time matching \[ p_a>0\Longrightarrow \theta_a=T(e) \] with distinct \(\theta_a\) permits at most one outcome. If \(T(e)\) equals no \(\theta_a\), no normalized preparation satisfies that matching rule. By contrast, a multi-outcome history experiment can be read at one later fixed event and interpreted through the separate history dictionary. \end{proposition} \begin{proof} There is at most one label equal to \(T(e)\), and probabilities sum to one. The history interpretation makes no equality claim between every \(\theta_a\) and the one apparatus coordinate. \end{proof} For event-code readout choose a regular laboratory event at \(T=T_*\) and prepare \(P_1\otimes\rho_S\) there. Its coordinate equals the incidence-time label, but its spatial position is not the apex. This is preinstalled calibration, not a signal arriving from \(\WW\). A history experiment may instead be read at \(T=T_*+4\ell/c\), after all four labeled singular-intersection times. A finite-duration reversible update performed afterward operates on metadata describing the prescribed zero germ. It does not physically cause an earlier singular incidence. The incidence was already part of the fixed classical family. \begin{proposition}[No universal clock return and no finite CCR] \label{prop:clocklimits} The channel \(\MM_C\) has no left inverse on its full input space. Finite matrices cannot satisfy \([T_C,H_C]=i\hbar I_C\) for \(\hbar\ne0\). Clock records do not uniquely determine a geometric dictionary. \end{proposition} \begin{proof} The distinct clock states \[ (\ket{\tau_0}+\ket{\tau_1})/\sqrt2,\qquad (\ket{\tau_0}-\ket{\tau_1})/\sqrt2 \] with the same payload have identical destructive outputs. Taking the trace of a commutator proves the second assertion. For the third, keep the finite channel fixed and change the declared time labels or local rod-frame dictionary. Its record law is unchanged. \end{proof} The time axis \(T\), metric \(\bar g\), and worldsheet \(\WW\) are background data not removed by consuming \(C\). \subsection{Optional relational conditioning, with its actual hypotheses} The following comparison is independent of the event table. It clarifies the connection with Page--Wootters conditioning \cite{PageWootters,CheungEn09}. \begin{proposition}[Conditioning becomes evolution only with covariance] \label{prop:relational} Let finite Hermitian \(H_C,H_S\) and a state \(\rho_{CS}\) satisfy \[ [H_C\otimes I+I\otimes H_S,\rho_{CS}]=0. \] Suppose a complete orthonormal clock basis satisfies, at its declared readings \(t_a\), \[ P_a=U_C(t_a)P_0U_C(t_a)^\dagger,\qquad U_C(t)=e^{-iH_Ct/\hbar}. \] Then \[ \sigma_{S,a}=U_S(t_a)\sigma_{S,0}U_S(t_a)^\dagger,\qquad U_S(t)=e^{-iH_St/\hbar}, \] where \(\sigma_{S,a}=D_a\rho D_a^\dagger\). All reading probabilities equal \(1/D\). The same evolution statement on \(S\otimes E\) requires stationarity of the full extension, not merely its \(CS\) marginal. \end{proposition} \begin{proof} Stationarity gives \[ (U_C^\dagger\otimes I)\rho(U_C\otimes I) =(I\otimes U_S)\rho(I\otimes U_S^\dagger). \] Insert the covariant \(P_a\) into the partial-trace expression for \(\sigma_{S,a}\) and use invariance of partial trace under clock conjugation. This gives the displayed equality. Its traces are equal and sum to one. The extended proof is the same precisely when the extended stationarity hypothesis holds. \end{proof} No covariance or stationarity assertion is made for the nonuniform event-time table \eqref{eq:evaluation-table}. Extraction alone does not supply it. The external schedule is retained throughout. %===================================================================== \section{Observation and recovery separations} \label{sec:observation} %===================================================================== The geometry-only rule acts identically on matter. Appending a clock instrument is a different operation. We retain the VII-E/VII-G distinction between record and residual interfaces \cite{CheungOperational}. \begin{theorem}[Record pullback versus residual compatibility] \label{thm:pullback} Let \(T:\BB(H)\to\BB(K)\) be a channel, \(\{P_z\}\) a projective resolution, \(q\) a coarse-graining, and \(\{\II_w\}\) an instrument on \(K\). Set \[ \Pi_w=\sum_{q(z)=w}P_z,\quad \Delta_w(X)=\sum_{q(z)=w}P_zXP_z,\quad E_w=\II_w^*(I). \] The recorded maps \[ \FF(X)=\sum_w\proj w\otimes T(\Delta_w(X)),\qquad \GG(X)=\sum_w\proj w\otimes\II_w(T(X)) \] have identical record laws on every state exactly when \[ T^*(E_w)=\Pi_w. \] They agree on recorded residual observables \(\sum_w\proj w\otimes B_w\), \(B_w\in\OO\), exactly when \[ \sum_{q(z)=w}P_zT^*(B)P_z =T^*(\II_w^*(B)) \quad(w,\ B\in\OO). \] The identities persist under untouched reference extension. \end{theorem} \begin{proof} Pair the record probabilities with the input state to obtain \(\Pi_w\) and \(T^*(E_w)\). Pair the \(w\)-residual block with \(B\) to obtain the two displayed adjoints. States separate Hermitian matrices; complex linearity and tensoring give the remaining conclusions. \end{proof} \begin{corollary}[An ideal residual gap survives accurate implementation] \label{cor:residual} Identical records can conceal residual outputs of trace distance \(1/2\). If two implementations have errors \(\varepsilon_f,\varepsilon_c\) on that witness, their mutual distance is at least \[ \frac12-\varepsilon_f-\varepsilon_c. \] \end{corollary} \begin{proof} Take one coarse outcome, \(T=\id\), and \(\II_*=\id\). Fine computational projection produces dephasing. On \(\rho_+=\proj+\), the residuals are \(\rho_+\) and \(I/2\). Their difference is \(X/2\), with eigenvalues \(1/2,-1/2\). Use \(d_{\rm tr}=\norm{\cdot}_1/2\), then the triangle inequality. \end{proof} The fine residual in this comparison is a represented pointer measurement returning the qubit. It is not a destructive return of that same measured qubit name. \begin{proposition}[Informative records exclude whole-input recovery] \label{prop:informative} For a finite recorded instrument \[ \NN(X)=\sum_w\proj w_R\otimes\Psi_w(X), \] if some effect \(\Psi_w^*(I)\) is nonscalar, no channel recovers every input in the full quantum algebra from the complete record-plus-residual output. \end{proposition} \begin{proof} Choose branch Kraus operators \(K_{w,\nu}\). Kraus operators of \(\NN\) are \(A_{w,\nu}=\ket w_R\otimes K_{w,\nu}\). If a recovery channel has Kraus operators \(B_j\), then \(B_jA_{w,\nu}\) is a Kraus family for the identity. The identity channel has rank-one Choi matrix, so each such operator is a scalar multiple of \(I\). Completeness of the \(B_j\) implies \[ A_{w,\nu}^\dagger A_{v,\kappa} =\lambda_{w,\nu;v,\kappa}I. \] In particular, summing the diagonal terms at fixed \(w\) makes \(\sum_\nu K_{w,\nu}^\dagger K_{w,\nu}\) scalar, a contradiction. \end{proof} This is the standard full-code correction mechanism \cite{KnillLaflamme,Watrous}. It concerns every unknown quantum input, not one expectation or an already classical label. It does not preclude recovery on a restricted code or correct transport of the surviving payload. \subsection{Fixed-table stability and its limit} Use half-diamond normalization \(d_\diamond(\Phi,\Psi)=\norm{\Phi-\Psi}_\diamond/2\). \begin{proposition}[Exact fixed-table comparison] \label{prop:table-stability} If the table probe \eqref{eq:table-probe} is replaced by one with probabilities \(\widetilde q_a\), on the same code and payload interfaces, then \[ d_\diamond(\mathcal P_q,\mathcal P_{\widetilde q}) =\max_a|q_a-\widetilde q_a|. \] Thus numerical approximation of this fixed finite table has a complete-channel limit. \end{proposition} \begin{proof} Clock pinching gives diagonal blocks \(X_{aa}\), and the two binary-record coefficients differ by \(\pm(q_a-\widetilde q_a)\). The amplified trace-norm difference is at most \(2\max_a|q_a-\widetilde q_a|\sum_a\norm{X_{aa}}_1\). Pinching is trace-norm contractive, giving the upper bound. Input \(P_a\otimes\rho_S\) attains the bound at a maximizing index. \end{proof} This convergence is not convergence of a finite-qubit field instrument. It does not authorize replacing the exact event guard by a continuous threshold or a rounded report. Proposition~\ref{prop:no-report-guard} remains in force. %===================================================================== \section{Schedule locality and certified physical containment} \label{sec:locality} %===================================================================== Let a finite metric site set \((X,d_{\rm comp})\) carry finite quantum--classical local algebras. At layer \(j\), let the complete channel factor over a partition \(\mathcal B_j\), with \(\diam_{d_{\rm comp}}B\le\Delta\). Every classical controller register read by a block is included in that block's input interface. For terminal region \(Y\), define \[ N_{n,n}(Y)=Y,\qquad N_{j-1,n}(Y)= \bigcup\{B\in\mathcal B_j:B\cap N_{j,n}(Y)\ne\varnothing\}. \] \begin{theorem}[Typed scheduled support] \label{thm:locality} A terminal observable on \(Y\) pulls back to an observable supported on \(N_{0,n}(Y)\), and \[ N_{j,n}(Y)\subseteq \{x:d_{\rm comp}(x,Y)\le(n-j)\Delta\}. \] Input states agreeing on that backward region, jointly with any retained reference, have equal corresponding output reductions. \end{theorem} \begin{proof} The adjoint of a trace-preserving block map fixes its identity. Blocks disjoint from the current backward support therefore cannot enlarge it. A block meeting it enlarges it only inside that block. Backward induction proves support containment. Each layer adds at most one block diameter, proving the metric bound. Pairing all terminal observables proves the reduction statement, also with reference observables included. \end{proof} The same argument works for prescribed gate envelopes rather than partitions, adjoining the full input envelope whenever its output is encountered in the backward graph. It fails if a supposedly local block reads an omitted remote controller. \begin{theorem}[Influence, schedule, and physical cone] \label{thm:physical} For a complete finite network, intervention influence is contained in scheduled ancestry. If every actual input-output dependency edge has future-causal ports in \((M,\bar g)\), scheduled ancestry is contained in the prescribed metric causal relation. Neither converse follows. \end{theorem} \begin{proof} Outside the backward network, replacement trace-preserving operations act only on identity factors of the pulled-back observable and cannot change its expectation. Compose the supplied causal paths along any directed dependency path to obtain physical containment. A replacement channel can ignore a scheduled input, showing that ancestry need not be influence. A causally related pair need not have an installed communication path, showing failure of the other converse. \end{proof} This reproves the finite support pattern compared with En05-A, En03-A, and VII-J \cite{CheungLocality}. The computational metric and block certificate are additional data; the physical diameter of the particle patch is not a substitute. Scheduler consistency does not derive this conical metric. %===================================================================== \section{Buffered regular realization and exact causal transport} \label{sec:transport} %===================================================================== \subsection{An explicit regular route} Choose sector-zero spatial ports \[ x_A=\ell(15/8,15/8,0),\qquad x_B=\ell(31/16,15/8,0). \] The segment has length \[ L=\ell/16. \] It lies strictly inside the horizontal collar \(7/2<\rho<4\) and away from the seam and worldsheet. A sufficiently thin regular tube around it is a single developing chart. For any chosen integer \(n\ge1\), install sites \[ x_j=x_A+\frac jn(x_B-x_A),\qquad j=0,\ldots,n, \] each with a \(p\)-qubit payload register \(S_j\). Initialize \(S_1,\ldots,S_n\) in one fixed blank state \(\omega\). All source and receiver site reserves are included in the initial laboratory data. At hop \(j\), admit a content-exchange communication channel between \(S_{j-1}\) and \(S_j\), with duration \(L/(nc)\). Its two transfer directions have null-related ports. This is a stipulated finite communication primitive, not an instantaneous spatially extended unitary. \begin{theorem}[Reference-preserving payload transport] \label{thm:transport} After hop \(j\), an arbitrary input \(\rho_{S_0E}\) has become \[ \rho_{S_jE}\otimes \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{S_i}. \] Thus the logical payload channel is exactly the identity, including every reference. Arrival is \(T_{\rm dep}+L/c\), and all communication dependencies lie inside the prescribed causal cone. \end{theorem} \begin{proof} The first exchange moves the whole payload factor and its reference correlations into \(S_1\), leaving a blank in \(S_0\). Repeat inductively. Each hop covers distance \(L/n\) in time \(L/(nc)\), so its transfer paths are null in the developing chart and causal after pullback. \end{proof} Location/history data can be included in the transported finite payload, or forwarded as separately classical registers. A classical control record is copied only in its designated classical basis and transmitted on an explicit causal wire. \begin{corollary}[Transport of the complete recorded instrument] \label{cor:transport-instrument} If the clock is read at the sender and its record is causally forwarded with the surviving payload, the receiver obtains \[ \sum_a\proj a_{R_B}\otimes \mathcal V\!\left(D_aXD_a^\dagger\right), \] where \(\mathcal V\) is the exact relocation identity. Any record-controlled operation at the receiver is performed only after both required inputs arrive. This is the same complete instrument with relocated output. \end{corollary} \begin{proof} Apply Theorem~\ref{thm:transport} with the classical record included in the reference. Forward that jointly classical record explicitly, then apply the same conditional maps at the receiver. Equality holds blockwise, including references. \end{proof} \begin{proposition}[No instantaneous remote SWAP] \label{prop:no-instant} A zero-duration SWAP or identity transmission between the two distinct simultaneous ports is not admitted. \end{proposition} \begin{proof} Orthogonal sender inputs would produce distinguishable receiver outputs, hence intervention influence. The simultaneous distinct endpoints are spacelike and violate Theorem~\ref{thm:physical}'s causal-port condition. \end{proof} \subsection{The buffer through the incidence parameter} Store the payload at the regular sender during the complete geometric window. At this port the marking crosses its static worldline at \(t=s+2\), hence at times \(1\le t\le3\). The event at the worldsheet occurs at \(t=0\). Neither the storage channel nor its port touches the defect. For the multi-outcome history experiment, complete clock readout at \(t=4\). Run the finite metadata circuit afterward. Give every elementary local gate a fixed positive duration \(\gamma\), and, if necessary, an additional positive interval for record packaging. Choose departure after their sum. A pointer used for a finite table probe is prepared and destroyed locally, with its own positive-duration slot. No logical identity allowing deferral changes this actual timing or memory ledger. \begin{proposition}[A physically causal buffered execution exists] \label{prop:buffer} The clock, reversible metadata circuit, storage, table probe, record forwarding, and target-adapted route admit one complete finite causal schedule in the regular tube. Its decoded geometric zero-germ update remains the exact update of Theorem~\ref{thm:reversible}. \end{proposition} \begin{proof} All local gate input and output ports are on the same static worldline and separated by positive time. The source table, exact guard word, and history reserves are preinstalled. Choose the departure time after the finite local gate list. The route uses the causal hops already proved. The decoder depends on the exact stored geometric parameter, not on the later physical execution time. \end{proof} The last distinction is substantive: the circuit realizes a finite representation of the stipulated event, not an apparatus causing or observing a singular gravitational process at \(t=0\). \subsection{Resource ledger} Let \[ P_{\rm pay}=p+2B+12n_{\rm m}, \] where \(p\) is the number of internal matter qubits and \(n_{\rm m}\) the number of original names whose current and history locations are represented. Here \(2B+12n_{\rm m}\) counts the current metadata and retained history after the reversible event; clean work is not transported. For a route with \(n\) hops, a sufficient reserve is: \[ \begin{array}{l|l} \text{resource}&\text{sufficient amount}\\ \hline \text{payload site registers}&(n+1)P_{\rm pay}\text{ qubits}\\ \text{event clean work at sender}&2B-2\text{ qubits}\\ \text{finite clock}&2\text{ qubits, then consumed}\\ \text{optional table pointer}&1\text{ qubit after clock readout}\\ \text{clock record buffers}&2(n+1)\text{ classical bits}\\ \text{optional binary probe record}&1\text{ classical bit}\\ \text{physical payload exchanges}&n\\ \text{classical record forwarding}&n\text{ links}\\ \text{metadata elementary gates}& (2^r-1)(4B-1)(8B-5)+6n_{\rm m} \end{array} \] Using the clock before the metadata circuit, and the pointer after clock consumption, gives the sufficient quantum peak \begin{equation} Q_{\rm peak}\le (n+1)P_{\rm pay}+\max\{2B-2,2\}. \label{eq:peak} \end{equation} The metadata registers are basis-coded when decoded as classical geometry; their quantum-carrier implementation does not make the cone a quantum variable. A purely classical implementation can reduce the charged quantum storage but is not needed for this bound. All blanks at unused sites are counted. Spectator and external-reference holdings are separate. No claim of optimal storage, physical energy cost, Landauer saturation, or thermal reversibility is made. \begin{proposition}[Complete implementation-error ledger] \label{prop:stability} For a fixed compatible sequence of ideal channels \(\Phi_1,\ldots,\Phi_N\) and implementations \(\widetilde\Phi_1,\ldots,\widetilde\Phi_N\), \[ d_\diamond(\widetilde\Phi_N\cdots\widetilde\Phi_1, \Phi_N\cdots\Phi_1) \le \min\left\{1,\sum_{j=1}^N d_\diamond(\widetilde\Phi_j,\Phi_j)\right\}. \] Records, histories, failures, and work outputs must remain on the same comparison interfaces. \end{proposition} \begin{proof} Subtract the compositions and replace one factor at a time. Each summand has one difference of corresponding factors with channels before and after it. Diamond-norm contraction under channel composition and the triangle inequality give the bound. Half-diamond distance between channels is at most one. \end{proof} This finite error estimate is not a continuum limit, a singular field estimate, or a uniform estimate after normalization of arbitrarily rare branches. %===================================================================== \section{Assembly and precise release} \label{sec:release} %===================================================================== \begin{theorem}[VII-F-ST local Grade S release] \label{thm:release} For the declared inputs and finite permissions, the following package exists: \begin{enumerate}[label=\textup{(\roman*)}] \item the prescribed conical \(3+1\) background, its singular worldsheet, finite developing atlas, and compact regular laboratory; \item an embedded uniformly spacelike product marking with exactly one transverse incidence-event parameter, changing the normal transverse particle link from three faces to six; \item exact coordinates, lengths, areas, one-sided laws, and nondegeneracy checks for every transverse cell in the compact certificate; \item one common positively primitive, reference-framed affine peripheral holonomy on the collar, core, outgoing star, and both zero germs; \item an exact geometry-only guard, spectator-blind location metadata transport, and a decoder-correct finite reversible implementation with explicit storage and gate bounds; \item a complete finite destructive-clock instrument, an exact four-row evaluation and table-probe package, and explicit event-time/history-label separation; \item a buffered causal execution and exact reference-preserving transport of the finite payload and its classical control record. \end{enumerate} These assertions satisfy Definition~\ref{def:grade}. They do not construct Grade E. \end{theorem} \begin{proof} Propositions~\ref{prop:geometry} and \ref{prop:nonGH}, together with Definition~\ref{def:support}, give (i). Theorem~\ref{thm:spacelike}, Theorem~\ref{thm:event}, and Corollary~\ref{cor:product} give (ii). Section~\ref{sec:cells} gives (iii). Theorems~\ref{thm:word} and \ref{thm:holonomy} give (iv). Section~\ref{sec:encoding} gives (v). Theorem~\ref{thm:instrument}, Proposition~\ref{prop:table-probe}, and Proposition~\ref{prop:calibration} give (vi). Section~\ref{sec:transport} gives (vii). The five Grade S clauses have therefore been supplied. No extraction of the background geometry from records occurs. \end{proof} \subsection{What is new in the adaptation} The source's transverse kinematic event is retained, not renamed as a new gravitational process. The additional result is an end-to-end finite certificate with: \begin{itemize} \item an explicit dimension correction from worldline to worldsheet; \item separate geometric and operational supports; \item complete selective-map and reference identities; \item an exact finite guarded decoder with a constructive gate bound; \item a finite evaluation instrument and complete-channel error ledger; \item a regular buffered causal realization with charged route resources. \end{itemize} The package can be used as a finite consistency benchmark for later proposed field acquisition or dynamics interfaces. It does not supply those missing interfaces. \subsection{Necessary limitations of the geometric claim} The admitted faces may be nonconvex and regular vertices need not be trivalent. This certificate is not an admission theorem for every stricter polygon calculus. The simultaneous three-to-six expansion is not claimed generic in the space of all slicings, and is not identified with one universal elementary transition catalogue. In \(3+1\) dimensions there is one event \emph{parameter} and an axial event locus, not an isolated singular event point. Only a transverse section has the finite vertex/edge/face counts. The axial product is stated explicitly; no compact axial taper or additional event-free interpolation is inferred. \subsection{Documentary comparisons, not incorporated premises} The supplied source comparisons have the following roles: \begin{itemize} \item I-B supplies the finite spacetime-interface pattern; En09-A supplies the conditioning/dynamics distinction \cite{CheungIB,CheungEn09}. \item VII-E, VII-G, and VII-E-ST distinguish records, residual instruments, and whole-input recovery. They do not derive this cone or turn slicing selection into gravitational dynamics \cite{CheungOperational}. \item VII-J, En03-A, and En05-A distinguish scheduled support, intervention influence, and independently certified physical causality \cite{CheungLocality}. \item VI-B and VII-A-ST are supported Grade S presentation comparisons. Their classical descent, source equations, and plateau calculations are not imported \cite{CheungSupported}. \item VIII-A/B/C concern preconstructed conical packets and separate source interfaces. Their packet families do not select this deficit, and a promised source token is not an acquisition instrument \cite{CheungVIII}. \item The OpenAI catalogue is attributed to OpenAI. It supplies no formal premise, cone, field instrument, clock, or verification of this adaptation \cite{OpenAIRecord}. \end{itemize} \subsection{Standing analytical and physical boundaries} The supplied XIV-H historical H2 ending is not discharged here. It is distinct from XIV-I's constructed internal I2 amplitude. That internal amplitude is not the complete characteristic operation still missing at I3/J2 \cite{CheungAnalytic}. No joint characteristic map, complete quantum remainder, transverse extension, singular contact compatibility theorem, output wavefront admission, or inherited quantum coefficient is constructed. The downstream quantum-admission exit called XV-B is not opened; this does not deny the existence of technical manuscripts with that identifier. XVI-N and its universal obligations are not discharged. Within the stated XVI-O prescription, \[ \mathrm{O.1}=\loc{NOT_CLOSED}. \] Separately scoped revisions using enlarged prescriptions do not update that specific statement or admit the inherited insertion \cite{CheungStops}. The quantum certificate remains \loc{RG-open}: no quantum renormalization separator is proved by this finite package. \statementbox{Standing refusals}{% Not spacetime emergence, Grade E, Einstein dynamics from records, a sourced Einstein--scalar solution, a global conical Cauchy/Green theorem, or an intrinsic conical field theory. Not a cosmological bounce, ultraviolet completion, radiative coefficient, quantum Ward/BRST/BV identity, Page curve, island, interior, vacuum-energy stability theorem, or gravity-generated-entanglement test. Not quantized relational time without external scheduling. Record compatibility is not residual agreement. An informative classical record is not recovery of every unknown quantum input. Matter-controlled slicing is not matter-driven gravitational dynamics. The clock encoding is not the geometric time axis. The event code is not a detector at the worldsheet. No inherited quantum coefficient is admitted.} %===================================================================== \appendix \section{Finite certificate checklist} \label{app:checklist} %===================================================================== \begin{center} \small \renewcommand{\arraystretch}{1.18} \begin{tabularx}{\linewidth}{@{}p{.28\linewidth}X@{}} \toprule Certificate item & Explicit witness\\ \midrule Prescribed deficit and spinless gluing & \(\mu=\pi/2\), \(J=(\diag(1,R(\pi/2),1),0)\).\\ Seam-compatible graph & \(\rho,h,k\) agree on each identified ray.\\ Uniform spacelike margin & \(1-\lambda^2=1/9\).\\ Strict ordering & \(1\le \Delta F/\Delta s\le2\).\\ Core--crest separation & \(\rho(C_i)\le30/11<3\).\\ Nonvanishing persistent face & \(\Area(V_i)\ge9\ell^2/11\).\\ Born edges & \(3\ell s/2,\ 3\ell\sqrt{41}s/11,\ 3\ell\sqrt{65}s/22\).\\ Born faces & \(9\ell^2s^2/11\), six copies.\\ Event guard & Exact \(m=0,\ j=0,\ \nu\ge1\), with canonical validity.\\ Primitive winding & Angular lift increment \(3\pi/2\), separately from \(J\).\\ Full affine matching & \(\mathscr A^{-1}(\mathscr A J\mathscr A^{-1}) \mathscr A=J\).\\ Code space & 55 transverse labels in six bits; exact dyadic parameter.\\ Event reserve & \(4B-2+12n_{\rm m}\), \(B=24b+d+r+85\).\\ Clock instrument & Kraus family \(\ket a_R(\bra{\tau_a}\otimes I)\).\\ Finite acquisition boundary & Known table input; no continuum-to-code extraction map.\\ Regular route & Collar segment of length \(\ell/16\).\\ Complete logical transport & \(\id\) on payload, including untouched references.\\ Physical timing & Positive local gate durations and \(n\) causal hops.\\ \bottomrule \end{tabularx} \end{center} \section{Optional registry-facing scope record} The following text is a proposed documentary record, not an assertion that any external registry or user interface was updated. \begin{verbatim} identifier: VII-F-ST grade: local Grade S problemConnectionsFor: ["time"] emergenceStatus: not constructed quantumCertificate: RG-open relationshipRole: separation relationshipDistinction: A documentary separation is not a proof transfer. scope: A supported classical transverse incidence certificate, its explicit product lift, and a finite destructive-clock encoding with reference-preserving causal transport. standingBoundaries: O.1 remains NOT_CLOSED in the stated XVI-O prescription. The downstream XV-B quantum-admission exit is not opened. XVI-N is not discharged. Historical H2 and full I3/J2 records are not discharged. No inherited quantum coefficient is admitted. \end{verbatim} Suitable emergence-walk wording is: \begin{quote} VII-F-ST realizes a finite readout and transport interface for one prescribed conical slicing certificate. It does not derive the cone, change the worldsheet, or measure an unknown singular geometry. \end{quote} %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Authorship.} The author field is intentionally empty for anonymous review. Authorized author and affiliation metadata are required for a signed submission. Citation of N.~Cheung, registry attribution, or citation of OpenAI does not assign authorship of this adaptation. \paragraph{Provenance.} The classical formulas are adapted from the identified VII-F source and reproved on the stated presentation. The I-B and companion records identify interface patterns and scope distinctions, not unprinted theorem premises. Some comparisons are available only through the version-specific passages supplied with the problem; no independent verification of their complete programmes is asserted. \paragraph{Verification and implementation.} This is one standard pdfLaTeX source with an internal bibliography. No successful pdfLaTeX execution, proof-assistant verification, numerical execution, physical circuit compilation, experiment, independent referee certification, or journal acceptance of this adaptation is asserted. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{CheungVIIF} N.~Cheung, \emph{A particle-supported slicing event on a fixed spinless cone. Manuscript VII-F}. \loc{VII-F.tex}, version 1, Enter Calculus Companion, unpublished manuscript. Relevant locators include \loc{eq:family}, \loc{prop:spacelike}, \loc{prop:lengths}, \loc{prop:areas}, \loc{thm:event}, \loc{prop:holonomy}, \loc{eq:Grule}, \loc{thm:certificate}, \loc{prop:particle-defects}, and \loc{prop:angle-audit}. \url{https://enter.grok.me/browse/VII-F}. \bibitem{CheungIB} N.~Cheung, \emph{I-B-v1: A Finite Enter Spacetime: Calibrated Destructive Time Readout on a Prescribed Conical Product}. \loc{I-B-v1.tex}, version 1, Enter Calculus Companion, supplied unpublished manuscript. The spacetime geometry and finite clock/transport permissions are prescribed inputs. \url{https://enter.grok.me/browse/I-B}. \bibitem{CheungEnter} N.~Cheung, ``The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,'' \emph{Frontiers in Physics} \textbf{14} (2026), article 1954943. Accepted 28 September 2026; the consulted publisher record identifies the final formatted version as forthcoming. DOI: \doi{10.3389/fphy.2026.1954943}. \bibitem{CheungI} N.~Cheung, \emph{Manuscript I}. \loc{I.tex}, version 2, Enter Calculus Companion, supplied unpublished manuscript. Finite measurement characterization within its stated strategy class and five requirements. \url{https://enter.grok.me/browse/I}. \bibitem{CheungPhilosophy} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2. Introductory essay, Enter Calculus Companion. \url{https://enter.grok.me/enter}. \bibitem{DJH} S.~Deser, R.~Jackiw, and G.~'t~Hooft, ``Three-dimensional Einstein gravity: Dynamics of flat space,'' \emph{Annals of Physics} \textbf{152} (1984), 220--235. DOI: \doi{10.1016/0003-4916(84)90085-X}. Classical conical context only; no \(3+1\)-dimensional source equation is imported. \bibitem{Hooft1993} G.~'t~Hooft, ``The evolution of gravitating point particles in \(2+1\) dimensions,'' \emph{Classical and Quantum Gravity} \textbf{10} (1993), 1023--1038. DOI: \doi{10.1088/0264-9381/10/5/019}. Polygon-transition context, not a premise of this construction. \bibitem{DaviesLewis} E.~B.~Davies and J.~T.~Lewis, ``An operational approach to quantum probability,'' \emph{Communications in Mathematical Physics} \textbf{17} (1970), 239--260. DOI: \doi{10.1007/BF01647093}. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author's book interface: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. \bibitem{Bennett} C.~H.~Bennett, ``Logical reversibility of computation,'' \emph{IBM Journal of Research and Development} \textbf{17} (1973), 525--532. DOI: \doi{10.1147/rd.176.0525}. \bibitem{PageWootters} D.~N.~Page and W.~K.~Wootters, ``Evolution without evolution: Dynamics described by stationary observables,'' \emph{Physical Review D} \textbf{27} (1983), 2885--2892. DOI: \doi{10.1103/PhysRevD.27.2885}. \bibitem{KnillLaflamme} E.~Knill and R.~Laflamme, ``Theory of quantum error-correcting codes,'' \emph{Physical Review A} \textbf{55} (1997), 900--911. DOI: \doi{10.1103/PhysRevA.55.900}. \bibitem{CheungEn09} N.~Cheung, \emph{En09-A: Realization of Relational Time by Destructive Enter on a Clock Register}. \loc{En09-A.tex}, version 1, Enter Calculus Companion, unpublished manuscript. Complete conditioning and the separate sampled-covariance and stationarity hypotheses. \url{https://enter.grok.me/browse/En09-A}. \bibitem{CheungOperational} N.~Cheung, \emph{VII-E}, \loc{VII-E.tex}, version 1; \emph{VII-G}, \loc{VII-G.tex}, version 1; and \emph{VII-E-ST}, \loc{VII-E-ST.tex}, version 1. Enter Calculus Companion, supplied unpublished source records. Observation boundaries, finite hybrid channels, and the record/residual/recovery distinctions. \url{https://enter.grok.me/browse/VII-E}; \url{https://enter.grok.me/browse/VII-G}; \url{https://enter.grok.me/browse/VII-E-ST}. \bibitem{CheungLocality} N.~Cheung, \emph{VII-J}, \loc{VII-J.tex}, version 1; \emph{En03-A}, \loc{En03-A.tex}, version 1; and \emph{En05-A}, \loc{En05-A.tex}, version 1. Enter Calculus Companion, supplied unpublished source records. Scheduled ancestry, intervention influence, causal embedding, and finite transport are separately specified. \url{https://enter.grok.me/browse/VII-J}; \url{https://enter.grok.me/browse/En03-A}; \url{https://enter.grok.me/browse/En05-A}. \bibitem{CheungSupported} N.~Cheung, \emph{VI-B}, \loc{VI-B-v1.tex}, version 1; and \emph{VII-A-ST}, \loc{VII-A-ST.tex}, version 1. Enter Calculus Companion, supplied unpublished manuscripts. Supported Grade S presentation comparisons only. \url{https://enter.grok.me/browse/VI-B}; \url{https://enter.grok.me/browse/VII-A-ST}. \bibitem{CheungVIII} N.~Cheung, \emph{VIII-A}, \emph{VIII-B}, and \emph{VIII-C}, version 3. Enter Calculus Companion, supplied unpublished source records. Fixed-presentation conical packets, source promises, and matter--geometry interface distinctions. No deficit-selection or dynamical law is transferred. \url{https://enter.grok.me/browse/VIII-A}; \url{https://enter.grok.me/browse/VIII-B}; \url{https://enter.grok.me/browse/VIII-C}. \bibitem{CheungAnalytic} N.~Cheung, \emph{XIV-H}, version 1; \emph{XIV-I}, version 1; \emph{XIV-J}, version 1; and technical \emph{XV-B}, version 2. Enter Calculus Companion, supplied unpublished source records. Historical H2, constructed internal I2, unresolved full I3/J2, and reconstruction/extension interfaces are distinct. \url{https://enter.grok.me/browse/XIV-H}; \url{https://enter.grok.me/browse/XIV-I}; \url{https://enter.grok.me/browse/XIV-J}; \url{https://enter.grok.me/browse/XV-B}. \bibitem{CheungStops} N.~Cheung, \emph{XVI-N}, revision r12; \emph{XVI-O}, revision o70-sync; and \emph{XVI-Q}, release Q11. Enter Calculus Companion, supplied unpublished source records. The version-specific inherited-insertion and XVI-O boundaries are not updated by separately scoped enlarged-prescription subsector statements. \url{https://enter.grok.me/browse/XVI-N}; \url{https://enter.grok.me/browse/XVI-O}; \url{https://enter.grok.me/browse/XVI-Q}. \bibitem{OpenAIRecord} OpenAI, \emph{OpenAI catalogue}. Separate documentary record supplied with the problem and recorded by the Enter Calculus Companion, 2026. Not a premise of this manuscript; no catalogue-family proof, formalization claim, or verification status is imported. \url{https://enter.grok.me}. \end{thebibliography} \end{document}