\documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{bm} \usepackage{booktabs,tabularx,longtable,array} \usepackage{enumitem} \usepackage{xcolor} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={IV-B: Represented Records with a Spacetime Enter Interface}, pdfsubject={Complete instruments, prescribed conical geometry, represented scalar-field models, and exact limits of transfer} } \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \setlength{\emergencystretch}{2em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Nzero}{\mathbb N_0} \newcommand{\HH}{\mathcal H} \newcommand{\Fock}{\Gamma_{\mathrm s}} \newcommand{\BB}{\mathcal B} \newcommand{\TT}{\mathcal T} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Dom}{\operatorname{Dom}} \newcommand{\Span}{\operatorname{Span}} \newcommand{\Repart}{\operatorname{Re}} \newcommand{\Impart}{\operatorname{Im}} \newcommand{\dev}{\operatorname{dev}} \newcommand{\Coh}{\operatorname{Coh}} \newcommand{\HS}{\mathrm{HS}} \newcommand{\op}{\mathrm{op}} \newcommand{\dd}{\,\mathrm d} \newcommand{\num}{\mathsf N} \newcommand{\PTT}{\mathsf P_{\mathrm{TT}}} \newcommand{\NW}{\mathrm{NW}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\!\bra{#2}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\ip}[2]{\langle #1,#2\rangle} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddia}{d_{\diamond}} \newcommand{\Mclock}{\mathcal M_C} \newcommand{\Ramsey}{\mathcal I} \newcommand{\rtrace}{\mathfrak r} \newcommand{\status}[1]{\textnormal{\texttt{#1}}} \newcommand{\scopebox}[2]{% \begin{center} \fbox{\begin{minipage}{0.94\linewidth} \textbf{#1}\par\smallskip #2 \end{minipage}} \end{center} } \title{\textbf{IV-B}\\[3pt] Represented Records with a Spacetime Enter Interface\\[5pt] \large Three separate scalar-field instruments on a prescribed conical background, with an explicit completion and transfer boundary} \author{} \date{} \begin{document} \maketitle \begin{abstract} We prove an exact interface adaptation of three separately represented binary measurements to a prescribed conical spacetime. A finite consumed clock context produces a classical time code and Ramsey outcome while retaining the complete conditional matter state, including arbitrary untouched references. The adaptation preserves the positive record contrasts and exact interpolation-off controls of the three represented models. Spectral-mask convergence at fixed particle-number cutoff is strengthened to convergence of the complete recorded channels in diamond norm; initial-edge removal in the unbounded model instead gives pointwise trace-norm convergence, without an unsupported uniform channel-norm claim. A separate target-adapted SWAP construction transports finite payloads exactly along prescribed local null segments, with an explicit register ledger. Two corrections are essential. Local flatness of a cone does not identify its global field representation or Newton inverse with the corresponding Minkowski objects. We therefore state the flat-completion prescription explicitly and prove its local pullback identities, rather than claiming an intrinsic quantum field theory on the entire cone. Likewise, assigning a time to a global controlled unitary does not establish a compact causal implementation. The causal theorem applies to the certified finite transport and control network, not to an unconstructed local realization of the global field apparatus. The contribution is thus an exact, reference-stable Grade-S interface construction with a precisely delimited geometric interpretation. Its potential usefulness is methodological: represented signals, conditional residuals, regulator errors, geometric calibration, and physical implementation can be compared without identifying their distinct mathematical objects. No priority claim is made for the underlying measurement identities. Records do not determine the metric, lapse, or deficit angle. No spacetime-emergence theorem, Einstein dynamics, gravity-generated-entanglement test, inherited gravitational quantum coefficient, or finite-qubit compilation is asserted. \end{abstract} \tableofcontents %===================================================================== \section{Scope, philosophy, and the corrected release} \label{sec:scope} %===================================================================== \subsection{Enter is an output interface} The operational philosophy of Enter distinguishes a consumed quantum name from a returned quantum system. Destructive readout returns a classical record and whatever other quantum factors the interface explicitly retains. Consumption is a statement about access and output type, not physical annihilation of every implementing degree of freedom. Quantum recovery is a different task: its output must return the specified quantum information, including the required reference correlations. Equality of channels also does not specify a thermal machine or a heat ledger. These distinctions are central to Cheung's account of Enter \cite{CheungEnter,CheungPhil,CheungI}. We use specified Born semantics and prove the channel identities needed here. We do not import a measurement-uniqueness theorem from a shortened list of its hypotheses. The spacetime interface follows the prescribed-background interpretation of I-B \cite{CheungIB}, not a reconstruction of geometry from records. The phrase \[ \text{``the fourth dimension is the factor Enter consumes''} \] will have only the following typed meaning: a designated finite clock encoding is consumed, whereas the three named spatial-carrier factors survive. It does \emph{not} mean that the coordinate \(T\), the manifold, or a geometric dimension is destroyed. \subsection{Two grades and one necessary subdivision} \begin{definition}[Grades used in this paper] \label{def:grades} A \emph{Grade-S interface adaptation} consists of: \begin{enumerate} \item a declared spacetime and local rods; \item separately specified represented measurement models; \item a finite destructive clock interface; \item explicit time, location, and causal-wire dictionaries; \item complete-channel identities and the stated regulator estimates. \end{enumerate} An \emph{intrinsic-cone field adaptation} additionally requires a global field representation, inverse prescriptions, domains, and measurement implementation on that same conical spacetime. \emph{Grade E} would require the relevant geometric structure and its calibration to be determined by operational data rather than supplied as inputs. \end{definition} The construction below achieves Grade S in the first, interface sense, using an explicit flat completion for the represented fields. It does not establish the additional intrinsic-cone field adaptation. This subdivision is necessary: the two meanings are not interchangeable. \scopebox{What has been debugged}{% The literal inference \[ \text{local conical flatness} \ \Longrightarrow\ \text{global transfer of all three Minkowski models} \] is invalid. So is \[ \text{a timestamp on a global Ramsey operation} \ \Longrightarrow\ \text{a compact causal system--probe implementation}. \] We prove the corrected interface result and leave these stronger connecting constructions explicitly unclaimed.} \subsection{Provenance and mathematical premises} The coefficient conventions of the represented models are those of IV, version 1 \cite{CheungIV}. They are restated below, and the operator, record, and interface arguments used in this paper are supplied here. I-B, En09-A, En03-A, En05-A, and En08-B are cited for provenance and comparison; their theorems are not silently substituted for the proofs below. Other companion manuscripts are documentary separations only. The external analytic-vector theorem of Nelson \cite{Nelson} is used, with its hypotheses checked. Page--Wootters conditioning \cite{PageWootters} is discussed only through the precise finite conditional-evolution statement proved in Section~\ref{sec:clock}. It is not a claim of externally unscheduled quantum evolution. %===================================================================== \section{The prescribed geometry and the completion boundary} \label{sec:geometry} %===================================================================== \subsection{Conical product, signature, and rods} Fix \[ 0<\mu<2\pi,\qquad \alpha=1-\frac{\mu}{2\pi},\qquad c>0. \] Let \[ C_\mu^\circ=(0,\infty)\times(\R/2\pi\mathbb Z), \qquad M=\R_T\times C_\mu^\circ\times\R_z, \] 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 time orientation is that of \(\partial_T\). \begin{proposition}[Elementary geometric facts] \label{prop:geometry} The metric has signature \((-,+,+,+)\). Static observers have proper-time increment \(\dd\tau=\dd T\). For a nonzero vector \(v=a\partial_T+w\), future causality is equivalent to \[ a>0,\qquad \norm{w}_h\le ca, \qquad h=\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2. \] On an angular interval with a fixed real lift of \(\phi\), the map \begin{equation} \dev(T,r,\phi,z) =\bigl(t=cT,\ y^1=r\cos(\alpha\phi),\ y^2=r\sin(\alpha\phi),\ y^3=z\bigr) \label{eq:dev} \end{equation} is a local isometry to \((\R^{1,3},\eta)\), where \(\eta=\operatorname{diag}(-1,1,1,1)\). \end{proposition} \begin{proof} The metric matrix is diagonal with one negative and three positive entries. The causal inequality is \(-c^2a^2+\norm{w}_h^2\le0\); a nonzero causal vector has \(a\ne0\). Along a static curve, \(-g/c^2=\dd T^2\). Finally, \[ (\dd y^1)^2+(\dd y^2)^2 =\dd r^2+\alpha^2r^2\dd\phi^2, \qquad \dd t=c\dd T, \] which proves the isometry statement. \end{proof} The completion adds the singular locus \[ \mathcal W=\R_T\times\{o\}\times\R_z. \] It is a two-dimensional worldsheet. Fixing \(z\) gives one worldline; the full defect is not a worldline. No smooth metric or ordinary pointwise Einstein tensor at the apex is used. The geometry and its interpretation agree with the prescribed branch of I-B \cite{CheungIB}. The conical and lapse comparisons in VIII-B and Em02-E remain separate source records \cite{CheungVIIIB,CheungEm02E}. \subsection{Why a global transfer does not follow} \begin{proposition}[Local isometry is insufficient] \label{prop:no-global-transfer} A local developing isometry, by itself, supplies neither: \begin{enumerate} \item a global Minkowski plane-wave or TT representation on \(M\); \item an intrinsic Newton inverse on a conical spatial slice; \item a local implementation of a globally specified controlled unitary. \end{enumerate} \end{proposition} \begin{proof} On the angular cover, changing \(\phi\) by \(2\pi\) rotates the developed spatial point by \(2\pi\alpha\). A generic function \(e^{ik\cdot y}\) is not invariant under that rotation. Its pullback therefore does not descend to a single-valued function on the cone. The same issue applies to tensor modes, with the additional tensor rotation. A global mode representation needs compatible gluing data. For the inverse statement, two smooth flat sources may agree on a developing neighborhood but differ outside it. Their Newton potentials on the neighborhood can differ by a nonconstant harmonic function. For example, a positive source bump outside the neighborhood produces a nonzero potential there. Consequently, restriction to the neighborhood does not determine the Newton response. For the implementation statement, a unitary on an abstract global Hilbert space has no specified spatial coupling, intermediate dynamics, or accessible local algebra. Adding a time label supplies none of these missing data. \end{proof} This proposition rules out the stated deductions, not every possible future cone quantization or local apparatus construction. \subsection{The explicit completion used here} \begin{definition}[Declared flat-completion prescription] \label{def:completion} Fix a regular developing patch \(U\subset M\), and write \(D=\dev|_U\). The analytic field models are defined on a separately declared auxiliary Minkowski spacetime \[ \widetilde M=\R_t\times\R_y^3. \] All global wave evolutions, Fourier modes, TT projectors, Newton inverses, states, and cutoffs are evaluated on \(\widetilde M\). A compact scalar test \(f_U\) with support in \(U\) corresponds to \(f=D_*f_U\), extended by zero outside \(D(U)\). Tensor tests are transported with their tensor type and the volume pairing. A represented field or response \(q\) has the patch interpretation \(D^*q\). \end{definition} This prescription is part of the model. It is not an assertion that \(\widetilde M\) is globally isometric to \(M\), or that different completions give identical answers. Every bounded spatial detector region needed below can be accommodated: a sufficiently distant portion of a developing angular sector contains a Euclidean ball of any prescribed finite radius. Translate the whole flat detector package into such a ball. Simultaneous translation of tests, modes, and coherent labels is unitary in the declared flat representations and does not change the numerical pairings. \begin{lemma}[Exact patch agreement] \label{lem:patch} Under Definition~\ref{def:completion}, compact pairings satisfy \[ \int_U f_U\,D^*q\dd\operatorname{vol}_g =\int_{\widetilde M} f q\dd^4x. \] The corresponding tensor identity also holds. Local differential equations and linearized curvature identities pull back through \(D\). A flat zero-data wave calculation agrees with its local calculation whenever the relevant causal domain is contained in \(D(U)\). \end{lemma} \begin{proof} The first assertions are change of variables and naturality of differential operators under an isometry. The volume form is \(\dd\operatorname{vol}_g=c\dd T\dd^3y=\dd t\dd^3y\). For the final assertion, subtract the two solutions on their common domain. The difference solves the homogeneous equation with zero Cauchy data; the local energy identity gives uniqueness. \end{proof} In Model 3, the precise response on \(U\) is therefore \begin{equation} j_U^{\Psi,\Xi} :=D^*\!\left(\mathcal R^\NW_{t_0} t^{\Psi,\Xi}_{\mathrm{complete}}\right). \label{eq:completed-response} \end{equation} The complete source on the right is part of the declared input. Equation~\eqref{eq:completed-response} is not a response determined by the restricted source alone. %===================================================================== \section{Complete binary instruments and quantitative transfer} \label{sec:ramsey} %===================================================================== All Hilbert spaces are complex and separable. Instruments act on trace-class operators; their adjoints act normally on bounded operators. Inner products are antilinear in the first entry. We use \(\hbar=1\) in the represented models. \subsection{The corrected Ramsey formula} For a self-adjoint operator \(A\), set \begin{equation} U_A=e^{i\tau A},\qquad K_s(A)=\frac12\left(I+s e^{-i\theta}U_A\right), \qquad s\in\{+1,-1\}. \label{eq:kraus} \end{equation} The phase in \eqref{eq:kraus} is \(e^{-i\theta}U_A\), not \(e^{-i\theta U_A}\). Define the complete recorded channel \begin{equation} \Ramsey_A(Z) =\sum_{s=\pm1}\kb{s}{s}_Y\otimes K_s(A)ZK_s(A)^*. \label{eq:ramsey-channel} \end{equation} \begin{lemma}[Ramsey identity and record law] \label{lem:ramsey} Equation~\eqref{eq:ramsey-channel} is CPTP and retains every untouched reference. Its outcome probabilities are \begin{equation} p_s(A;\rho) =\frac12+\frac{s}{2} \Repart\!\left(e^{-i\theta}\Tr(\rho e^{i\tau A})\right). \label{eq:ramsey-prob} \end{equation} It is implemented by preparing a probe in \(\ket+\), applying \[ W_A=\kb0{0}\otimes I+\kb1{1}\otimes U_A, \] and destructively reading the probe in the basis \[ \ket{s_\theta} =\frac{\ket0+s e^{i\theta}\ket1}{\sqrt2}. \] \end{lemma} \begin{proof} Direct expansion gives \(\sum_s K_s^*K_s=I\), which proves complete positivity and trace preservation. The same expansion gives \eqref{eq:ramsey-prob}. Moreover, \[ (\bra{s_\theta}\otimes I)W_A(\ket+\otimes I)=K_s(A). \] Tensoring these operator identities with any reference identity proves the complete statements. \end{proof} \subsection{A general positive-record certificate} \begin{lemma}[Mean separation implies a small-time contrast] \label{lem:contrast} Let \(\psi_0,\psi_1\in\Dom A\) be unit vectors, and put \[ d=\abs{\ip{\psi_1}{A\psi_1}-\ip{\psi_0}{A\psi_0}}>0, \qquad S=\norm{A\psi_0}^2+\norm{A\psi_1}^2. \] For \(\theta=\pi/2\) and \(0<\tau\le d/S\), \begin{equation} \abs{p_+(A;\psi_1)-p_+(A;\psi_0)} \ge\frac{\tau d}{4}>0. \label{eq:positive-record} \end{equation} \end{lemma} \begin{proof} The spectral theorem and \(\abs{e^{ix}-1-ix}\le x^2/2\) give \[ \abs{\ip{\psi}{e^{i\tau A}\psi} -1-i\tau\ip{\psi}{A\psi}} \le\frac{\tau^2}{2}\norm{A\psi}^2. \] At \(\theta=\pi/2\), the record reads the imaginary part. Subtracting the two expectations yields \[ \abs{\Delta p_+} \ge\frac{\tau d}{2}-\frac{\tau^2S}{4} \ge\frac{\tau d}{4}. \] \end{proof} \subsection{Complete-channel rather than record-only errors} Use \[ \dtr(\rho,\sigma)=\tfrac12\norm{\rho-\sigma}_1,\qquad \ddia(\Phi,\Psi)=\tfrac12\norm{\Phi-\Psi}_\diamond, \] where the diamond norm is understood on trace-class inputs and arbitrary reference extensions. \begin{proposition}[Uniform instrument transfer] \label{prop:instrument-error} For two unitaries \(U,V\), let \(\Ramsey_U,\Ramsey_V\) be the channels defined by \eqref{eq:kraus}--\eqref{eq:ramsey-channel}. Then \begin{align} \abs{p_s(U;\rho)-p_s(V;\rho)} &\le\frac12\norm{U-V}, \label{eq:prob-error}\\ \ddia(\Ramsey_U,\Ramsey_V) &\le\min\{1,\norm{U-V}/\sqrt2\}. \label{eq:instrument-error} \end{align} For bounded self-adjoint \(A,B\), \begin{equation} \norm{e^{i\tau A}-e^{i\tau B}} \le\abs{\tau}\norm{A-B}. \label{eq:duhamel} \end{equation} These estimates survive compatible CPTP pre- and post-composition. \end{proposition} \begin{proof} The probability bound follows from \eqref{eq:ramsey-prob}. For the complete bound, define isometries \[ V_U\psi=\sum_s\ket{s}\otimes K_s(U)\psi. \] Then \[ \norm{V_U-V_V}=\norm{U-V}/\sqrt2. \] For any reference extension and trace-class input \(Z\), \[ \norm{V_UZV_U^*-V_VZV_V^*}_1 \le2\norm{V_U-V_V}\norm{Z}_1. \] Pinching the first factor produces the recorded channels and cannot increase the norm. This proves \eqref{eq:instrument-error}. Differentiating \(e^{i(\tau-s)A}e^{isB}\) and integrating proves \eqref{eq:duhamel}; negative \(\tau\) follows by reversal. CPTP composition is a contraction for the complete comparison norm. \end{proof} In particular, a two-preparation contrast loses at most \(\norm{U-V}\) under this unitary replacement. This is a complete instrument estimate, not merely agreement of two effects. %===================================================================== \section{Model 1: the fixed TT plane and removable edge} \label{sec:model1} %===================================================================== \subsection{Classical coefficient and represented generator} Work first on the declared completion \(\widetilde M\), with \[ P=-\partial_t^2+\Delta-m^2,\qquad \Omega_m=\sqrt{-\Delta+m^2},\qquad m>0. \] Fix smooth real spatial TT waves \(e,r\) with Schwartz Cauchy data, spanning a nondegenerate real symplectic plane. They have no temporal components. For a free scalar \(\varphi\), define \begin{equation} S_{t_0}(a)\varphi(t) =\int_{t_0}^t \frac{\sin((t-s)\Omega_m)}{\Omega_m} \bigl(-a^{ij}(s)\partial_i\partial_j\varphi(s)\bigr) \dd s,\qquad a=e,r. \label{eq:scalar-response} \end{equation} This solves \(PS_{t_0}(a)\varphi=-L_a\varphi\), with zero correction data, where \(L_a\varphi=-a^{ij}\partial_i\partial_j\varphi\). For real \(f\in C_c^\infty((t_0,t_1)\times\R^3)\), write \[ \ell_{a,f}(\varphi)=\int f S_{t_0}(a)\varphi\dd^4x. \] Set \[ v_f(s)=\int_s^{t_1} \frac{\sin((t-s)\Omega_m)}{\Omega_m}f(t)\dd t,\qquad b_{a,f}=-\partial_i\partial_j(a^{ij}v_f). \] Integration by parts gives \[ \ell_{a,f}(\varphi)= \int_{t_0}^{t_1}\int b_{a,f}(s,x)\varphi(s,x)\dd x\dd s. \] Finite propagation makes the relevant spatial support compact on the finite interval. Let \(u_0,u_e,u_r\in L^2(\R^3)\) be the one-particle vectors representing \(\int f\varphi\), \(\ell_{e,f}\), and \(\ell_{r,f}\), respectively. Explicitly, if a real linear Cauchy functional is \(\int(\alpha\varphi_0+\beta\varpi_0)\dd x\), use \begin{equation} u=\frac{\widehat\alpha}{\sqrt{2\omega}} +i\sqrt{\frac{\omega}{2}}\,\widehat\beta, \qquad \omega(p)=\sqrt{\abs p^2+m^2}. \label{eq:oneparticle} \end{equation} For the response functional, \[ \alpha=\int_{t_0}^{t_1}\cos((s-t_0)\Omega_m)b_{a,f}(s)\dd s, \quad \beta=\int_{t_0}^{t_1} \frac{\sin((s-t_0)\Omega_m)}{\Omega_m}b_{a,f}(s)\dd s. \] All these vectors are Schwartz. On \[ \HH_1=L^2(\R)\otimes\Fock(L^2(\R^3)), \qquad Q=\frac{b+b^*}{\sqrt2},\qquad \Pi_0=-\frac{i(b-b^*)}{\sqrt2}, \] define \begin{equation} A^{(1)}(\lambda) =I\otimes\Phi(u_0) +\lambda\bigl(Q\otimes\Phi(u_e) +\Pi_0\otimes\Phi(u_r)\bigr), \qquad \Phi(u)=a(u)+a^*(u). \label{eq:A1} \end{equation} The real \(\lambda\) defines a numerical model. It is not evaluation of a formal nilpotent parameter at a nonzero real number. \begin{proposition}[Self-adjointness and active factors] \label{prop:A1} The operator \eqref{eq:A1} is essentially self-adjoint on the total finite-particle domain. Its closure contains \(\Dom\num\), and \[ \norm{A^{(1)}(\lambda)\Psi} \le C\norm{(\num+2)\Psi}. \] Only the gravitational oscillator and \(\Span_\C\{u_0,u_e,u_r\}\) are active. \end{proposition} \begin{proof} Creation and annihilation satisfy \[ \norm{a(u)\Psi}\le\norm u\norm{\num^{1/2}\Psi}, \qquad \norm{a^*(u)\Psi}\le\norm u\norm{(\num+1)^{1/2}\Psi}. \] Products of two such operators obey the stated number bound. Different species commute, so the polynomial is symmetric and changes total number by at most two. If \(\Psi\) has at most \(n\) particles, iteration gives \[ \norm{A^j\Psi} \le C^j\prod_{\ell=1}^j(n+2\ell)\norm\Psi \le (2C)^j\frac{(n+j)!}{n!}\norm\Psi. \] Thus every finite-particle vector is analytic for \(A\). Nelson's analytic-vector theorem \cite{Nelson} proves essential self-adjointness. Approximation by number truncations proves the domain inclusion and bound. Fock factorization over the displayed one-particle span and its orthogonal complement proves the final claim. \end{proof} \subsection{Edge removal and its topology} Choose smooth \(0\le\vartheta\le1\), with \(\vartheta(s)=0\) for \(s\le0\) and \(\vartheta(s)=1\) for \(s\ge1\). Replace \[ \mathbf1_{\{s\ge t_0\}}b_{a,f}(s) \quad\hbox{by}\quad \vartheta((s-t_0)/\varepsilon)b_{a,f}(s), \] and denote the resulting operator by \(A_\varepsilon^{(1)}\). Formula~\eqref{eq:oneparticle} and integration over an interval of length \(\varepsilon\) give \begin{equation} \norm{u_{a,\varepsilon}-u_a}\le C_a\varepsilon. \label{eq:edge-vector} \end{equation} Consequently, \[ \norm{(A_\varepsilon^{(1)}-A^{(1)})\Psi} \le D\varepsilon\norm{(\num+2)\Psi}, \] with uniform sector bounds for \(A_\varepsilon^{(1)}\). \begin{proposition}[Edge convergence of complete instruments] \label{prop:edge} For every \(\tau\in\R\), \[ e^{i\tau A_\varepsilon^{(1)}}\longrightarrow e^{i\tau A^{(1)}}\quad\hbox{strongly}. \] For every trace-class \(Z\) on \(\HH_1\otimes\HH_E\), with any untouched reference \(E\), \begin{equation} \norm{\bigl((\Ramsey_{A_\varepsilon^{(1)}} -\Ramsey_{A^{(1)}})\otimes\id_E\bigr)(Z)}_1 \longrightarrow0. \label{eq:edge-complete} \end{equation} No diamond-norm convergence is asserted. \end{proposition} \begin{proof} On the sector of at most \(n\) particles, telescope the powers in the analytic series. For \(2C\abs\tau<1\), the preceding sector estimates give \begin{equation} \norm{(e^{i\tau A_\varepsilon^{(1)}}-e^{i\tau A^{(1)}})\Psi} \le \frac{2D\varepsilon\abs\tau(n+1)} {(1-2C\abs\tau)^{n+2}}\norm\Psi. \label{eq:sector-edge} \end{equation} Finite-particle vectors are dense and all exponentials are unitary, so convergence is strong for these small times. Subdivision of an arbitrary time and finite telescoping prove strong convergence for every \(\tau\). The corresponding Kraus operators and their adjoints converge strongly. For finite-rank \(Z\), this implies trace-norm convergence of each Kraus block, also after tensoring with \(E\). Approximate a general trace-class \(Z\) by finite-rank operators and use the uniform channel bounds to obtain \eqref{eq:edge-complete}. \end{proof} For a coherent preparation of total mean number \(\nu\), an explicit unitary-vector error is obtained from \begin{equation} \mathcal E_\varepsilon(\nu;\tau) = \inf_{n\ge0}\left[ 2\sqrt{\Pr\{\operatorname{Pois}(\nu)>n\}} +\frac{2D\varepsilon\abs\tau(n+1)} {(1-2C\abs\tau)^{n+2}}\right]. \label{eq:coherent-edge} \end{equation} At fixed admissible \(\tau\), this tends to zero by first fixing a large \(n\), then taking \(\varepsilon\downarrow0\). \begin{remark}[Strong convergence is not a uniform channel estimate] For the simple family \(A_\varepsilon=\varepsilon Q\), the unitaries converge strongly to \(I\), but \(\norm{e^{i\tau\varepsilon Q}-I}=2\) whenever \(\tau\varepsilon\ne0\). Wave packets concentrated where the phase is near \(\pi\) also distinguish the corresponding \(\theta=0\) Ramsey records from those of \(I\) almost perfectly. This prevents promotion of a strong-limit proof to a uniform instrument-norm claim. \end{remark} \subsection{A nonzero record and its exact off-control} Translate so that \(e^{ij}(t_0,0)=E^{ij}\ne0\). Choose real compact scalar data \[ \varphi_*(x)=-\tfrac12 E_{ij}x^ix^j\eta_0(x), \qquad \varpi_*=0, \] where \(\eta_0=1\) near zero. Then \[ \partial_t^2 S_{t_0}(e)\varphi_*(t_0,0)=\abs E^2>0. \] A small future neighborhood therefore contains a nonnegative compact test \(f\) for which \[ d_1=\int f S_{t_0}(e)\varphi_*\dd^4x>0. \] Let \(z_*\) be the scalar coherent label for these data. The states \[ \Psi_0=\Coh(0)\otimes\Coh(z_*),\qquad \Psi_1=\Coh(1/\sqrt2)\otimes\Coh(z_*) \] have identical scalar factors, equal oscillator covariances, and \(\Delta\langle Q\rangle=1\), \(\Delta\langle\Pi_0\rangle=0\). Hence \[ \Delta\langle A^{(1)}(\lambda)\rangle=\lambda d_1. \] Their second moments are finite by Proposition~\ref{prop:A1}. \begin{theorem}[Model 1 record] \label{thm:model1} For every fixed \(\lambda\ne0\), some fixed \(\tau>0\) and \(\theta=\pi/2\) give \[ \abs{\Delta p_{+,\varepsilon}}\ge c_1-e_\varepsilon, \qquad c_1>0,\qquad e_\varepsilon\longrightarrow0. \] The limiting contrast is positive. At \(\lambda=0\), the two record laws agree exactly for every \(\varepsilon,\tau,\theta\). \end{theorem} \begin{proof} Apply Lemma~\ref{lem:contrast} with \(d=\abs\lambda d_1\). The edge error is at most \(\tfrac12[\mathcal E_\varepsilon(\nu_0;\tau) +\mathcal E_\varepsilon(\nu_1;\tau)]\). At \(\lambda=0\), the generator acts only on the common scalar factor. \end{proof} By Lemma~\ref{lem:patch}, all compact coefficient pairings have the same values in the declared conical patch. This is the precise, completion- dependent adaptation of Model 1, not a global TT field on the cone. %===================================================================== \section{Model 2: dense TT completion and fixed-number spectral removal} \label{sec:model2} %===================================================================== \subsection{Four legs and the distinction between coefficient norms} On the flat completion define \[ \mathfrak h_g= \left\{z\in L^2(\R^3;\operatorname{Sym}^2\C^3): p^iz_{ij}=0,\ z_i{}^i=0\ \text{a.e.}\right\}, \qquad \mathfrak h_\varphi=L^2(\R^3), \] and \[ \HH_2=\Fock(\mathfrak h_g)\otimes\Fock(\mathfrak h_\varphi). \] Let \(\PTT(p)\) be the orthogonal TT projection. Use modes \[ u_p(t,x)=\frac{e^{-i|p|(t-t_0)+ip\cdot x}} {(2\pi)^{3/2}\sqrt{2|p|}}, \qquad v_q(t,x)=\frac{e^{-iE_q(t-t_0)+iq\cdot x}} {(2\pi)^{3/2}\sqrt{2E_q}}, \quad E_q=\sqrt{|q|^2+m^2}, \] with a factor \(2\) in the TT field, as in IV \cite{CheungIV}. For \(u^+=u,u^-=\overline u\), and similarly for \(v\), define \begin{equation} H_{ij}^{\epsilon\delta}(p,q) =2\PTT_{ij,\ell m}(p)q_\ell q_m \int_{t_0}^{t_1}\int v_f(t,x)u_p^\epsilon(t,x)v_q^\delta(t,x) \dd x\dd t, \qquad \epsilon,\delta=\pm. \label{eq:fourlegs} \end{equation} The four operators are annihilation--annihilation, annihilation--creation, creation--annihilation, and creation--creation, with the reality relations supplied by real \(f\). Put \[ L=H^{-+}:\mathfrak h_\varphi\to\mathfrak h_g,\qquad P_2=H^{--}\in\mathfrak h_g\otimes\mathfrak h_\varphi. \] The tensor \(P_2\) is not the particle-number projection. Take smooth real even masks \(0\le a_R,b_R\le1\), equal to one for \(R^{-1}\le|p|\le R\) and \(|q|\le R\), respectively, and supported in the corresponding doubled regions. Set \[ H_R^{\epsilon\delta}(p,q)=a_R(p)b_R(q) H^{\epsilon\delta}(p,q). \] \begin{lemma}[Separate spectral estimates] \label{lem:spectral} The pair coefficient is Hilbert--Schmidt and the mixed coefficient is bounded. There are constants independent of the admitted masks such that \begin{equation} \norm{P_{2,R}-P_2}_{\HS}\le C_P R^{-1/2}, \qquad \norm{L_R-L}_{\op}\le C_LR^{-1}. \label{eq:spectral} \end{equation} \end{lemma} \begin{proof} Write \(\ell=t_1-t_0\), \[ F(s,k)=\int v_f(t_0+s,x)e^{ik\cdot x}\dd x,\qquad J(k,\nu)=\int_0^\ell F(s,k)e^{-i\nu s}\dd s. \] Spatial integration by parts and one temporal integration by parts, retaining the initial endpoint \(F(0,k)\), give \[ |J(k,\nu)|\le C_n\langle k\rangle^{-n}(1+|\nu|)^{-1}. \] For \(k=\epsilon p+\delta q\), transversality gives \(\PTT(p)(q\otimes q)=\PTT(p)(k\otimes k)\). After absorbing polynomial transfer factors into the arbitrary spatial decay, \begin{equation} |H^{\epsilon\delta}(p,q)| \le \frac{C_n\langle\epsilon p+\delta q\rangle^{-n}} {\sqrt{|p|E_q}\,[1+|\epsilon|p|+\delta E_q|]}. \label{eq:master} \end{equation} For pair signs this implies, after increasing \(n\), \[ |P_2(p,q)|^2 \le \frac{C_n\langle p+q\rangle^{-n}} {|p|\langle p\rangle^3}. \] The radial integral is finite. The gravitational low and high tails have squared integrals \(O(R^{-2})\) and \(O(R^{-1})\). The scalar high tail is \(O(R^{-1})\), by splitting \(|p|<1\) from \(|p|\ge1\) and transferring weights between \(p,q,p+q\). This proves the first estimate. For mixed signs, \eqref{eq:master} gives \[ |L(p,q)|\le C_n \begin{cases} |p|^{-1/2}\langle q\rangle^{-n},& |p|<1,\\ \langle p-q\rangle^{-n}\langle p\rangle^{-1},& |p|\ge1, \end{cases} \] and, on \(|p|\ge1\), also the bound with \(\langle q\rangle^{-1}\). The low part is Hilbert--Schmidt, with low-tail norm \(O(R^{-1})\). The two high-tail Schur integrals are \(O(R^{-1})\), using the appropriate \(p\)- or \(q\)-weighted bound; the remaining low-\(p\), high-\(q\) term decays faster. The Schur test proves the second estimate. \end{proof} The proof does not replace the mixed operator-norm estimate by an unproved Hilbert--Schmidt estimate. \subsection{Quadratic operators and the free-term correction} Identify the product Fock space with \(\Fock(\mathfrak h_g\oplus\mathfrak h_\varphi)\). Put \[ T_R=\begin{pmatrix}0&L_R\\L_R^*&0\end{pmatrix}, \] and let \(G_R\) be the symmetric tensor obtained by placing \(P_{2,R}\) in the two cross-species slots and adding its flip. Thus \(\norm{G_R}=\sqrt2\norm{P_{2,R}}_{\HS}\). Define the symmetric quadratic \[ B_R=\mathrm d\Gamma(T_R) +\tfrac12\bigl(a^*(G_R)+a(G_R)\bigr). \] With \(\num=\num_g+\num_\varphi\) and \(\Pi_M=\mathbf1_{\{\num\le M\}}\), sector estimates give \begin{align} \norm{B_R\Psi} &\le C_B\norm{(\num+2)\Psi}, \label{eq:B2graph}\\ \norm{\Pi_M(B_R-B_\infty)\Pi_M} &\le M\norm{L_R-L} +\sqrt{2M(M-1)}\,\norm{P_{2,R}-P_2}_{\HS}. \label{eq:B2compression} \end{align} For completeness, on the \(n\)-particle sector, \(\mathrm d\Gamma(T)\) has norm at most \(n\norm T\), while pair annihilation has norm at most \(\sqrt{n(n-1)}\norm G\). Pair creation is its adjoint. Summing the orthogonal output sectors proves \eqref{eq:B2graph}; compression and \(\norm{\Repart C}\le\norm C\) prove \eqref{eq:B2compression}. Define \begin{equation} F_{0,M} =\mathbf1_{\{\num_\varphi\le M\}}\Phi(u_0) \mathbf1_{\{\num_\varphi\le M\}}, \qquad A^{(2)}_{R,M}(\lambda) =I_g\otimes F_{0,M}+\lambda\Pi_MB_R\Pi_M. \label{eq:A2} \end{equation} Each generator is bounded self-adjoint. Its free scalar term is compressed by \(\num_\varphi\), not by total number. \begin{remark}[Why the separate free cutoff matters] Replacing the first term of \eqref{eq:A2} by \(\Pi_M(I_g\otimes\Phi(u_0))\Pi_M\) can make its record depend on gravitational occupation even at \(\lambda=0\). The scalar-only compression is required for the exact off-control. \end{remark} \begin{theorem}[Fixed-\(M\) complete spectral limit] \label{thm:spectral-channel} For fixed finite \(M\ge2\), put \[ \delta_M(R)=MC_LR^{-1} +\sqrt{2M(M-1)}C_PR^{-1/2}. \] Then \[ \norm{A^{(2)}_{R,M}-A^{(2)}_{\infty,M}} \le|\lambda|\delta_M(R), \] and \begin{equation} \ddia\!\left(\Ramsey_{A^{(2)}_{R,M}}, \Ramsey_{A^{(2)}_{\infty,M}}\right) \le \min\left\{1,\frac{|\tau\lambda|}{\sqrt2}\delta_M(R)\right\}. \label{eq:model2-complete} \end{equation} The estimate is uniform over states and references. \end{theorem} \begin{proof} Combine Lemma~\ref{lem:spectral}, \eqref{eq:B2compression}, and Proposition~\ref{prop:instrument-error}. \end{proof} No \(M\to\infty\) instrument limit is used. \subsection{A nonzero witness with a controlled number tail} Choose a smooth compact-annular TT coherent label whose classical field \(k_*\) is nonzero somewhere. Choose compact smooth scalar data so that \(L_{k_*}\varphi_*\) is nonzero there. The same Hessian construction used in Model 1 provides such data. Then \(\psi_*=S_{t_0}(k_*)\varphi_*\) is not zero. Choose a nonnegative compact bump \(b\) supported where it is nonzero and set \[ f_*=b\psi_*,\qquad d_2=\int b\psi_*^2\dd^4x>0. \] Let \(z_g,z_\varphi\) be the corresponding coherent labels and take \[ \Psi_0=\Omega_g\otimes\Coh(z_\varphi),\qquad \Psi_a=\Coh(az_g)\otimes\Coh(z_\varphi),\qquad a\ne0. \] Expanding the four legs gives exactly \[ \Delta\langle B_\infty\rangle=a d_2. \] For total coherent mean number \(\nu\), define \[ t_M(\nu)= \left[e^{-\nu}\sum_{n>M-2}(n+2)^2\frac{\nu^n}{n!}\right]^{1/2}. \] Number bandwidth two and \eqref{eq:B2graph} imply \[ \norm{(\Pi_MB_R\Pi_M-B_R)\Psi} \le2C_Bt_M(\nu). \] Therefore \[ \Delta\langle\Pi_MB_\infty\Pi_M\rangle =a d_2+\epsilon_M,\qquad |\epsilon_M|\le E_M,\qquad E_M\longrightarrow0. \] The same number estimates bound \(\norm{A^{(2)}_{R,M}\Psi_s}\) uniformly in \(R,M\). \begin{theorem}[Model 2 record] \label{thm:model2} For fixed \(a,\lambda\ne0\), one can choose a single \(\tau>0\), independent of \(R,M\), and \(\theta=\pi/2\), such that \[ |\Delta p_{+,R,M}| \ge c_2-e_M-e_R(M),\qquad c_2>0, \] where \[ e_M\longrightarrow0,\qquad e_R(M)\le|\tau\lambda|\delta_M(R)\longrightarrow0 \quad\text{at fixed }M. \] For sufficiently large but finite \(M\), the limiting contrast is positive. At \(\lambda=0\), the two record laws agree exactly for every \(R,M,\tau,\theta\). \end{theorem} \begin{proof} Let \(d=|\lambda a|d_2\), and let \(S_2\) be a common bound on the sum of the two generator second moments. Choose \(0<\tau\le d/S_2\). The proof of Lemma~\ref{lem:contrast} gives \[ |\Delta p_{+,\infty,M}| \ge\frac{\tau d}{4}-\frac{\tau|\lambda|E_M}{2}. \] Take \(c_2=\tau d/4\) and \(e_M=\tau|\lambda|E_M/2\). The two probability errors from the spectral replacement total at most \(e_R(M)\). Finally, \(A^{(2)}_{R,M}(0)=I_g\otimes F_{0,M}\), and the scalar preparations are identical. \end{proof} The conical patch interpretation uses Definition~\ref{def:completion}. The masks remain masks of that completion; they are not silently changed into intrinsic cone spectral projectors. %===================================================================== \section{Model 3: a specified sourced form and an exterior pairing} \label{sec:model3} %===================================================================== \subsection{The Newton--wave prescription} Translate \(t_0\) to zero. On the completion, let \(Q_{\mu\nu}\) be a smooth conserved symmetric source, Schwartz in space with all seminorms locally uniform in time. Define \[ (\Delta^{-1}b)(x) =-\frac1{4\pi}\int\frac{b(y)}{|x-y|}\dd y, \qquad \rho=Q_{00}(0),\quad J_i=Q_{0i}(0), \] and \[ a=-2\Delta^{-1}\rho,\qquad V_i=2\Delta^{-1}J_i. \] For trace-reversed data \(A_{\mu\nu},B_{\mu\nu}\), prescribe \[ A_{00}=a,\quad A_{0i}=-V_i,\quad A_{ij}=0, \qquad B_{00}=-\partial_iV_i,\quad B_{0i}=B_{ij}=0. \] With \(D_0=\sqrt{-\Delta}\), put \begin{equation} \overline j(t) =\cos(tD_0)A+\frac{\sin(tD_0)}{D_0}B +2\int_0^t\frac{\sin((t-s)D_0)}{D_0}Q(s)\dd s, \qquad j=\rtrace\overline j, \label{eq:NW} \end{equation} where \[ (\rtrace h)_{\mu\nu} =h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\eta^{\alpha\beta}h_{\alpha\beta}. \] \begin{proposition}[Response and propagated constraint] \label{prop:NW} Equation~\eqref{eq:NW} defines a smooth response \(\mathcal R^\NW Q=j\), continuous from suitable source Schwartz seminorms to compact smooth seminorms. It satisfies \[ \Box\overline j=-2Q,\qquad D_\nu(j):=\partial^\mu\overline j_{\mu\nu}=0,\qquad G^{(1)}(j)=Q. \] These equations are built into the prescribed response; they are not derived gravitational dynamics. \end{proposition} \begin{proof} The Newton multiplier \(-|p|^{-2}\) is locally integrable in three dimensions. Its differentiated multipliers and the wave propagators give smoothness and compact seminorm bounds. The Newton data can have Coulomb tails; square integrability of the metric itself is not needed. The wave formula gives \(\Box\overline j=-2Q\). Set \(d_\nu=-\dot{\overline j}_{0\nu} +\partial_i\overline j_{i\nu}\). The prescribed data give \(d(0)=0\). Since \(\Delta a=-2\rho\) and \(\Delta V_i=2J_i\), the wave equation also gives \(\dot d(0)=0\). Conservation gives \(\Box d=0\); uniqueness implies \(d=0\). Substitution into the linearized Einstein operator gives \(G^{(1)}(j)=-\tfrac12\Box\overline j=Q\). \end{proof} \subsection{A form, not an independent metric field} On \[ \HH_3=\Fock(L^2(\R^3)),\qquad \mathcal D_3=\Fock^{\mathrm{alg}}(\mathcal S(\R^3)), \] use the massive free scalar and the fixed vacuum-normal-ordered stress \[ :T_{\mu\nu}: =:\partial_\mu\widehat\varphi\,\partial_\nu\widehat\varphi: -\frac12\eta_{\mu\nu}: (\partial^\alpha\widehat\varphi\,\partial_\alpha\widehat\varphi +m^2\widehat\varphi^2):. \] For \(\Psi,\Xi\in\mathcal D_3\), its matrix element \(t_{\mu\nu}^{\Psi,\Xi}(x)\) is a finite sum of products of Schwartz wave packets and derivatives. Polarized Klein--Gordon conservation shows that it is conserved. For real compact symmetric \(J^{\mu\nu}\), define \begin{equation} j^\NW(J)[\Psi,\Xi] =\int J^{\mu\nu} \bigl(\mathcal R^\NW t^{\Psi,\Xi}\bigr)_{\mu\nu}\dd^4x. \label{eq:metric-form} \end{equation} The preceding seminorm estimates prove that this is a well-defined sesquilinear form, symmetric for real \(J\). Its conical patch interpretation is \eqref{eq:completed-response} and Lemma~\ref{lem:patch}. \subsection{Regulated quadratic generator} For clarity, the complete kernel prescription is recorded. Let \[ n_p=(2\pi)^{-3/2}(2\omega_p)^{-1/2}, \quad \ell_\mu(p)=(-\omega_p,p),\quad k=\sigma p+\delta q,\quad r=|k|,\quad \Omega=\sigma\omega_p+\delta\omega_q, \] and define \[ C_{\mu\nu}^{\sigma\delta}(p,q) =n_pn_q\left[ -\sigma\delta\,\ell_{(\mu}(p)\ell_{\nu)}(q) +\frac12\eta_{\mu\nu} \bigl(\sigma\delta\,\eta^{\alpha\beta} \ell_\alpha(p)\ell_\beta(q)-m^2\bigr)\right]. \] Apply the initial-data rule of \eqref{eq:NW} to \(C^{\sigma\delta}e^{-i\Omega t+ik\cdot x}\), obtaining its trace-reversed amplitude \(F^{\sigma\delta}(t;p,q)\). Explicitly, \[ A_{00}=\frac{2C_{00}}{r^2},\qquad A_{0i}=\frac{2C_{0i}}{r^2},\qquad A_{ij}=0, \quad B_{00}=ik_iA_{i0},\quad B_{0i}=B_{ij}=0, \] and \[ F(t)=\cos(rt)A+\frac{\sin(rt)}rB +2C\int_0^t\frac{\sin(r(t-s))}{r}e^{-i\Omega s}\dd s. \] Define \[ \mathcal K_J^{\sigma\delta}(p,q) =\int J^{\mu\nu}(t,x)e^{ik\cdot x} (\rtrace F^{\sigma\delta})_{\mu\nu}(t;p,q) \dd t\dd x. \] Choose a real even smooth compact mask \(\chi_\Lambda\), equal to one on \(|p|\le\Lambda\), and set \[ h_\Lambda(p,q)=2\chi_\Lambda(p)\chi_\Lambda(q) \mathcal K_J^{-+}(p,q), \qquad g_\Lambda(p,q)=2\chi_\Lambda(p)\chi_\Lambda(q) \mathcal K_J^{--}(p,q). \] \begin{lemma}[Pair and mixed transfer bounds] \label{lem:model3-kernels} At each fixed finite \(\Lambda\), \(g_\Lambda\) is a symmetric Hilbert--Schmidt tensor and \(h_\Lambda\) is bounded self-adjoint. More precisely, on the cutoff support, \[ |\mathcal K_J^{++}|+|\mathcal K_J^{--}| \le C_{J,\Lambda}(1+|p+q|^{-1}), \] and \[ |\mathcal K_J^{-+}|+|\mathcal K_J^{+-}| \le C_{J,\Lambda}(1+|p-q|^{-2}). \] No separate Newton infrared mask is required. \end{lemma} \begin{proof} For the pair sector, at \(q=-p\), \[ -\omega_p\omega_q-p\cdot q+m^2=0,\qquad \omega_pq_i+\omega_qp_i=0. \] The first expression also has vanishing first derivative there. On bounded momentum sets, smoothness of the massive dispersion relation therefore gives \(C_{00}^{++}=O(|p+q|^2)\) and \(C_{0i}^{++}=O(|p+q|)\). Thus the initial Newton amplitudes are respectively \(O(1)\) and \(O(|p+q|^{-1})\), and \(B_{00}=O(1)\). The wave factors are bounded on the finite time support, proving the pair bound. Its square is integrable in the three-dimensional transfer variable. In the mixed sector, the bounded numerators give at worst \(O(|p-q|^{-2})\). The sine-over-\(r\) factors are bounded on compact time intervals, so no stronger singularity occurs. On bounded momentum support, \[ \sup_p\int(1+|p-q|^{-2})\dd q<\infty, \] and likewise with \(p,q\) interchanged. The Schur test gives boundedness. Reality and polarization give the stated symmetries. \end{proof} Define on \(\Dom\num\) \[ B_\Lambda(J)=\mathrm d\Gamma(h_\Lambda) +\tfrac12\bigl(a^*(g_\Lambda)+a(g_\Lambda)\bigr). \] The sector estimates already used give \begin{equation} \norm{B_\Lambda(J)\Psi} \le C_\Lambda\norm{(\num+2)\Psi}, \qquad B_{\Lambda,M}(J)=\Pi_MB_\Lambda(J)\Pi_M \ \text{bounded self-adjoint}. \label{eq:B3} \end{equation} Neither \(\Pi_M\) nor the bounded momentum cutoff is asserted to have finite rank. Normal ordering gives the exact coherent identity \begin{equation} \ip{\Coh(z)}{B_\Lambda(J)\Coh(z)} =\int J^{\mu\nu} \bigl(\mathcal R^\NW T(\varphi_{\chi_\Lambda z})\bigr)_{\mu\nu} \dd^4x. \label{eq:coherent3} \end{equation} It follows by replacing the normal-ordered field factors by their coherent one-point functions. The graph bound justifies the expectation. \subsection{Exterior curvature certificate and finite cutoffs} Use \[ R^{(1)}_{1212}(h) =\partial_1\partial_2h_{12} -\tfrac12\partial_1^2h_{22} -\tfrac12\partial_2^2h_{11}. \] For real compact \(\psi\), set \[ J_\psi^{11}=-\tfrac12\partial_2^2\psi,\quad J_\psi^{22}=-\tfrac12\partial_1^2\psi,\quad J_\psi^{12}=J_\psi^{21} =\tfrac12\partial_1\partial_2\psi, \] with all other entries zero. Direct differentiation and integration by parts give \begin{equation} \partial_\mu J_\psi^{\mu\nu}=0,\qquad \int J_\psi^{\mu\nu}h_{\mu\nu}\dd^4x =\int\psi R^{(1)}_{1212}(h)\dd^4x. \label{eq:curvature-current} \end{equation} Choose nonzero real radial \(F\in C_c^\infty(\R^3)\), supported in \(|x|\le r_0\), and scalar data \((F,0)\). Their energy is \[ E=\frac12\int(|\nabla F|^2+m^2F^2)\dd x>0. \] The initial momentum density vanishes. Newton's radial formula and trace reversal give \[ j_{ij}(0,x)=\frac{E}{4\pi|x|}\delta_{ij} \quad (|x|>r_0). \] At \(x_*=(0,0,r_*)\), \(r_*>r_0\), \[ R^{(1)}_{1212}(j)(0,x_*)=\frac{E}{4\pi r_*^3}>0. \] Smoothness and finite propagation permit a small future neighborhood where the curvature remains positive and the uncut scalar vanishes. Choose a nonzero nonnegative \(\psi\) there. Then \[ d_3=\int\psi R^{(1)}_{1212} (\mathcal R^\NW T(\varphi_*))\dd^4x>0. \] Let \(z_*\) be the Schwartz coherent label for \((F,0)\). Since \(\chi_\Lambda z_*\to z_*\) in Schwartz topology, \eqref{eq:coherent3} and response continuity allow a sufficiently large \emph{finite} \(\Lambda\) with \[ \langle B_\Lambda(J_\psi)\rangle_{\Coh(z_*)}\ge3d_3/4. \] The weighted Poisson tail and \eqref{eq:B3} imply \[ \langle B_{\Lambda,M}(J_\psi)\rangle_{\Coh(z_*)} \longrightarrow \langle B_\Lambda(J_\psi)\rangle_{\Coh(z_*)}. \] For example, writing \[ T_M(\nu)^2=e^{-\nu}\sum_{n>M}(n+2)^2\frac{\nu^n}{n!}, \qquad \nu=\norm{z_*}^2, \] the expectation error is at most \(2C_\Lambda\sqrt{\nu^2+5\nu+4}\,T_M(\nu)\). Fix a finite \(M\) for which the compressed mean is at least \(d_3/2\). Its vacuum mean is exactly zero. \begin{theorem}[Model 3 record] \label{thm:model3} At the selected finite \(\Lambda,M\), the generator \[ A^{(3)}(\lambda)=\lambda B_{\Lambda,M}(J_\psi) \] has a positive vacuum--coherent Ramsey contrast for every \(\lambda\ne0\), at a sufficiently small fixed \(\tau>0\) and \(\theta=\pi/2\). At \(\lambda=0\), the record laws agree exactly. If the declared source response is multiplied by an additional classical strength \(\zeta\), the source-off control \(\zeta=0\) also gives exact agreement. \end{theorem} \begin{proof} The mean separation is at least \(|\lambda|d_3/2\), and both second moments are finite by boundedness. Apply Lemma~\ref{lem:contrast}. At either off-control, \(U=I\), so \(p_s=(1+s\cos\theta)/2\), independently of the preparation. \end{proof} The cutoff field is not spatially compact: momentum truncation creates tails. Thus exterior support separation is an uncut classical certificate; only its positive compact pairing has been transferred to the retained finite cutoffs. %===================================================================== \section{The finite destructive clock instrument} \label{sec:clock} %===================================================================== \subsection{Complete identity, references, and consumption} Let \(C\) have finite dimension \(d\), with orthonormal basis \(\{\ket{\tau_k}\}_{k=0}^{d-1}\) and projectors \(P_k=\kb{\tau_k}{\tau_k}\). Let \(X\) be any retained system, possibly one of the infinite-dimensional represented matter systems. \begin{definition}[Destructive Enter channel] \begin{equation} \Mclock(Z) =\sum_k\kb{k}{k}_R\otimes \Tr_C\!\left[(P_k\otimes I_X)Z(P_k\otimes I_X)\right]. \label{eq:enter} \end{equation} The output quantum context is \(X\), not \(C\otimes X\). \end{definition} \begin{theorem}[Complete clock extraction] \label{thm:instrument} The channel \eqref{eq:enter} is CPTP. With \(D_k=\bra{\tau_k}\otimes I_X\), it equals \[ \Mclock(Z)=\sum_k\kb{k}{k}\otimes D_kZD_k^*. \] For every untouched reference \(E\), this equality remains valid after tensoring with \(\id_E\). Moreover, \begin{equation} \Tr_R[(\Mclock\otimes\id_E)(Z)]=\Tr_C Z. \label{eq:drop-record} \end{equation} The conditional residual on a positive-probability branch is \(D_k\rho D_k^*/\Tr(D_k\rho D_k^*)\). \end{theorem} \begin{proof} The operators \(A_k=\ket{k}_R\otimes D_k\) satisfy \[ \sum_kA_k^*A_k=\sum_kP_k\otimes I_X=I. \] They therefore define a CPTP map. Expanding \(Z=\sum_{j,\ell}\kb{\tau_j}{\tau_\ell}\otimes Z_{j\ell}\) shows that both branch formulas equal \(Z_{kk}\). Summing gives \eqref{eq:drop-record}. The calculation is unchanged with \(X\) replaced by \(X\otimes E\). \end{proof} Conditional reference states can be steered by conditioning. Equation~\eqref{eq:drop-record} asserts invariance of the \emph{unconditional} surviving marginal, not of every normalized branch. The identity is the finite instrument comparison emphasized in En09-A \cite{CheungEn09}; here it has been proved for the retained possibly infinite-dimensional \(X\). \subsection{Three separations} \begin{proposition}[No return, no residual inference, no finite CCR] \label{prop:three-separations} For \(d\ge2\): \begin{enumerate} \item \(\Mclock\) has no universal left inverse on all clock--system states; \item equality of record effects does not imply equality of recorded residual instruments; \item finite matrices cannot satisfy \([T_C,H_C]=iI_C\). \end{enumerate} \end{proposition} \begin{proof} The two clock states \((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), tensored with the same residual, have the same image under \(\Mclock\). A left inverse would have to recover both distinct inputs from that image. For the second assertion, on a residual qubit compare \[ \mathcal A_s(Z)=\tfrac12 Z,\qquad \mathcal B_s(Z)=\tfrac12 XZX,\qquad s=\pm1. \] Both instruments have effects \(I/2\), but on \(\ket0\) their conditional residuals are \(\ket0\) and \(\ket1\). Thus their records agree on every input and their residuals need not. Finally, \(\Tr[T_C,H_C]=0\), whereas \(\Tr(iI_C)=id\ne0\). \end{proof} A fresh allocation after readout is a new quantum name, not return of a consumed name. The record-only clock channel is entanglement breaking on the clock input. The complete map on \(C\otimes X\) need not be entanglement breaking, because the retained \(X\) can preserve entanglement with a reference. \subsection{Refinement independence} \begin{proposition}[Destructive coarse-graining] \label{prop:refinement} Suppose mutually orthogonal projectors \(P_k\) are refined into orthonormal rank-one projectors \(P_{k\ell}\). Destructive fine readout followed by forgetting \(\ell\) has residual block \[ \sum_\ell\Tr_C(P_{k\ell}ZP_{k\ell}) =\Tr_C(P_kZP_k). \] It is independent of the chosen orthonormal basis within \(\operatorname{Ran}P_k\). \end{proposition} \begin{proof} Expand \(P_kZP_k\) in any orthonormal basis of its clock subspace. Partial trace removes all off-diagonal clock matrix units and sums the diagonal residual blocks. \end{proof} This is a destructive-readout statement. It is not an assertion that fine dephasing equals a nondestructive coarse Lüders channel when the clock itself is returned. \subsection{Conditional evolution requires additional hypotheses} \begin{proposition}[Finite covariant conditional evolution] \label{prop:relational} Let \(U_C(t),U_X(t)\) be prescribed unitary groups, and assume \[ (U_C(t)\otimes U_X(t))\rho (U_C(t)^*\otimes U_X(t)^*)=\rho \] at the relevant times. Suppose a complete rank-one reading basis obeys \[ P_k=U_C(t_k)P_0U_C(t_k)^*. \] Then its subnormalized residuals satisfy \[ \sigma_k=U_X(t_k)\sigma_0U_X(t_k)^*, \qquad p_k=1/d. \] Hence the normalized residuals have the corresponding unitary relation. \end{proposition} \begin{proof} Stationarity gives \[ (U_C(t)^*\otimes I)\rho(U_C(t)\otimes I) =(I\otimes U_X(t))\rho(I\otimes U_X(t)^*). \] Substitute the covariance formula for \(P_k\) into the partial trace. Unitary invariance of the clock trace gives the displayed relation. All traces \(p_k\) are equal and sum to one. \end{proof} This is the precise finite conditioning comparison associated with Page--Wootters and En09-A \cite{PageWootters,CheungEn09}. Stationarity alone is insufficient for an arbitrary reading basis. For a referenced conditional-evolution statement, stationarity must hold on the full referenced preparation. None of these assumptions is imposed automatically on the Ramsey preparations. %===================================================================== \section{The same-model dictionary and Ramsey--Enter composition} \label{sec:dictionary} %===================================================================== \subsection{Named factors and the three maps} For each model \(i=1,2,3\), retain the separate matter space \(\HH_i\). Add three finite named carrier factors \(L_1,L_2,L_3\). They are not a spatial tensor factorization of the continuum field. Their fixed locations mark the three developing rods. Use one total consumed context \[ C=C_T\otimes C_B,\qquad C_T\cong\C^{d_T},\qquad C_B\cong\C^2. \] Here \(C_T\) is finite time-code memory and \(C_B\) is the Ramsey probe. Taking \(d_T\) a power of two gives a finite qubit encoding. The retained context is \[ X_i=L_1\otimes L_2\otimes L_3\otimes\HH_i. \] Both clock/probe components are consumed by the complete readout. The dictionaries are: \begin{enumerate} \item \emph{Time map.} A fixed injective calibration \[ \gamma_T:k\longmapsto T_k. \] The binary outcome \(s\) is not another value of the event time. A separate history map \(\kappa\) may assign \(k\) to a labelled history event. \item \emph{Location map.} A map \(\ell\) sends each admitted finite carrier name and tick to a point of the regular spatial slice. Its three numerical rod coordinates are \(y\circ\ell\), using \eqref{eq:dev}. The field module \(\HH_i\) remains a global represented module; this map does not localize every one of its modes at the probe. \item \emph{Causal-wire map.} For every certified scheduled dependency \(a\to b\), \[ \ell_{\mathrm{event}}(b)\in J_g^+\!\left(\ell_{\mathrm{event}}(a)\right). \] Equivalently, an ancestor of an output belongs to the metric \emph{past} of that output. \end{enumerate} These are written maps in one declared model. They are not inferred from similarity of terminology between manuscripts. \begin{proposition}[Fixed-event calibration] \label{prop:event} If a clock is read at one fixed event \(e\), and \(\gamma_T(K)=T(e)\) almost surely with injective \(\gamma_T\), then \(K\) is deterministic. \end{proposition} \begin{proof} At most one \(k\) has \(T_k=T(e)\). Every other outcome must have probability zero. \end{proof} Consequently, a stochastic multi-time interpretation at one fixed event must be a history-label interpretation, or an explicitly imperfect clock calibration, not exact equality to several coordinates of that one event. \subsection{One complete composed channel} Prepare \(C_T\) in \(\ket{\tau_{k_0}}\), where \(T_{k_0}=T(e_i)\), and \(C_B\) in \(\ket+\). Apply \(W_{A_i}\) to \(C_B\otimes\HH_i\), leaving the rod factors untouched. Destructively read \(C\) in the basis \[ \ket{\tau_k}\otimes\ket{s_\theta}. \] \begin{theorem}[Ramsey--Enter identity] \label{thm:composition} For each of the three generators and every trace-class input \(Z\) on \(X_i\), the complete output is \begin{equation} \mathcal E_i(Z) = \sum_{s=\pm1} \kb{k_0}{k_0}_{R_T}\otimes\kb{s}{s}_Y \otimes(I_L\otimes K_s(A_i))Z(I_L\otimes K_s(A_i))^*. \label{eq:composed} \end{equation} This identity holds with every untouched reference. The time code is deterministic, the Ramsey outcome need not be, and all \(L_1,L_2,L_3\) factors survive. All contrast and convergence statements in Theorems~\ref{thm:model1}, \ref{thm:spectral-channel}, \ref{thm:model2}, and \ref{thm:model3} survive this attachment in their stated topologies. \end{theorem} \begin{proof} Apply Theorem~\ref{thm:instrument} to the two-factor clock and use Lemma~\ref{lem:ramsey}. The \(C_T\) bra annihilates every branch except \(k_0\); the \(C_B\) bra gives \(K_s(A_i)\). The rod identity and the reference identity tensor through the calculation. Attachment of a fixed classical code is an isometry on the comparison interface and changes no record probabilities. Compatible CPTP composition preserves the stated error bounds. \end{proof} For a preparation with several history blocks, the same calculation gives their subnormalized conditional blocks. It does not turn those history labels into the actual times of one fixed readout event. At \(\lambda=0\), Models 1 and 2 retain different gravitational preparations, so their complete \emph{matter} outputs need not agree. What agrees exactly is the declared binary record law. No arbitrary later gravitational-sensitive measurement is included in that off-control claim. \subsection{Dictionary nonuniqueness} \begin{proposition}[The complete records do not select the dictionary] \label{prop:nonunique} The finite recorded channels constructed here do not uniquely determine the rod frame, the lapse, or the global conical deficit. \end{proposition} \begin{proof} Translate or rotate the developing coordinates and all finite carrier locations. The matrices and the complete instruments are unchanged, while the rod numbers change. For the lapse comparison, retain the same coordinate events and channels but compare \(g_1=-c^2\dd T^2+h\) and \(g_2=-4c^2\dd T^2+h\). Every wire causal for \(g_1\) is causal for \(g_2\), but a static coordinate interval has different proper durations. The same channel data therefore do not fix the lapse calibration. Finally, two different deficits admit isometric regular balls. Install the same finite ports and the same flat-completion detector package in such balls. Their complete output channels coincide, although the global circumference-to-radius ratios differ. \end{proof} The lapse comparison is a non-identification test, not a replacement of the main unit-lapse freeze. These are the same kinds of distinctions recorded in I-B \cite{CheungIB}; the proof here does not assume a metric-reconstruction theorem. %===================================================================== \section{Certified transport and the precise locality statement} \label{sec:transport} %===================================================================== \subsection{Schedule past and metric containment} Consider finitely many finite carrier spaces at prescribed sites. At tick \(j\), let the complete channel factor over a partition \(\mathcal B_j\) of those sites. Include every classical control wire in this network. For terminal sites \(Y\), 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}[Scheduled influence bound] \label{thm:locality} The pulled-back terminal algebra satisfies \[ (\Phi_1^*\cdots\Phi_n^*)(\BB(\HH_Y)\otimes I) \subseteq \BB(\HH_{N_{0,n}(Y)})\otimes I. \] If every block dependency is certified metric causal, then each scheduled ancestor event belongs to the metric past of a terminal event. Equal input reductions on the scheduled past give equal output reductions. With references, equality of the corresponding joint input reductions gives equality of the joint output reductions. \end{theorem} \begin{proof} The adjoint of a channel is unital. A block disjoint from the current observable support therefore acts on an identity and cannot enlarge that support. Repeating this argument backwards proves the algebra inclusion. Equal reductions give equal expectations of every pulled-back observable. Tensoring with a reference proves the complete statement. Composing the certified causal dependencies gives a piecewise causal curve, proving metric containment. \end{proof} This is an upper bound on influence. A scheduled path can carry a replacement channel and no signal. It is not a derivation of the metric or an assertion that every ancestor is operationally detectable. The comparison with En03-A and En05-A is documentary \cite{CheungEn03,CheungEn05}. \subsection{Exact target-adapted null transport} Fix a convex developed spatial ball in \(U\), two distinct sites \(x_0,x_m\), and \[ L=\abs{y(x_m)-y(x_0)},\qquad p=(T_{\mathrm d},x_0),\qquad q=(T_{\mathrm d}+L/c,x_m). \] For an integer \(m\ge1\), install \[ y(x_j)=y(x_0)+\frac jm(y(x_m)-y(x_0)),\qquad a=L/m,\qquad \delta=L/(mc). \] At each site place \(A_j\cong\C^D\), with a fixed blank \(\sigma=\kb0{0}\), \(D\ge2\). At tick \(j\), swap \(A_{j-1}\) and \(A_j\). \begin{theorem}[Reference-preserving null transport] \label{thm:transport} For every payload--reference state \(\rho_{A_0E}\), after tick \(j\) the state is \[ \rho_{A_jE}\otimes \bigotimes_{\substack{0\le k\le m\\k\ne j}}\sigma_{A_k}, \] under the fixed register identifications. The destination channel is exactly \(\id_D\), including the reference. Arrival is at \(q\). The protocol uses \(m\) SWAP gates, \(m+1\) payload-sized registers, and \(m\) ticks, and every tick-boundary dependency is metric causal. \end{theorem} \begin{proof} The invariant holds initially. At tick \(j\), the destination factor is blank and independent. SWAP exchanges it with the entire current payload factor, including its correlations with \(E\), leaving the previous factor blank. Induction proves the invariant and the complete identity channel. Each active spatial separation is \(a=c\delta\), and the segment remains in the developing ball. The total duration is \(m\delta=L/c\). The resource counts follow from the construction. \end{proof} The target-adapted permission is essential. This is not exact arbitrary null transport on one previously fixed lattice. En08-B studies a different refinement contract \cite{CheungEn08B}; its resource formulas are not imported here. \subsection{Ledger and the boundary of the causal theorem} \begin{center} \begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}} \toprule Object & Charged resource or status\\ \midrule Route & \(m+1\) registers of dimension \(D\), \(m\) SWAPs, \(m\) ticks, duration \(L/c\).\\ Consumed context & Dimension \(2d_T\); all its quantum names are removed from the output context.\\ Spatial rods & The three named \(L_1,L_2,L_3\) factors remain.\\ Reference & Untouched; any external storage ledger remains external.\\ Represented field module & Infinite-dimensional ideal input; not priced as a finite-qubit program.\\ Energy, work, heat, reset & Not determined by the channel ledger.\\ \bottomrule \end{tabularx} \end{center} For \(D=2^p\), the route uses \((m+1)p\) carrier qubits. Add clock and rod counts only for disjoint simultaneously live names; do not count one factor twice. \scopebox{Global apparatus boundary}{% Theorems~\ref{thm:locality} and \ref{thm:transport} certify the finite carrier and control network. The global field operations \(W_{A_i}\) remain admitted global apparatuses. Their assignment to an externally scheduled experiment does not prove that their action on local field observables lies inside a compact spacetime region. No local causal-support theorem for those global apparatuses, no within-tick SWAP Hamiltonian, and no compact system--probe field interaction is inferred. A full causal field implementation would require an additional same-model construction.} In particular, the finite route transports finite payloads, not the entire \(\HH_i\). A finite number cutoff does not make that field module a finite payload. %===================================================================== \section{Manipulation of the consumed context} \label{sec:manipulation} %===================================================================== \begin{proposition}[Four admitted manipulations] \label{prop:manipulation} Within the declared interfaces: \begin{enumerate} \item a clock unitary \(V\) before Enter replaces the measured projectors by \(V^*P_kV\); \item a pre-readout coupling \(W\) on \(C\otimes X\) produces branch maps \(D_kW(\,\cdot\,)W^*D_k^*\); \item record-controlled continuation by channels \(\Lambda_k\) produces \[ \sum_k\kb{k}{k}\otimes\Lambda_k(D_kZD_k^*); \] \item fresh clock allocation permits another readout, but does not recover or reuse a consumed quantum name. \end{enumerate} These constructions are CPTP after retaining all branches and remain valid with untouched references. \end{proposition} \begin{proof} The first two claims follow by substituting \(VZV^*\) and \(WZW^*\) into \eqref{eq:enter}. For the third, the direct-sum channel \(\bigoplus_k\Lambda_k\) preserves the trace of each classical block and is completely positive. Fresh allocation is tensoring with a fixed new state; its name is distinct by definition. \end{proof} A general basis change is not a coordinate transformation. For example, a computational-basis clock state is deterministic in that basis and has a uniform law after a Hadamard basis change, while the event and all geometric coordinates can remain fixed. A coordinate relabelling does not perform that physical change of observable. The geometric readings of these manipulations are correspondingly limited: they select a clock observable, a conditional matter state, or a permitted future continuation. They do not change \eqref{eq:metric}, alter the defect dimension, or move the readout event into its own past. If a continuation retains the binary record, the already proved binary contrast remains available. If it discards or overwrites that record, no general contrast-preservation assertion is made. Model-specific cutoffs remain unchanged throughout. %===================================================================== \section{Documentary separations and optional certificates} \label{sec:separations} %===================================================================== \subsection{Classical descent, currents, and lapse} The classical functional is specified before representation. The quotient and descent problems of III do not follow from the existence of a represented instrument, and their equation spaces are not identified with the three present representations \cite{CheungIII}. The admissible-current and retarded-product constructions of V concern different objects \cite{CheungV}. In particular, a Newton--wave quadratic curvature pairing is not the unregulated cubic scalar correction \(S_{t_0}(\mathcal R^\NW T(\varphi))\varphi\). A classical matter-defined relational lapse, a finite clock record, an externally scheduled time, and a numerical interpolation parameter are distinct. Em02-E, XIX-A, and XX-A document their own prescriptions and non-identifications \cite{CheungEm02E,CheungXIXA,CheungXXA}. No map identifies those lapse constructions with the consumed clock factor of this paper. \subsection{Optional finite Ward residual: a separate interface} The finite local-Ward decomposition discussed in Oai282-A is a comparison, not a premise for Model 3 \cite{CheungOai282}. Its algebraic content can be stated independently. On a flat patch, let finite-matrix-valued distributions \(T_{\mu\nu}=T_{\nu\mu}\), \(L\), \(P_\nu\), \(S\), and \(C_X\) be defined on a common joint-test domain. Put \[ t_{\mu\nu} =T_{\mu\nu}+\tfrac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L, \] and \[ E_\nu=\partial^\mu T_{\mu\nu}-P_\nu,\qquad E_{\mathrm{tr}}=\eta^{\mu\nu}T_{\mu\nu}-\Box L-S,\qquad K_X=X^\nu P_\nu+\sigma_XS-C_X. \] For a conformal Killing field, \(\partial_{(\mu}X_{\nu)}=\sigma_X\eta_{\mu\nu}\). Direct differentiation gives, for every admitted compact joint test \(f\), \begin{equation} [\partial^\mu(X^\nu t_{\mu\nu})-C_X](f) =\sum_\nu E_\nu(X^\nu f) +E_{\mathrm{tr}}(\sigma_X f)+K_X(f). \label{eq:ward} \end{equation} Indeed the improvement has zero divergence and trace \(-\Box L\); the conformal Killing identity evaluates the differentiated \(X\) term. The triangle inequality gives the corresponding finite norm certificate. All insertion products and contacts in this statement are inputs. Equation~\eqref{eq:ward} supplies no missing product prescription, does not identify these matrices with \(B_{\Lambda,M}(J)\), and does not establish an untested Ward identity or Einstein dynamics. Number compression alone would not even provide the finite matrix space required by this optional certificate. \subsection{Recovery, islands, and gravity-generated entanglement} The Hawk-series recovery, operational-island, and operational-wormhole certificates concern separate accessible factors and complete complements \cite{CheungHawk}. No such factorization or certificate is constructed here. A binary contrast is not a Page curve, an interior, an island, or a wormhole. Similarly, the decoder and apparatus comparisons in the En02 and Em03 series are negative controls, not physical identifications \cite{CheungEn02Em03}. Reference preservation of SWAP does not prove entanglement generated by gravity. Model 1 changes gravitational preparation at fixed scalar preparation; its signal does not exclude every classical stochastic account. The OpenAI catalogue is a distinct documentary source \cite{OpenAICatalogue}. Every OpenAI citation in this paper is a separation edge: it supplies no premise, field representation, gravitational dictionary, implementation, or coefficient admission. %===================================================================== \section{Mathematical release and the unconstructed implications} \label{sec:release} %===================================================================== \begin{theorem}[IV-B interface release] \label{thm:release} Fix the conical geometry \eqref{eq:metric}, the developing patch, the flat-completion prescription, and the three independently defined represented models above. Then: \begin{enumerate} \item each model has the complete Ramsey--Enter channel \eqref{eq:composed}, with a consumed finite clock context, a classical record, surviving rods and matter, and arbitrary untouched references; \item the three positive-contrast conclusions hold in their stated, separate regulator regimes, together with their exact interpolation-off controls; \item Model 1 admits the stated edge removal with pointwise complete trace-norm convergence; Model 2 admits complete diamond-norm spectral convergence at each fixed finite \(M\); Model 3 retains its selected finite \(\Lambda,M\); \item compact coefficient and curvature pairings agree with their conical-patch readings under the declared completion; \item finite carrier/control networks with causal block certificates obey the schedule-past bound, and the target-adapted route implements exact complete null transport of a finite payload; \item the records do not uniquely determine the geometric dictionary, and neither a quantum return of the consumed clock nor a finite canonical commutator follows. \end{enumerate} Thus a Grade-S interface adaptation is constructed. An intrinsic-cone field adaptation and a compact causal implementation of the global apparatuses are not established. Grade E is not constructed. \end{theorem} \begin{proof} The operator and record assertions are Proposition~\ref{prop:A1}, Theorems~\ref{thm:model1}, \ref{thm:spectral-channel}, \ref{thm:model2}, and \ref{thm:model3}. Proposition~\ref{prop:edge} identifies the precise unbounded-model topology. Lemma~\ref{lem:patch} proves the geometric pairing statement. Theorems~\ref{thm:instrument} and \ref{thm:composition} prove the complete clock attachment. Theorems~\ref{thm:locality} and \ref{thm:transport} give the finite causal-wire and transport assertions on their explicitly certified interfaces. Propositions~\ref{prop:three-separations}, \ref{prop:event}, and \ref{prop:nonunique} establish the remaining separations. Proposition~\ref{prop:no-global-transfer} explains why the stronger field and implementation assertions are not deductions from these results. \end{proof} \subsection{The six stronger implications remain missing} In the terminology of I-B, the following stronger connecting constructions remain outside the release \cite{CheungIB}: \begin{enumerate} \item a chart reconstructed from records without a predeclared location map; \item lapse coefficients reconstructed without the geometric clock and background prescription; \item a Lorentz cone selected by scheduler consistency alone; \item a covariant residual stress tensor sourcing the same metric through a proved dynamics law; \item a native quantum-lapse construction with compatible joint regulator removal; \item a physical proper-time calibration derived from records, rather than supplied and checked against the declared metric and observer. \end{enumerate} The proved equality \(\dd\tau=\dd T\) for the specified static observers does not supply the last physical calibration. \subsection{Cutoff ledger} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.14\linewidth}p{.25\linewidth}X@{}} \toprule Model & Removed & Retained or not identified\\ \midrule 1 & Smooth initial edge \(\varepsilon\), in the proved topology & Fixed TT plane; numerical \(\lambda\); no field-sector limit.\\ 2 & Spectral masks \(R\), at fixed finite \(M\) & Number cutoff \(M\); TT representation; scalar-only free compression.\\ 3 & No instrument cutoff removed & Selected finite scalar momentum cutoff \(\Lambda\) and number cutoff \(M\); fixed normal ordering and Newton--wave prescription.\\ All & No common cutoff is defined & No identification of the representations, sources, or regulators between models.\\ \bottomrule \end{tabularx} \end{center} An expectation limit used to select a finite cutoff is not an instrument limit. Finite particle number is not finite one-particle dimension. A finite probe is not a finite-qubit representation of an entire infinite-dimensional field. \subsection{Standing stops and non-claims} This release does not: \begin{itemize} \item construct a full reduced Einstein--scalar phase space, a sourced graviton representation, an unregulated self-adjoint metric field, or the unregulated cubic scalar correction; \item construct an intrinsic global cone field theory, a compact system--probe coupling, or a microscopic causal realization of the global Ramsey controls; \item prove measurement uniqueness, or assert that the infinite-dimensional generators are primitives of a finite-qubit language; \item assert an \(M\to\infty\) instrument limit or identify the three representations, sources, normalizations, and cutoffs; \item derive the prescribed metric, Einstein dynamics, a quantum gravitational constraint algebra, or quantum relational evolution without an external schedule; \item construct spacetime emergence, holographic emergence, a Page curve, an island, an interior, or a gravity-generated-entanglement test; \item solve any of the nine standing problems in the supplied programme; \item infer heat, work, reset cost, or a Landauer ledger from channel equality. \end{itemize} The source records distinguishing the internal-amplitude advances from the missing inherited full characteristic map remain separate \cite{CheungXIVIJ,CheungXVB,CheungXVIN}. In particular, \[ \boxed{ \begin{gathered} \status{O.1 = NOT\_CLOSED},\\ \text{the downstream XV-B admission exit is not opened},\\ \text{XVI-N is not discharged},\\ \text{no inherited gravitational quantum coefficient is admitted}. \end{gathered}} \] The existence of a technical manuscript called XV-B is not denied; the statement concerns the downstream admission exit. No differential or Ward defect is applied here to an absent inherited coefficient. A represented pilot is not that coefficient's admission. \subsection{What is potentially useful and what is not new} The substantive combination is the following: exact destructive readout can be attached without losing the conditional field state, reference correlations, positive represented contrasts, or the distinction between their regulator topologies. In particular, fixed-\(M\) spectral estimates control the entire recorded instrument, not merely selected probabilities. The same formulation exposes two otherwise easy errors: treating a stochastic outcome as several exact times of one event, and treating a local developing chart as a global quantization theorem. These are precise interface and transfer statements. The elementary Ramsey, Kraus, partial-trace, and SWAP identities are not claimed to be new in isolation. No literature-wide novelty or experimental-impact claim is inferred from the internal manuscript comparison. %===================================================================== \appendix \section{Audit of the requested conversion} \label{app:audit} %===================================================================== \begin{longtable}{@{}p{.28\linewidth}p{.64\linewidth}@{}} \toprule Requested step & Exact result or correction\\ \midrule \endfirsthead \toprule Requested step & Exact result or correction\\ \midrule \endhead \bottomrule \endfoot Freeze the conical geometry & Completed in Proposition~\ref{prop:geometry}. The full defect is a worldsheet and unit lapse is retained.\\[4pt] Place the represented models on the cone & Completed as a declared-completion patch interpretation. Proposition~\ref{prop:no-global-transfer} prevents promotion to an intrinsic global-cone theory.\\[4pt] Evaluate the Newton--wave response on the conical background & The precise operation is \eqref{eq:completed-response}: compute the declared complete-source response, then pull it back. It satisfies the local pulled-back equations. An intrinsic cone inverse determined by local data is not asserted.\\[4pt] Use one consumed clock factor & The total finite context is \(C=C_T\otimes C_B\). Its time memory and Ramsey probe are both consumed. This avoids identifying a random binary signal with the exact coordinate time of one event.\\[4pt] Prove the three dictionaries & The maps are explicit in Section~\ref{sec:dictionary}. Agreement holds on the admitted ports, compact tests, and certified wires. Nonuniqueness is proved.\\[4pt] Prove complete measurement identities & Completed in Theorems~\ref{thm:instrument} and \ref{thm:composition}, with arbitrary untouched references.\\[4pt] Preserve positive records and controls & Completed separately for all three represented models. Record equality at zero interpolation is not confused with equality of their entire retained matter states.\\[4pt] Remove the admitted cutoffs & Only the listed cutoffs are removed. Model 1 gives pointwise complete trace-norm convergence. Model 2 gives diamond-norm spectral convergence at fixed \(M\). Model 3 retains finite \(\Lambda,M\).\\[4pt] Schedule global apparatuses inside the cone & External times can be assigned, but that assignment is not a local causal implementation theorem. The missing compact field coupling is not concealed by a timestamp.\\[4pt] Transport and locality & Exact complete finite-payload null transport and the causal-wire influence bound are proved. Neither theorem transports an entire unencoded field Hilbert space.\\[4pt] Manipulate the fourth factor & Completed as basis change, pre-readout coupling, classical continuation, and fresh allocation. No geometric time axis is consumed.\\[4pt] Release Grade S, not Grade E & Grade-S interface adaptation is constructed. Intrinsic-cone field transfer and compact causal apparatus realization remain separate missing bridges. Grade E and all standing analytic stops are unchanged.\\ \end{longtable} \section*{Declarations} \paragraph{Verification.} This is a single standard pdfLaTeX source with an internal bibliography. No successful pdfLaTeX build, proof-assistant check, numerical execution, hardware compilation, physical implementation, independent referee verification, or journal acceptance of this manuscript is asserted. \paragraph{Bibliographic status.} The Enter-series identifiers below distinguish unpublished source records and supplied excerpts from the separately published or accepted journal literature. No publication dates, DOIs, or stronger verification status are invented for the internal manuscripts. Descriptive annotations identify the comparison being made; they are not additional theorems imported into this paper. \paragraph{Conclusion.} The three represented instruments remain separate. The conical geometry remains prescribed. Enter consumes a finite clock encoding and returns its classical record together with the specified surviving matter. A \(3+1\) drawing is a visualization of this declared model, not a derivation of spacetime. %===================================================================== \begin{thebibliography}{99} %===================================================================== \bibitem{CheungEnter} N.~Cheung, \emph{The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement}. Frontiers in Physics \textbf{14} (2026), doi:\,\href{https://doi.org/10.3389/fphy.2026.1954943} {10.3389/fphy.2026.1954943}. Publisher abstract record: accepted 28 September 2026; the displayed record describes the final formatted version as forthcoming. Used for operational background, not as a field-theoretic premise. \bibitem{CheungPhil} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. EntPhil, version 2, introductory essay. \url{https://enter.grok.me/enter}. Documentary philosophical source, not a theorem premise. \bibitem{CheungI} N.~Cheung, \emph{Manuscript I}, version 2. Enter Calculus Companion. \url{https://enter.grok.me/browse/I}. Operational measurement characterization within its stated class; no uniqueness theorem is transferred here. \bibitem{CheungIB} N.~Cheung, \emph{I-B-v1: A Finite Enter Spacetime: Calibrated Destructive Time Readout on a Prescribed Conical Product}. Version 1, source file \texttt{I-B-v1.tex}. \url{https://enter.grok.me/browse/I-B}. Geometric and interface provenance; all assertions used in the present construction are restated and proved here. \bibitem{CheungIV} N.~Cheung, \emph{Represented records for three named scalar-field models: Finite-plane corrections, dense-TT corrections, and sourced curvature pairings}. Manuscript IV, version 1, source file \texttt{IV.tex}. \url{https://enter.grok.me/browse/IV}. Source of the three represented-model conventions and their distinct cutoff grammar. \bibitem{CheungEn09} N.~Cheung, \emph{Manuscript En09-A}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En09-A}. Supplied complete relational Enter instrument, covariance, stationarity, and refinement comparisons. Documentary comparison; the needed identities are proved in this paper. \bibitem{CheungEn03} N.~Cheung, \emph{Manuscript En03-A}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En03-A}. Supplied scheduled-block locality and record/residual interface comparisons; not an imported premise. \bibitem{CheungEn05} N.~Cheung, \emph{Manuscript En05-A}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En05-A}. Supplied schedule-past and metric-envelope comparisons; not a derivation of a metric and not an imported premise. \bibitem{CheungEn08B} N.~Cheung, \emph{En08-B: Resource-controlled operational approximation of null boundary transport}. Version 1. \url{https://enter.grok.me/browse/En08-B}. Different lattice-refinement transport contract; documentary only. \bibitem{CheungIII} N.~Cheung, \emph{Manuscript III}, version 2. Enter Calculus Companion. \url{https://enter.grok.me/browse/III}. Classical descent and equation-space separations; documentary only. \bibitem{CheungV} N.~Cheung, \emph{Manuscript V}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/V}. Admissible-current, retarded-product, and Newton--wave cubic comparisons; documentary only. \bibitem{CheungVIIIB} N.~Cheung, \emph{Manuscript VIII-B}, version 3. Enter Calculus Companion. \url{https://enter.grok.me/browse/VIII-B}. Conical-family, source, collar, and lapse non-identifications; documentary only. \bibitem{CheungEm02E} N.~Cheung, \emph{Manuscript Em02-E}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/Em02-E}. Supplied classical relational-lapse expansion on its own background; documentary only. \bibitem{CheungXIXA} N.~Cheung, \emph{Manuscript XIX-A}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/XIX-A}. Clock separation before coordinate identification; documentary only. \bibitem{CheungXXA} N.~Cheung, \emph{Manuscript XX-A}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/XX-A}. Classical original-rod lapse and separate quantum-admission boundary; documentary only. \bibitem{CheungOai282} N.~Cheung, \emph{Manuscript Oai282-A}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/Oai282-A}. Supplied finite local-Ward residual interface. A separation comparison, not a premise for Model 3 or Einstein dynamics. \bibitem{CheungHawk} N.~Cheung, \emph{Manuscripts Hawk04-C, Hawk04-D, Hawk05-A--Hawk05-C, Hawk06-A, Hawk07-A, and Hawk07-B}. Enter Calculus Companion; supplied source excerpts. Complete-complement, recovery, operational-island, and operational-wormhole comparisons on separately declared interfaces. Documentary negative bearings only. \bibitem{CheungEn02Em03} N.~Cheung, \emph{En02 and Em03 manuscript series}. Enter Calculus Companion; supplied version-specific excerpts. Decoder ledgers, packet and apparatus models, and entanglement-breaking channel comparisons. Documentary negative controls only; no model identification or theorem transfer. \bibitem{CheungXIVIJ} N.~Cheung, \emph{Manuscripts XIV-I and XIV-J}, versions 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/XIV-I}; \url{https://enter.grok.me/browse/XIV-J}. Internal-amplitude construction and the separate unresolved full characteristic-map and admission boundaries. Documentary only. \bibitem{CheungXVB} N.~Cheung, \emph{Manuscript XV-B}, version 2. Enter Calculus Companion; supplied text. \url{https://enter.grok.me/browse/XV-B}. Relative-lifting and reconstruction-descent interfaces; the supplied technical manuscript does not itself supply the inherited full graph or quantum admission. Documentary only. \bibitem{CheungXVIN} N.~Cheung, \emph{Manuscript XVI-N}, revision r12. Enter Calculus Companion; supplied conditional obstruction statement. \url{https://enter.grok.me/browse/XVI-N}. Not discharged or identified with the present measurement models. \bibitem{OpenAICatalogue} OpenAI, \emph{OpenAI mathematical manuscript catalogue}. Separate documentary collection displayed by the Enter Calculus Companion. \url{https://enter.grok.me/openai}. Catalogue attribution only. Not independently verified here and not a premise. No OpenAI family is identified with a Cheung manuscript or with the present represented instruments. \bibitem{Nelson} E.~Nelson, ``Analytic vectors,'' \emph{Annals of Mathematics}, second series, \textbf{70} (1959), 572--615. doi:\,\href{https://doi.org/10.2307/1970331}{10.2307/1970331}. External analytic-vector theorem used in Proposition~\ref{prop:A1} after explicit sector estimates. \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:\,\href{https://doi.org/10.1103/PhysRevD.27.2885} {10.1103/PhysRevD.27.2885}. External conditioning context; not a premise for geometric emergence or an unscheduled Ramsey implementation. \end{thebibliography} \end{document}