%===================================================================== % II-B.tex % % Finite contact-span certificates on a declared conical Enter spacetime % % Build: % pdflatex -interaction=nonstopmode -halt-on-error II-B.tex % pdflatex -interaction=nonstopmode -halt-on-error II-B.tex % % Self-contained: % - no external figures; % - no external bibliography database; % - no shell escape; % - the optional Python listing is not executed by LaTeX. % % Verification disclosure: % This source has not been compiled or its checker executed in this % interface. No laboratory implementation, proof-assistant verification, % independent vertex implementation, or journal acceptance is claimed. % % Principal corrections incorporated: % - stochastic history labels are separated from event-time readings; % - a two-qubit clock has four completely specified outcomes; % - Enter consumes clock names, not the geometric time coordinate; % - flat-space vertex labels are not global conical scattering states; % - a response-table probe is not a field-theoretic measurement bridge; % - a linear contact fit is not a convex mixture of instruments; % - remote record-dependent control requires a causal classical wire; % - contact coefficients remain constant across external labels; % - exact null transport uses a target-adapted endpoint-channel contract; % - technical XV-B manuscripts are distinguished from quantum admission. % % Directly accessed source records and journal metadata were consulted % on 16 October 2026. Other companion references are identified as % supplied source records, not independently verified publications. %===================================================================== \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{xcolor} \usepackage{listings} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={II-B: Finite contact-span certificates on a declared conical Enter spacetime}, pdfauthor={Ngo Cheung}, pdfsubject={A finite operational adaptation with prescribed geometry} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{3pt} \setlist{itemsep=3pt,topsep=5pt} \allowdisplaybreaks \numberwithin{equation}{section} \setcounter{secnumdepth}{2} \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} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\RR}{\mathbb R} \newcommand{\QQ}{\mathbb Q} \newcommand{\Cplx}{\mathbb C} \newcommand{\NN}{\mathbb N} \newcommand{\HH}{\mathcal H} \newcommand{\XX}{\mathcal X} \newcommand{\VV}{\mathcal V} \newcommand{\CC}{\mathcal C} \newcommand{\OO}{\mathcal O} \newcommand{\dd}{\mathrm d} \newcommand{\eps}{\epsilon} \newcommand{\EH}{\mathrm{EH}} \newcommand{\PF}{\mathrm{PF}} \newcommand{\sym}{\mathrm{sym}} \newcommand{\SEP}{\mathrm{SEP}} \newcommand{\GradeS}{\mathsf{Grade\ S}} \newcommand{\GradeE}{\mathsf{Grade\ E}} \newcommand{\Mclock}{\mathcal M_C} \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{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{#1}}} \newcommand{\SourceLabel}[1]{\texttt{\detokenize{#1}}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Contr}{Contr} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\Mat}{Mat} \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 } \lstset{ language=Python, basicstyle=\ttfamily\footnotesize, columns=fullflexible, keepspaces=true, breaklines=true, showstringspaces=false, frame=single, rulecolor=\color{black!30}, keywordstyle=\color{blue!50!black}, commentstyle=\color{black!55}, stringstyle=\color{green!35!black}, tabsize=4 } \title{\textbf{II-B: Finite contact-span certificates\\ on a declared conical Enter spacetime}\\[3pt] \large A Grade S operational adaptation} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We give a finite operational adaptation of the contact-span certificate of Manuscript II on the prescribed conical spacetime of I-B. The classical background, the flat-space response prescription, and the contact class remain separate inputs. We reproduce the finite algebraic proof: the three-label target is \(m_S=(0,16/3,16/3)^T\), the complete redundant catalogue of \(15120\) parity-even two-derivative quartic contacts has evaluated rank \(2\), two original columns have minor \(64\), and the constant coefficients \(2/3,2/3\) fit the target. A complete destructive two-qubit clock instrument supplies a classical code. An explicit dictionary assigns its four outcomes to the three external labels, with the fourth outcome completely specified rather than discarded. We prove preservation of the finite certificate under this dictionary. We also construct a bounded finite response-table probe whose centered binary instruments realize the evaluated response space on arbitrary reference-entangled inputs. This is a lookup-table realization, not a measurement construction for a gravitational field. A target-adapted SWAP chain transports the surviving payload with exact reference preservation and a finite resource ledger. Complete scheduled dependencies lie inside the prescribed metric cone. The resulting model satisfies the explicitly defined Grade S contract. The finite fit remains \(\SEP\)-finite: no global contact identity, derived geometry, sourced Einstein dynamics, continuum field limit, or inherited quantum coefficient is obtained. \end{abstract} \statementbox{Exact release}{% The construction consists of: \[ \boxed{ \begin{gathered} \text{prescribed conical geometry} +\text{complete destructive clock instrument} +\text{declared dictionary},\\ N_S\alpha_0=m_S,\qquad \rank N_S=\rank[N_S\mid m_S]=2,\\ \text{an exact finite table-probe realization} +\text{reference-preserving causal endpoint transport}. \end{gathered}} \] Here \(\GradeS\) is a contract defined in this paper, not an external accreditation. The algebraic status is \(\SEP\)-finite.} \statementbox{Interpretation boundary}{% Enter consumes a finite clock encoding, not the geometric time axis. A clock record becomes a contact label only after the dictionary is declared. A table-probe residual is not a stress tensor. The flat-space contact catalogue is not silently replaced by a curved-background or weighted catalogue.} \tableofcontents %===================================================================== \section{Introduction and the operational contract} \label{sec:introduction} %===================================================================== \subsection{Purpose} Manuscript II solves an exact finite contact-membership problem for one two-species massless spin-two tree \cite{CheungII}. Its conclusion concerns response functions evaluated on three specified external labels. I-B supplies a different object: a finite experiment with destructive clock readout on a prescribed conical Lorentzian product \cite{CheungIB}. The present paper constructs an explicit interface between these objects. It does not replace the action of II by an action on the cone. In particular, it does not assert that the globally defined plane waves or propagator of the flat-space calculation are scattering data on a conical spacetime. The interface has three parts: \begin{enumerate}[label=\textup{(\roman*)}] \item a classical dictionary from a clock code to an external label; \item a pullback of the already defined response functions along that dictionary; \item a finite operational probe realizing the resulting numerical table as a bounded response signature. \end{enumerate} The third part makes the operational model nonempty without pretending to construct a gravitational apparatus. \subsection{Enter as an interface discipline} Cheung's Enter calculus treats quantum names as access rights and destructive readout as an operation that consumes the measured names while returning a classical result \cite{CheungEnter,CheungI}. The introductory philosophy distinguishes destructive readout, quantum recovery, implementation, and geometric identification \cite{CheungPhil}. We retain that discipline. Agreement of record probabilities is weaker than agreement of instruments; a channel alone does not specify a thermal machine; and naming a register ``clock'' does not identify it with a continuum coordinate. We adopt finite Born and projective-update semantics. We do not rederive the separate measurement-characterization theorem from an abbreviated hypothesis list. The instrument terminology is consistent with the operational framework of Davies and Lewis and with finite completely positive maps \cite{DaviesLewis,Choi}. Every channel identity used below is proved directly. \subsection{Definition of the grade used here} \begin{definition}[Grade S operational adaptation] \label{def:grade} A \(\GradeS\) adaptation in this paper consists of: \begin{enumerate}[label=\textup{(S\arabic*)}] \item a prescribed Lorentzian background and a regular local rod chart; \item a finite named experiment, specified accessible algebras, a complete instruction tree, and an explicit resource ledger; \item a complete destructive clock instrument and a declared record, rod, and causal dictionary; \item exact evaluation of the frozen finite response certificate, together with a stated operational realization of its numerical signature; \item a reference-preserving transport channel and a proof that all admitted scheduled dependencies are causally embedded. \end{enumerate} The contract does not require the geometry, the response prescription, or the contact class to be reconstructed from records. \end{definition} The stronger phrase \(\GradeE\) will denote the unconstructed emergence target in which suitable operational data determine the relevant geometric and dynamical structures, with the comparison and uniqueness obligations explicitly supplied. Merely satisfying Definition~\ref{def:grade} does not satisfy that target. \subsection{Source roles} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.23\linewidth}X@{}} \toprule Source & Role and limit\\ \midrule II, version 2 \cite{CheungII} & Frozen classical response problem. Its finite certificate is reproved here.\\ \addlinespace I-B, version 1 \cite{CheungIB} & Prescribed conical model, calibration distinction, destructive instrument, and target-adapted transport architecture.\\ \addlinespace En09-A \cite{CheungEn09} & Clock conditioning and the separation of record, residual, and quantum return.\\ \addlinespace En03-A; En05-A \cite{CheungEn03,CheungEn05} & Declared access, spectator-blind control, and scheduled ancestry as an influence bound.\\ \addlinespace En08-B \cite{CheungEn08} & Comparison with transport approximations on a prescribed lattice family. No lattice resource formula is transferred here.\\ \addlinespace VII-F; VIII-B \cite{CheungCone} & Documentary conical-carrier comparisons, not transferred cell counts or source laws.\\ \addlinespace XIX-A; Em02-E; XX-A \cite{CheungXIX,CheungEm02E,CheungXXA} & Separation of time objects and an optional classical lapse-composition comparison.\\ \bottomrule \end{tabularx} \end{center} No companion theorem is needed as an unstated premise. The finite algebraic and operational assertions used in the release are defined and proved below. %===================================================================== \section{The prescribed conical geometry} \label{sec:geometry} %===================================================================== \subsection{Regular part and completion} Fix \[ 0<\mu<2\pi,\qquad \alpha=1-\frac{\mu}{2\pi},\qquad c>0. \] Let \[ C_\mu^\circ=(0,\infty)\times(\RR/2\pi\mathbb Z) \] with spatial metric \[ h_C=\dd r^2+\alpha^2r^2\dd\phi^2. \] Its metric completion has an apex \(o\). Set \[ M=\RR_T\times C_\mu^\circ\times\RR_z, \qquad \overline M=\RR_T\times C_\mu\times\RR_z, \] and prescribe \begin{equation} g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2. \label{eq:metric} \end{equation} The singular completion locus is \[ \mathcal W=\RR_T\times\{o\}\times\RR_z. \] It is a worldsheet. Fixing \(z\) gives one worldline. We work on the smooth regular part \(M\). We do not assign an ordinary pointwise curvature tensor at the apex. This is the prescribed-background convention of I-B \cite{CheungIB}. \begin{lemma}[Local rods, time orientation, and causal segments] \label{lem:local-causal} There is a regular spatial neighborhood \(U\) with a developing chart \[ \chi:U\longrightarrow V\subset\RR^3 \] such that \(V\) is convex and \[ g|_{\RR\times U}=-c^2\dd T^2+\sum_{a=1}^3(\dd x^a)^2, \qquad x=\chi(r,\phi,z). \] Choose \(\partial_T\) future directed. Then: \begin{enumerate}[label=\textup{(\roman*)}] \item a future-causal tangent \(v=a\partial_T+w\ne0\) satisfies \[ a>0,\qquad |w|_h\le ca; \] \item events \((T,x)\), \((T+\Delta T,x')\) in this chart are joined by a future-causal straight segment whenever \[ \Delta T>0,\qquad \norm{x'-x}_2\le c\Delta T; \] \item along a static observer, \(\dd\tau=\dd T\). \end{enumerate} \end{lemma} \begin{proof} On a sufficiently small lifted angular sector put \[ x^1=r\cos(\alpha\phi),\qquad x^2=r\sin(\alpha\phi),\qquad x^3=z. \] Its spatial metric is Euclidean. Restrict to the inverse image of a sufficiently small convex ball avoiding the apex and the sector boundary. The causal inequality follows from \[ g(v,v)=-c^2a^2+|w|_h^2\le0. \] The straight segment has constant developed velocity \((x'-x)/\Delta T\), proving the second assertion. For a static curve, \(g=-c^2\dd T^2\), proving the third. \end{proof} The natural orthonormal frame on this patch is \[ e_0=c^{-1}\partial_T,\qquad e_a=\partial_{x^a}. \] Thus the auxiliary Minkowski convention used below is compatible with the local frame: \[ \eta=\diag(-1,1,1,1). \] This compatibility identifies tensor components at declared ports. It does not construct global field solutions on the cone. \subsection{Geometry remains input data} The circumference of \(r=R\) is \(2\pi\alpha R\). Hence the global deficit is \(\mu\). Locally, every sufficiently small regular patch is flat. Neither fact is inferred from a quantum record. No source map, metric-update rule, or matter action on \((M,g)\) is included in the geometric specification. In particular, the two species of the following algebraic problem do not become two dynamically coupled metrics on \(M\). %===================================================================== \section{The frozen response problem} \label{sec:algebra} %===================================================================== \subsection{Auxiliary flat-space domain} All response functions in this section are defined on an auxiliary flat-space domain \(\XX\). Fourier waves are \(e^{ik\cdot x}\), all momenta are incoming, and index operations use \(\eta\). A label specifies: \begin{enumerate}[label=\textup{(\roman*)}] \item species \(J_i\in\{1,2\}\), \(i=1,\ldots,4\); \item nonzero real null momenta satisfying \[ \sum_i k_i=0,\qquad s=(k_1+k_2)^2,\quad t=(k_1+k_3)^2,\quad u=(k_1+k_4)^2,\qquad stu\ne0; \] \item one replaced leg \(j\) and a real vector \(\xi_j\); \item real symmetric physical tensors \(\eps_i\), \(i\ne j\), and timelike references \(t_i\), satisfying \[ k_i^\mu\eps_{i\mu\nu}=0,\qquad t_i^\mu\eps_{i\mu\nu}=0,\qquad \eta^{\mu\nu}\eps_{i\mu\nu}=0. \] \end{enumerate} The replaced tensor is \begin{equation} G_{\mu\nu}(k,\xi)=k_\mu\xi_\nu+k_\nu\xi_\mu. \label{eq:G} \end{equation} No condition \(k\cdot\xi=0\) is imposed. Simultaneous permutations of all external data are allowed. These are formal vertex-evaluation labels. Their later assignment to local rods does not assert that they are compact wave packets, conical asymptotic states, or apparatus preparations. \subsection{Pauli--Fierz normalization} For one species define \begin{align} S_2[h]=\int\dd^4x\,\bigg( &-\frac12\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu} +\partial_\mu h^{\mu\nu}\partial^\rho h_{\rho\nu} \nonumber\\ &-\partial_\mu h^{\mu\nu}\partial_\nu h +\frac12\partial_\lambda h\,\partial^\lambda h \bigg), \qquad h=\eta^{\mu\nu}h_{\mu\nu}. \label{eq:PF} \end{align} This is the massless spin-two quadratic normalization \cite{FierzPauli}. Its Hessian is \begin{align} (K_0h)_{\mu\nu} ={}&\Box h_{\mu\nu} -\partial_\mu\partial^\rho h_{\rho\nu} -\partial_\nu\partial^\rho h_{\rho\mu} +\partial_\mu\partial_\nu h \nonumber\\ &+\eta_{\mu\nu} \left(\partial^\rho\partial^\sigma h_{\rho\sigma}-\Box h\right). \label{eq:K0} \end{align} Use the fixed propagator coefficient \[ D(p)=-\frac{P}{p^2}, \qquad P_{\mu\nu,\rho\sigma} =\frac12\left( \eta_{\mu\rho}\eta_{\nu\sigma} +\eta_{\mu\sigma}\eta_{\nu\rho} -\eta_{\mu\nu}\eta_{\rho\sigma}\right). \] \begin{lemma}[The inverse needed by the tree] \label{lem:inverse} For \(p^2\ne0\) and \(p^\mu J_{\mu\nu}=0\), \[ K_0(p)D(p)J=J. \] Also \(K_0G(k,\xi)=0\). \end{lemma} \begin{proof} Write \(J_{\rm tr}=\eta^{\mu\nu}J_{\mu\nu}\). For \(h=DJ\), \[ h_{\mu\nu} =-\frac{J_{\mu\nu}-\eta_{\mu\nu}J_{\rm tr}/2}{p^2}, \quad h=\frac{J_{\rm tr}}{p^2}, \quad p^\mu h_{\mu\nu}=\frac{p_\nu J_{\rm tr}}{2p^2}. \] Substitution into the Fourier transform of \eqref{eq:K0} cancels the trace and longitudinal terms. The remainder is \(J\). Direct substitution of \eqref{eq:G} proves the second identity. \end{proof} No inverse identity on arbitrary nonconserved tensors is used. \subsection{Einstein coefficients and species} Fix the curvature sign and coefficients by the first-derivative density \begin{equation} \frac{2}{\kappa^2}\sqrt{-g}\,g^{\mu\nu} \left( \Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\alpha} -\Gamma^\alpha_{\mu\nu}\Gamma^\beta_{\alpha\beta} \right), \qquad g=\eta+\kappa h, \label{eq:GG} \end{equation} whose integrated expansion, modulo divergences, is \[ S_2[h]+\kappa E_3[h]+\kappa^2E_4[h]+O(\kappa^3). \] For homogeneous \(E_n\), set \[ E_n^{[n]}=\frac1{n!}D^nE_n[0], \qquad v_n=D^nE_n[0]. \] Thus \(v_n(u_1,\ldots,u_n)\) is the coefficient of \(z_1\cdots z_n\) in \(E_n[\sum_i z_i u_i]\). There is no additional factorial. Let \(d_{IJK}\) be real and completely symmetric. Define \[ S_3(d)=\sum_{I,J,K}d_{IJK}E_3^{[3]}(h^I,h^J,h^K) \] and \[ T_{IJKL}(d)=\frac13\sum_M \left( d_{IJM}d_{KLM}+d_{IKM}d_{JLM}+d_{ILM}d_{JKM} \right). \] The reference quartic is \[ S_{4,\sym}(d)= \sum_{I,J,K,L}T_{IJKL}(d)E_4^{[4]}(h^I,h^J,h^K,h^L). \] With an interaction parameter \(\lambda\), the locked action is \begin{equation} S(d,\theta)= S_2[h^1]+S_2[h^2]+\lambda S_3(d) +\lambda^2\left(S_{4,\sym}(d)+\sum_\ell\theta_\ell\OO_\ell\right). \label{eq:action} \end{equation} Every \(\theta_\ell\) is constant on \(\XX\). Einstein self-coupling and local deformation theory provide context \cite{Deser,BarnichHenneaux,BoulangerEtAl}; no classification theorem from those works replaces the calculation below. \subsection{Ward reduction and the target function} For a gauge parameter \(\zeta\), let \[ G_\zeta=\partial\zeta+(\partial\zeta)^T, \] \[ (R_\zeta h)_{\mu\nu} =\zeta^\rho\partial_\rho h_{\mu\nu} +h_{\rho\nu}\partial_\mu\zeta^\rho +h_{\mu\rho}\partial_\nu\zeta^\rho. \] The variation \(\delta h=G_\zeta+\kappa R_\zeta h\) is the scaled Lie derivative of the metric. Invariance of the integrated Einstein density modulo a divergence gives, after polarization, \begin{align} v_3(G_\zeta,u,v) &=-\pair{K_0u}{R_\zeta v} -\pair{K_0v}{R_\zeta u}, \label{eq:ward3}\\ v_4(G_\zeta,u,v,w) &=-v_3(R_\zeta u,v,w)-v_3(u,R_\zeta v,w) -v_3(u,v,R_\zeta w). \label{eq:ward4} \end{align} The pairings mean integrated coefficients or stripped conserved-momentum coefficients. The identities follow by differentiating \(\delta_0E_3+\delta_1S_2=0\) and \(\delta_0E_4+\delta_1E_3=0\), respectively. Define the cubic current by \[ \pair{J(v,w)}{z}=v_3(z,v,w). \] For free \(v,w\), \eqref{eq:ward3} with arbitrary \(\zeta\) implies conservation of \(J(v,w)\). Lemma~\ref{lem:inverse} therefore applies at the physical end of each exchange. The stripped tree prescription is \[ \mathscr T_4 =V_4-\sum_{\{ij|kl\}\in\{12|34,13|24,14|23\}} V_3(i,j,\cdot)D(k_i+k_j)V_3(\cdot,k,l). \] The common momentum delta, \(\lambda^2\), and overall Feynman factor \(i\) are removed. Take leg one replaced. Put \[ (\mathscr R_{1i})_{\mu\nu} =(\xi_1\cdot k_i)\eps_{i\mu\nu} +k_{1\mu}\xi_1^\rho\eps_{i\rho\nu} +k_{1\nu}\xi_1^\rho\eps_{i\mu\rho}, \] and define \[ W_2=v_3(\mathscr R_{12},\eps_3,\eps_4),\quad W_3=v_3(\mathscr R_{13},\eps_2,\eps_4),\quad W_4=v_3(\mathscr R_{14},\eps_2,\eps_3). \] The momentum of \(\mathscr R_{1i}\) is \(k_1+k_i\); that insertion is generally off shell. Write \[ D_s=\sum_Md_{J_1J_2M}d_{J_3J_4M}, \quad D_t=\sum_Md_{J_1J_3M}d_{J_2J_4M}, \quad D_u=\sum_Md_{J_1J_4M}d_{J_2J_3M}, \] and \(\overline D=(D_s+D_t+D_u)/3\). \begin{proposition}[Reference response] \label{prop:reference} The single-longitudinal reference response is \begin{equation} F(d)=(D_s-\overline D)W_2 +(D_t-\overline D)W_3+(D_u-\overline D)W_4. \label{eq:F} \end{equation} \end{proposition} \begin{proof} Equations~\eqref{eq:ward3} and Lemma~\ref{lem:inverse} give, channel by channel, \[ v_3(G_\zeta,u,DJ(v,w)) =-v_3(R_\zeta u,v,w). \] The tree minus sign therefore makes the exchange contributions \(D_sW_2+D_tW_3+D_uW_4\). The quartic contributes \(-\overline D(W_2+W_3+W_4)\) by \eqref{eq:ward4}. \end{proof} Define \[ q(d)=d_{112}^2+d_{122}^2-d_{111}d_{122}-d_{112}d_{222}, \] and let \(d_*\) have \(d_{112}=1\) and all other independent components zero. Then \(q(d_*)=1\). Put \(m=F(d_*)\). For species \(1122\) or \(2211\), \[ D_s=d_{111}d_{122}+d_{112}d_{222},\qquad D_t=D_u=d_{112}^2+d_{122}^2, \] so \begin{equation} F(d)=q(d)\frac{-2W_2+W_3+W_4}{3}. \label{eq:qfactor} \end{equation} Other two-and-two placements permute the channel coefficients. For \(1111,2222,1112,1222\), up to permutation, the three channel coefficients agree, so \(F(d)=0\). Consequently the complete response has the global factorization \begin{equation} r(d,\theta;\beta)=q(d)m(\beta)+\sum_\ell\theta_\ell n_\ell(\beta), \qquad \beta\in\XX. \label{eq:response-factorization} \end{equation} This factorization is not a global contact identity. %===================================================================== \section{Complete contacts and the exact three-row certificate} \label{sec:certificate} %===================================================================== \subsection{Contact class and three-clause completeness} Let \(\VV\) be the real vector space of local quartic density classes with exactly two derivatives, constant coefficients, no extra fields, no inverse operators, and no background tensors other than \(\eta\) and the orientation tensor. The densities are translation invariant, Lorentz scalar, and parity even. The quotient is by total derivatives and genuine tensor identities, not by equations of motion or field redefinitions. For a species word \(\mathbf I=(I_1,I_2,I_3,I_4)\in\{1,2\}^4\) and a perfect matching \(P\) of ten slots, define \begin{align} \OO_{\mathbf I,P}=\int\dd^4x\, &(\partial_{\lambda_1}h^{I_1}_{\lambda_3\lambda_4}) (\partial_{\lambda_2}h^{I_2}_{\lambda_5\lambda_6}) h^{I_3}_{\lambda_7\lambda_8} h^{I_4}_{\lambda_9\lambda_{10}} \nonumber\\ &\hspace{2em}\times \prod_{\{u,v\}\in P}\eta^{\lambda_u\lambda_v}. \label{eq:raw} \end{align} There are \[ L=2^4(10-1)!!=16\cdot945=15120 \] raw labels. \begin{definition}[Complete response catalogue] \label{def:complete} A response catalogue is complete when: \begin{enumerate}[label=\textup{(\roman*)}] \item every allowed response change is a listed linear combination with coefficients constant on the evaluation domain; \item every listed direction is allowed, or its permitted coefficient subspace is explicitly specified; \item admitted representative changes are either covered by those combinations or proved response-null under the same prescription. \end{enumerate} \end{definition} \begin{lemma}[Coverage and descent] \label{lem:coverage} The raw contacts span \(\VV\), every raw contact is allowed, and their evaluation descends to \(\VV\). Thus the catalogue satisfies all three clauses of Definition~\ref{def:complete}, with unrestricted coefficient space \(\RR^{15120}\). \end{lemma} \begin{proof} If both derivatives act on one factor, integrate one derivative by parts. This gives a sum with derivatives on distinct factors, modulo a divergence. Relabel those factors into positions one and two. The remaining ten tensor slots are completely paired by metrics and hence occur in \eqref{eq:raw}. Every even collection of orientation tensors reduces to metric pairings by repeated use of the determinant identity. Species words exhaust all assignments. Conversely each raw expression has the declared locality, field degree, derivative degree, covariance, and parity. After differentiation, a total derivative has the overall factor \(i\sum_j k_{j\mu}=0\). Tensor identities vanish before and after evaluation. Hence action-equivalent representatives have identical responses. \end{proof} The response map \(\mathcal R:\VV\to\RR^\XX\) need not be injective. Its image is \[ \CC=\Span_{\RR}\{n_{\mathbf I,P}\}. \] Action equivalence and equality in the response quotient \(\RR^\XX/\CC\) are different notions. \subsection{The evaluator} For covariant momenta \(p,q\), define \[ \Contr_P(p,q;e_1,e_2,e_3,e_4) = \sum_{\lambda_1,\ldots,\lambda_{10}=0}^3 p_{\lambda_1}q_{\lambda_2} \prod_{r=1}^4(e_r)_{\lambda_{2r+1}\lambda_{2r+2}} \prod_{\{u,v\}\in P}\eta^{\lambda_u\lambda_v}. \] The response of a raw contact is \begin{equation} n_{\mathbf I,P}(\beta) = -\sum_{\sigma\in S_4} \left(\prod_{r=1}^4\delta_{I_r,J_{\sigma(r)}}\right) \Contr_P\bigl( k_{\sigma(1)},k_{\sigma(2)}; e_{\sigma(1)},e_{\sigma(2)},e_{\sigma(3)},e_{\sigma(4)} \bigr). \label{eq:evaluator} \end{equation} One \(e_i\) is \(G\). Substitution of four distinctly labelled waves into \eqref{eq:raw} and extraction of their product gives this formula. The two derivatives contribute \(i^2=-1\). \subsection{Three admissible labels} Use upper-index momenta \begin{equation} \begin{aligned} k_1&=(1,0,0,1),& k_2&=(1,0,0,-1),\\ k_3&=(-1,-1,0,0),& k_4&=(-1,1,0,0), \end{aligned} \label{eq:momenta} \end{equation} and covariant tensors \[ A=\diag(0,1,-1,0),\qquad B=\diag(0,0,-1,1). \] Leg one is replaced, the physical tensors are \(A,B,B\), and their references are \(t_0=(1,0,0,0)\). \begin{center} \begin{tabular}{@{}ccc@{}} \toprule Label & Species & \(\xi_1\)\\ \midrule \(\beta_0\)&\(1122\)&\((0,1,0,0)\)\\ \(\beta_+\)&\(1122\)&\((1,0,0,0)\)\\ \(\beta_-\)&\(2211\)&\((1,0,0,0)\)\\ \bottomrule \end{tabular} \end{center} Set \(S=(\beta_0,\beta_+,\beta_-)\). Each momentum is nonzero and null; they sum to zero, and \[ s=-4,\qquad t=u=2. \] The tensors are symmetric, trace free, temporally zero, and transverse to their corresponding momenta. Thus all three labels belong to \(\XX\). \begin{lemma}[The blind row] \label{lem:blind} At \(\beta_0\), \[ m(\beta_0)=0,\qquad n_\ell(\beta_0)=0 \quad\hbox{for every raw contact}. \] \end{lemma} \begin{proof} The reflection \[ \mathsf S=\diag(1,-1,1,1) \] fixes \(k_1,k_2,A,t_0\), interchanges \(k_3,k_4\), preserves \(B\), and negates the gauge vector. Follow the reflection by interchange of all data of legs three and four. Their species and physical tensors agree. A parity-even scalar response is unchanged by these operations but is linear in the replaced tensor, which has been negated. It therefore equals its negative. This applies to each raw contact and to the complete reference tree. \end{proof} \subsection{Derivation of the reference value} \begin{lemma}[Restricted Einstein cubic] \label{lem:cubic} On the linear subspace \[ h_{\mu\nu}=\diag(0,H_1,H_2,H_3),\qquad H_1+H_2+H_3=0, \] the cubic coefficient is, modulo a divergence, \[ E_3=-\int\dd^4x\, \left( H_1\dot H_2\dot H_3+ H_2\dot H_1\dot H_3+ H_3\dot H_1\dot H_2 \right). \] Consequently, for incoming \(p+q+r=0\), \begin{equation} v_3(A_p,B_q,B_r) =-2(p^0q^0+p^0r^0+q^0r^0) =(p^0)^2+(q^0)^2+(r^0)^2. \label{eq:ABB} \end{equation} No on-shell hypothesis on \(p,q,r\) is required. \end{lemma} \begin{proof} Set \(q_i=1+\kappa H_i\), \(q_0=1\), and \(\phi_i=\frac12\log q_i\), \(\phi_0=0\). For a diagonal metric the nonzero Christoffel patterns are \[ \Gamma^a_{aa}=\partial_a\phi_a,\qquad \Gamma^a_{ab}=\partial_b\phi_a,\qquad \Gamma^a_{bb}=-\frac{g_{bb}}{g_{aa}}\partial_a\phi_b \quad(a\ne b). \] Substitution into \eqref{eq:GG} gives \[ \sqrt{q_1q_2q_3} \sum_{a=0}^3\frac{\eta^{aa}}{q_a} \sum_{\substack{1\le i0\), the conditional state is \(\sigma_k/p_k\). All identities hold after tensoring with \(\id_E\). The clock factor \(C\) is absent from the returned quantum context. \end{theorem} \begin{proof} The destructive bra for outcome \(k\) is \(\bra{k}B_C=\bra{\tau_k}\). The Kraus operator including its record is \(L_k=\ket{k}_R\otimes D_k\), and \[ \sum_kL_k^\dagger L_k =\sum_kP_k\otimes I_X=I_{CX}. \] This proves complete positivity and trace preservation. Tensoring the Kraus operators with \(I_E\) proves the reference statement. Their codomain contains \(X\), not \(C\). \end{proof} No normalized state is assigned to an outcome of probability zero. This complete instrument is the finite extraction interface discussed in En09-A \cite{CheungEn09}. \begin{proposition}[No return, and record is not residual] \label{prop:no-return} The clock instrument has no universal left inverse on its full input space. Its classical marginal does not determine the surviving state. \end{proposition} \begin{proof} The clock states \[ \ket{\pm}=\frac{\ket{\tau_0}\pm\ket{\tau_1}}{\sqrt2} \] with the same fixed \(X\) input have identical outputs under \(\Mclock\), although their input density operators differ. A left inverse cannot recover both. For the second assertion, take \(P_0\otimes\rho_X\) and \(P_0\otimes\rho'_X\) with \(\rho_X\ne\rho'_X\). Both return the same deterministic classical record but different residual states. \end{proof} Coherent recording, a nondestructive instrument retaining \(C\), and \eqref{eq:instrument} are different interfaces. Discarding the retained \(C\) from the nondestructive instrument recovers \eqref{eq:instrument}; it does not identify their larger output interfaces. %===================================================================== \section{The three dictionaries and calibration} \label{sec:dictionary} %===================================================================== \subsection{Record-to-label map} Fix a sender \(x_A\in U\), a readout time \(T_d\), and \(\tau_*>0\). Define four distinct labels \[ t_k=T_d-(4-k)\tau_*, \qquad k=0,1,2,3, \] and history events \[ \kappa(k)=(t_k,x_A). \] The readout itself occurs at \[ e_r=(T_d,x_A). \] The contact-label dictionary is \begin{equation} \delta(0)=\beta_0,\qquad \delta(1)=\beta_+,\qquad \delta(2)=\beta_-,\qquad \delta(3)=\beta_0. \label{eq:delta} \end{equation} The fourth outcome is a second label for the blind row, not an unreported failure. The stored code is \(k\). Its time interpretation is \(t_k\); its response interpretation is \(\delta(k)\). These are two different maps. \begin{definition}[Event-time and history-label calibration] \label{def:calibration} For distinct \(t_k\): \begin{enumerate}[label=\textup{(\roman*)}] \item exact event-time calibration at a fixed reading event \(e\) means \[ p_k>0\Longrightarrow t_k=T(e); \] \item history-label calibration means that a supplied event map \(\kappa\) satisfies \(T(\kappa(k))=t_k\). A reading at \(e_r\) labels \(\kappa(k)\), not necessarily \(e_r\). \end{enumerate} \end{definition} This distinction is necessary in I-B's corrected calibration \cite{CheungIB}. \begin{proposition}[A random reading at one event is not several event times] \label{prop:fixed-event} Exact event-time calibration at a fixed event, with distinct \(t_k\), forces a deterministic outcome. If no \(t_k=T(e)\), no normalized exactly calibrated input exists. \end{proposition} \begin{proof} At most one index \(j\) can satisfy \(t_j=T(e)\). Calibration and normalization force \(p_j=1\). For \(Q=I-P_j\otimes I_X\), positivity gives \[ 0=\Tr(Q\rho) =\norm{Q\rho^{1/2}}_{\rm HS}^2. \] Thus \(\rho=(P_j\otimes I_X)\rho(P_j\otimes I_X)\). The converse is immediate. \end{proof} Accordingly, the nontrivial stochastic experiment in this paper uses history-label calibration. \subsection{An inhabited finite preparation} For any probability vector \(p=(p_0,\ldots,p_3)\) and fixed unitaries \(U_k\) on \(X\), define an isometry \[ W_p\ket{\psi} =\sum_{k=0}^3\sqrt{p_k}\ket{\tau_k}_C\otimes U_k\ket{\psi}_X. \] Orthogonality of the clock basis gives \(W_p^\dagger W_p=I_X\). It can be prepared by a fixed clock-state preparation and a clock-controlled unitary, all at the sender. For any referenced input \(\rho_{XE}\), \[ (\Mclock\otimes\id_E)(W_p\rho_{XE}W_p^\dagger) = \sum_kp_k\proj{k}_R\otimes (U_k\otimes I_E)\rho_{XE}(U_k^\dagger\otimes I_E). \] Thus the conditional states and their probabilities are exact. A default preparation is \[ p=(1/3,1/3,1/3,0),\qquad U_k=I_X. \] All three distinct contact rows then have positive probability. Choosing \(U_k=e^{-iH(t_k-t_0)}\) would install sampled conditional evolution for a declared finite Hamiltonian \(H\). That is an engineered correlation, not a derivation of an external-time-free dynamics or of a Hamiltonian constraint. Page--Wootters conditioning is a useful comparison \cite{PageWootters,CheungEn09}, not an additional premise. \subsection{Name-to-rod map} At every cut a surviving carrier name \(a\) has a prescribed spatial position \(x(a)\in U\). Its rod coordinates are \[ \mathfrak r(a)=\chi(x(a)). \] The same declared frame \((e_0,e_1,e_2,e_3)\) interprets the numerical tensor components of \(\delta(k)\). In particular, the four momenta of a label are auxiliary components in a common local frame. They are not four quantum names and are not four globally propagating states on the cone. \subsection{Schedule-past map} The complete port dependency graph has a location map \[ \iota:\{\hbox{scheduled ports}\}\longrightarrow M. \] Every directed dependency edge must connect a future-causal pair, with an explicit local route or endpoint certificate. Consequently an ancestor of an output is mapped into its metric past. Section~\ref{sec:locality} proves the corresponding influence bound. \begin{proposition}[Agreement on the admitted finite events] \label{prop:agreement} For the declared preparation, labels, rods, and causally embedded schedule: \begin{enumerate}[label=\textup{(\roman*)}] \item the record \(k\) labels the event \(\kappa(k)\) with \(T(\kappa(k))=t_k\) and the external datum \(\delta(k)\); \item each accessible surviving name has its prescribed rod coordinates; \item every scheduled ancestor of an output lies in the output's metric past. \end{enumerate} \end{proposition} \begin{proof} The first two statements are the specified maps evaluated on their domains. Their operational probabilities and residuals are Theorem~\ref{thm:instrument}. The third follows by concatenating the certified causal edges. \end{proof} This is exact agreement within a declared model, not a statistical calibration procedure that discovers the dictionary from records. \subsection{Nonuniqueness} \begin{proposition}[The clock does not select its contact dictionary] \label{prop:label-nonunique} The same complete clock instrument is compatible with different record-to-contact assignments. \end{proposition} \begin{proof} Keep \(\Mclock\), its basis, locations, and input unchanged. Replace \(\delta\) by \[ \delta'(0)=\beta_+,\quad \delta'(1)=\beta_0,\quad \delta'(2)=\beta_-,\quad \delta'(3)=\beta_0. \] Nothing in \eqref{eq:instrument} changes. On the input \(P_0\otimes\rho_X\), however, \[ m(\delta(0))=0,\qquad m(\delta'(0))=16/3. \] Therefore the clock channel does not determine the label assignment. A later dictionary-dependent probe would be a different declared continuation. \end{proof} %===================================================================== \section{Pullback of the complete finite certificate} \label{sec:pullback} %===================================================================== \subsection{Deterministic label pullback} For any finite code set \(\mathcal K\) and map \(\delta:\mathcal K\to S\), let \[ A_\delta\in\{0,1\}^{|\mathcal K|\times3}, \qquad (A_\delta)_{ki}=\mathbf1_{\{\delta(k)=\beta_i\}}, \] where \(\beta_i\) ranges over \(\beta_0,\beta_+,\beta_-\). Define \[ N^\delta=A_\delta N_S,\qquad m^\delta=A_\delta m_S. \] \begin{theorem}[Certificate preservation under a surjective dictionary] \label{thm:pullback} If \(\delta\) is surjective, then \[ \rank N^\delta=\rank[N^\delta\mid m^\delta]=2, \qquad N^\delta\alpha_0=m^\delta, \] and \[ \ker_{\rm right}N^\delta=\ker_{\rm right}N_S. \] The complete left kernel is \begin{equation} \ker_{\rm left}N^\delta = \left\{z: \sum_{\delta(k)=\beta_+}z_k=0,\quad \sum_{\delta(k)=\beta_-}z_k=0 \right\}. \label{eq:pullback-left} \end{equation} Every such annihilator also annihilates \(m^\delta\). \end{theorem} \begin{proof} A surjective dictionary makes \(A_\delta:\RR^3\to \RR^{|\mathcal K|}\) injective: each original coordinate is read by at least one code. Hence it preserves the ranks and the right kernel. The original pivot columns become \(8\) times the indicator vectors of the \(\beta_+\) and \(\beta_-\) fibers. Their annihilator equations are precisely \eqref{eq:pullback-left}. They span the complete image, so the equations are sufficient. The target has coefficient \(16/3\) on those two indicator vectors and is therefore annihilated by the same rows. \end{proof} For \eqref{eq:delta}, \[ A_\delta= \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ 1&0&0 \end{pmatrix}, \qquad m^\delta=\begin{pmatrix}0\\16/3\\16/3\\0\end{pmatrix}, \] and \[ \ker_{\rm left}N^\delta =\Span\{(1,0,0,0),(0,0,0,1)\}. \] The affine family of four-code fits still has raw-coordinate dimension \(15118\). \subsection{What sampling can lose} A row that has zero probability in one preparation is not thereby tested in that run. If only one nonzero row is sampled, the observed weighted image can have smaller rank. For a known probability vector \(p\), a record-retaining weighted table is \[ \diag(p)N^\delta. \] Its rank equals two whenever the \(\beta_+\) and \(\beta_-\) fibers each receive positive total probability. The default preparation has this property. Keeping the complete code distinguishes rowwise information from the single scalar average \[ \sum_kp_k\,r(d,\theta;\delta(k)). \] That average generally loses information. None of these classical row operations weights the action density or changes the integration-by-parts quotient of Section~\ref{sec:certificate}. %===================================================================== \section{An exact finite response-table instrument} \label{sec:probe} %===================================================================== \subsection{Why a separate probe is needed} The instrument \(\Mclock\) supplies a label and surviving matter. It does not, by itself, make \(m(\beta)\) or \(n_\ell(\beta)\) an expectation value of that matter. We now construct one explicit finite comparison probe. Its inputs are the already calculated table entries. The construction is deliberately a table realization, not a physical derivation of those entries from a spin-two interaction. \subsection{A bounded encoding} Set \(B_*=768\), and let \(Q\) be a fresh qubit in \(\ket0\). For a real value \(|v|\le B_*\), define \[ a_v=\sqrt{\frac{1+v/B_*}{2}}, \qquad b_v=\sqrt{\frac{1-v/B_*}{2}}, \] \[ V_v= \begin{pmatrix} a_v&-b_v\\ b_v&a_v \end{pmatrix}, \qquad \ket{\psi_v}=V_v\ket0. \] Then \(V_v\) is unitary and, for \(Z=\diag(1,-1)\), \begin{equation} \bra{\psi_v}B_*Z\ket{\psi_v}=v. \label{eq:score} \end{equation} Let \(f\) be one setting in the finite list \[ \mathcal F=\{0,m,n_1,\ldots,n_{15120}\}. \] Lemma~\ref{lem:bound} and \eqref{eq:target-value} ensure \(|f(\delta(k))|\le B_*\). After clock consumption, apply \(V_{f(\delta(k))}\) to \(Q\), controlled by the classical record at the sender. The resulting complete preparation instrument is \begin{equation} \mathcal P_f(\rho) = \sum_{k=0}^3\proj{k}_R\otimes \sigma_k\otimes\proj{\psi_{f(\delta(k))}}_Q. \label{eq:probe-prep} \end{equation} For a reference, \(\sigma_k\) is the joint \(XE\) block. This uses a finite gate library: there are only finitely many specified values. Exact gates are admitted mathematical primitives of the model. No compilation into an unspecified hardware gate set is claimed. \subsection{Complete binary instruments and centered response} Destructively read \(Q\) in the \(Z\) basis and retain the binary value \(y\in\{+1,-1\}\) in a classical register \(Y\). The complete resulting instrument is \begin{equation} \mathcal B_f(\rho) = \sum_{k,y}\proj{k}_R\otimes\proj{y}_Y\otimes \frac{1+yf(\delta(k))/B_*}{2}\,\sigma_k. \label{eq:binary} \end{equation} Define its centered response map \[ \Delta_f=\mathcal B_f-\mathcal B_0. \] Thus \begin{equation} \Delta_f(\rho) = \sum_{k,y}\proj{k}_R\otimes\proj{y}_Y\otimes \frac{y f(\delta(k))}{2B_*}\,\sigma_k. \label{eq:centered} \end{equation} A centered response map is a linear difference of channels, not generally a completely positive map. \begin{theorem}[Exact operational embedding of the finite response space] \label{thm:operational-embedding} The following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item Equations~\eqref{eq:probe-prep} and \eqref{eq:binary} define complete instruments on every input, with every untouched reference. \item For \(p_k>0\), \[ \mathbb E[B_*y\mid k,f]=f(\delta(k)). \] \item The map taking a three-row real vector \(v\) to the linear map given by the right side of \eqref{eq:centered} is injective when \(\delta\) is surjective. \item The centered operational contact image has rank two and \begin{equation} \Delta_m =\frac23\Delta_{n_{\ell_a}}+\frac23\Delta_{n_{\ell_b}}. \label{eq:operational-fit} \end{equation} Its raw-coordinate fitting family is \(\alpha_0+\ker_{\rm right}N_S\). \end{enumerate} \end{theorem} \begin{proof} The first instrument is obtained from a complete clock instrument, a fresh allocation, and a classically controlled unitary. The second is obtained by complete destructive readout of \(Q\). Alternatively, each displayed branch is completely positive and the branch probabilities sum to \(\Tr\rho\). The same arguments apply with \(I_E\) appended. Equation~\eqref{eq:score} gives (ii). For (iii), if \(v_i\ne0\), choose a code \(k\) with \(\delta(k)=\beta_i\) and input \(P_k\otimes\rho_X\). The \(y=+1\) block of the centered output is nonzero. Thus the induced linear map has trivial kernel. Formula~\eqref{eq:centered} is linear in \(v\). Applying this injective linear map to Theorem~\ref{thm:frozen} proves (iv), including equality of the coefficient kernels. \end{proof} \begin{remark}[The fit is not a mixture of instruments] \label{rem:not-convex} Equation~\eqref{eq:operational-fit} does not assert \[ \mathcal B_m= \frac23\mathcal B_{n_{\ell_a}} +\frac23\mathcal B_{n_{\ell_b}}. \] The right side of this false equation has total trace \(4/3\). The correct identity is the centered one, equivalently \[ \mathcal B_m= \frac23\mathcal B_{n_{\ell_a}} +\frac23\mathcal B_{n_{\ell_b}} -\frac13\mathcal B_0. \] This is an affine linear identity, not an instruction to implement a negative-probability mixture. The physical instrument \(\mathcal B_m\) is implemented directly by its own declared controlled gates. \end{remark} \subsection{Arbitrary fixed contact coefficients} For fixed \(d,\theta\), the three values \[ v_i=q(d)m(\beta_i)+(N_S\theta)_i \] are finite. Choosing a separate bound \[ B_{d,\theta}\ge\max\{1,\abs{v_0},\abs{v_+},\abs{v_-}\} \] constructs a direct probe for that fixed response table. The common \(B_*\) is used for the finite list \(\mathcal F\). Arbitrary linear combinations of its centered maps remain well-defined linear maps, but do not automatically yield a physical channel with that same bound. Physical positivity and the algebraic response span must not be conflated. In particular, the fixed choice \[ \theta_{\ell_a}=\theta_{\ell_b}=-\frac23q(d), \qquad \theta_\ell=0\ \hbox{otherwise}, \] gives \(r_S(d,\theta)=0\). Its directly constructed response probe is then the baseline binary instrument. This says nothing about gauge consistency of the corrected action beyond the three selected tests. \subsection{The missing field-measurement bridge} \begin{definition}[Additional physical matching hypothesis] \label{def:matching} A physical interpretation as a measurement of the locked tree would require a separately specified field representation, admitted states and external preparations, a system--apparatus interaction, and a complete instrument comparison whose calibrated response equals the locked function on the stated input class, with a stated error topology and bound. \end{definition} No such hypothesis is proved here. The probe contains the table as control data; it does not extract the table from an unknown gravitational interaction. Likewise, none of \eqref{eq:probe-prep}--\eqref{eq:centered} defines a tensor-valued spacetime distribution, a variational stress tensor, or an Einstein source. Equality of these finite response maps is not a stress-tensor identity. %===================================================================== \section{Complete locality and spectator independence} \label{sec:locality} %===================================================================== \subsection{Scheduled channels} Use finitely many sites in the convex developing patch. At each tick a partition \(\mathcal P_s\) of the reserved site registers is supplied. Each block \(b\in\mathcal P_s\) has a complete channel. Classical controller and record registers are included in the network. If a block has input time \(T_s\) and output time \(T_{s+1}\), require \begin{equation} \diam_{\chi}b\le c(T_{s+1}-T_s). \label{eq:block} \end{equation} All classical dependencies must either be preinstalled data or explicit wires satisfying the same causal requirement. Finite adaptive trees can be unrolled into this reserved network, using local classical flags and identity operations on inactive branches. For a terminal register set \(Y\), define its scheduled ancestors backward by \[ N_n(Y)=Y, \qquad N_s(Y)= \bigcup\{b\in\mathcal P_s:b\cap N_{s+1}(Y)\ne\varnothing\}. \] \begin{theorem}[Complete schedule locality and metric containment] \label{thm:locality} Let \(\Phi\) be the complete scheduled channel. Then \[ \Phi^*(\mathcal A(Y))\subseteq\mathcal A(N_0(Y)). \] The output on \(Y\) depends only on the initial reduction to \(N_0(Y)\). With a retained reference \(E\), it depends only on the joint initial reduction to \(N_0(Y)E\). Replacing a complete operation outside the backward dependency region leaves the terminal law unchanged. Every scheduled ancestor event is in the metric past of the output event. \end{theorem} \begin{proof} An output observable is the identity on any block disjoint from its current support. The adjoint of a trace-preserving channel is unital, so that block remains an identity. Other blocks enlarge support only to their union. Backward induction proves the adjoint inclusion. To obtain a reduced channel, append any fixed state on the initial complement, apply the complete channel, and reduce to \(Y\). The adjoint inclusion proves independence of that appended state and agreement with the actual output even for correlated inputs. Tensoring the argument with \(I_E\) proves the joint reference assertion. A replacement outside the dependency region acts on an identity in the backward calculation. Finally, each dependency pair is future causal by \eqref{eq:block} and Lemma~\ref{lem:local-causal}. Concatenate the segments. \end{proof} This is the local argument underlying the schedule comparisons of En03-A and En05-A \cite{CheungEn03,CheungEn05}. It does not require the initial state to factor across a cone boundary. \begin{remark}[Influence is not normalized postselection] The theorem concerns complete channels and retained records. A normalized conditional state is not a trace-preserving map. A change in a selectively normalized state, especially in the presence of pre-existing entanglement, is not by itself a superluminal signaling protocol. \end{remark} Scheduled ancestry is an upper bound, not necessarily the minimal influence region. For example, an identity channel assigned to a two-site block has the block's formal ancestry but no cross-site signal. \subsection{Spectator-blind control} \begin{proposition}[Inaccessible references do not change the certificate] \label{prop:spectator} Suppose all controllers use only the declared code, fixed setting, geometry, accessible data, and resource permissions, and every operation extends by the identity on an inaccessible reference \(E\). Then: \begin{enumerate}[label=\textup{(\roman*)}] \item adding \(E\) does not change \(\delta\), the catalogue, or any matrix entry; \item the instrument, centered-response, and transport identities remain valid on arbitrary reference-entangled inputs; \item equal joint inputs on the declared dependency interface give equal corresponding joint outputs. \end{enumerate} \end{proposition} \begin{proof} No control function has an input from \(E\), so its selected operation and classical table remain unchanged. All relevant identities are identities of linear maps and hence remain identities after tensoring with \(\id_E\). The last assertion follows from Theorem~\ref{thm:locality}. \end{proof} Spectator blindness is a condition on the complete control specification, not a claim that inaccessible systems cease to exist. Their storage is retained in a separate ledger. %===================================================================== \section{Reference-preserving target-adapted transport} \label{sec:transport} %===================================================================== \subsection{The chain and its permission contract} Fix distinct \(x_A,x_B\in U\), and let \[ L_{AB}=\norm{\chi(x_B)-\chi(x_A)}_2. \] The source event is \((T_d,x_A)\); the prescribed null destination is \[ e_B=(T_d+L_{AB}/c,x_B). \] Choose \(m\ge1\), and install the target-adapted sites \[ \chi(x_j)=\chi(x_A) +\frac jm\bigl(\chi(x_B)-\chi(x_A)\bigr), \qquad 0\le j\le m. \] Thus \(x_0=x_A\), \(x_m=x_B\). Set \[ a=\frac{L_{AB}}m,\qquad \Delta T=\frac{L_{AB}}{mc}. \] Each site carries a \(p\)-qubit payload register \(A_j\). The payload may be the surviving \(XQ\) of \eqref{eq:probe-prep}, so \(p=p_X+1\). Prepare \(A_1,\ldots,A_m\) in a fixed blank state \(\omega=\proj{0\cdots0}\), independently of \(A_0E\). At tick \(j\), apply SWAP to \(A_{j-1}A_j\), and identities elsewhere. A two-site SWAP across the tick is an admitted block primitive with causally embedded endpoint dependencies. No continuous-time Hamiltonian implementing that primitive within the tick is asserted. \begin{theorem}[Exact complete transport] \label{thm:transport} For every state \(\rho_{A_0E}\), after tick \(j\) the state is, under the fixed register identifications, \[ \rho_{A_jE}\otimes \bigotimes_{\substack{0\le i\le m\\i\ne j}}\omega_{A_i}. \] At the destination: \begin{enumerate}[label=\textup{(\roman*)}] \item the logical channel is exactly \(\id_{2^p}\), including every untouched reference; \item all non-output route registers are jointly blank and payload independent; \item arrival is at \(e_B\); \item every scheduled dependency is metric causal; \item the protocol uses \(m\) register SWAPs, \(m\) ticks, and \(m+1\) payload-sized registers. \end{enumerate} \end{theorem} \begin{proof} The assertion at \(j=0\) is the preparation. At tick \(j\), \(A_j\) is blank and independent; SWAP exchanges it with the entire \(A_{j-1}\) tensor factor, including its correlations with \(E\). The previous site becomes blank. This proves the invariant by induction. The total time is \(m\Delta T=L_{AB}/c\). Each active pair has separation \(a=c\Delta T\), so Lemma~\ref{lem:local-causal} certifies all its endpoint dependencies. The counts follow from the printed schedule. \end{proof} This is the target-adapted construction used in I-B \cite{CheungIB}. En08-B's lattice-family approximation has a different spatial permission contract \cite{CheungEn08}. Exactness here does not mean that every null target is exactly attained on a previously fixed lattice. \subsection{Composition with readout and the table probe} Let \(\mathcal T_m\) be the route channel with blanks included, and let \(\iota_A:XQ\to A_m\) be the fixed output identification. For arbitrary \(\rho_{CXE}\), the complete output of readout, local controlled probe preparation, and routing is \begin{equation} \sum_k\proj{k}_{R_A}\otimes (\iota_A\otimes\id_E) \left(\sigma_{XE,k}\otimes \proj{\psi_{f(\delta(k))}}_Q\right) \otimes\bigotimes_{j=0}^{m-1}\omega_{A_j}. \label{eq:composed} \end{equation} This follows by applying Theorem~\ref{thm:transport} to each subnormalized block and summing. Reading the transported \(Q\) in the \(Z\) basis gives \eqref{eq:binary}, with the residual \(X\) relocated to \(x_B\). Consequently the centered operational fit \eqref{eq:operational-fit} survives transport exactly. The clock record \(R_A\) remains at the sender in \eqref{eq:composed}. A joint-state formula is not instantaneous delivery of that record. A receiver operation depending on \(k\) requires a separately declared causal record wire, or the record-controlled operation must be performed before departure. Our controlled probe preparation is performed at the sender. The terminal \(Z\) readout itself is independent of \(k\). Joint records can be collected later in a common causal future. \subsection{Resource ledger and implementation errors} With a two-qubit clock and \(p\)-qubit payload registers, all route blanks may be reserved before readout. The laboratory quantum peak is at most \begin{equation} Q_{\rm lab}=2+p(m+1). \label{eq:peak} \end{equation} After clock consumption it is at most \(p(m+1)\). There are two clock-record bits and, after the terminal probe readout, one additional binary record. An \(r\)-qubit untouched reference contributes \(r\) to the separate external-storage ledger. Preinstalled tables, locations, settings, and the route schedule are classical resources. In a compatible implementation comparison, define the half-diamond distance \[ d_\diamond(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \] If the preparation and final decoding errors are at most \(\epsilon_{\rm in},\epsilon_{\rm out}\), and the complete tick errors are at most \(\epsilon_j\), then \[ d_\diamond(\widetilde\Phi,\Phi) \le \min\left\{1,\epsilon_{\rm in} +\sum_{j=1}^m\epsilon_j+\epsilon_{\rm out}\right\}. \] Indeed, replace one compatible factor at a time in the composed channel and use diamond-norm contraction under channel composition. This is a conditional error ledger, not an assertion that implementation errors have been measured. %===================================================================== \section{Permitted manipulations and their limits} \label{sec:manipulations} %===================================================================== \begin{proposition}[Clock basis changes] \label{prop:basis} Changing the clock reading basis changes the pointer observable and generally the clock instrument. It does not change the coordinate function \(T\), rods, or metric. \end{proposition} \begin{proof} For a unitary \(U\) on \(C\), the new projectors are \(UP_kU^\dagger\). Unless this is an appropriate relabelling of the old projectors, the branch effects and conditional residuals change. The classical components \(M,g,T,\chi\) do not occur as outputs of this operation and remain fixed. \end{proof} \begin{proposition}[Pre-read coupling is a declared interaction] \label{prop:coupling} A unitary \(U_{CX}\) before consumption replaces the instrument by \(\Mclock\circ\Ad_{U_{CX}}\). This may change the record law and residual states, but supplies no metric back-reaction. \end{proposition} \begin{proof} Compose the Kraus operators with \(U_{CX}\). Completeness is preserved because \(U_{CX}\) is unitary. No component of the specified geometry has a state-dependent update equation. \end{proof} The absence of an update rule is not a theorem excluding every future matter-driven geometry. It identifies what this model does and does not define. \begin{proposition}[Record control and constant contact coefficients] \label{prop:constant} Classical control that chooses the evaluation label \(\delta(k)\) does not enlarge the frozen contact class, provided the setting and contact coefficient vector are fixed across labels. Allowing an independently selected \(\theta(k)\) changes the response contract. \end{proposition} \begin{proof} For fixed \(\theta\), the evaluated correction is \[ (N^\delta\theta)_k =\sum_\ell\theta_\ell n_\ell(\delta(k)). \] These are evaluations of one allowed constant-coefficient combination. By contrast, \[ \sum_\ell\theta_\ell(k)n_\ell(\delta(k)) \] uses row-dependent coefficients and need not be the restriction of any single combination on \(\XX\). The original quantifier has been replaced. \end{proof} The table setting \(f\) in Section~\ref{sec:probe} is preinstalled task data for a run. It is not chosen separately after each row to manufacture a fit. \begin{proposition}[Fresh allocation and finite clocks] \label{prop:fresh} Fresh clock allocation creates a new named reading event. It does not produce a continuum limit of the contact catalogue. Moreover no finite clock admits a canonical commutator \([T_C,H_C]=iI_C\). \end{proposition} \begin{proof} Fresh allocation extends the finite instruction tree and live-name ledger. It changes neither the field degree nor derivative bound defining \(\VV\), so it changes no catalogue entry. For the second assertion, \(\Tr[T_C,H_C]=0\), whereas \(\Tr(iI_C)=4i\ne0\). \end{proof} The geometric time axis remains present after every such reading. A finite Hilbert-space factor is not the fourth coordinate of a manifold. The precise statement is that the model has three local spatial rods and a separately supplied time coordinate, to which finite clock records are calibrated. %===================================================================== \section{Non-identifications and absence of geometric reconstruction} \label{sec:nonidentifications} %===================================================================== \subsection{Finite records do not determine the global cone} \begin{proposition}[A same-channel geometric countermodel] \label{prop:geometry-nonunique} The complete finite experiment just constructed does not determine the global conical deficit. \end{proposition} \begin{proof} Choose two deficits \(\mu_1\ne\mu_2\). Their regular products each contain a developing patch isometric to the same sufficiently small Euclidean spatial ball times a time interval. Install identical names, matrices, table values, clock calibrations, and target-adapted routes in these two patches. Every finite channel and record law is identical, since all operations use the same local data. The completed backgrounds nevertheless have different marked circumference-to-radius ratios \(2\pi-\mu_1\) and \(2\pi-\mu_2\). Thus the same finite operational data do not determine the global deficit. \end{proof} This is a counterexample to reconstruction from the data of this experiment, not a universal impossibility theorem for all possible geometric measurements. The same distinction applies to lapse comparisons. In an enlarged comparison class \[ g_N=-c^2N^2\dd T^2+h \] with constant \(N=1\) or \(N=2\), the same edges satisfying the unit-lapse causal bound remain causal and the same matrices yield the same channels. Static proper durations differ by a factor of two. The main release still fixes \(N=1\); this comparison does not silently change that input. \subsection{Separate time objects} The following objects are not identified: \begin{enumerate}[label=\textup{(\roman*)}] \item the continuum coordinate \(T\); \item the finite pointer \(T_C=\sum_k t_kP_k\); \item a finite history-event map \(\kappa\); \item a conical slicing parameter; \item a harmonic relational scalar; \item an original-rod relational lapse; \item a prescribed density ramp. \end{enumerate} The distinction parallels XIX-A \cite{CheungXIX}. For example, the scalar \(T\) is harmonic for both constant-lapse metrics above, while \[ -g_N^{-1}(\dd T,\dd T)=\frac1{c^2N^2} \] depends on \(N\). The scalar profile alone does not determine the lapse. For the unit-lapse static history events chosen here, \(t_k-t_j\) equals their geometric static proper-time interval. That equality follows from the supplied metric and event map. It is not inferred from the numerical code, and the individual code does not determine a proper-time origin. \subsection{Corrections table} \begingroup \small \renewcommand{\arraystretch}{1.12} \begin{longtable}{@{}p{.30\linewidth}p{.64\linewidth}@{}} \toprule Potential identification & Correction in II-B\\ \midrule \endfirsthead \toprule Potential identification & Correction in II-B\\ \midrule \endhead A finite fit is a global identity & The certificate concerns one three-label family. Global constant-coefficient membership remains undecided here.\\ \addlinespace Action and response equivalence coincide & Evaluation descends to action classes but can have a nontrivial kernel; the response quotient is a different quotient.\\ \addlinespace A clock record is already a contact label & The map \(\delta\) is additional declared data, with an explicit nonuniqueness counterexample.\\ \addlinespace One random readout gives several times of one event & Distinct exact event-time labels force a deterministic outcome. The stochastic construction uses history labels.\\ \addlinespace Enter destroys the time axis & It consumes finite quantum names. \(M,g,T\), and the causal relation remain present.\\ \addlinespace A clock basis change is a coordinate transformation & It generally changes the pointer observable and instrument, not the geometry.\\ \addlinespace Three labels need a three-dimensional qubit factor & The model uses two clock qubits and specifies all four outcomes; the fourth maps to the blind label.\\ \addlinespace Flat-space labels are conical scattering states & They remain auxiliary vertex labels interpreted in a declared local frame; no global scattering bridge is constructed.\\ \addlinespace A table probe measures the gravitational tree & The probe is controlled by the computed table. A field--apparatus matching theorem would be additional.\\ \addlinespace A contact fit is a mixture of channels & The valid operational identity uses centered instrument differences, not a non-normalized positive mixture.\\ \addlinespace The residual response is a stress tensor & No tensor-valued source map or variational matter action is supplied by the finite table instrument.\\ \addlinespace Weighting preserves the old quotient & A nonconstant density weight can turn a divergence into a nonzero response term; a switched problem needs a new contract.\\ \addlinespace A joint output formula delivers the clock record remotely & Remote use of \(k\) requires a causal classical wire or local record-controlled processing before departure.\\ \addlinespace A scheduled cone is the exact influence region & It is an upper bound for complete channels, not necessarily a minimal region and not a postselection theorem.\\ \addlinespace Exact null transport is a microscopic Hamiltonian model & The proof is a finite endpoint-channel and causal-dependency certificate on a target-adapted chain.\\ \addlinespace Increasing the number of registers gives a field continuum limit & Each experiment is finite; no field algebra, convergence theorem, or detector scaling limit follows from the register count.\\ \bottomrule \end{longtable} \endgroup %===================================================================== \section{Release theorem and standing boundaries} \label{sec:release} %===================================================================== \begin{theorem}[II-B Grade S release] \label{thm:release} For the frozen action, external domain, and contact class of Sections~\ref{sec:algebra}--\ref{sec:certificate}, the prescribed background \eqref{eq:metric}, and the declared finite experiment: \begin{enumerate}[label=\textup{(\roman*)}] \item the complete three-row contact certificate is unchanged: \[ m_S=(0,16/3,16/3)^T,\qquad \rank N_S=\rank[N_S\mid m_S]=2, \] with original minor \(64\), fit coefficients \(2/3,2/3\), and left kernel \(\Span\{(1,0,0)\}\); \item destructive Enter supplies the complete four-code instrument \eqref{eq:instrument}, consuming both clock names; \item the dictionary supplies history-event labels and the external labels of II, and its pullback preserves the finite fit and complete response image; \item the finite table-probe instruments realize that response space as centered binary instruments, with exact correctness on every referenced input; \item the target-adapted route transports the surviving payload by the identity logical channel, with the resource ledger \eqref{eq:peak}; \item complete scheduled influence is contained in the prescribed metric causal relation. \end{enumerate} Thus the construction satisfies Definition~\ref{def:grade}. Its contact status remains \(\SEP\)-finite. \end{theorem} \begin{proof} Item (i) is Theorem~\ref{thm:frozen}. Theorem~\ref{thm:instrument} proves (ii). Definitions~\ref{def:calibration}, \eqref{eq:delta}, Proposition~\ref{prop:agreement}, and Theorem~\ref{thm:pullback} prove (iii). Theorem~\ref{thm:operational-embedding} proves (iv). Theorem~\ref{thm:transport} and \eqref{eq:composed} prove (v). Theorem~\ref{thm:locality} proves (vi). All names, settings, locations, gates, records, and route resources have been specified. These are precisely the five clauses of the Grade S contract. No additional global contact argument has been supplied. \end{proof} \subsection{What remains unconstructed} The theorem does not construct: \begin{enumerate}[label=\textup{(\roman*)}] \item a metric, global deficit, lapse, or Lorentz cone derived from records alone; \item a contact class derived from the clock instrument; \item a global identity \(m=\sum_\ell\alpha_\ell n_\ell\) or a global contact separator; \item a field-theoretic implementation of the spin-two response or the matching hypothesis of Definition~\ref{def:matching}; \item a sourced Einstein equation, a stress-tensor dictionary, or a source-dependent energy-condition theorem; \item microscopic within-tick transport dynamics or a quantum field-theoretic continuum limit; \item a full characteristic quantum product, a joint section, the corresponding transverse and contact estimates, or its quantum output wavefront theorem; \item a heat, work, apparatus-reset, or thermodynamic ledger. \end{enumerate} The last item is not repaired by the fact that the instrument is destructive. Thermodynamic erasure requires a physical memory, preparation, bath, and control account; a channel name does not supply them \cite{Landauer,CheungPhil}. \subsection{Analytic non-transfer} The constructed internal amplitude of XIV-I and the partial supported-sector reference of XIV-J concern different analytic objects from the finite matrix here \cite{CheungXIVI,CheungXIVJ}. A finite linear certificate can compare already defined candidates in a completely parameterized correction class. It cannot define an absent distributional product. The reconstruction-descent and extension interfaces of technical XV-B, version 2, likewise do not identify their correction spaces with the \(15120\)-label catalogue \cite{CheungXVB}. The existence of that technical manuscript is distinct from opening the inherited downstream quantum-admission exit. For this release, the supplied standing entries are unchanged: \[ \text{O.1}=\SourceLabel{NOT_CLOSED},\qquad \text{XV-B admission exit: not opened},\qquad \text{XVI-N: not discharged}. \] These are release-specific documentary boundaries, not an independent audit of every version or differently scoped subsector. In particular, they do not deny the separate scoped two-point report recorded in XVI-Q, Q11 \cite{CheungStops}. No inherited quantum coefficient is admitted. No universal nonexistence theorem for all future characteristic constructions is asserted. \subsection{Conclusion} The adaptation proves an exact compatibility statement: \[ \boxed{ \begin{gathered} \text{a destructive clock record} \xrightarrow{\ \delta\ } \text{a frozen external label} \xrightarrow{\ \text{evaluation}\ } \text{a finite response value},\\ \Delta_m=\frac23\Delta_{n_{\ell_a}} +\frac23\Delta_{n_{\ell_b}},\\ \text{surviving payload transport}=\id \quad\text{with every untouched reference}. \end{gathered}} \] The arrows are supplied maps, not emergent identifications. The metric remains prescribed, the contact class remains an algebraic input, and the field-measurement bridge remains separate. %===================================================================== \appendix \section{Optional exact reconstruction checker} \label{app:checker} %===================================================================== The following standard-library Python listing reconstructs the raw contact matrix from the stated sparse tensors. It checks the blind row, species complementation, the original pivots and minor, the three-row and four-code fits, and the stated finite bounds. Its target calculation uses the proved restricted cubic formula. It is not an independent Einstein-vertex implementation. The listing has not been executed in this interface and is not a premise of the proofs. Slots are zero-based in the code. Matchings are generated by pairing the smallest remaining slot with every possible partner and recursing. This gives each matching exactly once. \begin{lstlisting} from fractions import Fraction from itertools import permutations, product from math import prod ETA = (-1, 1, 1, 1) PERMS = tuple(permutations(range(4))) KUP = ( ( 1, 0, 0, 1), ( 1, 0, 0, -1), (-1,-1, 0, 0), (-1, 1, 0, 0), ) A = ( (0,0, 0,0), (0,1, 0,0), (0,0,-1,0), (0,0, 0,0), ) B = ( (0,0, 0,0), (0,0, 0,0), (0,0,-1,0), (0,0, 0,1), ) def lower(v): return tuple(ETA[a] * v[a] for a in range(4)) def dot(v, w): return sum(ETA[a] * v[a] * w[a] for a in range(4)) def gauge(k, xi): kl, xl = lower(k), lower(xi) return tuple( tuple(kl[a]*xl[b] + kl[b]*xl[a] for b in range(4)) for a in range(4) ) def sparse_vector(v): return tuple(((a,), v[a]) for a in range(4) if v[a] != 0) def sparse_tensor(H): return tuple( ((a,b), H[a][b]) for a in range(4) for b in range(4) if H[a][b] != 0 ) def matchings(slots): slots = tuple(slots) if not slots: yield () return a = slots[0] for j in range(1, len(slots)): b = slots[j] rest = slots[1:j] + slots[j+1:] for tail in matchings(rest): yield ((a,b),) + tail P_LIST = tuple(matchings(range(10))) I_LIST = tuple(product((1,2), repeat=4)) LABELS = tuple((I,P) for I in I_LIST for P in P_LIST) assert len(P_LIST) == len(set(P_LIST)) == 945 assert len(LABELS) == 15120 K = tuple(sparse_vector(lower(k)) for k in KUP) GT = sparse_tensor(gauge(KUP[0], (1,0,0,0))) GX = sparse_tensor(gauge(KUP[0], (0,1,0,0))) AS = sparse_tensor(A) BS = sparse_tensor(B) H_TIME = (GT, AS, BS, BS) H_BLIND = (GX, AS, BS, BS) J_PLUS = (1,1,2,2) J_MINUS = (2,2,1,1) assert all(any(k) and dot(k,k) == 0 for k in KUP) assert all(sum(k[a] for k in KUP) == 0 for a in range(4)) def contract(P, factors): total = 0 for chosen in product(*factors): indices = tuple( a for index_tuple, value in chosen for a in index_tuple ) assert len(indices) == 10 if any(indices[u] != indices[v] for u,v in P): continue value = prod(value for index_tuple, value in chosen) value *= prod(ETA[indices[u]] for u,v in P) total += value return total def raw(I, P, J, H): if sorted(I) != sorted(J): return 0 total = 0 for sigma in PERMS: if any(I[r] != J[sigma[r]] for r in range(4)): continue factors = ( K[sigma[0]], K[sigma[1]], H[sigma[0]], H[sigma[1]], H[sigma[2]], H[sigma[3]], ) total -= contract(P, factors) return total P0 = ((0,1),(2,3),(4,6),(5,8),(7,9)) LA = ((1,1,2,2), P0) LB = ((2,2,1,1), P0) def v_abb(p0, q0, r0): assert p0 + q0 + r0 == 0 return -2*(p0*q0 + p0*r0 + q0*r0) # Uses the proved restricted Einstein cubic. W = (-v_abb(2,-1,-1), v_abb(1,0,-1), v_abb(1,0,-1)) assert W == (-6,2,2) target_value = Fraction(-2*W[0] + W[1] + W[2], 3) assert target_value == Fraction(16,3) target = (Fraction(0), target_value, target_value) def reconstruct(): plus = {} blind = {} for label in LABELS: I, P = label plus[label] = raw(I, P, J_PLUS, H_TIME) blind[label] = raw(I, P, J_PLUS, H_BLIND) assert blind[label] == 0 assert abs(plus[label]) <= 768 columns = {} for I, P in LABELS: Ibar = tuple(3-i for i in I) minus = raw(I, P, J_MINUS, H_TIME) assert minus == plus[(Ibar,P)] assert abs(minus) <= 768 columns[(I,P)] = (0, plus[(I,P)], minus) ca, cb = columns[LA], columns[LB] assert ca == (0,8,0) assert cb == (0,0,8) # An actual minor of original columns. minor = ca[1]*cb[2] - cb[1]*ca[2] assert minor == 64 fit = tuple( Fraction(2,3)*ca[i] + Fraction(2,3)*cb[i] for i in range(3) ) assert fit == target # delta(0),delta(1),delta(2),delta(3) = 0,+,-,0. row_map = (0,1,2,0) target4 = tuple(target[i] for i in row_map) ca4 = tuple(ca[i] for i in row_map) cb4 = tuple(cb[i] for i in row_map) assert target4 == ( Fraction(0), Fraction(16,3), Fraction(16,3), Fraction(0) ) assert tuple( Fraction(2,3)*ca4[i] + Fraction(2,3)*cb4[i] for i in range(4) ) == target4 # Analytic upper bound: every first entry is zero. # Analytic lower bound: original minor 64. # Augmented rank: the explicit fit above. return columns, minor if __name__ == "__main__": columns, minor = reconstruct() print("shape =", (3, len(columns))) print("target =", target) print("original minor =", minor) print("status = SEP-finite; declared Grade S model") \end{lstlisting} %===================================================================== \section{Optional classical lapse-composition interface} \label{app:lapse} %===================================================================== This appendix is a calculus statement for a separately supplied classical geometry. It is not used in the conical release. Suppose a scalar lapse prescription and a coordinate map are given with expansions \[ \mathcal N_\varepsilon =N_0+\varepsilon N_1+\frac{\varepsilon^2}{2}N_2+O(\varepsilon^3), \] \[ Y_\varepsilon^\mu(x) =x^\mu+\varepsilon a_1^\mu(x) +\frac{\varepsilon^2}{2}a_2^\mu(x)+O(\varepsilon^3). \] Assume \(N_0=N_0(x^0)\), and write \[ a_1^0=u_1,\qquad a_1^i=-Y_1^i,\qquad a_2^0=u_2-2Y_1^i\partial_i u_1. \] Define the relationally composed scalar \(n_\varepsilon=\mathcal N_\varepsilon\circ Y_\varepsilon\). \begin{proposition}[Classical second-order composition] \label{prop:lapse} Its first two coefficients are \[ n_1=N_1+\dot N_0u_1, \] \[ n_2= N_2+2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2 -2Y_1^i\partial_i n_1. \] The second spatial coefficient \(a_2^i\) does not occur. \end{proposition} \begin{proof} Taylor expansion gives \[ n_2=N_2+2a_1^\mu\partial_\mu N_1 +a_2^\mu\partial_\mu N_0 +a_1^\mu a_1^\nu\partial_\mu\partial_\nu N_0. \] Use the stated temporal dependence of \(N_0\) and substitute the coefficients of \(a_1,a_2\). The spatial terms are \[ -2Y_1^i\partial_iN_1 -2\dot N_0Y_1^i\partial_i u_1 =-2Y_1^i\partial_i n_1. \] Also \(\partial_iN_0=0\), so \(a_2^i\) drops out. \end{proof} This is the composition pattern recorded in Em02-E and the original-rod comparison of XX-A \cite{CheungEm02E,CheungXXA}. Calling \(\mathcal N_\varepsilon\) a lapse requires its own underlying metric, timelike branch, clock calibration, and rod prescription. A lapse scalar does not supply the spatial metric and shift of a complete geometry. No formula in this appendix is computed from \(N_S\), a clock record, or the table probe. No quantum composite lapse, renormalization, or characteristic admission follows from the Taylor expansion. %===================================================================== \section*{Declarations and provenance} \addcontentsline{toc}{section}{Declarations and provenance} \paragraph{Mathematical contribution.} The adaptation is the explicitly defined finite operational interface, its injective centered-instrument realization, and its composition with the frozen contact certificate and causal endpoint transport. The four algebraic quantities \(16/3\), \(8\), \(64\), and \(2/3\) are reproduced from Manuscript II, not presented as a new global contact result. \paragraph{Model status.} The finite experiment has a prescribed continuum classical background. Its quantum factors and instruction tree are finite. No field apparatus, microscopic transport Hamiltonian, continuum quantum field theory, or spacetime-emergence theorem is claimed. \paragraph{Verification.} No successful \texttt{pdflatex} execution, checker execution, independent Einstein-vertex implementation, proof-assistant check, numerical experiment, laboratory realization, referee approval, or journal acceptance is asserted. The checker is optional and does not replace the analytic proofs. \paragraph{Bibliographic status.} The publisher record for Cheung's Enter article and the linked I-B and philosophy records were directly consulted. The accepted Enter article is cited separately from unpublished companion manuscripts. Other companion citations identify supplied source records and versions. A source citation is not independent verification of the cited proof. \paragraph{Documentary negative controls.} The separately attributed OpenAI catalogue is documentary only \cite{CatalogueRecord}. No catalogue family, reported formalization, authorship record, or physical interpretation is used as a premise or verified by this paper. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{CheungEnter} N.~Cheung. \newblock The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement. \newblock \emph{Frontiers in Physics}, volume 14, 2026. \newblock doi:\ \doi{10.3389/fphy.2026.1954943}. \newblock Publisher accepted-article record: accepted 28 September 2026; the directly consulted record listed the final formatted version as forthcoming. \newblock \url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}. \bibitem{CheungPhil} N.~Cheung. \newblock \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \newblock \SourceLabel{EntPhil-v2.tex}, version 2. Introductory essay, Enter Calculus Companion. \newblock \url{https://enter.grok.me/enter}. \newblock Documentary orientation, not a theorem premise. \bibitem{CheungI} N.~Cheung. \newblock \emph{I}, \SourceLabel{I.tex}, version 2. \newblock Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied source record: destructive measurement characterization within its stated structural class. \newblock \url{https://enter.grok.me/}. \bibitem{CheungII} N.~Cheung. \newblock \emph{Finite contact-span certificates for a two-species spin-two tree}. \newblock Manuscript II, \SourceLabel{II.tex}, version 2. \newblock Supplied complete manuscript source, Enter Calculus Companion. \newblock Relevant items: \SourceLabel{thm:encoding}, \SourceLabel{thm:rank}, \SourceLabel{thm:certificate}, \SourceLabel{cor:affine-fits}, and \SourceLabel{thm:alternative}. \newblock \url{https://enter.grok.me/}. \bibitem{CheungIB} N.~Cheung. \newblock \emph{I-B-v1: A Finite Enter Spacetime: Calibrated Destructive Time Readout on a Prescribed Conical Product}. \newblock \SourceLabel{I-B-v1.tex}, version 1. Unpublished registered manuscript, Enter Calculus Companion. \newblock Relevant items: \SourceLabel{def:experiment}, \SourceLabel{thm:instrument}, \SourceLabel{def:calibration}, \SourceLabel{prop:fixed-event}, \SourceLabel{lem:local-causal}, \SourceLabel{thm:transport}, and \SourceLabel{prop:interfaces}. \newblock \url{https://enter.grok.me/browse/I-B}. \bibitem{CheungEn09} N.~Cheung. \newblock \emph{Realization of Relational Time by Destructive Enter on a Clock Register}. \newblock \SourceLabel{En09-A.tex}, version 1. Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied source records: complete clock instrument, no-return, record-not-residual, and finite-commutator distinctions. \newblock \url{https://enter.grok.me/browse/En09-A}. \bibitem{CheungEn03} N.~Cheung. \newblock \emph{Observation Boundaries for Destructive Quantum Readout and External Geometry}. \newblock \SourceLabel{En03-A.tex}, version 1. Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied source record: four declared interface structures, complete observation boundaries, and scheduled locality. \newblock \url{https://enter.grok.me/browse/En03-A}. \bibitem{CheungEn05} N.~Cheung. \newblock \emph{En05-A}, \SourceLabel{En05-A.tex}, version 1. \newblock Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied locator: \SourceLabel{thm:cone}; scheduled ancestry and its metric envelope are upper bounds. \newblock \url{https://enter.grok.me/}. \bibitem{CheungEn08} N.~Cheung. \newblock \emph{Resource-controlled operational approximation of null boundary transport}. \newblock \SourceLabel{En08-B.tex}, version 1. Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied locator: \SourceLabel{thm:main}. \newblock \url{https://enter.grok.me/browse/En08-B}. \bibitem{CheungCone} N.~Cheung. \newblock \emph{VII-F}, \SourceLabel{VII-F.tex}, version 1; and \emph{VIII-B}, \SourceLabel{VIII-B-v3.tex}, version 3. \newblock Unpublished registered manuscripts, Enter Calculus Companion. \newblock Supplied comparison records: particle-supported conical slicing and the variable-deficit geometric dictionary. \newblock No cell-count or source-law transfer is used here. \newblock \url{https://enter.grok.me/}. \bibitem{CheungXIX} N.~Cheung. \newblock \emph{XIX-A}, \SourceLabel{XIX-A.tex}, version 1. \newblock Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied locator: \SourceLabel{thm:main}, clock separation before coordinate identification. \newblock \url{https://enter.grok.me/}. \bibitem{CheungEm02E} N.~Cheung. \newblock \emph{Em02-E}, \SourceLabel{Em02-E-v1.tex}, version 1. \newblock Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied locator: \SourceLabel{thm:lapse}, classical relational lapse through second order. \newblock \url{https://enter.grok.me/}. \bibitem{CheungXXA} N.~Cheung. \newblock \emph{Original-Rod Lapse: Classical Freeze and Quantum Admission Contract}. \newblock \SourceLabel{XX-A.tex}, version 1. Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied classical comparison and non-transfer records. \newblock \url{https://enter.grok.me/}. \bibitem{FierzPauli} M.~Fierz and W.~Pauli. \newblock On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. \newblock \emph{Proceedings of the Royal Society of London A}, 173:211--232, 1939. \newblock doi:\ \doi{10.1098/rspa.1939.0140}. \bibitem{Deser} S.~Deser. \newblock Self-interaction and gauge invariance. \newblock \emph{General Relativity and Gravitation}, 1:9--18, 1970. \newblock doi:\ \doi{10.1007/BF00759198}. \newblock Author's repost: \url{https://arxiv.org/abs/gr-qc/0411023}. \bibitem{BarnichHenneaux} G.~Barnich and M.~Henneaux. \newblock Consistent couplings between fields with a gauge freedom and deformations of the master equation. \newblock \emph{Physics Letters B}, 311:123--129, 1993. \newblock doi:\ \doi{10.1016/0370-2693(93)90544-R}. \newblock \url{https://arxiv.org/abs/hep-th/9304057}. \bibitem{BoulangerEtAl} N.~Boulanger, T.~Damour, L.~Gualtieri, and M.~Henneaux. \newblock Inconsistency of interacting, multi-graviton theories. \newblock \emph{Nuclear Physics B}, 597:127--171, 2001. \newblock doi:\ \doi{10.1016/S0550-3213(00)00718-5}. \newblock \url{https://arxiv.org/abs/hep-th/0007220}. \bibitem{DaviesLewis} E.~B.~Davies and J.~T.~Lewis. \newblock An operational approach to quantum probability. \newblock \emph{Communications in Mathematical Physics}, 17:239--260, 1970. \newblock doi:\ \doi{10.1007/BF01647093}. \bibitem{Choi} M.-D.~Choi. \newblock Completely positive linear maps on complex matrices. \newblock \emph{Linear Algebra and its Applications}, 10(3):285--290, 1975. \newblock doi:\ \doi{10.1016/0024-3795(75)90075-0}. \bibitem{PageWootters} D.~N.~Page and W.~K.~Wootters. \newblock Evolution without evolution: Dynamics described by stationary observables. \newblock \emph{Physical Review D}, 27:2885--2892, 1983. \newblock doi:\ \doi{10.1103/PhysRevD.27.2885}. \bibitem{Landauer} R.~Landauer. \newblock Irreversibility and heat generation in the computing process. \newblock \emph{IBM Journal of Research and Development}, 5(3):183--191, 1961. \newblock doi:\ \doi{10.1147/rd.53.0183}. \newblock Thermodynamic comparison only; no heat bound is derived from the finite channel in this paper. \bibitem{CheungXIVI} N.~Cheung. \newblock \emph{XIV-I}, \SourceLabel{XIV-I.tex}, version 1. \newblock Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied locators: \SourceLabel{thm:I2}, \SourceLabel{thm:I3}, and \SourceLabel{thm:release}. \newblock Constructed internal amplitude and unconstructed full joint characteristic map are separate objects. \newblock \url{https://enter.grok.me/}. \bibitem{CheungXIVJ} N.~Cheung. \newblock \emph{XIV-J}, \SourceLabel{XIV-J.tex}, version 1. \newblock Unpublished registered manuscript, Enter Calculus Companion. \newblock Supplied locators: \SourceLabel{prop:descent}, \SourceLabel{thm:reference}, \SourceLabel{thm:J2}, and \SourceLabel{cor:noadmission}. \newblock \url{https://enter.grok.me/}. \bibitem{CheungXVB} N.~Cheung. \newblock \emph{XV-B}, \SourceLabel{XV-B.tex}, version 2. \newblock Unpublished technical manuscript, Enter Calculus Companion. \newblock Supplied locators: \SourceLabel{thm:lifting}, \SourceLabel{thm:jetnull}, and \SourceLabel{thm:release}. \newblock Technical lifting and extension interfaces do not by themselves admit the inherited quantum coefficient. \newblock \url{https://enter.grok.me/}. \bibitem{CheungStops} N.~Cheung. \newblock \emph{XVI-N}, \SourceLabel{XVI-N.tex}, revision r12; \emph{XVI-O}, revision o70-sync; and \emph{XVI-Q}, release Q11. \newblock Unpublished supplied source records, Enter Calculus Companion. \newblock The conditional insertion obstruction, the inherited \SourceLabel{NOT_CLOSED} entry, and the separately scoped two-point report have different hypothesis packages. \newblock These documentary comparisons do not update one another through the present paper. \newblock \url{https://enter.grok.me/}. \bibitem{CatalogueRecord} OpenAI. \newblock \emph{OpenAI catalogue}. \newblock Separately attributed catalogue source record reproduced in the Enter Calculus Companion. \newblock Supplied documentary material only. No catalogue proof, count, formalization claim, or family result is used as a premise or independently verified here. \newblock \url{https://enter.grok.me/}. \end{thebibliography} \end{document}